252 files changed, 217816 insertions, 0 deletions

diff --git a/gsl-1.9/doc/12-cities.eps b/gsl-1.9/doc/12-cities.eps
new file mode 100644
index 0000000..f2b8726
--- /dev/null
+++ b/gsl-1.9/doc/12-cities.eps
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+119 MLine
+End
+
+%%PageTrailer
+End %I eop
+showpage
+
+%%Trailer
+end
+%%EOF
diff --git a/gsl-1.9/doc/ChangeLog b/gsl-1.9/doc/ChangeLog
new file mode 100644
index 0000000..09cd473
--- /dev/null
+++ b/gsl-1.9/doc/ChangeLog
@@ -0,0 +1,62 @@
+2006-02-09  Brian Gough  <bjg@network-theory.co.uk>
+
+	* Makefile.am: disable pdf, as it would require maintaining a
+	separate set of figure files.
+
+Mon Apr  7 12:54:30 MDT 2003  G. Jungman
+
+    * specfunc-gamma.texi: added entry for gsl_sf_gamma_inc_(),
+    gsl_sf_gamma_inc()
+
+
+2000-10-26  Mark Galassi  <rosalia@lanl.gov>
+
+	* gsl-ref.texi: small changes to update the author list.
+
+Mon Oct 23 21:49:25 2000  Brian Gough  <bjg@network-theory.co.uk>
+
+	* usage.texi: added info on portability functions
+
+Sat Jul 15 23:52:59 2000  Brian Gough  <bjg@network-theory.co.uk>
+
+	* Makefile.am (EXTRA_DIST): added gsl-design.texi to the
+ 	distributed files
+
+Thu May 18 12:11:24 2000  Brian Gough  <bjg@network-theory.co.uk>
+
+	* gsl-ref.texi (Preliminaries): changed the phrase "collection of
+ 	routines for numerical analysis" to "... numerical computing", as
+ 	on the web page. 
+	
+Wed Apr 26 17:46:48 2000  Brian Gough  <bjg@network-theory.co.uk>
+
+	* math.texi: added a chapter on the elementary functions and
+ 	constants in gsl_math.h
+
+Tue Apr  4 21:45:27 2000  Brian Gough  <bjg@network-theory.co.uk>
+
+	* upgraded texinfo.tex to texinfo-4.0
+
+2000-04-01  Mark Galassi  <rosalia@lanl.gov>
+
+	* roots.texi (Providing the function to solve): fixed a typo,
+	THANKS Dave Morrison.
+
+1999-12-03  Mark Galassi  <rosalia@lanl.gov>
+
+	* gsl-ref.texi, gsl-design.texi: updated my email address.
+
+	* gsl-design.texi: minor changes.
+
+Tue Feb 16 11:36:03 1999  Brian Gough  <bjg@netsci.freeserve.co.uk>
+
+	* statistics.texi: removed the multiple descriptions of double,
+ 	int versions of the functions. Now we just describe the double
+ 	versions and note the existence of the corresponding versions for
+ 	other types.
+
+	* ChangeLog: added a changelog file for the documentation,
+ 	although it is not required by the GNU Coding standards. I use the
+ 	Changelogs to keep track of pending changes that I need to commit
+ 	to repository.
+
diff --git a/gsl-1.9/doc/Makefile.am b/gsl-1.9/doc/Makefile.am
new file mode 100644
index 0000000..12095a5
--- /dev/null
+++ b/gsl-1.9/doc/Makefile.am
@@ -0,0 +1,21 @@
+## Process this file with automake to produce Makefile.in
+
+info_TEXINFOS = gsl-ref.texi
+noinst_TEXINFOS = gsl-design.texi
+gsl_ref_TEXINFOS = err.texi cblas.texi blas.texi min.texi fft.texi rng.texi randist.texi roots.texi statistics.texi specfunc.texi specfunc-airy.texi specfunc-bessel.texi specfunc-clausen.texi specfunc-coulomb.texi specfunc-coupling.texi specfunc-dawson.texi specfunc-debye.texi specfunc-dilog.texi specfunc-elementary.texi specfunc-ellint.texi specfunc-elljac.texi specfunc-erf.texi specfunc-exp.texi specfunc-expint.texi specfunc-fermi-dirac.texi specfunc-gamma.texi specfunc-gegenbauer.texi specfunc-hyperg.texi specfunc-lambert.texi specfunc-laguerre.texi specfunc-legendre.texi specfunc-log.texi specfunc-mathieu.texi specfunc-pow-int.texi specfunc-psi.texi specfunc-synchrotron.texi specfunc-transport.texi specfunc-trig.texi specfunc-zeta.texi siman.texi vectors.texi debug.texi histogram.texi ode-initval.texi integration.texi ieee754.texi montecarlo.texi sum.texi intro.texi usage.texi dwt.texi dht.texi interp.texi poly.texi linalg.texi eigen.texi multiroots.texi sort.texi permutation.texi combination.texi complex.texi math.texi fitting.texi multifit.texi const.texi ntuple.texi diff.texi qrng.texi cheb.texi multimin.texi gpl.texi fdl.texi autoconf.texi freemanuals.texi bspline.texi
+
+man_MANS = gsl.3 gsl-config.1 gsl-randist.1 gsl-histogram.1
+
+figures = multimin.eps siman-test.eps siman-energy.eps 12-cities.eps initial-route.eps final-route.eps fft-complex-radix2-f.eps fft-complex-radix2-t.eps fft-complex-radix2.eps fft-real-mixedradix.eps roots-bisection.eps roots-false-position.eps roots-newtons-method.eps roots-secant-method.eps histogram.eps histogram2d.eps min-interval.eps fit-wlinear.eps fit-wlinear2.eps fit-exp.eps ntuple.eps qrng.eps cheb.eps vdp.eps interp2.eps rand-beta.tex rand-binomial.tex rand-cauchy.tex rand-chisq.tex rand-erlang.tex rand-exponential.tex rand-fdist.tex rand-flat.tex rand-gamma.tex rand-gaussian.tex rand-geometric.tex rand-laplace.tex rand-logarithmic.tex rand-logistic.tex rand-lognormal.tex rand-pareto.tex rand-poisson.tex rand-hypergeometric.tex rand-nbinomial.tex rand-pascal.tex rand-bivariate-gaussian.tex rand-rayleigh.tex rand-rayleigh-tail.tex rand-tdist.tex rand-weibull.tex random-walk.tex randplots.gnp rand-exppow.tex rand-levy.tex rand-levyskew.tex rand-gumbel.tex rand-bernoulli.tex rand-gaussian-tail.tex rand-gumbel1.tex rand-gumbel2.tex landau.dat rand-landau.tex dwt-orig.eps dwt-samp.eps interpp2.eps bspline.eps
+
+examples_src = examples/blas.c examples/block.c examples/cblas.c examples/cdf.c examples/cheb.c examples/combination.c examples/const.c examples/demo_fn.c examples/diff.c examples/eigen.c examples/fft.c examples/fftmr.c examples/fftreal.c examples/fitting.c examples/fitting2.c examples/fitting3.c examples/histogram.c examples/histogram2d.c examples/ieee.c examples/ieeeround.c examples/integration.c examples/interp.c examples/intro.c examples/linalglu.c examples/matrix.c examples/matrixw.c examples/min.c examples/monte.c examples/ntupler.c examples/ntuplew.c examples/ode-initval.c examples/odefixed.c examples/permseq.c examples/permshuffle.c examples/polyroots.c examples/qrng.c examples/randpoisson.c examples/randwalk.c examples/rng.c examples/rngunif.c examples/rootnewt.c examples/roots.c examples/siman.c examples/sortsmall.c examples/specfun.c examples/specfun_e.c examples/stat.c examples/statsort.c examples/sum.c examples/vector.c examples/vectorr.c examples/vectorview.c examples/vectorw.c examples/demo_fn.h examples/dwt.c examples/expfit.c examples/nlfit.c examples/interpp.c examples/eigen_nonsymm.c examples/bspline.c
+
+examples_out = examples/blas.out examples/block.out examples/cblas.out examples/cdf.out examples/combination.out examples/const.out examples/diff.out examples/integration.out examples/intro.out examples/linalglu.out examples/min.out examples/polyroots.out examples/randpoisson.2.out examples/randpoisson.out examples/rng.out examples/rngunif.2.out examples/rngunif.out examples/sortsmall.out examples/specfun.out examples/specfun_e.out examples/stat.out examples/statsort.out examples/sum.out examples/vectorview.out examples/ecg.dat examples/dwt.dat 
+
+noinst_DATA = $(examples_src) $(examples_out) $(figures)
+
+EXTRA_DIST = $(man_MANS) $(noinst_DATA) gsl-design.texi fftalgorithms.tex fftalgorithms.bib algorithm.sty algorithmic.sty calc.sty
+
+# pdf disabled, use postscript and ps2pdf
+.PHONY: pdf
+pdf:
diff --git a/gsl-1.9/doc/Makefile.in b/gsl-1.9/doc/Makefile.in
new file mode 100644
index 0000000..2c470b7
--- /dev/null
+++ b/gsl-1.9/doc/Makefile.in
@@ -0,0 +1,674 @@
+# Makefile.in generated by automake 1.9.6 from Makefile.am.
+# @configure_input@
+
+# Copyright (C) 1994, 1995, 1996, 1997, 1998, 1999, 2000, 2001, 2002,
+# 2003, 2004, 2005  Free Software Foundation, Inc.
+# This Makefile.in is free software; the Free Software Foundation
+# gives unlimited permission to copy and/or distribute it,
+# with or without modifications, as long as this notice is preserved.
+
+# This program is distributed in the hope that it will be useful,
+# but WITHOUT ANY WARRANTY, to the extent permitted by law; without
+# even the implied warranty of MERCHANTABILITY or FITNESS FOR A
+# PARTICULAR PURPOSE.
+
+@SET_MAKE@
+
+srcdir = @srcdir@
+top_srcdir = @top_srcdir@
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+top_builddir = ..
+am__cd = CDPATH="$${ZSH_VERSION+.}$(PATH_SEPARATOR)" && cd
+INSTALL = @INSTALL@
+install_sh_DATA = $(install_sh) -c -m 644
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+install_sh_SCRIPT = $(install_sh) -c
+INSTALL_HEADER = $(INSTALL_DATA)
+transform = $(program_transform_name)
+NORMAL_INSTALL = :
+PRE_INSTALL = :
+POST_INSTALL = :
+NORMAL_UNINSTALL = :
+PRE_UNINSTALL = :
+POST_UNINSTALL = :
+build_triplet = @build@
+host_triplet = @host@
+subdir = doc
+DIST_COMMON = $(gsl_ref_TEXINFOS) $(srcdir)/Makefile.am \
+	$(srcdir)/Makefile.in $(srcdir)/stamp-vti \
+	$(srcdir)/version-ref.texi ChangeLog mdate-sh texinfo.tex
+ACLOCAL_M4 = $(top_srcdir)/aclocal.m4
+am__aclocal_m4_deps = $(top_srcdir)/configure.ac
+am__configure_deps = $(am__aclocal_m4_deps) $(CONFIGURE_DEPENDENCIES) \
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+mkinstalldirs = $(SHELL) $(top_srcdir)/mkinstalldirs
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+CONFIG_CLEAN_FILES =
+depcomp =
+am__depfiles_maybe =
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+PDFS = gsl-ref.pdf
+PSS = gsl-ref.ps
+HTMLS = gsl-ref.html
+TEXINFOS = gsl-ref.texi
+TEXI2DVI = texi2dvi
+TEXI2PDF = $(TEXI2DVI) --pdf --batch
+MAKEINFOHTML = $(MAKEINFO) --html
+AM_MAKEINFOHTMLFLAGS = $(AM_MAKEINFOFLAGS)
+DVIPS = dvips
+am__installdirs = "$(DESTDIR)$(infodir)" "$(DESTDIR)$(man1dir)" \
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+MANS = $(man_MANS)
+DATA = $(noinst_DATA)
+DISTFILES = $(DIST_COMMON) $(DIST_SOURCES) $(TEXINFOS) $(EXTRA_DIST)
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+HAVE_AIX_IEEE_INTERFACE = @HAVE_AIX_IEEE_INTERFACE@
+HAVE_DARWIN86_IEEE_INTERFACE = @HAVE_DARWIN86_IEEE_INTERFACE@
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+HAVE_GNUM68K_IEEE_INTERFACE = @HAVE_GNUM68K_IEEE_INTERFACE@
+HAVE_GNUPPC_IEEE_INTERFACE = @HAVE_GNUPPC_IEEE_INTERFACE@
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+HAVE_GNUX86_IEEE_INTERFACE = @HAVE_GNUX86_IEEE_INTERFACE@
+HAVE_HPUX11_IEEE_INTERFACE = @HAVE_HPUX11_IEEE_INTERFACE@
+HAVE_HPUX_IEEE_INTERFACE = @HAVE_HPUX_IEEE_INTERFACE@
+HAVE_IEEE_COMPARISONS = @HAVE_IEEE_COMPARISONS@
+HAVE_IEEE_DENORMALS = @HAVE_IEEE_DENORMALS@
+HAVE_INLINE = @HAVE_INLINE@
+HAVE_IRIX_IEEE_INTERFACE = @HAVE_IRIX_IEEE_INTERFACE@
+HAVE_NETBSD_IEEE_INTERFACE = @HAVE_NETBSD_IEEE_INTERFACE@
+HAVE_OPENBSD_IEEE_INTERFACE = @HAVE_OPENBSD_IEEE_INTERFACE@
+HAVE_OS2EMX_IEEE_INTERFACE = @HAVE_OS2EMX_IEEE_INTERFACE@
+HAVE_PRINTF_LONGDOUBLE = @HAVE_PRINTF_LONGDOUBLE@
+HAVE_SOLARIS_IEEE_INTERFACE = @HAVE_SOLARIS_IEEE_INTERFACE@
+HAVE_SUNOS4_IEEE_INTERFACE = @HAVE_SUNOS4_IEEE_INTERFACE@
+HAVE_TRU64_IEEE_INTERFACE = @HAVE_TRU64_IEEE_INTERFACE@
+INSTALL_DATA = @INSTALL_DATA@
+INSTALL_PROGRAM = @INSTALL_PROGRAM@
+INSTALL_SCRIPT = @INSTALL_SCRIPT@
+INSTALL_STRIP_PROGRAM = @INSTALL_STRIP_PROGRAM@
+LDFLAGS = @LDFLAGS@
+LIBOBJS = @LIBOBJS@
+LIBS = @LIBS@
+LIBTOOL = @LIBTOOL@
+LN_S = @LN_S@
+LTLIBOBJS = @LTLIBOBJS@
+MAINT = @MAINT@
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+build_vendor = @build_vendor@
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+exec_prefix = @exec_prefix@
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+sharedstatedir = @sharedstatedir@
+sysconfdir = @sysconfdir@
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+info_TEXINFOS = gsl-ref.texi
+noinst_TEXINFOS = gsl-design.texi
+gsl_ref_TEXINFOS = err.texi cblas.texi blas.texi min.texi fft.texi \
+	rng.texi randist.texi roots.texi statistics.texi specfunc.texi \
+	specfunc-airy.texi specfunc-bessel.texi specfunc-clausen.texi \
+	specfunc-coulomb.texi specfunc-coupling.texi \
+	specfunc-dawson.texi specfunc-debye.texi specfunc-dilog.texi \
+	specfunc-elementary.texi specfunc-ellint.texi \
+	specfunc-elljac.texi specfunc-erf.texi specfunc-exp.texi \
+	specfunc-expint.texi specfunc-fermi-dirac.texi \
+	specfunc-gamma.texi specfunc-gegenbauer.texi \
+	specfunc-hyperg.texi specfunc-lambert.texi \
+	specfunc-laguerre.texi specfunc-legendre.texi \
+	specfunc-log.texi specfunc-mathieu.texi specfunc-pow-int.texi \
+	specfunc-psi.texi specfunc-synchrotron.texi \
+	specfunc-transport.texi specfunc-trig.texi specfunc-zeta.texi \
+	siman.texi vectors.texi debug.texi histogram.texi \
+	ode-initval.texi integration.texi ieee754.texi montecarlo.texi \
+	sum.texi intro.texi usage.texi dwt.texi dht.texi interp.texi \
+	poly.texi linalg.texi eigen.texi multiroots.texi sort.texi \
+	permutation.texi combination.texi complex.texi math.texi \
+	fitting.texi multifit.texi const.texi ntuple.texi diff.texi \
+	qrng.texi cheb.texi multimin.texi gpl.texi fdl.texi \
+	autoconf.texi freemanuals.texi bspline.texi
+man_MANS = gsl.3 gsl-config.1 gsl-randist.1 gsl-histogram.1
+figures = multimin.eps siman-test.eps siman-energy.eps 12-cities.eps \
+	initial-route.eps final-route.eps fft-complex-radix2-f.eps \
+	fft-complex-radix2-t.eps fft-complex-radix2.eps \
+	fft-real-mixedradix.eps roots-bisection.eps \
+	roots-false-position.eps roots-newtons-method.eps \
+	roots-secant-method.eps histogram.eps histogram2d.eps \
+	min-interval.eps fit-wlinear.eps fit-wlinear2.eps fit-exp.eps \
+	ntuple.eps qrng.eps cheb.eps vdp.eps interp2.eps rand-beta.tex \
+	rand-binomial.tex rand-cauchy.tex rand-chisq.tex \
+	rand-erlang.tex rand-exponential.tex rand-fdist.tex \
+	rand-flat.tex rand-gamma.tex rand-gaussian.tex \
+	rand-geometric.tex rand-laplace.tex rand-logarithmic.tex \
+	rand-logistic.tex rand-lognormal.tex rand-pareto.tex \
+	rand-poisson.tex rand-hypergeometric.tex rand-nbinomial.tex \
+	rand-pascal.tex rand-bivariate-gaussian.tex rand-rayleigh.tex \
+	rand-rayleigh-tail.tex rand-tdist.tex rand-weibull.tex \
+	random-walk.tex randplots.gnp rand-exppow.tex rand-levy.tex \
+	rand-levyskew.tex rand-gumbel.tex rand-bernoulli.tex \
+	rand-gaussian-tail.tex rand-gumbel1.tex rand-gumbel2.tex \
+	landau.dat rand-landau.tex dwt-orig.eps dwt-samp.eps \
+	interpp2.eps bspline.eps
+examples_src = examples/blas.c examples/block.c examples/cblas.c \
+	examples/cdf.c examples/cheb.c examples/combination.c \
+	examples/const.c examples/demo_fn.c examples/diff.c \
+	examples/eigen.c examples/fft.c examples/fftmr.c \
+	examples/fftreal.c examples/fitting.c examples/fitting2.c \
+	examples/fitting3.c examples/histogram.c \
+	examples/histogram2d.c examples/ieee.c examples/ieeeround.c \
+	examples/integration.c examples/interp.c examples/intro.c \
+	examples/linalglu.c examples/matrix.c examples/matrixw.c \
+	examples/min.c examples/monte.c examples/ntupler.c \
+	examples/ntuplew.c examples/ode-initval.c examples/odefixed.c \
+	examples/permseq.c examples/permshuffle.c examples/polyroots.c \
+	examples/qrng.c examples/randpoisson.c examples/randwalk.c \
+	examples/rng.c examples/rngunif.c examples/rootnewt.c \
+	examples/roots.c examples/siman.c examples/sortsmall.c \
+	examples/specfun.c examples/specfun_e.c examples/stat.c \
+	examples/statsort.c examples/sum.c examples/vector.c \
+	examples/vectorr.c examples/vectorview.c examples/vectorw.c \
+	examples/demo_fn.h examples/dwt.c examples/expfit.c \
+	examples/nlfit.c examples/interpp.c examples/eigen_nonsymm.c \
+	examples/bspline.c
+examples_out = examples/blas.out examples/block.out examples/cblas.out examples/cdf.out examples/combination.out examples/const.out examples/diff.out examples/integration.out examples/intro.out examples/linalglu.out examples/min.out examples/polyroots.out examples/randpoisson.2.out examples/randpoisson.out examples/rng.out examples/rngunif.2.out examples/rngunif.out examples/sortsmall.out examples/specfun.out examples/specfun_e.out examples/stat.out examples/statsort.out examples/sum.out examples/vectorview.out examples/ecg.dat examples/dwt.dat 
+noinst_DATA = $(examples_src) $(examples_out) $(figures)
+EXTRA_DIST = $(man_MANS) $(noinst_DATA) gsl-design.texi fftalgorithms.tex fftalgorithms.bib algorithm.sty algorithmic.sty calc.sty
+all: all-am
+
+.SUFFIXES:
+.SUFFIXES: .dvi .html .info .pdf .ps .texi
+$(srcdir)/Makefile.in: @MAINTAINER_MODE_TRUE@ $(srcdir)/Makefile.am  $(am__configure_deps)
+	@for dep in $?; do \
+	  case '$(am__configure_deps)' in \
+	    *$$dep*) \
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+		&& exit 0; \
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+
+maintainer-clean-aminfo:
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+	    *.3*) list="$$list $$i" ;; \
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+	  case "$$ext" in \
+	    3*) ;; \
+	    *) ext='3' ;; \
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+	  $(INSTALL_DATA) "$$file" "$(DESTDIR)$(man3dir)/$$inst"; \
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+	for i in $$l2; do \
+	  case "$$i" in \
+	    *.3*) list="$$list $$i" ;; \
+	  esac; \
+	done; \
+	for i in $$list; do \
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+	  case "$$ext" in \
+	    3*) ;; \
+	    *) ext='3' ;; \
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+tags: TAGS
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+	    || exit 1; \
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+	    echo "INSTALL_PROGRAM_ENV=STRIPPROG='$(STRIP)'"` install
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+	      echo " $(INSTALL_DATA) '$$ifile' '$(DESTDIR)$(infodir)/$$relfile'"; \
+	      $(INSTALL_DATA) "$$ifile" "$(DESTDIR)$(infodir)/$$relfile"; \
+	    else : ; fi; \
+	  done; \
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+	@$(POST_INSTALL)
+	@if (install-info --version && \
+	     install-info --version 2>&1 | sed 1q | grep -i -v debian) >/dev/null 2>&1; then \
+	  list='$(INFO_DEPS)'; \
+	  for file in $$list; do \
+	    relfile=`echo "$$file" | sed 's|^.*/||'`; \
+	    echo " install-info --info-dir='$(DESTDIR)$(infodir)' '$(DESTDIR)$(infodir)/$$relfile'";\
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diff --git a/gsl-1.9/doc/algorithm.sty b/gsl-1.9/doc/algorithm.sty
new file mode 100644
index 0000000..843e3d5
--- /dev/null
+++ b/gsl-1.9/doc/algorithm.sty
@@ -0,0 +1,79 @@
+% ALGORITHM STYLE -- Released 8 April 1996
+%    for LaTeX-2e
+% Copyright -- 1994 Peter Williams
+% E-mail Peter.Williams@dsto.defence.gov.au
+\NeedsTeXFormat{LaTeX2e}
+\ProvidesPackage{algorithm}
+\typeout{Document Style `algorithm' - floating environment}
+
+\RequirePackage{float}
+\RequirePackage{ifthen}
+\newcommand{\ALG@within}{nothing}
+\newboolean{ALG@within}
+\setboolean{ALG@within}{false}
+\newcommand{\ALG@floatstyle}{ruled}
+\newcommand{\ALG@name}{Algorithm}
+\newcommand{\listalgorithmname}{List of \ALG@name s}
+
+% Declare Options
+% first appearance
+\DeclareOption{plain}{
+  \renewcommand{\ALG@floatstyle}{plain}
+}
+\DeclareOption{ruled}{
+  \renewcommand{\ALG@floatstyle}{ruled}
+}
+\DeclareOption{boxed}{
+  \renewcommand{\ALG@floatstyle}{boxed}
+}
+% then numbering convention
+\DeclareOption{part}{
+  \renewcommand{\ALG@within}{part}
+  \setboolean{ALG@within}{true}
+}
+\DeclareOption{chapter}{
+  \renewcommand{\ALG@within}{chapter}
+  \setboolean{ALG@within}{true}
+}
+\DeclareOption{section}{
+  \renewcommand{\ALG@within}{section}
+  \setboolean{ALG@within}{true}
+}
+\DeclareOption{subsection}{
+  \renewcommand{\ALG@within}{subsection}
+  \setboolean{ALG@within}{true}
+}
+\DeclareOption{subsubsection}{
+  \renewcommand{\ALG@within}{subsubsection}
+  \setboolean{ALG@within}{true}
+}
+\DeclareOption{nothing}{
+  \renewcommand{\ALG@within}{nothing}
+  \setboolean{ALG@within}{true}
+}
+\DeclareOption*{\edef\ALG@name{\CurrentOption}}
+
+% ALGORITHM
+%
+\ProcessOptions
+\floatstyle{\ALG@floatstyle}
+\ifthenelse{\boolean{ALG@within}}{
+  \ifthenelse{\equal{\ALG@within}{part}}
+     {\newfloat{algorithm}{htbp}{loa}[part]}{}
+  \ifthenelse{\equal{\ALG@within}{chapter}}
+     {\newfloat{algorithm}{htbp}{loa}[chapter]}{}
+  \ifthenelse{\equal{\ALG@within}{section}}
+     {\newfloat{algorithm}{htbp}{loa}[section]}{}
+  \ifthenelse{\equal{\ALG@within}{subsection}}
+     {\newfloat{algorithm}{htbp}{loa}[subsection]}{}
+  \ifthenelse{\equal{\ALG@within}{subsubsection}}
+     {\newfloat{algorithm}{htbp}{loa}[subsubsection]}{}
+  \ifthenelse{\equal{\ALG@within}{nothing}}
+     {\newfloat{algorithm}{htbp}{loa}}{}
+}{
+  \newfloat{algorithm}{htbp}{loa}
+}
+\floatname{algorithm}{\ALG@name}
+
+\newcommand{\listofalgorithms}{\listof{algorithm}{\listalgorithmname}}
+
diff --git a/gsl-1.9/doc/algorithmic.sty b/gsl-1.9/doc/algorithmic.sty
new file mode 100644
index 0000000..68f18aa
--- /dev/null
+++ b/gsl-1.9/doc/algorithmic.sty
@@ -0,0 +1,158 @@
+% ALGORITHMIC STYLE -- Released 8 APRIL 1996
+%    for LaTeX version 2e
+% Copyright -- 1994 Peter Williams
+% E-mail PeterWilliams@dsto.defence.gov.au
+\NeedsTeXFormat{LaTeX2e}
+\ProvidesPackage{algorithmic}
+\typeout{Document Style `algorithmic' - environment}
+%
+\RequirePackage{ifthen}
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diff --git a/gsl-1.9/doc/autoconf.texi b/gsl-1.9/doc/autoconf.texi
new file mode 100644
index 0000000..8f8344a
--- /dev/null
+++ b/gsl-1.9/doc/autoconf.texi
@@ -0,0 +1,132 @@
+@cindex autoconf, using with GSL
+
+For applications using @code{autoconf} the standard macro
+@code{AC_CHECK_LIB} can be used to link with GSL automatically
+from a @code{configure} script.  The library itself depends on the
+presence of a @sc{cblas} and math library as well, so these must also be
+located before linking with the main @code{libgsl} file.  The following
+commands should be placed in the @file{configure.ac} file to perform
+these tests,
+
+@example
+AC_CHECK_LIB(m,main)
+AC_CHECK_LIB(gslcblas,main)
+AC_CHECK_LIB(gsl,main)
+@end example
+
+@noindent
+It is important to check for @code{libm} and @code{libgslcblas} before
+@code{libgsl}, otherwise the tests will fail.  Assuming the libraries
+are found the output during the configure stage looks like this,
+
+@example
+checking for main in -lm... yes
+checking for main in -lgslcblas... yes
+checking for main in -lgsl... yes
+@end example
+
+@noindent
+If the library is found then the tests will define the macros
+@code{HAVE_LIBGSL}, @code{HAVE_LIBGSLCBLAS}, @code{HAVE_LIBM} and add
+the options @code{-lgsl -lgslcblas -lm} to the variable @code{LIBS}.
+
+The tests above will find any version of the library.  They are suitable
+for general use, where the versions of the functions are not important.
+An alternative macro is available in the file @file{gsl.m4} to test for
+a specific version of the library.  To use this macro simply add the
+following line to your @file{configure.in} file instead of the tests
+above:
+
+@example
+AM_PATH_GSL(GSL_VERSION,
+           [action-if-found],
+           [action-if-not-found])
+@end example
+
+@noindent
+The argument @code{GSL_VERSION} should be the two or three digit
+@sc{major.minor} or @sc{major.minor.micro} version number of the release
+you require. A suitable choice for @code{action-if-not-found} is,
+
+@example
+AC_MSG_ERROR(could not find required version of GSL)
+@end example
+
+@noindent
+Then you can add the variables @code{GSL_LIBS} and @code{GSL_CFLAGS} to
+your Makefile.am files to obtain the correct compiler flags.
+@code{GSL_LIBS} is equal to the output of the @code{gsl-config --libs}
+command and @code{GSL_CFLAGS} is equal to @code{gsl-config --cflags}
+command. For example,
+
+@example
+libfoo_la_LDFLAGS = -lfoo $(GSL_LIBS) -lgslcblas
+@end example
+
+@noindent
+Note that the macro @code{AM_PATH_GSL} needs to use the C compiler so it
+should appear in the @file{configure.in} file before the macro
+@code{AC_LANG_CPLUSPLUS} for programs that use C++.
+
+To test for @code{inline} the following test should be placed in your
+@file{configure.in} file,
+
+@example
+AC_C_INLINE
+
+if test "$ac_cv_c_inline" != no ; then
+  AC_DEFINE(HAVE_INLINE,1)
+  AC_SUBST(HAVE_INLINE)
+fi
+@end example
+
+@noindent
+and the macro will then be defined in the compilation flags or by
+including the file @file{config.h} before any library headers.  
+
+The following autoconf test will check for @code{extern inline},
+
+@smallexample
+dnl Check for "extern inline", using a modified version
+dnl of the test for AC_C_INLINE from acspecific.mt
+dnl
+AC_CACHE_CHECK([for extern inline], ac_cv_c_extern_inline,
+[ac_cv_c_extern_inline=no
+AC_TRY_COMPILE([extern $ac_cv_c_inline double foo(double x);
+extern $ac_cv_c_inline double foo(double x) @{ return x+1.0; @};
+double foo (double x) @{ return x + 1.0; @};], 
+[  foo(1.0)  ],
+[ac_cv_c_extern_inline="yes"])
+])
+
+if test "$ac_cv_c_extern_inline" != no ; then
+  AC_DEFINE(HAVE_INLINE,1)
+  AC_SUBST(HAVE_INLINE)
+fi
+@end smallexample
+
+The substitution of portability functions can be made automatically if
+you use @code{autoconf}. For example, to test whether the BSD function
+@code{hypot} is available you can include the following line in the
+configure file @file{configure.in} for your application,
+
+@example
+AC_CHECK_FUNCS(hypot)
+@end example
+
+@noindent
+and place the following macro definitions in the file
+@file{config.h.in},
+
+@example
+/* Substitute gsl_hypot for missing system hypot */
+
+#ifndef HAVE_HYPOT
+#define hypot gsl_hypot
+#endif
+@end example
+
+@noindent
+The application source files can then use the include command
+@code{#include <config.h>} to substitute @code{gsl_hypot} for each
+occurrence of @code{hypot} when @code{hypot} is not available.
diff --git a/gsl-1.9/doc/blas.texi b/gsl-1.9/doc/blas.texi
new file mode 100644
index 0000000..65ad95b
--- /dev/null
+++ b/gsl-1.9/doc/blas.texi
@@ -0,0 +1,686 @@
+@cindex linear algebra, BLAS
+@cindex matrix, operations
+@cindex vector, operations
+@cindex BLAS
+@cindex CBLAS
+@cindex Basic Linear Algebra Subroutines (BLAS)
+
+The Basic Linear Algebra Subprograms (@sc{blas}) define a set of fundamental
+operations on vectors and matrices which can be used to create optimized
+higher-level linear algebra functionality.
+
+The library provides a low-level layer which corresponds directly to the
+C-language @sc{blas} standard, referred to here as ``@sc{cblas}'', and a
+higher-level interface for operations on GSL vectors and matrices.
+Users who are interested in simple operations on GSL vector and matrix
+objects should use the high-level layer, which is declared in the file
+@code{gsl_blas.h}.  This should satisfy the needs of most users.  Note
+that GSL matrices are implemented using dense-storage so the interface
+only includes the corresponding dense-storage @sc{blas} functions.  The full
+@sc{blas} functionality for band-format and packed-format matrices is
+available through the low-level @sc{cblas} interface.
+
+The interface for the @code{gsl_cblas} layer is specified in the file
+@code{gsl_cblas.h}.  This interface corresponds to the @sc{blas} Technical
+Forum's draft standard for the C interface to legacy @sc{blas}
+implementations. Users who have access to other conforming @sc{cblas}
+implementations can use these in place of the version provided by the
+library.  Note that users who have only a Fortran @sc{blas} library can
+use a @sc{cblas} conformant wrapper to convert it into a @sc{cblas}
+library.  A reference @sc{cblas} wrapper for legacy Fortran
+implementations exists as part of the draft @sc{cblas} standard and can
+be obtained from Netlib.  The complete set of @sc{cblas} functions is
+listed in an appendix (@pxref{GSL CBLAS Library}).
+
+There are three levels of @sc{blas} operations,
+
+@table @b
+@item Level 1
+Vector operations, e.g. @math{y = \alpha x + y}
+@item Level 2
+Matrix-vector operations, e.g. @math{y = \alpha A x + \beta y}
+@item Level 3
+Matrix-matrix operations, e.g. @math{C = \alpha A B + C}
+@end table
+
+@noindent
+Each routine has a name which specifies the operation, the type of
+matrices involved and their precisions.  Some of the most common
+operations and their names are given below,
+
+@table @b
+@item DOT
+scalar product, @math{x^T y}
+@item AXPY
+vector sum, @math{\alpha x + y}
+@item MV
+matrix-vector product, @math{A x}
+@item SV
+matrix-vector solve, @math{inv(A) x}
+@item MM
+matrix-matrix product, @math{A B}
+@item SM
+matrix-matrix solve, @math{inv(A) B}
+@end table
+
+@noindent
+The types of matrices are,
+
+@table @b
+@item GE
+general
+@item GB
+general band
+@item SY
+symmetric
+@item SB
+symmetric band
+@item SP
+symmetric packed
+@item HE
+hermitian
+@item HB
+hermitian band
+@item HP
+hermitian packed
+@item TR
+triangular 
+@item TB
+triangular band
+@item TP
+triangular packed
+@end table
+
+@noindent
+Each operation is defined for four precisions,
+
+@table @b
+@item S
+single real
+@item D
+double real
+@item C
+single complex
+@item Z
+double complex
+@end table
+
+@noindent
+Thus, for example, the name @sc{sgemm} stands for ``single-precision
+general matrix-matrix multiply'' and @sc{zgemm} stands for
+``double-precision complex matrix-matrix multiply''.
+
+@menu
+* GSL BLAS Interface::          
+* BLAS Examples::               
+* BLAS References and Further Reading::  
+@end menu
+
+@node GSL BLAS Interface
+@section GSL BLAS Interface
+
+GSL provides dense vector and matrix objects, based on the relevant
+built-in types.  The library provides an interface to the @sc{blas}
+operations which apply to these objects.  The interface to this
+functionality is given in the file @code{gsl_blas.h}.
+
+@comment CblasNoTrans, CblasTrans, CblasConjTrans
+@comment CblasUpper, CblasLower
+@comment CblasNonUnit, CblasUnit
+@comment CblasLeft, CblasRight
+
+@menu
+* Level 1 GSL BLAS Interface::  
+* Level 2 GSL BLAS Interface::  
+* Level 3 GSL BLAS Interface::  
+@end menu
+
+@node Level 1 GSL BLAS Interface
+@subsection Level 1 
+
+@deftypefun int gsl_blas_sdsdot (float @var{alpha}, const gsl_vector_float * @var{x}, const gsl_vector_float * @var{y}, float * @var{result})
+@cindex DOT, Level-1 BLAS
+This function computes the sum @math{\alpha + x^T y} for the vectors
+@var{x} and @var{y}, returning the result in @var{result}.
+@end deftypefun
+
+@deftypefun int gsl_blas_sdot (const gsl_vector_float * @var{x}, const gsl_vector_float * @var{y}, float * @var{result})
+@deftypefunx int gsl_blas_dsdot (const gsl_vector_float * @var{x}, const gsl_vector_float * @var{y}, double * @var{result})
+@deftypefunx int gsl_blas_ddot (const gsl_vector * @var{x}, const gsl_vector * @var{y}, double * @var{result})
+These functions compute the scalar product @math{x^T y} for the vectors
+@var{x} and @var{y}, returning the result in @var{result}.
+@end deftypefun
+
+@deftypefun int gsl_blas_cdotu (const gsl_vector_complex_float * @var{x}, const gsl_vector_complex_float * @var{y}, gsl_complex_float * @var{dotu})
+@deftypefunx int gsl_blas_zdotu (const gsl_vector_complex * @var{x}, const gsl_vector_complex * @var{y}, gsl_complex * @var{dotu})
+These functions compute the complex scalar product @math{x^T y} for the
+vectors @var{x} and @var{y}, returning the result in @var{result}
+@end deftypefun
+
+@deftypefun int gsl_blas_cdotc (const gsl_vector_complex_float * @var{x}, const gsl_vector_complex_float * @var{y}, gsl_complex_float * @var{dotc})
+@deftypefunx int gsl_blas_zdotc (const gsl_vector_complex * @var{x}, const gsl_vector_complex * @var{y}, gsl_complex * @var{dotc})
+These functions compute the complex conjugate scalar product @math{x^H
+y} for the vectors @var{x} and @var{y}, returning the result in
+@var{result}
+@end deftypefun
+
+@deftypefun float gsl_blas_snrm2 (const gsl_vector_float * @var{x})
+@deftypefunx double gsl_blas_dnrm2 (const gsl_vector * @var{x})
+@cindex NRM2, Level-1 BLAS
+These functions compute the Euclidean norm 
+@c{$||x||_2 = \sqrt{\sum x_i^2}$}
+@math{||x||_2 = \sqrt @{\sum x_i^2@}} of the vector @var{x}.
+@end deftypefun
+
+@deftypefun float gsl_blas_scnrm2 (const gsl_vector_complex_float * @var{x})
+@deftypefunx double gsl_blas_dznrm2 (const gsl_vector_complex * @var{x})
+These functions compute the Euclidean norm of the complex vector @var{x},
+@tex
+\beforedisplay
+$$
+||x||_2 = \sqrt{\sum (\Re(x_i)^2 + \Im(x_i)^2)}.
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+||x||_2 = \sqrt @{\sum (\Re(x_i)^2 + \Im(x_i)^2)@}.
+@end example
+@end ifinfo
+@end deftypefun
+
+@deftypefun float gsl_blas_sasum (const gsl_vector_float * @var{x})
+@deftypefunx double gsl_blas_dasum (const gsl_vector * @var{x})
+@cindex ASUM, Level-1 BLAS
+These functions compute the absolute sum @math{\sum |x_i|} of the
+elements of the vector @var{x}.
+@end deftypefun
+
+@deftypefun float gsl_blas_scasum (const gsl_vector_complex_float * @var{x})
+@deftypefunx double gsl_blas_dzasum (const gsl_vector_complex * @var{x})
+These functions compute the sum of the magnitudes of the real and
+imaginary parts of the complex vector @var{x}, 
+@c{$\sum \left( |\Re(x_i)| + |\Im(x_i)| \right)$}
+@math{\sum |\Re(x_i)| + |\Im(x_i)|}.
+@end deftypefun
+
+@deftypefun CBLAS_INDEX_t gsl_blas_isamax (const gsl_vector_float * @var{x})
+@deftypefunx CBLAS_INDEX_t gsl_blas_idamax (const gsl_vector * @var{x})
+@deftypefunx CBLAS_INDEX_t gsl_blas_icamax (const gsl_vector_complex_float * @var{x})
+@deftypefunx CBLAS_INDEX_t gsl_blas_izamax (const gsl_vector_complex * @var{x})
+@cindex AMAX, Level-1 BLAS
+These functions return the index of the largest element of the vector
+@var{x}. The largest element is determined by its absolute magnitude for
+real vectors and by the sum of the magnitudes of the real and imaginary
+parts @math{|\Re(x_i)| + |\Im(x_i)|} for complex vectors.  If the
+largest value occurs several times then the index of the first
+occurrence is returned.
+@end deftypefun
+
+@deftypefun int gsl_blas_sswap (gsl_vector_float * @var{x}, gsl_vector_float * @var{y})
+@deftypefunx int gsl_blas_dswap (gsl_vector * @var{x}, gsl_vector * @var{y})
+@deftypefunx int gsl_blas_cswap (gsl_vector_complex_float * @var{x}, gsl_vector_complex_float * @var{y})
+@deftypefunx int gsl_blas_zswap (gsl_vector_complex * @var{x}, gsl_vector_complex * @var{y})
+@cindex SWAP, Level-1 BLAS
+These functions exchange the elements of the vectors @var{x} and @var{y}.
+@end deftypefun
+
+@deftypefun int gsl_blas_scopy (const gsl_vector_float * @var{x}, gsl_vector_float * @var{y})
+@deftypefunx int gsl_blas_dcopy (const gsl_vector * @var{x}, gsl_vector * @var{y})
+@deftypefunx int gsl_blas_ccopy (const gsl_vector_complex_float * @var{x}, gsl_vector_complex_float * @var{y})
+@deftypefunx int gsl_blas_zcopy (const gsl_vector_complex * @var{x}, gsl_vector_complex * @var{y})
+@cindex COPY, Level-1 BLAS
+These functions copy the elements of the vector @var{x} into the vector
+@var{y}.
+@end deftypefun
+
+
+@deftypefun int gsl_blas_saxpy (float @var{alpha}, const gsl_vector_float * @var{x}, gsl_vector_float * @var{y})
+@deftypefunx int gsl_blas_daxpy (double @var{alpha}, const gsl_vector * @var{x}, gsl_vector * @var{y})
+@deftypefunx int gsl_blas_caxpy (const gsl_complex_float @var{alpha}, const gsl_vector_complex_float * @var{x}, gsl_vector_complex_float * @var{y})
+@deftypefunx int gsl_blas_zaxpy (const gsl_complex @var{alpha}, const gsl_vector_complex * @var{x}, gsl_vector_complex * @var{y})
+@cindex AXPY, Level-1 BLAS
+@cindex DAXPY, Level-1 BLAS
+@cindex SAXPY, Level-1 BLAS
+These functions compute the sum @math{y = \alpha x + y} for the vectors
+@var{x} and @var{y}.
+@end deftypefun
+
+@deftypefun void gsl_blas_sscal (float @var{alpha}, gsl_vector_float * @var{x})
+@deftypefunx void gsl_blas_dscal (double @var{alpha}, gsl_vector * @var{x})
+@deftypefunx void gsl_blas_cscal (const gsl_complex_float @var{alpha}, gsl_vector_complex_float * @var{x})
+@deftypefunx void gsl_blas_zscal (const gsl_complex @var{alpha}, gsl_vector_complex * @var{x})
+@deftypefunx void gsl_blas_csscal (float @var{alpha}, gsl_vector_complex_float * @var{x})
+@deftypefunx void gsl_blas_zdscal (double @var{alpha}, gsl_vector_complex * @var{x})
+@cindex SCAL, Level-1 BLAS
+These functions rescale the vector @var{x} by the multiplicative factor
+@var{alpha}.
+@end deftypefun
+
+@deftypefun int gsl_blas_srotg (float @var{a}[], float @var{b}[], float @var{c}[], float @var{s}[])
+@deftypefunx int gsl_blas_drotg (double @var{a}[], double @var{b}[], double @var{c}[], double @var{s}[])
+@cindex ROTG, Level-1 BLAS
+@cindex Givens Rotation, BLAS
+These functions compute a Givens rotation @math{(c,s)} which zeroes the
+vector @math{(a,b)},
+@tex
+\beforedisplay
+$$
+\left(
+\matrix{c&s\cr
+-s&c\cr}
+\right)
+\left(
+\matrix{a\cr
+b\cr}
+\right)
+=
+\left(
+\matrix{r'\cr
+0\cr}
+\right)
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+[  c  s ] [ a ] = [ r ]
+[ -s  c ] [ b ]   [ 0 ]
+@end example
+
+@end ifinfo
+@noindent
+The variables @var{a} and @var{b} are overwritten by the routine.
+@end deftypefun
+
+@deftypefun int gsl_blas_srot (gsl_vector_float * @var{x}, gsl_vector_float * @var{y}, float @var{c}, float @var{s})
+@deftypefunx int gsl_blas_drot (gsl_vector * @var{x}, gsl_vector * @var{y}, const double @var{c}, const double @var{s})
+These functions apply a Givens rotation @math{(x', y') = (c x + s y, -s
+x + c y)} to the vectors @var{x}, @var{y}.
+@end deftypefun
+
+@deftypefun int gsl_blas_srotmg (float @var{d1}[], float @var{d2}[], float @var{b1}[], float @var{b2}, float @var{P}[])
+@deftypefunx int gsl_blas_drotmg (double @var{d1}[], double @var{d2}[], double @var{b1}[], double @var{b2}, double @var{P}[])
+@cindex Modified Givens Rotation, BLAS
+@cindex Givens Rotation, Modified, BLAS
+These functions compute a modified Givens transformation.  The modified
+Givens transformation is defined in the original Level-1 @sc{blas}
+specification, given in the references.
+@end deftypefun
+
+@deftypefun int gsl_blas_srotm (gsl_vector_float * @var{x}, gsl_vector_float * @var{y}, const float @var{P}[])
+@deftypefunx int gsl_blas_drotm (gsl_vector * @var{x}, gsl_vector * @var{y}, const double @var{P}[])
+These functions apply a modified Givens transformation.  
+@end deftypefun
+
+@node Level 2 GSL BLAS Interface
+@subsection Level 2 
+
+@deftypefun int gsl_blas_sgemv (CBLAS_TRANSPOSE_t @var{TransA}, float @var{alpha}, const gsl_matrix_float * @var{A}, const gsl_vector_float * @var{x}, float @var{beta}, gsl_vector_float * @var{y})
+@deftypefunx int gsl_blas_dgemv (CBLAS_TRANSPOSE_t @var{TransA}, double @var{alpha}, const gsl_matrix * @var{A}, const gsl_vector * @var{x}, double @var{beta}, gsl_vector * @var{y})
+@deftypefunx int gsl_blas_cgemv (CBLAS_TRANSPOSE_t @var{TransA}, const gsl_complex_float @var{alpha}, const gsl_matrix_complex_float * @var{A}, const gsl_vector_complex_float * @var{x}, const gsl_complex_float @var{beta}, gsl_vector_complex_float * @var{y})
+@deftypefunx int gsl_blas_zgemv (CBLAS_TRANSPOSE_t @var{TransA}, const gsl_complex @var{alpha}, const gsl_matrix_complex * @var{A}, const gsl_vector_complex * @var{x}, const gsl_complex @var{beta}, gsl_vector_complex * @var{y})
+@cindex GEMV, Level-2 BLAS
+These functions compute the matrix-vector product and sum @math{y =
+\alpha op(A) x + \beta y}, where @math{op(A) = A},
+@math{A^T}, @math{A^H} for @var{TransA} = @code{CblasNoTrans},
+@code{CblasTrans}, @code{CblasConjTrans}.
+@end deftypefun
+
+
+@deftypefun int gsl_blas_strmv (CBLAS_UPLO_t @var{Uplo}, CBLAS_TRANSPOSE_t @var{TransA}, CBLAS_DIAG_t @var{Diag}, const gsl_matrix_float * @var{A}, gsl_vector_float * @var{x})
+@deftypefunx int gsl_blas_dtrmv (CBLAS_UPLO_t @var{Uplo}, CBLAS_TRANSPOSE_t @var{TransA}, CBLAS_DIAG_t @var{Diag}, const gsl_matrix * @var{A}, gsl_vector * @var{x})
+@deftypefunx int gsl_blas_ctrmv (CBLAS_UPLO_t @var{Uplo}, CBLAS_TRANSPOSE_t @var{TransA}, CBLAS_DIAG_t @var{Diag}, const gsl_matrix_complex_float * @var{A}, gsl_vector_complex_float * @var{x})
+@deftypefunx int gsl_blas_ztrmv (CBLAS_UPLO_t @var{Uplo}, CBLAS_TRANSPOSE_t @var{TransA}, CBLAS_DIAG_t @var{Diag}, const gsl_matrix_complex * @var{A}, gsl_vector_complex * @var{x})
+@cindex TRMV, Level-2 BLAS
+These functions compute the matrix-vector product 
+@math{x = op(A) x} for the triangular matrix @var{A}, where
+@math{op(A) = A}, @math{A^T}, @math{A^H} for @var{TransA} =
+@code{CblasNoTrans}, @code{CblasTrans}, @code{CblasConjTrans}.  When
+@var{Uplo} is @code{CblasUpper} then the upper triangle of @var{A} is
+used, and when @var{Uplo} is @code{CblasLower} then the lower triangle
+of @var{A} is used.  If @var{Diag} is @code{CblasNonUnit} then the
+diagonal of the matrix is used, but if @var{Diag} is @code{CblasUnit}
+then the diagonal elements of the matrix @var{A} are taken as unity and
+are not referenced.
+@end deftypefun
+
+
+@deftypefun int gsl_blas_strsv (CBLAS_UPLO_t @var{Uplo}, CBLAS_TRANSPOSE_t @var{TransA}, CBLAS_DIAG_t @var{Diag}, const gsl_matrix_float * @var{A}, gsl_vector_float * @var{x})
+@deftypefunx int gsl_blas_dtrsv (CBLAS_UPLO_t @var{Uplo}, CBLAS_TRANSPOSE_t @var{TransA}, CBLAS_DIAG_t @var{Diag}, const gsl_matrix * @var{A}, gsl_vector * @var{x})
+@deftypefunx int gsl_blas_ctrsv (CBLAS_UPLO_t @var{Uplo}, CBLAS_TRANSPOSE_t @var{TransA}, CBLAS_DIAG_t @var{Diag}, const gsl_matrix_complex_float * @var{A}, gsl_vector_complex_float * @var{x})
+@deftypefunx int gsl_blas_ztrsv (CBLAS_UPLO_t @var{Uplo}, CBLAS_TRANSPOSE_t @var{TransA}, CBLAS_DIAG_t @var{Diag}, const gsl_matrix_complex * @var{A}, gsl_vector_complex * @var{x})
+@cindex TRSV, Level-2 BLAS
+These functions compute @math{inv(op(A)) x} for @var{x}, where
+@math{op(A) = A}, @math{A^T}, @math{A^H} for @var{TransA} =
+@code{CblasNoTrans}, @code{CblasTrans}, @code{CblasConjTrans}.  When
+@var{Uplo} is @code{CblasUpper} then the upper triangle of @var{A} is
+used, and when @var{Uplo} is @code{CblasLower} then the lower triangle
+of @var{A} is used.  If @var{Diag} is @code{CblasNonUnit} then the
+diagonal of the matrix is used, but if @var{Diag} is @code{CblasUnit}
+then the diagonal elements of the matrix @var{A} are taken as unity and
+are not referenced.
+@end deftypefun
+
+
+@deftypefun int gsl_blas_ssymv (CBLAS_UPLO_t @var{Uplo}, float @var{alpha}, const gsl_matrix_float * @var{A}, const gsl_vector_float * @var{x}, float @var{beta}, gsl_vector_float * @var{y})
+@deftypefunx int gsl_blas_dsymv (CBLAS_UPLO_t @var{Uplo}, double @var{alpha}, const gsl_matrix * @var{A}, const gsl_vector * @var{x}, double @var{beta}, gsl_vector * @var{y})
+@cindex SYMV, Level-2 BLAS
+These functions compute the matrix-vector product and sum @math{y =
+\alpha A x + \beta y} for the symmetric matrix @var{A}.  Since the
+matrix @var{A} is symmetric only its upper half or lower half need to be
+stored.  When @var{Uplo} is @code{CblasUpper} then the upper triangle
+and diagonal of @var{A} are used, and when @var{Uplo} is
+@code{CblasLower} then the lower triangle and diagonal of @var{A} are
+used.
+@end deftypefun
+
+@deftypefun int gsl_blas_chemv (CBLAS_UPLO_t @var{Uplo}, const gsl_complex_float @var{alpha}, const gsl_matrix_complex_float * @var{A}, const gsl_vector_complex_float * @var{x}, const gsl_complex_float @var{beta}, gsl_vector_complex_float * @var{y})
+@deftypefunx int gsl_blas_zhemv (CBLAS_UPLO_t @var{Uplo}, const gsl_complex @var{alpha}, const gsl_matrix_complex * @var{A}, const gsl_vector_complex * @var{x}, const gsl_complex @var{beta}, gsl_vector_complex * @var{y})
+@cindex HEMV, Level-2 BLAS
+These functions compute the matrix-vector product and sum @math{y =
+\alpha A x + \beta y} for the hermitian matrix @var{A}.  Since the
+matrix @var{A} is hermitian only its upper half or lower half need to be
+stored.  When @var{Uplo} is @code{CblasUpper} then the upper triangle
+and diagonal of @var{A} are used, and when @var{Uplo} is
+@code{CblasLower} then the lower triangle and diagonal of @var{A} are
+used.  The imaginary elements of the diagonal are automatically assumed
+to be zero and are not referenced.
+@end deftypefun
+
+@deftypefun int gsl_blas_sger (float @var{alpha}, const gsl_vector_float * @var{x}, const gsl_vector_float * @var{y}, gsl_matrix_float * @var{A})
+@deftypefunx int gsl_blas_dger (double @var{alpha}, const gsl_vector * @var{x}, const gsl_vector * @var{y}, gsl_matrix * @var{A})
+@deftypefunx int gsl_blas_cgeru (const gsl_complex_float @var{alpha}, const gsl_vector_complex_float * @var{x}, const gsl_vector_complex_float * @var{y}, gsl_matrix_complex_float * @var{A})
+@deftypefunx int gsl_blas_zgeru (const gsl_complex @var{alpha}, const gsl_vector_complex * @var{x}, const gsl_vector_complex * @var{y}, gsl_matrix_complex * @var{A})
+@cindex GER, Level-2 BLAS
+@cindex GERU, Level-2 BLAS
+These functions compute the rank-1 update @math{A = \alpha x y^T + A} of
+the matrix @var{A}.
+@end deftypefun
+
+@deftypefun int gsl_blas_cgerc (const gsl_complex_float @var{alpha}, const gsl_vector_complex_float * @var{x}, const gsl_vector_complex_float * @var{y}, gsl_matrix_complex_float * @var{A})
+@deftypefunx int gsl_blas_zgerc (const gsl_complex @var{alpha}, const gsl_vector_complex * @var{x}, const gsl_vector_complex * @var{y}, gsl_matrix_complex * @var{A})
+@cindex GERC, Level-2 BLAS
+These functions compute the conjugate rank-1 update @math{A = \alpha x
+y^H + A} of the matrix @var{A}.
+@end deftypefun
+
+@deftypefun int gsl_blas_ssyr (CBLAS_UPLO_t @var{Uplo}, float @var{alpha}, const gsl_vector_float * @var{x}, gsl_matrix_float * @var{A})
+@deftypefunx int gsl_blas_dsyr (CBLAS_UPLO_t @var{Uplo}, double @var{alpha}, const gsl_vector * @var{x}, gsl_matrix * @var{A})
+@cindex SYR, Level-2 BLAS
+These functions compute the symmetric rank-1 update @math{A = \alpha x
+x^T + A} of the symmetric matrix @var{A}.  Since the matrix @var{A} is
+symmetric only its upper half or lower half need to be stored.  When
+@var{Uplo} is @code{CblasUpper} then the upper triangle and diagonal of
+@var{A} are used, and when @var{Uplo} is @code{CblasLower} then the
+lower triangle and diagonal of @var{A} are used.
+@end deftypefun
+
+@deftypefun int gsl_blas_cher (CBLAS_UPLO_t @var{Uplo}, float @var{alpha}, const gsl_vector_complex_float * @var{x}, gsl_matrix_complex_float * @var{A})
+@deftypefunx int gsl_blas_zher (CBLAS_UPLO_t @var{Uplo}, double @var{alpha}, const gsl_vector_complex * @var{x}, gsl_matrix_complex * @var{A})
+@cindex HER, Level-2 BLAS
+These functions compute the hermitian rank-1 update @math{A = \alpha x
+x^H + A} of the hermitian matrix @var{A}.  Since the matrix @var{A} is
+hermitian only its upper half or lower half need to be stored.  When
+@var{Uplo} is @code{CblasUpper} then the upper triangle and diagonal of
+@var{A} are used, and when @var{Uplo} is @code{CblasLower} then the
+lower triangle and diagonal of @var{A} are used.  The imaginary elements
+of the diagonal are automatically set to zero.
+@end deftypefun
+
+@deftypefun int gsl_blas_ssyr2 (CBLAS_UPLO_t @var{Uplo}, float @var{alpha}, const gsl_vector_float * @var{x}, const gsl_vector_float * @var{y}, gsl_matrix_float * @var{A})
+@deftypefunx int gsl_blas_dsyr2 (CBLAS_UPLO_t @var{Uplo}, double @var{alpha}, const gsl_vector * @var{x}, const gsl_vector * @var{y}, gsl_matrix * @var{A})
+@cindex SYR2, Level-2 BLAS
+These functions compute the symmetric rank-2 update @math{A = \alpha x
+y^T + \alpha y x^T + A} of the symmetric matrix @var{A}.  Since the
+matrix @var{A} is symmetric only its upper half or lower half need to be
+stored.  When @var{Uplo} is @code{CblasUpper} then the upper triangle
+and diagonal of @var{A} are used, and when @var{Uplo} is
+@code{CblasLower} then the lower triangle and diagonal of @var{A} are
+used.
+@end deftypefun
+
+@deftypefun int gsl_blas_cher2 (CBLAS_UPLO_t @var{Uplo}, const gsl_complex_float @var{alpha}, const gsl_vector_complex_float * @var{x}, const gsl_vector_complex_float * @var{y}, gsl_matrix_complex_float * @var{A})
+@deftypefunx int gsl_blas_zher2 (CBLAS_UPLO_t @var{Uplo}, const gsl_complex @var{alpha}, const gsl_vector_complex * @var{x}, const gsl_vector_complex * @var{y}, gsl_matrix_complex * @var{A})
+@cindex HER2, Level-2 BLAS
+These functions compute the hermitian rank-2 update @math{A = \alpha x
+y^H + \alpha^* y x^H A} of the hermitian matrix @var{A}.  Since the
+matrix @var{A} is hermitian only its upper half or lower half need to be
+stored.  When @var{Uplo} is @code{CblasUpper} then the upper triangle
+and diagonal of @var{A} are used, and when @var{Uplo} is
+@code{CblasLower} then the lower triangle and diagonal of @var{A} are
+used.  The imaginary elements of the diagonal are automatically set to zero.
+@end deftypefun
+
+@node Level 3 GSL BLAS Interface
+@subsection Level 3
+
+
+@deftypefun int gsl_blas_sgemm (CBLAS_TRANSPOSE_t @var{TransA}, CBLAS_TRANSPOSE_t @var{TransB}, float @var{alpha}, const gsl_matrix_float * @var{A}, const gsl_matrix_float * @var{B}, float @var{beta}, gsl_matrix_float * @var{C})
+@deftypefunx int gsl_blas_dgemm (CBLAS_TRANSPOSE_t @var{TransA}, CBLAS_TRANSPOSE_t @var{TransB}, double @var{alpha}, const gsl_matrix * @var{A}, const gsl_matrix * @var{B}, double @var{beta}, gsl_matrix * @var{C})
+@deftypefunx int gsl_blas_cgemm (CBLAS_TRANSPOSE_t @var{TransA}, CBLAS_TRANSPOSE_t @var{TransB}, const gsl_complex_float @var{alpha}, const gsl_matrix_complex_float * @var{A}, const gsl_matrix_complex_float * @var{B}, const gsl_complex_float @var{beta}, gsl_matrix_complex_float * @var{C})
+@deftypefunx int gsl_blas_zgemm (CBLAS_TRANSPOSE_t @var{TransA}, CBLAS_TRANSPOSE_t @var{TransB}, const gsl_complex @var{alpha}, const gsl_matrix_complex * @var{A}, const gsl_matrix_complex * @var{B}, const gsl_complex @var{beta}, gsl_matrix_complex * @var{C})
+@cindex GEMM, Level-3 BLAS
+These functions compute the matrix-matrix product and sum @math{C =
+\alpha op(A) op(B) + \beta C} where @math{op(A) = A}, @math{A^T},
+@math{A^H} for @var{TransA} = @code{CblasNoTrans}, @code{CblasTrans},
+@code{CblasConjTrans} and similarly for the parameter @var{TransB}.
+@end deftypefun
+
+
+@deftypefun int gsl_blas_ssymm (CBLAS_SIDE_t @var{Side}, CBLAS_UPLO_t @var{Uplo}, float @var{alpha}, const gsl_matrix_float * @var{A}, const gsl_matrix_float * @var{B}, float @var{beta}, gsl_matrix_float * @var{C})
+@deftypefunx int gsl_blas_dsymm (CBLAS_SIDE_t @var{Side}, CBLAS_UPLO_t @var{Uplo}, double @var{alpha}, const gsl_matrix * @var{A}, const gsl_matrix * @var{B}, double @var{beta}, gsl_matrix * @var{C})
+@deftypefunx int gsl_blas_csymm (CBLAS_SIDE_t @var{Side}, CBLAS_UPLO_t @var{Uplo}, const gsl_complex_float @var{alpha}, const gsl_matrix_complex_float * @var{A}, const gsl_matrix_complex_float * @var{B}, const gsl_complex_float @var{beta}, gsl_matrix_complex_float * @var{C})
+@deftypefunx int gsl_blas_zsymm (CBLAS_SIDE_t @var{Side}, CBLAS_UPLO_t @var{Uplo}, const gsl_complex @var{alpha}, const gsl_matrix_complex * @var{A}, const gsl_matrix_complex * @var{B}, const gsl_complex @var{beta}, gsl_matrix_complex * @var{C})
+@cindex SYMM, Level-3 BLAS
+These functions compute the matrix-matrix product and sum @math{C =
+\alpha A B + \beta C} for @var{Side} is @code{CblasLeft} and @math{C =
+\alpha B A + \beta C} for @var{Side} is @code{CblasRight}, where the
+matrix @var{A} is symmetric.  When @var{Uplo} is @code{CblasUpper} then
+the upper triangle and diagonal of @var{A} are used, and when @var{Uplo}
+is @code{CblasLower} then the lower triangle and diagonal of @var{A} are
+used.
+@end deftypefun
+
+@deftypefun int gsl_blas_chemm (CBLAS_SIDE_t @var{Side}, CBLAS_UPLO_t @var{Uplo}, const gsl_complex_float @var{alpha}, const gsl_matrix_complex_float * @var{A}, const gsl_matrix_complex_float * @var{B}, const gsl_complex_float @var{beta}, gsl_matrix_complex_float * @var{C})
+@deftypefunx int gsl_blas_zhemm (CBLAS_SIDE_t @var{Side}, CBLAS_UPLO_t @var{Uplo}, const gsl_complex @var{alpha}, const gsl_matrix_complex * @var{A}, const gsl_matrix_complex * @var{B}, const gsl_complex @var{beta}, gsl_matrix_complex * @var{C})
+@cindex HEMM, Level-3 BLAS
+These functions compute the matrix-matrix product and sum @math{C =
+\alpha A B + \beta C} for @var{Side} is @code{CblasLeft} and @math{C =
+\alpha B A + \beta C} for @var{Side} is @code{CblasRight}, where the
+matrix @var{A} is hermitian.  When @var{Uplo} is @code{CblasUpper} then
+the upper triangle and diagonal of @var{A} are used, and when @var{Uplo}
+is @code{CblasLower} then the lower triangle and diagonal of @var{A} are
+used.  The imaginary elements of the diagonal are automatically set to
+zero.
+@end deftypefun
+
+@deftypefun int gsl_blas_strmm (CBLAS_SIDE_t @var{Side}, CBLAS_UPLO_t @var{Uplo}, CBLAS_TRANSPOSE_t @var{TransA}, CBLAS_DIAG_t @var{Diag}, float @var{alpha}, const gsl_matrix_float * @var{A}, gsl_matrix_float * @var{B})
+@deftypefunx int gsl_blas_dtrmm (CBLAS_SIDE_t @var{Side}, CBLAS_UPLO_t @var{Uplo}, CBLAS_TRANSPOSE_t @var{TransA}, CBLAS_DIAG_t @var{Diag}, double @var{alpha}, const gsl_matrix * @var{A}, gsl_matrix * @var{B})
+@deftypefunx int gsl_blas_ctrmm (CBLAS_SIDE_t @var{Side}, CBLAS_UPLO_t @var{Uplo}, CBLAS_TRANSPOSE_t @var{TransA}, CBLAS_DIAG_t @var{Diag}, const gsl_complex_float @var{alpha}, const gsl_matrix_complex_float * @var{A}, gsl_matrix_complex_float * @var{B})
+@deftypefunx int gsl_blas_ztrmm (CBLAS_SIDE_t @var{Side}, CBLAS_UPLO_t @var{Uplo}, CBLAS_TRANSPOSE_t @var{TransA}, CBLAS_DIAG_t @var{Diag}, const gsl_complex @var{alpha}, const gsl_matrix_complex * @var{A}, gsl_matrix_complex * @var{B})
+@cindex TRMM, Level-3 BLAS
+These functions compute the matrix-matrix product @math{B = \alpha op(A)
+B} for @var{Side} is @code{CblasLeft} and @math{B = \alpha B op(A)} for
+@var{Side} is @code{CblasRight}.  The matrix @var{A} is triangular and
+@math{op(A) = A}, @math{A^T}, @math{A^H} for @var{TransA} =
+@code{CblasNoTrans}, @code{CblasTrans}, @code{CblasConjTrans}. When
+@var{Uplo} is @code{CblasUpper} then the upper triangle of @var{A} is
+used, and when @var{Uplo} is @code{CblasLower} then the lower triangle
+of @var{A} is used.  If @var{Diag} is @code{CblasNonUnit} then the
+diagonal of @var{A} is used, but if @var{Diag} is @code{CblasUnit} then
+the diagonal elements of the matrix @var{A} are taken as unity and are
+not referenced.
+@end deftypefun
+
+
+@deftypefun int gsl_blas_strsm (CBLAS_SIDE_t @var{Side}, CBLAS_UPLO_t @var{Uplo}, CBLAS_TRANSPOSE_t @var{TransA}, CBLAS_DIAG_t @var{Diag}, float @var{alpha}, const gsl_matrix_float * @var{A}, gsl_matrix_float * @var{B})
+@deftypefunx int gsl_blas_dtrsm (CBLAS_SIDE_t @var{Side}, CBLAS_UPLO_t @var{Uplo}, CBLAS_TRANSPOSE_t @var{TransA}, CBLAS_DIAG_t @var{Diag}, double @var{alpha}, const gsl_matrix * @var{A}, gsl_matrix * @var{B})
+@deftypefunx int gsl_blas_ctrsm (CBLAS_SIDE_t @var{Side}, CBLAS_UPLO_t @var{Uplo}, CBLAS_TRANSPOSE_t @var{TransA}, CBLAS_DIAG_t @var{Diag}, const gsl_complex_float @var{alpha}, const gsl_matrix_complex_float * @var{A}, gsl_matrix_complex_float * @var{B})
+@deftypefunx int gsl_blas_ztrsm (CBLAS_SIDE_t @var{Side}, CBLAS_UPLO_t @var{Uplo}, CBLAS_TRANSPOSE_t @var{TransA}, CBLAS_DIAG_t @var{Diag}, const gsl_complex @var{alpha}, const gsl_matrix_complex * @var{A}, gsl_matrix_complex * @var{B})
+@cindex TRSM, Level-3 BLAS
+These functions compute the inverse-matrix matrix product 
+@math{B = \alpha op(inv(A))B} for @var{Side} is 
+@code{CblasLeft} and @math{B = \alpha B op(inv(A))} for
+@var{Side} is @code{CblasRight}.  The matrix @var{A} is triangular and
+@math{op(A) = A}, @math{A^T}, @math{A^H} for @var{TransA} =
+@code{CblasNoTrans}, @code{CblasTrans}, @code{CblasConjTrans}. When
+@var{Uplo} is @code{CblasUpper} then the upper triangle of @var{A} is
+used, and when @var{Uplo} is @code{CblasLower} then the lower triangle
+of @var{A} is used.  If @var{Diag} is @code{CblasNonUnit} then the
+diagonal of @var{A} is used, but if @var{Diag} is @code{CblasUnit} then
+the diagonal elements of the matrix @var{A} are taken as unity and are
+not referenced.
+@end deftypefun
+
+@deftypefun int gsl_blas_ssyrk (CBLAS_UPLO_t @var{Uplo}, CBLAS_TRANSPOSE_t @var{Trans}, float @var{alpha}, const gsl_matrix_float * @var{A}, float @var{beta}, gsl_matrix_float * @var{C})
+@deftypefunx int gsl_blas_dsyrk (CBLAS_UPLO_t @var{Uplo}, CBLAS_TRANSPOSE_t @var{Trans}, double @var{alpha}, const gsl_matrix * @var{A}, double @var{beta}, gsl_matrix * @var{C})
+@deftypefunx int gsl_blas_csyrk (CBLAS_UPLO_t @var{Uplo}, CBLAS_TRANSPOSE_t @var{Trans}, const gsl_complex_float @var{alpha}, const gsl_matrix_complex_float * @var{A}, const gsl_complex_float @var{beta}, gsl_matrix_complex_float * @var{C})
+@deftypefunx int gsl_blas_zsyrk (CBLAS_UPLO_t @var{Uplo}, CBLAS_TRANSPOSE_t @var{Trans}, const gsl_complex @var{alpha}, const gsl_matrix_complex * @var{A}, const gsl_complex @var{beta}, gsl_matrix_complex * @var{C})
+@cindex SYRK, Level-3 BLAS
+These functions compute a rank-k update of the symmetric matrix @var{C},
+@math{C = \alpha A A^T + \beta C} when @var{Trans} is
+@code{CblasNoTrans} and @math{C = \alpha A^T A + \beta C} when
+@var{Trans} is @code{CblasTrans}.  Since the matrix @var{C} is symmetric
+only its upper half or lower half need to be stored.  When @var{Uplo} is
+@code{CblasUpper} then the upper triangle and diagonal of @var{C} are
+used, and when @var{Uplo} is @code{CblasLower} then the lower triangle
+and diagonal of @var{C} are used.
+@end deftypefun
+
+@deftypefun int gsl_blas_cherk (CBLAS_UPLO_t @var{Uplo}, CBLAS_TRANSPOSE_t @var{Trans}, float @var{alpha}, const gsl_matrix_complex_float * @var{A}, float @var{beta}, gsl_matrix_complex_float * @var{C})
+@deftypefunx int gsl_blas_zherk (CBLAS_UPLO_t @var{Uplo}, CBLAS_TRANSPOSE_t @var{Trans}, double @var{alpha}, const gsl_matrix_complex * @var{A}, double @var{beta}, gsl_matrix_complex * @var{C})
+@cindex HERK, Level-3 BLAS
+These functions compute a rank-k update of the hermitian matrix @var{C},
+@math{C = \alpha A A^H + \beta C} when @var{Trans} is
+@code{CblasNoTrans} and @math{C = \alpha A^H A + \beta C} when
+@var{Trans} is @code{CblasTrans}.  Since the matrix @var{C} is hermitian
+only its upper half or lower half need to be stored.  When @var{Uplo} is
+@code{CblasUpper} then the upper triangle and diagonal of @var{C} are
+used, and when @var{Uplo} is @code{CblasLower} then the lower triangle
+and diagonal of @var{C} are used.  The imaginary elements of the
+diagonal are automatically set to zero.
+@end deftypefun
+
+@deftypefun int gsl_blas_ssyr2k (CBLAS_UPLO_t @var{Uplo}, CBLAS_TRANSPOSE_t @var{Trans}, float @var{alpha}, const gsl_matrix_float * @var{A}, const gsl_matrix_float * @var{B}, float @var{beta}, gsl_matrix_float * @var{C})
+@deftypefunx int gsl_blas_dsyr2k (CBLAS_UPLO_t @var{Uplo}, CBLAS_TRANSPOSE_t @var{Trans}, double @var{alpha}, const gsl_matrix * @var{A}, const gsl_matrix * @var{B}, double @var{beta}, gsl_matrix * @var{C})
+@deftypefunx int gsl_blas_csyr2k (CBLAS_UPLO_t @var{Uplo}, CBLAS_TRANSPOSE_t @var{Trans}, const gsl_complex_float @var{alpha}, const gsl_matrix_complex_float * @var{A}, const gsl_matrix_complex_float * @var{B}, const gsl_complex_float @var{beta}, gsl_matrix_complex_float * @var{C})
+@deftypefunx int gsl_blas_zsyr2k (CBLAS_UPLO_t @var{Uplo}, CBLAS_TRANSPOSE_t @var{Trans}, const gsl_complex @var{alpha}, const gsl_matrix_complex * @var{A}, const gsl_matrix_complex * @var{B}, const gsl_complex @var{beta}, gsl_matrix_complex * @var{C})
+@cindex SYR2K, Level-3 BLAS
+These functions compute a rank-2k update of the symmetric matrix @var{C},
+@math{C = \alpha A B^T + \alpha B A^T + \beta C} when @var{Trans} is
+@code{CblasNoTrans} and @math{C = \alpha A^T B + \alpha B^T A + \beta C} when
+@var{Trans} is @code{CblasTrans}.  Since the matrix @var{C} is symmetric
+only its upper half or lower half need to be stored.  When @var{Uplo} is
+@code{CblasUpper} then the upper triangle and diagonal of @var{C} are
+used, and when @var{Uplo} is @code{CblasLower} then the lower triangle
+and diagonal of @var{C} are used.
+@end deftypefun
+
+@deftypefun int gsl_blas_cher2k (CBLAS_UPLO_t @var{Uplo}, CBLAS_TRANSPOSE_t @var{Trans}, const gsl_complex_float @var{alpha}, const gsl_matrix_complex_float * @var{A}, const gsl_matrix_complex_float * @var{B}, float @var{beta}, gsl_matrix_complex_float * @var{C})
+@deftypefunx int gsl_blas_zher2k (CBLAS_UPLO_t @var{Uplo}, CBLAS_TRANSPOSE_t @var{Trans}, const gsl_complex @var{alpha}, const gsl_matrix_complex * @var{A}, const gsl_matrix_complex * @var{B}, double @var{beta}, gsl_matrix_complex * @var{C})
+@cindex HER2K, Level-3 BLAS
+These functions compute a rank-2k update of the hermitian matrix @var{C},
+@math{C = \alpha A B^H + \alpha^* B A^H + \beta C} when @var{Trans} is
+@code{CblasNoTrans} and @math{C = \alpha A^H B + \alpha^* B^H A + \beta C} when
+@var{Trans} is @code{CblasConjTrans}.  Since the matrix @var{C} is hermitian
+only its upper half or lower half need to be stored.  When @var{Uplo} is
+@code{CblasUpper} then the upper triangle and diagonal of @var{C} are
+used, and when @var{Uplo} is @code{CblasLower} then the lower triangle
+and diagonal of @var{C} are used.  The imaginary elements of the
+diagonal are automatically set to zero.
+@end deftypefun
+
+@node BLAS Examples
+@section Examples
+
+The following program computes the product of two matrices using the
+Level-3 @sc{blas} function @sc{dgemm},
+@tex
+\beforedisplay
+$$
+\left(
+\matrix{0.11&0.12&0.13\cr
+0.21&0.22&0.23\cr}
+\right)
+\left(
+\matrix{1011&1012\cr
+1021&1022\cr
+1031&1031\cr}
+\right)
+=
+\left(
+\matrix{367.76&368.12\cr
+674.06&674.72\cr}
+\right)
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+[ 0.11 0.12 0.13 ]  [ 1011 1012 ]     [ 367.76 368.12 ]
+[ 0.21 0.22 0.23 ]  [ 1021 1022 ]  =  [ 674.06 674.72 ]
+                    [ 1031 1032 ]
+@end example
+
+@end ifinfo
+@noindent
+The matrices are stored in row major order, according to the C convention 
+for arrays.
+
+@example
+@verbatiminclude examples/blas.c
+@end example
+
+@noindent
+Here is the output from the program,
+
+@example
+$ ./a.out
+@verbatiminclude examples/blas.out
+@end example
+
+@node BLAS References and Further Reading
+@section References and Further Reading
+
+Information on the @sc{blas} standards, including both the legacy and
+draft interface standards, is available online from the @sc{blas}
+Homepage and @sc{blas} Technical Forum web-site.
+
+@itemize @asis
+@item
+@cite{BLAS Homepage} @*
+@uref{http://www.netlib.org/blas/}
+@item
+@cite{BLAS Technical Forum} @*
+@uref{http://www.netlib.org/cgi-bin/checkout/blast/blast.pl}
+@end itemize
+
+@noindent
+The following papers contain the specifications for Level 1, Level 2 and
+Level 3 @sc{blas}.
+
+@itemize @asis
+@item
+C. Lawson, R. Hanson, D. Kincaid, F. Krogh, ``Basic Linear Algebra
+Subprograms for Fortran Usage'', @cite{ACM Transactions on Mathematical
+Software}, Vol.@: 5 (1979), Pages 308--325.
+
+@item
+J.J. Dongarra, J. DuCroz, S. Hammarling, R. Hanson, ``An Extended Set of
+Fortran Basic Linear Algebra Subprograms'', @cite{ACM Transactions on
+Mathematical Software}, Vol.@: 14, No.@: 1 (1988), Pages 1--32.
+
+@item
+J.J. Dongarra, I. Duff, J. DuCroz, S. Hammarling, ``A Set of
+Level 3 Basic Linear Algebra Subprograms'', @cite{ACM Transactions on
+Mathematical Software}, Vol.@: 16 (1990), Pages 1--28.
+@end itemize
+
+@noindent
+Postscript versions of the latter two papers are available from
+@uref{http://www.netlib.org/blas/}. A @sc{cblas} wrapper for Fortran @sc{blas}
+libraries is available from the same location.
diff --git a/gsl-1.9/doc/bspline.eps b/gsl-1.9/doc/bspline.eps
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+9078 4978
+9124 4919
+9170 4864
+9216 4812
+151 MLine
+End
+
+%%PageTrailer
+End %I eop
+showpage
+
+%%Trailer
+end
+%%EOF
diff --git a/gsl-1.9/doc/bspline.texi b/gsl-1.9/doc/bspline.texi
new file mode 100644
index 0000000..718e280
--- /dev/null
+++ b/gsl-1.9/doc/bspline.texi
@@ -0,0 +1,158 @@
+@cindex basis splines, B-splines
+@cindex splines, basis
+
+This chapter describes functions for the computation of smoothing
+basis splines (B-splines). The header file @file{gsl_bspline.h}
+contains prototypes for the bspline functions and related declarations.
+
+@menu
+* Overview of B-splines::
+* Initializing the B-splines solver::
+* Constructing the knots vector::
+* Evaluation of B-spline basis functions::
+* Example programs for B-splines::
+* References and Further Reading::
+@end menu
+
+@node Overview of B-splines
+@section Overview
+@cindex basis splines, overview
+
+B-splines are commonly used as basis functions to fit smoothing
+curves to large data sets. To do this, the abscissa axis is
+broken up into some number of intervals, where the endpoints
+of each interval are called @dfn{breakpoints}. These breakpoints
+are then converted to @dfn{knots} by imposing various continuity
+and smoothness conditions at each interface. Given a nondecreasing
+knot vector @math{t = \{t_0, t_1, \dots, t_{n+k-1}\}}, the
+@math{n} basis splines of order @math{k} are defined by
+@tex
+\beforedisplay
+$$
+B_{i,1}(x) = \left\{\matrix{1, & t_i \le x < t_{i+1}\cr
+                            0, & else}\right.
+$$
+$$
+B_{i,k}(x) = \left[(x - t_i)/(t_{i+k-1} - t_i)\right] B_{i,k-1}(x) +
+             \left[(t_{i+k} - x)/(t_{i+k} - t_{i+1})\right] B_{i+1,k-1}(x)
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+B_(i,1)(x) = (1, t_i <= x < t_(i+1)
+             (0, else
+B_(i,k)(x) = [(x - t_i)/(t_(i+k-1) - t_i)] B_(i,k-1)(x) + [(t_(i+k) - x)/(t_(i+k) - t_(i+1))] B_(i+1,k-1)(x)
+@end example
+
+@end ifinfo
+for @math{i = 0, \dots, n-1}. The common case of cubic B-splines
+is given by @math{k = 4}. The above recurrence relation can be
+evaluated in a numerically stable way by the de Boor algorithm.
+
+If we define appropriate knots on an interval @math{[a,b]} then
+the B-spline basis functions form a complete set on that interval.
+Therefore we can expand a smoothing function as
+@tex
+\beforedisplay
+$$
+f(x) = \sum_{i=0}^{n-1} c_i B_{i,k}(x)
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+f(x) = \sum_i c_i B_(i,k)(x)
+@end example
+
+@end ifinfo
+given enough @math{(x_j, f(x_j))} data pairs. The @math{c_i} can
+be readily obtained from a least-squares fit.
+
+@node Initializing the B-splines solver
+@section Initializing the B-splines solver
+@cindex basis splines, initializing
+
+@deftypefun {gsl_bspline_workspace *} gsl_bspline_alloc (const size_t @var{k}, const size_t @var{nbreak})
+This function allocates a workspace for computing B-splines of order
+@var{k}. The number of breakpoints is given by @var{nbreak}. This
+leads to @math{n = nbreak + k - 2} basis functions. Cubic B-splines
+are specified by @math{k = 4}. The size of the workspace is
+@math{O(5k + nbreak)}.
+@end deftypefun
+
+@deftypefun void gsl_bspline_free (gsl_bspline_workspace * @var{w})
+This function frees the memory associated with the workspace @var{w}.
+@end deftypefun
+
+@node Constructing the knots vector
+@section Constructing the knots vector
+@cindex knots
+
+@deftypefun int gsl_bspline_knots (const gsl_vector * @var{breakpts}, gsl_bspline_workspace * @var{w})
+This function computes the knots associated with the given breakpoints
+and stores them internally in @code{w->knots}.
+@end deftypefun
+
+@deftypefun int gsl_bspline_knots_uniform (const double a, const double b, gsl_bspline_workspace * @var{w})
+This function assumes uniformly spaced breakpoints on @math{[a,b]}
+and constructs the corresponding knot vector using the previously
+specified @var{nbreak} parameter. The knots are stored in
+@code{w->knots}.
+@end deftypefun
+
+@node Evaluation of B-spline basis functions
+@section Evaluation of B-splines
+@cindex basis splines, evaluation
+
+@deftypefun int gsl_bspline_eval (const double @var{x}, gsl_vector * @var{B}, gsl_bspline_workspace * @var{w})
+This function evaluates all B-spline basis functions at the position
+@var{x} and stores them in @var{B}, so that the @math{i}th element
+of @var{B} is @math{B_i(x)}. @var{B} must be of length
+@math{n = nbreak + k - 2}. This value is also stored in @code{w->n}.
+It is far more efficient to compute all of the basis functions at
+once than to compute them individually, due to the nature of the
+defining recurrence relation.
+@end deftypefun
+
+@node Example programs for B-splines
+@section Example programs for B-splines
+@cindex basis splines, examples
+
+The following program computes a linear least squares fit to data using
+cubic B-spline basis functions with uniform breakpoints. The data is
+generated from the curve @math{y(x) = \cos{(x)} \exp{(-0.1 x)}} on
+@math{[0, 15]} with gaussian noise added.
+
+@example
+@verbatiminclude examples/bspline.c
+@end example
+
+The output can be plotted with @sc{gnu} @code{graph}.
+
+@example
+$ ./a.out > bspline.dat
+$ graph -T ps -X x -Y y -x 0 15 -y -1 1.3 < bspline.dat > bspline.ps
+@end example
+
+@iftex
+@sp 1
+@center @image{bspline,3.4in}
+@end iftex
+
+@node References and Further Reading
+@section References and Further Reading
+
+Further information on the algorithms described in this section can be
+found in the following book,
+
+@itemize @asis
+C. de Boor, @cite{A Practical Guide to Splines} (1978), Springer-Verlag,
+ISBN 0-387-90356-9.
+@end itemize
+
+@noindent
+A large collection of B-spline routines is available in the
+@sc{pppack} library available at @uref{http://www.netlib.org/pppack}.
diff --git a/gsl-1.9/doc/calc.sty b/gsl-1.9/doc/calc.sty
new file mode 100644
index 0000000..086273e
--- /dev/null
+++ b/gsl-1.9/doc/calc.sty
@@ -0,0 +1,150 @@
+%% 
+%% This is file `calc.sty',
+%% generated with the docstrip utility.
+%% 
+%% The original source files were:
+%% 
+%% calc.dtx  (with options: `package')
+
+%% File `calc.dtx'.
+%% Copyright (C) 1992--1995 Kresten Krab Thorup and Frank Jensen.
+%% All rights reserved.
+
+\def\fileversion{v4.0c (TEST)}
+\def\filedate{1995/04/10}
+
+%% \CharacterTable
+%%  {Upper-case    \A\B\C\D\E\F\G\H\I\J\K\L\M\N\O\P\Q\R\S\T\U\V\W\X\Y\Z
+%%   Lower-case    \a\b\c\d\e\f\g\h\i\j\k\l\m\n\o\p\q\r\s\t\u\v\w\x\y\z
+%%   Digits        \0\1\2\3\4\5\6\7\8\9
+%%   Exclamation   \!     Double quote  \"     Hash (number) \#
+%%   Dollar        \$     Percent       \%     Ampersand     \&
+%%   Acute accent  \'     Left paren    \(     Right paren   \)
+%%   Asterisk      \*     Plus          \+     Comma         \,
+%%   Minus         \-     Point         \.     Solidus       \/
+%%   Colon         \:     Semicolon     \;     Less than     \<
+%%   Equals        \=     Greater than  \>     Question mark \?
+%%   Commercial at \@     Left bracket  \[     Backslash     \\
+%%   Right bracket \]     Circumflex    \^     Underscore    \_
+%%   Grave accent  \`     Left brace    \{     Vertical bar  \|
+%%   Right brace   \}     Tilde         \~}
+\NeedsTeXFormat{LaTeX2e}
+\ProvidesPackage{calc}[\filedate\space\fileversion]
+\typeout{Package: `calc' \fileversion\space <\filedate> (KKT and FJ)}
+\def\calc@assign@generic#1#2#3#4{\let\calc@A#1\let\calc@B#2%
+    \expandafter\calc@open\expandafter(#4!%
+    \global\calc@A\calc@B\endgroup#3\calc@B}
+\def\calc@assign@count{\calc@assign@generic\calc@Acount\calc@Bcount}
+\def\calc@assign@dimen{\calc@assign@generic\calc@Adimen\calc@Bdimen}
+\def\calc@assign@skip{\calc@assign@generic\calc@Askip\calc@Bskip}
+\newcount\calc@Acount   \newcount\calc@Bcount
+\newdimen\calc@Adimen   \newdimen\calc@Bdimen
+\newskip\calc@Askip     \newskip\calc@Bskip
+\def\setcounter#1#2{\@ifundefined{c@#1}{\@nocounterr{#1}}%
+   {\calc@assign@count{\global\csname c@#1\endcsname}{#2}}}
+\def\addtocounter#1#2{\@ifundefined{c@#1}{\@nocounterr{#1}}%
+   {\calc@assign@count{\global\advance\csname c@#1\endcsname}{#2}}}
+\DeclareRobustCommand\setlength{\calc@assign@skip}
+\DeclareRobustCommand\addtolength[1]{\calc@assign@skip{\advance#1}}
+\def\calc@pre@scan#1{%
+   \ifx(#1%
+       \let\calc@next\calc@open
+   \else
+       \let\calc@next\calc@numeric
+   \fi
+   \calc@next#1}
+\def\calc@open({\begingroup\aftergroup\calc@initB
+   \begingroup\aftergroup\calc@initB
+   \calc@pre@scan}
+\def\calc@initB{\calc@B\calc@A}
+\def\calc@numeric{\afterassignment\calc@post@scan \global\calc@A}
+\def\calc@post@scan#1{%
+   \ifx#1!\let\calc@next\endgroup \else
+    \ifx#1+\let\calc@next\calc@add \else
+     \ifx#1-\let\calc@next\calc@subtract \else
+      \ifx#1*\let\calc@next\calc@multiplyx \else
+       \ifx#1/\let\calc@next\calc@dividex \else
+        \ifx#1)\let\calc@next\calc@close \else \calc@error#1%
+        \fi
+       \fi
+      \fi
+     \fi
+    \fi
+   \fi
+   \calc@next}
+\def\calc@add{\calc@generic@add\calc@addAtoB}
+\def\calc@subtract{\calc@generic@add\calc@subtractAfromB}
+\def\calc@generic@add#1{\endgroup\global\calc@A\calc@B\endgroup
+   \begingroup\aftergroup#1\begingroup\aftergroup\calc@initB
+   \calc@pre@scan}
+\def\calc@addAtoB{\advance\calc@B\calc@A}
+\def\calc@subtractAfromB{\advance\calc@B-\calc@A}
+\def\calc@multiplyx#1{\def\calc@tmp{#1}%
+   \ifx\calc@tmp\calc@ratio@x \let\calc@next\calc@ratio@multiply \else
+      \ifx\calc@tmp\calc@real@x \let\calc@next\calc@real@multiply \else
+         \let\calc@next\calc@multiply
+      \fi
+   \fi
+   \calc@next#1}
+\def\calc@dividex#1{\def\calc@tmp{#1}%
+   \ifx\calc@tmp\calc@ratio@x \let\calc@next\calc@ratio@divide \else
+      \ifx\calc@tmp\calc@real@x \let\calc@next\calc@real@divide \else
+         \let\calc@next\calc@divide
+      \fi
+   \fi
+   \calc@next#1}
+\def\calc@ratio@x{\ratio}
+\def\calc@real@x{\real}
+\def\calc@multiply{\calc@generic@multiply\calc@multiplyBbyA}
+\def\calc@divide{\calc@generic@multiply\calc@divideBbyA}
+\def\calc@generic@multiply#1{\endgroup\begingroup
+   \let\calc@A\calc@Acount \let\calc@B\calc@Bcount
+   \aftergroup#1\calc@pre@scan}
+\def\calc@multiplyBbyA{\multiply\calc@B\calc@Acount}
+\def\calc@divideBbyA{\divide\calc@B\calc@Acount}
+\def\calc@close
+   {\endgroup\global\calc@A\calc@B
+    \endgroup\global\calc@A\calc@B
+    \calc@post@scan}
+\def\calc@ratio@multiply\ratio{\calc@ratio@evaluate}
+\def\calc@ratio@divide\ratio#1#2{\calc@ratio@evaluate{#2}{#1}}
+\let\calc@numerator=\calc@Bcount
+\newcount\calc@denominator
+\def\calc@ratio@evaluate#1#2{%
+   \endgroup\begingroup
+      \calc@assign@dimen\calc@numerator{#1}%
+      \calc@assign@dimen\calc@denominator{#2}%
+      \gdef\calc@the@ratio{}%
+      \ifnum\calc@numerator<0 \calc@numerator-\calc@numerator
+         \gdef\calc@the@ratio{-}%
+      \fi
+      \ifnum\calc@denominator<0 \calc@denominator-\calc@denominator
+         \xdef\calc@the@ratio{\calc@the@ratio-}%
+      \fi
+      \calc@Acount\calc@numerator
+      \divide\calc@Acount\calc@denominator
+      \xdef\calc@the@ratio{\calc@the@ratio\number\calc@Acount.}%
+      \calc@next@digit \calc@next@digit \calc@next@digit
+      \calc@next@digit \calc@next@digit \calc@next@digit
+   \endgroup
+   \calc@multiply@by@real\calc@the@ratio
+   \begingroup
+   \calc@post@scan}
+\def\calc@next@digit{%
+      \multiply\calc@Acount\calc@denominator
+      \advance\calc@numerator -\calc@Acount
+      \multiply\calc@numerator 10
+      \calc@Acount\calc@numerator
+      \divide\calc@Acount\calc@denominator
+      \xdef\calc@the@ratio{\calc@the@ratio\number\calc@Acount}}
+\def\calc@multiply@by@real#1{\calc@Bdimen #1\calc@B \calc@B\calc@Bdimen}
+\def\calc@real@multiply\real#1{\endgroup
+   \calc@multiply@by@real{#1}\begingroup
+   \calc@post@scan}
+\def\calc@real@divide\real#1{\calc@ratio@evaluate{1pt}{#1pt}}
+\def\calc@error#1{%
+   \errhelp{Calc error: I expected to see one of: + - * / )}%
+   \errmessage{Invalid character `#1' in arithmetic expression}}
+\endinput
+%% 
+%% End of file `calc.sty'.
diff --git a/gsl-1.9/doc/cblas.texi b/gsl-1.9/doc/cblas.texi
new file mode 100644
index 0000000..75bf7a7
--- /dev/null
+++ b/gsl-1.9/doc/cblas.texi
@@ -0,0 +1,514 @@
+@cindex Low-level CBLAS
+@cindex CBLAS, Low-level interface
+@cindex BLAS, Low-level C interface
+@cindex Basic Linear Algebra Subroutines (BLAS)
+The prototypes for the low-level @sc{cblas} functions are declared in
+the file @code{gsl_cblas.h}.  For the definition of the functions
+consult the documentation available from Netlib (@pxref{BLAS References
+and Further Reading}).
+
+@menu
+* Level 1 CBLAS Functions::     
+* Level 2 CBLAS Functions::     
+* Level 3 CBLAS Functions::     
+* GSL CBLAS Examples::          
+@end menu
+
+@node Level 1 CBLAS Functions
+@section Level 1 
+
+@deftypefun float cblas_sdsdot (const int @var{N}, const float @var{alpha}, const float * @var{x}, const int @var{incx}, const float * @var{y}, const int @var{incy})
+@end deftypefun
+
+@deftypefun double cblas_dsdot (const int @var{N}, const float * @var{x}, const int @var{incx}, const float * @var{y}, const int @var{incy})
+@end deftypefun
+
+@deftypefun float cblas_sdot (const int @var{N}, const float * @var{x}, const int @var{incx}, const float * @var{y}, const int @var{incy})
+@end deftypefun
+
+@deftypefun double cblas_ddot (const int @var{N}, const double * @var{x}, const int @var{incx}, const double * @var{y}, const int @var{incy})
+@end deftypefun
+
+@deftypefun void cblas_cdotu_sub (const int @var{N}, const void * @var{x}, const int @var{incx}, const void * @var{y}, const int @var{incy}, void * @var{dotu})
+@end deftypefun
+
+@deftypefun void cblas_cdotc_sub (const int @var{N}, const void * @var{x}, const int @var{incx}, const void * @var{y}, const int @var{incy}, void * @var{dotc})
+@end deftypefun
+
+@deftypefun void cblas_zdotu_sub (const int @var{N}, const void * @var{x}, const int @var{incx}, const void * @var{y}, const int @var{incy}, void * @var{dotu})
+@end deftypefun
+
+@deftypefun void cblas_zdotc_sub (const int @var{N}, const void * @var{x}, const int @var{incx}, const void * @var{y}, const int @var{incy}, void * @var{dotc})
+@end deftypefun
+
+@deftypefun float cblas_snrm2 (const int @var{N}, const float * @var{x}, const int @var{incx})
+@end deftypefun
+
+@deftypefun float cblas_sasum (const int @var{N}, const float * @var{x}, const int @var{incx})
+@end deftypefun
+
+@deftypefun double cblas_dnrm2 (const int @var{N}, const double * @var{x}, const int @var{incx})
+@end deftypefun
+
+@deftypefun double cblas_dasum (const int @var{N}, const double * @var{x}, const int @var{incx})
+@end deftypefun
+
+@deftypefun float cblas_scnrm2 (const int @var{N}, const void * @var{x}, const int @var{incx})
+@end deftypefun
+
+@deftypefun float cblas_scasum (const int @var{N}, const void * @var{x}, const int @var{incx})
+@end deftypefun
+
+@deftypefun double cblas_dznrm2 (const int @var{N}, const void * @var{x}, const int @var{incx})
+@end deftypefun
+
+@deftypefun double cblas_dzasum (const int @var{N}, const void * @var{x}, const int @var{incx})
+@end deftypefun
+
+@deftypefun CBLAS_INDEX cblas_isamax (const int @var{N}, const float * @var{x}, const int @var{incx})
+@end deftypefun
+
+@deftypefun CBLAS_INDEX cblas_idamax (const int @var{N}, const double * @var{x}, const int @var{incx})
+@end deftypefun
+
+@deftypefun CBLAS_INDEX cblas_icamax (const int @var{N}, const void * @var{x}, const int @var{incx})
+@end deftypefun
+
+@deftypefun CBLAS_INDEX cblas_izamax (const int @var{N}, const void * @var{x}, const int @var{incx})
+@end deftypefun
+
+@deftypefun void cblas_sswap (const int @var{N}, float * @var{x}, const int @var{incx}, float * @var{y}, const int @var{incy})
+@end deftypefun
+
+@deftypefun void cblas_scopy (const int @var{N}, const float * @var{x}, const int @var{incx}, float * @var{y}, const int @var{incy})
+@end deftypefun
+
+@deftypefun void cblas_saxpy (const int @var{N}, const float @var{alpha}, const float * @var{x}, const int @var{incx}, float * @var{y}, const int @var{incy})
+@end deftypefun
+
+@deftypefun void cblas_dswap (const int @var{N}, double * @var{x}, const int @var{incx}, double * @var{y}, const int @var{incy})
+@end deftypefun
+
+@deftypefun void cblas_dcopy (const int @var{N}, const double * @var{x}, const int @var{incx}, double * @var{y}, const int @var{incy})
+@end deftypefun
+
+@deftypefun void cblas_daxpy (const int @var{N}, const double @var{alpha}, const double * @var{x}, const int @var{incx}, double * @var{y}, const int @var{incy})
+@end deftypefun
+
+@deftypefun void cblas_cswap (const int @var{N}, void * @var{x}, const int @var{incx}, void * @var{y}, const int @var{incy})
+@end deftypefun
+
+@deftypefun void cblas_ccopy (const int @var{N}, const void * @var{x}, const int @var{incx}, void * @var{y}, const int @var{incy})
+@end deftypefun
+
+@deftypefun void cblas_caxpy (const int @var{N}, const void * @var{alpha}, const void * @var{x}, const int @var{incx}, void * @var{y}, const int @var{incy})
+@end deftypefun
+
+@deftypefun void cblas_zswap (const int @var{N}, void * @var{x}, const int @var{incx}, void * @var{y}, const int @var{incy})
+@end deftypefun
+
+@deftypefun void cblas_zcopy (const int @var{N}, const void * @var{x}, const int @var{incx}, void * @var{y}, const int @var{incy})
+@end deftypefun
+
+@deftypefun void cblas_zaxpy (const int @var{N}, const void * @var{alpha}, const void * @var{x}, const int @var{incx}, void * @var{y}, const int @var{incy})
+@end deftypefun
+
+@deftypefun void cblas_srotg (float * @var{a}, float * @var{b}, float * @var{c}, float * @var{s})
+@end deftypefun
+
+@deftypefun void cblas_srotmg (float * @var{d1}, float * @var{d2}, float * @var{b1}, const float @var{b2}, float * @var{P})
+@end deftypefun
+
+@deftypefun void cblas_srot (const int @var{N}, float * @var{x}, const int @var{incx}, float * @var{y}, const int @var{incy}, const float @var{c}, const float @var{s})
+@end deftypefun
+
+@deftypefun void cblas_srotm (const int @var{N}, float * @var{x}, const int @var{incx}, float * @var{y}, const int @var{incy}, const float * @var{P})
+@end deftypefun
+
+@deftypefun void cblas_drotg (double * @var{a}, double * @var{b}, double * @var{c}, double * @var{s})
+@end deftypefun
+
+@deftypefun void cblas_drotmg (double * @var{d1}, double * @var{d2}, double * @var{b1}, const double @var{b2}, double * @var{P})
+@end deftypefun
+
+@deftypefun void cblas_drot (const int @var{N}, double * @var{x}, const int @var{incx}, double * @var{y}, const int @var{incy}, const double @var{c}, const double @var{s})
+@end deftypefun
+
+@deftypefun void cblas_drotm (const int @var{N}, double * @var{x}, const int @var{incx}, double * @var{y}, const int @var{incy}, const double * @var{P})
+@end deftypefun
+
+@deftypefun void cblas_sscal (const int @var{N}, const float @var{alpha}, float * @var{x}, const int @var{incx})
+@end deftypefun
+
+@deftypefun void cblas_dscal (const int @var{N}, const double @var{alpha}, double * @var{x}, const int @var{incx})
+@end deftypefun
+
+@deftypefun void cblas_cscal (const int @var{N}, const void * @var{alpha}, void * @var{x}, const int @var{incx})
+@end deftypefun
+
+@deftypefun void cblas_zscal (const int @var{N}, const void * @var{alpha}, void * @var{x}, const int @var{incx})
+@end deftypefun
+
+@deftypefun void cblas_csscal (const int @var{N}, const float @var{alpha}, void * @var{x}, const int @var{incx})
+@end deftypefun
+
+@deftypefun void cblas_zdscal (const int @var{N}, const double @var{alpha}, void * @var{x}, const int @var{incx})
+@end deftypefun
+
+@node Level 2 CBLAS Functions
+@section Level 2 
+
+@deftypefun void cblas_sgemv (const enum CBLAS_ORDER @var{order}, const enum CBLAS_TRANSPOSE @var{TransA}, const int @var{M}, const int @var{N}, const float @var{alpha}, const float * @var{A}, const int @var{lda}, const float * @var{x}, const int @var{incx}, const float @var{beta}, float * @var{y}, const int @var{incy})
+@end deftypefun
+
+@deftypefun void cblas_sgbmv (const enum CBLAS_ORDER @var{order}, const enum CBLAS_TRANSPOSE @var{TransA}, const int @var{M}, const int @var{N}, const int @var{KL}, const int @var{KU}, const float @var{alpha}, const float * @var{A}, const int @var{lda}, const float * @var{x}, const int @var{incx}, const float @var{beta}, float * @var{y}, const int @var{incy})
+@end deftypefun
+
+@deftypefun void cblas_strmv (const enum CBLAS_ORDER @var{order}, const enum CBLAS_UPLO @var{Uplo}, const enum CBLAS_TRANSPOSE @var{TransA}, const enum CBLAS_DIAG @var{Diag}, const int @var{N}, const float * @var{A}, const int @var{lda}, float * @var{x}, const int @var{incx})
+@end deftypefun
+
+@deftypefun void cblas_stbmv (const enum CBLAS_ORDER @var{order}, const enum CBLAS_UPLO @var{Uplo}, const enum CBLAS_TRANSPOSE @var{TransA}, const enum CBLAS_DIAG @var{Diag}, const int @var{N}, const int @var{K}, const float * @var{A}, const int @var{lda}, float * @var{x}, const int @var{incx})
+@end deftypefun
+
+@deftypefun void cblas_stpmv (const enum CBLAS_ORDER @var{order}, const enum CBLAS_UPLO @var{Uplo}, const enum CBLAS_TRANSPOSE @var{TransA}, const enum CBLAS_DIAG @var{Diag}, const int @var{N}, const float * @var{Ap}, float * @var{x}, const int @var{incx})
+@end deftypefun
+
+@deftypefun void cblas_strsv (const enum CBLAS_ORDER @var{order}, const enum CBLAS_UPLO @var{Uplo}, const enum CBLAS_TRANSPOSE @var{TransA}, const enum CBLAS_DIAG @var{Diag}, const int @var{N}, const float * @var{A}, const int @var{lda}, float * @var{x}, const int @var{incx})
+@end deftypefun
+
+@deftypefun void cblas_stbsv (const enum CBLAS_ORDER @var{order}, const enum CBLAS_UPLO @var{Uplo}, const enum CBLAS_TRANSPOSE @var{TransA}, const enum CBLAS_DIAG @var{Diag}, const int @var{N}, const int @var{K}, const float * @var{A}, const int @var{lda}, float * @var{x}, const int @var{incx})
+@end deftypefun
+
+@deftypefun void cblas_stpsv (const enum CBLAS_ORDER @var{order}, const enum CBLAS_UPLO @var{Uplo}, const enum CBLAS_TRANSPOSE @var{TransA}, const enum CBLAS_DIAG @var{Diag}, const int @var{N}, const float * @var{Ap}, float * @var{x}, const int @var{incx})
+@end deftypefun
+
+@deftypefun void cblas_dgemv (const enum CBLAS_ORDER @var{order}, const enum CBLAS_TRANSPOSE @var{TransA}, const int @var{M}, const int @var{N}, const double @var{alpha}, const double * @var{A}, const int @var{lda}, const double * @var{x}, const int @var{incx}, const double @var{beta}, double * @var{y}, const int @var{incy})
+@end deftypefun
+
+@deftypefun void cblas_dgbmv (const enum CBLAS_ORDER @var{order}, const enum CBLAS_TRANSPOSE @var{TransA}, const int @var{M}, const int @var{N}, const int @var{KL}, const int @var{KU}, const double @var{alpha}, const double * @var{A}, const int @var{lda}, const double * @var{x}, const int @var{incx}, const double @var{beta}, double * @var{y}, const int @var{incy})
+@end deftypefun
+
+@deftypefun void cblas_dtrmv (const enum CBLAS_ORDER @var{order}, const enum CBLAS_UPLO @var{Uplo}, const enum CBLAS_TRANSPOSE @var{TransA}, const enum CBLAS_DIAG @var{Diag}, const int @var{N}, const double * @var{A}, const int @var{lda}, double * @var{x}, const int @var{incx})
+@end deftypefun
+
+@deftypefun void cblas_dtbmv (const enum CBLAS_ORDER @var{order}, const enum CBLAS_UPLO @var{Uplo}, const enum CBLAS_TRANSPOSE @var{TransA}, const enum CBLAS_DIAG @var{Diag}, const int @var{N}, const int @var{K}, const double * @var{A}, const int @var{lda}, double * @var{x}, const int @var{incx})
+@end deftypefun
+
+@deftypefun void cblas_dtpmv (const enum CBLAS_ORDER @var{order}, const enum CBLAS_UPLO @var{Uplo}, const enum CBLAS_TRANSPOSE @var{TransA}, const enum CBLAS_DIAG @var{Diag}, const int @var{N}, const double * @var{Ap}, double * @var{x}, const int @var{incx})
+@end deftypefun
+
+@deftypefun void cblas_dtrsv (const enum CBLAS_ORDER @var{order}, const enum CBLAS_UPLO @var{Uplo}, const enum CBLAS_TRANSPOSE @var{TransA}, const enum CBLAS_DIAG @var{Diag}, const int @var{N}, const double * @var{A}, const int @var{lda}, double * @var{x}, const int @var{incx})
+@end deftypefun
+
+@deftypefun void cblas_dtbsv (const enum CBLAS_ORDER @var{order}, const enum CBLAS_UPLO @var{Uplo}, const enum CBLAS_TRANSPOSE @var{TransA}, const enum CBLAS_DIAG @var{Diag}, const int @var{N}, const int @var{K}, const double * @var{A}, const int @var{lda}, double * @var{x}, const int @var{incx})
+@end deftypefun
+
+@deftypefun void cblas_dtpsv (const enum CBLAS_ORDER @var{order}, const enum CBLAS_UPLO @var{Uplo}, const enum CBLAS_TRANSPOSE @var{TransA}, const enum CBLAS_DIAG @var{Diag}, const int @var{N}, const double * @var{Ap}, double * @var{x}, const int @var{incx})
+@end deftypefun
+
+@deftypefun void cblas_cgemv (const enum CBLAS_ORDER @var{order}, const enum CBLAS_TRANSPOSE @var{TransA}, const int @var{M}, const int @var{N}, const void * @var{alpha}, const void * @var{A}, const int @var{lda}, const void * @var{x}, const int @var{incx}, const void * @var{beta}, void * @var{y}, const int @var{incy})
+@end deftypefun
+
+@deftypefun void cblas_cgbmv (const enum CBLAS_ORDER @var{order}, const enum CBLAS_TRANSPOSE @var{TransA}, const int @var{M}, const int @var{N}, const int @var{KL}, const int @var{KU}, const void * @var{alpha}, const void * @var{A}, const int @var{lda}, const void * @var{x}, const int @var{incx}, const void * @var{beta}, void * @var{y}, const int @var{incy})
+@end deftypefun
+
+@deftypefun void cblas_ctrmv (const enum CBLAS_ORDER @var{order}, const enum CBLAS_UPLO @var{Uplo}, const enum CBLAS_TRANSPOSE @var{TransA}, const enum CBLAS_DIAG @var{Diag}, const int @var{N}, const void * @var{A}, const int @var{lda}, void * @var{x}, const int @var{incx})
+@end deftypefun
+
+@deftypefun void cblas_ctbmv (const enum CBLAS_ORDER @var{order}, const enum CBLAS_UPLO @var{Uplo}, const enum CBLAS_TRANSPOSE @var{TransA}, const enum CBLAS_DIAG @var{Diag}, const int @var{N}, const int @var{K}, const void * @var{A}, const int @var{lda}, void * @var{x}, const int @var{incx})
+@end deftypefun
+
+@deftypefun void cblas_ctpmv (const enum CBLAS_ORDER @var{order}, const enum CBLAS_UPLO @var{Uplo}, const enum CBLAS_TRANSPOSE @var{TransA}, const enum CBLAS_DIAG @var{Diag}, const int @var{N}, const void * @var{Ap}, void * @var{x}, const int @var{incx})
+@end deftypefun
+
+@deftypefun void cblas_ctrsv (const enum CBLAS_ORDER @var{order}, const enum CBLAS_UPLO @var{Uplo}, const enum CBLAS_TRANSPOSE @var{TransA}, const enum CBLAS_DIAG @var{Diag}, const int @var{N}, const void * @var{A}, const int @var{lda}, void * @var{x}, const int @var{incx})
+@end deftypefun
+
+@deftypefun void cblas_ctbsv (const enum CBLAS_ORDER @var{order}, const enum CBLAS_UPLO @var{Uplo}, const enum CBLAS_TRANSPOSE @var{TransA}, const enum CBLAS_DIAG @var{Diag}, const int @var{N}, const int @var{K}, const void * @var{A}, const int @var{lda}, void * @var{x}, const int @var{incx})
+@end deftypefun
+
+@deftypefun void cblas_ctpsv (const enum CBLAS_ORDER @var{order}, const enum CBLAS_UPLO @var{Uplo}, const enum CBLAS_TRANSPOSE @var{TransA}, const enum CBLAS_DIAG @var{Diag}, const int @var{N}, const void * @var{Ap}, void * @var{x}, const int @var{incx})
+@end deftypefun
+
+@deftypefun void cblas_zgemv (const enum CBLAS_ORDER @var{order}, const enum CBLAS_TRANSPOSE @var{TransA}, const int @var{M}, const int @var{N}, const void * @var{alpha}, const void * @var{A}, const int @var{lda}, const void * @var{x}, const int @var{incx}, const void * @var{beta}, void * @var{y}, const int @var{incy})
+@end deftypefun
+
+@deftypefun void cblas_zgbmv (const enum CBLAS_ORDER @var{order}, const enum CBLAS_TRANSPOSE @var{TransA}, const int @var{M}, const int @var{N}, const int @var{KL}, const int @var{KU}, const void * @var{alpha}, const void * @var{A}, const int @var{lda}, const void * @var{x}, const int @var{incx}, const void * @var{beta}, void * @var{y}, const int @var{incy})
+@end deftypefun
+
+@deftypefun void cblas_ztrmv (const enum CBLAS_ORDER @var{order}, const enum CBLAS_UPLO @var{Uplo}, const enum CBLAS_TRANSPOSE @var{TransA}, const enum CBLAS_DIAG @var{Diag}, const int @var{N}, const void * @var{A}, const int @var{lda}, void * @var{x}, const int @var{incx})
+@end deftypefun
+
+@deftypefun void cblas_ztbmv (const enum CBLAS_ORDER @var{order}, const enum CBLAS_UPLO @var{Uplo}, const enum CBLAS_TRANSPOSE @var{TransA}, const enum CBLAS_DIAG @var{Diag}, const int @var{N}, const int @var{K}, const void * @var{A}, const int @var{lda}, void * @var{x}, const int @var{incx})
+@end deftypefun
+
+@deftypefun void cblas_ztpmv (const enum CBLAS_ORDER @var{order}, const enum CBLAS_UPLO @var{Uplo}, const enum CBLAS_TRANSPOSE @var{TransA}, const enum CBLAS_DIAG @var{Diag}, const int @var{N}, const void * @var{Ap}, void * @var{x}, const int @var{incx})
+@end deftypefun
+
+@deftypefun void cblas_ztrsv (const enum CBLAS_ORDER @var{order}, const enum CBLAS_UPLO @var{Uplo}, const enum CBLAS_TRANSPOSE @var{TransA}, const enum CBLAS_DIAG @var{Diag}, const int @var{N}, const void * @var{A}, const int @var{lda}, void * @var{x}, const int @var{incx})
+@end deftypefun
+
+@deftypefun void cblas_ztbsv (const enum CBLAS_ORDER @var{order}, const enum CBLAS_UPLO @var{Uplo}, const enum CBLAS_TRANSPOSE @var{TransA}, const enum CBLAS_DIAG @var{Diag}, const int @var{N}, const int @var{K}, const void * @var{A}, const int @var{lda}, void * @var{x}, const int @var{incx})
+@end deftypefun
+
+@deftypefun void cblas_ztpsv (const enum CBLAS_ORDER @var{order}, const enum CBLAS_UPLO @var{Uplo}, const enum CBLAS_TRANSPOSE @var{TransA}, const enum CBLAS_DIAG @var{Diag}, const int @var{N}, const void * @var{Ap}, void * @var{x}, const int @var{incx})
+@end deftypefun
+
+@deftypefun void cblas_ssymv (const enum CBLAS_ORDER @var{order}, const enum CBLAS_UPLO @var{Uplo}, const int @var{N}, const float @var{alpha}, const float * @var{A}, const int @var{lda}, const float * @var{x}, const int @var{incx}, const float @var{beta}, float * @var{y}, const int @var{incy})
+@end deftypefun
+
+@deftypefun void cblas_ssbmv (const enum CBLAS_ORDER @var{order}, const enum CBLAS_UPLO @var{Uplo}, const int @var{N}, const int @var{K}, const float @var{alpha}, const float * @var{A}, const int @var{lda}, const float * @var{x}, const int @var{incx}, const float @var{beta}, float * @var{y}, const int @var{incy})
+@end deftypefun
+
+@deftypefun void cblas_sspmv (const enum CBLAS_ORDER @var{order}, const enum CBLAS_UPLO @var{Uplo}, const int @var{N}, const float @var{alpha}, const float * @var{Ap}, const float * @var{x}, const int @var{incx}, const float @var{beta}, float * @var{y}, const int @var{incy})
+@end deftypefun
+
+@deftypefun void cblas_sger (const enum CBLAS_ORDER @var{order}, const int @var{M}, const int @var{N}, const float @var{alpha}, const float * @var{x}, const int @var{incx}, const float * @var{y}, const int @var{incy}, float * @var{A}, const int @var{lda})
+@end deftypefun
+
+@deftypefun void cblas_ssyr (const enum CBLAS_ORDER @var{order}, const enum CBLAS_UPLO @var{Uplo}, const int @var{N}, const float @var{alpha}, const float * @var{x}, const int @var{incx}, float * @var{A}, const int @var{lda})
+@end deftypefun
+
+@deftypefun void cblas_sspr (const enum CBLAS_ORDER @var{order}, const enum CBLAS_UPLO @var{Uplo}, const int @var{N}, const float @var{alpha}, const float * @var{x}, const int @var{incx}, float * @var{Ap})
+@end deftypefun
+
+@deftypefun void cblas_ssyr2 (const enum CBLAS_ORDER @var{order}, const enum CBLAS_UPLO @var{Uplo}, const int @var{N}, const float @var{alpha}, const float * @var{x}, const int @var{incx}, const float * @var{y}, const int @var{incy}, float * @var{A}, const int @var{lda})
+@end deftypefun
+
+@deftypefun void cblas_sspr2 (const enum CBLAS_ORDER @var{order}, const enum CBLAS_UPLO @var{Uplo}, const int @var{N}, const float @var{alpha}, const float * @var{x}, const int @var{incx}, const float * @var{y}, const int @var{incy}, float * @var{A})
+@end deftypefun
+
+@deftypefun void cblas_dsymv (const enum CBLAS_ORDER @var{order}, const enum CBLAS_UPLO @var{Uplo}, const int @var{N}, const double @var{alpha}, const double * @var{A}, const int @var{lda}, const double * @var{x}, const int @var{incx}, const double @var{beta}, double * @var{y}, const int @var{incy})
+@end deftypefun
+
+@deftypefun void cblas_dsbmv (const enum CBLAS_ORDER @var{order}, const enum CBLAS_UPLO @var{Uplo}, const int @var{N}, const int @var{K}, const double @var{alpha}, const double * @var{A}, const int @var{lda}, const double * @var{x}, const int @var{incx}, const double @var{beta}, double * @var{y}, const int @var{incy})
+@end deftypefun
+
+@deftypefun void cblas_dspmv (const enum CBLAS_ORDER @var{order}, const enum CBLAS_UPLO @var{Uplo}, const int @var{N}, const double @var{alpha}, const double * @var{Ap}, const double * @var{x}, const int @var{incx}, const double @var{beta}, double * @var{y}, const int @var{incy})
+@end deftypefun
+
+@deftypefun void cblas_dger (const enum CBLAS_ORDER @var{order}, const int @var{M}, const int @var{N}, const double @var{alpha}, const double * @var{x}, const int @var{incx}, const double * @var{y}, const int @var{incy}, double * @var{A}, const int @var{lda})
+@end deftypefun
+
+@deftypefun void cblas_dsyr (const enum CBLAS_ORDER @var{order}, const enum CBLAS_UPLO @var{Uplo}, const int @var{N}, const double @var{alpha}, const double * @var{x}, const int @var{incx}, double * @var{A}, const int @var{lda})
+@end deftypefun
+
+@deftypefun void cblas_dspr (const enum CBLAS_ORDER @var{order}, const enum CBLAS_UPLO @var{Uplo}, const int @var{N}, const double @var{alpha}, const double * @var{x}, const int @var{incx}, double * @var{Ap})
+@end deftypefun
+
+@deftypefun void cblas_dsyr2 (const enum CBLAS_ORDER @var{order}, const enum CBLAS_UPLO @var{Uplo}, const int @var{N}, const double @var{alpha}, const double * @var{x}, const int @var{incx}, const double * @var{y}, const int @var{incy}, double * @var{A}, const int @var{lda})
+@end deftypefun
+
+@deftypefun void cblas_dspr2 (const enum CBLAS_ORDER @var{order}, const enum CBLAS_UPLO @var{Uplo}, const int @var{N}, const double @var{alpha}, const double * @var{x}, const int @var{incx}, const double * @var{y}, const int @var{incy}, double * @var{A})
+@end deftypefun
+
+@deftypefun void cblas_chemv (const enum CBLAS_ORDER @var{order}, const enum CBLAS_UPLO @var{Uplo}, const int @var{N}, const void * @var{alpha}, const void * @var{A}, const int @var{lda}, const void * @var{x}, const int @var{incx}, const void * @var{beta}, void * @var{y}, const int @var{incy})
+@end deftypefun
+
+@deftypefun void cblas_chbmv (const enum CBLAS_ORDER @var{order}, const enum CBLAS_UPLO @var{Uplo}, const int @var{N}, const int @var{K}, const void * @var{alpha}, const void * @var{A}, const int @var{lda}, const void * @var{x}, const int @var{incx}, const void * @var{beta}, void * @var{y}, const int @var{incy})
+@end deftypefun
+
+@deftypefun void cblas_chpmv (const enum CBLAS_ORDER @var{order}, const enum CBLAS_UPLO @var{Uplo}, const int @var{N}, const void * @var{alpha}, const void * @var{Ap}, const void * @var{x}, const int @var{incx}, const void * @var{beta}, void * @var{y}, const int @var{incy})
+@end deftypefun
+
+@deftypefun void cblas_cgeru (const enum CBLAS_ORDER @var{order}, const int @var{M}, const int @var{N}, const void * @var{alpha}, const void * @var{x}, const int @var{incx}, const void * @var{y}, const int @var{incy}, void * @var{A}, const int @var{lda})
+@end deftypefun
+
+@deftypefun void cblas_cgerc (const enum CBLAS_ORDER @var{order}, const int @var{M}, const int @var{N}, const void * @var{alpha}, const void * @var{x}, const int @var{incx}, const void * @var{y}, const int @var{incy}, void * @var{A}, const int @var{lda})
+@end deftypefun
+
+@deftypefun void cblas_cher (const enum CBLAS_ORDER @var{order}, const enum CBLAS_UPLO @var{Uplo}, const int @var{N}, const float @var{alpha}, const void * @var{x}, const int @var{incx}, void * @var{A}, const int @var{lda})
+@end deftypefun
+
+@deftypefun void cblas_chpr (const enum CBLAS_ORDER @var{order}, const enum CBLAS_UPLO @var{Uplo}, const int @var{N}, const float @var{alpha}, const void * @var{x}, const int @var{incx}, void * @var{A})
+@end deftypefun
+
+@deftypefun void cblas_cher2 (const enum CBLAS_ORDER @var{order}, const enum CBLAS_UPLO @var{Uplo}, const int @var{N}, const void * @var{alpha}, const void * @var{x}, const int @var{incx}, const void * @var{y}, const int @var{incy}, void * @var{A}, const int @var{lda})
+@end deftypefun
+
+@deftypefun void cblas_chpr2 (const enum CBLAS_ORDER @var{order}, const enum CBLAS_UPLO @var{Uplo}, const int @var{N}, const void * @var{alpha}, const void * @var{x}, const int @var{incx}, const void * @var{y}, const int @var{incy}, void * @var{Ap})
+@end deftypefun
+
+@deftypefun void cblas_zhemv (const enum CBLAS_ORDER @var{order}, const enum CBLAS_UPLO @var{Uplo}, const int @var{N}, const void * @var{alpha}, const void * @var{A}, const int @var{lda}, const void * @var{x}, const int @var{incx}, const void * @var{beta}, void * @var{y}, const int @var{incy})
+@end deftypefun
+
+@deftypefun void cblas_zhbmv (const enum CBLAS_ORDER @var{order}, const enum CBLAS_UPLO @var{Uplo}, const int @var{N}, const int @var{K}, const void * @var{alpha}, const void * @var{A}, const int @var{lda}, const void * @var{x}, const int @var{incx}, const void * @var{beta}, void * @var{y}, const int @var{incy})
+@end deftypefun
+
+@deftypefun void cblas_zhpmv (const enum CBLAS_ORDER @var{order}, const enum CBLAS_UPLO @var{Uplo}, const int @var{N}, const void * @var{alpha}, const void * @var{Ap}, const void * @var{x}, const int @var{incx}, const void * @var{beta}, void * @var{y}, const int @var{incy})
+@end deftypefun
+
+@deftypefun void cblas_zgeru (const enum CBLAS_ORDER @var{order}, const int @var{M}, const int @var{N}, const void * @var{alpha}, const void * @var{x}, const int @var{incx}, const void * @var{y}, const int @var{incy}, void * @var{A}, const int @var{lda})
+@end deftypefun
+
+@deftypefun void cblas_zgerc (const enum CBLAS_ORDER @var{order}, const int @var{M}, const int @var{N}, const void * @var{alpha}, const void * @var{x}, const int @var{incx}, const void * @var{y}, const int @var{incy}, void * @var{A}, const int @var{lda})
+@end deftypefun
+
+@deftypefun void cblas_zher (const enum CBLAS_ORDER @var{order}, const enum CBLAS_UPLO @var{Uplo}, const int @var{N}, const double @var{alpha}, const void * @var{x}, const int @var{incx}, void * @var{A}, const int @var{lda})
+@end deftypefun
+
+@deftypefun void cblas_zhpr (const enum CBLAS_ORDER @var{order}, const enum CBLAS_UPLO @var{Uplo}, const int @var{N}, const double @var{alpha}, const void * @var{x}, const int @var{incx}, void * @var{A})
+@end deftypefun
+
+@deftypefun void cblas_zher2 (const enum CBLAS_ORDER @var{order}, const enum CBLAS_UPLO @var{Uplo}, const int @var{N}, const void * @var{alpha}, const void * @var{x}, const int @var{incx}, const void * @var{y}, const int @var{incy}, void * @var{A}, const int @var{lda})
+@end deftypefun
+
+@deftypefun void cblas_zhpr2 (const enum CBLAS_ORDER @var{order}, const enum CBLAS_UPLO @var{Uplo}, const int @var{N}, const void * @var{alpha}, const void * @var{x}, const int @var{incx}, const void * @var{y}, const int @var{incy}, void * @var{Ap})
+@end deftypefun
+
+@node Level 3 CBLAS Functions
+@section Level 3 
+
+
+@deftypefun void cblas_sgemm (const enum CBLAS_ORDER @var{Order}, const enum CBLAS_TRANSPOSE @var{TransA}, const enum CBLAS_TRANSPOSE @var{TransB}, const int @var{M}, const int @var{N}, const int @var{K}, const float @var{alpha}, const float * @var{A}, const int @var{lda}, const float * @var{B}, const int @var{ldb}, const float @var{beta}, float * @var{C}, const int @var{ldc})
+@end deftypefun
+
+@deftypefun void cblas_ssymm (const enum CBLAS_ORDER @var{Order}, const enum CBLAS_SIDE @var{Side}, const enum CBLAS_UPLO @var{Uplo}, const int @var{M}, const int @var{N}, const float @var{alpha}, const float * @var{A}, const int @var{lda}, const float * @var{B}, const int @var{ldb}, const float @var{beta}, float * @var{C}, const int @var{ldc})
+@end deftypefun
+
+@deftypefun void cblas_ssyrk (const enum CBLAS_ORDER @var{Order}, const enum CBLAS_UPLO @var{Uplo}, const enum CBLAS_TRANSPOSE @var{Trans}, const int @var{N}, const int @var{K}, const float @var{alpha}, const float * @var{A}, const int @var{lda}, const float @var{beta}, float * @var{C}, const int @var{ldc})
+@end deftypefun
+
+@deftypefun void cblas_ssyr2k (const enum CBLAS_ORDER @var{Order}, const enum CBLAS_UPLO @var{Uplo}, const enum CBLAS_TRANSPOSE @var{Trans}, const int @var{N}, const int @var{K}, const float @var{alpha}, const float * @var{A}, const int @var{lda}, const float * @var{B}, const int @var{ldb}, const float @var{beta}, float * @var{C}, const int @var{ldc})
+@end deftypefun
+
+@deftypefun void cblas_strmm (const enum CBLAS_ORDER @var{Order}, const enum CBLAS_SIDE @var{Side}, const enum CBLAS_UPLO @var{Uplo}, const enum CBLAS_TRANSPOSE @var{TransA}, const enum CBLAS_DIAG @var{Diag}, const int @var{M}, const int @var{N}, const float @var{alpha}, const float * @var{A}, const int @var{lda}, float * @var{B}, const int @var{ldb})
+@end deftypefun
+
+@deftypefun void cblas_strsm (const enum CBLAS_ORDER @var{Order}, const enum CBLAS_SIDE @var{Side}, const enum CBLAS_UPLO @var{Uplo}, const enum CBLAS_TRANSPOSE @var{TransA}, const enum CBLAS_DIAG @var{Diag}, const int @var{M}, const int @var{N}, const float @var{alpha}, const float * @var{A}, const int @var{lda}, float * @var{B}, const int @var{ldb})
+@end deftypefun
+
+@deftypefun void cblas_dgemm (const enum CBLAS_ORDER @var{Order}, const enum CBLAS_TRANSPOSE @var{TransA}, const enum CBLAS_TRANSPOSE @var{TransB}, const int @var{M}, const int @var{N}, const int @var{K}, const double @var{alpha}, const double * @var{A}, const int @var{lda}, const double * @var{B}, const int @var{ldb}, const double @var{beta}, double * @var{C}, const int @var{ldc})
+@end deftypefun
+
+@deftypefun void cblas_dsymm (const enum CBLAS_ORDER @var{Order}, const enum CBLAS_SIDE @var{Side}, const enum CBLAS_UPLO @var{Uplo}, const int @var{M}, const int @var{N}, const double @var{alpha}, const double * @var{A}, const int @var{lda}, const double * @var{B}, const int @var{ldb}, const double @var{beta}, double * @var{C}, const int @var{ldc})
+@end deftypefun
+
+@deftypefun void cblas_dsyrk (const enum CBLAS_ORDER @var{Order}, const enum CBLAS_UPLO @var{Uplo}, const enum CBLAS_TRANSPOSE @var{Trans}, const int @var{N}, const int @var{K}, const double @var{alpha}, const double * @var{A}, const int @var{lda}, const double @var{beta}, double * @var{C}, const int @var{ldc})
+@end deftypefun
+
+@deftypefun void cblas_dsyr2k (const enum CBLAS_ORDER @var{Order}, const enum CBLAS_UPLO @var{Uplo}, const enum CBLAS_TRANSPOSE @var{Trans}, const int @var{N}, const int @var{K}, const double @var{alpha}, const double * @var{A}, const int @var{lda}, const double * @var{B}, const int @var{ldb}, const double @var{beta}, double * @var{C}, const int @var{ldc})
+@end deftypefun
+
+@deftypefun void cblas_dtrmm (const enum CBLAS_ORDER @var{Order}, const enum CBLAS_SIDE @var{Side}, const enum CBLAS_UPLO @var{Uplo}, const enum CBLAS_TRANSPOSE @var{TransA}, const enum CBLAS_DIAG @var{Diag}, const int @var{M}, const int @var{N}, const double @var{alpha}, const double * @var{A}, const int @var{lda}, double * @var{B}, const int @var{ldb})
+@end deftypefun
+
+@deftypefun void cblas_dtrsm (const enum CBLAS_ORDER @var{Order}, const enum CBLAS_SIDE @var{Side}, const enum CBLAS_UPLO @var{Uplo}, const enum CBLAS_TRANSPOSE @var{TransA}, const enum CBLAS_DIAG @var{Diag}, const int @var{M}, const int @var{N}, const double @var{alpha}, const double * @var{A}, const int @var{lda}, double * @var{B}, const int @var{ldb})
+@end deftypefun
+
+@deftypefun void cblas_cgemm (const enum CBLAS_ORDER @var{Order}, const enum CBLAS_TRANSPOSE @var{TransA}, const enum CBLAS_TRANSPOSE @var{TransB}, const int @var{M}, const int @var{N}, const int @var{K}, const void * @var{alpha}, const void * @var{A}, const int @var{lda}, const void * @var{B}, const int @var{ldb}, const void * @var{beta}, void * @var{C}, const int @var{ldc})
+@end deftypefun
+
+@deftypefun void cblas_csymm (const enum CBLAS_ORDER @var{Order}, const enum CBLAS_SIDE @var{Side}, const enum CBLAS_UPLO @var{Uplo}, const int @var{M}, const int @var{N}, const void * @var{alpha}, const void * @var{A}, const int @var{lda}, const void * @var{B}, const int @var{ldb}, const void * @var{beta}, void * @var{C}, const int @var{ldc})
+@end deftypefun
+
+@deftypefun void cblas_csyrk (const enum CBLAS_ORDER @var{Order}, const enum CBLAS_UPLO @var{Uplo}, const enum CBLAS_TRANSPOSE @var{Trans}, const int @var{N}, const int @var{K}, const void * @var{alpha}, const void * @var{A}, const int @var{lda}, const void * @var{beta}, void * @var{C}, const int @var{ldc})
+@end deftypefun
+
+@deftypefun void cblas_csyr2k (const enum CBLAS_ORDER @var{Order}, const enum CBLAS_UPLO @var{Uplo}, const enum CBLAS_TRANSPOSE @var{Trans}, const int @var{N}, const int @var{K}, const void * @var{alpha}, const void * @var{A}, const int @var{lda}, const void * @var{B}, const int @var{ldb}, const void * @var{beta}, void * @var{C}, const int @var{ldc})
+@end deftypefun
+
+@deftypefun void cblas_ctrmm (const enum CBLAS_ORDER @var{Order}, const enum CBLAS_SIDE @var{Side}, const enum CBLAS_UPLO @var{Uplo}, const enum CBLAS_TRANSPOSE @var{TransA}, const enum CBLAS_DIAG @var{Diag}, const int @var{M}, const int @var{N}, const void * @var{alpha}, const void * @var{A}, const int @var{lda}, void * @var{B}, const int @var{ldb})
+@end deftypefun
+
+@deftypefun void cblas_ctrsm (const enum CBLAS_ORDER @var{Order}, const enum CBLAS_SIDE @var{Side}, const enum CBLAS_UPLO @var{Uplo}, const enum CBLAS_TRANSPOSE @var{TransA}, const enum CBLAS_DIAG @var{Diag}, const int @var{M}, const int @var{N}, const void * @var{alpha}, const void * @var{A}, const int @var{lda}, void * @var{B}, const int @var{ldb})
+@end deftypefun
+
+@deftypefun void cblas_zgemm (const enum CBLAS_ORDER @var{Order}, const enum CBLAS_TRANSPOSE @var{TransA}, const enum CBLAS_TRANSPOSE @var{TransB}, const int @var{M}, const int @var{N}, const int @var{K}, const void * @var{alpha}, const void * @var{A}, const int @var{lda}, const void * @var{B}, const int @var{ldb}, const void * @var{beta}, void * @var{C}, const int @var{ldc})
+@end deftypefun
+
+@deftypefun void cblas_zsymm (const enum CBLAS_ORDER @var{Order}, const enum CBLAS_SIDE @var{Side}, const enum CBLAS_UPLO @var{Uplo}, const int @var{M}, const int @var{N}, const void * @var{alpha}, const void * @var{A}, const int @var{lda}, const void * @var{B}, const int @var{ldb}, const void * @var{beta}, void * @var{C}, const int @var{ldc})
+@end deftypefun
+
+@deftypefun void cblas_zsyrk (const enum CBLAS_ORDER @var{Order}, const enum CBLAS_UPLO @var{Uplo}, const enum CBLAS_TRANSPOSE @var{Trans}, const int @var{N}, const int @var{K}, const void * @var{alpha}, const void * @var{A}, const int @var{lda}, const void * @var{beta}, void * @var{C}, const int @var{ldc})
+@end deftypefun
+
+@deftypefun void cblas_zsyr2k (const enum CBLAS_ORDER @var{Order}, const enum CBLAS_UPLO @var{Uplo}, const enum CBLAS_TRANSPOSE @var{Trans}, const int @var{N}, const int @var{K}, const void * @var{alpha}, const void * @var{A}, const int @var{lda}, const void * @var{B}, const int @var{ldb}, const void * @var{beta}, void * @var{C}, const int @var{ldc})
+@end deftypefun
+
+@deftypefun void cblas_ztrmm (const enum CBLAS_ORDER @var{Order}, const enum CBLAS_SIDE @var{Side}, const enum CBLAS_UPLO @var{Uplo}, const enum CBLAS_TRANSPOSE @var{TransA}, const enum CBLAS_DIAG @var{Diag}, const int @var{M}, const int @var{N}, const void * @var{alpha}, const void * @var{A}, const int @var{lda}, void * @var{B}, const int @var{ldb})
+@end deftypefun
+
+@deftypefun void cblas_ztrsm (const enum CBLAS_ORDER @var{Order}, const enum CBLAS_SIDE @var{Side}, const enum CBLAS_UPLO @var{Uplo}, const enum CBLAS_TRANSPOSE @var{TransA}, const enum CBLAS_DIAG @var{Diag}, const int @var{M}, const int @var{N}, const void * @var{alpha}, const void * @var{A}, const int @var{lda}, void * @var{B}, const int @var{ldb})
+@end deftypefun
+
+@deftypefun void cblas_chemm (const enum CBLAS_ORDER @var{Order}, const enum CBLAS_SIDE @var{Side}, const enum CBLAS_UPLO @var{Uplo}, const int @var{M}, const int @var{N}, const void * @var{alpha}, const void * @var{A}, const int @var{lda}, const void * @var{B}, const int @var{ldb}, const void * @var{beta}, void * @var{C}, const int @var{ldc})
+@end deftypefun
+
+@deftypefun void cblas_cherk (const enum CBLAS_ORDER @var{Order}, const enum CBLAS_UPLO @var{Uplo}, const enum CBLAS_TRANSPOSE @var{Trans}, const int @var{N}, const int @var{K}, const float @var{alpha}, const void * @var{A}, const int @var{lda}, const float @var{beta}, void * @var{C}, const int @var{ldc})
+@end deftypefun
+
+@deftypefun void cblas_cher2k (const enum CBLAS_ORDER @var{Order}, const enum CBLAS_UPLO @var{Uplo}, const enum CBLAS_TRANSPOSE @var{Trans}, const int @var{N}, const int @var{K}, const void * @var{alpha}, const void * @var{A}, const int @var{lda}, const void * @var{B}, const int @var{ldb}, const float @var{beta}, void * @var{C}, const int @var{ldc})
+@end deftypefun
+
+@deftypefun void cblas_zhemm (const enum CBLAS_ORDER @var{Order}, const enum CBLAS_SIDE @var{Side}, const enum CBLAS_UPLO @var{Uplo}, const int @var{M}, const int @var{N}, const void * @var{alpha}, const void * @var{A}, const int @var{lda}, const void * @var{B}, const int @var{ldb}, const void * @var{beta}, void * @var{C}, const int @var{ldc})
+@end deftypefun
+
+@deftypefun void cblas_zherk (const enum CBLAS_ORDER @var{Order}, const enum CBLAS_UPLO @var{Uplo}, const enum CBLAS_TRANSPOSE @var{Trans}, const int @var{N}, const int @var{K}, const double @var{alpha}, const void * @var{A}, const int @var{lda}, const double @var{beta}, void * @var{C}, const int @var{ldc})
+@end deftypefun
+
+@deftypefun void cblas_zher2k (const enum CBLAS_ORDER @var{Order}, const enum CBLAS_UPLO @var{Uplo}, const enum CBLAS_TRANSPOSE @var{Trans}, const int @var{N}, const int @var{K}, const void * @var{alpha}, const void * @var{A}, const int @var{lda}, const void * @var{B}, const int @var{ldb}, const double @var{beta}, void * @var{C}, const int @var{ldc})
+@end deftypefun
+
+@deftypefun void cblas_xerbla (int @var{p}, const char * @var{rout}, const char * @var{form}, ...)
+@end deftypefun
+
+@node GSL CBLAS Examples
+@section Examples
+
+The following program computes the product of two matrices using the
+Level-3 @sc{blas} function @sc{sgemm},
+@tex
+\beforedisplay
+$$
+\left(
+\matrix{0.11&0.12&0.13\cr
+0.21&0.22&0.23\cr}
+\right)
+\left(
+\matrix{1011&1012\cr
+1021&1022\cr
+1031&1031\cr}
+\right)
+=
+\left(
+\matrix{367.76&368.12\cr
+674.06&674.72\cr}
+\right)
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+[ 0.11 0.12 0.13 ]  [ 1011 1012 ]     [ 367.76 368.12 ]
+[ 0.21 0.22 0.23 ]  [ 1021 1022 ]  =  [ 674.06 674.72 ]
+                    [ 1031 1032 ]
+@end example
+
+@end ifinfo
+@noindent
+The matrices are stored in row major order but could be stored in column
+major order if the first argument of the call to @code{cblas_sgemm} was
+changed to @code{CblasColMajor}.
+
+@example
+@verbatiminclude examples/cblas.c
+@end example
+
+@noindent
+To compile the program use the following command line,
+
+@example
+$ gcc -Wall demo.c -lgslcblas
+@end example
+
+@noindent
+There is no need to link with the main library @code{-lgsl} in this
+case as the @sc{cblas} library is an independent unit. Here is the output
+from the program,
+
+@example
+$ ./a.out
+@verbatiminclude examples/cblas.out
+@end example
diff --git a/gsl-1.9/doc/cheb.eps b/gsl-1.9/doc/cheb.eps
new file mode 100644
index 0000000..6e868af
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+stroke
+grestore
+end
+showpage
+%%Trailer
+%%DocumentFonts: Helvetica
diff --git a/gsl-1.9/doc/cheb.texi b/gsl-1.9/doc/cheb.texi
new file mode 100644
index 0000000..0d97a86
--- /dev/null
+++ b/gsl-1.9/doc/cheb.texi
@@ -0,0 +1,177 @@
+@cindex Chebyshev series
+@cindex fitting, using Chebyshev polynomials
+@cindex interpolation, using Chebyshev polynomials
+
+This chapter describes routines for computing Chebyshev approximations
+to univariate functions.  A Chebyshev approximation is a truncation of
+the series @math{f(x) = \sum c_n T_n(x)}, where the Chebyshev
+polynomials @math{T_n(x) = \cos(n \arccos x)} provide an orthogonal
+basis of polynomials on the interval @math{[-1,1]} with the weight
+function @c{$1 / \sqrt{1-x^2}$} 
+@math{1 / \sqrt@{1-x^2@}}.  The first few Chebyshev polynomials are,
+@math{T_0(x) = 1}, @math{T_1(x) = x}, @math{T_2(x) = 2 x^2 - 1}.
+For further information see Abramowitz & Stegun, Chapter 22. 
+
+The functions described in this chapter are declared in the header file
+@file{gsl_chebyshev.h}.
+
+@menu
+* Chebyshev Definitions::       
+* Creation and Calculation of Chebyshev Series::  
+* Chebyshev Series Evaluation::  
+* Derivatives and Integrals::   
+* Chebyshev Approximation examples::  
+* Chebyshev Approximation References and Further Reading::  
+@end menu
+
+@node Chebyshev Definitions
+@section Definitions
+
+A Chebyshev series  is stored using the following structure,
+
+@example
+typedef struct
+@{
+  double * c;   /* coefficients  c[0] .. c[order] */
+  int order;    /* order of expansion             */
+  double a;     /* lower interval point           */
+  double b;     /* upper interval point           */
+  ...
+@} gsl_cheb_series
+@end example
+
+@noindent
+The approximation is made over the range @math{[a,b]} using
+@var{order}+1 terms, including the coefficient @math{c[0]}.  The series
+is computed using the following convention,
+@tex
+\beforedisplay
+$$
+f(x) = {c_0 \over 2} + \sum_{n=1} c_n T_n(x)
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+f(x) = (c_0 / 2) + \sum_@{n=1@} c_n T_n(x)
+@end example
+
+@end ifinfo
+@noindent
+which is needed when accessing the coefficients directly.
+
+@node Creation and Calculation of Chebyshev Series
+@section Creation and Calculation of Chebyshev Series
+
+@deftypefun {gsl_cheb_series *} gsl_cheb_alloc (const size_t @var{n})
+This function allocates space for a Chebyshev series of order @var{n}
+and returns a pointer to a new @code{gsl_cheb_series} struct.
+@end deftypefun
+
+@deftypefun void gsl_cheb_free (gsl_cheb_series * @var{cs})
+This function frees a previously allocated Chebyshev series @var{cs}.
+@end deftypefun
+
+@deftypefun int gsl_cheb_init (gsl_cheb_series * @var{cs}, const gsl_function * @var{f}, const double @var{a}, const double @var{b})
+This function computes the Chebyshev approximation @var{cs} for the
+function @var{f} over the range @math{(a,b)} to the previously specified
+order.  The computation of the Chebyshev approximation is an
+@math{O(n^2)} process, and requires @math{n} function evaluations.
+@end deftypefun
+
+@node Chebyshev Series Evaluation
+@section Chebyshev Series Evaluation
+
+@deftypefun double gsl_cheb_eval (const gsl_cheb_series * @var{cs}, double @var{x})
+This function evaluates the Chebyshev series @var{cs} at a given point @var{x}.
+@end deftypefun
+
+@deftypefun int gsl_cheb_eval_err (const gsl_cheb_series * @var{cs}, const double @var{x}, double * @var{result}, double * @var{abserr})
+This function computes the Chebyshev series @var{cs} at a given point
+@var{x}, estimating both the series @var{result} and its absolute error
+@var{abserr}.  The error estimate is made from the first neglected term
+in the series.
+@end deftypefun
+
+@deftypefun double gsl_cheb_eval_n (const gsl_cheb_series * @var{cs}, size_t @var{order}, double @var{x})
+This function evaluates the Chebyshev series @var{cs} at a given point
+@var{n}, to (at most) the given order @var{order}.
+@end deftypefun
+
+@deftypefun int gsl_cheb_eval_n_err (const gsl_cheb_series * @var{cs}, const size_t @var{order}, const double @var{x}, double * @var{result}, double * @var{abserr})
+This function evaluates a Chebyshev series @var{cs} at a given point
+@var{x}, estimating both the series @var{result} and its absolute error
+@var{abserr}, to (at most) the given order @var{order}.  The error
+estimate is made from the first neglected term in the series.
+@end deftypefun
+
+@comment @deftypefun double gsl_cheb_eval_mode (const gsl_cheb_series * @var{cs}, double @var{x}, gsl_mode_t @var{mode})
+@comment @end deftypefun
+
+@comment @deftypefun int gsl_cheb_eval_mode_err (const gsl_cheb_series * @var{cs}, const double @var{x}, gsl_mode_t @var{mode}, double * @var{result}, double * @var{abserr})
+@comment Evaluate a Chebyshev series at a given point, using the default
+@comment order for double precision mode(s) and the single precision
+@comment order for other modes.
+@comment @end deftypefun
+
+
+@node Derivatives and Integrals
+@section Derivatives and Integrals
+
+The following functions allow a Chebyshev series to be differentiated or
+integrated, producing a new Chebyshev series.  Note that the error
+estimate produced by evaluating the derivative series will be
+underestimated due to the contribution of higher order terms being
+neglected.
+
+@deftypefun int gsl_cheb_calc_deriv (gsl_cheb_series * @var{deriv}, const gsl_cheb_series * @var{cs})
+This function computes the derivative of the series @var{cs}, storing
+the derivative coefficients in the previously allocated @var{deriv}.
+The two series @var{cs} and @var{deriv} must have been allocated with
+the same order.
+@end deftypefun
+
+@deftypefun int gsl_cheb_calc_integ (gsl_cheb_series * @var{integ}, const gsl_cheb_series * @var{cs})
+This function computes the integral of the series @var{cs}, storing the
+integral coefficients in the previously allocated @var{integ}.  The two
+series @var{cs} and @var{integ} must have been allocated with the same
+order.  The lower limit of the integration is taken to be the left hand
+end of the range @var{a}.
+@end deftypefun
+
+@node Chebyshev Approximation examples
+@section Examples
+
+The following example program computes Chebyshev approximations to a
+step function.  This is an extremely difficult approximation to make,
+due to the discontinuity, and was chosen as an example where
+approximation error is visible.  For smooth functions the Chebyshev
+approximation converges extremely rapidly and errors would not be
+visible.
+
+@example
+@verbatiminclude examples/cheb.c
+@end example
+
+@noindent
+The output from the program gives the original function, 10-th order
+approximation and 40-th order approximation, all sampled at intervals of
+0.001 in @math{x}.
+
+@iftex
+@sp 1
+@center @image{cheb,3.4in}
+@end iftex
+
+@node Chebyshev Approximation References and Further Reading
+@section References and Further Reading
+
+The following paper describes the use of Chebyshev series,
+
+@itemize @asis
+@item 
+R. Broucke, ``Ten Subroutines for the Manipulation of Chebyshev Series
+[C1] (Algorithm 446)''. @cite{Communications of the ACM} 16(4), 254--256
+(1973)
+@end itemize
diff --git a/gsl-1.9/doc/combination.texi b/gsl-1.9/doc/combination.texi
new file mode 100644
index 0000000..d9ef7e1
--- /dev/null
+++ b/gsl-1.9/doc/combination.texi
@@ -0,0 +1,221 @@
+@cindex combinations
+
+This chapter describes functions for creating and manipulating
+combinations. A combination @math{c} is represented by an array of
+@math{k} integers in the range 0 to @math{n-1}, where each value
+@math{c_i} occurs at most once.  The combination @math{c} corresponds to
+indices of @math{k} elements chosen from an @math{n} element vector.
+Combinations are useful for iterating over all @math{k}-element subsets
+of a set.
+
+The functions described in this chapter are defined in the header file
+@file{gsl_combination.h}.
+
+@menu
+* The Combination struct::      
+* Combination allocation::      
+* Accessing combination elements::  
+* Combination properties::      
+* Combination functions::       
+* Reading and writing combinations::  
+* Combination Examples::        
+* Combination References and Further Reading::
+@end menu
+
+@node The Combination struct
+@section The Combination struct
+
+A combination is defined by a structure containing three components, the
+values of @math{n} and @math{k}, and a pointer to the combination array.
+The elements of the combination array are all of type @code{size_t}, and
+are stored in increasing order.  The @code{gsl_combination} structure
+looks like this,
+
+@example
+typedef struct
+@{
+  size_t n;
+  size_t k;
+  size_t *data;
+@} gsl_combination;
+@end example
+@comment
+
+@noindent
+
+@node Combination allocation
+@section Combination allocation
+
+@deftypefun {gsl_combination *} gsl_combination_alloc (size_t @var{n}, size_t @var{k})
+This function allocates memory for a new combination with parameters
+@var{n}, @var{k}.  The combination is not initialized and its elements
+are undefined.  Use the function @code{gsl_combination_calloc} if you
+want to create a combination which is initialized to the
+lexicographically first combination. A null pointer is returned if
+insufficient memory is available to create the combination.
+@end deftypefun
+
+@deftypefun {gsl_combination *} gsl_combination_calloc (size_t @var{n}, size_t @var{k})
+This function allocates memory for a new combination with parameters
+@var{n}, @var{k} and initializes it to the lexicographically first
+combination. A null pointer is returned if insufficient memory is
+available to create the combination.
+@end deftypefun
+
+@deftypefun void gsl_combination_init_first (gsl_combination * @var{c})
+This function initializes the combination @var{c} to the
+lexicographically first combination, i.e.  @math{(0,1,2,@dots{},k-1)}.
+@end deftypefun
+
+@deftypefun void gsl_combination_init_last (gsl_combination * @var{c})
+This function initializes the combination @var{c} to the
+lexicographically last combination, i.e.  @math{(n-k,n-k+1,@dots{},n-1)}.
+@end deftypefun
+
+@deftypefun void gsl_combination_free (gsl_combination * @var{c})
+This function frees all the memory used by the combination @var{c}.
+@end deftypefun
+
+@deftypefun int gsl_combination_memcpy (gsl_combination * @var{dest}, const gsl_combination * @var{src})
+This function copies the elements of the combination @var{src} into the
+combination @var{dest}.  The two combinations must have the same size.
+@end deftypefun
+
+
+@node Accessing combination elements
+@section Accessing combination elements
+
+The following function can be used to access the elements of a combination.
+
+@deftypefun size_t gsl_combination_get (const gsl_combination * @var{c}, const size_t @var{i})
+This function returns the value of the @var{i}-th element of the
+combination @var{c}.  If @var{i} lies outside the allowed range of 0 to
+@math{@var{k}-1} then the error handler is invoked and 0 is returned.
+@end deftypefun
+
+@node Combination properties
+@section Combination properties
+
+@deftypefun size_t gsl_combination_n (const gsl_combination * @var{c})
+This function returns the range (@math{n}) of the combination @var{c}.
+@end deftypefun
+
+@deftypefun size_t gsl_combination_k (const gsl_combination * @var{c})
+This function returns the number of elements (@math{k}) in the combination @var{c}.
+@end deftypefun
+
+@deftypefun {size_t *} gsl_combination_data (const gsl_combination * @var{c})
+This function returns a pointer to the array of elements in the
+combination @var{c}.
+@end deftypefun
+
+@deftypefun int gsl_combination_valid (gsl_combination * @var{c})
+@cindex checking combination for validity
+@cindex testing combination for validity
+This function checks that the combination @var{c} is valid.  The @var{k}
+elements should lie in the range 0 to @math{@var{n}-1}, with each
+value occurring once at most and in increasing order.
+@end deftypefun
+
+@node Combination functions
+@section Combination functions
+
+@deftypefun int gsl_combination_next (gsl_combination * @var{c})
+@cindex iterating through combinations
+This function advances the combination @var{c} to the next combination
+in lexicographic order and returns @code{GSL_SUCCESS}.  If no further
+combinations are available it returns @code{GSL_FAILURE} and leaves
+@var{c} unmodified.  Starting with the first combination and
+repeatedly applying this function will iterate through all possible
+combinations of a given order.
+@end deftypefun
+
+@deftypefun int gsl_combination_prev (gsl_combination * @var{c})
+This function steps backwards from the combination @var{c} to the
+previous combination in lexicographic order, returning
+@code{GSL_SUCCESS}.  If no previous combination is available it returns
+@code{GSL_FAILURE} and leaves @var{c} unmodified.
+@end deftypefun
+
+
+@node Reading and writing combinations
+@section Reading and writing combinations
+
+The library provides functions for reading and writing combinations to a
+file as binary data or formatted text.
+
+@deftypefun int gsl_combination_fwrite (FILE * @var{stream}, const gsl_combination * @var{c})
+This function writes the elements of the combination @var{c} to the
+stream @var{stream} in binary format.  The function returns
+@code{GSL_EFAILED} if there was a problem writing to the file.  Since the
+data is written in the native binary format it may not be portable
+between different architectures.
+@end deftypefun
+
+@deftypefun int gsl_combination_fread (FILE * @var{stream}, gsl_combination * @var{c})
+This function reads elements from the open stream @var{stream} into the
+combination @var{c} in binary format.  The combination @var{c} must be
+preallocated with correct values of @math{n} and @math{k} since the
+function uses the size of @var{c} to determine how many bytes to read.
+The function returns @code{GSL_EFAILED} if there was a problem reading
+from the file.  The data is assumed to have been written in the native
+binary format on the same architecture.
+@end deftypefun
+
+@deftypefun int gsl_combination_fprintf (FILE * @var{stream}, const gsl_combination * @var{c}, const char * @var{format})
+This function writes the elements of the combination @var{c}
+line-by-line to the stream @var{stream} using the format specifier
+@var{format}, which should be suitable for a type of @var{size_t}.  On a
+GNU system the type modifier @code{Z} represents @code{size_t}, so
+@code{"%Zu\n"} is a suitable format.  The function returns
+@code{GSL_EFAILED} if there was a problem writing to the file.
+@end deftypefun
+
+@deftypefun int gsl_combination_fscanf (FILE * @var{stream}, gsl_combination * @var{c})
+This function reads formatted data from the stream @var{stream} into the
+combination @var{c}.  The combination @var{c} must be preallocated with
+correct values of @math{n} and @math{k} since the function uses the size of @var{c} to
+determine how many numbers to read.  The function returns
+@code{GSL_EFAILED} if there was a problem reading from the file.
+@end deftypefun
+
+
+@node Combination Examples
+@section Examples
+The example program below prints all subsets of the set
+@math{@{0,1,2,3@}} ordered by size.  Subsets of the same size are
+ordered lexicographically.
+
+@example
+@verbatiminclude examples/combination.c
+@end example
+
+@noindent
+Here is the output from the program,
+
+@example
+$ ./a.out 
+@verbatiminclude examples/combination.out
+@end example
+
+@noindent
+All 16 subsets are generated, and the subsets of each size are sorted
+lexicographically.
+
+
+@node Combination References and Further Reading
+@section References and Further Reading
+
+@noindent
+Further information on combinations can be found in,
+
+@itemize @asis
+@item
+Donald L. Kreher, Douglas R. Stinson, @cite{Combinatorial Algorithms:
+Generation, Enumeration and Search}, 1998, CRC Press LLC, ISBN
+084933988X
+@end itemize
+
+@noindent
+
+
diff --git a/gsl-1.9/doc/complex.texi b/gsl-1.9/doc/complex.texi
new file mode 100644
index 0000000..b84a7ac
--- /dev/null
+++ b/gsl-1.9/doc/complex.texi
@@ -0,0 +1,493 @@
+@cindex complex numbers
+
+The functions described in this chapter provide support for complex
+numbers.  The algorithms take care to avoid unnecessary intermediate
+underflows and overflows, allowing the functions to be evaluated over 
+as much of the complex plane as possible. 
+
+@comment FIXME: this still needs to be
+@comment done for the csc,sec,cot,csch,sech,coth functions
+
+For multiple-valued functions the branch cuts have been chosen to follow
+the conventions of Abramowitz and Stegun in the @cite{Handbook of
+Mathematical Functions}. The functions return principal values which are
+the same as those in GNU Calc, which in turn are the same as those in
+@cite{Common Lisp, The Language (Second Edition)}@footnote{Note that the
+first edition uses different definitions.} and the HP-28/48 series of
+calculators.
+
+The complex types are defined in the header file @file{gsl_complex.h},
+while the corresponding complex functions and arithmetic operations are
+defined in @file{gsl_complex_math.h}.
+
+@menu
+* Complex numbers::             
+* Properties of complex numbers::  
+* Complex arithmetic operators::  
+* Elementary Complex Functions::  
+* Complex Trigonometric Functions::  
+* Inverse Complex Trigonometric Functions::  
+* Complex Hyperbolic Functions::  
+* Inverse Complex Hyperbolic Functions::  
+* Complex Number References and Further Reading::  
+@end menu
+
+@node Complex numbers
+@section Complex numbers
+@cindex representations of complex numbers
+@cindex polar form of complex numbers
+@tpindex gsl_complex
+
+Complex numbers are represented using the type @code{gsl_complex}. The
+internal representation of this type may vary across platforms and
+should not be accessed directly. The functions and macros described
+below allow complex numbers to be manipulated in a portable way.
+
+For reference, the default form of the @code{gsl_complex} type is
+given by the following struct,
+
+@example
+typedef struct
+@{
+  double dat[2];
+@} gsl_complex;
+@end example
+
+@noindent
+The real and imaginary part are stored in contiguous elements of a two
+element array. This eliminates any padding between the real and
+imaginary parts, @code{dat[0]} and @code{dat[1]}, allowing the struct to
+be mapped correctly onto packed complex arrays.
+
+@deftypefun gsl_complex gsl_complex_rect (double @var{x}, double @var{y})
+This function uses the rectangular cartesian components
+(@var{x},@var{y}) to return the complex number @math{z = x + i y}.
+@end deftypefun
+
+@deftypefun gsl_complex gsl_complex_polar (double @var{r}, double @var{theta})
+This function returns the complex number @math{z = r \exp(i \theta) = r
+(\cos(\theta) + i \sin(\theta))} from the polar representation
+(@var{r},@var{theta}).
+@end deftypefun
+
+@defmac GSL_REAL (@var{z})
+@defmacx GSL_IMAG (@var{z})
+These macros return the real and imaginary parts of the complex number
+@var{z}.
+@end defmac
+
+@defmac GSL_SET_COMPLEX (@var{zp}, @var{x}, @var{y})
+This macro uses the cartesian components (@var{x},@var{y}) to set the
+real and imaginary parts of the complex number pointed to by @var{zp}.
+For example,
+
+@example
+GSL_SET_COMPLEX(&z, 3, 4)
+@end example
+
+@noindent
+sets @var{z} to be @math{3 + 4i}.
+@end defmac
+
+@defmac GSL_SET_REAL (@var{zp},@var{x})
+@defmacx GSL_SET_IMAG (@var{zp},@var{y})
+These macros allow the real and imaginary parts of the complex number
+pointed to by @var{zp} to be set independently.
+@end defmac
+
+@node Properties of complex numbers
+@section Properties of complex numbers
+
+@deftypefun double gsl_complex_arg (gsl_complex @var{z})
+@cindex argument of complex number 
+This function returns the argument of the complex number @var{z},
+@math{\arg(z)}, where @c{$-\pi < \arg(z) \leq \pi$}
+@math{-\pi < \arg(z) <= \pi}.
+@end deftypefun
+
+@deftypefun double gsl_complex_abs (gsl_complex @var{z})
+@cindex magnitude of complex number 
+This function returns the magnitude of the complex number @var{z}, @math{|z|}.
+@end deftypefun
+
+@deftypefun double gsl_complex_abs2 (gsl_complex @var{z})
+This function returns the squared magnitude of the complex number
+@var{z}, @math{|z|^2}.
+@end deftypefun
+
+@deftypefun double gsl_complex_logabs (gsl_complex @var{z})
+This function returns the natural logarithm of the magnitude of the
+complex number @var{z}, @math{\log|z|}.  It allows an accurate
+evaluation of @math{\log|z|} when @math{|z|} is close to one. The direct
+evaluation of @code{log(gsl_complex_abs(z))} would lead to a loss of
+precision in this case.
+@end deftypefun
+
+
+@node Complex arithmetic operators
+@section Complex arithmetic operators
+@cindex complex arithmetic
+
+@deftypefun gsl_complex gsl_complex_add (gsl_complex @var{a}, gsl_complex @var{b})
+This function returns the sum of the complex numbers @var{a} and
+@var{b}, @math{z=a+b}.
+@end deftypefun
+
+@deftypefun gsl_complex gsl_complex_sub (gsl_complex @var{a}, gsl_complex @var{b})
+This function returns the difference of the complex numbers @var{a} and
+@var{b}, @math{z=a-b}.
+@end deftypefun
+
+@deftypefun gsl_complex gsl_complex_mul (gsl_complex @var{a}, gsl_complex @var{b})
+This function returns the product of the complex numbers @var{a} and
+@var{b}, @math{z=ab}.
+@end deftypefun
+
+@deftypefun gsl_complex gsl_complex_div (gsl_complex @var{a}, gsl_complex @var{b})
+This function returns the quotient of the complex numbers @var{a} and
+@var{b}, @math{z=a/b}.
+@end deftypefun
+
+
+@deftypefun gsl_complex gsl_complex_add_real (gsl_complex @var{a}, double @var{x})
+This function returns the sum of the complex number @var{a} and the
+real number @var{x}, @math{z=a+x}.
+@end deftypefun
+
+@deftypefun gsl_complex gsl_complex_sub_real (gsl_complex @var{a}, double @var{x})
+This function returns the difference of the complex number @var{a} and the
+real number @var{x}, @math{z=a-x}.
+@end deftypefun
+
+@deftypefun gsl_complex gsl_complex_mul_real (gsl_complex @var{a}, double @var{x})
+This function returns the product of the complex number @var{a} and the
+real number @var{x}, @math{z=ax}.
+@end deftypefun
+
+@deftypefun gsl_complex gsl_complex_div_real (gsl_complex @var{a}, double @var{x})
+This function returns the quotient of the complex number @var{a} and the
+real number @var{x}, @math{z=a/x}.
+@end deftypefun
+
+@deftypefun gsl_complex gsl_complex_add_imag (gsl_complex @var{a}, double @var{y})
+This function returns the sum of the complex number @var{a} and the
+imaginary number @math{i}@var{y}, @math{z=a+iy}.
+@end deftypefun
+
+@deftypefun gsl_complex gsl_complex_sub_imag (gsl_complex @var{a}, double @var{y})
+This function returns the difference of the complex number @var{a} and the
+imaginary number @math{i}@var{y}, @math{z=a-iy}.
+@end deftypefun
+
+@deftypefun gsl_complex gsl_complex_mul_imag (gsl_complex @var{a}, double @var{y})
+This function returns the product of the complex number @var{a} and the
+imaginary number @math{i}@var{y}, @math{z=a*(iy)}.
+@end deftypefun
+
+@deftypefun gsl_complex gsl_complex_div_imag (gsl_complex @var{a}, double @var{y})
+This function returns the quotient of the complex number @var{a} and the
+imaginary number @math{i}@var{y}, @math{z=a/(iy)}.
+@end deftypefun
+
+@deftypefun gsl_complex gsl_complex_conjugate (gsl_complex @var{z})
+@cindex conjugate of complex number
+This function returns the complex conjugate of the complex number
+@var{z}, @math{z^* = x - i y}.
+@end deftypefun
+
+@deftypefun gsl_complex gsl_complex_inverse (gsl_complex @var{z})
+This function returns the inverse, or reciprocal, of the complex number
+@var{z}, @math{1/z = (x - i y)/(x^2 + y^2)}.
+@end deftypefun
+
+@deftypefun gsl_complex gsl_complex_negative (gsl_complex @var{z})
+This function returns the negative of the complex number
+@var{z}, @math{-z = (-x) + i(-y)}.
+@end deftypefun
+
+
+@node Elementary Complex Functions
+@section Elementary Complex Functions
+
+@deftypefun gsl_complex gsl_complex_sqrt (gsl_complex @var{z})
+@cindex square root of complex number
+This function returns the square root of the complex number @var{z},
+@math{\sqrt z}. The branch cut is the negative real axis. The result
+always lies in the right half of the complex plane.
+@end deftypefun
+
+@deftypefun gsl_complex gsl_complex_sqrt_real (double @var{x})
+This function returns the complex square root of the real number
+@var{x}, where @var{x} may be negative.
+@end deftypefun
+
+
+@deftypefun gsl_complex gsl_complex_pow (gsl_complex @var{z}, gsl_complex @var{a})
+@cindex power of complex number
+@cindex exponentiation of complex number
+The function returns the complex number @var{z} raised to the complex
+power @var{a}, @math{z^a}. This is computed as @math{\exp(\log(z)*a)}
+using complex logarithms and complex exponentials.
+@end deftypefun
+
+@deftypefun gsl_complex gsl_complex_pow_real (gsl_complex @var{z}, double @var{x})
+This function returns the complex number @var{z} raised to the real
+power @var{x}, @math{z^x}.
+@end deftypefun
+
+
+@deftypefun gsl_complex gsl_complex_exp (gsl_complex @var{z})
+This function returns the complex exponential of the complex number
+@var{z}, @math{\exp(z)}.
+@end deftypefun
+
+@deftypefun gsl_complex gsl_complex_log (gsl_complex @var{z})
+@cindex logarithm of complex number
+This function returns the complex natural logarithm (base @math{e}) of
+the complex number @var{z}, @math{\log(z)}.  The branch cut is the
+negative real axis. 
+@end deftypefun
+
+@deftypefun gsl_complex gsl_complex_log10 (gsl_complex @var{z})
+This function returns the complex base-10 logarithm of
+the complex number @var{z}, @c{$\log_{10}(z)$}
+@math{\log_10 (z)}.
+@end deftypefun
+
+@deftypefun gsl_complex gsl_complex_log_b (gsl_complex @var{z}, gsl_complex @var{b})
+This function returns the complex base-@var{b} logarithm of the complex
+number @var{z}, @math{\log_b(z)}. This quantity is computed as the ratio
+@math{\log(z)/\log(b)}.
+@end deftypefun
+
+
+@node Complex Trigonometric Functions
+@section Complex Trigonometric Functions
+@cindex trigonometric functions of complex numbers
+
+@deftypefun gsl_complex gsl_complex_sin (gsl_complex @var{z})
+@cindex sin, of complex number
+This function returns the complex sine of the complex number @var{z},
+@math{\sin(z) = (\exp(iz) - \exp(-iz))/(2i)}.
+@end deftypefun
+
+@deftypefun gsl_complex gsl_complex_cos (gsl_complex @var{z})
+@cindex cosine of complex number
+This function returns the complex cosine of the complex number @var{z},
+@math{\cos(z) = (\exp(iz) + \exp(-iz))/2}.
+@end deftypefun
+
+@deftypefun gsl_complex gsl_complex_tan (gsl_complex @var{z})
+@cindex tangent of complex number
+This function returns the complex tangent of the complex number @var{z},
+@math{\tan(z) = \sin(z)/\cos(z)}.
+@end deftypefun
+
+@deftypefun gsl_complex gsl_complex_sec (gsl_complex @var{z})
+This function returns the complex secant of the complex number @var{z},
+@math{\sec(z) = 1/\cos(z)}.
+@end deftypefun
+
+@deftypefun gsl_complex gsl_complex_csc (gsl_complex @var{z})
+This function returns the complex cosecant of the complex number @var{z},
+@math{\csc(z) = 1/\sin(z)}.
+@end deftypefun
+
+@deftypefun gsl_complex gsl_complex_cot (gsl_complex @var{z})
+This function returns the complex cotangent of the complex number @var{z},
+@math{\cot(z) = 1/\tan(z)}.
+@end deftypefun
+
+
+@node Inverse Complex Trigonometric Functions
+@section Inverse Complex Trigonometric Functions
+@cindex inverse complex trigonometric functions
+
+@deftypefun gsl_complex gsl_complex_arcsin (gsl_complex @var{z})
+This function returns the complex arcsine of the complex number @var{z},
+@math{\arcsin(z)}. The branch cuts are on the real axis, less than @math{-1}
+and greater than @math{1}.
+@end deftypefun
+
+@deftypefun gsl_complex gsl_complex_arcsin_real (double @var{z})
+This function returns the complex arcsine of the real number @var{z},
+@math{\arcsin(z)}. For @math{z} between @math{-1} and @math{1}, the
+function returns a real value in the range @math{[-\pi/2,\pi/2]}. For
+@math{z} less than @math{-1} the result has a real part of @math{-\pi/2}
+and a positive imaginary part.  For @math{z} greater than @math{1} the
+result has a real part of @math{\pi/2} and a negative imaginary part.
+@end deftypefun
+
+@deftypefun gsl_complex gsl_complex_arccos (gsl_complex @var{z})
+This function returns the complex arccosine of the complex number @var{z},
+@math{\arccos(z)}. The branch cuts are on the real axis, less than @math{-1}
+and greater than @math{1}.
+@end deftypefun
+
+@deftypefun gsl_complex gsl_complex_arccos_real (double @var{z})
+This function returns the complex arccosine of the real number @var{z},
+@math{\arccos(z)}. For @math{z} between @math{-1} and @math{1}, the
+function returns a real value in the range @math{[0,\pi]}. For @math{z}
+less than @math{-1} the result has a real part of @math{\pi} and a
+negative imaginary part.  For @math{z} greater than @math{1} the result
+is purely imaginary and positive.
+@end deftypefun
+
+@deftypefun gsl_complex gsl_complex_arctan (gsl_complex @var{z})
+This function returns the complex arctangent of the complex number
+@var{z}, @math{\arctan(z)}. The branch cuts are on the imaginary axis,
+below @math{-i} and above @math{i}.
+@end deftypefun
+
+@deftypefun gsl_complex gsl_complex_arcsec (gsl_complex @var{z})
+This function returns the complex arcsecant of the complex number @var{z},
+@math{\arcsec(z) = \arccos(1/z)}. 
+@end deftypefun
+
+@deftypefun gsl_complex gsl_complex_arcsec_real (double @var{z})
+This function returns the complex arcsecant of the real number @var{z},
+@math{\arcsec(z) = \arccos(1/z)}. 
+@end deftypefun
+
+@deftypefun gsl_complex gsl_complex_arccsc (gsl_complex @var{z})
+This function returns the complex arccosecant of the complex number @var{z},
+@math{\arccsc(z) = \arcsin(1/z)}. 
+@end deftypefun
+
+@deftypefun gsl_complex gsl_complex_arccsc_real (double @var{z})
+This function returns the complex arccosecant of the real number @var{z},
+@math{\arccsc(z) = \arcsin(1/z)}. 
+@end deftypefun
+
+@deftypefun gsl_complex gsl_complex_arccot (gsl_complex @var{z})
+This function returns the complex arccotangent of the complex number @var{z},
+@math{\arccot(z) = \arctan(1/z)}. 
+@end deftypefun
+
+
+@node Complex Hyperbolic Functions
+@section Complex Hyperbolic Functions
+@cindex hyperbolic functions, complex numbers
+
+@deftypefun gsl_complex gsl_complex_sinh (gsl_complex @var{z})
+This function returns the complex hyperbolic sine of the complex number
+@var{z}, @math{\sinh(z) = (\exp(z) - \exp(-z))/2}.
+@end deftypefun
+
+@deftypefun gsl_complex gsl_complex_cosh (gsl_complex @var{z})
+This function returns the complex hyperbolic cosine of the complex number
+@var{z}, @math{\cosh(z) = (\exp(z) + \exp(-z))/2}.
+@end deftypefun
+
+@deftypefun gsl_complex gsl_complex_tanh (gsl_complex @var{z})
+This function returns the complex hyperbolic tangent of the complex number
+@var{z}, @math{\tanh(z) = \sinh(z)/\cosh(z)}.
+@end deftypefun
+
+
+@deftypefun gsl_complex gsl_complex_sech (gsl_complex @var{z})
+This function returns the complex hyperbolic secant of the complex
+number @var{z}, @math{\sech(z) = 1/\cosh(z)}.
+@end deftypefun
+
+@deftypefun gsl_complex gsl_complex_csch (gsl_complex @var{z})
+This function returns the complex hyperbolic cosecant of the complex
+number @var{z}, @math{\csch(z) = 1/\sinh(z)}.
+@end deftypefun
+
+@deftypefun gsl_complex gsl_complex_coth (gsl_complex @var{z})
+This function returns the complex hyperbolic cotangent of the complex
+number @var{z}, @math{\coth(z) = 1/\tanh(z)}.
+@end deftypefun
+
+
+@node Inverse Complex Hyperbolic Functions
+@section Inverse Complex Hyperbolic Functions
+@cindex inverse hyperbolic functions, complex numbers
+
+@deftypefun gsl_complex gsl_complex_arcsinh (gsl_complex @var{z})
+This function returns the complex hyperbolic arcsine of the
+complex number @var{z}, @math{\arcsinh(z)}.  The branch cuts are on the
+imaginary axis, below @math{-i} and above @math{i}.
+@end deftypefun
+
+@deftypefun gsl_complex gsl_complex_arccosh (gsl_complex @var{z})
+This function returns the complex hyperbolic arccosine of the complex
+number @var{z}, @math{\arccosh(z)}.  The branch cut is on the real
+axis, less than @math{1}.  Note that in this case we use the negative
+square root in formula 4.6.21 of Abramowitz & Stegun giving
+@c{$\arccosh(z)=\log(z-\sqrt{z^2-1})$}
+@math{\arccosh(z)=\log(z-\sqrt@{z^2-1@})}.
+@end deftypefun
+
+@deftypefun gsl_complex gsl_complex_arccosh_real (double @var{z})
+This function returns the complex hyperbolic arccosine of
+the real number @var{z}, @math{\arccosh(z)}.
+@end deftypefun
+
+@deftypefun gsl_complex gsl_complex_arctanh (gsl_complex @var{z})
+This function returns the complex hyperbolic arctangent of the complex
+number @var{z}, @math{\arctanh(z)}.  The branch cuts are on the real
+axis, less than @math{-1} and greater than @math{1}.
+@end deftypefun
+
+@deftypefun gsl_complex gsl_complex_arctanh_real (double @var{z})
+This function returns the complex hyperbolic arctangent of the real
+number @var{z}, @math{\arctanh(z)}.
+@end deftypefun
+
+@deftypefun gsl_complex gsl_complex_arcsech (gsl_complex @var{z})
+This function returns the complex hyperbolic arcsecant of the complex
+number @var{z}, @math{\arcsech(z) = \arccosh(1/z)}.
+@end deftypefun
+
+@deftypefun gsl_complex gsl_complex_arccsch (gsl_complex @var{z})
+This function returns the complex hyperbolic arccosecant of the complex
+number @var{z}, @math{\arccsch(z) = \arcsin(1/z)}.
+@end deftypefun
+
+@deftypefun gsl_complex gsl_complex_arccoth (gsl_complex @var{z})
+This function returns the complex hyperbolic arccotangent of the complex
+number @var{z}, @math{\arccoth(z) = \arctanh(1/z)}.
+@end deftypefun
+
+@node Complex Number References and Further Reading
+@section References and Further Reading
+
+The implementations of the elementary and trigonometric functions are
+based on the following papers,
+
+@itemize @asis
+@item
+T. E. Hull, Thomas F. Fairgrieve, Ping Tak Peter Tang,
+``Implementing Complex Elementary Functions Using Exception
+Handling'', @cite{ACM Transactions on Mathematical Software}, Volume 20
+(1994), pp 215--244, Corrigenda, p553
+
+@item
+T. E. Hull, Thomas F. Fairgrieve, Ping Tak Peter Tang,
+``Implementing the complex arcsin and arccosine functions using exception
+handling'', @cite{ACM Transactions on Mathematical Software}, Volume 23
+(1997) pp 299--335
+@end itemize
+
+@noindent
+The general formulas and details of branch cuts can be found in the
+following books,
+
+@itemize @asis
+@item
+Abramowitz and Stegun, @cite{Handbook of Mathematical Functions},
+``Circular Functions in Terms of Real and Imaginary Parts'', Formulas
+4.3.55--58,
+``Inverse Circular Functions in Terms of Real and Imaginary Parts'',
+Formulas 4.4.37--39,
+``Hyperbolic Functions in Terms of Real and Imaginary Parts'',
+Formulas 4.5.49--52,
+``Inverse Hyperbolic Functions---relation to Inverse Circular Functions'',
+Formulas 4.6.14--19.
+
+@item
+Dave Gillespie, @cite{Calc Manual}, Free Software Foundation, ISBN
+1-882114-18-3
+@end itemize
diff --git a/gsl-1.9/doc/const.texi b/gsl-1.9/doc/const.texi
new file mode 100644
index 0000000..aed2935
--- /dev/null
+++ b/gsl-1.9/doc/const.texi
@@ -0,0 +1,575 @@
+@cindex physical constants
+@cindex constants, physical
+@cindex conversion of units
+@cindex units, conversion of
+This chapter describes macros for the values of physical constants, such
+as the speed of light, @math{c}, and gravitational constant, @math{G}.
+The values are available in different unit systems, including the
+standard MKSA system (meters, kilograms, seconds, amperes) and the CGSM
+system (centimeters, grams, seconds, gauss), which is commonly used in
+Astronomy.
+
+The definitions of constants in the MKSA system are available in the file
+@file{gsl_const_mksa.h}.  The constants in the CGSM system are defined in
+@file{gsl_const_cgsm.h}.  Dimensionless constants, such as the fine
+structure constant, which are pure numbers are defined in
+@file{gsl_const_num.h}.
+
+@menu
+* Fundamental Constants::       
+* Astronomy and Astrophysics::  
+* Atomic and Nuclear Physics::  
+* Measurement of Time::         
+* Imperial Units ::             
+* Speed and Nautical Units::    
+* Printers Units::              
+* Volume Area and Length::      
+* Mass and Weight ::            
+* Thermal Energy and Power::    
+* Pressure::                    
+* Viscosity::                   
+* Light and Illumination::      
+* Radioactivity::               
+* Force and Energy::            
+* Prefixes::                    
+* Physical Constant Examples::  
+* Physical Constant References and Further Reading::  
+@end menu
+
+The full list of constants is described briefly below.  Consult the
+header files themselves for the values of the constants used in the
+library.
+
+@node Fundamental Constants
+@section Fundamental Constants
+@cindex fundamental constants
+@cindex constants, fundamental
+@table @code
+@item GSL_CONST_MKSA_SPEED_OF_LIGHT
+The speed of light in vacuum, @math{c}.
+
+@item GSL_CONST_MKSA_VACUUM_PERMEABILITY
+The permeability of free space, @math{\mu_0}. This constant is defined
+in the MKSA system only.
+
+@item GSL_CONST_MKSA_VACUUM_PERMITTIVITY
+The permittivity of free space, @math{\epsilon_0}.  This constant is
+defined in the MKSA system only.
+
+@item GSL_CONST_MKSA_PLANCKS_CONSTANT_H
+Planck's constant, @math{h}.
+
+@item GSL_CONST_MKSA_PLANCKS_CONSTANT_HBAR
+Planck's constant divided by @math{2\pi}, @math{\hbar}.
+
+@item GSL_CONST_NUM_AVOGADRO
+Avogadro's number, @math{N_a}.
+
+@item GSL_CONST_MKSA_FARADAY
+The molar charge of 1 Faraday.
+
+@item GSL_CONST_MKSA_BOLTZMANN
+The Boltzmann constant, @math{k}.
+
+@item GSL_CONST_MKSA_MOLAR_GAS
+The molar gas constant, @math{R_0}.
+
+@item GSL_CONST_MKSA_STANDARD_GAS_VOLUME
+The standard gas volume, @math{V_0}.
+
+@item GSL_CONST_MKSA_STEFAN_BOLTZMANN_CONSTANT
+The Stefan-Boltzmann radiation constant, @math{\sigma}.
+
+@item GSL_CONST_MKSA_GAUSS
+The magnetic field of 1 Gauss.
+@end table
+
+@node Astronomy and Astrophysics
+@section Astronomy and Astrophysics
+@cindex astronomical constants
+@table @code
+@item GSL_CONST_MKSA_ASTRONOMICAL_UNIT
+The length of 1 astronomical unit (mean earth-sun distance), @math{au}.
+
+@item GSL_CONST_MKSA_GRAVITATIONAL_CONSTANT
+The gravitational constant, @math{G}.
+
+@item GSL_CONST_MKSA_LIGHT_YEAR
+The distance of 1 light-year, @math{ly}.
+
+@item GSL_CONST_MKSA_PARSEC
+The distance of 1 parsec, @math{pc}.
+
+@item GSL_CONST_MKSA_GRAV_ACCEL
+The standard gravitational acceleration on Earth, @math{g}.
+
+@item GSL_CONST_MKSA_SOLAR_MASS
+The mass of the Sun.
+@end table
+
+@node Atomic and Nuclear Physics
+@section Atomic and Nuclear Physics
+@cindex atomic physics, constants
+@cindex nuclear physics, constants
+@table @code
+@item GSL_CONST_MKSA_ELECTRON_CHARGE
+The charge of the electron, @math{e}.
+
+@item GSL_CONST_MKSA_ELECTRON_VOLT
+The energy of 1 electron volt, @math{eV}.
+
+@item GSL_CONST_MKSA_UNIFIED_ATOMIC_MASS
+The unified atomic mass, @math{amu}.
+
+@item GSL_CONST_MKSA_MASS_ELECTRON
+The mass of the electron, @math{m_e}.
+
+@item GSL_CONST_MKSA_MASS_MUON
+The mass of the muon, @math{m_\mu}.
+
+@item GSL_CONST_MKSA_MASS_PROTON
+The mass of the proton, @math{m_p}.
+
+@item GSL_CONST_MKSA_MASS_NEUTRON
+The mass of the neutron, @math{m_n}.
+
+@item GSL_CONST_NUM_FINE_STRUCTURE
+The electromagnetic fine structure constant @math{\alpha}.
+
+@item GSL_CONST_MKSA_RYDBERG
+The Rydberg constant, @math{Ry}, in units of energy.  This is related to
+the Rydberg inverse wavelength @math{R} by @math{Ry = h c R}.
+
+@item GSL_CONST_MKSA_BOHR_RADIUS
+The Bohr radius, @math{a_0}.
+
+@item GSL_CONST_MKSA_ANGSTROM
+The length of 1 angstrom.
+
+@item GSL_CONST_MKSA_BARN
+The area of 1 barn.
+
+@item GSL_CONST_MKSA_BOHR_MAGNETON
+The Bohr Magneton, @math{\mu_B}.
+
+@item GSL_CONST_MKSA_NUCLEAR_MAGNETON
+The Nuclear Magneton, @math{\mu_N}.
+
+@item GSL_CONST_MKSA_ELECTRON_MAGNETIC_MOMENT
+The absolute value of the magnetic moment of the electron, @math{\mu_e}.
+The physical magnetic moment of the electron is negative.
+
+@item GSL_CONST_MKSA_PROTON_MAGNETIC_MOMENT
+The magnetic moment of the proton, @math{\mu_p}.
+
+@item GSL_CONST_MKSA_THOMSON_CROSS_SECTION
+The Thomson cross section, @math{\sigma_T}.
+
+@item GSL_CONST_MKSA_DEBYE
+The electric dipole moment of 1 Debye, @math{D}.
+@end table
+
+
+@node Measurement of Time
+@section Measurement of Time
+@cindex time units
+@table @code
+@item GSL_CONST_MKSA_MINUTE
+The number of seconds in 1 minute.
+
+@item GSL_CONST_MKSA_HOUR
+The number of seconds in 1 hour.
+
+@item GSL_CONST_MKSA_DAY
+The number of seconds in 1 day.
+
+@item GSL_CONST_MKSA_WEEK
+The number of seconds in 1 week.
+@end table
+
+
+@node Imperial Units 
+@section Imperial Units
+@cindex imperial units
+@cindex units, imperial
+@table @code
+@item GSL_CONST_MKSA_INCH
+The length of 1 inch.
+
+@item GSL_CONST_MKSA_FOOT
+The length of 1 foot.
+
+@item GSL_CONST_MKSA_YARD
+The length of 1 yard.
+
+@item GSL_CONST_MKSA_MILE
+The length of 1 mile.
+
+@item GSL_CONST_MKSA_MIL
+The length of 1 mil (1/1000th of an inch).
+@end table
+
+
+@node Speed and Nautical Units
+@section Speed and Nautical Units
+@cindex nautical units
+
+@table @code
+@item GSL_CONST_MKSA_KILOMETERS_PER_HOUR
+The speed of 1 kilometer per hour.
+
+@item GSL_CONST_MKSA_MILES_PER_HOUR
+The speed of 1 mile per hour.
+
+@item GSL_CONST_MKSA_NAUTICAL_MILE
+The length of 1 nautical mile.
+
+@item GSL_CONST_MKSA_FATHOM
+The length of 1 fathom.
+
+@item GSL_CONST_MKSA_KNOT
+The speed of 1 knot.
+@end table
+
+
+@node Printers Units
+@section Printers Units
+@cindex printers units
+
+@table @code
+@item GSL_CONST_MKSA_POINT
+The length of 1 printer's point (1/72 inch).
+
+@item GSL_CONST_MKSA_TEXPOINT
+The length of 1 TeX point (1/72.27 inch).
+@end table
+
+
+@node Volume Area and Length
+@section Volume, Area and Length
+@cindex volume units
+
+@table @code
+@item GSL_CONST_MKSA_MICRON
+The length of 1 micron.
+
+@item GSL_CONST_MKSA_HECTARE
+The area of 1 hectare.
+
+@item GSL_CONST_MKSA_ACRE
+The area of 1 acre.
+
+@item GSL_CONST_MKSA_LITER
+The volume of 1 liter.
+
+@item GSL_CONST_MKSA_US_GALLON
+The volume of 1 US gallon.
+
+@item GSL_CONST_MKSA_CANADIAN_GALLON
+The volume of 1 Canadian gallon.
+
+@item GSL_CONST_MKSA_UK_GALLON
+The volume of 1 UK gallon.
+
+@item GSL_CONST_MKSA_QUART
+The volume of 1 quart.
+
+@item GSL_CONST_MKSA_PINT
+The volume of 1 pint.
+@end table
+
+
+@comment @node Cookery
+@comment @section Cookery
+@comment @commentindex cookery units
+
+@comment @table @commentode
+@comment @item GSL_CONST_MKSA_CUP
+@comment The volume of 1 cup.
+
+@comment @item GSL_CONST_MKSA_FLUID_OUNCE
+@comment The volume of 1 fluid ounce.
+
+@comment @item GSL_CONST_MKSA_TABLESPOON
+@comment The volume of 1 tablespoon.
+
+@comment @item GSL_CONST_MKSA_TEASPOON
+@comment The volume of 1 teaspoon.
+@comment @end table
+
+
+@node Mass and Weight 
+@section Mass and Weight
+@cindex mass, units of
+@cindex weight, units of
+@table @code
+@item GSL_CONST_MKSA_POUND_MASS
+The mass of 1 pound.
+
+@item GSL_CONST_MKSA_OUNCE_MASS
+The mass of 1 ounce.
+
+@item GSL_CONST_MKSA_TON
+The mass of 1 ton.
+
+@item GSL_CONST_MKSA_METRIC_TON
+The mass of 1 metric ton (1000 kg).
+
+@item GSL_CONST_MKSA_UK_TON
+The mass of 1 UK ton.
+
+@item GSL_CONST_MKSA_TROY_OUNCE
+The mass of 1 troy ounce.
+
+@item GSL_CONST_MKSA_CARAT
+The mass of 1 carat.
+
+@item GSL_CONST_MKSA_GRAM_FORCE
+The force of 1 gram weight.
+
+@item GSL_CONST_MKSA_POUND_FORCE
+The force of 1 pound weight.
+
+@item GSL_CONST_MKSA_KILOPOUND_FORCE
+The force of 1 kilopound weight.
+
+@item GSL_CONST_MKSA_POUNDAL
+The force of 1 poundal.
+@end table
+
+
+@node Thermal Energy and Power
+@section Thermal Energy and Power
+@cindex energy, units of
+@cindex power, units of
+@cindex thermal energy, units of
+@table @code
+@item GSL_CONST_MKSA_CALORIE
+The energy of 1 calorie.
+
+@item GSL_CONST_MKSA_BTU
+The energy of 1 British Thermal Unit, @math{btu}.
+
+@item GSL_CONST_MKSA_THERM
+The energy of 1 Therm.
+
+@item GSL_CONST_MKSA_HORSEPOWER
+The power of 1 horsepower.
+@end table
+
+
+@node Pressure
+@section Pressure
+@cindex pressure, units of
+@table @code
+@item GSL_CONST_MKSA_BAR
+The pressure of 1 bar.
+
+@item GSL_CONST_MKSA_STD_ATMOSPHERE
+The pressure of 1 standard atmosphere.
+
+@item GSL_CONST_MKSA_TORR
+The pressure of 1 torr.
+
+@item GSL_CONST_MKSA_METER_OF_MERCURY
+The pressure of 1 meter of mercury.
+
+@item GSL_CONST_MKSA_INCH_OF_MERCURY
+The pressure of 1 inch of mercury.
+
+@item GSL_CONST_MKSA_INCH_OF_WATER
+The pressure of 1 inch of water.
+
+@item GSL_CONST_MKSA_PSI
+The pressure of 1 pound per square inch.
+@end table
+
+@node Viscosity
+@section Viscosity
+@cindex viscosity, units of
+@table @code
+@item GSL_CONST_MKSA_POISE
+The dynamic viscosity of 1 poise.
+
+@item GSL_CONST_MKSA_STOKES
+The kinematic viscosity of 1 stokes.
+@end table
+
+
+@node Light and Illumination
+@section Light and Illumination
+@cindex light, units of
+@cindex illumination, units of
+
+@table @code
+@item GSL_CONST_MKSA_STILB
+The luminance of 1 stilb.
+
+@item GSL_CONST_MKSA_LUMEN
+The luminous flux of 1 lumen.
+
+@item GSL_CONST_MKSA_LUX
+The illuminance of 1 lux.
+
+@item GSL_CONST_MKSA_PHOT
+The illuminance of 1 phot.
+
+@item GSL_CONST_MKSA_FOOTCANDLE
+The illuminance of 1 footcandle.
+
+@item GSL_CONST_MKSA_LAMBERT
+The luminance of 1 lambert.
+
+@item GSL_CONST_MKSA_FOOTLAMBERT
+The luminance of 1 footlambert.
+@end table
+
+
+@node Radioactivity
+@section Radioactivity
+@cindex radioactivity, units of
+@table @code
+@item GSL_CONST_MKSA_CURIE
+The activity of 1 curie.
+
+@item GSL_CONST_MKSA_ROENTGEN
+The exposure of 1 roentgen.
+
+@item GSL_CONST_MKSA_RAD
+The absorbed dose of 1 rad.
+@end table
+
+
+@node Force and Energy
+@section Force and Energy
+@cindex force and energy, units of
+@table @code
+@item GSL_CONST_MKSA_NEWTON
+The SI unit of force, 1 Newton.
+
+@item GSL_CONST_MKSA_DYNE
+The force of 1 Dyne = @c{$10^{-5}$}
+@math{10^-5} Newton.
+
+@item GSL_CONST_MKSA_JOULE
+The SI unit of energy, 1 Joule.
+
+@item GSL_CONST_MKSA_ERG 
+The energy 1 erg = @c{$10^{-7}$} 
+@math{10^-7} Joule.
+@end table
+
+
+@node Prefixes
+@section Prefixes
+@cindex prefixes
+@cindex constants, prefixes
+
+These constants are dimensionless scaling factors.
+
+@table @code
+@item GSL_CONST_NUM_YOTTA
+@c{$10^{24}$}
+@math{10^24}
+
+@item GSL_CONST_NUM_ZETTA
+@c{$10^{21}$}
+@math{10^21}
+
+@item GSL_CONST_NUM_EXA
+@c{$10^{18}$}
+@math{10^18}
+
+@item GSL_CONST_NUM_PETA
+@c{$10^{15}$}
+@math{10^15}
+
+@item GSL_CONST_NUM_TERA
+@c{$10^{12}$}
+@math{10^12}
+
+@item GSL_CONST_NUM_GIGA
+@math{10^9}
+
+@item GSL_CONST_NUM_MEGA
+@math{10^6}
+
+@item GSL_CONST_NUM_KILO
+@math{10^3}
+
+@item GSL_CONST_NUM_MILLI
+@c{$10^{-3}$}
+@math{10^-3}
+
+@item GSL_CONST_NUM_MICRO
+@c{$10^{-6}$}
+@math{10^-6}
+
+@item GSL_CONST_NUM_NANO
+@c{$10^{-9}$}
+@math{10^-9}
+
+@item GSL_CONST_NUM_PICO
+@c{$10^{-12}$}
+@math{10^-12}
+
+@item GSL_CONST_NUM_FEMTO
+@c{$10^{-15}$}
+@math{10^-15}
+
+@item GSL_CONST_NUM_ATTO
+@c{$10^{-18}$}
+@math{10^-18}
+
+@item GSL_CONST_NUM_ZEPTO
+@c{$10^{-21}$}
+@math{10^-21}
+
+@item GSL_CONST_NUM_YOCTO
+@c{$10^{-24}$}
+@math{10^-24}
+@end table
+
+@node Physical Constant Examples
+@section Examples
+
+The following program demonstrates the use of the physical constants in
+a calculation.  In this case, the goal is to calculate the range of
+light-travel times from Earth to Mars.
+
+The required data is the average distance of each planet from the Sun in
+astronomical units (the eccentricities and inclinations of the orbits
+will be neglected for the purposes of this calculation).  The average
+radius of the orbit of Mars is 1.52 astronomical units, and for the
+orbit of Earth it is 1 astronomical unit (by definition).  These values
+are combined with the MKSA values of the constants for the speed of
+light and the length of an astronomical unit to produce a result for the
+shortest and longest light-travel times in seconds.  The figures are
+converted into minutes before being displayed.
+
+@example
+@verbatiminclude examples/const.c
+@end example
+
+@noindent
+Here is the output from the program,
+
+@example
+@verbatiminclude examples/const.out
+@end example
+
+@node Physical Constant References and Further Reading
+@section References and Further Reading
+
+The authoritative sources for physical constants are the 2002 CODATA
+recommended values, published in the articles below. Further information
+on the values of physical constants is also available from the cited
+articles and the NIST website. 
+
+@itemize @asis
+@item Journal of Physical and Chemical Reference Data, 28(6), 1713-1852, 1999
+@item Reviews of Modern Physics, 72(2), 351-495, 2000
+@item @uref{http://www.physics.nist.gov/cuu/Constants/index.html}
+@item @uref{http://physics.nist.gov/Pubs/SP811/appenB9.html}
+@end itemize
+
diff --git a/gsl-1.9/doc/debug.texi b/gsl-1.9/doc/debug.texi
new file mode 100644
index 0000000..2cf0065
--- /dev/null
+++ b/gsl-1.9/doc/debug.texi
@@ -0,0 +1,389 @@
+This chapter describes some tips and tricks for debugging numerical
+programs which use GSL.
+
+@menu
+* Using gdb::                   
+* Examining floating point registers::  
+* Handling floating point exceptions::  
+* GCC warning options for numerical programs::  
+* Debugging References::        
+@end menu
+
+@node Using gdb
+@section Using gdb
+@cindex gdb
+@cindex debugging numerical programs
+@cindex breakpoints
+Any errors reported by the library are passed to the function
+@code{gsl_error}.  By running your programs under gdb and setting a
+breakpoint in this function you can automatically catch any library
+errors.  You can add a breakpoint for every session by putting
+
+@example
+break gsl_error
+@end example
+@comment 
+
+@noindent
+into your @file{.gdbinit} file in the directory where your program is
+started.  
+
+If the breakpoint catches an error then you can use a backtrace
+(@code{bt}) to see the call-tree, and the arguments which possibly
+caused the error.  By moving up into the calling function you can
+investigate the values of variables at that point.  Here is an example
+from the program @code{fft/test_trap}, which contains the following
+line,
+
+@smallexample
+status = gsl_fft_complex_wavetable_alloc (0, &complex_wavetable);
+@end smallexample
+
+@noindent
+The function @code{gsl_fft_complex_wavetable_alloc} takes the length of
+an FFT as its first argument.  When this line is executed an error will
+be generated because the length of an FFT is not allowed to be zero.
+
+To debug this problem we start @code{gdb}, using the file
+@file{.gdbinit} to define a breakpoint in @code{gsl_error},
+
+@smallexample
+$ gdb test_trap
+
+GDB is free software and you are welcome to distribute copies
+of it under certain conditions; type "show copying" to see
+the conditions.  There is absolutely no warranty for GDB;
+type "show warranty" for details.  GDB 4.16 (i586-debian-linux), 
+Copyright 1996 Free Software Foundation, Inc.
+
+Breakpoint 1 at 0x8050b1e: file error.c, line 14.
+@end smallexample
+
+@noindent
+When we run the program this breakpoint catches the error and shows the
+reason for it. 
+
+@smallexample
+(gdb) run
+Starting program: test_trap 
+
+Breakpoint 1, gsl_error (reason=0x8052b0d 
+    "length n must be positive integer", 
+    file=0x8052b04 "c_init.c", line=108, gsl_errno=1) 
+    at error.c:14
+14        if (gsl_error_handler) 
+@end smallexample
+@comment 
+
+@noindent
+The first argument of @code{gsl_error} is always a string describing the
+error.  Now we can look at the backtrace to see what caused the problem,
+
+@smallexample
+(gdb) bt
+#0  gsl_error (reason=0x8052b0d 
+    "length n must be positive integer", 
+    file=0x8052b04 "c_init.c", line=108, gsl_errno=1)
+    at error.c:14
+#1  0x8049376 in gsl_fft_complex_wavetable_alloc (n=0,
+    wavetable=0xbffff778) at c_init.c:108
+#2  0x8048a00 in main (argc=1, argv=0xbffff9bc) 
+    at test_trap.c:94
+#3  0x80488be in ___crt_dummy__ ()
+@end smallexample
+@comment 
+
+@noindent
+We can see that the error was generated in the function
+@code{gsl_fft_complex_wavetable_alloc} when it was called with an
+argument of @var{n=0}.  The original call came from line 94 in the
+file @file{test_trap.c}.
+
+By moving up to the level of the original call we can find the line that
+caused the error,
+
+@smallexample
+(gdb) up
+#1  0x8049376 in gsl_fft_complex_wavetable_alloc (n=0,
+    wavetable=0xbffff778) at c_init.c:108
+108   GSL_ERROR ("length n must be positive integer", GSL_EDOM);
+(gdb) up
+#2  0x8048a00 in main (argc=1, argv=0xbffff9bc) 
+    at test_trap.c:94
+94    status = gsl_fft_complex_wavetable_alloc (0,
+        &complex_wavetable);
+@end smallexample
+@comment 
+
+@noindent
+Thus we have found the line that caused the problem.  From this point we
+could also print out the values of other variables such as
+@code{complex_wavetable}.
+
+@node Examining floating point registers
+@section Examining floating point registers
+
+The contents of floating point registers can be examined using the
+command @code{info float} (on supported platforms).
+
+@smallexample
+(gdb) info float
+     st0: 0xc4018b895aa17a945000  Valid Normal -7.838871e+308
+     st1: 0x3ff9ea3f50e4d7275000  Valid Normal 0.0285946
+     st2: 0x3fe790c64ce27dad4800  Valid Normal 6.7415931e-08
+     st3: 0x3ffaa3ef0df6607d7800  Spec  Normal 0.0400229
+     st4: 0x3c028000000000000000  Valid Normal 4.4501477e-308
+     st5: 0x3ffef5412c22219d9000  Zero  Normal 0.9580257
+     st6: 0x3fff8000000000000000  Valid Normal 1
+     st7: 0xc4028b65a1f6d243c800  Valid Normal -1.566206e+309
+   fctrl: 0x0272 53 bit; NEAR; mask DENOR UNDER LOS;
+   fstat: 0xb9ba flags 0001; top 7; excep DENOR OVERF UNDER LOS
+    ftag: 0x3fff
+     fip: 0x08048b5c
+     fcs: 0x051a0023
+  fopoff: 0x08086820
+  fopsel: 0x002b
+@end smallexample
+
+@noindent
+Individual registers can be examined using the variables @var{$reg},
+where @var{reg} is the register name.
+
+@smallexample
+(gdb) p $st1 
+$1 = 0.02859464454261210347719
+@end smallexample
+
+@node Handling floating point exceptions
+@section Handling floating point exceptions
+
+It is possible to stop the program whenever a @code{SIGFPE} floating
+point exception occurs.  This can be useful for finding the cause of an
+unexpected infinity or @code{NaN}.  The current handler settings can be
+shown with the command @code{info signal SIGFPE}.
+
+@smallexample
+(gdb) info signal SIGFPE
+Signal  Stop  Print  Pass to program Description
+SIGFPE  Yes   Yes    Yes             Arithmetic exception
+@end smallexample
+
+@noindent
+Unless the program uses a signal handler the default setting should be
+changed so that SIGFPE is not passed to the program, as this would cause
+it to exit.  The command @code{handle SIGFPE stop nopass} prevents this.
+
+@smallexample
+(gdb) handle SIGFPE stop nopass
+Signal  Stop  Print  Pass to program Description
+SIGFPE  Yes   Yes    No              Arithmetic exception
+@end smallexample
+
+@noindent
+Depending on the platform it may be necessary to instruct the kernel to
+generate signals for floating point exceptions.  For programs using GSL
+this can be achieved using the @code{GSL_IEEE_MODE} environment variable
+in conjunction with the function @code{gsl_ieee_env_setup} as described
+in @pxref{IEEE floating-point arithmetic}.
+
+@example
+(gdb) set env GSL_IEEE_MODE=double-precision
+@end example
+
+
+@node GCC warning options for numerical programs
+@section GCC warning options for numerical programs
+@cindex warning options
+@cindex gcc warning options
+
+Writing reliable numerical programs in C requires great care.  The
+following GCC warning options are recommended when compiling numerical
+programs:
+
+@comment Uninitialized variables, conversions to and from integers or from
+@comment signed to unsigned integers can all cause hard-to-find problems.  For
+@comment many non-numerical programs compiling with @code{gcc}'s warning option
+@comment @code{-Wall} provides a good check against common errors.  However, for
+@comment numerical programs @code{-Wall} is not enough. 
+
+@comment If you are unconvinced take a look at this program which contains an
+@comment error that can occur in numerical code,
+
+@comment @example
+@comment #include <math.h>
+@comment #include <stdio.h>
+
+@comment double f (int x);
+
+@comment int main ()
+@comment @{
+@comment   double a = 1.5;
+@comment   double y = f(a);
+@comment   printf("a = %g, sqrt(a) = %g\n", a, y);  
+@comment   return 0;
+@comment @}
+
+@comment double f(x) @{
+@comment   return sqrt(x);
+@comment @}
+@comment @end example
+
+@comment @noindent
+@comment This code compiles cleanly with @code{-Wall} but produces some strange
+@comment output,
+
+@comment @example
+@comment bash$ gcc -Wall tmp.c -lm
+@comment bash$ ./a.out 
+@comment a = 1.5, sqrt(a) = 1
+@comment @end example
+
+@comment @noindent
+@comment Note that adding @code{-ansi} does not help here, since the program does
+@comment not contain any invalid constructs.  What is happening is that the
+@comment prototype for the function @code{f(int x)} is not consistent with the
+@comment function call @code{f(y)}, where @code{y} is a floating point
+@comment number.  This results in the argument being silently converted to an
+@comment integer.  This is valid C, but in a numerical program it also likely to
+@comment be a programming error so we would like to be warned about it. (If we
+@comment genuinely wanted to convert @code{y} to an integer then we could use an
+@comment explicit cast, @code{(int)y}).  
+
+@comment Fortunately GCC provides many additional warnings which can alert you to
+@comment problems such as this.  You just have to remember to use them.  Here is a
+@comment set of recommended warning options for numerical programs.
+
+@example
+gcc -ansi -pedantic -Werror -Wall -W 
+  -Wmissing-prototypes -Wstrict-prototypes 
+  -Wtraditional -Wconversion -Wshadow
+  -Wpointer-arith -Wcast-qual -Wcast-align 
+  -Wwrite-strings -Wnested-externs 
+  -fshort-enums -fno-common -Dinline= -g -O2
+@end example
+
+@noindent
+For details of each option consult the manual @cite{Using and Porting
+GCC}.  The following table gives a brief explanation of what types of
+errors these options catch.
+
+@table @code
+@item -ansi -pedantic 
+Use ANSI C, and reject any non-ANSI extensions.  These flags help in
+writing portable programs that will compile on other systems.
+@item -Werror 
+Consider warnings to be errors, so that compilation stops.  This prevents
+warnings from scrolling off the top of the screen and being lost.  You
+won't be able to compile the program until it is completely
+warning-free.
+@item -Wall 
+This turns on a set of warnings for common programming problems.  You
+need @code{-Wall}, but it is not enough on its own.
+@item -O2
+Turn on optimization.  The warnings for uninitialized variables in
+@code{-Wall} rely on the optimizer to analyze the code.  If there is no
+optimization then these warnings aren't generated.
+@item -W 
+This turns on some extra warnings not included in @code{-Wall}, such as
+missing return values and comparisons between signed and unsigned
+integers.
+@item -Wmissing-prototypes -Wstrict-prototypes 
+Warn if there are any missing or inconsistent prototypes.  Without
+prototypes it is harder to detect problems with incorrect arguments. 
+@item -Wtraditional 
+This warns about certain constructs that behave differently in
+traditional and ANSI C. Whether the traditional or ANSI interpretation
+is used might be unpredictable on other compilers. 
+@item -Wconversion 
+The main use of this option is to warn about conversions from signed to
+unsigned integers.  For example, @code{unsigned int x = -1}.  If you need
+to perform such a conversion you can use an explicit cast.
+@item -Wshadow
+This warns whenever a local variable shadows another local variable.  If
+two variables have the same name then it is a potential source of
+confusion.
+@item -Wpointer-arith -Wcast-qual -Wcast-align 
+These options warn if you try to do pointer arithmetic for types which
+don't have a size, such as @code{void}, if you remove a @code{const}
+cast from a pointer, or if you cast a pointer to a type which has a
+different size, causing an invalid alignment.
+@item -Wwrite-strings
+This option gives string constants a @code{const} qualifier so that it
+will be a compile-time error to attempt to overwrite them.
+@item -fshort-enums 
+This option makes the type of @code{enum} as short as possible.  Normally
+this makes an @code{enum} different from an @code{int}.  Consequently any
+attempts to assign a pointer-to-int to a pointer-to-enum will generate a
+cast-alignment warning.
+@item -fno-common
+This option prevents global variables being simultaneously defined in
+different object files (you get an error at link time).  Such a variable
+should be defined in one file and referred to in other files with an
+@code{extern} declaration.
+@item -Wnested-externs 
+This warns if an @code{extern} declaration is encountered within a
+function.
+@item -Dinline= 
+The @code{inline} keyword is not part of ANSI C. Thus if you want to use
+@code{-ansi} with a program which uses inline functions you can use this
+preprocessor definition to remove the @code{inline} keywords.
+@item -g 
+It always makes sense to put debugging symbols in the executable so that
+you can debug it using @code{gdb}.  The only effect of debugging symbols
+is to increase the size of the file, and you can use the @code{strip}
+command to remove them later if necessary.
+@end table
+
+@comment For comparison, this is what happens when the test program above is
+@comment compiled with these options.
+
+@comment @example
+@comment bash$ gcc -ansi -pedantic -Werror -W -Wall -Wtraditional 
+@comment -Wconversion -Wshadow -Wpointer-arith -Wcast-qual -Wcast-align 
+@comment -Wwrite-strings -Waggregate-return -Wstrict-prototypes -fshort-enums 
+@comment -fno-common -Wmissing-prototypes -Wnested-externs -Dinline= 
+@comment -g -O4 tmp.c 
+@comment cc1: warnings being treated as errors
+@comment tmp.c:7: warning: function declaration isn't a prototype
+@comment tmp.c: In function `main':
+@comment tmp.c:9: warning: passing arg 1 of `f' as integer rather than floating 
+@comment due to prototype
+@comment tmp.c: In function `f':
+@comment tmp.c:14: warning: type of `x' defaults to `int'
+@comment tmp.c:15: warning: passing arg 1 of `sqrt' as floating rather than integer 
+@comment due to prototype
+@comment make: *** [tmp] Error 1
+@comment @end example
+
+@comment @noindent
+@comment The error in the prototype is flagged, plus the fact that we should have
+@comment defined main as @code{int main (void)} in ANSI C. Clearly there is some
+@comment work to do before this program is ready to run.
+
+@node Debugging References
+@section References and Further Reading
+
+The following books are essential reading for anyone writing and
+debugging numerical programs with @sc{gcc} and @sc{gdb}.
+
+@itemize @asis
+@item
+R.M. Stallman, @cite{Using and Porting GNU CC}, Free Software
+Foundation, ISBN 1882114388
+
+@item
+R.M. Stallman, R.H. Pesch, @cite{Debugging with GDB: The GNU
+Source-Level Debugger}, Free Software Foundation, ISBN 1882114779
+@end itemize
+
+@noindent
+For a tutorial introduction to the GNU C Compiler and related programs,
+see 
+
+@itemize @asis
+@item
+B.J. Gough, @cite{An Introduction to GCC}, Network Theory Ltd, ISBN
+0954161793
+@end itemize
+
+
diff --git a/gsl-1.9/doc/dht.texi b/gsl-1.9/doc/dht.texi
new file mode 100644
index 0000000..7f9f25f
--- /dev/null
+++ b/gsl-1.9/doc/dht.texi
@@ -0,0 +1,152 @@
+@cindex discrete Hankel transforms
+@cindex Hankel transforms, discrete
+@cindex transforms, Hankel
+This chapter describes functions for performing Discrete Hankel
+Transforms (DHTs).  The functions are declared in the header file
+@file{gsl_dht.h}.
+
+@menu
+* Discrete Hankel Transform Definition::  
+* Discrete Hankel Transform Functions::  
+* Discrete Hankel Transform References::  
+@end menu
+
+@node Discrete Hankel Transform Definition
+@section Definitions
+
+The discrete Hankel transform acts on a vector of sampled data, where
+the samples are assumed to have been taken at points related to the
+zeroes of a Bessel function of fixed order; compare this to the case of
+the discrete Fourier transform, where samples are taken at points
+related to the zeroes of the sine or cosine function.
+
+Specifically, let @math{f(t)} be a function on the unit interval.
+Then the finite @math{\nu}-Hankel transform of @math{f(t)} is defined
+to be the set of numbers @math{g_m} given by,
+@tex
+\beforedisplay
+$$
+g_m = \int_0^1 t dt\, J_\nu(j_{\nu,m}t) f(t),
+$$
+\afterdisplay
+@end tex
+@ifinfo
+@example
+g_m = \int_0^1 t dt J_\nu(j_(\nu,m)t) f(t),
+@end example
+
+@end ifinfo
+@noindent
+so that,
+@tex
+\beforedisplay
+$$
+f(t) = \sum_{m=1}^\infty {{2 J_\nu(j_{\nu,m}x)}\over{J_{\nu+1}(j_{\nu,m})^2}} g_m.
+$$
+\afterdisplay
+@end tex
+@ifinfo
+@example
+f(t) = \sum_@{m=1@}^\infty (2 J_\nu(j_(\nu,m)x) / J_(\nu+1)(j_(\nu,m))^2) g_m.
+@end example
+
+@end ifinfo
+@noindent
+Suppose that @math{f} is band-limited in the sense that
+@math{g_m=0} for @math{m > M}. Then we have the following
+fundamental sampling theorem.
+@tex
+\beforedisplay
+$$
+g_m = {{2}\over{j_{\nu,M}^2}}
+      \sum_{k=1}^{M-1} f\left({{j_{\nu,k}}\over{j_{\nu,M}}}\right)
+          {{J_\nu(j_{\nu,m} j_{\nu,k} / j_{\nu,M})}\over{J_{\nu+1}(j_{\nu,k})^2}}.
+$$
+\afterdisplay
+@end tex
+@ifinfo
+@example
+g_m = (2 / j_(\nu,M)^2)
+      \sum_@{k=1@}^@{M-1@} f(j_(\nu,k)/j_(\nu,M))
+          (J_\nu(j_(\nu,m) j_(\nu,k) / j_(\nu,M)) / J_(\nu+1)(j_(\nu,k))^2).
+@end example
+
+@end ifinfo
+@noindent
+It is this discrete expression which defines the discrete Hankel
+transform. The kernel in the summation above defines the matrix of the
+@math{\nu}-Hankel transform of size @math{M-1}.  The coefficients of
+this matrix, being dependent on @math{\nu} and @math{M}, must be
+precomputed and stored; the @code{gsl_dht} object encapsulates this
+data.  The allocation function @code{gsl_dht_alloc} returns a
+@code{gsl_dht} object which must be properly initialized with
+@code{gsl_dht_init} before it can be used to perform transforms on data
+sample vectors, for fixed @math{\nu} and @math{M}, using the
+@code{gsl_dht_apply} function. The implementation allows a scaling of
+the fundamental interval, for convenience, so that one can assume the
+function is defined on the interval @math{[0,X]}, rather than the unit
+interval.
+
+Notice that by assumption @math{f(t)} vanishes at the endpoints
+of the interval, consistent with the inversion formula
+and the sampling formula given above. Therefore, this transform
+corresponds to an orthogonal expansion in eigenfunctions
+of the Dirichlet problem for the Bessel differential equation.
+
+
+@node Discrete Hankel Transform Functions
+@section Functions
+
+@deftypefun {gsl_dht *} gsl_dht_alloc (size_t @var{size})
+This function allocates a Discrete Hankel transform object of size
+@var{size}.
+@end deftypefun
+
+@deftypefun int gsl_dht_init (gsl_dht * @var{t}, double @var{nu}, double @var{xmax})
+This function initializes the transform @var{t} for the given values of
+@var{nu} and @var{x}.
+@end deftypefun
+
+@deftypefun {gsl_dht *} gsl_dht_new (size_t @var{size}, double @var{nu}, double @var{xmax})
+This function allocates a Discrete Hankel transform object of size
+@var{size} and initializes it for the given values of @var{nu} and
+@var{x}.
+@end deftypefun
+
+@deftypefun void gsl_dht_free (gsl_dht * @var{t})
+This function frees the transform @var{t}.
+@end deftypefun
+
+@deftypefun int gsl_dht_apply (const gsl_dht * @var{t}, double * @var{f_in}, double * @var{f_out})
+This function applies the transform @var{t} to the array @var{f_in}
+whose size is equal to the size of the transform.  The result is stored
+in the array @var{f_out} which must be of the same length.
+@end deftypefun
+
+@deftypefun double gsl_dht_x_sample (const gsl_dht * @var{t}, int @var{n})
+This function returns the value of the @var{n}-th sample point in the unit interval,
+@c{${({j_{\nu,n+1}} / {j_{\nu,M}}}) X$}
+@math{(j_@{\nu,n+1@}/j_@{\nu,M@}) X}. These are the
+points where the function @math{f(t)} is assumed to be sampled.
+@end deftypefun
+
+@deftypefun double gsl_dht_k_sample (const gsl_dht * @var{t}, int @var{n})
+This function returns the value of the @var{n}-th sample point in ``k-space'',
+@c{${{j_{\nu,n+1}} / X}$}
+@math{j_@{\nu,n+1@}/X}.
+@end deftypefun
+
+@node Discrete Hankel Transform References
+@section References and Further Reading
+
+The algorithms used by these functions are described in the following papers,
+
+@itemize @asis
+@item
+H. Fisk Johnson, Comp.@: Phys.@: Comm.@: 43, 181 (1987).
+@end itemize
+
+@itemize @asis
+@item
+D. Lemoine, J. Chem.@: Phys.@: 101, 3936 (1994).
+@end itemize
diff --git a/gsl-1.9/doc/diff.texi b/gsl-1.9/doc/diff.texi
new file mode 100644
index 0000000..ba11ebc
--- /dev/null
+++ b/gsl-1.9/doc/diff.texi
@@ -0,0 +1,105 @@
+@cindex differentiation of functions, numeric
+@cindex functions, numerical differentiation
+@cindex derivatives, calculating numerically
+@cindex numerical derivatives
+@cindex slope, see numerical derivative
+
+The functions described in this chapter compute numerical derivatives by
+finite differencing.  An adaptive algorithm is used to find the best
+choice of finite difference and to estimate the error in the derivative.
+These functions are declared in the header file @file{gsl_deriv.h}.
+
+@menu
+* Numerical Differentiation functions::  
+* Numerical Differentiation Examples::  
+* Numerical Differentiation References::  
+@end menu
+
+@node Numerical Differentiation functions
+@section Functions
+
+@deftypefun int gsl_deriv_central (const gsl_function * @var{f}, double @var{x}, double @var{h}, double * @var{result}, double * @var{abserr})
+This function computes the numerical derivative of the function @var{f}
+at the point @var{x} using an adaptive central difference algorithm with
+a step-size of @var{h}.   The derivative is returned in @var{result} and an
+estimate of its absolute error is returned in @var{abserr}.
+
+The initial value of @var{h} is used to estimate an optimal step-size,
+based on the scaling of the truncation error and round-off error in the
+derivative calculation.  The derivative is computed using a 5-point rule
+for equally spaced abscissae at @math{x-h}, @math{x-h/2}, @math{x},
+@math{x+h/2}, @math{x+h}, with an error estimate taken from the difference
+between the 5-point rule and the corresponding 3-point rule @math{x-h},
+@math{x}, @math{x+h}.  Note that the value of the function at @math{x}
+does not contribute to the derivative calculation, so only 4-points are
+actually used.
+@end deftypefun
+
+@deftypefun int gsl_deriv_forward (const gsl_function * @var{f}, double @var{x}, double @var{h}, double * @var{result}, double * @var{abserr})
+This function computes the numerical derivative of the function @var{f}
+at the point @var{x} using an adaptive forward difference algorithm with
+a step-size of @var{h}. The function is evaluated only at points greater
+than @var{x}, and never at @var{x} itself.  The derivative is returned in
+@var{result} and an estimate of its absolute error is returned in
+@var{abserr}.  This function should be used if @math{f(x)} has a
+discontinuity at @var{x}, or is undefined for values less than @var{x}.
+
+The initial value of @var{h} is used to estimate an optimal step-size,
+based on the scaling of the truncation error and round-off error in the
+derivative calculation.  The derivative at @math{x} is computed using an
+``open'' 4-point rule for equally spaced abscissae at @math{x+h/4},
+@math{x+h/2}, @math{x+3h/4}, @math{x+h}, with an error estimate taken
+from the difference between the 4-point rule and the corresponding
+2-point rule @math{x+h/2}, @math{x+h}. 
+@end deftypefun
+
+@deftypefun int gsl_deriv_backward (const gsl_function * @var{f}, double @var{x},  double @var{h}, double * @var{result}, double * @var{abserr})
+This function computes the numerical derivative of the function @var{f}
+at the point @var{x} using an adaptive backward difference algorithm
+with a step-size of @var{h}. The function is evaluated only at points
+less than @var{x}, and never at @var{x} itself.  The derivative is
+returned in @var{result} and an estimate of its absolute error is
+returned in @var{abserr}.  This function should be used if @math{f(x)}
+has a discontinuity at @var{x}, or is undefined for values greater than
+@var{x}.
+
+This function is equivalent to calling @code{gsl_deriv_forward} with a
+negative step-size.
+@end deftypefun
+
+@node Numerical Differentiation Examples
+@section Examples
+
+The following code estimates the derivative of the function 
+@c{$f(x) = x^{3/2}$}
+@math{f(x) = x^@{3/2@}} 
+at @math{x=2} and at @math{x=0}.  The function @math{f(x)} is
+undefined for @math{x<0} so the derivative at @math{x=0} is computed
+using @code{gsl_deriv_forward}.
+
+@example
+@verbatiminclude examples/diff.c
+@end example
+
+@noindent
+Here is the output of the program,
+
+@example
+$ ./a.out
+@verbatiminclude examples/diff.out
+@end example
+
+@node Numerical Differentiation References
+@section References and Further Reading
+
+The algorithms used by these functions are described in the following sources:
+
+@itemize @asis
+@item
+Abramowitz and Stegun, @cite{Handbook of Mathematical Functions},
+Section 25.3.4, and Table 25.5 (Coefficients for Differentiation).
+
+@item
+S.D. Conte and Carl de Boor, @cite{Elementary Numerical Analysis: An
+Algorithmic Approach}, McGraw-Hill, 1972.
+@end itemize
diff --git a/gsl-1.9/doc/dwt-orig.eps b/gsl-1.9/doc/dwt-orig.eps
new file mode 100644
index 0000000..2bf4de7
--- /dev/null
+++ b/gsl-1.9/doc/dwt-orig.eps
@@ -0,0 +1,2657 @@
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diff --git a/gsl-1.9/doc/dwt.texi b/gsl-1.9/doc/dwt.texi
new file mode 100644
index 0000000..fa80b19
--- /dev/null
+++ b/gsl-1.9/doc/dwt.texi
@@ -0,0 +1,449 @@
+@cindex DWT, see wavelet transforms
+@cindex wavelet transforms
+@cindex transforms, wavelet
+
+This chapter describes functions for performing Discrete Wavelet
+Transforms (DWTs).  The library includes wavelets for real data in both
+one and two dimensions.  The wavelet functions are declared in the header
+files @file{gsl_wavelet.h} and @file{gsl_wavelet2d.h}.
+
+@menu
+* DWT Definitions::             
+* DWT Initialization::          
+* DWT Transform Functions::     
+* DWT Examples::                
+* DWT References::              
+@end menu
+
+@node DWT Definitions
+@section Definitions
+@cindex DWT, mathematical definition
+
+The continuous wavelet transform and its inverse are defined by
+the relations,
+@iftex
+@tex
+\beforedisplay
+$$
+w(s, \tau) = \int_{-\infty}^\infty f(t) * \psi^*_{s,\tau}(t) dt
+$$
+\afterdisplay
+@end tex
+@end iftex
+@ifinfo
+
+@example
+w(s,\tau) = \int f(t) * \psi^*_@{s,\tau@}(t) dt
+@end example
+
+@end ifinfo
+@noindent
+and,
+@tex
+\beforedisplay
+$$
+f(t) = \int_0^\infty ds \int_{-\infty}^\infty w(s, \tau) * \psi_{s,\tau}(t) d\tau
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+f(t) = \int \int_@{-\infty@}^\infty w(s, \tau) * \psi_@{s,\tau@}(t) d\tau ds
+@end example
+
+@end ifinfo
+@noindent
+where the basis functions @c{$\psi_{s,\tau}$}
+@math{\psi_@{s,\tau@}} are obtained by scaling
+and translation from a single function, referred to as the @dfn{mother
+wavelet}.
+
+The discrete version of the wavelet transform acts on equally-spaced
+samples, with fixed scaling and translation steps (@math{s},
+@math{\tau}).  The frequency and time axes are sampled @dfn{dyadically}
+on scales of @math{2^j} through a level parameter @math{j}.
+@c  The wavelet @math{\psi}
+@c  can be expressed in terms of a scaling function @math{\varphi},
+@c
+@c  @tex
+@c  \beforedisplay
+@c  $$
+@c  \psi(2^{j-1},t) = \sum_{k=0}^{2^j-1} g_j(k) * \bar{\varphi}(2^j t-k)
+@c  $$
+@c  \afterdisplay
+@c  @end tex
+@c  @ifinfo
+@c  @example
+@c  \psi(2^@{j-1@},t) = \sum_@{k=0@}^@{2^j-1@} g_j(k) * \bar@{\varphi@}(2^j t-k)
+@c  @end example
+@c  @end ifinfo
+@c  @noindent
+@c  and
+@c
+@c  @tex
+@c  \beforedisplay
+@c  $$
+@c  \varphi(2^{j-1},t) = \sum_{k=0}^{2^j-1} h_j(k) * \bar{\varphi}(2^j t-k)
+@c  $$
+@c  \afterdisplay
+@c  @end tex
+@c  @ifinfo
+@c  @example
+@c  \varphi(2^@{j-1@},t) = \sum_@{k=0@}^@{2^j-1@} h_j(k) * \bar@{\varphi@}(2^j t-k)
+@c  @end example
+@c  @end ifinfo
+@c  @noindent
+@c  The functions @math{\psi} and @math{\varphi} are related through the
+@c  coefficients
+@c  @c{$g_{n} = (-1)^n h_{L-1-n}$}
+@c  @math{g_@{n@} = (-1)^n h_@{L-1-n@}}
+@c  for @c{$n=0 \dots L-1$}
+@c  @math{n=0 ... L-1},
+@c  where @math{L} is the total number of coefficients.  The two sets of
+@c  coefficients @math{h_j} and @math{g_i} define the scaling function and
+@c the wavelet.  
+The resulting family of functions @c{$\{\psi_{j,n}\}$}
+@math{@{\psi_@{j,n@}@}}
+constitutes an orthonormal
+basis for square-integrable signals.  
+
+The discrete wavelet transform is an @math{O(N)} algorithm, and is also
+referred to as the @dfn{fast wavelet transform}.
+
+@node DWT Initialization
+@section Initialization
+@cindex DWT initialization
+
+The @code{gsl_wavelet} structure contains the filter coefficients
+defining the wavelet and any associated offset parameters.
+
+@deftypefun {gsl_wavelet *} gsl_wavelet_alloc (const gsl_wavelet_type * @var{T}, size_t @var{k})
+This function allocates and initializes a wavelet object of type
+@var{T}.  The parameter @var{k} selects the specific member of the
+wavelet family.  A null pointer is returned if insufficient memory is
+available or if a unsupported member is selected.
+@end deftypefun
+
+The following wavelet types are implemented:
+
+@deffn {Wavelet} gsl_wavelet_daubechies
+@deffnx {Wavelet} gsl_wavelet_daubechies_centered
+@cindex Daubechies wavelets
+@cindex maximal phase, Daubechies wavelets
+The is the Daubechies wavelet family of maximum phase with @math{k/2}
+vanishing moments.  The implemented wavelets are 
+@c{$k=4, 6, \dots, 20$}
+@math{k=4, 6, @dots{}, 20}, with @var{k} even.
+@end deffn
+
+@deffn {Wavelet} gsl_wavelet_haar
+@deffnx {Wavelet} gsl_wavelet_haar_centered
+@cindex Haar wavelets
+This is the Haar wavelet.  The only valid choice of @math{k} for the
+Haar wavelet is @math{k=2}.
+@end deffn
+
+@deffn {Wavelet} gsl_wavelet_bspline
+@deffnx {Wavelet} gsl_wavelet_bspline_centered
+@cindex biorthogonal wavelets
+@cindex B-spline wavelets
+This is the biorthogonal B-spline wavelet family of order @math{(i,j)}.  
+The implemented values of @math{k = 100*i + j} are 103, 105, 202, 204,
+206, 208, 301, 303, 305 307, 309.
+@end deffn
+
+@noindent
+The centered forms of the wavelets align the coefficients of the various
+sub-bands on edges.  Thus the resulting visualization of the
+coefficients of the wavelet transform in the phase plane is easier to
+understand.
+
+@deftypefun {const char *} gsl_wavelet_name (const gsl_wavelet * @var{w})
+This function returns a pointer to the name of the wavelet family for
+@var{w}.
+@end deftypefun
+
+@c  @deftypefun {void} gsl_wavelet_print (const gsl_wavelet * @var{w})
+@c  This function prints the filter coefficients (@code{**h1}, @code{**g1}, @code{**h2}, @code{**g2}) of the wavelet object @var{w}.
+@c  @end deftypefun
+
+@deftypefun {void} gsl_wavelet_free (gsl_wavelet * @var{w})
+This function frees the wavelet object @var{w}.
+@end deftypefun
+
+The @code{gsl_wavelet_workspace} structure contains scratch space of the
+same size as the input data and is used to hold intermediate results
+during the transform.
+
+@deftypefun {gsl_wavelet_workspace *} gsl_wavelet_workspace_alloc (size_t @var{n})
+This function allocates a workspace for the discrete wavelet transform.
+To perform a one-dimensional transform on @var{n} elements, a workspace
+of size @var{n} must be provided.  For two-dimensional transforms of
+@var{n}-by-@var{n} matrices it is sufficient to allocate a workspace of
+size @var{n}, since the transform operates on individual rows and
+columns.
+@end deftypefun
+
+@deftypefun {void} gsl_wavelet_workspace_free (gsl_wavelet_workspace * @var{work})
+This function frees the allocated workspace @var{work}.
+@end deftypefun
+
+@node DWT Transform Functions
+@section Transform Functions
+
+This sections describes the actual functions performing the discrete
+wavelet transform.  Note that the transforms use periodic boundary
+conditions.  If the signal is not periodic in the sample length then
+spurious coefficients will appear at the beginning and end of each level
+of the transform.
+
+@menu
+* DWT in one dimension::        
+* DWT in two dimension::        
+@end menu
+
+@node DWT in one dimension
+@subsection Wavelet transforms in one dimension
+@cindex DWT, one dimensional
+
+@deftypefun int gsl_wavelet_transform (const gsl_wavelet * @var{w}, double * @var{data}, size_t @var{stride}, size_t @var{n}, gsl_wavelet_direction @var{dir}, gsl_wavelet_workspace * @var{work})
+@deftypefunx int gsl_wavelet_transform_forward (const gsl_wavelet * @var{w}, double * @var{data}, size_t @var{stride}, size_t @var{n}, gsl_wavelet_workspace * @var{work})
+@deftypefunx int gsl_wavelet_transform_inverse (const gsl_wavelet * @var{w}, double * @var{data}, size_t @var{stride}, size_t @var{n}, gsl_wavelet_workspace * @var{work})
+
+These functions compute in-place forward and inverse discrete wavelet
+transforms of length @var{n} with stride @var{stride} on the array
+@var{data}. The length of the transform @var{n} is restricted to powers
+of two.  For the @code{transform} version of the function the argument
+@var{dir} can be either @code{forward} (@math{+1}) or @code{backward}
+(@math{-1}).  A workspace @var{work} of length @var{n} must be provided.
+
+For the forward transform, the elements of the original array are 
+replaced by the discrete wavelet
+transform @c{$f_i \rightarrow w_{j,k}$}
+@math{f_i -> w_@{j,k@}} 
+in a packed triangular storage layout, 
+where @var{j} is the index of the level 
+@c{$j = 0 \dots J-1$}
+@math{j = 0 ... J-1}
+and
+@var{k} is the index of the coefficient within each level,
+@c{$k = 0 \dots 2^j - 1$}
+@math{k = 0 ... (2^j)-1}.  
+The total number of levels is @math{J = \log_2(n)}.  The output data
+has the following form,
+@tex
+\beforedisplay
+$$
+(s_{-1,0}, d_{0,0}, d_{1,0}, d_{1,1}, d_{2,0},\cdots, d_{j,k},\cdots, d_{J-1,2^{J-1} - 1}) 
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+(s_@{-1,0@}, d_@{0,0@}, d_@{1,0@}, d_@{1,1@}, d_@{2,0@}, ..., 
+  d_@{j,k@}, ..., d_@{J-1,2^@{J-1@}-1@}) 
+@end example
+
+@end ifinfo
+@noindent
+where the first element is the smoothing coefficient @c{$s_{-1,0}$}
+@math{s_@{-1,0@}}, followed by the detail coefficients @c{$d_{j,k}$}
+@math{d_@{j,k@}} for each level
+@math{j}.  The backward transform inverts these coefficients to obtain 
+the original data.
+
+These functions return a status of @code{GSL_SUCCESS} upon successful
+completion.  @code{GSL_EINVAL} is returned if @var{n} is not an integer
+power of 2 or if insufficient workspace is provided.
+@end deftypefun
+
+@node DWT in two dimension
+@subsection Wavelet transforms in two dimension
+@cindex DWT, two dimensional
+
+The library provides functions to perform two-dimensional discrete
+wavelet transforms on square matrices.  The matrix dimensions must be an
+integer power of two.  There are two possible orderings of the rows and
+columns in the two-dimensional wavelet transform, referred to as the
+``standard'' and ``non-standard'' forms.
+
+The ``standard'' transform performs a complete discrete wavelet
+transform on the rows of the matrix, followed by a separate complete
+discrete wavelet transform on the columns of the resulting
+row-transformed matrix.  This procedure uses the same ordering as a
+two-dimensional fourier transform.
+
+The ``non-standard'' transform is performed in interleaved passes on the
+rows and columns of the matrix for each level of the transform.  The
+first level of the transform is applied to the matrix rows, and then to
+the matrix columns.  This procedure is then repeated across the rows and
+columns of the data for the subsequent levels of the transform, until
+the full discrete wavelet transform is complete.  The non-standard form
+of the discrete wavelet transform is typically used in image analysis.
+
+The functions described in this section are declared in the header file
+@file{gsl_wavelet2d.h}.
+
+@deftypefun {int} gsl_wavelet2d_transform (const gsl_wavelet * @var{w}, double * @var{data}, size_t @var{tda}, size_t @var{size1}, size_t @var{size2}, gsl_wavelet_direction @var{dir}, gsl_wavelet_workspace * @var{work})
+@deftypefunx {int} gsl_wavelet2d_transform_forward (const gsl_wavelet * @var{w}, double * @var{data}, size_t @var{tda}, size_t @var{size1}, size_t @var{size2}, gsl_wavelet_workspace * @var{work})
+@deftypefunx {int} gsl_wavelet2d_transform_inverse (const gsl_wavelet * @var{w}, double * @var{data}, size_t @var{tda}, size_t @var{size1}, size_t @var{size2}, gsl_wavelet_workspace * @var{work})
+
+These functions compute two-dimensional in-place forward and inverse
+discrete wavelet transforms in standard and non-standard forms on the
+array @var{data} stored in row-major form with dimensions @var{size1}
+and @var{size2} and physical row length @var{tda}.  The dimensions must
+be equal (square matrix) and are restricted to powers of two.  For the
+@code{transform} version of the function the argument @var{dir} can be
+either @code{forward} (@math{+1}) or @code{backward} (@math{-1}).  A
+workspace @var{work} of the appropriate size must be provided.  On exit,
+the appropriate elements of the array @var{data} are replaced by their
+two-dimensional wavelet transform.
+
+The functions return a status of @code{GSL_SUCCESS} upon successful
+completion.  @code{GSL_EINVAL} is returned if @var{size1} and
+@var{size2} are not equal and integer powers of 2, or if insufficient
+workspace is provided.
+@end deftypefun
+
+@deftypefun {int} gsl_wavelet2d_transform_matrix (const gsl_wavelet * @var{w}, gsl_matrix * @var{m}, gsl_wavelet_direction @var{dir}, gsl_wavelet_workspace * @var{work})
+@deftypefunx {int} gsl_wavelet2d_transform_matrix_forward (const gsl_wavelet * @var{w}, gsl_matrix * @var{m}, gsl_wavelet_workspace * @var{work})
+@deftypefunx {int} gsl_wavelet2d_transform_matrix_inverse (const gsl_wavelet * @var{w}, gsl_matrix * @var{m}, gsl_wavelet_workspace * @var{work})
+These functions compute the two-dimensional in-place wavelet transform
+on a matrix @var{a}.
+@end deftypefun
+
+@deftypefun {int} gsl_wavelet2d_nstransform (const gsl_wavelet * @var{w}, double * @var{data}, size_t @var{tda}, size_t @var{size1}, size_t @var{size2}, gsl_wavelet_direction @var{dir}, gsl_wavelet_workspace * @var{work})
+@deftypefunx {int} gsl_wavelet2d_nstransform_forward (const gsl_wavelet * @var{w}, double * @var{data}, size_t @var{tda}, size_t @var{size1}, size_t @var{size2}, gsl_wavelet_workspace * @var{work})
+@deftypefunx {int} gsl_wavelet2d_nstransform_inverse (const gsl_wavelet * @var{w}, double * @var{data}, size_t @var{tda}, size_t @var{size1}, size_t @var{size2}, gsl_wavelet_workspace * @var{work})
+These functions compute the two-dimensional wavelet transform in
+non-standard form.
+@end deftypefun
+
+@deftypefun {int} gsl_wavelet2d_nstransform_matrix (const gsl_wavelet * @var{w}, gsl_matrix * @var{m}, gsl_wavelet_direction @var{dir}, gsl_wavelet_workspace * @var{work})
+@deftypefunx {int} gsl_wavelet2d_nstransform_matrix_forward (const gsl_wavelet * @var{w}, gsl_matrix * @var{m}, gsl_wavelet_workspace * @var{work})
+@deftypefunx {int} gsl_wavelet2d_nstransform_matrix_inverse (const gsl_wavelet * @var{w}, gsl_matrix * @var{m}, gsl_wavelet_workspace * @var{work})
+These functions compute the non-standard form of the two-dimensional
+in-place wavelet transform on a matrix @var{a}.
+@end deftypefun
+
+@node DWT Examples
+@section Examples
+
+The following program demonstrates the use of the one-dimensional
+wavelet transform functions.  It computes an approximation to an input
+signal (of length 256) using the 20 largest components of the wavelet
+transform, while setting the others to zero.
+
+@example
+@verbatiminclude examples/dwt.c
+@end example
+
+@noindent
+The output can be used with the @sc{gnu} plotutils @code{graph} program,
+
+@example
+$ ./a.out ecg.dat > dwt.dat
+$ graph -T ps -x 0 256 32 -h 0.3 -a dwt.dat > dwt.ps
+@end example
+
+@iftex
+The graphs below show an original and compressed version of a sample ECG
+recording from the MIT-BIH Arrhythmia Database, part of the PhysioNet
+archive of public-domain of medical datasets.
+
+@sp 1
+@center @image{dwt-orig,3.4in} 
+@center @image{dwt-samp,3.4in} 
+@quotation
+Original (upper) and wavelet-compressed (lower) ECG signals, using the
+20 largest components of the Daubechies(4) discrete wavelet transform.
+@end quotation
+@end iftex
+
+@node DWT References
+@section References and Further Reading
+
+The mathematical background to wavelet transforms is covered in the
+original lectures by Daubechies,
+
+@itemize @asis
+@item
+Ingrid Daubechies.
+Ten Lectures on Wavelets.
+@cite{CBMS-NSF Regional Conference Series in Applied Mathematics} (1992), 
+SIAM, ISBN 0898712742.
+@end itemize
+
+@noindent
+An easy to read introduction to the subject with an emphasis on the
+application of the wavelet transform in various branches of science is,
+
+@itemize @asis
+@item
+Paul S. Addison. @cite{The Illustrated Wavelet Transform Handbook}.
+Institute of Physics Publishing (2002), ISBN 0750306920.
+@end itemize
+
+@noindent
+For extensive coverage of signal analysis by wavelets, wavelet packets
+and local cosine bases see,
+
+@itemize @asis
+@item
+S. G. Mallat.  @cite{A wavelet tour of signal processing} (Second
+edition). Academic Press (1999), ISBN 012466606X.
+@end itemize
+
+@noindent
+The concept of multiresolution analysis underlying the wavelet transform
+is described in,
+
+@itemize @asis
+@item
+S. G. Mallat.
+Multiresolution Approximations and Wavelet Orthonormal Bases of L@math{^2}(R).
+@cite{Transactions of the American Mathematical Society}, 315(1), 1989, 69--87.
+@end itemize
+
+@itemize @asis
+@item
+S. G. Mallat.
+A Theory for Multiresolution Signal Decomposition---The Wavelet Representation.
+@cite{IEEE Transactions on Pattern Analysis and Machine Intelligence}, 11, 1989,
+674--693. 
+@end itemize
+
+@noindent
+The coefficients for the individual wavelet families implemented by the
+library can be found in the following papers,
+
+@itemize @asis
+@item
+I. Daubechies.
+Orthonormal Bases of Compactly Supported Wavelets.
+@cite{Communications on Pure and Applied Mathematics}, 41 (1988) 909--996.
+@end itemize
+
+@itemize @asis
+@item
+A. Cohen, I. Daubechies, and J.-C. Feauveau.
+Biorthogonal Bases of Compactly Supported Wavelets.
+@cite{Communications on Pure and Applied Mathematics}, 45 (1992)
+485--560.
+@end itemize
+
+@noindent
+The PhysioNet archive of physiological datasets can be found online at
+@uref{http://www.physionet.org/} and is described in the following
+paper,
+
+@itemize @asis
+@item
+Goldberger et al.  
+PhysioBank, PhysioToolkit, and PhysioNet: Components
+of a New Research Resource for Complex Physiologic
+Signals. 
+@cite{Circulation} 101(23):e215-e220 2000.
+@end itemize
diff --git a/gsl-1.9/doc/eigen.texi b/gsl-1.9/doc/eigen.texi
new file mode 100644
index 0000000..e2c291a
--- /dev/null
+++ b/gsl-1.9/doc/eigen.texi
@@ -0,0 +1,462 @@
+@cindex eigenvalues and eigenvectors
+This chapter describes functions for computing eigenvalues and
+eigenvectors of matrices.  There are routines for real symmetric,
+real nonsymmetric, and complex hermitian matrices. Eigenvalues can
+be computed with or without eigenvectors. The hermitian matrix algorithms
+used are symmetric bidiagonalization followed by QR reduction. The
+nonsymmetric algorithm is the Francis QR double-shift.
+
+@cindex LAPACK, recommended for linear algebra
+These routines are intended for ``small'' systems where simple algorithms are
+acceptable.  Anyone interested in finding eigenvalues and eigenvectors of
+large matrices will want to use the sophisticated routines found in
+@sc{lapack}. The Fortran version of @sc{lapack} is recommended as the
+standard package for large-scale linear algebra.
+
+The functions described in this chapter are declared in the header file
+@file{gsl_eigen.h}.
+
+@menu
+* Real Symmetric Matrices::     
+* Complex Hermitian Matrices::  
+* Real Nonsymmetric Matrices::     
+* Sorting Eigenvalues and Eigenvectors::  
+* Eigenvalue and Eigenvector Examples::  
+* Eigenvalue and Eigenvector References::  
+@end menu
+
+@node Real Symmetric Matrices
+@section Real Symmetric Matrices
+@cindex symmetric matrix, real, eigensystem
+@cindex real symmetric matrix, eigensystem
+
+@deftypefun {gsl_eigen_symm_workspace *} gsl_eigen_symm_alloc (const size_t @var{n})
+This function allocates a workspace for computing eigenvalues of
+@var{n}-by-@var{n} real symmetric matrices.  The size of the workspace
+is @math{O(2n)}.
+@end deftypefun
+
+@deftypefun void gsl_eigen_symm_free (gsl_eigen_symm_workspace * @var{w})
+This function frees the memory associated with the workspace @var{w}.
+@end deftypefun
+
+@deftypefun int gsl_eigen_symm (gsl_matrix * @var{A}, gsl_vector * @var{eval}, gsl_eigen_symm_workspace * @var{w})
+This function computes the eigenvalues of the real symmetric matrix
+@var{A}.  Additional workspace of the appropriate size must be provided
+in @var{w}.  The diagonal and lower triangular part of @var{A} are
+destroyed during the computation, but the strict upper triangular part
+is not referenced.  The eigenvalues are stored in the vector @var{eval}
+and are unordered.
+@end deftypefun
+
+@deftypefun {gsl_eigen_symmv_workspace *} gsl_eigen_symmv_alloc (const size_t @var{n})
+This function allocates a workspace for computing eigenvalues and
+eigenvectors of @var{n}-by-@var{n} real symmetric matrices.  The size of
+the workspace is @math{O(4n)}.
+@end deftypefun
+
+@deftypefun void gsl_eigen_symmv_free (gsl_eigen_symmv_workspace * @var{w})
+This function frees the memory associated with the workspace @var{w}.
+@end deftypefun
+
+@deftypefun int gsl_eigen_symmv (gsl_matrix * @var{A}, gsl_vector * @var{eval}, gsl_matrix * @var{evec}, gsl_eigen_symmv_workspace * @var{w})
+This function computes the eigenvalues and eigenvectors of the real
+symmetric matrix @var{A}.  Additional workspace of the appropriate size
+must be provided in @var{w}.  The diagonal and lower triangular part of
+@var{A} are destroyed during the computation, but the strict upper
+triangular part is not referenced.  The eigenvalues are stored in the
+vector @var{eval} and are unordered.  The corresponding eigenvectors are
+stored in the columns of the matrix @var{evec}.  For example, the
+eigenvector in the first column corresponds to the first eigenvalue.
+The eigenvectors are guaranteed to be mutually orthogonal and normalised
+to unit magnitude.
+@end deftypefun
+
+@node Complex Hermitian Matrices
+@section Complex Hermitian Matrices
+
+@cindex hermitian matrix, complex, eigensystem
+@cindex complex hermitian matrix, eigensystem
+
+@deftypefun {gsl_eigen_herm_workspace *} gsl_eigen_herm_alloc (const size_t @var{n})
+This function allocates a workspace for computing eigenvalues of
+@var{n}-by-@var{n} complex hermitian matrices.  The size of the workspace
+is @math{O(3n)}.
+@end deftypefun
+
+@deftypefun void gsl_eigen_herm_free (gsl_eigen_herm_workspace * @var{w})
+This function frees the memory associated with the workspace @var{w}.
+@end deftypefun
+
+@deftypefun int gsl_eigen_herm (gsl_matrix_complex * @var{A}, gsl_vector * @var{eval}, gsl_eigen_herm_workspace * @var{w})
+This function computes the eigenvalues of the complex hermitian matrix
+@var{A}.  Additional workspace of the appropriate size must be provided
+in @var{w}.  The diagonal and lower triangular part of @var{A} are
+destroyed during the computation, but the strict upper triangular part
+is not referenced.  The imaginary parts of the diagonal are assumed to be
+zero and are not referenced. The eigenvalues are stored in the vector
+@var{eval} and are unordered.
+@end deftypefun
+
+@deftypefun {gsl_eigen_hermv_workspace *} gsl_eigen_hermv_alloc (const size_t @var{n})
+This function allocates a workspace for computing eigenvalues and
+eigenvectors of @var{n}-by-@var{n} complex hermitian matrices.  The size of
+the workspace is @math{O(5n)}.
+@end deftypefun
+
+@deftypefun void gsl_eigen_hermv_free (gsl_eigen_hermv_workspace * @var{w})
+This function frees the memory associated with the workspace @var{w}.
+@end deftypefun
+
+@deftypefun int gsl_eigen_hermv (gsl_matrix_complex * @var{A}, gsl_vector * @var{eval}, gsl_matrix_complex * @var{evec}, gsl_eigen_hermv_workspace * @var{w})
+This function computes the eigenvalues and eigenvectors of the complex
+hermitian matrix @var{A}.  Additional workspace of the appropriate size
+must be provided in @var{w}.  The diagonal and lower triangular part of
+@var{A} are destroyed during the computation, but the strict upper
+triangular part is not referenced. The imaginary parts of the diagonal
+are assumed to be zero and are not referenced.  The eigenvalues are
+stored in the vector @var{eval} and are unordered.  The corresponding
+complex eigenvectors are stored in the columns of the matrix @var{evec}.
+For example, the eigenvector in the first column corresponds to the
+first eigenvalue.  The eigenvectors are guaranteed to be mutually
+orthogonal and normalised to unit magnitude.
+@end deftypefun
+
+@node Real Nonsymmetric Matrices
+@section Real Nonsymmetric Matrices
+@cindex nonsymmetric matrix, real, eigensystem
+@cindex real nonsymmetric matrix, eigensystem
+
+The solution of the real nonsymmetric eigensystem problem for a
+matrix @math{A} involves computing the Schur decomposition
+@tex
+\beforedisplay
+$$
+A = Z T Z^T
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+A = Z T Z^T
+@end example
+
+@end ifinfo
+where @math{Z} is an orthogonal matrix of Schur vectors and @math{T},
+the Schur form, is quasi upper triangular with diagonal
+@math{1}-by-@math{1} blocks which are real eigenvalues of @math{A}, and
+diagonal @math{2}-by-@math{2} blocks whose eigenvalues are complex
+conjugate eigenvalues of @math{A}. The algorithm used is the double
+shift Francis method.
+
+@deftypefun {gsl_eigen_nonsymm_workspace *} gsl_eigen_nonsymm_alloc (const size_t @var{n})
+This function allocates a workspace for computing eigenvalues of
+@var{n}-by-@var{n} real nonsymmetric matrices. The size of the workspace
+is @math{O(2n)}.
+@end deftypefun
+
+@deftypefun void gsl_eigen_nonsymm_free (gsl_eigen_nonsymm_workspace * @var{w})
+This function frees the memory associated with the workspace @var{w}.
+@end deftypefun
+
+@deftypefun void gsl_eigen_nonsymm_params (const int @var{compute_t}, const int @var{balance}, gsl_eigen_nonsymm_workspace * @var{w})
+This function sets some parameters which determine how the eigenvalue
+problem is solved in subsequent calls to @code{gsl_eigen_nonsymm}.
+
+If @var{compute_t} is set to 1, the full Schur form @math{T} will be
+computed by @code{gsl_eigen_nonsymm}. If it is set to 0,
+@math{T} will not be computed (this is the default setting). Computing
+the full Schur form @math{T} requires approximately 1.5-2 times the
+number of flops.
+
+If @var{balance} is set to 1, a balancing transformation is applied
+to the matrix prior to computing eigenvalues. This transformation is
+designed to make the rows and columns of the matrix have comparable
+norms, and can result in more accurate eigenvalues for matrices
+whose entries vary widely in magnitude. See @ref{Balancing} for more
+information. Note that the balancing transformation does not preserve
+the orthogonality of the Schur vectors, so if you wish to compute the
+Schur vectors with @code{gsl_eigen_nonsymm_Z} you will obtain the Schur
+vectors of the balanced matrix instead of the original matrix. The
+relationship will be
+@tex
+\beforedisplay
+$$
+T = Q^t D^{-1} A D Q
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+T = Q^t D^(-1) A D Q
+@end example
+
+@end ifinfo
+@noindent
+where @var{Q} is the matrix of Schur vectors for the balanced matrix, and
+@var{D} is the balancing transformation. Then @code{gsl_eigen_nonsymm_Z}
+will compute a matrix @var{Z} which satisfies
+@tex
+\beforedisplay
+$$
+T = Z^{-1} A Z
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+T = Z^(-1) A Z
+@end example
+
+@end ifinfo
+@noindent
+with @math{Z = D Q}. Note that @var{Z} will not be orthogonal. For
+this reason, balancing is not performed by default.
+@end deftypefun
+
+@deftypefun int gsl_eigen_nonsymm (gsl_matrix * @var{A}, gsl_vector_complex * @var{eval}, gsl_eigen_nonsymm_workspace * @var{w})
+This function computes the eigenvalues of the real nonsymmetric matrix
+@var{A} and stores them in the vector @var{eval}. If @math{T} is
+desired, it is stored in the upper portion of @var{A} on output.
+Otherwise, on output, the diagonal of @var{A} will contain the
+@math{1}-by-@math{1} real eigenvalues and @math{2}-by-@math{2}
+complex conjugate eigenvalue systems, and the rest of @var{A} is
+destroyed. In rare cases, this function will fail to find all
+eigenvalues. If this happens, an error code is returned
+and the number of converged eigenvalues is stored in @code{w->n_evals}.
+The converged eigenvalues are stored in the beginning of @var{eval}.
+@end deftypefun
+
+@deftypefun int gsl_eigen_nonsymm_Z (gsl_matrix * @var{A}, gsl_vector_complex * @var{eval}, gsl_matrix * @var{Z}, gsl_eigen_nonsymm_workspace * @var{w})
+This function is identical to @code{gsl_eigen_nonsymm} except it also
+computes the Schur vectors and stores them into @var{Z}.
+@end deftypefun
+
+@deftypefun {gsl_eigen_nonsymmv_workspace *} gsl_eigen_nonsymmv_alloc (const size_t @var{n})
+This function allocates a workspace for computing eigenvalues and
+eigenvectors of @var{n}-by-@var{n} real nonsymmetric matrices. The
+size of the workspace is @math{O(5n)}.
+@end deftypefun
+
+@deftypefun void gsl_eigen_nonsymmv_free (gsl_eigen_nonsymmv_workspace * @var{w})
+This function frees the memory associated with the workspace @var{w}.
+@end deftypefun
+
+@deftypefun int gsl_eigen_nonsymmv (gsl_matrix * @var{A}, gsl_vector_complex * @var{eval}, gsl_matrix_complex * @var{evec}, gsl_eigen_nonsymmv_workspace * @var{w})
+This function computes eigenvalues and right eigenvectors of the
+@var{n}-by-@var{n} real nonsymmetric matrix @var{A}. It first calls
+@code{gsl_eigen_nonsymm} to compute the eigenvalues, Schur form @math{T}, and
+Schur vectors. Then it finds eigenvectors of @math{T} and backtransforms
+them using the Schur vectors. The Schur vectors are destroyed in the
+process, but can be saved by using @code{gsl_eigen_nonsymmv_Z}. The
+computed eigenvectors are normalized to have Euclidean norm 1. On
+output, the upper portion of @var{A} contains the Schur form
+@math{T}. If @code{gsl_eigen_nonsymm} fails, no eigenvectors are
+computed, and an error code is returned.
+@end deftypefun
+
+@deftypefun int gsl_eigen_nonsymmv_Z (gsl_matrix * @var{A}, gsl_vector_complex * @var{eval}, gsl_matrix_complex * @var{evec}, gsl_matrix * @var{Z}, gsl_eigen_nonsymmv_workspace * @var{w})
+This function is identical to @code{gsl_eigen_nonsymmv} except it also saves
+the Schur vectors into @var{Z}.
+@end deftypefun
+
+@node Sorting Eigenvalues and Eigenvectors
+@section Sorting Eigenvalues and Eigenvectors
+@cindex sorting eigenvalues and eigenvectors
+
+@deftypefun int gsl_eigen_symmv_sort (gsl_vector * @var{eval}, gsl_matrix * @var{evec}, gsl_eigen_sort_t @var{sort_type})
+This function simultaneously sorts the eigenvalues stored in the vector
+@var{eval} and the corresponding real eigenvectors stored in the columns
+of the matrix @var{evec} into ascending or descending order according to
+the value of the parameter @var{sort_type},
+
+@table @code
+@item GSL_EIGEN_SORT_VAL_ASC
+ascending order in numerical value
+@item GSL_EIGEN_SORT_VAL_DESC
+descending order in numerical value
+@item GSL_EIGEN_SORT_ABS_ASC
+ascending order in magnitude
+@item GSL_EIGEN_SORT_ABS_DESC
+descending order in magnitude
+@end table
+
+@end deftypefun
+
+@deftypefun int gsl_eigen_hermv_sort (gsl_vector * @var{eval}, gsl_matrix_complex * @var{evec}, gsl_eigen_sort_t @var{sort_type})
+This function simultaneously sorts the eigenvalues stored in the vector
+@var{eval} and the corresponding complex eigenvectors stored in the
+columns of the matrix @var{evec} into ascending or descending order
+according to the value of the parameter @var{sort_type} as shown above.
+@end deftypefun
+
+@deftypefun int gsl_eigen_nonsymmv_sort (gsl_vector_complex * @var{eval}, gsl_matrix_complex * @var{evec}, gsl_eigen_sort_t @var{sort_type})
+This function simultaneously sorts the eigenvalues stored in the vector
+@var{eval} and the corresponding complex eigenvectors stored in the
+columns of the matrix @var{evec} into ascending or descending order
+according to the value of the parameter @var{sort_type} as shown above.
+Only GSL_EIGEN_SORT_ABS_ASC and GSL_EIGEN_SORT_ABS_DESC are supported
+due to the eigenvalues being complex.
+@end deftypefun
+
+
+@comment @deftypefun int gsl_eigen_jacobi (gsl_matrix * @var{matrix}, gsl_vector * @var{eval}, gsl_matrix * @var{evec}, unsigned int @var{max_rot}, unsigned int * @var{nrot})
+@comment This function finds the eigenvectors and eigenvalues of a real symmetric
+@comment matrix by Jacobi iteration. The data in the input matrix is destroyed.
+@comment @end deftypefun
+
+@comment @deftypefun int gsl_la_invert_jacobi (const gsl_matrix * @var{matrix}, gsl_matrix * @var{ainv}, unsigned int @var{max_rot})
+@comment Invert a matrix by Jacobi iteration.
+@comment @end deftypefun
+
+@comment @deftypefun int gsl_eigen_sort (gsl_vector * @var{eval}, gsl_matrix * @var{evec}, gsl_eigen_sort_t @var{sort_type})
+@comment This functions sorts the eigensystem results based on eigenvalues.
+@comment Sorts in order of increasing value or increasing
+@comment absolute value, depending on the value of
+@comment @var{sort_type}, which can be @code{GSL_EIGEN_SORT_VALUE}
+@comment or @code{GSL_EIGEN_SORT_ABSVALUE}.
+@comment @end deftypefun
+
+@node Eigenvalue and Eigenvector Examples
+@section Examples
+
+The following program computes the eigenvalues and eigenvectors of the 4-th order Hilbert matrix, @math{H(i,j) = 1/(i + j + 1)}.
+
+@example
+@verbatiminclude examples/eigen.c
+@end example
+
+@noindent
+Here is the beginning of the output from the program,
+
+@example
+$ ./a.out 
+eigenvalue = 9.67023e-05
+eigenvector = 
+-0.0291933
+0.328712
+-0.791411
+0.514553
+...
+@end example
+
+@noindent
+This can be compared with the corresponding output from @sc{gnu octave},
+
+@example
+octave> [v,d] = eig(hilb(4));
+octave> diag(d)  
+ans =
+
+   9.6702e-05
+   6.7383e-03
+   1.6914e-01
+   1.5002e+00
+
+octave> v 
+v =
+
+   0.029193   0.179186  -0.582076   0.792608
+  -0.328712  -0.741918   0.370502   0.451923
+   0.791411   0.100228   0.509579   0.322416
+  -0.514553   0.638283   0.514048   0.252161
+@end example
+
+@noindent
+Note that the eigenvectors can differ by a change of sign, since the
+sign of an eigenvector is arbitrary.
+
+The following program illustrates the use of the nonsymmetric
+eigensolver, by computing the eigenvalues and eigenvectors of
+the Vandermonde matrix @math{V(x;i,j) = x_i^{n - j}} with
+@math{x = (-1,-2,3,4)}.
+
+@example
+@verbatiminclude examples/eigen_nonsymm.c
+@end example
+
+@noindent
+Here is the beginning of the output from the program,
+
+@example
+$ ./a.out 
+eigenvalue = -6.41391 + 0i
+eigenvector = 
+-0.0998822 + 0i
+-0.111251 + 0i
+0.292501 + 0i
+0.944505 + 0i
+eigenvalue = 5.54555 + 3.08545i
+eigenvector = 
+-0.043487 + -0.0076308i
+0.0642377 + -0.142127i
+-0.515253 + 0.0405118i
+-0.840592 + -0.00148565i
+...
+@end example
+
+@noindent
+This can be compared with the corresponding output from @sc{gnu octave},
+
+@example
+octave> [v,d] = eig(vander([-1 -2 3 4]));
+octave> diag(d)
+ans =
+
+  -6.4139 + 0.0000i
+   5.5456 + 3.0854i
+   5.5456 - 3.0854i
+   2.3228 + 0.0000i
+
+octave> v
+v =
+
+ Columns 1 through 3:
+
+  -0.09988 + 0.00000i  -0.04350 - 0.00755i  -0.04350 + 0.00755i
+  -0.11125 + 0.00000i   0.06399 - 0.14224i   0.06399 + 0.14224i
+   0.29250 + 0.00000i  -0.51518 + 0.04142i  -0.51518 - 0.04142i
+   0.94451 + 0.00000i  -0.84059 + 0.00000i  -0.84059 - 0.00000i
+
+ Column 4:
+
+  -0.14493 + 0.00000i
+   0.35660 + 0.00000i
+   0.91937 + 0.00000i
+   0.08118 + 0.00000i
+
+@end example
+Note that the eigenvectors corresponding to the eigenvalue
+@math{5.54555 + 3.08545i} are slightly different. This is because
+they differ by the multiplicative constant
+@math{0.9999984 + 0.0017674i} which has magnitude 1.
+
+@node Eigenvalue and Eigenvector References
+@section References and Further Reading
+
+Further information on the algorithms described in this section can be
+found in the following book,
+
+@itemize @asis
+@item
+G. H. Golub, C. F. Van Loan, @cite{Matrix Computations} (3rd Ed, 1996),
+Johns Hopkins University Press, ISBN 0-8018-5414-8.
+@end itemize
+
+@noindent
+The @sc{lapack} library is described in,
+
+@itemize @asis
+@item
+@cite{LAPACK Users' Guide} (Third Edition, 1999), Published by SIAM,
+ISBN 0-89871-447-8.
+
+@uref{http://www.netlib.org/lapack} 
+@end itemize
+
+@noindent
+The @sc{lapack} source code can be found at the website above along with
+an online copy of the users guide.
diff --git a/gsl-1.9/doc/err.texi b/gsl-1.9/doc/err.texi
new file mode 100644
index 0000000..a687120
--- /dev/null
+++ b/gsl-1.9/doc/err.texi
@@ -0,0 +1,306 @@
+@cindex error handling
+This chapter describes the way that GSL functions report and handle
+errors.  By examining the status information returned by every function
+you can determine whether it succeeded or failed, and if it failed you
+can find out what the precise cause of failure was.  You can also define
+your own error handling functions to modify the default behavior of the
+library.
+
+The functions described in this section are declared in the header file
+@file{gsl_errno.h}.
+
+@menu
+* Error Reporting::             
+* Error Codes::                 
+* Error Handlers::              
+* Using GSL error reporting in your own functions::  
+* Error Reporting Examples::    
+@end menu
+
+@node Error Reporting
+@section Error Reporting
+
+The library follows the thread-safe error reporting conventions of the
+@sc{posix} Threads library.  Functions return a non-zero error code to
+indicate an error and @code{0} to indicate success.
+
+@example
+int status = gsl_function (...)
+
+if (status) @{ /* an error occurred */
+  .....       
+  /* status value specifies the type of error */
+@}
+@end example
+
+The routines report an error whenever they cannot perform the task
+requested of them.  For example, a root-finding function would return a
+non-zero error code if could not converge to the requested accuracy, or
+exceeded a limit on the number of iterations.  Situations like this are
+a normal occurrence when using any mathematical library and you should
+check the return status of the functions that you call.
+
+Whenever a routine reports an error the return value specifies the type
+of error.  The return value is analogous to the value of the variable
+@code{errno} in the C library.  The caller can examine the return code
+and decide what action to take, including ignoring the error if it is
+not considered serious.
+
+In addition to reporting errors by return codes the library also has an
+error handler function @code{gsl_error}.  This function is called by
+other library functions when they report an error, just before they
+return to the caller.  The default behavior of the error handler is to
+print a message and abort the program,
+
+@example
+gsl: file.c:67: ERROR: invalid argument supplied by user
+Default GSL error handler invoked.
+Aborted
+@end example
+
+The purpose of the @code{gsl_error} handler is to provide a function
+where a breakpoint can be set that will catch library errors when
+running under the debugger.  It is not intended for use in production
+programs, which should handle any errors using the return codes.
+
+@node Error Codes
+@section Error Codes
+@cindex error codes, reserved
+The error code numbers returned by library functions are defined in
+the file @file{gsl_errno.h}.  They all have the prefix @code{GSL_} and
+expand to non-zero constant integer values. Error codes above 1024 are
+reserved for applications, and are not used by the library.  Many of
+the error codes use the same base name as the corresponding error code
+in the C library.  Here are some of the most common error codes,
+
+@cindex error codes
+@deftypefn {Macro} int GSL_EDOM
+Domain error; used by mathematical functions when an argument value does
+not fall into the domain over which the function is defined (like
+EDOM in the C library)
+@end deftypefn
+
+@deftypefn {Macro} int GSL_ERANGE
+Range error; used by mathematical functions when the result value is not
+representable because of overflow or underflow (like ERANGE in the C
+library)
+@end deftypefn
+
+@deftypefn {Macro} int GSL_ENOMEM
+No memory available.  The system cannot allocate more virtual memory
+because its capacity is full (like ENOMEM in the C library).  This error
+is reported when a GSL routine encounters problems when trying to
+allocate memory with @code{malloc}.
+@end deftypefn
+
+@deftypefn {Macro} int GSL_EINVAL
+Invalid argument.  This is used to indicate various kinds of problems
+with passing the wrong argument to a library function (like EINVAL in the C
+library). 
+@end deftypefn
+
+The error codes can be converted into an error message using the
+function @code{gsl_strerror}.
+
+@deftypefun {const char *} gsl_strerror (const int @var{gsl_errno})
+This function returns a pointer to a string describing the error code
+@var{gsl_errno}. For example,
+
+@example
+printf ("error: %s\n", gsl_strerror (status));
+@end example
+
+@noindent
+would print an error message like @code{error: output range error} for a
+status value of @code{GSL_ERANGE}.
+@end deftypefun
+
+@node Error Handlers
+@section Error Handlers
+@cindex Error handlers
+
+The default behavior of the GSL error handler is to print a short
+message and call @code{abort}.  When this default is in use programs
+will stop with a core-dump whenever a library routine reports an error.
+This is intended as a fail-safe default for programs which do not check
+the return status of library routines (we don't encourage you to write
+programs this way).
+
+If you turn off the default error handler it is your responsibility to
+check the return values of routines and handle them yourself.  You can
+also customize the error behavior by providing a new error handler. For
+example, an alternative error handler could log all errors to a file,
+ignore certain error conditions (such as underflows), or start the
+debugger and attach it to the current process when an error occurs.
+
+All GSL error handlers have the type @code{gsl_error_handler_t}, which is
+defined in @file{gsl_errno.h},
+
+@deftp {Data Type} gsl_error_handler_t
+
+This is the type of GSL error handler functions.  An error handler will
+be passed four arguments which specify the reason for the error (a
+string), the name of the source file in which it occurred (also a
+string), the line number in that file (an integer) and the error number
+(an integer).  The source file and line number are set at compile time
+using the @code{__FILE__} and @code{__LINE__} directives in the
+preprocessor.  An error handler function returns type @code{void}.
+Error handler functions should be defined like this,
+
+@example
+void handler (const char * reason, 
+              const char * file, 
+              int line, 
+              int gsl_errno)
+@end example
+@end deftp
+@comment 
+
+@noindent
+To request the use of your own error handler you need to call the
+function @code{gsl_set_error_handler} which is also declared in
+@file{gsl_errno.h},
+
+@deftypefun {gsl_error_handler_t *} gsl_set_error_handler (gsl_error_handler_t @var{new_handler})
+
+This function sets a new error handler, @var{new_handler}, for the GSL
+library routines.  The previous handler is returned (so that you can
+restore it later).  Note that the pointer to a user defined error
+handler function is stored in a static variable, so there can be only
+one error handler per program.  This function should be not be used in
+multi-threaded programs except to set up a program-wide error handler
+from a master thread.  The following example shows how to set and
+restore a new error handler,
+
+@example
+/* save original handler, install new handler */
+old_handler = gsl_set_error_handler (&my_handler); 
+
+/* code uses new handler */
+.....     
+
+/* restore original handler */
+gsl_set_error_handler (old_handler); 
+@end example
+
+@noindent
+To use the default behavior (@code{abort} on error) set the error
+handler to @code{NULL},
+
+@example
+old_handler = gsl_set_error_handler (NULL); 
+@end example
+@end deftypefun
+
+@deftypefun {gsl_error_handler_t *} gsl_set_error_handler_off ()
+This function turns off the error handler by defining an error handler
+which does nothing. This will cause the program to continue after any
+error, so the return values from any library routines must be checked.
+This is the recommended behavior for production programs.  The previous
+handler is returned (so that you can restore it later).
+@end deftypefun
+
+The error behavior can be changed for specific applications by
+recompiling the library with a customized definition of the
+@code{GSL_ERROR} macro in the file @file{gsl_errno.h}.
+
+@node Using GSL error reporting in your own functions
+@section Using GSL error reporting in your own functions
+@cindex error handling macros
+If you are writing numerical functions in a program which also uses GSL
+code you may find it convenient to adopt the same error reporting
+conventions as in the library.
+
+To report an error you need to call the function @code{gsl_error} with a
+string describing the error and then return an appropriate error code
+from @code{gsl_errno.h}, or a special value, such as @code{NaN}.  For
+convenience the file @file{gsl_errno.h} defines two macros which carry
+out these steps:
+
+@deffn {Macro} GSL_ERROR (@var{reason}, @var{gsl_errno})
+
+This macro reports an error using the GSL conventions and returns a
+status value of @code{gsl_errno}.  It expands to the following code fragment,
+
+@example
+gsl_error (reason, __FILE__, __LINE__, gsl_errno);
+return gsl_errno;
+@end example
+
+@noindent
+The macro definition in @file{gsl_errno.h} actually wraps the code
+in a @code{do @{ ... @} while (0)} block to prevent possible
+parsing problems.
+@end deffn
+
+Here is an example of how the macro could be used to report that a
+routine did not achieve a requested tolerance.  To report the error the
+routine needs to return the error code @code{GSL_ETOL}.
+
+@example
+if (residual > tolerance) 
+  @{
+    GSL_ERROR("residual exceeds tolerance", GSL_ETOL);
+  @}
+@end example
+
+@deffn {Macro} GSL_ERROR_VAL (@var{reason}, @var{gsl_errno}, @var{value})
+
+This macro is the same as @code{GSL_ERROR} but returns a user-defined
+value of @var{value} instead of an error code.  It can be used for
+mathematical functions that return a floating point value.
+@end deffn
+
+The following example shows how to return a @code{NaN} at a mathematical
+singularity using the @code{GSL_ERROR_VAL} macro,
+
+@example
+if (x == 0) 
+  @{
+    GSL_ERROR_VAL("argument lies on singularity", 
+                  GSL_ERANGE, GSL_NAN);
+  @}
+@end example
+
+
+@node Error Reporting Examples
+@section Examples
+
+Here is an example of some code which checks the return value of a
+function where an error might be reported,
+
+@example
+#include <stdio.h>
+#include <gsl/gsl_errno.h>
+#include <gsl/gsl_fft_complex.h>
+
+...
+  int status;
+  size_t n = 37;
+
+  gsl_set_error_handler_off();
+
+  status = gsl_fft_complex_radix2_forward (data, n);
+
+  if (status) @{
+    if (status == GSL_EINVAL) @{
+       fprintf (stderr, "invalid argument, n=%d\n", n);
+    @} else @{
+       fprintf (stderr, "failed, gsl_errno=%d\n", 
+                        status);
+    @}
+    exit (-1);
+  @}
+...
+@end example
+@comment 
+
+@noindent
+The function @code{gsl_fft_complex_radix2} only accepts integer lengths
+which are a power of two.  If the variable @code{n} is not a power of
+two then the call to the library function will return @code{GSL_EINVAL},
+indicating that the length argument is invalid.  The function call to
+@code{gsl_set_error_handler_off} stops the default error handler from
+aborting the program.  The @code{else} clause catches any other possible
+errors.
+
diff --git a/gsl-1.9/doc/examples/blas.c b/gsl-1.9/doc/examples/blas.c
new file mode 100644
index 0000000..ab4b39b
--- /dev/null
+++ b/gsl-1.9/doc/examples/blas.c
@@ -0,0 +1,31 @@
+#include <stdio.h>
+#include <gsl/gsl_blas.h>
+
+int
+main (void)
+{
+  double a[] = { 0.11, 0.12, 0.13,
+                 0.21, 0.22, 0.23 };
+
+  double b[] = { 1011, 1012,
+                 1021, 1022,
+                 1031, 1032 };
+
+  double c[] = { 0.00, 0.00,
+                 0.00, 0.00 };
+
+  gsl_matrix_view A = gsl_matrix_view_array(a, 2, 3);
+  gsl_matrix_view B = gsl_matrix_view_array(b, 3, 2);
+  gsl_matrix_view C = gsl_matrix_view_array(c, 2, 2);
+
+  /* Compute C = A B */
+
+  gsl_blas_dgemm (CblasNoTrans, CblasNoTrans,
+                  1.0, &A.matrix, &B.matrix,
+                  0.0, &C.matrix);
+
+  printf ("[ %g, %g\n", c[0], c[1]);
+  printf ("  %g, %g ]\n", c[2], c[3]);
+
+  return 0;  
+}
diff --git a/gsl-1.9/doc/examples/blas.out b/gsl-1.9/doc/examples/blas.out
new file mode 100644
index 0000000..e8ec960
--- /dev/null
+++ b/gsl-1.9/doc/examples/blas.out
@@ -0,0 +1,2 @@
+[ 367.76, 368.12
+  674.06, 674.72 ]
\ No newline at end of file
diff --git a/gsl-1.9/doc/examples/block.c b/gsl-1.9/doc/examples/block.c
new file mode 100644
index 0000000..53f62a1
--- /dev/null
+++ b/gsl-1.9/doc/examples/block.c
@@ -0,0 +1,14 @@
+#include <stdio.h>
+#include <gsl/gsl_block.h>
+
+int
+main (void)
+{
+  gsl_block * b = gsl_block_alloc (100);
+  
+  printf ("length of block = %u\n", b->size);
+  printf ("block data address = %#x\n", b->data);
+
+  gsl_block_free (b);
+  return 0;
+}
diff --git a/gsl-1.9/doc/examples/block.out b/gsl-1.9/doc/examples/block.out
new file mode 100644
index 0000000..1f84cd0
--- /dev/null
+++ b/gsl-1.9/doc/examples/block.out
@@ -0,0 +1,2 @@
+length of block = 100
+block data address = 0x804b0d8
diff --git a/gsl-1.9/doc/examples/bspline.c b/gsl-1.9/doc/examples/bspline.c
new file mode 100644
index 0000000..0d6a584
--- /dev/null
+++ b/gsl-1.9/doc/examples/bspline.c
@@ -0,0 +1,116 @@
+#include <stdio.h>
+#include <stdlib.h>
+#include <math.h>
+#include <gsl/gsl_bspline.h>
+#include <gsl/gsl_multifit.h>
+#include <gsl/gsl_rng.h>
+#include <gsl/gsl_randist.h>
+
+/* number of data points to fit */
+#define N        200
+
+/* number of fit coefficients */
+#define NCOEFFS  8
+
+/* nbreak = ncoeffs + 2 - k = ncoeffs - 2 since k = 4 */
+#define NBREAK   (NCOEFFS - 2)
+
+int
+main (void)
+{
+  const size_t n = N;
+  const size_t ncoeffs = NCOEFFS;
+  const size_t nbreak = NBREAK;
+  size_t i, j;
+  gsl_bspline_workspace *bw;
+  gsl_vector *B;
+  double dy;
+  gsl_rng *r;
+  gsl_vector *c, *w;
+  gsl_vector *x, *y;
+  gsl_matrix *X, *cov;
+  gsl_multifit_linear_workspace *mw;
+  double chisq;
+
+  gsl_rng_env_setup();
+  r = gsl_rng_alloc(gsl_rng_default);
+
+  /* allocate a cubic bspline workspace (k = 4) */
+  bw = gsl_bspline_alloc(4, nbreak);
+  B = gsl_vector_alloc(ncoeffs);
+
+  x = gsl_vector_alloc(n);
+  y = gsl_vector_alloc(n);
+  X = gsl_matrix_alloc(n, ncoeffs);
+  c = gsl_vector_alloc(ncoeffs);
+  w = gsl_vector_alloc(n);
+  cov = gsl_matrix_alloc(ncoeffs, ncoeffs);
+  mw = gsl_multifit_linear_alloc(n, ncoeffs);
+
+  printf("#m=0,S=0\n");
+  /* this is the data to be fitted */
+  for (i = 0; i < n; ++i)
+    {
+      double sigma;
+      double xi = (15.0/(N-1)) * i;
+      double yi = cos(xi) * exp(-0.1 * xi);
+
+      sigma = 0.1;
+      dy = gsl_ran_gaussian(r, sigma);
+      yi += dy;
+
+      gsl_vector_set(x, i, xi);
+      gsl_vector_set(y, i, yi);
+      gsl_vector_set(w, i, 1.0 / (sigma*sigma));
+
+      printf("%f %f\n", xi, yi);
+    }
+
+  /* use uniform breakpoints on [0, 15] */
+  gsl_bspline_knots_uniform(0.0, 15.0, bw);
+
+  /* construct the fit matrix X */
+  for (i = 0; i < n; ++i)
+    {
+      double xi = gsl_vector_get(x, i);
+
+      /* compute B_j(xi) for all j */
+      gsl_bspline_eval(xi, B, bw);
+
+      /* fill in row i of X */
+      for (j = 0; j < ncoeffs; ++j)
+        {
+          double Bj = gsl_vector_get(B, j);
+          gsl_matrix_set(X, i, j, Bj);
+        }
+    }
+
+  /* do the fit */
+  gsl_multifit_wlinear(X, w, y, c, cov, &chisq, mw);
+
+  /* output the smoothed curve */
+  {
+    double xi, yi, yerr;
+
+    printf("#m=1,S=0\n");
+    for (xi = 0.0; xi < 15.0; xi += 0.1)
+      {
+        gsl_bspline_eval(xi, B, bw);
+        gsl_multifit_linear_est(B, c, cov, &yi, &yerr);
+        printf("%f %f\n", xi, yi);
+      }
+  }
+
+  gsl_rng_free(r);
+  gsl_bspline_free(bw);
+  gsl_vector_free(B);
+  gsl_vector_free(x);
+  gsl_vector_free(y);
+  gsl_matrix_free(X);
+  gsl_vector_free(c);
+  gsl_vector_free(w);
+  gsl_matrix_free(cov);
+  gsl_multifit_linear_free(mw);
+
+  return 0;
+} /* main() */
diff --git a/gsl-1.9/doc/examples/cblas.c b/gsl-1.9/doc/examples/cblas.c
new file mode 100644
index 0000000..cb6b31a
--- /dev/null
+++ b/gsl-1.9/doc/examples/cblas.c
@@ -0,0 +1,33 @@
+#include <stdio.h>
+#include <gsl/gsl_cblas.h>
+
+int
+main (void)
+{
+  int lda = 3;
+
+  float A[] = { 0.11, 0.12, 0.13,
+                0.21, 0.22, 0.23 };
+
+  int ldb = 2;
+  
+  float B[] = { 1011, 1012,
+                1021, 1022,
+                1031, 1032 };
+
+  int ldc = 2;
+
+  float C[] = { 0.00, 0.00,
+                0.00, 0.00 };
+
+  /* Compute C = A B */
+
+  cblas_sgemm (CblasRowMajor, 
+               CblasNoTrans, CblasNoTrans, 2, 2, 3,
+               1.0, A, lda, B, ldb, 0.0, C, ldc);
+
+  printf ("[ %g, %g\n", C[0], C[1]);
+  printf ("  %g, %g ]\n", C[2], C[3]);
+
+  return 0;  
+}
diff --git a/gsl-1.9/doc/examples/cblas.out b/gsl-1.9/doc/examples/cblas.out
new file mode 100644
index 0000000..a0f51d7
--- /dev/null
+++ b/gsl-1.9/doc/examples/cblas.out
@@ -0,0 +1,2 @@
+[ 367.76, 368.12
+  674.06, 674.72 ]
diff --git a/gsl-1.9/doc/examples/cdf.c b/gsl-1.9/doc/examples/cdf.c
new file mode 100644
index 0000000..968eb96
--- /dev/null
+++ b/gsl-1.9/doc/examples/cdf.c
@@ -0,0 +1,23 @@
+#include <stdio.h>
+#include <gsl/gsl_cdf.h>
+
+int
+main (void)
+{
+  double P, Q;
+  double x = 2.0;
+
+  P = gsl_cdf_ugaussian_P (x);
+  printf ("prob(x < %f) = %f\n", x, P);
+
+  Q = gsl_cdf_ugaussian_Q (x);
+  printf ("prob(x > %f) = %f\n", x, Q);
+
+  x = gsl_cdf_ugaussian_Pinv (P);
+  printf ("Pinv(%f) = %f\n", P, x);
+
+  x = gsl_cdf_ugaussian_Qinv (Q);
+  printf ("Qinv(%f) = %f\n", Q, x);
+
+  return 0;
+}
diff --git a/gsl-1.9/doc/examples/cdf.out b/gsl-1.9/doc/examples/cdf.out
new file mode 100644
index 0000000..f04b75b
--- /dev/null
+++ b/gsl-1.9/doc/examples/cdf.out
@@ -0,0 +1,4 @@
+prob(x < 2.000000) = 0.977250
+prob(x > 2.000000) = 0.022750
+Pinv(0.977250) = 2.000000
+Qinv(0.022750) = 2.000000
diff --git a/gsl-1.9/doc/examples/cheb.c b/gsl-1.9/doc/examples/cheb.c
new file mode 100644
index 0000000..58a0c54
--- /dev/null
+++ b/gsl-1.9/doc/examples/cheb.c
@@ -0,0 +1,40 @@
+#include <stdio.h>
+#include <gsl/gsl_math.h>
+#include <gsl/gsl_chebyshev.h>
+
+double
+f (double x, void *p)
+{
+  if (x < 0.5)
+    return 0.25;
+  else
+    return 0.75;
+}
+
+int
+main (void)
+{
+  int i, n = 10000; 
+
+  gsl_cheb_series *cs = gsl_cheb_alloc (40);
+
+  gsl_function F;
+
+  F.function = f;
+  F.params = 0;
+
+  gsl_cheb_init (cs, &F, 0.0, 1.0);
+
+  for (i = 0; i < n; i++)
+    {
+      double x = i / (double)n;
+      double r10 = gsl_cheb_eval_n (cs, 10, x);
+      double r40 = gsl_cheb_eval (cs, x);
+      printf ("%g %g %g %g\n", 
+              x, GSL_FN_EVAL (&F, x), r10, r40);
+    }
+
+  gsl_cheb_free (cs);
+
+  return 0;
+}
diff --git a/gsl-1.9/doc/examples/combination.c b/gsl-1.9/doc/examples/combination.c
new file mode 100644
index 0000000..708174f
--- /dev/null
+++ b/gsl-1.9/doc/examples/combination.c
@@ -0,0 +1,25 @@
+#include <stdio.h>
+#include <gsl/gsl_combination.h>
+
+int 
+main (void) 
+{
+  gsl_combination * c;
+  size_t i;
+
+  printf ("All subsets of {0,1,2,3} by size:\n") ;
+  for (i = 0; i <= 4; i++)
+    {
+      c = gsl_combination_calloc (4, i);
+      do
+        {
+          printf ("{");
+          gsl_combination_fprintf (stdout, c, " %u");
+          printf (" }\n");
+        }
+      while (gsl_combination_next (c) == GSL_SUCCESS);
+      gsl_combination_free (c);
+    }
+
+  return 0;
+}
diff --git a/gsl-1.9/doc/examples/combination.out b/gsl-1.9/doc/examples/combination.out
new file mode 100644
index 0000000..a9050ab
--- /dev/null
+++ b/gsl-1.9/doc/examples/combination.out
@@ -0,0 +1,17 @@
+All subsets of {0,1,2,3} by size:
+{ }
+{ 0 }
+{ 1 }
+{ 2 }
+{ 3 }
+{ 0 1 }
+{ 0 2 }
+{ 0 3 }
+{ 1 2 }
+{ 1 3 }
+{ 2 3 }
+{ 0 1 2 }
+{ 0 1 3 }
+{ 0 2 3 }
+{ 1 2 3 }
+{ 0 1 2 3 }
diff --git a/gsl-1.9/doc/examples/const.c b/gsl-1.9/doc/examples/const.c
new file mode 100644
index 0000000..47718c5
--- /dev/null
+++ b/gsl-1.9/doc/examples/const.c
@@ -0,0 +1,25 @@
+#include <stdio.h>
+#include <gsl/gsl_const_mksa.h>
+
+int
+main (void)
+{
+  double c  = GSL_CONST_MKSA_SPEED_OF_LIGHT;
+  double au = GSL_CONST_MKSA_ASTRONOMICAL_UNIT;
+  double minutes = GSL_CONST_MKSA_MINUTE;
+
+  /* distance stored in meters */
+  double r_earth = 1.00 * au;  
+  double r_mars  = 1.52 * au;
+
+  double t_min, t_max;
+
+  t_min = (r_mars - r_earth) / c;
+  t_max = (r_mars + r_earth) / c;
+
+  printf ("light travel time from Earth to Mars:\n");
+  printf ("minimum = %.1f minutes\n", t_min / minutes);
+  printf ("maximum = %.1f minutes\n", t_max / minutes);
+
+  return 0;
+}
diff --git a/gsl-1.9/doc/examples/const.out b/gsl-1.9/doc/examples/const.out
new file mode 100644
index 0000000..6552a3d
--- /dev/null
+++ b/gsl-1.9/doc/examples/const.out
@@ -0,0 +1,3 @@
+light travel time from Earth to Mars:
+minimum = 4.3 minutes
+maximum = 21.0 minutes
diff --git a/gsl-1.9/doc/examples/demo_fn.c b/gsl-1.9/doc/examples/demo_fn.c
new file mode 100644
index 0000000..4d8a552
--- /dev/null
+++ b/gsl-1.9/doc/examples/demo_fn.c
@@ -0,0 +1,40 @@
+double
+quadratic (double x, void *params)
+{
+  struct quadratic_params *p 
+    = (struct quadratic_params *) params;
+
+  double a = p->a;
+  double b = p->b;
+  double c = p->c;
+
+  return (a * x + b) * x + c;
+}
+
+double
+quadratic_deriv (double x, void *params)
+{
+  struct quadratic_params *p 
+    = (struct quadratic_params *) params;
+
+  double a = p->a;
+  double b = p->b;
+  double c = p->c;
+
+  return 2.0 * a * x + b;
+}
+
+void
+quadratic_fdf (double x, void *params, 
+               double *y, double *dy)
+{
+  struct quadratic_params *p 
+    = (struct quadratic_params *) params;
+
+  double a = p->a;
+  double b = p->b;
+  double c = p->c;
+
+  *y = (a * x + b) * x + c;
+  *dy = 2.0 * a * x + b;
+}
diff --git a/gsl-1.9/doc/examples/demo_fn.h b/gsl-1.9/doc/examples/demo_fn.h
new file mode 100644
index 0000000..77aeea1
--- /dev/null
+++ b/gsl-1.9/doc/examples/demo_fn.h
@@ -0,0 +1,9 @@
+struct quadratic_params
+  {
+    double a, b, c;
+  };
+
+double quadratic (double x, void *params);
+double quadratic_deriv (double x, void *params);
+void quadratic_fdf (double x, void *params, 
+                    double *y, double *dy);
diff --git a/gsl-1.9/doc/examples/diff.c b/gsl-1.9/doc/examples/diff.c
new file mode 100644
index 0000000..b32e193
--- /dev/null
+++ b/gsl-1.9/doc/examples/diff.c
@@ -0,0 +1,32 @@
+#include <stdio.h>
+#include <gsl/gsl_math.h>
+#include <gsl/gsl_deriv.h>
+
+double f (double x, void * params)
+{
+  return pow (x, 1.5);
+}
+
+int
+main (void)
+{
+  gsl_function F;
+  double result, abserr;
+
+  F.function = &f;
+  F.params = 0;
+
+  printf ("f(x) = x^(3/2)\n");
+
+  gsl_deriv_central (&F, 2.0, 1e-8, &result, &abserr);
+  printf ("x = 2.0\n");
+  printf ("f'(x) = %.10f +/- %.10f\n", result, abserr);
+  printf ("exact = %.10f\n\n", 1.5 * sqrt(2.0));
+
+  gsl_deriv_forward (&F, 0.0, 1e-8, &result, &abserr);
+  printf ("x = 0.0\n");
+  printf ("f'(x) = %.10f +/- %.10f\n", result, abserr);
+  printf ("exact = %.10f\n", 0.0);
+
+  return 0;
+}
diff --git a/gsl-1.9/doc/examples/diff.out b/gsl-1.9/doc/examples/diff.out
new file mode 100644
index 0000000..dab42d5
--- /dev/null
+++ b/gsl-1.9/doc/examples/diff.out
@@ -0,0 +1,8 @@
+f(x) = x^(3/2)
+x = 2.0
+f'(x) = 2.1213203120 +/- 0.0000004064
+exact = 2.1213203436
+
+x = 0.0
+f'(x) = 0.0000000160 +/- 0.0000000339
+exact = 0.0000000000
diff --git a/gsl-1.9/doc/examples/dwt.c b/gsl-1.9/doc/examples/dwt.c
new file mode 100644
index 0000000..062b2c6
--- /dev/null
+++ b/gsl-1.9/doc/examples/dwt.c
@@ -0,0 +1,54 @@
+#include <stdio.h>
+#include <math.h>
+#include <gsl/gsl_sort.h>
+#include <gsl/gsl_wavelet.h>
+
+int
+main (int argc, char **argv)
+{
+  int i, n = 256, nc = 20;
+  double *data = malloc (n * sizeof (double));
+  double *abscoeff = malloc (n * sizeof (double));
+  size_t *p = malloc (n * sizeof (size_t));
+
+  FILE * f;
+  gsl_wavelet *w;
+  gsl_wavelet_workspace *work;
+
+  w = gsl_wavelet_alloc (gsl_wavelet_daubechies, 4);
+  work = gsl_wavelet_workspace_alloc (n);
+
+  f = fopen (argv[1], "r");
+  for (i = 0; i < n; i++)
+    {
+      fscanf (f, "%lg", &data[i]);
+    }
+  fclose (f);
+
+  gsl_wavelet_transform_forward (w, data, 1, n, work);
+
+  for (i = 0; i < n; i++)
+    {
+      abscoeff[i] = fabs (data[i]);
+    }
+  
+  gsl_sort_index (p, abscoeff, 1, n);
+  
+  for (i = 0; (i + nc) < n; i++)
+    data[p[i]] = 0;
+  
+  gsl_wavelet_transform_inverse (w, data, 1, n, work);
+  
+  for (i = 0; i < n; i++)
+    {
+      printf ("%g\n", data[i]);
+    }
+  
+  gsl_wavelet_free (w);
+  gsl_wavelet_workspace_free (work);
+
+  free (data);
+  free (abscoeff);
+  free (p);
+  return 0;
+}
diff --git a/gsl-1.9/doc/examples/dwt.dat b/gsl-1.9/doc/examples/dwt.dat
new file mode 100644
index 0000000..e7ff47b
--- /dev/null
+++ b/gsl-1.9/doc/examples/dwt.dat
@@ -0,0 +1,256 @@
+0.0167729
+0.031888
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diff --git a/gsl-1.9/doc/examples/ecg.dat b/gsl-1.9/doc/examples/ecg.dat
new file mode 100644
index 0000000..5ea824a
--- /dev/null
+++ b/gsl-1.9/doc/examples/ecg.dat
@@ -0,0 +1,256 @@
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diff --git a/gsl-1.9/doc/examples/eigen.c b/gsl-1.9/doc/examples/eigen.c
new file mode 100644
index 0000000..a24c912
--- /dev/null
+++ b/gsl-1.9/doc/examples/eigen.c
@@ -0,0 +1,50 @@
+#include <stdio.h>
+#include <gsl/gsl_math.h>
+#include <gsl/gsl_eigen.h>
+
+int
+main (void)
+{
+  double data[] = { 1.0  , 1/2.0, 1/3.0, 1/4.0,
+                    1/2.0, 1/3.0, 1/4.0, 1/5.0,
+                    1/3.0, 1/4.0, 1/5.0, 1/6.0,
+                    1/4.0, 1/5.0, 1/6.0, 1/7.0 };
+
+  gsl_matrix_view m 
+    = gsl_matrix_view_array (data, 4, 4);
+
+  gsl_vector *eval = gsl_vector_alloc (4);
+  gsl_matrix *evec = gsl_matrix_alloc (4, 4);
+
+  gsl_eigen_symmv_workspace * w = 
+    gsl_eigen_symmv_alloc (4);
+  
+  gsl_eigen_symmv (&m.matrix, eval, evec, w);
+
+  gsl_eigen_symmv_free (w);
+
+  gsl_eigen_symmv_sort (eval, evec, 
+                        GSL_EIGEN_SORT_ABS_ASC);
+  
+  {
+    int i;
+
+    for (i = 0; i < 4; i++)
+      {
+        double eval_i 
+           = gsl_vector_get (eval, i);
+        gsl_vector_view evec_i 
+           = gsl_matrix_column (evec, i);
+
+        printf ("eigenvalue = %g\n", eval_i);
+        printf ("eigenvector = \n");
+        gsl_vector_fprintf (stdout, 
+                            &evec_i.vector, "%g");
+      }
+  }
+
+  gsl_vector_free (eval);
+  gsl_matrix_free (evec);
+
+  return 0;
+}
diff --git a/gsl-1.9/doc/examples/eigen_nonsymm.c b/gsl-1.9/doc/examples/eigen_nonsymm.c
new file mode 100644
index 0000000..ccbef2d
--- /dev/null
+++ b/gsl-1.9/doc/examples/eigen_nonsymm.c
@@ -0,0 +1,54 @@
+#include <stdio.h>
+#include <gsl/gsl_math.h>
+#include <gsl/gsl_eigen.h>
+
+int
+main (void)
+{
+  double data[] = { -1.0, 1.0, -1.0, 1.0,
+                    -8.0, 4.0, -2.0, 1.0,
+                    27.0, 9.0, 3.0, 1.0,
+                    64.0, 16.0, 4.0, 1.0 };
+
+  gsl_matrix_view m 
+    = gsl_matrix_view_array (data, 4, 4);
+
+  gsl_vector_complex *eval = gsl_vector_complex_alloc (4);
+  gsl_matrix_complex *evec = gsl_matrix_complex_alloc (4, 4);
+
+  gsl_eigen_nonsymmv_workspace * w = 
+    gsl_eigen_nonsymmv_alloc (4);
+  
+  gsl_eigen_nonsymmv (&m.matrix, eval, evec, w);
+
+  gsl_eigen_nonsymmv_free (w);
+
+  gsl_eigen_nonsymmv_sort (eval, evec, 
+                           GSL_EIGEN_SORT_ABS_DESC);
+  
+  {
+    int i, j;
+
+    for (i = 0; i < 4; i++)
+      {
+        gsl_complex eval_i 
+           = gsl_vector_complex_get (eval, i);
+        gsl_vector_complex_view evec_i 
+           = gsl_matrix_complex_column (evec, i);
+
+        printf ("eigenvalue = %g + %gi\n",
+                GSL_REAL(eval_i), GSL_IMAG(eval_i));
+        printf ("eigenvector = \n");
+        for (j = 0; j < 4; ++j)
+          {
+            gsl_complex z = gsl_vector_complex_get(&evec_i.vector, j);
+            printf("%g + %gi\n", GSL_REAL(z), GSL_IMAG(z));
+          }
+      }
+  }
+
+  gsl_vector_complex_free(eval);
+  gsl_matrix_complex_free(evec);
+
+  return 0;
+}
diff --git a/gsl-1.9/doc/examples/expfit.c b/gsl-1.9/doc/examples/expfit.c
new file mode 100644
index 0000000..f94cf2b
--- /dev/null
+++ b/gsl-1.9/doc/examples/expfit.c
@@ -0,0 +1,70 @@
+/* expfit.c -- model functions for exponential + background */
+
+struct data {
+  size_t n;
+  double * y;
+  double * sigma;
+};
+
+int
+expb_f (const gsl_vector * x, void *data, 
+        gsl_vector * f)
+{
+  size_t n = ((struct data *)data)->n;
+  double *y = ((struct data *)data)->y;
+  double *sigma = ((struct data *) data)->sigma;
+
+  double A = gsl_vector_get (x, 0);
+  double lambda = gsl_vector_get (x, 1);
+  double b = gsl_vector_get (x, 2);
+
+  size_t i;
+
+  for (i = 0; i < n; i++)
+    {
+      /* Model Yi = A * exp(-lambda * i) + b */
+      double t = i;
+      double Yi = A * exp (-lambda * t) + b;
+      gsl_vector_set (f, i, (Yi - y[i])/sigma[i]);
+    }
+
+  return GSL_SUCCESS;
+}
+
+int
+expb_df (const gsl_vector * x, void *data, 
+         gsl_matrix * J)
+{
+  size_t n = ((struct data *)data)->n;
+  double *sigma = ((struct data *) data)->sigma;
+
+  double A = gsl_vector_get (x, 0);
+  double lambda = gsl_vector_get (x, 1);
+
+  size_t i;
+
+  for (i = 0; i < n; i++)
+    {
+      /* Jacobian matrix J(i,j) = dfi / dxj, */
+      /* where fi = (Yi - yi)/sigma[i],      */
+      /*       Yi = A * exp(-lambda * i) + b  */
+      /* and the xj are the parameters (A,lambda,b) */
+      double t = i;
+      double s = sigma[i];
+      double e = exp(-lambda * t);
+      gsl_matrix_set (J, i, 0, e/s); 
+      gsl_matrix_set (J, i, 1, -t * A * e/s);
+      gsl_matrix_set (J, i, 2, 1/s);
+    }
+  return GSL_SUCCESS;
+}
+
+int
+expb_fdf (const gsl_vector * x, void *data,
+          gsl_vector * f, gsl_matrix * J)
+{
+  expb_f (x, data, f);
+  expb_df (x, data, J);
+
+  return GSL_SUCCESS;
+}
diff --git a/gsl-1.9/doc/examples/fft.c b/gsl-1.9/doc/examples/fft.c
new file mode 100644
index 0000000..aa00242
--- /dev/null
+++ b/gsl-1.9/doc/examples/fft.c
@@ -0,0 +1,43 @@
+#include <stdio.h>
+#include <math.h>
+#include <gsl/gsl_errno.h>
+#include <gsl/gsl_fft_complex.h>
+
+#define REAL(z,i) ((z)[2*(i)])
+#define IMAG(z,i) ((z)[2*(i)+1])
+
+int
+main (void)
+{
+  int i; double data[2*128];
+
+  for (i = 0; i < 128; i++)
+    {
+       REAL(data,i) = 0.0; IMAG(data,i) = 0.0;
+    }
+
+  REAL(data,0) = 1.0;
+
+  for (i = 1; i <= 10; i++)
+    {
+       REAL(data,i) = REAL(data,128-i) = 1.0;
+    }
+
+  for (i = 0; i < 128; i++)
+    {
+      printf ("%d %e %e\n", i, 
+              REAL(data,i), IMAG(data,i));
+    }
+  printf ("\n");
+
+  gsl_fft_complex_radix2_forward (data, 1, 128);
+
+  for (i = 0; i < 128; i++)
+    {
+      printf ("%d %e %e\n", i, 
+              REAL(data,i)/sqrt(128), 
+              IMAG(data,i)/sqrt(128));
+    }
+
+  return 0;
+}
diff --git a/gsl-1.9/doc/examples/fftmr.c b/gsl-1.9/doc/examples/fftmr.c
new file mode 100644
index 0000000..0a43e0a
--- /dev/null
+++ b/gsl-1.9/doc/examples/fftmr.c
@@ -0,0 +1,60 @@
+#include <stdio.h>
+#include <math.h>
+#include <gsl/gsl_errno.h>
+#include <gsl/gsl_fft_complex.h>
+
+#define REAL(z,i) ((z)[2*(i)])
+#define IMAG(z,i) ((z)[2*(i)+1])
+
+int
+main (void)
+{
+  int i;
+  const int n = 630;
+  double data[2*n];
+
+  gsl_fft_complex_wavetable * wavetable;
+  gsl_fft_complex_workspace * workspace;
+
+  for (i = 0; i < n; i++)
+    {
+      REAL(data,i) = 0.0;
+      IMAG(data,i) = 0.0;
+    }
+
+  data[0] = 1.0;
+
+  for (i = 1; i <= 10; i++)
+    {
+      REAL(data,i) = REAL(data,n-i) = 1.0;
+    }
+
+  for (i = 0; i < n; i++)
+    {
+      printf ("%d: %e %e\n", i, REAL(data,i), 
+                                IMAG(data,i));
+    }
+  printf ("\n");
+
+  wavetable = gsl_fft_complex_wavetable_alloc (n);
+  workspace = gsl_fft_complex_workspace_alloc (n);
+
+  for (i = 0; i < wavetable->nf; i++)
+    {
+       printf ("# factor %d: %d\n", i, 
+               wavetable->factor[i]);
+    }
+
+  gsl_fft_complex_forward (data, 1, n, 
+                           wavetable, workspace);
+
+  for (i = 0; i < n; i++)
+    {
+      printf ("%d: %e %e\n", i, REAL(data,i), 
+                                IMAG(data,i));
+    }
+
+  gsl_fft_complex_wavetable_free (wavetable);
+  gsl_fft_complex_workspace_free (workspace);
+  return 0;
+}
diff --git a/gsl-1.9/doc/examples/fftreal.c b/gsl-1.9/doc/examples/fftreal.c
new file mode 100644
index 0000000..ec34458
--- /dev/null
+++ b/gsl-1.9/doc/examples/fftreal.c
@@ -0,0 +1,59 @@
+#include <stdio.h>
+#include <math.h>
+#include <gsl/gsl_errno.h>
+#include <gsl/gsl_fft_real.h>
+#include <gsl/gsl_fft_halfcomplex.h>
+
+int
+main (void)
+{
+  int i, n = 100;
+  double data[n];
+
+  gsl_fft_real_wavetable * real;
+  gsl_fft_halfcomplex_wavetable * hc;
+  gsl_fft_real_workspace * work;
+
+  for (i = 0; i < n; i++)
+    {
+      data[i] = 0.0;
+    }
+
+  for (i = n / 3; i < 2 * n / 3; i++)
+    {
+      data[i] = 1.0;
+    }
+
+  for (i = 0; i < n; i++)
+    {
+      printf ("%d: %e\n", i, data[i]);
+    }
+  printf ("\n");
+
+  work = gsl_fft_real_workspace_alloc (n);
+  real = gsl_fft_real_wavetable_alloc (n);
+
+  gsl_fft_real_transform (data, 1, n, 
+                          real, work);
+
+  gsl_fft_real_wavetable_free (real);
+
+  for (i = 11; i < n; i++)
+    {
+      data[i] = 0;
+    }
+
+  hc = gsl_fft_halfcomplex_wavetable_alloc (n);
+
+  gsl_fft_halfcomplex_inverse (data, 1, n, 
+                               hc, work);
+  gsl_fft_halfcomplex_wavetable_free (hc);
+
+  for (i = 0; i < n; i++)
+    {
+      printf ("%d: %e\n", i, data[i]);
+    }
+
+  gsl_fft_real_workspace_free (work);
+  return 0;
+}
diff --git a/gsl-1.9/doc/examples/fitting.c b/gsl-1.9/doc/examples/fitting.c
new file mode 100644
index 0000000..5c36969
--- /dev/null
+++ b/gsl-1.9/doc/examples/fitting.c
@@ -0,0 +1,45 @@
+#include <stdio.h>
+#include <gsl/gsl_fit.h>
+
+int
+main (void)
+{
+  int i, n = 4;
+  double x[4] = { 1970, 1980, 1990, 2000 };
+  double y[4] = {   12,   11,   14,   13 };
+  double w[4] = {  0.1,  0.2,  0.3,  0.4 };
+
+  double c0, c1, cov00, cov01, cov11, chisq;
+
+  gsl_fit_wlinear (x, 1, w, 1, y, 1, n, 
+                   &c0, &c1, &cov00, &cov01, &cov11, 
+                   &chisq);
+
+  printf ("# best fit: Y = %g + %g X\n", c0, c1);
+  printf ("# covariance matrix:\n");
+  printf ("# [ %g, %g\n#   %g, %g]\n", 
+          cov00, cov01, cov01, cov11);
+  printf ("# chisq = %g\n", chisq);
+
+  for (i = 0; i < n; i++)
+    printf ("data: %g %g %g\n", 
+                   x[i], y[i], 1/sqrt(w[i]));
+
+  printf ("\n");
+
+  for (i = -30; i < 130; i++)
+    {
+      double xf = x[0] + (i/100.0) * (x[n-1] - x[0]);
+      double yf, yf_err;
+
+      gsl_fit_linear_est (xf, 
+                          c0, c1, 
+                          cov00, cov01, cov11, 
+                          &yf, &yf_err);
+
+      printf ("fit: %g %g\n", xf, yf);
+      printf ("hi : %g %g\n", xf, yf + yf_err);
+      printf ("lo : %g %g\n", xf, yf - yf_err);
+    }
+  return 0;
+}
diff --git a/gsl-1.9/doc/examples/fitting2.c b/gsl-1.9/doc/examples/fitting2.c
new file mode 100644
index 0000000..43f8c06
--- /dev/null
+++ b/gsl-1.9/doc/examples/fitting2.c
@@ -0,0 +1,80 @@
+#include <stdio.h>
+#include <gsl/gsl_multifit.h>
+
+int
+main (int argc, char **argv)
+{
+  int i, n;
+  double xi, yi, ei, chisq;
+  gsl_matrix *X, *cov;
+  gsl_vector *y, *w, *c;
+
+  if (argc != 2)
+    {
+      fprintf (stderr,"usage: fit n < data\n");
+      exit (-1);
+    }
+
+  n = atoi (argv[1]);
+
+  X = gsl_matrix_alloc (n, 3);
+  y = gsl_vector_alloc (n);
+  w = gsl_vector_alloc (n);
+
+  c = gsl_vector_alloc (3);
+  cov = gsl_matrix_alloc (3, 3);
+
+  for (i = 0; i < n; i++)
+    {
+      int count = fscanf (stdin, "%lg %lg %lg",
+                          &xi, &yi, &ei);
+
+      if (count != 3)
+        {
+          fprintf (stderr, "error reading file\n");
+          exit (-1);
+        }
+
+      printf ("%g %g +/- %g\n", xi, yi, ei);
+      
+      gsl_matrix_set (X, i, 0, 1.0);
+      gsl_matrix_set (X, i, 1, xi);
+      gsl_matrix_set (X, i, 2, xi*xi);
+      
+      gsl_vector_set (y, i, yi);
+      gsl_vector_set (w, i, 1.0/(ei*ei));
+    }
+
+  {
+    gsl_multifit_linear_workspace * work 
+      = gsl_multifit_linear_alloc (n, 3);
+    gsl_multifit_wlinear (X, w, y, c, cov,
+                          &chisq, work);
+    gsl_multifit_linear_free (work);
+  }
+
+#define C(i) (gsl_vector_get(c,(i)))
+#define COV(i,j) (gsl_matrix_get(cov,(i),(j)))
+
+  {
+    printf ("# best fit: Y = %g + %g X + %g X^2\n", 
+            C(0), C(1), C(2));
+
+    printf ("# covariance matrix:\n");
+    printf ("[ %+.5e, %+.5e, %+.5e  \n",
+               COV(0,0), COV(0,1), COV(0,2));
+    printf ("  %+.5e, %+.5e, %+.5e  \n", 
+               COV(1,0), COV(1,1), COV(1,2));
+    printf ("  %+.5e, %+.5e, %+.5e ]\n", 
+               COV(2,0), COV(2,1), COV(2,2));
+    printf ("# chisq = %g\n", chisq);
+  }
+
+  gsl_matrix_free (X);
+  gsl_vector_free (y);
+  gsl_vector_free (w);
+  gsl_vector_free (c);
+  gsl_matrix_free (cov);
+
+  return 0;
+}
diff --git a/gsl-1.9/doc/examples/fitting3.c b/gsl-1.9/doc/examples/fitting3.c
new file mode 100644
index 0000000..a3f1e32
--- /dev/null
+++ b/gsl-1.9/doc/examples/fitting3.c
@@ -0,0 +1,29 @@
+#include <stdio.h>
+#include <math.h>
+#include <gsl/gsl_randist.h>
+
+int
+main (void)
+{
+  double x;
+  const gsl_rng_type * T;
+  gsl_rng * r;
+  
+  gsl_rng_env_setup ();
+  
+  T = gsl_rng_default;
+  r = gsl_rng_alloc (T);
+
+  for (x = 0.1; x < 2; x+= 0.1)
+    {
+      double y0 = exp (x);
+      double sigma = 0.1 * y0;
+      double dy = gsl_ran_gaussian (r, sigma);
+
+      printf ("%g %g %g\n", x, y0 + dy, sigma);
+    }
+
+  gsl_rng_free(r);
+
+  return 0;
+}
diff --git a/gsl-1.9/doc/examples/histogram.c b/gsl-1.9/doc/examples/histogram.c
new file mode 100644
index 0000000..a3c5150
--- /dev/null
+++ b/gsl-1.9/doc/examples/histogram.c
@@ -0,0 +1,37 @@
+#include <stdio.h>
+#include <stdlib.h>
+#include <gsl/gsl_histogram.h>
+
+int
+main (int argc, char **argv)
+{
+  double a, b;
+  size_t n;
+
+  if (argc != 4)
+    {
+      printf ("Usage: gsl-histogram xmin xmax n\n"
+              "Computes a histogram of the data "
+              "on stdin using n bins from xmin "
+              "to xmax\n");
+      exit (0);
+    }
+
+  a = atof (argv[1]);
+  b = atof (argv[2]);
+  n = atoi (argv[3]);
+
+  {
+    double x;
+    gsl_histogram * h = gsl_histogram_alloc (n);
+    gsl_histogram_set_ranges_uniform (h, a, b);
+
+    while (fscanf (stdin, "%lg", &x) == 1)
+      {
+        gsl_histogram_increment (h, x);
+      }
+    gsl_histogram_fprintf (stdout, h, "%g", "%g");
+    gsl_histogram_free (h);
+  }
+  exit (0);
+}
diff --git a/gsl-1.9/doc/examples/histogram2d.c b/gsl-1.9/doc/examples/histogram2d.c
new file mode 100644
index 0000000..3fcabb4
--- /dev/null
+++ b/gsl-1.9/doc/examples/histogram2d.c
@@ -0,0 +1,50 @@
+#include <stdio.h>
+#include <gsl/gsl_rng.h>
+#include <gsl/gsl_histogram2d.h>
+
+int
+main (void)
+{
+  const gsl_rng_type * T;
+  gsl_rng * r;
+
+  gsl_histogram2d * h = gsl_histogram2d_alloc (10, 10);
+
+  gsl_histogram2d_set_ranges_uniform (h, 
+                                      0.0, 1.0,
+                                      0.0, 1.0);
+
+  gsl_histogram2d_accumulate (h, 0.3, 0.3, 1);
+  gsl_histogram2d_accumulate (h, 0.8, 0.1, 5);
+  gsl_histogram2d_accumulate (h, 0.7, 0.9, 0.5);
+
+  gsl_rng_env_setup ();
+  
+  T = gsl_rng_default;
+  r = gsl_rng_alloc (T);
+
+  {
+    int i;
+    gsl_histogram2d_pdf * p 
+      = gsl_histogram2d_pdf_alloc (h->nx, h->ny);
+    
+    gsl_histogram2d_pdf_init (p, h);
+
+    for (i = 0; i < 1000; i++) {
+      double x, y;
+      double u = gsl_rng_uniform (r);
+      double v = gsl_rng_uniform (r);
+       
+      gsl_histogram2d_pdf_sample (p, u, v, &x, &y);
+      
+      printf ("%g %g\n", x, y);
+    }
+
+    gsl_histogram2d_pdf_free (p);
+  }
+
+  gsl_histogram2d_free (h);
+  gsl_rng_free (r);
+
+  return 0;
+}
diff --git a/gsl-1.9/doc/examples/ieee.c b/gsl-1.9/doc/examples/ieee.c
new file mode 100644
index 0000000..c7f1110
--- /dev/null
+++ b/gsl-1.9/doc/examples/ieee.c
@@ -0,0 +1,22 @@
+#include <stdio.h>
+#include <gsl/gsl_ieee_utils.h>
+
+int
+main (void) 
+{
+  float f = 1.0/3.0;
+  double d = 1.0/3.0;
+
+  double fd = f; /* promote from float to double */
+  
+  printf (" f="); gsl_ieee_printf_float(&f); 
+  printf ("\n");
+
+  printf ("fd="); gsl_ieee_printf_double(&fd); 
+  printf ("\n");
+
+  printf (" d="); gsl_ieee_printf_double(&d); 
+  printf ("\n");
+
+  return 0;
+}
diff --git a/gsl-1.9/doc/examples/ieeeround.c b/gsl-1.9/doc/examples/ieeeround.c
new file mode 100644
index 0000000..04f837d
--- /dev/null
+++ b/gsl-1.9/doc/examples/ieeeround.c
@@ -0,0 +1,30 @@
+#include <stdio.h>
+#include <gsl/gsl_math.h>
+#include <gsl/gsl_ieee_utils.h>
+
+int
+main (void)
+{
+  double x = 1, oldsum = 0, sum = 0; 
+  int i = 0;
+
+  gsl_ieee_env_setup (); /* read GSL_IEEE_MODE */
+
+  do 
+    {
+      i++;
+      
+      oldsum = sum;
+      sum += x;
+      x = x / i;
+      
+      printf ("i=%2d sum=%.18f error=%g\n",
+              i, sum, sum - M_E);
+
+      if (i > 30)
+         break;
+    }  
+  while (sum != oldsum);
+
+  return 0;
+}
diff --git a/gsl-1.9/doc/examples/integration.c b/gsl-1.9/doc/examples/integration.c
new file mode 100644
index 0000000..a8f2450
--- /dev/null
+++ b/gsl-1.9/doc/examples/integration.c
@@ -0,0 +1,37 @@
+#include <stdio.h>
+#include <math.h>
+#include <gsl/gsl_integration.h>
+
+double f (double x, void * params) {
+  double alpha = *(double *) params;
+  double f = log(alpha*x) / sqrt(x);
+  return f;
+}
+
+int
+main (void)
+{
+  gsl_integration_workspace * w 
+    = gsl_integration_workspace_alloc (1000);
+  
+  double result, error;
+  double expected = -4.0;
+  double alpha = 1.0;
+
+  gsl_function F;
+  F.function = &f;
+  F.params = &alpha;
+
+  gsl_integration_qags (&F, 0, 1, 0, 1e-7, 1000,
+                        w, &result, &error); 
+
+  printf ("result          = % .18f\n", result);
+  printf ("exact result    = % .18f\n", expected);
+  printf ("estimated error = % .18f\n", error);
+  printf ("actual error    = % .18f\n", result - expected);
+  printf ("intervals =  %d\n", w->size);
+
+  gsl_integration_workspace_free (w);
+
+  return 0;
+}
diff --git a/gsl-1.9/doc/examples/integration.out b/gsl-1.9/doc/examples/integration.out
new file mode 100644
index 0000000..4758ebf
--- /dev/null
+++ b/gsl-1.9/doc/examples/integration.out
@@ -0,0 +1,5 @@
+result          = -3.999999999999973799
+exact result    = -4.000000000000000000
+estimated error =  0.000000000000246025
+actual error    =  0.000000000000026201
+intervals =  8
diff --git a/gsl-1.9/doc/examples/interp.c b/gsl-1.9/doc/examples/interp.c
new file mode 100644
index 0000000..5c6e8fc
--- /dev/null
+++ b/gsl-1.9/doc/examples/interp.c
@@ -0,0 +1,41 @@
+#include <stdlib.h>
+#include <stdio.h>
+#include <math.h>
+#include <gsl/gsl_errno.h>
+#include <gsl/gsl_spline.h>
+
+int
+main (void)
+{
+  int i;
+  double xi, yi, x[10], y[10];
+
+  printf ("#m=0,S=2\n");
+
+  for (i = 0; i < 10; i++)
+    {
+      x[i] = i + 0.5 * sin (i);
+      y[i] = i + cos (i * i);
+      printf ("%g %g\n", x[i], y[i]);
+    }
+
+  printf ("#m=1,S=0\n");
+
+  {
+    gsl_interp_accel *acc 
+      = gsl_interp_accel_alloc ();
+    gsl_spline *spline 
+      = gsl_spline_alloc (gsl_interp_cspline, 10);
+
+    gsl_spline_init (spline, x, y, 10);
+
+    for (xi = x[0]; xi < x[9]; xi += 0.01)
+      {
+        yi = gsl_spline_eval (spline, xi, acc);
+        printf ("%g %g\n", xi, yi);
+      }
+    gsl_spline_free (spline);
+    gsl_interp_accel_free (acc);
+  }
+  return 0;
+}
diff --git a/gsl-1.9/doc/examples/interpp.c b/gsl-1.9/doc/examples/interpp.c
new file mode 100644
index 0000000..8ffdbcc
--- /dev/null
+++ b/gsl-1.9/doc/examples/interpp.c
@@ -0,0 +1,40 @@
+#include <stdlib.h>
+#include <stdio.h>
+#include <math.h>
+#include <gsl/gsl_errno.h>
+#include <gsl/gsl_spline.h>
+
+int
+main (void)
+{
+  int N = 4;
+  double x[4] = {0.00, 0.10,  0.27,  0.30};
+  double y[4] = {0.15, 0.70, -0.10,  0.15}; /* Note: first = last 
+                                               for periodic data */
+
+  gsl_interp_accel *acc = gsl_interp_accel_alloc ();
+  const gsl_interp_type *t = gsl_interp_cspline_periodic; 
+  gsl_spline *spline = gsl_spline_alloc (t, N);
+
+  int i; double xi, yi;
+
+  printf ("#m=0,S=5\n");
+  for (i = 0; i < N; i++)
+    {
+      printf ("%g %g\n", x[i], y[i]);
+    }
+
+  printf ("#m=1,S=0\n");
+  gsl_spline_init (spline, x, y, N);
+
+  for (i = 0; i <= 100; i++)
+    {
+      xi = (1 - i / 100.0) * x[0] + (i / 100.0) * x[N-1];
+      yi = gsl_spline_eval (spline, xi, acc);
+      printf ("%g %g\n", xi, yi);
+    }
+  
+  gsl_spline_free (spline);
+  gsl_interp_accel_free (acc);
+  return 0;
+}
diff --git a/gsl-1.9/doc/examples/intro.c b/gsl-1.9/doc/examples/intro.c
new file mode 100644
index 0000000..bd28482
--- /dev/null
+++ b/gsl-1.9/doc/examples/intro.c
@@ -0,0 +1,11 @@
+#include <stdio.h>
+#include <gsl/gsl_sf_bessel.h>
+
+int
+main (void)
+{
+  double x = 5.0;
+  double y = gsl_sf_bessel_J0 (x);
+  printf ("J0(%g) = %.18e\n", x, y);
+  return 0;
+}
diff --git a/gsl-1.9/doc/examples/intro.out b/gsl-1.9/doc/examples/intro.out
new file mode 100644
index 0000000..ca264dd
--- /dev/null
+++ b/gsl-1.9/doc/examples/intro.out
@@ -0,0 +1 @@
+J0(5) = -1.775967713143382920e-01
diff --git a/gsl-1.9/doc/examples/linalglu.c b/gsl-1.9/doc/examples/linalglu.c
new file mode 100644
index 0000000..61b5293
--- /dev/null
+++ b/gsl-1.9/doc/examples/linalglu.c
@@ -0,0 +1,36 @@
+#include <stdio.h>
+#include <gsl/gsl_linalg.h>
+
+int
+main (void)
+{
+  double a_data[] = { 0.18, 0.60, 0.57, 0.96,
+                      0.41, 0.24, 0.99, 0.58,
+                      0.14, 0.30, 0.97, 0.66,
+                      0.51, 0.13, 0.19, 0.85 };
+
+  double b_data[] = { 1.0, 2.0, 3.0, 4.0 };
+
+  gsl_matrix_view m 
+    = gsl_matrix_view_array (a_data, 4, 4);
+
+  gsl_vector_view b
+    = gsl_vector_view_array (b_data, 4);
+
+  gsl_vector *x = gsl_vector_alloc (4);
+  
+  int s;
+
+  gsl_permutation * p = gsl_permutation_alloc (4);
+
+  gsl_linalg_LU_decomp (&m.matrix, p, &s);
+
+  gsl_linalg_LU_solve (&m.matrix, p, &b.vector, x);
+
+  printf ("x = \n");
+  gsl_vector_fprintf (stdout, x, "%g");
+
+  gsl_permutation_free (p);
+  gsl_vector_free (x);
+  return 0;
+}
diff --git a/gsl-1.9/doc/examples/linalglu.out b/gsl-1.9/doc/examples/linalglu.out
new file mode 100644
index 0000000..406f59a
--- /dev/null
+++ b/gsl-1.9/doc/examples/linalglu.out
@@ -0,0 +1,4 @@
+x = -4.05205
+-12.6056
+1.66091
+8.69377
diff --git a/gsl-1.9/doc/examples/matrix.c b/gsl-1.9/doc/examples/matrix.c
new file mode 100644
index 0000000..7704687
--- /dev/null
+++ b/gsl-1.9/doc/examples/matrix.c
@@ -0,0 +1,22 @@
+#include <stdio.h>
+#include <gsl/gsl_matrix.h>
+
+int
+main (void)
+{
+  int i, j; 
+  gsl_matrix * m = gsl_matrix_alloc (10, 3);
+  
+  for (i = 0; i < 10; i++)
+    for (j = 0; j < 3; j++)
+      gsl_matrix_set (m, i, j, 0.23 + 100*i + j);
+  
+  for (i = 0; i < 100; i++)  /* OUT OF RANGE ERROR */
+    for (j = 0; j < 3; j++)
+      printf ("m(%d,%d) = %g\n", i, j, 
+              gsl_matrix_get (m, i, j));
+
+  gsl_matrix_free (m);
+
+  return 0;
+}
diff --git a/gsl-1.9/doc/examples/matrixw.c b/gsl-1.9/doc/examples/matrixw.c
new file mode 100644
index 0000000..b05ffbd
--- /dev/null
+++ b/gsl-1.9/doc/examples/matrixw.c
@@ -0,0 +1,40 @@
+#include <stdio.h>
+#include <gsl/gsl_matrix.h>
+
+int
+main (void)
+{
+  int i, j, k = 0; 
+  gsl_matrix * m = gsl_matrix_alloc (100, 100);
+  gsl_matrix * a = gsl_matrix_alloc (100, 100);
+  
+  for (i = 0; i < 100; i++)
+    for (j = 0; j < 100; j++)
+      gsl_matrix_set (m, i, j, 0.23 + i + j);
+
+  {  
+     FILE * f = fopen ("test.dat", "wb");
+     gsl_matrix_fwrite (f, m);
+     fclose (f);
+  }
+
+  {  
+     FILE * f = fopen ("test.dat", "rb");
+     gsl_matrix_fread (f, a);
+     fclose (f);
+  }
+
+  for (i = 0; i < 100; i++)
+    for (j = 0; j < 100; j++)
+      {
+        double mij = gsl_matrix_get (m, i, j);
+        double aij = gsl_matrix_get (a, i, j);
+        if (mij != aij) k++;
+      }
+
+  gsl_matrix_free (m);
+  gsl_matrix_free (a);
+
+  printf ("differences = %d (should be zero)\n", k);
+  return (k > 0);
+}
diff --git a/gsl-1.9/doc/examples/min.c b/gsl-1.9/doc/examples/min.c
new file mode 100644
index 0000000..ebba625
--- /dev/null
+++ b/gsl-1.9/doc/examples/min.c
@@ -0,0 +1,65 @@
+#include <stdio.h>
+#include <gsl/gsl_errno.h>
+#include <gsl/gsl_math.h>
+#include <gsl/gsl_min.h>
+
+double fn1 (double x, void * params)
+{
+  return cos(x) + 1.0;
+}
+
+int
+main (void)
+{
+  int status;
+  int iter = 0, max_iter = 100;
+  const gsl_min_fminimizer_type *T;
+  gsl_min_fminimizer *s;
+  double m = 2.0, m_expected = M_PI;
+  double a = 0.0, b = 6.0;
+  gsl_function F;
+
+  F.function = &fn1;
+  F.params = 0;
+
+  T = gsl_min_fminimizer_brent;
+  s = gsl_min_fminimizer_alloc (T);
+  gsl_min_fminimizer_set (s, &F, m, a, b);
+
+  printf ("using %s method\n",
+          gsl_min_fminimizer_name (s));
+
+  printf ("%5s [%9s, %9s] %9s %10s %9s\n",
+          "iter", "lower", "upper", "min",
+          "err", "err(est)");
+
+  printf ("%5d [%.7f, %.7f] %.7f %+.7f %.7f\n",
+          iter, a, b,
+          m, m - m_expected, b - a);
+
+  do
+    {
+      iter++;
+      status = gsl_min_fminimizer_iterate (s);
+
+      m = gsl_min_fminimizer_x_minimum (s);
+      a = gsl_min_fminimizer_x_lower (s);
+      b = gsl_min_fminimizer_x_upper (s);
+
+      status 
+        = gsl_min_test_interval (a, b, 0.001, 0.0);
+
+      if (status == GSL_SUCCESS)
+        printf ("Converged:\n");
+
+      printf ("%5d [%.7f, %.7f] "
+              "%.7f %.7f %+.7f %.7f\n",
+              iter, a, b,
+              m, m_expected, m - m_expected, b - a);
+    }
+  while (status == GSL_CONTINUE && iter < max_iter);
+
+  gsl_min_fminimizer_free (s);
+
+  return status;
+}
diff --git a/gsl-1.9/doc/examples/min.out b/gsl-1.9/doc/examples/min.out
new file mode 100644
index 0000000..93d02fc
--- /dev/null
+++ b/gsl-1.9/doc/examples/min.out
@@ -0,0 +1,13 @@
+    0 [0.0000000, 6.0000000] 2.0000000 -1.1415927 6.0000000
+    1 [2.0000000, 6.0000000] 3.2758640 +0.1342713 4.0000000
+    2 [2.0000000, 3.2831929] 3.2758640 +0.1342713 1.2831929
+    3 [2.8689068, 3.2831929] 3.2758640 +0.1342713 0.4142862
+    4 [2.8689068, 3.2831929] 3.2758640 +0.1342713 0.4142862
+    5 [2.8689068, 3.2758640] 3.1460585 +0.0044658 0.4069572
+    6 [3.1346075, 3.2758640] 3.1460585 +0.0044658 0.1412565
+    7 [3.1346075, 3.1874620] 3.1460585 +0.0044658 0.0528545
+    8 [3.1346075, 3.1460585] 3.1460585 +0.0044658 0.0114510
+    9 [3.1346075, 3.1460585] 3.1424060 +0.0008133 0.0114510
+   10 [3.1346075, 3.1424060] 3.1415885 -0.0000041 0.0077985
+Converged:                            
+   11 [3.1415885, 3.1424060] 3.1415927 -0.0000000 0.0008175
diff --git a/gsl-1.9/doc/examples/monte.c b/gsl-1.9/doc/examples/monte.c
new file mode 100644
index 0000000..2c23747
--- /dev/null
+++ b/gsl-1.9/doc/examples/monte.c
@@ -0,0 +1,106 @@
+#include <stdlib.h>
+#include <gsl/gsl_math.h>
+#include <gsl/gsl_monte.h>
+#include <gsl/gsl_monte_plain.h>
+#include <gsl/gsl_monte_miser.h>
+#include <gsl/gsl_monte_vegas.h>
+
+/* Computation of the integral,
+
+      I = int (dx dy dz)/(2pi)^3  1/(1-cos(x)cos(y)cos(z))
+
+   over (-pi,-pi,-pi) to (+pi, +pi, +pi).  The exact answer
+   is Gamma(1/4)^4/(4 pi^3).  This example is taken from
+   C.Itzykson, J.M.Drouffe, "Statistical Field Theory -
+   Volume 1", Section 1.1, p21, which cites the original
+   paper M.L.Glasser, I.J.Zucker, Proc.Natl.Acad.Sci.USA 74
+   1800 (1977) */
+
+/* For simplicity we compute the integral over the region 
+   (0,0,0) -> (pi,pi,pi) and multiply by 8 */
+
+double exact = 1.3932039296856768591842462603255;
+
+double
+g (double *k, size_t dim, void *params)
+{
+  double A = 1.0 / (M_PI * M_PI * M_PI);
+  return A / (1.0 - cos (k[0]) * cos (k[1]) * cos (k[2]));
+}
+
+void
+display_results (char *title, double result, double error)
+{
+  printf ("%s ==================\n", title);
+  printf ("result = % .6f\n", result);
+  printf ("sigma  = % .6f\n", error);
+  printf ("exact  = % .6f\n", exact);
+  printf ("error  = % .6f = %.1g sigma\n", result - exact,
+          fabs (result - exact) / error);
+}
+
+int
+main (void)
+{
+  double res, err;
+
+  double xl[3] = { 0, 0, 0 };
+  double xu[3] = { M_PI, M_PI, M_PI };
+
+  const gsl_rng_type *T;
+  gsl_rng *r;
+
+  gsl_monte_function G = { &g, 3, 0 };
+
+  size_t calls = 500000;
+
+  gsl_rng_env_setup ();
+
+  T = gsl_rng_default;
+  r = gsl_rng_alloc (T);
+
+  {
+    gsl_monte_plain_state *s = gsl_monte_plain_alloc (3);
+    gsl_monte_plain_integrate (&G, xl, xu, 3, calls, r, s, 
+                               &res, &err);
+    gsl_monte_plain_free (s);
+
+    display_results ("plain", res, err);
+  }
+
+  {
+    gsl_monte_miser_state *s = gsl_monte_miser_alloc (3);
+    gsl_monte_miser_integrate (&G, xl, xu, 3, calls, r, s,
+                               &res, &err);
+    gsl_monte_miser_free (s);
+
+    display_results ("miser", res, err);
+  }
+
+  {
+    gsl_monte_vegas_state *s = gsl_monte_vegas_alloc (3);
+
+    gsl_monte_vegas_integrate (&G, xl, xu, 3, 10000, r, s,
+                               &res, &err);
+    display_results ("vegas warm-up", res, err);
+
+    printf ("converging...\n");
+
+    do
+      {
+        gsl_monte_vegas_integrate (&G, xl, xu, 3, calls/5, r, s,
+                                   &res, &err);
+        printf ("result = % .6f sigma = % .6f "
+                "chisq/dof = %.1f\n", res, err, s->chisq);
+      }
+    while (fabs (s->chisq - 1.0) > 0.5);
+
+    display_results ("vegas final", res, err);
+
+    gsl_monte_vegas_free (s);
+  }
+
+  gsl_rng_free (r);
+
+  return 0;
+}
diff --git a/gsl-1.9/doc/examples/nlfit.c b/gsl-1.9/doc/examples/nlfit.c
new file mode 100644
index 0000000..49b4137
--- /dev/null
+++ b/gsl-1.9/doc/examples/nlfit.c
@@ -0,0 +1,115 @@
+#include <stdlib.h>
+#include <stdio.h>
+#include <gsl/gsl_rng.h>
+#include <gsl/gsl_randist.h>
+#include <gsl/gsl_vector.h>
+#include <gsl/gsl_blas.h>
+#include <gsl/gsl_multifit_nlin.h>
+
+#include "expfit.c"
+
+#define N 40
+
+void print_state (size_t iter, gsl_multifit_fdfsolver * s);
+
+int
+main (void)
+{
+  const gsl_multifit_fdfsolver_type *T;
+  gsl_multifit_fdfsolver *s;
+  int status;
+  unsigned int i, iter = 0;
+  const size_t n = N;
+  const size_t p = 3;
+
+  gsl_matrix *covar = gsl_matrix_alloc (p, p);
+  double y[N], sigma[N];
+  struct data d = { n, y, sigma};
+  gsl_multifit_function_fdf f;
+  double x_init[3] = { 1.0, 0.0, 0.0 };
+  gsl_vector_view x = gsl_vector_view_array (x_init, p);
+  const gsl_rng_type * type;
+  gsl_rng * r;
+
+  gsl_rng_env_setup();
+
+  type = gsl_rng_default;
+  r = gsl_rng_alloc (type);
+
+  f.f = &expb_f;
+  f.df = &expb_df;
+  f.fdf = &expb_fdf;
+  f.n = n;
+  f.p = p;
+  f.params = &d;
+
+  /* This is the data to be fitted */
+
+  for (i = 0; i < n; i++)
+    {
+      double t = i;
+      y[i] = 1.0 + 5 * exp (-0.1 * t) 
+                 + gsl_ran_gaussian (r, 0.1);
+      sigma[i] = 0.1;
+      printf ("data: %u %g %g\n", i, y[i], sigma[i]);
+    };
+
+  T = gsl_multifit_fdfsolver_lmsder;
+  s = gsl_multifit_fdfsolver_alloc (T, n, p);
+  gsl_multifit_fdfsolver_set (s, &f, &x.vector);
+
+  print_state (iter, s);
+
+  do
+    {
+      iter++;
+      status = gsl_multifit_fdfsolver_iterate (s);
+
+      printf ("status = %s\n", gsl_strerror (status));
+
+      print_state (iter, s);
+
+      if (status)
+        break;
+
+      status = gsl_multifit_test_delta (s->dx, s->x,
+                                        1e-4, 1e-4);
+    }
+  while (status == GSL_CONTINUE && iter < 500);
+
+  gsl_multifit_covar (s->J, 0.0, covar);
+
+#define FIT(i) gsl_vector_get(s->x, i)
+#define ERR(i) sqrt(gsl_matrix_get(covar,i,i))
+
+  { 
+    double chi = gsl_blas_dnrm2(s->f);
+    double dof = n - p;
+    double c = GSL_MAX_DBL(1, chi / sqrt(dof)); 
+
+    printf("chisq/dof = %g\n",  pow(chi, 2.0) / dof);
+
+    printf ("A      = %.5f +/- %.5f\n", FIT(0), c*ERR(0));
+    printf ("lambda = %.5f +/- %.5f\n", FIT(1), c*ERR(1));
+    printf ("b      = %.5f +/- %.5f\n", FIT(2), c*ERR(2));
+  }
+
+  printf ("status = %s\n", gsl_strerror (status));
+
+  gsl_multifit_fdfsolver_free (s);
+  gsl_matrix_free (covar);
+  gsl_rng_free (r);
+  return 0;
+}
+
+void
+print_state (size_t iter, gsl_multifit_fdfsolver * s)
+{
+  printf ("iter: %3u x = % 15.8f % 15.8f % 15.8f "
+          "|f(x)| = %g\n",
+          iter,
+          gsl_vector_get (s->x, 0), 
+          gsl_vector_get (s->x, 1),
+          gsl_vector_get (s->x, 2), 
+          gsl_blas_dnrm2 (s->f));
+}
diff --git a/gsl-1.9/doc/examples/ntupler.c b/gsl-1.9/doc/examples/ntupler.c
new file mode 100644
index 0000000..a926522
--- /dev/null
+++ b/gsl-1.9/doc/examples/ntupler.c
@@ -0,0 +1,72 @@
+#include <math.h>
+#include <gsl/gsl_ntuple.h>
+#include <gsl/gsl_histogram.h>
+
+struct data
+{
+  double x;
+  double y;
+  double z;
+};
+
+int sel_func (void *ntuple_data, void *params);
+double val_func (void *ntuple_data, void *params);
+
+int
+main (void)
+{
+  struct data ntuple_row;
+
+  gsl_ntuple *ntuple 
+    = gsl_ntuple_open ("test.dat", &ntuple_row,
+                       sizeof (ntuple_row));
+  double lower = 1.5;
+
+  gsl_ntuple_select_fn S;
+  gsl_ntuple_value_fn V;
+
+  gsl_histogram *h = gsl_histogram_alloc (100);
+  gsl_histogram_set_ranges_uniform(h, 0.0, 10.0);
+
+  S.function = &sel_func;
+  S.params = &lower;
+
+  V.function = &val_func;
+  V.params = 0;
+
+  gsl_ntuple_project (h, ntuple, &V, &S);
+  gsl_histogram_fprintf (stdout, h, "%f", "%f");
+  gsl_histogram_free (h);
+  gsl_ntuple_close (ntuple);
+
+  return 0;
+}
+
+int
+sel_func (void *ntuple_data, void *params)
+{
+  struct data * data = (struct data *) ntuple_data;  
+  double x, y, z, E2, scale;
+  scale = *(double *) params;
+  
+  x = data->x;
+  y = data->y;
+  z = data->z;
+
+  E2 = x * x + y * y + z * z;
+
+  return E2 > scale;
+}
+
+double
+val_func (void *ntuple_data, void *params)
+{
+  struct data * data = (struct data *) ntuple_data;  
+  double x, y, z;
+
+  x = data->x;
+  y = data->y;
+  z = data->z;
+
+  return x * x + y * y + z * z;
+}
diff --git a/gsl-1.9/doc/examples/ntuplew.c b/gsl-1.9/doc/examples/ntuplew.c
new file mode 100644
index 0000000..82f6fa4
--- /dev/null
+++ b/gsl-1.9/doc/examples/ntuplew.c
@@ -0,0 +1,43 @@
+#include <gsl/gsl_ntuple.h>
+#include <gsl/gsl_rng.h>
+#include <gsl/gsl_randist.h>
+
+struct data
+{
+  double x;
+  double y;
+  double z;
+};
+
+int
+main (void)
+{
+  const gsl_rng_type * T;
+  gsl_rng * r;
+
+  struct data ntuple_row;
+  int i;
+
+  gsl_ntuple *ntuple 
+    = gsl_ntuple_create ("test.dat", &ntuple_row, 
+                         sizeof (ntuple_row));
+
+  gsl_rng_env_setup ();
+
+  T = gsl_rng_default; 
+  r = gsl_rng_alloc (T);
+
+  for (i = 0; i < 10000; i++)
+    {
+      ntuple_row.x = gsl_ran_ugaussian (r);
+      ntuple_row.y = gsl_ran_ugaussian (r);
+      ntuple_row.z = gsl_ran_ugaussian (r);
+      
+      gsl_ntuple_write (ntuple);
+    }
+  
+  gsl_ntuple_close (ntuple);
+  gsl_rng_free (r);
+
+  return 0;
+}
diff --git a/gsl-1.9/doc/examples/ode-initval.c b/gsl-1.9/doc/examples/ode-initval.c
new file mode 100644
index 0000000..8a8f794
--- /dev/null
+++ b/gsl-1.9/doc/examples/ode-initval.c
@@ -0,0 +1,70 @@
+#include <stdio.h>
+#include <gsl/gsl_errno.h>
+#include <gsl/gsl_matrix.h>
+#include <gsl/gsl_odeiv.h>
+
+int
+func (double t, const double y[], double f[],
+      void *params)
+{
+  double mu = *(double *)params;
+  f[0] = y[1];
+  f[1] = -y[0] - mu*y[1]*(y[0]*y[0] - 1);
+  return GSL_SUCCESS;
+}
+
+int
+jac (double t, const double y[], double *dfdy, 
+     double dfdt[], void *params)
+{
+  double mu = *(double *)params;
+  gsl_matrix_view dfdy_mat 
+    = gsl_matrix_view_array (dfdy, 2, 2);
+  gsl_matrix * m = &dfdy_mat.matrix; 
+  gsl_matrix_set (m, 0, 0, 0.0);
+  gsl_matrix_set (m, 0, 1, 1.0);
+  gsl_matrix_set (m, 1, 0, -2.0*mu*y[0]*y[1] - 1.0);
+  gsl_matrix_set (m, 1, 1, -mu*(y[0]*y[0] - 1.0));
+  dfdt[0] = 0.0;
+  dfdt[1] = 0.0;
+  return GSL_SUCCESS;
+}
+
+int
+main (void)
+{
+  const gsl_odeiv_step_type * T 
+    = gsl_odeiv_step_rk8pd;
+
+  gsl_odeiv_step * s 
+    = gsl_odeiv_step_alloc (T, 2);
+  gsl_odeiv_control * c 
+    = gsl_odeiv_control_y_new (1e-6, 0.0);
+  gsl_odeiv_evolve * e 
+    = gsl_odeiv_evolve_alloc (2);
+
+  double mu = 10;
+  gsl_odeiv_system sys = {func, jac, 2, &mu};
+
+  double t = 0.0, t1 = 100.0;
+  double h = 1e-6;
+  double y[2] = { 1.0, 0.0 };
+
+  while (t < t1)
+    {
+      int status = gsl_odeiv_evolve_apply (e, c, s,
+                                           &sys, 
+                                           &t, t1,
+                                           &h, y);
+
+      if (status != GSL_SUCCESS)
+          break;
+
+      printf ("%.5e %.5e %.5e\n", t, y[0], y[1]);
+    }
+
+  gsl_odeiv_evolve_free (e);
+  gsl_odeiv_control_free (c);
+  gsl_odeiv_step_free (s);
+  return 0;
+}
diff --git a/gsl-1.9/doc/examples/odefixed.c b/gsl-1.9/doc/examples/odefixed.c
new file mode 100644
index 0000000..3c81bf3
--- /dev/null
+++ b/gsl-1.9/doc/examples/odefixed.c
@@ -0,0 +1,42 @@
+int
+main (void)
+{
+  const gsl_odeiv_step_type * T 
+    = gsl_odeiv_step_rk4;
+
+  gsl_odeiv_step * s 
+    = gsl_odeiv_step_alloc (T, 2);
+
+  double mu = 10;
+  gsl_odeiv_system sys = {func, jac, 2, &mu};
+
+  double t = 0.0, t1 = 100.0;
+  double h = 1e-2;
+  double y[2] = { 1.0, 0.0 }, y_err[2];
+  double dydt_in[2], dydt_out[2];
+
+  /* initialise dydt_in from system parameters */
+  GSL_ODEIV_FN_EVAL(&sys, t, y, dydt_in);
+
+  while (t < t1)
+    {
+      int status = gsl_odeiv_step_apply (s, t, h, 
+                                         y, y_err, 
+                                         dydt_in, 
+                                         dydt_out, 
+                                         &sys);
+
+      if (status != GSL_SUCCESS)
+          break;
+
+      dydt_in[0] = dydt_out[0];
+      dydt_in[1] = dydt_out[1];
+
+      t += h;
+
+      printf ("%.5e %.5e %.5e\n", t, y[0], y[1]);
+    }
+
+  gsl_odeiv_step_free (s);
+  return 0;
+}
diff --git a/gsl-1.9/doc/examples/permseq.c b/gsl-1.9/doc/examples/permseq.c
new file mode 100644
index 0000000..08ebf63
--- /dev/null
+++ b/gsl-1.9/doc/examples/permseq.c
@@ -0,0 +1,21 @@
+#include <stdio.h>
+#include <gsl/gsl_permutation.h>
+
+int
+main (void) 
+{
+  gsl_permutation * p = gsl_permutation_alloc (3);
+
+  gsl_permutation_init (p);
+
+  do 
+   {
+      gsl_permutation_fprintf (stdout, p, " %u");
+      printf ("\n");
+   }
+  while (gsl_permutation_next(p) == GSL_SUCCESS);
+
+  gsl_permutation_free (p);
+
+  return 0;
+}
diff --git a/gsl-1.9/doc/examples/permshuffle.c b/gsl-1.9/doc/examples/permshuffle.c
new file mode 100644
index 0000000..7247864
--- /dev/null
+++ b/gsl-1.9/doc/examples/permshuffle.c
@@ -0,0 +1,40 @@
+#include <stdio.h>
+#include <gsl/gsl_rng.h>
+#include <gsl/gsl_randist.h>
+#include <gsl/gsl_permutation.h>
+
+int
+main (void) 
+{
+  const size_t N = 10;
+  const gsl_rng_type * T;
+  gsl_rng * r;
+
+  gsl_permutation * p = gsl_permutation_alloc (N);
+  gsl_permutation * q = gsl_permutation_alloc (N);
+
+  gsl_rng_env_setup();
+  T = gsl_rng_default;
+  r = gsl_rng_alloc (T);
+
+  printf ("initial permutation:");  
+  gsl_permutation_init (p);
+  gsl_permutation_fprintf (stdout, p, " %u");
+  printf ("\n");
+
+  printf (" random permutation:");  
+  gsl_ran_shuffle (r, p->data, N, sizeof(size_t));
+  gsl_permutation_fprintf (stdout, p, " %u");
+  printf ("\n");
+
+  printf ("inverse permutation:");  
+  gsl_permutation_inverse (q, p);
+  gsl_permutation_fprintf (stdout, q, " %u");
+  printf ("\n");
+
+  gsl_permutation_free (p);
+  gsl_permutation_free (q);
+  gsl_rng_free (r);
+
+  return 0;
+}
diff --git a/gsl-1.9/doc/examples/polyroots.c b/gsl-1.9/doc/examples/polyroots.c
new file mode 100644
index 0000000..6debbd5
--- /dev/null
+++ b/gsl-1.9/doc/examples/polyroots.c
@@ -0,0 +1,26 @@
+#include <stdio.h>
+#include <gsl/gsl_poly.h>
+
+int
+main (void)
+{
+  int i;
+  /* coefficients of P(x) =  -1 + x^5  */
+  double a[6] = { -1, 0, 0, 0, 0, 1 };  
+  double z[10];
+
+  gsl_poly_complex_workspace * w 
+      = gsl_poly_complex_workspace_alloc (6);
+  
+  gsl_poly_complex_solve (a, 6, w, z);
+
+  gsl_poly_complex_workspace_free (w);
+
+  for (i = 0; i < 5; i++)
+    {
+      printf ("z%d = %+.18f %+.18f\n", 
+              i, z[2*i], z[2*i+1]);
+    }
+
+  return 0;
+}
diff --git a/gsl-1.9/doc/examples/polyroots.out b/gsl-1.9/doc/examples/polyroots.out
new file mode 100644
index 0000000..253fe11
--- /dev/null
+++ b/gsl-1.9/doc/examples/polyroots.out
@@ -0,0 +1,5 @@
+z0 = -0.809016994374947451 +0.587785252292473137
+z1 = -0.809016994374947451 -0.587785252292473137
+z2 = +0.309016994374947451 +0.951056516295153642
+z3 = +0.309016994374947451 -0.951056516295153642
+z4 = +1.000000000000000000 +0.000000000000000000
diff --git a/gsl-1.9/doc/examples/qrng.c b/gsl-1.9/doc/examples/qrng.c
new file mode 100644
index 0000000..5275209
--- /dev/null
+++ b/gsl-1.9/doc/examples/qrng.c
@@ -0,0 +1,19 @@
+#include <stdio.h>
+#include <gsl/gsl_qrng.h>
+
+int
+main (void)
+{
+  int i;
+  gsl_qrng * q = gsl_qrng_alloc (gsl_qrng_sobol, 2);
+
+  for (i = 0; i < 1024; i++)
+    {
+      double v[2];
+      gsl_qrng_get (q, v);
+      printf ("%.5f %.5f\n", v[0], v[1]);
+    }
+
+  gsl_qrng_free (q);
+  return 0;
+}
diff --git a/gsl-1.9/doc/examples/randpoisson.2.out b/gsl-1.9/doc/examples/randpoisson.2.out
new file mode 100644
index 0000000..fbe36a2
--- /dev/null
+++ b/gsl-1.9/doc/examples/randpoisson.2.out
@@ -0,0 +1,2 @@
+GSL_RNG_SEED=123
+ 4 5 6 3 3 1 4 2 5 5
diff --git a/gsl-1.9/doc/examples/randpoisson.c b/gsl-1.9/doc/examples/randpoisson.c
new file mode 100644
index 0000000..ef69b72
--- /dev/null
+++ b/gsl-1.9/doc/examples/randpoisson.c
@@ -0,0 +1,35 @@
+#include <stdio.h>
+#include <gsl/gsl_rng.h>
+#include <gsl/gsl_randist.h>
+
+int
+main (void)
+{
+  const gsl_rng_type * T;
+  gsl_rng * r;
+
+  int i, n = 10;
+  double mu = 3.0;
+
+  /* create a generator chosen by the 
+     environment variable GSL_RNG_TYPE */
+
+  gsl_rng_env_setup();
+
+  T = gsl_rng_default;
+  r = gsl_rng_alloc (T);
+
+  /* print n random variates chosen from 
+     the poisson distribution with mean 
+     parameter mu */
+
+  for (i = 0; i < n; i++) 
+    {
+      unsigned int k = gsl_ran_poisson (r, mu);
+      printf (" %u", k);
+    }
+
+  printf ("\n");
+  gsl_rng_free (r);
+  return 0;
+}
diff --git a/gsl-1.9/doc/examples/randpoisson.out b/gsl-1.9/doc/examples/randpoisson.out
new file mode 100644
index 0000000..1639cca
--- /dev/null
+++ b/gsl-1.9/doc/examples/randpoisson.out
@@ -0,0 +1 @@
+ 2 5 5 2 1 0 3 4 1 1
diff --git a/gsl-1.9/doc/examples/randwalk.c b/gsl-1.9/doc/examples/randwalk.c
new file mode 100644
index 0000000..588aa50
--- /dev/null
+++ b/gsl-1.9/doc/examples/randwalk.c
@@ -0,0 +1,29 @@
+#include <stdio.h>
+#include <gsl/gsl_rng.h>
+#include <gsl/gsl_randist.h>
+
+int
+main (void)
+{
+  int i;
+  double x = 0, y = 0, dx, dy;
+
+  const gsl_rng_type * T;
+  gsl_rng * r;
+
+  gsl_rng_env_setup();
+  T = gsl_rng_default;
+  r = gsl_rng_alloc (T);
+
+  printf ("%g %g\n", x, y);
+
+  for (i = 0; i < 10; i++)
+    {
+      gsl_ran_dir_2d (r, &dx, &dy);
+      x += dx; y += dy; 
+      printf ("%g %g\n", x, y);
+    }
+
+  gsl_rng_free (r);
+  return 0;
+}
diff --git a/gsl-1.9/doc/examples/rng.c b/gsl-1.9/doc/examples/rng.c
new file mode 100644
index 0000000..6330264
--- /dev/null
+++ b/gsl-1.9/doc/examples/rng.c
@@ -0,0 +1,22 @@
+#include <stdio.h>
+#include <gsl/gsl_rng.h>
+
+gsl_rng * r;  /* global generator */
+
+int
+main (void)
+{
+  const gsl_rng_type * T;
+
+  gsl_rng_env_setup();
+
+  T = gsl_rng_default;
+  r = gsl_rng_alloc (T);
+  
+  printf ("generator type: %s\n", gsl_rng_name (r));
+  printf ("seed = %lu\n", gsl_rng_default_seed);
+  printf ("first value = %lu\n", gsl_rng_get (r));
+
+  gsl_rng_free (r);
+  return 0;
+}
diff --git a/gsl-1.9/doc/examples/rng.out b/gsl-1.9/doc/examples/rng.out
new file mode 100644
index 0000000..00f33a3
--- /dev/null
+++ b/gsl-1.9/doc/examples/rng.out
@@ -0,0 +1,3 @@
+generator type: mt19937
+seed = 0
+first value = 4293858116
diff --git a/gsl-1.9/doc/examples/rngunif.2.out b/gsl-1.9/doc/examples/rngunif.2.out
new file mode 100644
index 0000000..d04fccd
--- /dev/null
+++ b/gsl-1.9/doc/examples/rngunif.2.out
@@ -0,0 +1,12 @@
+GSL_RNG_TYPE=mrg
+GSL_RNG_SEED=123
+0.33050
+0.86631
+0.32982
+0.67620
+0.53391
+0.06457
+0.16847
+0.70229
+0.04371
+0.86374
diff --git a/gsl-1.9/doc/examples/rngunif.c b/gsl-1.9/doc/examples/rngunif.c
new file mode 100644
index 0000000..ca4b2db
--- /dev/null
+++ b/gsl-1.9/doc/examples/rngunif.c
@@ -0,0 +1,26 @@
+#include <stdio.h>
+#include <gsl/gsl_rng.h>
+
+int
+main (void)
+{
+  const gsl_rng_type * T;
+  gsl_rng * r;
+
+  int i, n = 10;
+
+  gsl_rng_env_setup();
+
+  T = gsl_rng_default;
+  r = gsl_rng_alloc (T);
+
+  for (i = 0; i < n; i++) 
+    {
+      double u = gsl_rng_uniform (r);
+      printf ("%.5f\n", u);
+    }
+
+  gsl_rng_free (r);
+
+  return 0;
+}
diff --git a/gsl-1.9/doc/examples/rngunif.out b/gsl-1.9/doc/examples/rngunif.out
new file mode 100644
index 0000000..b4049ef
--- /dev/null
+++ b/gsl-1.9/doc/examples/rngunif.out
@@ -0,0 +1,10 @@
+0.99974
+0.16291
+0.28262
+0.94720
+0.23166
+0.48497
+0.95748
+0.74431
+0.54004
+0.73995
diff --git a/gsl-1.9/doc/examples/rootnewt.c b/gsl-1.9/doc/examples/rootnewt.c
new file mode 100644
index 0000000..fea5f11
--- /dev/null
+++ b/gsl-1.9/doc/examples/rootnewt.c
@@ -0,0 +1,52 @@
+#include <stdio.h>
+#include <gsl/gsl_errno.h>
+#include <gsl/gsl_math.h>
+#include <gsl/gsl_roots.h>
+
+#include "demo_fn.h"
+#include "demo_fn.c"
+
+int
+main (void)
+{
+  int status;
+  int iter = 0, max_iter = 100;
+  const gsl_root_fdfsolver_type *T;
+  gsl_root_fdfsolver *s;
+  double x0, x = 5.0, r_expected = sqrt (5.0);
+  gsl_function_fdf FDF;
+  struct quadratic_params params = {1.0, 0.0, -5.0};
+
+  FDF.f = &quadratic;
+  FDF.df = &quadratic_deriv;
+  FDF.fdf = &quadratic_fdf;
+  FDF.params = &params;
+
+  T = gsl_root_fdfsolver_newton;
+  s = gsl_root_fdfsolver_alloc (T);
+  gsl_root_fdfsolver_set (s, &FDF, x);
+
+  printf ("using %s method\n", 
+          gsl_root_fdfsolver_name (s));
+
+  printf ("%-5s %10s %10s %10s\n",
+          "iter", "root", "err", "err(est)");
+  do
+    {
+      iter++;
+      status = gsl_root_fdfsolver_iterate (s);
+      x0 = x;
+      x = gsl_root_fdfsolver_root (s);
+      status = gsl_root_test_delta (x, x0, 0, 1e-3);
+
+      if (status == GSL_SUCCESS)
+        printf ("Converged:\n");
+
+      printf ("%5d %10.7f %+10.7f %10.7f\n",
+              iter, x, x - r_expected, x - x0);
+    }
+  while (status == GSL_CONTINUE && iter < max_iter);
+
+  gsl_root_fdfsolver_free (s);
+  return status;
+}
diff --git a/gsl-1.9/doc/examples/roots.c b/gsl-1.9/doc/examples/roots.c
new file mode 100644
index 0000000..7854fbc
--- /dev/null
+++ b/gsl-1.9/doc/examples/roots.c
@@ -0,0 +1,58 @@
+#include <stdio.h>
+#include <gsl/gsl_errno.h>
+#include <gsl/gsl_math.h>
+#include <gsl/gsl_roots.h>
+
+#include "demo_fn.h"
+#include "demo_fn.c"
+
+int
+main (void)
+{
+  int status;
+  int iter = 0, max_iter = 100;
+  const gsl_root_fsolver_type *T;
+  gsl_root_fsolver *s;
+  double r = 0, r_expected = sqrt (5.0);
+  double x_lo = 0.0, x_hi = 5.0;
+  gsl_function F;
+  struct quadratic_params params = {1.0, 0.0, -5.0};
+
+  F.function = &quadratic;
+  F.params = &params;
+
+  T = gsl_root_fsolver_brent;
+  s = gsl_root_fsolver_alloc (T);
+  gsl_root_fsolver_set (s, &F, x_lo, x_hi);
+
+  printf ("using %s method\n", 
+          gsl_root_fsolver_name (s));
+
+  printf ("%5s [%9s, %9s] %9s %10s %9s\n",
+          "iter", "lower", "upper", "root", 
+          "err", "err(est)");
+
+  do
+    {
+      iter++;
+      status = gsl_root_fsolver_iterate (s);
+      r = gsl_root_fsolver_root (s);
+      x_lo = gsl_root_fsolver_x_lower (s);
+      x_hi = gsl_root_fsolver_x_upper (s);
+      status = gsl_root_test_interval (x_lo, x_hi,
+                                       0, 0.001);
+
+      if (status == GSL_SUCCESS)
+        printf ("Converged:\n");
+
+      printf ("%5d [%.7f, %.7f] %.7f %+.7f %.7f\n",
+              iter, x_lo, x_hi,
+              r, r - r_expected, 
+              x_hi - x_lo);
+    }
+  while (status == GSL_CONTINUE && iter < max_iter);
+
+  gsl_root_fsolver_free (s);
+
+  return status;
+}
diff --git a/gsl-1.9/doc/examples/siman.c b/gsl-1.9/doc/examples/siman.c
new file mode 100644
index 0000000..50f08e0
--- /dev/null
+++ b/gsl-1.9/doc/examples/siman.c
@@ -0,0 +1,82 @@
+#include <math.h>
+#include <stdlib.h>
+#include <string.h>
+#include <gsl/gsl_siman.h>
+
+/* set up parameters for this simulated annealing run */
+
+/* how many points do we try before stepping */
+#define N_TRIES 200             
+
+/* how many iterations for each T? */
+#define ITERS_FIXED_T 1000
+
+/* max step size in random walk */
+#define STEP_SIZE 1.0            
+
+/* Boltzmann constant */
+#define K 1.0                   
+
+/* initial temperature */
+#define T_INITIAL 0.008         
+
+/* damping factor for temperature */
+#define MU_T 1.003              
+#define T_MIN 2.0e-6
+
+gsl_siman_params_t params 
+  = {N_TRIES, ITERS_FIXED_T, STEP_SIZE,
+     K, T_INITIAL, MU_T, T_MIN};
+
+/* now some functions to test in one dimension */
+double E1(void *xp)
+{
+  double x = * ((double *) xp);
+
+  return exp(-pow((x-1.0),2.0))*sin(8*x);
+}
+
+double M1(void *xp, void *yp)
+{
+  double x = *((double *) xp);
+  double y = *((double *) yp);
+
+  return fabs(x - y);
+}
+
+void S1(const gsl_rng * r, void *xp, double step_size)
+{
+  double old_x = *((double *) xp);
+  double new_x;
+
+  double u = gsl_rng_uniform(r);
+  new_x = u * 2 * step_size - step_size + old_x;
+
+  memcpy(xp, &new_x, sizeof(new_x));
+}
+
+void P1(void *xp)
+{
+  printf ("%12g", *((double *) xp));
+}
+
+int
+main(int argc, char *argv[])
+{
+  const gsl_rng_type * T;
+  gsl_rng * r;
+
+  double x_initial = 15.5;
+
+  gsl_rng_env_setup();
+
+  T = gsl_rng_default;
+  r = gsl_rng_alloc(T);
+
+  gsl_siman_solve(r, &x_initial, E1, S1, M1, P1,
+                  NULL, NULL, NULL, 
+                  sizeof(double), params);
+
+  gsl_rng_free (r);
+  return 0;
+}
diff --git a/gsl-1.9/doc/examples/sortsmall.c b/gsl-1.9/doc/examples/sortsmall.c
new file mode 100644
index 0000000..5b72517
--- /dev/null
+++ b/gsl-1.9/doc/examples/sortsmall.c
@@ -0,0 +1,38 @@
+#include <gsl/gsl_rng.h>
+#include <gsl/gsl_sort_double.h>
+
+int
+main (void)
+{
+  const gsl_rng_type * T;
+  gsl_rng * r;
+
+  size_t i, k = 5, N = 100000;
+
+  double * x = malloc (N * sizeof(double));
+  double * small = malloc (k * sizeof(double));
+
+  gsl_rng_env_setup();
+
+  T = gsl_rng_default;
+  r = gsl_rng_alloc (T);
+
+  for (i = 0; i < N; i++)
+    {
+      x[i] = gsl_rng_uniform(r);
+    }
+
+  gsl_sort_smallest (small, k, x, 1, N);
+
+  printf ("%d smallest values from %d\n", k, N);
+
+  for (i = 0; i < k; i++)
+    {
+      printf ("%d: %.18f\n", i, small[i]);
+    }
+
+  free (x);
+  free (small);
+  gsl_rng_free (r);
+  return 0;
+}
diff --git a/gsl-1.9/doc/examples/sortsmall.out b/gsl-1.9/doc/examples/sortsmall.out
new file mode 100644
index 0000000..b8c7c93
--- /dev/null
+++ b/gsl-1.9/doc/examples/sortsmall.out
@@ -0,0 +1,6 @@
+5 smallest values from 100000
+0: 0.000003489200025797
+1: 0.000008199829608202
+2: 0.000008953968062997
+3: 0.000010712770745158
+4: 0.000033531803637743
diff --git a/gsl-1.9/doc/examples/specfun.c b/gsl-1.9/doc/examples/specfun.c
new file mode 100644
index 0000000..7c1a9b0
--- /dev/null
+++ b/gsl-1.9/doc/examples/specfun.c
@@ -0,0 +1,15 @@
+#include <stdio.h>
+#include <gsl/gsl_sf_bessel.h>
+
+int
+main (void)
+{
+  double x = 5.0;
+  double expected = -0.17759677131433830434739701;
+  
+  double y = gsl_sf_bessel_J0 (x);
+
+  printf ("J0(5.0) = %.18f\n", y);
+  printf ("exact   = %.18f\n", expected);
+  return 0;
+}
diff --git a/gsl-1.9/doc/examples/specfun.out b/gsl-1.9/doc/examples/specfun.out
new file mode 100644
index 0000000..dbe2443
--- /dev/null
+++ b/gsl-1.9/doc/examples/specfun.out
@@ -0,0 +1,2 @@
+J0(5.0) = -0.177596771314338292
+exact   = -0.177596771314338292
diff --git a/gsl-1.9/doc/examples/specfun_e.c b/gsl-1.9/doc/examples/specfun_e.c
new file mode 100644
index 0000000..4f41f2a
--- /dev/null
+++ b/gsl-1.9/doc/examples/specfun_e.c
@@ -0,0 +1,21 @@
+#include <stdio.h>
+#include <gsl/gsl_errno.h>
+#include <gsl/gsl_sf_bessel.h>
+
+int
+main (void)
+{
+  double x = 5.0;
+  gsl_sf_result result;
+
+  double expected = -0.17759677131433830434739701;
+  
+  int status = gsl_sf_bessel_J0_e (x, &result);
+
+  printf ("status  = %s\n", gsl_strerror(status));
+  printf ("J0(5.0) = %.18f\n"
+          "      +/- % .18f\n", 
+          result.val, result.err);
+  printf ("exact   = %.18f\n", expected);
+  return status;
+}
diff --git a/gsl-1.9/doc/examples/specfun_e.out b/gsl-1.9/doc/examples/specfun_e.out
new file mode 100644
index 0000000..61e1314
--- /dev/null
+++ b/gsl-1.9/doc/examples/specfun_e.out
@@ -0,0 +1,4 @@
+status  = success
+J0(5.0) = -0.177596771314338292 
+      +/-  0.000000000000000193
+exact   = -0.177596771314338292
diff --git a/gsl-1.9/doc/examples/stat.c b/gsl-1.9/doc/examples/stat.c
new file mode 100644
index 0000000..253f704
--- /dev/null
+++ b/gsl-1.9/doc/examples/stat.c
@@ -0,0 +1,23 @@
+#include <stdio.h>
+#include <gsl/gsl_statistics.h>
+
+int
+main(void)
+{
+  double data[5] = {17.2, 18.1, 16.5, 18.3, 12.6};
+  double mean, variance, largest, smallest;
+
+  mean     = gsl_stats_mean(data, 1, 5);
+  variance = gsl_stats_variance(data, 1, 5);
+  largest  = gsl_stats_max(data, 1, 5);
+  smallest = gsl_stats_min(data, 1, 5);
+
+  printf ("The dataset is %g, %g, %g, %g, %g\n",
+         data[0], data[1], data[2], data[3], data[4]);
+
+  printf ("The sample mean is %g\n", mean);
+  printf ("The estimated variance is %g\n", variance);
+  printf ("The largest value is %g\n", largest);
+  printf ("The smallest value is %g\n", smallest);
+  return 0;
+}
diff --git a/gsl-1.9/doc/examples/stat.out b/gsl-1.9/doc/examples/stat.out
new file mode 100644
index 0000000..232081d
--- /dev/null
+++ b/gsl-1.9/doc/examples/stat.out
@@ -0,0 +1,5 @@
+The dataset is 17.2, 18.1, 16.5, 18.3, 12.6
+The sample mean is 16.54
+The estimated variance is 4.2984
+The largest value is 18.3
+The smallest value is 12.6
diff --git a/gsl-1.9/doc/examples/statsort.c b/gsl-1.9/doc/examples/statsort.c
new file mode 100644
index 0000000..2780f67
--- /dev/null
+++ b/gsl-1.9/doc/examples/statsort.c
@@ -0,0 +1,36 @@
+#include <stdio.h>
+#include <gsl/gsl_sort.h>
+#include <gsl/gsl_statistics.h>
+
+int
+main(void)
+{
+  double data[5] = {17.2, 18.1, 16.5, 18.3, 12.6};
+  double median, upperq, lowerq;
+
+  printf ("Original dataset:  %g, %g, %g, %g, %g\n",
+         data[0], data[1], data[2], data[3], data[4]);
+
+  gsl_sort (data, 1, 5);
+
+  printf ("Sorted dataset: %g, %g, %g, %g, %g\n",
+         data[0], data[1], data[2], data[3], data[4]);
+
+  median 
+    = gsl_stats_median_from_sorted_data (data, 
+                                         1, 5);
+
+  upperq 
+    = gsl_stats_quantile_from_sorted_data (data, 
+                                           1, 5,
+                                           0.75);
+  lowerq 
+    = gsl_stats_quantile_from_sorted_data (data, 
+                                           1, 5,
+                                           0.25);
+
+  printf ("The median is %g\n", median);
+  printf ("The upper quartile is %g\n", upperq);
+  printf ("The lower quartile is %g\n", lowerq);
+  return 0;
+}
diff --git a/gsl-1.9/doc/examples/statsort.out b/gsl-1.9/doc/examples/statsort.out
new file mode 100644
index 0000000..628522c
--- /dev/null
+++ b/gsl-1.9/doc/examples/statsort.out
@@ -0,0 +1,5 @@
+Original dataset: 17.2, 18.1, 16.5, 18.3, 12.6
+Sorted dataset: 12.6, 16.5, 17.2, 18.1, 18.3
+The median is 17.2
+The upper quartile is 18.1
+The lower quartile is 16.5
diff --git a/gsl-1.9/doc/examples/sum.c b/gsl-1.9/doc/examples/sum.c
new file mode 100644
index 0000000..c0237ea
--- /dev/null
+++ b/gsl-1.9/doc/examples/sum.c
@@ -0,0 +1,47 @@
+#include <stdio.h>
+#include <gsl/gsl_math.h>
+#include <gsl/gsl_sum.h>
+
+#define N 20
+
+int
+main (void)
+{
+  double t[N];
+  double sum_accel, err;
+  double sum = 0;
+  int n;
+  
+  gsl_sum_levin_u_workspace * w 
+    = gsl_sum_levin_u_alloc (N);
+
+  const double zeta_2 = M_PI * M_PI / 6.0;
+  
+  /* terms for zeta(2) = \sum_{n=1}^{\infty} 1/n^2 */
+
+  for (n = 0; n < N; n++)
+    {
+      double np1 = n + 1.0;
+      t[n] = 1.0 / (np1 * np1);
+      sum += t[n];
+    }
+  
+  gsl_sum_levin_u_accel (t, N, w, &sum_accel, &err);
+
+  printf ("term-by-term sum = % .16f using %d terms\n", 
+          sum, N);
+
+  printf ("term-by-term sum = % .16f using %d terms\n", 
+          w->sum_plain, w->terms_used);
+
+  printf ("exact value      = % .16f\n", zeta_2);
+  printf ("accelerated sum  = % .16f using %d terms\n", 
+          sum_accel, w->terms_used);
+
+  printf ("estimated error  = % .16f\n", err);
+  printf ("actual error     = % .16f\n", 
+          sum_accel - zeta_2);
+
+  gsl_sum_levin_u_free (w);
+  return 0;
+}
diff --git a/gsl-1.9/doc/examples/sum.out b/gsl-1.9/doc/examples/sum.out
new file mode 100644
index 0000000..28089e9
--- /dev/null
+++ b/gsl-1.9/doc/examples/sum.out
@@ -0,0 +1,6 @@
+term-by-term sum =  1.5961632439130233 using 20 terms
+term-by-term sum =  1.5759958390005426 using 13 terms
+exact value      =  1.6449340668482264
+accelerated sum  =  1.6449340668166479 using 13 terms
+estimated error  =  0.0000000000508580
+actual error     = -0.0000000000315785
diff --git a/gsl-1.9/doc/examples/vector.c b/gsl-1.9/doc/examples/vector.c
new file mode 100644
index 0000000..27462b0
--- /dev/null
+++ b/gsl-1.9/doc/examples/vector.c
@@ -0,0 +1,22 @@
+#include <stdio.h>
+#include <gsl/gsl_vector.h>
+
+int
+main (void)
+{
+  int i;
+  gsl_vector * v = gsl_vector_alloc (3);
+  
+  for (i = 0; i < 3; i++)
+    {
+      gsl_vector_set (v, i, 1.23 + i);
+    }
+  
+  for (i = 0; i < 100; i++) /* OUT OF RANGE ERROR */
+    {
+      printf ("v_%d = %g\n", i, gsl_vector_get (v, i));
+    }
+
+  gsl_vector_free (v);
+  return 0;
+}
diff --git a/gsl-1.9/doc/examples/vectorr.c b/gsl-1.9/doc/examples/vectorr.c
new file mode 100644
index 0000000..203986b
--- /dev/null
+++ b/gsl-1.9/doc/examples/vectorr.c
@@ -0,0 +1,23 @@
+#include <stdio.h>
+#include <gsl/gsl_vector.h>
+
+int
+main (void)
+{
+  int i; 
+  gsl_vector * v = gsl_vector_alloc (10);
+
+  {  
+     FILE * f = fopen ("test.dat", "r");
+     gsl_vector_fscanf (f, v);
+     fclose (f);
+  }
+
+  for (i = 0; i < 10; i++)
+    {
+      printf ("%g\n", gsl_vector_get(v, i));
+    }
+
+  gsl_vector_free (v);
+  return 0;
+}
diff --git a/gsl-1.9/doc/examples/vectorview.c b/gsl-1.9/doc/examples/vectorview.c
new file mode 100644
index 0000000..9225c28
--- /dev/null
+++ b/gsl-1.9/doc/examples/vectorview.c
@@ -0,0 +1,30 @@
+#include <math.h>
+#include <stdio.h>
+#include <gsl/gsl_matrix.h>
+#include <gsl/gsl_blas.h>
+
+int
+main (void)
+{
+  size_t i,j;
+
+  gsl_matrix *m = gsl_matrix_alloc (10, 10);
+
+  for (i = 0; i < 10; i++)
+    for (j = 0; j < 10; j++)
+      gsl_matrix_set (m, i, j, sin (i) + cos (j));
+
+  for (j = 0; j < 10; j++)
+    {
+      gsl_vector_view column = gsl_matrix_column (m, j);
+      double d;
+
+      d = gsl_blas_dnrm2 (&column.vector);
+
+      printf ("matrix column %d, norm = %g\n", j, d);
+    }
+
+  gsl_matrix_free (m);
+
+  return 0;
+}
diff --git a/gsl-1.9/doc/examples/vectorview.out b/gsl-1.9/doc/examples/vectorview.out
new file mode 100644
index 0000000..0138c24
--- /dev/null
+++ b/gsl-1.9/doc/examples/vectorview.out
@@ -0,0 +1,10 @@
+matrix column 0, norm = 4.31461
+matrix column 1, norm = 3.1205
+matrix column 2, norm = 2.19316
+matrix column 3, norm = 3.26114
+matrix column 4, norm = 2.53416
+matrix column 5, norm = 2.57281
+matrix column 6, norm = 4.20469
+matrix column 7, norm = 3.65202
+matrix column 8, norm = 2.08524
+matrix column 9, norm = 3.07313
diff --git a/gsl-1.9/doc/examples/vectorw.c b/gsl-1.9/doc/examples/vectorw.c
new file mode 100644
index 0000000..f25468b
--- /dev/null
+++ b/gsl-1.9/doc/examples/vectorw.c
@@ -0,0 +1,23 @@
+#include <stdio.h>
+#include <gsl/gsl_vector.h>
+
+int
+main (void)
+{
+  int i; 
+  gsl_vector * v = gsl_vector_alloc (100);
+  
+  for (i = 0; i < 100; i++)
+    {
+      gsl_vector_set (v, i, 1.23 + i);
+    }
+
+  {  
+     FILE * f = fopen ("test.dat", "w");
+     gsl_vector_fprintf (f, v, "%.5g");
+     fclose (f);
+  }
+
+  gsl_vector_free (v);
+  return 0;
+}
diff --git a/gsl-1.9/doc/fdl.texi b/gsl-1.9/doc/fdl.texi
new file mode 100644
index 0000000..f63a200
--- /dev/null
+++ b/gsl-1.9/doc/fdl.texi
@@ -0,0 +1,455 @@
+@cindex FDL, GNU Free Documentation License
+@center Version 1.2, November 2002
+@iftex
+@smallerfonts @rm
+@end iftex
+
+@display
+Copyright @copyright{} 2000,2001,2002 Free Software Foundation, Inc.
+51 Franklin St, Fifth Floor, Boston, MA  02110-1301, USA
+
+Everyone is permitted to copy and distribute verbatim copies of this license 
+document, but changing it is not allowed.
+@end display
+
+@enumerate 0
+@item
+PREAMBLE
+
+The purpose of this License is to make a manual, textbook, or other
+functional and useful document @dfn{free} in the sense of freedom: to
+assure everyone the effective freedom to copy and redistribute it,
+with or without modifying it, either commercially or noncommercially.
+Secondarily, this License preserves for the author and publisher a way
+to get credit for their work, while not being considered responsible
+for modifications made by others.
+
+This License is a kind of ``copyleft'', which means that derivative
+works of the document must themselves be free in the same sense.  It
+complements the GNU General Public License, which is a copyleft
+license designed for free software.
+
+We have designed this License in order to use it for manuals for free
+software, because free software needs free documentation: a free
+program should come with manuals providing the same freedoms that the
+software does.  But this License is not limited to software manuals;
+it can be used for any textual work, regardless of subject matter or
+whether it is published as a printed book.  We recommend this License
+principally for works whose purpose is instruction or reference.
+
+@item
+APPLICABILITY AND DEFINITIONS
+
+This License applies to any manual or other work, in any medium, that
+contains a notice placed by the copyright holder saying it can be
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+The ``Title Page'' means, for a printed book, the title page itself,
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+
+The Document may include Warranty Disclaimers next to the notice which
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+VERBATIM COPYING
+
+You may copy and distribute the Document in any medium, either
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+
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+
+@item
+COPYING IN QUANTITY
+
+If you publish printed copies (or copies in media that commonly have
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+legibly, you should put the first ones listed (as many as fit
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+
+If you publish or distribute Opaque copies of the Document numbering
+more than 100, you must either include a machine-readable Transparent
+copy along with each Opaque copy, or state in or with each Opaque copy
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+that this Transparent copy will remain thus accessible at the stated
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+
+It is requested, but not required, that you contact the authors of the
+Document well before redistributing any large number of copies, to give
+them a chance to provide you with an updated version of the Document.
+
+@item
+MODIFICATIONS
+
+You may copy and distribute a Modified Version of the Document under
+the conditions of sections 2 and 3 above, provided that you release
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+and modification of the Modified Version to whoever possesses a copy
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+
+@enumerate A
+@item
+Use in the Title Page (and on the covers, if any) a title distinct
+from that of the Document, and from those of previous versions
+(which should, if there were any, be listed in the History section
+of the Document).  You may use the same title as a previous version
+if the original publisher of that version gives permission.
+
+@item
+List on the Title Page, as authors, one or more persons or entities
+responsible for authorship of the modifications in the Modified
+Version, together with at least five of the principal authors of the
+Document (all of its principal authors, if it has fewer than five),
+unless they release you from this requirement.
+
+@item
+State on the Title page the name of the publisher of the
+Modified Version, as the publisher.
+
+@item
+Preserve all the copyright notices of the Document.
+
+@item
+Add an appropriate copyright notice for your modifications
+adjacent to the other copyright notices.
+
+@item
+Include, immediately after the copyright notices, a license notice
+giving the public permission to use the Modified Version under the
+terms of this License, in the form shown in the Addendum below.
+
+@item
+Preserve in that license notice the full lists of Invariant Sections
+and required Cover Texts given in the Document's license notice.
+
+@item
+Include an unaltered copy of this License.
+
+@item
+Preserve the section Entitled ``History'', Preserve its Title, and add
+to it an item stating at least the title, year, new authors, and
+publisher of the Modified Version as given on the Title Page.  If
+there is no section Entitled ``History'' in the Document, create one
+stating the title, year, authors, and publisher of the Document as
+given on its Title Page, then add an item describing the Modified
+Version as stated in the previous sentence.
+
+@item
+Preserve the network location, if any, given in the Document for
+public access to a Transparent copy of the Document, and likewise
+the network locations given in the Document for previous versions
+it was based on.  These may be placed in the ``History'' section.
+You may omit a network location for a work that was published at
+least four years before the Document itself, or if the original
+publisher of the version it refers to gives permission.
+
+@item
+For any section Entitled ``Acknowledgements'' or ``Dedications'', Preserve
+the Title of the section, and preserve in the section all the
+substance and tone of each of the contributor acknowledgements and/or
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+
+@item
+Preserve all the Invariant Sections of the Document,
+unaltered in their text and in their titles.  Section numbers
+or the equivalent are not considered part of the section titles.
+
+@item
+Delete any section Entitled ``Endorsements''.  Such a section
+may not be included in the Modified Version.
+
+@item
+Do not retitle any existing section to be Entitled ``Endorsements'' or
+to conflict in title with any Invariant Section.
+
+@item
+Preserve any Warranty Disclaimers.
+@end enumerate
+
+If the Modified Version includes new front-matter sections or
+appendices that qualify as Secondary Sections and contain no material
+copied from the Document, you may at your option designate some or all
+of these sections as invariant.  To do this, add their titles to the
+list of Invariant Sections in the Modified Version's license notice.
+These titles must be distinct from any other section titles.
+
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+nothing but endorsements of your Modified Version by various
+parties---for example, statements of peer review or that the text has
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+
+You may add a passage of up to five words as a Front-Cover Text, and a
+passage of up to 25 words as a Back-Cover Text, to the end of the list
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+you may not add another; but you may replace the old one, on explicit
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+
+The author(s) and publisher(s) of the Document do not by this License
+give permission to use their names for publicity for or to assert or
+imply endorsement of any Modified Version.
+
+@item
+COMBINING DOCUMENTS
+
+You may combine the Document with other documents released under this
+License, under the terms defined in section 4 above for modified
+versions, provided that you include in the combination all of the
+Invariant Sections of all of the original documents, unmodified, and
+list them all as Invariant Sections of your combined work in its
+license notice, and that you preserve all their Warranty Disclaimers.
+
+The combined work need only contain one copy of this License, and
+multiple identical Invariant Sections may be replaced with a single
+copy.  If there are multiple Invariant Sections with the same name but
+different contents, make the title of each such section unique by
+adding at the end of it, in parentheses, the name of the original
+author or publisher of that section if known, or else a unique number.
+Make the same adjustment to the section titles in the list of
+Invariant Sections in the license notice of the combined work.
+
+In the combination, you must combine any sections Entitled ``History''
+in the various original documents, forming one section Entitled
+``History''; likewise combine any sections Entitled ``Acknowledgements'',
+and any sections Entitled ``Dedications''.  You must delete all
+sections Entitled ``Endorsements.''
+
+@item
+COLLECTIONS OF DOCUMENTS
+
+You may make a collection consisting of the Document and other documents
+released under this License, and replace the individual copies of this
+License in the various documents with a single copy that is included in
+the collection, provided that you follow the rules of this License for
+verbatim copying of each of the documents in all other respects.
+
+You may extract a single document from such a collection, and distribute
+it individually under this License, provided you insert a copy of this
+License into the extracted document, and follow this License in all
+other respects regarding verbatim copying of that document.
+
+@item
+AGGREGATION WITH INDEPENDENT WORKS
+
+A compilation of the Document or its derivatives with other separate
+and independent documents or works, in or on a volume of a storage or
+distribution medium, is called an ``aggregate'' if the copyright
+resulting from the compilation is not used to limit the legal rights
+of the compilation's users beyond what the individual works permit.
+When the Document is included in an aggregate, this License does not
+apply to the other works in the aggregate which are not themselves
+derivative works of the Document.
+
+If the Cover Text requirement of section 3 is applicable to these
+copies of the Document, then if the Document is less than one half of
+the entire aggregate, the Document's Cover Texts may be placed on
+covers that bracket the Document within the aggregate, or the
+electronic equivalent of covers if the Document is in electronic form.
+Otherwise they must appear on printed covers that bracket the whole
+aggregate.
+
+@item
+TRANSLATION
+
+Translation is considered a kind of modification, so you may
+distribute translations of the Document under the terms of section 4.
+Replacing Invariant Sections with translations requires special
+permission from their copyright holders, but you may include
+translations of some or all Invariant Sections in addition to the
+original versions of these Invariant Sections.  You may include a
+translation of this License, and all the license notices in the
+Document, and any Warranty Disclaimers, provided that you also include
+the original English version of this License and the original versions
+of those notices and disclaimers.  In case of a disagreement between
+the translation and the original version of this License or a notice
+or disclaimer, the original version will prevail.
+
+If a section in the Document is Entitled ``Acknowledgements'',
+``Dedications'', or ``History'', the requirement (section 4) to Preserve
+its Title (section 1) will typically require changing the actual
+title.
+
+@item
+TERMINATION
+
+You may not copy, modify, sublicense, or distribute the Document except
+as expressly provided for under this License.  Any other attempt to
+copy, modify, sublicense or distribute the Document is void, and will
+automatically terminate your rights under this License.  However,
+parties who have received copies, or rights, from you under this
+License will not have their licenses terminated so long as such
+parties remain in full compliance.
+
+@item
+FUTURE REVISIONS OF THIS LICENSE
+
+The Free Software Foundation may publish new, revised versions
+of the GNU Free Documentation License from time to time.  Such new
+versions will be similar in spirit to the present version, but may
+differ in detail to address new problems or concerns.  See
+@uref{http://www.gnu.org/copyleft/}.
+
+Each version of the License is given a distinguishing version number.
+If the Document specifies that a particular numbered version of this
+License ``or any later version'' applies to it, you have the option of
+following the terms and conditions either of that specified version or
+of any later version that has been published (not as a draft) by the
+Free Software Foundation.  If the Document does not specify a version
+number of this License, you may choose any version ever published (not
+as a draft) by the Free Software Foundation.
+@end enumerate
+
+@page
+@heading ADDENDUM: How to use this License for your documents
+
+To use this License in a document you have written, include a copy of
+the License in the document and put the following copyright and
+license notices just after the title page:
+
+@smallexample
+@group
+  Copyright (C) @var{year} @var{your name}.  
+  Permission is granted to copy, distribute and/or modify
+  this document under the terms of the GNU Free
+  Documentation License, Version 1.2 or any later version
+  published by the Free Software Foundation; with no
+  Invariant Sections, no Front-Cover Texts, and no
+  Back-Cover Texts.  A copy of the license is included in
+  the section entitled ``GNU Free Documentation License''.
+@end group
+@end smallexample
+
+If you have Invariant Sections, Front-Cover Texts and Back-Cover Texts,
+replace the ``with...Texts.'' line with this:
+
+@smallexample
+@group
+  with the Invariant Sections being @var{list their
+  titles}, with the Front-Cover Texts being @var{list}, and
+  with the Back-Cover Texts being @var{list}.
+@end group
+@end smallexample
+
+If you have Invariant Sections without Cover Texts, or some other
+combination of the three, merge those two alternatives to suit the
+situation.
+
+If your document contains nontrivial examples of program code, we
+recommend releasing these examples in parallel under your choice of
+free software license, such as the GNU General Public License,
+to permit their use in free software.
+
+@iftex
+@textfonts @rm
+@end iftex
+@c Local Variables:
+@c ispell-local-pdict: "ispell-dict"
+@c End:
+
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diff --git a/gsl-1.9/doc/fft.texi b/gsl-1.9/doc/fft.texi
new file mode 100644
index 0000000..16cc5a9
--- /dev/null
+++ b/gsl-1.9/doc/fft.texi
@@ -0,0 +1,1001 @@
+@cindex FFT
+@cindex Fast Fourier Transforms, see FFT
+@cindex Fourier Transforms, see FFT
+@cindex Discrete Fourier Transforms, see FFT
+@cindex DFTs, see FFT
+
+This chapter describes functions for performing Fast Fourier Transforms
+(FFTs).  The library includes radix-2 routines (for lengths which are a
+power of two) and mixed-radix routines (which work for any length).  For
+efficiency there are separate versions of the routines for real data and
+for complex data.  The mixed-radix routines are a reimplementation of the
+@sc{fftpack} library of Paul Swarztrauber.  Fortran code for @sc{fftpack} is
+available on Netlib (@sc{fftpack} also includes some routines for sine and
+cosine transforms but these are currently not available in GSL).  For
+details and derivations of the underlying algorithms consult the
+document @cite{GSL FFT Algorithms} (@pxref{FFT References and Further Reading})
+
+@menu
+* Mathematical Definitions::    
+* Overview of complex data FFTs::  
+* Radix-2 FFT routines for complex data::  
+* Mixed-radix FFT routines for complex data::  
+* Overview of real data FFTs::  
+* Radix-2 FFT routines for real data::  
+* Mixed-radix FFT routines for real data::  
+* FFT References and Further Reading::  
+@end menu
+
+@node Mathematical Definitions
+@section Mathematical Definitions 
+@cindex FFT mathematical definition
+
+Fast Fourier Transforms are efficient algorithms for
+calculating the discrete fourier transform (DFT),
+@tex
+\beforedisplay
+$$
+x_j = \sum_{k=0}^{N-1} z_k \exp(-2\pi i j k / N) 
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+x_j = \sum_@{k=0@}^@{N-1@} z_k \exp(-2\pi i j k / N) 
+@end example
+@end ifinfo
+
+The DFT usually arises as an approximation to the continuous fourier
+transform when functions are sampled at discrete intervals in space or
+time.  The naive evaluation of the discrete fourier transform is a
+matrix-vector multiplication 
+@c{$W\vec{z}$}
+@math{W\vec@{z@}}. A general matrix-vector multiplication takes
+@math{O(N^2)} operations for @math{N} data-points.  Fast fourier
+transform algorithms use a divide-and-conquer strategy to factorize the
+matrix @math{W} into smaller sub-matrices, corresponding to the integer
+factors of the length @math{N}.  If @math{N} can be factorized into a
+product of integers
+@c{$f_1 f_2 \ldots f_n$} 
+@math{f_1 f_2 ... f_n} then the DFT can be computed in @math{O(N \sum
+f_i)} operations.  For a radix-2 FFT this gives an operation count of
+@math{O(N \log_2 N)}.
+
+All the FFT functions offer three types of transform: forwards, inverse
+and backwards, based on the same mathematical definitions.  The
+definition of the @dfn{forward fourier transform},
+@c{$x = \hbox{FFT}(z)$}
+@math{x = FFT(z)}, is,
+@tex
+\beforedisplay
+$$
+x_j = \sum_{k=0}^{N-1} z_k \exp(-2\pi i j k / N) 
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+x_j = \sum_@{k=0@}^@{N-1@} z_k \exp(-2\pi i j k / N) 
+@end example
+
+@end ifinfo
+@noindent
+and the definition of the @dfn{inverse fourier transform},
+@c{$x = \hbox{IFFT}(z)$}
+@math{x = IFFT(z)}, is,
+@tex
+\beforedisplay
+$$
+z_j = {1 \over N} \sum_{k=0}^{N-1} x_k \exp(2\pi i j k / N).
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+z_j = @{1 \over N@} \sum_@{k=0@}^@{N-1@} x_k \exp(2\pi i j k / N).
+@end example
+
+@end ifinfo
+@noindent
+The factor of @math{1/N} makes this a true inverse.  For example, a call
+to @code{gsl_fft_complex_forward} followed by a call to
+@code{gsl_fft_complex_inverse} should return the original data (within
+numerical errors).
+
+In general there are two possible choices for the sign of the
+exponential in the transform/ inverse-transform pair. GSL follows the
+same convention as @sc{fftpack}, using a negative exponential for the forward
+transform.  The advantage of this convention is that the inverse
+transform recreates the original function with simple fourier
+synthesis.  Numerical Recipes uses the opposite convention, a positive
+exponential in the forward transform.
+
+The @dfn{backwards FFT} is simply our terminology for an unscaled
+version of the inverse FFT,
+@tex
+\beforedisplay
+$$
+z^{backwards}_j = \sum_{k=0}^{N-1} x_k \exp(2\pi i j k / N).
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+z^@{backwards@}_j = \sum_@{k=0@}^@{N-1@} x_k \exp(2\pi i j k / N).
+@end example
+
+@end ifinfo
+@noindent
+When the overall scale of the result is unimportant it is often
+convenient to use the backwards FFT instead of the inverse to save
+unnecessary divisions.
+
+@node Overview of complex data FFTs
+@section Overview of complex data FFTs
+@cindex FFT, complex data
+
+The inputs and outputs for the complex FFT routines are @dfn{packed
+arrays} of floating point numbers.  In a packed array the real and
+imaginary parts of each complex number are placed in alternate
+neighboring elements.  For example, the following definition of a packed
+array of length 6,
+
+@example
+double x[3*2];
+gsl_complex_packed_array data = x;
+@end example
+
+@noindent
+can be used to hold an array of three complex numbers, @code{z[3]}, in
+the following way,
+
+@example
+data[0] = Re(z[0])
+data[1] = Im(z[0])
+data[2] = Re(z[1])
+data[3] = Im(z[1])
+data[4] = Re(z[2])
+data[5] = Im(z[2])
+@end example
+
+@noindent
+The array indices for the data have the same ordering as those
+in the definition of the DFT---i.e. there are no index transformations
+or permutations of the data.
+
+A @dfn{stride} parameter allows the user to perform transforms on the
+elements @code{z[stride*i]} instead of @code{z[i]}.  A stride greater
+than 1 can be used to take an in-place FFT of the column of a matrix. A
+stride of 1 accesses the array without any additional spacing between
+elements.  
+
+To perform an FFT on a vector argument, such as @code{gsl_vector_complex
+* v}, use the following definitions (or their equivalents) when calling
+the functions described in this chapter:
+
+@example
+gsl_complex_packed_array data = v->data;
+size_t stride = v->stride;
+size_t n = v->size;
+@end example
+
+For physical applications it is important to remember that the index
+appearing in the DFT does not correspond directly to a physical
+frequency.  If the time-step of the DFT is @math{\Delta} then the
+frequency-domain includes both positive and negative frequencies,
+ranging from @math{-1/(2\Delta)} through 0 to @math{+1/(2\Delta)}.  The
+positive frequencies are stored from the beginning of the array up to
+the middle, and the negative frequencies are stored backwards from the
+end of the array.
+
+Here is a table which shows the layout of the array @var{data}, and the
+correspondence between the time-domain data @math{z}, and the
+frequency-domain data @math{x}.
+
+@example
+index    z               x = FFT(z)
+
+0        z(t = 0)        x(f = 0)
+1        z(t = 1)        x(f = 1/(N Delta))
+2        z(t = 2)        x(f = 2/(N Delta))
+.        ........        ..................
+N/2      z(t = N/2)      x(f = +1/(2 Delta),
+                               -1/(2 Delta))
+.        ........        ..................
+N-3      z(t = N-3)      x(f = -3/(N Delta))
+N-2      z(t = N-2)      x(f = -2/(N Delta))
+N-1      z(t = N-1)      x(f = -1/(N Delta))
+@end example
+
+@noindent
+When @math{N} is even the location @math{N/2} contains the most positive
+and negative frequencies (@math{+1/(2 \Delta)}, @math{-1/(2 \Delta)})
+which are equivalent.  If @math{N} is odd then general structure of the
+table above still applies, but @math{N/2} does not appear.
+
+
+@node Radix-2 FFT routines for complex data
+@section Radix-2 FFT routines for complex data
+@cindex FFT of complex data, radix-2 algorithm
+@cindex Radix-2 FFT, complex data
+
+The radix-2 algorithms described in this section are simple and compact,
+although not necessarily the most efficient.  They use the Cooley-Tukey
+algorithm to compute in-place complex FFTs for lengths which are a power
+of 2---no additional storage is required.  The corresponding
+self-sorting mixed-radix routines offer better performance at the
+expense of requiring additional working space.
+
+All the functions described in this section are declared in the header file @file{gsl_fft_complex.h}.
+
+@deftypefun int gsl_fft_complex_radix2_forward (gsl_complex_packed_array @var{data}, size_t @var{stride}, size_t @var{n})
+
+@deftypefunx int gsl_fft_complex_radix2_transform (gsl_complex_packed_array @var{data}, size_t @var{stride}, size_t @var{n}, gsl_fft_direction @var{sign})
+
+@deftypefunx int gsl_fft_complex_radix2_backward (gsl_complex_packed_array @var{data}, size_t @var{stride}, size_t @var{n})
+
+@deftypefunx int gsl_fft_complex_radix2_inverse (gsl_complex_packed_array @var{data}, size_t @var{stride}, size_t @var{n})
+
+These functions compute forward, backward and inverse FFTs of length
+@var{n} with stride @var{stride}, on the packed complex array @var{data}
+using an in-place radix-2 decimation-in-time algorithm.  The length of
+the transform @var{n} is restricted to powers of two.  For the
+@code{transform} version of the function the @var{sign} argument can be
+either @code{forward} (@math{-1}) or @code{backward} (@math{+1}).
+
+The functions return a value of @code{GSL_SUCCESS} if no errors were
+detected, or @code{GSL_EDOM} if the length of the data @var{n} is not a
+power of two.
+@end deftypefun
+
+@deftypefun int gsl_fft_complex_radix2_dif_forward (gsl_complex_packed_array @var{data}, size_t @var{stride}, size_t @var{n})
+
+@deftypefunx int gsl_fft_complex_radix2_dif_transform (gsl_complex_packed_array @var{data}, size_t @var{stride}, size_t @var{n}, gsl_fft_direction @var{sign})
+
+@deftypefunx int gsl_fft_complex_radix2_dif_backward (gsl_complex_packed_array @var{data}, size_t @var{stride}, size_t @var{n})
+
+@deftypefunx int gsl_fft_complex_radix2_dif_inverse (gsl_complex_packed_array @var{data}, size_t @var{stride}, size_t @var{n})
+
+These are decimation-in-frequency versions of the radix-2 FFT functions.
+
+@end deftypefun
+
+
+@comment @node Example of using radix-2 FFT routines for complex data
+@comment @subsection Example of using radix-2 FFT routines for complex data
+
+Here is an example program which computes the FFT of a short pulse in a
+sample of length 128.  To make the resulting fourier transform real the
+pulse is defined for equal positive and negative times (@math{-10}
+@dots{} @math{10}), where the negative times wrap around the end of the
+array.
+
+@example
+@verbatiminclude examples/fft.c
+@end example
+
+@noindent
+Note that we have assumed that the program is using the default error
+handler (which calls @code{abort} for any errors).  If you are not using
+a safe error handler you would need to check the return status of
+@code{gsl_fft_complex_radix2_forward}.
+
+The transformed data is rescaled by @math{1/\sqrt N} so that it fits on
+the same plot as the input.  Only the real part is shown, by the choice
+of the input data the imaginary part is zero.  Allowing for the
+wrap-around of negative times at @math{t=128}, and working in units of
+@math{k/N}, the DFT approximates the continuum fourier transform, giving
+a modulated sine function.
+@iftex
+@tex
+\beforedisplay
+$$
+\int_{-a}^{+a} e^{-2 \pi i k x} dx = {\sin(2\pi k a) \over\pi k}
+$$
+\afterdisplay
+@end tex
+
+@sp 1
+@center @image{fft-complex-radix2-t,2.8in} 
+@center @image{fft-complex-radix2-f,2.8in}
+@quotation
+A pulse and its discrete fourier transform, output from
+the example program.
+@end quotation
+@end iftex
+
+@node Mixed-radix FFT routines for complex data
+@section Mixed-radix FFT routines for complex data
+@cindex FFT of complex data, mixed-radix algorithm
+@cindex Mixed-radix FFT, complex data
+
+This section describes mixed-radix FFT algorithms for complex data.  The
+mixed-radix functions work for FFTs of any length.  They are a
+reimplementation of Paul Swarztrauber's Fortran @sc{fftpack} library.
+The theory is explained in the review article @cite{Self-sorting
+Mixed-radix FFTs} by Clive Temperton.  The routines here use the same
+indexing scheme and basic algorithms as @sc{fftpack}.
+
+The mixed-radix algorithm is based on sub-transform modules---highly
+optimized small length FFTs which are combined to create larger FFTs.
+There are efficient modules for factors of 2, 3, 4, 5, 6 and 7.  The
+modules for the composite factors of 4 and 6 are faster than combining
+the modules for @math{2*2} and @math{2*3}.
+
+For factors which are not implemented as modules there is a fall-back to
+a general length-@math{n} module which uses Singleton's method for
+efficiently computing a DFT. This module is @math{O(n^2)}, and slower
+than a dedicated module would be but works for any length @math{n}.  Of
+course, lengths which use the general length-@math{n} module will still
+be factorized as much as possible.  For example, a length of 143 will be
+factorized into @math{11*13}.  Large prime factors are the worst case
+scenario, e.g. as found in @math{n=2*3*99991}, and should be avoided
+because their @math{O(n^2)} scaling will dominate the run-time (consult
+the document @cite{GSL FFT Algorithms} included in the GSL distribution
+if you encounter this problem).
+
+The mixed-radix initialization function @code{gsl_fft_complex_wavetable_alloc}
+returns the list of factors chosen by the library for a given length
+@math{N}.  It can be used to check how well the length has been
+factorized, and estimate the run-time.  To a first approximation the
+run-time scales as @math{N \sum f_i}, where the @math{f_i} are the
+factors of @math{N}.  For programs under user control you may wish to
+issue a warning that the transform will be slow when the length is
+poorly factorized.  If you frequently encounter data lengths which
+cannot be factorized using the existing small-prime modules consult
+@cite{GSL FFT Algorithms} for details on adding support for other
+factors.
+
+@comment First, the space for the trigonometric lookup tables and scratch area is
+@comment allocated by a call to one of the @code{alloc} functions.  We
+@comment call the combination of factorization, scratch space and trigonometric
+@comment lookup arrays a @dfn{wavetable}.  It contains the sine and cosine
+@comment waveforms for the all the frequencies that will be used in the FFT.
+
+@comment The wavetable is initialized by a call to the corresponding @code{init}
+@comment function.  It factorizes the data length, using the implemented
+@comment subtransforms as preferred factors wherever possible.  The trigonometric
+@comment lookup table for the chosen factorization is also computed.
+
+@comment An FFT is computed by a call to one of the @code{forward},
+@comment @code{backward} or @code{inverse} functions, with the data, length and
+@comment wavetable as arguments.
+
+All the functions described in this section are declared in the header
+file @file{gsl_fft_complex.h}.
+
+@deftypefun {gsl_fft_complex_wavetable *} gsl_fft_complex_wavetable_alloc (size_t @var{n})
+This function prepares a trigonometric lookup table for a complex FFT of
+length @var{n}. The function returns a pointer to the newly allocated
+@code{gsl_fft_complex_wavetable} if no errors were detected, and a null
+pointer in the case of error.  The length @var{n} is factorized into a
+product of subtransforms, and the factors and their trigonometric
+coefficients are stored in the wavetable. The trigonometric coefficients
+are computed using direct calls to @code{sin} and @code{cos}, for
+accuracy.  Recursion relations could be used to compute the lookup table
+faster, but if an application performs many FFTs of the same length then
+this computation is a one-off overhead which does not affect the final
+throughput.
+
+The wavetable structure can be used repeatedly for any transform of the
+same length.  The table is not modified by calls to any of the other FFT
+functions.  The same wavetable can be used for both forward and backward
+(or inverse) transforms of a given length.
+@end deftypefun
+
+@deftypefun void gsl_fft_complex_wavetable_free (gsl_fft_complex_wavetable * @var{wavetable})
+This function frees the memory associated with the wavetable
+@var{wavetable}.  The wavetable can be freed if no further FFTs of the
+same length will be needed.
+@end deftypefun
+
+@noindent
+These functions operate on a @code{gsl_fft_complex_wavetable} structure
+which contains internal parameters for the FFT.  It is not necessary to
+set any of the components directly but it can sometimes be useful to
+examine them.  For example, the chosen factorization of the FFT length
+is given and can be used to provide an estimate of the run-time or
+numerical error. The wavetable structure is declared in the header file
+@file{gsl_fft_complex.h}.
+
+@deftp {Data Type} gsl_fft_complex_wavetable
+This is a structure that holds the factorization and trigonometric
+lookup tables for the mixed radix fft algorithm.  It has the following
+components:
+
+@table @code
+@item size_t n
+This is the number of complex data points
+
+@item size_t nf
+This is the number of factors that the length @code{n} was decomposed into.
+
+@item size_t factor[64]
+This is the array of factors.  Only the first @code{nf} elements are
+used. 
+
+@comment (FIXME: This is a fixed length array and therefore probably in
+@comment violation of the GNU Coding Standards).
+
+@item gsl_complex * trig
+This is a pointer to a preallocated trigonometric lookup table of
+@code{n} complex elements.
+
+@item gsl_complex * twiddle[64]
+This is an array of pointers into @code{trig}, giving the twiddle
+factors for each pass.
+@end table
+@end deftp
+
+@noindent
+The mixed radix algorithms require additional working space to hold
+the intermediate steps of the transform.
+
+@deftypefun {gsl_fft_complex_workspace *} gsl_fft_complex_workspace_alloc (size_t @var{n})
+This function allocates a workspace for a complex transform of length
+@var{n}.
+@end deftypefun
+
+@deftypefun void gsl_fft_complex_workspace_free (gsl_fft_complex_workspace * @var{workspace})
+This function frees the memory associated with the workspace
+@var{workspace}. The workspace can be freed if no further FFTs of the
+same length will be needed.
+@end deftypefun
+
+@comment @deftp {Data Type} gsl_fft_complex_workspace
+@comment This is a structure that holds the workspace for the mixed radix fft
+@comment algorithm.  It has the following components:
+@comment
+@comment @table @code
+@comment @item gsl_complex * scratch
+@comment This is a pointer to a workspace of @code{n} complex elements,
+@comment capable of holding intermediate copies of the original data set.
+@comment @end table
+@comment @end deftp
+
+@noindent
+The following functions compute the transform,
+
+@deftypefun int gsl_fft_complex_forward (gsl_complex_packed_array @var{data}, size_t @var{stride}, size_t @var{n}, const gsl_fft_complex_wavetable * @var{wavetable}, gsl_fft_complex_workspace * @var{work})
+@deftypefunx int gsl_fft_complex_transform (gsl_complex_packed_array @var{data}, size_t @var{stride}, size_t @var{n}, const gsl_fft_complex_wavetable * @var{wavetable}, gsl_fft_complex_workspace * @var{work}, gsl_fft_direction @var{sign})
+@deftypefunx int gsl_fft_complex_backward (gsl_complex_packed_array @var{data}, size_t @var{stride}, size_t @var{n}, const gsl_fft_complex_wavetable * @var{wavetable}, gsl_fft_complex_workspace * @var{work})
+@deftypefunx int gsl_fft_complex_inverse (gsl_complex_packed_array @var{data}, size_t @var{stride}, size_t @var{n}, const gsl_fft_complex_wavetable * @var{wavetable}, gsl_fft_complex_workspace * @var{work})
+
+These functions compute forward, backward and inverse FFTs of length
+@var{n} with stride @var{stride}, on the packed complex array
+@var{data}, using a mixed radix decimation-in-frequency algorithm.
+There is no restriction on the length @var{n}.  Efficient modules are
+provided for subtransforms of length 2, 3, 4, 5, 6 and 7.  Any remaining
+factors are computed with a slow, @math{O(n^2)}, general-@math{n}
+module. The caller must supply a @var{wavetable} containing the
+trigonometric lookup tables and a workspace @var{work}.  For the
+@code{transform} version of the function the @var{sign} argument can be
+either @code{forward} (@math{-1}) or @code{backward} (@math{+1}).
+
+The functions return a value of @code{0} if no errors were detected. The
+following @code{gsl_errno} conditions are defined for these functions:
+
+@table @code
+@item GSL_EDOM
+The length of the data @var{n} is not a positive integer (i.e. @var{n}
+is zero).
+
+@item GSL_EINVAL
+The length of the data @var{n} and the length used to compute the given
+@var{wavetable} do not match.
+@end table
+@end deftypefun
+
+@comment @node Example of using mixed-radix FFT routines for complex data
+@comment @subsection Example of using mixed-radix FFT routines for complex data
+
+Here is an example program which computes the FFT of a short pulse in a
+sample of length 630 (@math{=2*3*3*5*7}) using the mixed-radix
+algorithm.
+
+@example
+@verbatiminclude examples/fftmr.c
+@end example
+
+@noindent
+Note that we have assumed that the program is using the default
+@code{gsl} error handler (which calls @code{abort} for any errors).  If
+you are not using a safe error handler you would need to check the
+return status of all the @code{gsl} routines.
+
+@node Overview of real data FFTs
+@section Overview of real data FFTs
+@cindex FFT of real data
+The functions for real data are similar to those for complex data.
+However, there is an important difference between forward and inverse
+transforms.  The fourier transform of a real sequence is not real.  It is
+a complex sequence with a special symmetry:
+@tex
+\beforedisplay
+$$
+z_k = z_{N-k}^*
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+z_k = z_@{N-k@}^*
+@end example
+
+@end ifinfo
+@noindent
+A sequence with this symmetry is called @dfn{conjugate-complex} or
+@dfn{half-complex}.  This different structure requires different
+storage layouts for the forward transform (from real to half-complex)
+and inverse transform (from half-complex back to real).  As a
+consequence the routines are divided into two sets: functions in
+@code{gsl_fft_real} which operate on real sequences and functions in
+@code{gsl_fft_halfcomplex} which operate on half-complex sequences.
+
+Functions in @code{gsl_fft_real} compute the frequency coefficients of a
+real sequence.  The half-complex coefficients @math{c} of a real sequence
+@math{x} are given by fourier analysis,
+@tex
+\beforedisplay
+$$
+c_k = \sum_{j=0}^{N-1} x_j \exp(-2 \pi i j k /N)
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+c_k = \sum_@{j=0@}^@{N-1@} x_j \exp(-2 \pi i j k /N)
+@end example
+
+@end ifinfo
+@noindent
+Functions in @code{gsl_fft_halfcomplex} compute inverse or backwards
+transforms.  They reconstruct real sequences by fourier synthesis from
+their half-complex frequency coefficients, @math{c},
+@tex
+\beforedisplay
+$$
+x_j = {1 \over N} \sum_{k=0}^{N-1} c_k \exp(2 \pi i j k /N)
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+x_j = @{1 \over N@} \sum_@{k=0@}^@{N-1@} c_k \exp(2 \pi i j k /N)
+@end example
+
+@end ifinfo
+@noindent
+The symmetry of the half-complex sequence implies that only half of the
+complex numbers in the output need to be stored.  The remaining half can
+be reconstructed using the half-complex symmetry condition. This works
+for all lengths, even and odd---when the length is even the middle value
+where @math{k=N/2} is also real.  Thus only @var{N} real numbers are
+required to store the half-complex sequence, and the transform of a real
+sequence can be stored in the same size array as the original data.
+
+The precise storage arrangements depend on the algorithm, and are
+different for radix-2 and mixed-radix routines.  The radix-2 function
+operates in-place, which constrains the locations where each element can
+be stored.  The restriction forces real and imaginary parts to be stored
+far apart.  The mixed-radix algorithm does not have this restriction, and
+it stores the real and imaginary parts of a given term in neighboring
+locations (which is desirable for better locality of memory accesses).
+
+@node Radix-2 FFT routines for real data
+@section Radix-2 FFT routines for real data
+@cindex FFT of real data, radix-2 algorithm
+@cindex Radix-2 FFT for real data
+
+This section describes radix-2 FFT algorithms for real data.  They use
+the Cooley-Tukey algorithm to compute in-place FFTs for lengths which
+are a power of 2. 
+
+The radix-2 FFT functions for real data are declared in the header files
+@file{gsl_fft_real.h} 
+
+@deftypefun int gsl_fft_real_radix2_transform (double @var{data}[], size_t @var{stride}, size_t @var{n})
+
+This function computes an in-place radix-2 FFT of length @var{n} and
+stride @var{stride} on the real array @var{data}.  The output is a
+half-complex sequence, which is stored in-place.  The arrangement of the
+half-complex terms uses the following scheme: for @math{k < N/2} the
+real part of the @math{k}-th term is stored in location @math{k}, and
+the corresponding imaginary part is stored in location @math{N-k}.  Terms
+with @math{k > N/2} can be reconstructed using the symmetry 
+@c{$z_k = z^*_{N-k}$}
+@math{z_k = z^*_@{N-k@}}. 
+The terms for @math{k=0} and @math{k=N/2} are both purely
+real, and count as a special case.  Their real parts are stored in
+locations @math{0} and @math{N/2} respectively, while their imaginary
+parts which are zero are not stored.
+
+The following table shows the correspondence between the output
+@var{data} and the equivalent results obtained by considering the input
+data as a complex sequence with zero imaginary part,
+
+@example
+complex[0].real    =    data[0] 
+complex[0].imag    =    0 
+complex[1].real    =    data[1] 
+complex[1].imag    =    data[N-1]
+...............         ................
+complex[k].real    =    data[k]
+complex[k].imag    =    data[N-k] 
+...............         ................
+complex[N/2].real  =    data[N/2]
+complex[N/2].imag  =    0
+...............         ................
+complex[k'].real   =    data[k]        k' = N - k
+complex[k'].imag   =   -data[N-k] 
+...............         ................
+complex[N-1].real  =    data[1]
+complex[N-1].imag  =   -data[N-1]
+@end example
+@noindent
+Note that the output data can be converted into the full complex
+sequence using the function @code{gsl_fft_halfcomplex_unpack}
+described in the next section.
+
+@end deftypefun
+The radix-2 FFT functions for halfcomplex data are declared in the
+header file @file{gsl_fft_halfcomplex.h}.
+
+@deftypefun int gsl_fft_halfcomplex_radix2_inverse (double @var{data}[], size_t @var{stride}, size_t @var{n})
+@deftypefunx int gsl_fft_halfcomplex_radix2_backward (double @var{data}[], size_t @var{stride}, size_t @var{n})
+
+These functions compute the inverse or backwards in-place radix-2 FFT of
+length @var{n} and stride @var{stride} on the half-complex sequence
+@var{data} stored according the output scheme used by
+@code{gsl_fft_real_radix2}.  The result is a real array stored in natural
+order.
+
+@end deftypefun
+
+
+
+@node Mixed-radix FFT routines for real data
+@section Mixed-radix FFT routines for real data
+@cindex FFT of real data, mixed-radix algorithm
+@cindex Mixed-radix FFT, real data
+
+This section describes mixed-radix FFT algorithms for real data.  The
+mixed-radix functions work for FFTs of any length.  They are a
+reimplementation of the real-FFT routines in the Fortran @sc{fftpack} library
+by Paul Swarztrauber.  The theory behind the algorithm is explained in
+the article @cite{Fast Mixed-Radix Real Fourier Transforms} by Clive
+Temperton.  The routines here use the same indexing scheme and basic
+algorithms as @sc{fftpack}.
+
+The functions use the @sc{fftpack} storage convention for half-complex
+sequences.  In this convention the half-complex transform of a real
+sequence is stored with frequencies in increasing order, starting at
+zero, with the real and imaginary parts of each frequency in neighboring
+locations.  When a value is known to be real the imaginary part is not
+stored.  The imaginary part of the zero-frequency component is never
+stored.  It is known to be zero (since the zero frequency component is
+simply the sum of the input data (all real)).  For a sequence of even
+length the imaginary part of the frequency @math{n/2} is not stored
+either, since the symmetry 
+@c{$z_k = z_{N-k}^*$}
+@math{z_k = z_@{N-k@}^*} implies that this is
+purely real too.
+
+The storage scheme is best shown by some examples.  The table below
+shows the output for an odd-length sequence, @math{n=5}.  The two columns
+give the correspondence between the 5 values in the half-complex
+sequence returned by @code{gsl_fft_real_transform}, @var{halfcomplex}[] and the
+values @var{complex}[] that would be returned if the same real input
+sequence were passed to @code{gsl_fft_complex_backward} as a complex
+sequence (with imaginary parts set to @code{0}),
+
+@example
+complex[0].real  =  halfcomplex[0] 
+complex[0].imag  =  0
+complex[1].real  =  halfcomplex[1] 
+complex[1].imag  =  halfcomplex[2]
+complex[2].real  =  halfcomplex[3]
+complex[2].imag  =  halfcomplex[4]
+complex[3].real  =  halfcomplex[3]
+complex[3].imag  = -halfcomplex[4]
+complex[4].real  =  halfcomplex[1]
+complex[4].imag  = -halfcomplex[2]
+@end example
+
+@noindent
+The upper elements of the @var{complex} array, @code{complex[3]} and
+@code{complex[4]} are filled in using the symmetry condition.  The
+imaginary part of the zero-frequency term @code{complex[0].imag} is
+known to be zero by the symmetry.
+
+The next table shows the output for an even-length sequence, @math{n=6}
+In the even case there are two values which are purely real,
+
+@example
+complex[0].real  =  halfcomplex[0]
+complex[0].imag  =  0
+complex[1].real  =  halfcomplex[1] 
+complex[1].imag  =  halfcomplex[2] 
+complex[2].real  =  halfcomplex[3] 
+complex[2].imag  =  halfcomplex[4] 
+complex[3].real  =  halfcomplex[5] 
+complex[3].imag  =  0 
+complex[4].real  =  halfcomplex[3] 
+complex[4].imag  = -halfcomplex[4]
+complex[5].real  =  halfcomplex[1] 
+complex[5].imag  = -halfcomplex[2] 
+@end example
+
+@noindent
+The upper elements of the @var{complex} array, @code{complex[4]} and
+@code{complex[5]} are filled in using the symmetry condition.  Both
+@code{complex[0].imag} and @code{complex[3].imag} are known to be zero.
+
+All these functions are declared in the header files
+@file{gsl_fft_real.h} and @file{gsl_fft_halfcomplex.h}.
+
+@deftypefun {gsl_fft_real_wavetable *} gsl_fft_real_wavetable_alloc (size_t @var{n})
+@deftypefunx {gsl_fft_halfcomplex_wavetable *} gsl_fft_halfcomplex_wavetable_alloc (size_t @var{n})
+These functions prepare trigonometric lookup tables for an FFT of size
+@math{n} real elements.  The functions return a pointer to the newly
+allocated struct if no errors were detected, and a null pointer in the
+case of error.  The length @var{n} is factorized into a product of
+subtransforms, and the factors and their trigonometric coefficients are
+stored in the wavetable. The trigonometric coefficients are computed
+using direct calls to @code{sin} and @code{cos}, for accuracy.
+Recursion relations could be used to compute the lookup table faster,
+but if an application performs many FFTs of the same length then
+computing the wavetable is a one-off overhead which does not affect the
+final throughput.
+
+The wavetable structure can be used repeatedly for any transform of the
+same length.  The table is not modified by calls to any of the other FFT
+functions.  The appropriate type of wavetable must be used for forward
+real or inverse half-complex transforms.
+@end deftypefun
+
+@deftypefun void gsl_fft_real_wavetable_free (gsl_fft_real_wavetable * @var{wavetable})
+@deftypefunx void gsl_fft_halfcomplex_wavetable_free (gsl_fft_halfcomplex_wavetable * @var{wavetable})
+These functions free the memory associated with the wavetable
+@var{wavetable}. The wavetable can be freed if no further FFTs of the
+same length will be needed.
+@end deftypefun
+
+@noindent
+The mixed radix algorithms require additional working space to hold
+the intermediate steps of the transform,
+
+@deftypefun {gsl_fft_real_workspace *} gsl_fft_real_workspace_alloc (size_t @var{n})
+This function allocates a workspace for a real transform of length
+@var{n}.  The same workspace can be used for both forward real and inverse
+halfcomplex transforms.
+@end deftypefun
+
+@deftypefun void gsl_fft_real_workspace_free (gsl_fft_real_workspace * @var{workspace})
+This function frees the memory associated with the workspace
+@var{workspace}. The workspace can be freed if no further FFTs of the
+same length will be needed.
+@end deftypefun
+
+@noindent
+The following functions compute the transforms of real and half-complex
+data,
+
+@deftypefun int gsl_fft_real_transform (double @var{data}[], size_t @var{stride}, size_t @var{n}, const gsl_fft_real_wavetable * @var{wavetable}, gsl_fft_real_workspace * @var{work})
+@deftypefunx int gsl_fft_halfcomplex_transform (double @var{data}[], size_t @var{stride}, size_t @var{n}, const gsl_fft_halfcomplex_wavetable * @var{wavetable}, gsl_fft_real_workspace * @var{work})
+These functions compute the FFT of @var{data}, a real or half-complex
+array of length @var{n}, using a mixed radix decimation-in-frequency
+algorithm.  For @code{gsl_fft_real_transform} @var{data} is an array of
+time-ordered real data.  For @code{gsl_fft_halfcomplex_transform}
+@var{data} contains fourier coefficients in the half-complex ordering
+described above.  There is no restriction on the length @var{n}.
+Efficient modules are provided for subtransforms of length 2, 3, 4 and
+5.  Any remaining factors are computed with a slow, @math{O(n^2)},
+general-n module.  The caller must supply a @var{wavetable} containing
+trigonometric lookup tables and a workspace @var{work}. 
+@end deftypefun
+
+@deftypefun int gsl_fft_real_unpack (const double @var{real_coefficient}[], gsl_complex_packed_array @var{complex_coefficient}, size_t @var{stride}, size_t @var{n})
+
+This function converts a single real array, @var{real_coefficient} into
+an equivalent complex array, @var{complex_coefficient}, (with imaginary
+part set to zero), suitable for @code{gsl_fft_complex} routines.  The
+algorithm for the conversion is simply,
+
+@example
+for (i = 0; i < n; i++)
+  @{
+    complex_coefficient[i].real 
+      = real_coefficient[i];
+    complex_coefficient[i].imag 
+      = 0.0;
+  @}
+@end example
+@end deftypefun
+
+@deftypefun int gsl_fft_halfcomplex_unpack (const double @var{halfcomplex_coefficient}[], gsl_complex_packed_array @var{complex_coefficient}, size_t @var{stride}, size_t @var{n})
+
+This function converts @var{halfcomplex_coefficient}, an array of
+half-complex coefficients as returned by @code{gsl_fft_real_transform}, into an
+ordinary complex array, @var{complex_coefficient}.  It fills in the
+complex array using the symmetry 
+@c{$z_k = z_{N-k}^*$}
+@math{z_k = z_@{N-k@}^*}
+to reconstruct the redundant elements.  The algorithm for the conversion
+is,
+
+@example  
+complex_coefficient[0].real 
+  = halfcomplex_coefficient[0];
+complex_coefficient[0].imag 
+  = 0.0;
+
+for (i = 1; i < n - i; i++)
+  @{
+    double hc_real 
+      = halfcomplex_coefficient[2 * i - 1];
+    double hc_imag 
+      = halfcomplex_coefficient[2 * i];
+    complex_coefficient[i].real = hc_real;
+    complex_coefficient[i].imag = hc_imag;
+    complex_coefficient[n - i].real = hc_real;
+    complex_coefficient[n - i].imag = -hc_imag;
+  @}
+
+if (i == n - i)
+  @{
+    complex_coefficient[i].real 
+      = halfcomplex_coefficient[n - 1];
+    complex_coefficient[i].imag 
+      = 0.0;
+  @}
+@end example
+@end deftypefun
+
+@comment @node Example of using mixed-radix FFT routines for real data
+@comment @subsection Example of using mixed-radix FFT routines for real data
+
+Here is an example program using @code{gsl_fft_real_transform} and
+@code{gsl_fft_halfcomplex_inverse}.  It generates a real signal in the
+shape of a square pulse.  The pulse is fourier transformed to frequency
+space, and all but the lowest ten frequency components are removed from
+the array of fourier coefficients returned by
+@code{gsl_fft_real_transform}.
+
+The remaining fourier coefficients are transformed back to the
+time-domain, to give a filtered version of the square pulse.  Since
+fourier coefficients are stored using the half-complex symmetry both
+positive and negative frequencies are removed and the final filtered
+signal is also real.
+
+@example
+@verbatiminclude examples/fftreal.c
+@end example
+
+@iftex
+@sp 1
+@center @image{fft-real-mixedradix,3.4in}
+
+@center Low-pass filtered version of a real pulse, 
+@center output from the example program.
+@end iftex
+
+@node FFT References and Further Reading
+@section References and Further Reading
+
+A good starting point for learning more about the FFT is the review
+article @cite{Fast Fourier Transforms: A Tutorial Review and A State of
+the Art} by Duhamel and Vetterli,
+
+@itemize @asis
+@item
+P. Duhamel and M. Vetterli.
+Fast fourier transforms: A tutorial review and a state of the art.
+@cite{Signal Processing}, 19:259--299, 1990.
+@end itemize
+
+@noindent
+To find out about the algorithms used in the GSL routines you may want
+to consult the document @cite{GSL FFT Algorithms} (it is included
+in GSL, as @file{doc/fftalgorithms.tex}).  This has general information
+on FFTs and explicit derivations of the implementation for each
+routine.  There are also references to the relevant literature.  For
+convenience some of the more important references are reproduced below.
+
+@noindent
+There are several introductory books on the FFT with example programs,
+such as @cite{The Fast Fourier Transform} by Brigham and @cite{DFT/FFT
+and Convolution Algorithms} by Burrus and Parks,
+
+@itemize @asis 
+@item
+E. Oran Brigham.
+@cite{The Fast Fourier Transform}.
+Prentice Hall, 1974.
+
+@item
+C. S. Burrus and T. W. Parks.
+@cite{DFT/FFT and Convolution Algorithms}.
+Wiley, 1984.
+@end itemize
+
+@noindent
+Both these introductory books cover the radix-2 FFT in some detail.
+The mixed-radix algorithm at the heart of the @sc{fftpack} routines is
+reviewed in Clive Temperton's paper,
+
+@itemize @asis 
+@item
+Clive Temperton.
+Self-sorting mixed-radix fast fourier transforms.
+@cite{Journal of Computational Physics}, 52(1):1--23, 1983.
+@end itemize
+
+@noindent
+The derivation of FFTs for real-valued data is explained in the
+following two articles,
+
+@itemize @asis
+@item
+Henrik V. Sorenson, Douglas L. Jones, Michael T. Heideman, and C. Sidney
+Burrus.
+Real-valued fast fourier transform algorithms.
+@cite{IEEE Transactions on Acoustics, Speech, and Signal Processing},
+ASSP-35(6):849--863, 1987.
+
+@item
+Clive Temperton.
+Fast mixed-radix real fourier transforms.
+@cite{Journal of Computational Physics}, 52:340--350, 1983.
+@end itemize
+
+@noindent
+In 1979 the IEEE published a compendium of carefully-reviewed Fortran
+FFT programs in @cite{Programs for Digital Signal Processing}.  It is a
+useful reference for implementations of many different FFT
+algorithms,
+
+@itemize @asis 
+@item
+Digital Signal Processing Committee and IEEE Acoustics, Speech, and Signal
+Processing Committee, editors.
+@cite{Programs for Digital Signal Processing}.
+IEEE Press, 1979.
+@end itemize
+@comment  @noindent
+@comment  There is also an annotated bibliography of papers on the FFT and related
+@comment  topics by Burrus,
+
+@comment  @itemize @asis 
+@comment  @item C. S. Burrus.  Notes on the FFT.
+@comment  @end itemize
+@comment  @noindent
+@comment The notes are available from @url{http://www-dsp.rice.edu/res/fft/fftnote.asc}.
+
+@noindent
+For large-scale FFT work we recommend the use of the dedicated FFTW library
+by Frigo and Johnson.  The FFTW library is self-optimizing---it
+automatically tunes itself for each hardware platform in order to
+achieve maximum performance.  It is available under the GNU GPL.
+
+@itemize @asis
+@item
+FFTW Website, @uref{http://www.fftw.org/}
+@end itemize
+
+@noindent
+The source code for @sc{fftpack} is available from Netlib,
+
+@itemize @asis
+@item
+FFTPACK, @uref{http://www.netlib.org/fftpack/}
+@end itemize
+
+
diff --git a/gsl-1.9/doc/fftalgorithms.bib b/gsl-1.9/doc/fftalgorithms.bib
new file mode 100644
index 0000000..1a8506e
--- /dev/null
+++ b/gsl-1.9/doc/fftalgorithms.bib
@@ -0,0 +1,240 @@
+@string{jcp = {Journal of Computational Physics}}
+@string{assp = {IEEE Transactions on Acoustics, Speech, and Signal Processing}}
+
+
+@Article{mehalic85,
+  author = 	 {Mehalic and Rustan and Route},
+  title = {Effects of Architecture Implementation on {DFT} Algorithm
+Performance},
+  journal = 	 assp,
+  year = 	 1985,
+  volume =	 {ASP-33},
+  pages =	 {684-693}
+}
+
+@Article{temperton83,
+  author = 	 {Clive Temperton},
+  title = 	 {Self-Sorting Mixed-Radix Fast Fourier Transforms},
+  journal = 	 {Journal of Computational Physics},
+  year = 	 1983,
+  volume =	 52,
+  number =	 1,
+  pages =	 {1-23}
+}
+
+@Article{symfft,
+   title= 	{Symmetric {FFT}s},
+   author=      {Paul N. Swarztrauber},
+   journal=     {Mathematics of Computation},
+   volume=      {47},
+   number=      {185},
+   pages =      {323-346},
+   year =       {1986}
+}
+
+@Article{temperton83real,
+  author = 	 {Clive Temperton},
+  title = 	 {Fast Mixed-Radix Real Fourier Transforms},
+  journal = 	 {Journal of Computational Physics},
+  year = 	 1983,
+  volume =	 52,
+  pages =	 {340-350}
+}
+
+@Article{temperton85,
+  author = {Clive Temperton},
+  title = {Implementation of a Self-Sorting In-Place Prime Factor {FFT} 
+           Algorithm},
+  journal = 	 {Journal of Computational Physics},
+  year = 	 1985,
+  volume =	 58,
+  pages =	 {283-299}
+}
+
+@Article{temperton83pfa,
+  author = 	 {Clive Temperton},
+  title = 	 {A Note on Prime Factor {FFT} Algorithms},
+  journal = 	 {Journal of Computational Physics},
+  year = 	 1983,
+  volume =	 58,
+  pages =	 {198-204}
+}
+
+@Article{burrus87real,
+  author = 	 {Henrik V. Sorenson and Douglas L. Jones and Michael T. Heideman and C. Sidney Burrus},
+  title = 	 {Real-Valued Fast Fourier Transform Algorithms},
+  journal = 	 assp,
+  year = 	 1987,
+  volume =	 {ASSP-35},
+  number =	 6,
+  pages =	 {849-863}
+}
+
+@Article{sorenson86,
+  author = 	 {Henrik V. Sorenson and Michael T. Heideman and C. Sidney Burrus},
+  title = 	 {On Computing the Split-Radix {FFT} },
+  journal = 	 assp,
+  year = 	 1986,
+  volume =	 {ASSP-34},
+  number =	 1,
+  pages =	 {152-156}
+}
+
+@Article{duhamel86,
+  author = 	 {Pierre Duhamel},
+  title = 	 {Implementation of Split-Radix {FFT} Algorithms for Complex, Real, and Real-Symmetric Data},
+  journal = 	 assp,
+  year = 	 1986,
+  volume =	 {ASSP-34},
+  number =	 2,
+  pages =	 {285-295}
+}
+
+@Article{visscher96,
+  author = 	 {P. B. Visscher},
+  title = 	 {The {FFT}: Fourier Transforming One Bit at a Time},
+  journal = 	 {Computers in Physics},
+  year = 	 1996,
+  volume =	 10,
+  number =	 5,
+  month =	 {Sep/Oct},
+  pages =	 {438-443}
+}
+
+@Article{uniyal94:_trans_real_value_sequen,
+  author = 	 {P. R. Uniyal},
+  title = 	 {Transforming Real-Value Sequences: Fast Fourier versus Fast Hartley Tranform Algorithms},
+  journal = 	 {IEEE Transactions on Signal Processing},
+  year = 	 1994,
+  volume =	 42,
+  number =	 11,
+  pages =	 {3249-3254 }
+}
+
+
+
+@Article{duhamel90,
+  author = 	 {P. Duhamel and M. Vetterli},
+  title = 	 {Fast Fourier Transforms: A Tutorial Review and A State of the Art},
+  journal = 	 {Signal Processing},
+  year = 	 1990,
+  volume =	 19,
+  pages =	 {259-299}
+}
+
+@Article{rodriguez89,
+  author = 	 {Jeffrey J. Rodriguez},
+  title = 	 {An Improved {FFT} Digit-Reversal Algorithm},
+  journal = 	 assp,
+  year = 	 1989,
+  volume =	 37,
+  number =	 8,
+  pages =	 {1298-1300}
+}
+
+@Article{rosel89,
+  author = {Petr R\"osel},
+  title = {Timing of some bit reversal algorithms},
+  journal = {Signal Processing},
+  year = 1989,
+  volume = 18,
+  pages = {425-433}
+}
+
+
+@Book{brigham74,
+  author = 	 {E. Oran Brigham},
+  title = 	 {The Fast Fourier Transform},
+  publisher = 	 {Prentice Hall},
+  year = 	 1974,
+  isbn =	 {0-13-307496-X}
+}
+
+
+
+
+@Book{elliott82,
+  author = 	 {Douglas F. Elliott and K. Ramamohan Rao},
+  title = 	 {Fast transforms: algorithms, analyses, applications},
+  publisher = 	 {Academic Press},
+  year = 	 1982,
+  note =	 {This book does not contain actual code, but covers the
+more advanced mathematics and number theory needed in the derivation
+of fast transforms.},
+  isbn =         {0-12-237080-5}
+}
+
+@Article{raderprimes,
+  author = 	 {Charles M. Rader},
+  title = 	 {Discrete Fourier Transform when the number of data
+                  samples is prime},
+  journal = 	 {IEEE Proceedings},
+  year = 	 1968,
+  volume =	 56,
+  number =	 6,
+  pages =	 {1107-1108}
+}
+
+@Book{mcclellan79,
+  author = 	 {James H. McClellan and Charles M. Rader},
+  title = 	 {Number Theory in Digital Signal Processing},
+  publisher = 	 {Prentice-Hall},
+  year = 	 1979,
+  isbn =         {ISBN 0-13-627349-1}
+}
+
+@Book{burrus84,
+  author = 	 {C. S. Burrus and T. W. Parks},
+  title = 	 {DFT/FFT and Convolution Algorithms},
+  publisher = 	 {Wiley},
+  year = 	 1984,
+  isbn =         {0-471-81932-8}
+}
+
+
+
+@Book{smith95,
+  author = 	 {Winthrop W. Smith and Joanne M. Smith},
+  title = 	 {Handbook of Real-Time Fast Fourier Transforms},
+  publisher = 	 {IEEE Press},
+  year = 	 1995,
+  isbn =         {0-7803-1091-8}
+}
+
+
+@Book{committee79,
+  title = 	 {Programs for Digital Signal Processing},
+  publisher = 	 {IEEE Press},
+  year = 	 1979,
+  editor =	 {Digital Signal Processing Committee and {IEEE Acoustics,
+Speech, and Signal Processing Committee}},
+  isbn =         {0-87942-127-4 (pb) 0-87942-128-2 (hb)}
+}
+
+@Book{blahut,
+  author = 	 {Richard E. Blahut},
+  title = 	 {Fast Algorithms for Digital Signal Processing},
+  publisher = 	 {Addison-Wesley},
+  year = 	 1984,
+  isbn = {0-201-10155-6}
+}
+
+@Unpublished{burrus-note,
+  author = 	 {C. S. Burrus},
+  title = 	 {Notes  on  the  {FFT}},
+  note =	 {Available from http://www-dsp.rice.edu/res/fft/fftnote.asc}
+}
+
+
+@Article{singleton,
+  author = 	 {Richard C. Singleton},
+  title = 	 {An Algorithm for Computing the Mixed Radix Fast
+                  Fourier Transform},
+  journal = 	 {IEEE Transactions on Audio and Electroacoustics},
+  year = 	 1969,
+  volume =	 {AU-17},
+  number =	 2,
+  month =	 {June},
+  pages =	 {93-103}
+}
+
diff --git a/gsl-1.9/doc/fftalgorithms.tex b/gsl-1.9/doc/fftalgorithms.tex
new file mode 100644
index 0000000..4f4a725
--- /dev/null
+++ b/gsl-1.9/doc/fftalgorithms.tex
@@ -0,0 +1,3296 @@
+\documentclass[fleqn,12pt]{article}
+%
+\setlength{\oddsidemargin}{-0.25in}
+\setlength{\textwidth}{7.0in}
+\setlength{\topmargin}{-0.25in}
+\setlength{\textheight}{9.5in}
+%
+\usepackage{algorithmic}
+\newenvironment{algorithm}{\begin{quote} %\vspace{1em}
+\begin{algorithmic}\samepage}{\end{algorithmic} %\vspace{1em} 
+\end{quote}}
+\newcommand{\Real}{\mathop{\mathrm{Re}}}
+\newcommand{\Imag}{\mathop{\mathrm{Im}}}
+
+\begin{document}
+\title{FFT Algorithms}
+\author{Brian Gough, {\tt bjg@network-theory.co.uk}}
+\date{May 1997}
+\maketitle
+
+\section{Introduction}
+
+Fast Fourier Transforms (FFTs) are efficient algorithms for
+calculating the discrete fourier transform (DFT),
+%
+\begin{eqnarray}
+h_a &=& \mathrm{DFT}(g_b) \\
+    &=& \sum_{b=0}^{N-1} g_b \exp(-2\pi i a b /N) \qquad 0 \leq a \leq N-1\\
+    &=& \sum_{b=0}^{N-1} g_b W_N^{ab} \qquad W_N= \exp(-2\pi i/N)
+\end{eqnarray}
+%
+The DFT usually arises as an approximation to the continuous fourier
+transform when functions are sampled at discrete intervals in space or
+time. The naive evaluation of the discrete fourier transform is a
+matrix-vector multiplication ${\mathbf W}\vec{g}$, and would take
+$O(N^2)$ operations for $N$ data-points. The general principle of the
+Fast Fourier Transform algorithms is to use a divide-and-conquer
+strategy to factorize the matrix $W$ into smaller sub-matrices,
+typically reducing the operation count to $O(N \sum f_i)$ if $N$ can
+be factorized into smaller integers, $N=f_1 f_2 \dots f_n$.
+
+This chapter explains the algorithms used in the GSL FFT routines
+and provides some information on how to extend them. To learn more about
+the FFT you should read the review article {\em Fast Fourier
+Transforms: A Tutorial Review and A State of the Art} by Duhamel and
+Vetterli~\cite{duhamel90}. There are several introductory books on the
+FFT with example programs, such as {\em The Fast Fourier Transform} by
+Brigham~\cite{brigham74} and {\em DFT/FFT and Convolution Algorithms}
+by Burrus and Parks~\cite{burrus84}. In 1979 the IEEE published a
+compendium of carefully-reviewed Fortran FFT programs in {\em Programs
+for Digital Signal Processing}~\cite{committee79} which is a useful
+reference for implementations of many different FFT algorithms. If you
+are interested in using DSPs then the {\em Handbook of Real-Time Fast
+Fourier Transforms}~\cite{smith95} provides detailed information on
+the algorithms and hardware needed to design, build and test DSP
+applications. Many FFT algorithms rely on results from number theory.
+These results are covered in the books {\em Fast transforms:
+algorithms, analyses, applications}, by Elliott and
+Rao~\cite{elliott82}, {\em Fast Algorithms for Digital Signal
+Processing} by Blahut~\cite{blahut} and {\em Number Theory in Digital
+Signal Processing} by McClellan and Rader~\cite{mcclellan79}. There is
+also an annotated bibliography of papers on the FFT and related topics
+by Burrus~\cite{burrus-note}.
+
+\section{Families of FFT algorithms}
+%
+There are two main families of FFT algorithms: the Cooley-Tukey
+algorithm and the Prime Factor algorithm. These differ in the way they
+map the full FFT into smaller sub-transforms. Of the Cooley-Tukey
+algorithms there are two types of routine in common use: mixed-radix
+(general-$N$) algorithms and radix-2 (power of 2) algorithms. Each
+type of algorithm can be further classified by additional characteristics,
+such as whether it operates in-place or uses additional scratch space,
+whether its output is in a sorted or scrambled order, and whether it
+uses decimation-in-time or -frequency iterations.
+
+Mixed-radix algorithms work by factorizing the data vector into
+shorter lengths. These can then be transformed by small-$N$ FFTs.
+Typical programs include FFTs for small prime factors, such as 2, 3,
+5, \dots which are highly optimized. The small-$N$ FFT modules act as
+building blocks and can be multiplied together to make longer
+transforms. By combining a reasonable set of modules it is possible to
+compute FFTs of many different lengths. If the small-$N$ modules are
+supplemented by an $O(N^2)$ general-$N$ module then an FFT of any
+length can be computed, in principle. Of course, any lengths which
+contain large prime factors would perform only as $O(N^2)$.
+
+Radix-2 algorithms, or ``power of two'' algorithms, are simplified
+versions of the mixed-radix algorithm. They are restricted to lengths
+which are a power of two. The basic radix-2 FFT module only involves
+addition and subtraction, so the algorithms are very simple. Radix-2
+algorithms have been the subject of much research into optimizing the
+FFT. Many of the most efficient radix-2 routines are based on the
+``split-radix'' algorithm. This is actually a hybrid which combines
+the best parts of both radix-2 and radix-4 (``power of 4'')
+algorithms~\cite{sorenson86,duhamel86}.
+
+The prime factor algorithm (PFA) is an alternative form of general-$N$
+algorithm based on a different way of recombining small-$N$ FFT
+modules~\cite{temperton83pfa,temperton85}. It has a very simple indexing
+scheme which makes it attractive. However it only works in the case
+where all factors are mutually prime. This requirement makes it more
+suitable as a specialized algorithm for given lengths.
+
+
+\begin{subsection}{FFTs of prime lengths}
+%
+  Large prime lengths cannot be handled efficiently by any of these
+algorithms. However it may still possible to compute a DFT, by using
+results from number theory. Rader showed that it is possible to
+convert a length-$p$ FFT (where $p$ is prime) into a convolution of
+length-($p-1$).  There is a simple identity between the convolution of
+length $N$ and the FFT of the same length, so if $p-1$ is easily
+factorizable this allows the convolution to be computed efficiently
+via the FFT. The idea is presented in the original paper by
+Rader~\cite{raderprimes} (also reprinted in~\cite{mcclellan79}), but
+for more details see the theoretical books mentioned earlier.
+\end{subsection}
+
+%\subsection{In-place algorithms vs algorithms using scratch space}
+%%
+%For simple algorithms, such as the Radix-2 algorithms, the FFT can be
+%performed in-place, without additional memory. For this to be possible
+%the index calculations must be simple enough that the calculation of
+%the result to be stored in index $k$ can be carried out at the same
+%time as the data in $k$ is used, so that no temporary storage is
+%needed.
+
+%The mixed-radix algorithm is too complicated for this. It requires an
+%area of scratch space sufficient to hold intermediate copies of the
+%input data.
+
+%\subsection{Self-sorting algorithms vs scrambling algorithms}
+%%
+%A self-sorting algorithm At each iteration of the FFT internal permutations are included
+%which leave the final iteration in its natural order.
+
+
+%The mixed-radix algorithm can be made self-sorting. The additional
+%scratch space helps here, although it is in fact possible to design
+%self-sorting algorithms which do not require scratch
+%space. 
+
+
+%The in-place radix-2 algorithm is not self-sorting. The data is
+%permuted, and transformed into bit-reversed order. Note that
+%bit-reversal only refers to the order of the array, not the values for
+%each index which are of course not changed. More precisely, the data
+%stored in location $[b_n \dots b_2 b_1 b_0]$ is moved to location
+%$[b_0 b_1 b_2 \dots b_n]$, where $b_0 \dots b_n$ are the raw bits of a
+%given index. Placing the data in bit reversed order is easily done,
+%using right shifts to extract the least significant bits of the index
+%and left-shifts to feed them into a register for the bit-reversed
+%location. Simple radix-2 FFT usually recompute the bit reverse
+%ordering in the naive way for every call. For repeated calls it might
+%be worthwhile to precompute the permutation and cache it. There are
+%also more sophisticated algorithms which reduce the number of
+%operations needed to perform bit-reversal~\cite{bit-reversal}.
+
+
+%\begin{subsection}{Decimation-in-time (DIT) vs Decimation-in-Frequency (DIF)}
+
+%\end{subsection}
+
+
+\subsection{Optimization}
+%
+There is no such thing as the single fastest FFT {\em algorithm}. FFT
+algorithms involve a mixture of floating point calculations, integer
+arithmetic and memory access. Each of these operations will have
+different relative speeds on different platforms.  The performance of
+an algorithm is a function of the hardware it is implemented on.  The
+goal of optimization is thus to choose the algorithm best suited to
+the characteristics of a given hardware platform.
+
+For example, the Winograd Fourier Transform (WFTA) is an algorithm
+which is designed to reduce the number of floating point
+multiplications in the FFT. However, it does this at the expense of
+using many more additions and data transfers than other algorithms.
+As a consequence the WFTA might be a good candidate algorithm for
+machines where data transfers occupy a negligible time relative to
+floating point arithmetic. However on most modern machines, where the
+speed of data transfers is comparable to or slower than floating point
+operations, it would be outperformed by an algorithm which used a
+better mix of operations (i.e. more floating point operations but
+fewer data transfers).
+
+For a study of this sort of effect in detail, comparing the different
+algorithms on different platforms consult the paper {\it Effects of
+Architecture Implementation on DFT Algorithm Performance} by Mehalic,
+Rustan and Route~\cite{mehalic85}. The paper was written in the early
+1980's and has data for super- and mini-computers which you are
+unlikely to see today, except in a museum. However, the methodology is
+still valid and it would be interesting to see similar results for
+present day computers.
+
+
+\section{FFT Concepts}
+%
+Factorization is the key principle of the mixed-radix FFT divide-and-conquer
+strategy. If $N$ can be factorized into a product of $n_f$ integers,
+%
+\begin{equation}
+N = f_1 f_2 ... f_{n_f} ,
+\end{equation}
+%
+then the FFT itself can be divided into smaller FFTs for each factor.
+More precisely, an FFT of length $N$ can be broken up into,
+%
+\begin{quote}
+\begin{tabular}{l}
+$(N/f_1)$ FFTs of length $f_1$, \\
+$(N/f_2)$ FFTs of length $f_2$, \\
+\dots \\
+$(N/f_{n_f})$ FFTs of length $f_{n_f}$. 
+\end{tabular}
+\end{quote}
+%
+The total number of operations for these sub-operations will be $O(
+N(f_1 + f_2 + ... + f_{n_f}))$. When the factors of $N$ are all small
+integers this will be substantially less than $O(N^2)$. For example,
+when $N$ is a power of 2 an FFT of length $N=2^m$ can be reduced to $m
+N/2$ FFTs of length 2, or $O(N\log_2 N)$ operations.  Here is a
+demonstration which shows this:
+
+We start with the full DFT,
+%
+\begin{eqnarray}
+h_a &=& \sum_{b=0}^{N-1} g_b W_N^{ab}       \qquad W_N=\exp(-2\pi i/N)
+\end{eqnarray}
+%
+and split the sum into even and odd terms,
+%
+\begin{eqnarray}
+\phantom{h_a}
+%   &=& \left(\sum_{b=0,2,4,\dots} + \sum_{b=1,3,5,\dots}\right) g_b W_N^{ab}\\
+   &=& \sum_{b=0}^{N/2-1} g_{2b} W_N^{a(2b)} + 
+      \sum_{b=0}^{N/2-1} g_{2b+1} W_N^{a(2b+1)}.
+\end{eqnarray}
+%
+This converts the original DFT of length $N$ into two DFTs of length
+$N/2$,
+%
+\begin{equation}
+h_a = \sum_{b=0}^{N/2-1} g_{2b} W_{(N/2)}^{ab} + 
+      W_N^a \sum_{b=0}^{N/2-1} g_{2b+1} W_{(N/2)}^{ab} 
+\end{equation}
+%
+The first term is a DFT of the even elements of $g$. The second term
+is a DFT of the odd elements of $g$, premultiplied by an exponential
+factor $W_N^k$ (known as a {\em twiddle factor}).
+%
+\begin{equation}
+\mathrm{DFT}(h)  =  \mathrm{DFT}(g_{even}) + W_N^k \mathrm{DFT}(g_{odd})
+\end{equation}
+%
+By splitting the DFT into its even and odd parts we have reduced the
+operation count from $N^2$ (for a DFT of length $N$) to $2 (N/2)^2$
+(for two DFTs of length $N/2$). The cost of the splitting is that we
+need an additional $O(N)$ operations to multiply by the twiddle factor
+$W_N^k$ and recombine the two sums.
+
+We can repeat the splitting procedure recursively $\log_2 N$ times
+until the full DFT is reduced to DFTs of single terms. The DFT of a
+single value is just the identity operation, which costs nothing.
+However since $O(N)$ operations were needed at each stage to recombine
+the even and odd parts the total number of operations to obtain the
+full DFT is $O(N \log_2 N)$. If we had used a length which was a
+product of factors $f_1$, $f_2$, $\dots$ we could have split the sum
+in a similar way. First we would split terms corresponding to the
+factor $f_1$, instead of the even and odd terms corresponding to a
+factor of two.  Then we would repeat this procedure for the subsequent
+factors. This would lead to a final operation count of $O(N \sum
+f_i)$.
+
+This procedure gives some motivation for why the number of operations
+in a DFT can in principle be reduced from $O(N^2)$ to $O(N \sum f_i)$.
+It does not give a good explanation of how to implement the algorithm
+in practice which is what we shall do in the next section.
+
+\section{Radix-2 Algorithms}
+%
+For radix-2 FFTs it is natural to write array indices in binary form
+because the length of the data is a power of two. This is nicely
+explained in the article {\em The FFT: Fourier Transforming One Bit at
+a Time} by P.B. Visscher~\cite{visscher96}. A binary representation
+for indices is the key to deriving the simplest efficient radix-2
+algorithms.
+
+We can write an index $b$ ($0 \leq b < 2^{n-1}$) in binary
+representation like this,
+%
+\begin{equation}
+b = [b_{n-1} \dots b_1 b_0] = 2^{n-1}b_{n-1} + \dots + 2 b_1 + b_0 .
+\end{equation}
+%
+Each of the $b_0, b_1, \dots, b_{n-1}$ are the bits (either 0 or 1) of
+$b$.
+
+Using this notation the original definition of the DFT
+can be rewritten as a sum over the bits of $b$,
+%
+\begin{equation} 
+h(a) = \sum_{b=0}^{N-1} g_b \exp(-2\pi i a b /N)
+\end{equation}
+%
+to give an equivalent summation like this,
+%
+\begin{equation} 
+h([a_{n-1} \dots a_1 a_0]) = 
+\sum_{b_0=0}^{1} 
+\sum_{b_1=0}^{1} 
+\dots
+\sum_{b_{n-1}=0}^{1} 
+ g([b_{n-1} \dots b_1 b_0]) W_N^{ab}
+\end{equation}
+%
+where the bits of $a$ are $a=[a_{n-1} \dots a_1 a_0]$. 
+
+To reduce the number of operations in the sum we will use the
+periodicity of the exponential term,
+%
+\begin{equation}
+W_N^{x+N} = W_N^{x}.
+\end{equation}
+%
+Most of the products $ab$ in $W_N^{ab}$ are greater than $N$. By
+making use of this periodicity they can all be collapsed down into the
+range $0\dots N-1$. This allows us to reduce the number of operations
+by combining common terms, modulo $N$. Using this idea we can derive
+decimation-in-time or decimation-in-frequency algorithms, depending on
+how we break the DFT summation down into common terms. We'll first
+consider the decimation-in-time algorithm.
+
+\subsection{Radix-2 Decimation-in-Time (DIT)}
+%
+To derive the the decimation-in-time algorithm we start by separating
+out the most significant bit of the index $b$,
+%
+\begin{equation}
+[b_{n-1} \dots b_1 b_0] = 2^{n-1}b_{n-1} + [b_{n-2} \dots b_1 b_0]
+\end{equation}
+%
+Now we can evaluate the innermost sum of the DFT without any
+dependence on the remaining bits of $b$ in the exponential,
+%
+\begin{eqnarray} 
+h([a_{n-1} \dots a_1 a_0]) &=& 
+\sum_{b_0=0}^{1} 
+\sum_{b_1=0}^{1} 
+\dots
+\sum_{b_{n-1}=0}^{1} 
+ g(b) 
+W_N^{a(2^{n-1}b_{n-1}+[b_{n-2} \dots b_1 b_0])} \\
+ &=& 
+\sum_{b_0=0}^{1} 
+\sum_{b_1=0}^{1} 
+\dots
+\sum_{b_{n-2}=0}^{1} 
+W_N^{a[b_{n-2} \dots b_1 b_0]}
+\sum_{b_{n-1}=0}^{1} 
+ g(b) 
+W_N^{a(2^{n-1}b_{n-1})}
+\end{eqnarray}
+%
+Looking at the term $W_N^{a(2^{n-1}b_{n-1})}$ we see that we can also
+remove most of the dependence on $a$ as well, by using the periodicity of the
+exponential,
+%
+\begin{eqnarray}
+W_N^{a(2^{n-1}b_{n-1})} &=&
+\exp(-2\pi i [a_{n-1} \dots a_1 a_0] 2^{n-1} b_{n-1}/ 2^n )\\
+&=& \exp(-2\pi i [a_{n-1} \dots a_1 a_0] b_{n-1}/ 2 )\\
+&=& \exp(-2\pi i ( 2^{n-2}a_{n-1} + \dots + a_1 + (a_0/2)) b_{n-1} )\\
+&=& \exp(-2\pi i a_0 b_{n-1}/2 ) \\
+&=& W_2^{a_0 b_{n-1}}
+\end{eqnarray}
+%
+Thus the innermost exponential term simplifies so that it only
+involves the highest order bit of $b$ and the lowest order bit of $a$,
+and the sum can be reduced to,
+%
+\begin{equation}
+h([a_{n-1} \dots a_1 a_0])
+= 
+\sum_{b_0=0}^{1} 
+\sum_{b_1=0}^{1} 
+\dots
+\sum_{b_{n-2}=0}^{1} 
+W_N^{a[b_{n-2} \dots b_1 b_0]}
+\sum_{b_{n-1}=0}^{1} 
+ g(b) 
+W_2^{a_0 b_{n-1}}.
+\end{equation}
+%
+We can repeat this this procedure for the next most significant bit of
+$b$, $b_{n-2}$, using a similar identity,
+%
+\begin{eqnarray}
+W_N^{a(2^{n-2}b_{n-2})} 
+&=& \exp(-2\pi i [a_{n-1} \dots a_1 a_0] 2^{n-2} b_{n-2}/ 2^n )\\
+&=& W_4^{ [a_1 a_0] b_{n-2}}.
+\end{eqnarray}
+%
+to give a formula with even less dependence on the bits of $a$, 
+%
+\begin{equation}
+h([a_{n-1} \dots a_1 a_0])
+= 
+\sum_{b_0=0}^{1} 
+\sum_{b_1=0}^{1} 
+\dots
+\sum_{b_{n-3}=0}^{1} 
+W_N^{a[b_{n-3} \dots b_1 b_0]}
+\sum_{b_{n-2}=0}^{1} 
+W_4^{[a_1 a_0] b_{n-2}}
+\sum_{b_{n-1}=0}^{1} 
+ g(b) 
+W_2^{a_0 b_{n-1}}.
+\end{equation}
+%
+If we repeat the process for all the remaining bits we obtain a
+simplified DFT formula which is the basis of the radix-2
+decimation-in-time algorithm,
+%
+\begin{eqnarray}
+h([a_{n-1} \dots a_1 a_0]) &=& 
+\sum_{b_0=0}^{1} 
+W_{N}^{[a_{n-1} \dots a_1 a_0]b_0} 
+%\sum_{b_1=0}^{1} 
+%W_{N/2}^{[a_{n-1} \dots a_1 a_0]b_1} 
+\dots
+\sum_{b_{n-2}=0}^{1} 
+W_4^{ [a_1 a_0] b_{n-2}}
+\sum_{b_{n-1}=0}^{1} 
+W_2^{a_0 b_{n-1}}
+g(b)
+\end{eqnarray}
+%
+To convert the formula to an algorithm we expand out the sum
+recursively, evaluating each of the intermediate summations, which we
+denote by $g_1$, $g_2$, \dots, $g_n$,
+%
+\begin{eqnarray}
+g_1(a_0,  b_{n-2}, b_{n-3}, \dots, b_1, b_0) 
+&=& 
+\sum_{b_{n-1}=0}^{1} 
+W_2^{a_0 b_{n-1}}
+g([b_{n-1}  b_{n-2}  b_{n-3}  \dots  b_1  b_0]) \\
+g_2(a_0, {}_{\phantom{-2}} a_{1}, b_{n-3}, \dots, b_1, b_0) 
+&=& 
+\sum_{b_{n-2}=0}^{1} 
+W_4^{[a_1 a_0] b_{n-2}}
+g_1(a_0, b_{n-2}, b_{n-3}, \dots, b_1, b_0) \\
+g_3(a_0, {}_{\phantom{-2}} a_{1}, {}_{\phantom{-3}} a_{2}, \dots, b_1, b_0) 
+&=& 
+\sum_{b_{n-3}=0}^{1} 
+W_8^{[a_2 a_1 a_0] b_{n-2}}
+g_2(a_0, a_1, b_{n-3}, \dots, b_1, b_0) \\
+\dots &=& \dots \\
+g_n(a_0, a_{1}, a_{2}, \dots, a_{n-2}, a_{n-1}) 
+&=&
+\sum_{b_{0}=0}^{1} 
+W_N^{[a_{n-1} \dots a_1 a_0]b_0}
+g_{n-1}(a_0, a_1, a_2, \dots, a_{n-2}, b_0) 
+\end{eqnarray}
+%
+After the final sum, we can obtain the transform $h$ from $g_n$,
+%
+\begin{equation}
+h([a_{n-1} \dots a_1 a_0]) 
+= 
+g_n(a_0, a_1, \dots, a_{n-1}) 
+\end{equation}
+% 
+Note that we left the storage arrangements of the intermediate sums
+unspecified by using the bits as function arguments and not as an
+index. The storage of intermediate sums is different for the
+decimation-in-time and decimation-in-frequency algorithms.
+
+Before deciding on the best storage scheme we'll show that the results
+of each stage, $g_1$, $g_2$, \dots, can be carried out {\em
+in-place}. For example, in the case of $g_1$, the inputs are,
+%
+\begin{equation}
+g([\underline{b_{n-1}} b_{n-2} b_{n-3} \dots b_1 b_0])
+\end{equation}
+%
+for $b_{n-1}=(0,1)$, and the corresponding outputs are,
+%
+\begin{equation}
+g_1(\underline{a_0},b_{n-2}, b_{n-3}, \dots, b_1, b_0)
+\end{equation}
+%
+for $a_0=(0,1)$.  It's clear that if we hold $b_{n-2}, b_{n-3}, \dots,
+b_1, b_0$ fixed and compute the sum over $b_{n-1}$ in memory for both
+values of $a_0 = 0,1$ then we can store the result for $a_0=0$ in the
+location which originally had $b_0=0$ and the result for $a_0=1$ in
+the location which originally had $b_0=1$. The two inputs and two
+outputs are known as {\em dual node pairs}. At each stage of the
+calculation the sums for each dual node pair are independent of the
+others. It is this property which allows an in-place calculation.
+
+So for an in-place pass our storage has to be arranged so that the two
+outputs $g_1(a_0,\dots)$ overwrite the two input terms
+$g([b_{n-1},\dots])$. Note that the order of $a$ is reversed from the
+natural order of $b$. i.e. the least significant bit of $a$
+replaces the most significant bit of $b$. This is inconvenient
+because $a$ occurs in its natural order in all the exponentials,
+$W^{ab}$. We could keep track of both $a$ and its bit-reverse,
+$a^{\mathit bit-reversed}$ at all times but there is a neat trick
+which avoids this: if we bit-reverse the order of the input data $g$
+before we start the calculation we can also bit-reverse the order of
+$a$ when storing intermediate results. Since the storage involving $a$
+was originally in bit-reversed order the switch in the input $g$ now
+allows us to use normal ordered storage for $a$, the same ordering
+that occurs in the exponential factors.
+
+This is complicated to explain, so here is an example of the 4 passes
+needed for an $N=16$ decimation-in-time FFT, with the initial data
+stored in bit-reversed order,
+%
+\begin{eqnarray}
+g_1([b_0 b_1 b_2 a_0]) 
+&=& 
+\sum_{b_3=0}^{1} W_2^{a_0 b_3} g([b_0 b_1 b_2 b_3])
+\\
+g_2([b_0 b_1 a_1 a_0]) 
+&=& 
+\sum_{b_2=0}^{1} W_4^{[a_1 a_0] b_2} g_1([b_0 b_1 b_2 a_0])
+\\
+g_3([b_0 a_2 a_1 a_0]) 
+&=& 
+\sum_{b_1=0}^{1} W_8^{[a_2 a_1 a_0] b_1} g_2([b_0 b_1 a_1 a_0])
+\\
+h(a) = g_4([a_3 a_2 a_1 a_0]) 
+&=& 
+\sum_{b_0=0}^{1} W_{16}^{[a_3 a_2 a_1 a_0] b_0} g_3([b_0 a_2 a_1 a_0])
+\end{eqnarray}
+%
+We compensate for the bit reversal of the input data by accessing $g$
+with the bit-reversed form of $b$ in the first stage. This ensures
+that we are still carrying out the same calculation, using the same
+data, and not accessing different values. Only single bits of $b$ ever
+occur in the exponential so we never need the bit-reversed form of
+$b$.
+
+Let's examine the third pass in detail,
+%
+\begin{equation}
+g_3([b_0 a_2 a_1 a_0]) 
+=
+\sum_{b_1=0}^{1} W_8^{[a_2 a_1 a_0] b_1} g_2([b_0 b_1 a_1 a_0])
+\end{equation}
+%
+First note that only one bit, $b_1$, varies in each summation.  The
+other bits of $b$ ($b_0$) and of $a$ ($a_1 a_0$) are essentially
+``spectators'' -- we must loop over all combinations of these bits and
+carry out the same basic calculation for each, remembering to update
+the exponentials involving $W_8$ appropriately.  If we are storing the
+results in-place (with $g_3$ overwriting $g_2$ we will need to compute
+the sums involving $b_1=0,1$ and $a_2=0,1$ simultaneously.
+%
+\begin{equation}
+\left(
+\begin{array}{c}
+g_3([b_0 0 a_1 a_0]) \vphantom{W_8^{[]}} \\
+g_3([b_0 1 a_1 a_0]) \vphantom{W_8^{[]}} 
+\end{array}
+\right)
+=
+\left(
+\begin{array}{c}
+g_2([b_0 0 a_1 a_0]) + W_8^{[0 a_1 a_0]} g_2([b_2 1 a_1 a_0]) \\
+g_2([b_0 0 a_1 a_0]) + W_8^{[1 a_1 a_0]} g_2([b_2 1 a_1 a_0])
+\end{array}
+\right)
+\end{equation}
+%
+We can write this in a more symmetric form by simplifying the exponential,
+%
+\begin{equation}
+W_8^{[a_2 a_1 a_0]} 
+= W_8^{4 a_2 + [a_1 a_0]} 
+= (-1)^{a_2} W_8^{[a_1 a_0]}
+\end{equation}
+%
+\begin{equation}
+\left(
+\begin{array}{c}
+g_3([b_0 0 a_1 a_0]) \vphantom{W_8^{[]}} \\
+g_3([b_0 1 a_1 a_0]) \vphantom{W_8^{[]}} 
+\end{array}
+\right)
+=
+\left(
+\begin{array}{c}
+g_2([b_0 0 a_1 a_0]) + W_8^{[a_1 a_0]} g_2([b_2 1 a_1 a_0]) \\
+g_2([b_0 0 a_1 a_0]) - W_8^{[a_1 a_0]} g_2([b_2 1 a_1 a_0])
+\end{array}
+\right)
+\end{equation}
+%
+The exponentials $W_8^{[a_1 a_0]}$ are referred to as {\em twiddle
+factors}. The form of this calculation, a symmetrical sum and
+difference involving a twiddle factor is called {\em a butterfly}.
+It is often shown diagrammatically, and in the case $b_0=a_0=a_1=0$
+would be drawn like this,
+%
+\begin{center}
+\setlength{\unitlength}{1cm}
+\begin{picture}(9,4)
+%
+%\put(0,0){\line(1,0){8}}
+%\put(0,0){\line(0,1){4}}
+%\put(8,4){\line(0,-1){4}}
+%\put(8,4){\line(-1,0){8}}
+%
+\put(0,1){$g_2(4)$} \put(4.5,1){$g_3(4)=g_2(0) - W^a_8 g_2(4)$}
+\put(0,3){$g_2(0)$} \put(4.5,3){$g_3(0)=g_2(0) + W^a_8 g_2(4)$}
+\put(1,1){\vector(1,0){0.5}}
+\put(1.5,1){\line(1,0){0.5}}
+\put(1.5,0.5){$W^a_8$}
+\put(1,3){\vector(1,0){0.5}}\put(1.5,3){\line(1,0){0.5}}
+\put(2,1){\circle*{0.1}}
+\put(2,3){\circle*{0.1}}
+\put(2,1){\vector(1,1){2}} 
+\put(2,1){\vector(1,0){1}} 
+\put(3,1){\line(1,0){1}}
+\put(3,0.5){$-1$}
+\put(2,3){\vector(1,-1){2}} 
+\put(2,3){\vector(1,0){1}} 
+\put(3,3){\line(1,0){1}}
+\put(4,1){\circle*{0.1}}
+\put(4,3){\circle*{0.1}}
+\end{picture}
+\end{center}
+%
+The inputs are shown on the left and the outputs on the right. The
+outputs are computed by multiplying the incoming lines by their
+accompanying factors (shown next to the lines) and summing the results
+at each node.
+
+In general, denoting the bit for dual-node pairs by $\Delta$ and the
+remaining bits of $a$ and $b$ by ${\hat a}$ and ${\hat b}$, the
+butterfly is,
+%
+\begin{equation}
+\left(
+\begin{array}{c}
+g({\hat b} + {\hat a}) \\
+g({\hat b} + \Delta + {\hat a}) \\
+\end{array}
+\right)
+\leftarrow
+\left(
+\begin{array}{c}
+g({\hat b} + {\hat a}) + W_{2\Delta}^{\hat a} g({\hat b} + \Delta + {\hat a})\\
+g({\hat b} + {\hat a}) - W_{2\Delta}^{\hat a} g({\hat b} + \Delta + {\hat a})
+\end{array}
+\right)
+\end{equation}
+%
+where ${\hat a}$ runs from $0 \dots \Delta-1$ and ${\hat b}$ runs
+through $0 \times 2\Delta$, $1\times 2\Delta$, $\dots$, $(N/\Delta -
+1)2\Delta$. The value of $\Delta$ is 1 on the first pass, 2 on the
+second pass and $2^{n-1}$ on the $n$-th pass.  Each pass requires
+$N/2$ in-place computations, each involving two input locations and
+two output locations.
+
+In the example above $\Delta = [100] = 4$, ${\hat a} = [a_1 a_0]$ and
+${\hat b} = [b_0 0 0 0]$.
+
+This leads to the canonical radix-2 decimation-in-time FFT algorithm
+for $2^n$ data points stored in the array $g(0) \dots g(2^n-1)$.
+%
+\begin{algorithm}
+\STATE bit-reverse ordering of $g$
+\STATE {$\Delta \Leftarrow 1$}
+\FOR {$\mbox{pass} = 1 \dots n$}
+  \STATE {$W \Leftarrow \exp(-2 \pi i / 2\Delta)$}
+  \FOR {$(a = 0 ; a < \Delta ; a{++})$}
+    \FOR{$(b = 0 ; b < N ; b {+=} 2*\Delta)$}
+        \STATE{$t_0 \Leftarrow g(b+a) + W^a g(b+\Delta+a)$}
+        \STATE{$t_1 \Leftarrow g(b+a) - W^a g(b+\Delta+a)$}
+        \STATE{$g(b+a) \Leftarrow t_0$}
+        \STATE{$g(b+\Delta+a) \Leftarrow t_1$}
+    \ENDFOR
+  \ENDFOR
+  \STATE{$\Delta \Leftarrow 2\Delta$}
+\ENDFOR
+\end{algorithm}
+%
+%This algorithm appears in Numerical Recipes as the routine {\tt
+%four1}, where the implementation is attributed to N.M. Brenner.
+%
+\subsection{Details of the Implementation}
+It is straightforward to implement a simple radix-2 decimation-in-time
+routine from the algorithm above. Some optimizations can be made by
+pulling the special case of $a=0$ out of the loop over $a$, to avoid
+unnecessary multiplications when $W^a=1$,
+%
+\begin{algorithm}
+    \FOR{$(b = 0 ; b < N ; b {+=} 2*\Delta)$}
+        \STATE{$t_0 \Leftarrow g(b) + g(b+\Delta)$}
+        \STATE{$t_1 \Leftarrow g(b) - g(b+\Delta)$}
+        \STATE{$g(b) \Leftarrow t_0$}
+        \STATE{$g(b+\Delta) \Leftarrow t_1$}
+    \ENDFOR
+\end{algorithm}
+%
+There are several algorithms for doing fast bit-reversal. We use the
+Gold-Rader algorithm, which is simple and does not require any working
+space,
+%
+\begin{algorithm}
+\FOR{$i = 0 \dots n - 2$}
+        \STATE {$ k = n / 2 $}
+        \IF {$i < j$}
+                \STATE {swap $g(i)$ and $g(j)$}
+        \ENDIF
+
+        \WHILE {$k \leq j$}
+                \STATE{$j \Leftarrow j - k$} 
+                \STATE{$k \Leftarrow k / 2$} 
+        \ENDWHILE
+      
+      \STATE{$j \Leftarrow j + k$}
+\ENDFOR
+\end{algorithm}
+%
+The Gold-Rader algorithm is typically twice as fast as a naive
+bit-reversal algorithm (where the bit reversal is carried out by
+left-shifts and right-shifts on the index).  The library also has a
+routine for the Rodriguez bit reversal algorithm, which also does not
+require any working space~\cite{rodriguez89}. There are faster bit
+reversal algorithms, but they all use additional scratch
+space~\cite{rosel89}.
+
+Within the loop for $a$ we can compute $W^a$  using a trigonometric
+recursion relation,
+%
+\begin{eqnarray}
+W^{a+1} &=& W W^a \\
+        &=& (\cos(2\pi/2\Delta) + i \sin(2\pi/2\Delta)) W^a
+\end{eqnarray}
+%
+This requires only $2 \log_2 N$ trigonometric calls, to compute the
+initial values of $\exp(2\pi i /2\Delta)$ for each pass.
+
+\subsection{Radix-2 Decimation-in-Frequency (DIF)}
+%
+To derive the decimation-in-frequency algorithm we start by separating
+out the lowest order bit of the index $a$. Here is an example for the
+decimation-in-frequency $N=16$ DFT.
+%
+\begin{eqnarray}
+W_{16}^{[a_3 a_2 a_1 a_0][b_3 b_2 b_1 b_0]} 
+&=&
+W_{16}^{[a_3 a_2 a_1 a_0][b_2 b_1 b_0]} W_{16}^{[a_3 a_2 a_1 a_0] [b_3
+0 0 0]} \\
+&=&
+W_8^{[a_3 a_2 a_1][b_2 b_1 b_0]} W_{16}^{a_0 [b_2 b_1 b_0]} W_2^{a_0
+b_3} \\
+&=&
+W_8^{[a_3 a_2 a_1][b_2 b_1 b_0]} W_{16}^{a_0 [b_2 b_1 b_0]} (-1)^{a_0 b_3}
+\end{eqnarray}
+%
+By repeating the same type of the expansion on the term,
+%
+\begin{equation}
+W_8^{[a_3 a_2 a_1][b_2 b_1 b_0]}
+\end{equation}
+%
+we can reduce the transform to an alternative simple form,
+%
+\begin{equation}
+h(a) = 
+\sum_{b_0=0}^1 (-1)^{a_3 b_0} W_4^{a_2 b_0}
+\sum_{b_1=0}^1 (-1)^{a_2 b_1} W_8^{a_1 [b_1 b_0]}
+\sum_{b_2=0}^1 (-1)^{a_1 b_2} W_{16}^{a_0 [b_2 b_1 b_0]}
+\sum_{b_3=0}^1 (-1)^{a_0 b_3} g(b)
+\end{equation}
+%
+To implement this we can again write the sum recursively. In this case
+we do not have the problem of the order of $a$ being bit reversed --
+the calculation can be done in-place using the natural ordering of
+$a$ and $b$,
+%
+\begin{eqnarray}
+g_1([a_0 b_2 b_1 b_0]) 
+&=&
+W_{16}^{a_0 [b_2 b_1 b_0]} 
+\sum_{b_3=0}^1 (-1)^{a_0 b_3} g([b_3 b_2 b_1 b_0]) \\
+g_2([a_0 a_1 b_1 b_0]) 
+&=&
+W_{8}^{a_1 [b_1 b_0]} 
+\sum_{b_2=0}^1 (-1)^{a_1 b_2} g_1([a_0 b_2 b_1 b_0]) \\
+g_3([a_0 a_1 a_2 b_0]) 
+&=&
+W_{4}^{a_2 b_0} 
+\sum_{b_1=0}^1 (-1)^{a_2 b_1} g_2([a_0 a_1 b_1 b_0]) \\
+h(a)
+= 
+g_4([a_0 a_1 a_2 a_3]) 
+&=&
+\sum_{b_0=0}^1 (-1)^{a_3 b_0} g_3([a_0 a_1 a_2 b_0])
+\end{eqnarray}
+%
+The final pass leaves the data for $h(a)$ in bit-reversed order, but
+this is easily fixed by a final bit-reversal of the ordering.
+
+The basic in-place calculation or butterfly for each pass is slightly
+different from the decimation-in-time version,
+%
+\begin{equation}
+\left(
+\begin{array}{c}
+g({\hat a} + {\hat b}) \\
+g({\hat a} + \Delta + {\hat b}) \\
+\end{array}
+\right)
+\leftarrow
+\left(
+\begin{array}{c}
+g({\hat a} + {\hat b}) +  g({\hat a} + \Delta + {\hat b})\\
+W_{\Delta}^{\hat b} 
+\left( g({\hat a} + {\hat b}) -  g({\hat a} + \Delta + {\hat b}) \right)
+\end{array}
+\right)
+\end{equation}
+%
+In each pass ${\hat b}$ runs from $0 \dots \Delta-1$ and ${\hat
+a}$ runs from $0, 2\Delta, \dots, (N/\Delta -1) \Delta$. On the first
+pass we start with $\Delta=16$, and on subsequent passes $\Delta$ takes
+the values $8, 4, \dots, 1$.
+
+This leads to the canonical radix-2 decimation-in-frequency FFT
+algorithm for $2^n$ data points stored in the array $g(0) \dots
+g(2^n-1)$.
+%
+\begin{algorithm}
+\STATE {$\Delta \Leftarrow 2^{n-1}$}
+\FOR {$\mbox{pass} = 1 \dots n$}
+  \STATE {$W \Leftarrow \exp(-2 \pi i / 2\Delta)$}
+  \FOR {$(b = 0 ; b < \Delta ; b++)$}
+    \FOR{$(a = 0 ; a < N ; a += 2*\Delta)$}
+        \STATE{$t_0 \Leftarrow g(b+a) + g(a+\Delta+b)$}
+        \STATE{$t_1 \Leftarrow W^b \left( g(a+b) - g(a+\Delta+b) \right)$}
+        \STATE{$g(a+b) \Leftarrow t_0$}
+        \STATE{$g(a+\Delta+b) \Leftarrow t_1$}
+    \ENDFOR
+  \ENDFOR
+  \STATE{$\Delta \Leftarrow \Delta/2$}
+\ENDFOR
+\STATE bit-reverse ordering of $g$
+\end{algorithm}
+%
+
+\section{Self-Sorting Mixed-Radix Complex FFTs}
+%
+This section is based on the review article {\em Self-sorting
+Mixed-Radix Fast Fourier Transforms} by Clive
+Temperton~\cite{temperton83}. You should consult his article for full
+details of all the possible algorithms (there are many
+variations). Here I have annotated the derivation of the simplest
+mixed-radix decimation-in-frequency algorithm.
+
+For general-$N$ FFT algorithms the simple binary-notation of radix-2
+algorithms is no longer useful.  The mixed-radix FFT has to be built
+up using products of matrices acting on a data vector.  The aim is to
+take the full DFT matrix $W_N$ and factor it into a set of small,
+sparse matrices corresponding to each factor of $N$.
+
+
+We'll denote the components of matrices using either subscripts or
+function notation,
+%
+\begin{equation}
+M_{ij} = M(i,j)
+\end{equation}
+%
+with (C-like) indices running from 0 to $N-1$.  Matrix products will be 
+denoted using square brackets,
+%
+\begin{equation}
+[AB]_{ij} = \sum_{k} A_{ik} B_{kj}
+\end{equation}
+%
+%
+Three special matrices will be needed in the mixed-radix factorization
+of the DFT: the identity matrix, $I$, a permutation matrix, $P$ and a
+matrix of twiddle factors, $D$, as well as the normal DFT matrices
+$W_n$.
+
+We write the identity matrix of order $r$ as $I_r(n,m)$,
+%
+\begin{equation}
+I_r(n,m) = \delta_{nm}
+\end{equation}
+%
+for $0 \leq n,m \leq r-1$.
+
+We also need to define a permutation matrix $P^a_b$ that performs
+digit reversal of the ordering of a vector. If the index of a vector
+$j= 0\dots N-1$ is factorized into $j = la +m$, with $0 \leq l \leq
+b-1$ and $0 \leq m \leq a-1$ then the operation of the matrix $P$ will
+exchange positions $la+m$ and $mb+l$ in the vector (this sort of
+digit-reversal is the generalization of bit-reversal to a number
+system with exponents $a$ and $b$).
+
+In mathematical terms $P$ is a square matrix of size $ab \times ab$
+with the property,
+%
+\begin{eqnarray}
+P^a_b(j,k) &=& 1 ~\mbox{if}~ j=ra+s ~\mbox{and}~ k=sb+r \\
+           &=& 0 ~\mbox{otherwise}
+\end{eqnarray}
+%
+
+Finally the FFT algorithm needs a matrix of twiddle factors, $D^a_b$,
+for the trigonometric sums. $D^a_b$ is a diagonal square matrix of
+size $ab \times ab$ with the definition,
+%
+\begin{eqnarray}
+D^a_b(j,k) &=& \omega^{sr}_{ab} ~\mbox{if}~ j=k=sb+r \\
+           &=& 0 ~\mbox{otherwise}
+\end{eqnarray}
+%
+where $\omega_{ab} = e^{-2\pi i/ab}$.
+
+
+\subsection{The Kronecker Product}
+The Kronecker matrix product plays an important role in all the
+algorithms for combining operations on different subspaces. The
+Kronecker product $A \otimes B$ of two square matrices $A$ and $B$, of
+sizes $a \times a$ and $b \times b$ respectively, is a square matrix
+of size $a b \times a b$, defined as,
+%
+\begin{equation}
+[A \otimes B] (tb+u, rb+s) = A(t,r) B(u,s)
+\end{equation}
+%
+where $0 \leq u,s < b$ and $0 \leq t,r < a$.  Let's examine a specific
+example. If we take a $2 \times 2$ matrix and a $3
+\times 3$ matrix,
+%
+\begin{equation}
+\begin{array}{ll}
+A = 
+\left(
+\begin{array}{cc}
+a_{11} & a_{12} \\
+a_{21} & a_{22} 
+\end{array}
+\right)
+&
+B =
+\left(
+\begin{array}{ccc}
+b_{11} & b_{12} & b_{13} \\
+b_{21} & b_{22} & b_{23} \\
+b_{31} & b_{32} & b_{33} 
+\end{array}
+\right)
+\end{array}
+\end{equation}
+%
+then the Kronecker product $A \otimes B$ is,
+%
+\begin{eqnarray}
+A \otimes B &= &
+\left(
+\begin{array}{cc}
+a_{11} B & a_{12} B \\
+a_{12} B & a_{22} B
+\end{array}
+\right) \\
+ &=&
+\left(
+\begin{array}{cccccc}
+a_{11} b_{11} & a_{11} b_{12} & a_{11} b_{13} &
+  a_{12} b_{11} & a_{12} b_{12} & a_{12} b_{13} \\
+a_{11} b_{21} & a_{11} b_{22} & a_{11} b_{23} &
+  a_{12} b_{21} & a_{12} b_{22} & a_{12} b_{23} \\
+a_{11} b_{31} & a_{11} b_{32} & a_{11} b_{33} &
+  a_{12} b_{31} & a_{12} b_{32} & a_{12} b_{33} \\
+a_{21} b_{11} & a_{21} b_{12} & a_{21} b_{13} &
+  a_{22} b_{11} & a_{22} b_{12} & a_{22} b_{13} \\
+a_{21} b_{21} & a_{21} b_{22} & a_{21} b_{23} &
+  a_{22} b_{21} & a_{22} b_{22} & a_{22} b_{23} \\
+a_{21} b_{31} & a_{21} b_{32} & a_{21} b_{33} &
+  a_{22} b_{31} & a_{22} b_{32} & a_{22} b_{33}
+\end{array}
+\right)
+\end{eqnarray}
+%
+When the Kronecker product $A \otimes B$ acts on a vector of length
+$ab$, each matrix operates on a different subspace of the vector.
+Writing the index $i$ as $i=t b + u$, with $0\leq u \leq b-1$
+and $0\leq t\leq a$, we can see this explicitly by looking at components,
+%
+\begin{eqnarray}
+[(A \otimes B) v]_{(tb+u)}
+& = & \sum_{t'=0}^{a-1} \sum_{u'=0}^{b-1} 
+        [A \otimes B]_{(tb+u,t'b+u')} v_{t'b+u'} \\
+& = & \sum_{t'u'} A_{tt'} B_{uu'} v_{t'b+u'} 
+\end{eqnarray}
+%
+The matrix $B$ operates on the ``index'' $u'$, for all values of $t'$, and
+the matrix $A$ operates on the ``index'' $t'$, for all values of $u'$.
+%
+The most important property needed for deriving the FFT factorization
+is that the matrix product of two Kronecker products is the Kronecker
+product of the two matrix products,
+%
+\begin{equation}
+(A \otimes B)(C \otimes D) = (AC \otimes BD)
+\end{equation}
+%
+This follows straightforwardly from the original definition of the
+Kronecker product.
+
+\subsection{Two factor case, $N=ab$}
+%
+First consider the simplest possibility, where the data length $N$ can
+be divided into two factors, $N=ab$. The aim is to reduce the DFT
+matrix $W_N$ into simpler matrices corresponding to each factor. To
+make the derivation easier we will start from the known factorization
+and verify it (the initial factorization can be guessed by
+generalizing from simple cases). Here is the factorization we are
+going to prove,
+%
+\begin{equation}
+W_{ab} = (W_b \otimes I_a) P^a_b D^a_b (W_a \otimes I_b).
+\end{equation}
+%
+We can check it by expanding the product into components,
+%
+\begin{eqnarray}
+\lefteqn{[(W_b \otimes I_a) P^a_b D^a_b (W_a \otimes I_b)](la+m,rb+s)}  \\
+& = &
+\sum_{u=0}^{b-1} \sum_{t=0}^{a-1}
+[(W_b \otimes I_a)](la+m,ua+t) [P^a_b D^a_b (W_a \otimes I_b)](ua+t,rb+s)
+\end{eqnarray}
+%
+where we have split the indices to match the Kronecker product $0 \leq
+m, r \leq a$, $0 \leq l, s \leq b$.  The first term in the sum can
+easily be reduced to its component form,
+%
+\begin{eqnarray}
+[(W_b \otimes I_a)](la+m,ua+t) 
+&=& W_b(l,u) I_a(m,t) \\
+&=& \omega_b^{lu} \delta_{mt}
+\end{eqnarray}
+%
+The second term is more complicated. We can expand the Kronecker
+product like this,
+\begin{eqnarray}
+(W_a \otimes I_b)(tb+u,rb+s)
+&=& W_a(t,r) I_a(u,s) \\
+&=& \omega_a^{tr} \delta_{us}
+\end{eqnarray}
+%
+and use this term to build up the product, $P^a_b D^a_b (W_a \otimes
+I_b)$. We first multiply by $D^a_b$,
+%
+\begin{equation}
+[D^a_b (W_a \otimes I_b)](tb+u,rb+s) 
+= 
+\omega^{tu}_{ab} \omega^{tr}_{a} \delta_{su}
+\end{equation}
+%
+and then apply the permutation matrix, $P^a_b$, which digit-reverses
+the ordering of the first index, to obtain,
+%
+\begin{equation}
+[P^a_b D^a_b (W_a \otimes I_b)](ua+t,rb+s) 
+= 
+\omega^{tu}_{ab} \omega^{tr}_{a} \delta_{su}
+\end{equation}
+%
+Combining the two terms in the matrix product we can obtain the full
+expansion in terms of the exponential $\omega$,
+%
+\begin{eqnarray}
+[(W_b \otimes I_a) P^a_b D^a_b (W_a \otimes I_b)](la+m,rb+s)
+&=&
+\sum_{u=0}^{b-1} \sum_{t=0}^{a-1}
+\omega_b^{lu} \delta_{mt} \omega^{tu}_{ab} \omega^{tr}_{a} \delta_{su}
+\end{eqnarray}
+%
+If we evaluate this sum explicitly we can make the connection between
+the product involving $W_a$ and $W_b$ (above) and the expansion of the
+full DFT matrix $W_{ab}$,
+%
+\begin{eqnarray}
+\sum_{u=0}^{b-1} \sum_{t=0}^{a-1}
+\omega_b^{lu} \delta_{mt} \omega^{tu}_{ab} \omega^{tr}_{a} \delta_{su} 
+&=& \omega^{ls}_b \omega^{ms}_{ab} \omega^{mr}_a \\
+&=& \omega^{als + ms +bmr}_{ab} \\
+&=& \omega^{als + ms +bmr}_{ab} \omega^{lrab}_{ab} \quad\mbox{using~} \omega^{ab}_{ab} =1\\
+&=& \omega^{(la+m)(rb+s)}_{ab} \\
+&=& W_{ab}(la+m,rb+s)
+\end{eqnarray}
+% 
+The final line shows that matrix product given above is identical to
+the full two-factor DFT matrix, $W_{ab}$.
+%
+Thus the full DFT matrix $W_{ab}$ for two factors $a$, $b$ can be
+broken down into a product of sub-transforms, $W_a$ and $W_b$, plus
+permutations, $P$, and twiddle factors, $D$, according to the formula,
+%
+\begin{equation}
+W_{ab} = (W_b \otimes I_a) P^a_b D^a_b (W_a \otimes I_b).
+\end{equation}
+%
+This relation is the foundation of the general-$N$ mixed-radix FFT algorithm.
+
+\subsection{Three factor case, $N=abc$}
+%
+The result for the two-factor expansion can easily be generalized to
+three factors. We first consider $abc$ as being a product of two
+factors $a$ and $(bc)$, and then further expand the product $(bc)$ into
+$b$ and $c$. The first step of the expansion looks like this,
+%
+\begin{eqnarray}
+W_{abc} &=& W_{a(bc)}\\
+&=& (W_{bc} \otimes I_a) P^a_{bc} D^a_{bc} (W_a \otimes I_{bc}) .
+\end{eqnarray}
+%
+And after using the two-factor result to expand out $W_{bc}$ we obtain
+the factorization of $W_{abc}$,
+%
+\begin{eqnarray}
+W_{abc} &=& (((W_c \otimes I_b) P^b_c D^b_c (W_b \otimes I_c)) \otimes I_a )
+P^a_{bc} D^a_{bc} (W_a \otimes I_{bc}) \\
+&=& (W_c \otimes I_{ab}) (P^b_c D^b_c \otimes I_a) (W_b \otimes I_{ac}) P^a_{bc} D^a_{bc} (W_a \otimes I_c)
+\end{eqnarray}
+%
+We can write this factorization in a product form, with one term for
+each factor,
+%
+\begin{equation}
+W_{abc} = T_3 T_2 T_1
+\end{equation}
+%
+where we read off $T_1$, $T_2$ and $T_3$,
+%
+\begin{eqnarray}
+T_1 &=& P^a_{bc} D^a_{bc} (W_a \otimes I_{bc}) \\
+T_2 &=& (P^b_c D^b_c \otimes I_a) (W_b \otimes I_{ac}) \\
+T_3 &=& (W_c \otimes I_{ab} ) 
+\end{eqnarray}
+%
+
+
+\subsection{General case, $N=f_1 f_2 \dots f_{n_f}$}
+%
+If we continue the procedure that we have used for two- and
+three-factors then a general pattern begins to emerge in the
+factorization of $W_{f_1 f_2 \dots f_{n_f}}$. To see the beginning of
+the pattern we can rewrite the three factor case as,
+%
+\begin{eqnarray}
+T_1 &=& (P^a_{bc} D^a_{bc} \otimes I_1) (W_a \otimes I_{bc}) \\
+T_2 &=& (P^b_c D^b_c \otimes I_a) (W_b \otimes I_{ac}) \\
+T_3 &=& (P^c_1 D^c_1 \otimes I_{ab}) (W_c \otimes I_{ab} ) 
+\end{eqnarray}
+%
+using the special cases $P^c_1 = D^c_1 = I_c$.
+%
+In general, we can write the factorization of $W_N$ for $N= f_1 f_2
+\dots f_{n_f}$ as,
+%       
+\begin{equation}
+W_N = T_{n_f} \dots T_2 T_1
+\end{equation}
+%
+where the matrix factors $T_i$ are,
+%
+\begin{equation}
+T_i = (P^{f_i}_{q_i} D^{f_i}_{q_i} \otimes I_{p_{i-1}}) ( W_{f_i}
+\otimes I_{m_i})
+\end{equation}
+%
+We have defined the following three additional variables $p$, $q$ and
+$m$ to denote different partial products of the factors,
+%
+\begin{eqnarray}
+p_i &=& f_1 f_2 \dots f_i \quad (p_0 = 1)  \\
+q_i &=& N / p_i  \\
+m_i &=& N / f_i 
+\end{eqnarray}
+%
+Note that the FFT modules $W$ are applied before the permutations $P$,
+which makes this a decimation-in-frequency algorithm.
+
+\subsection{Implementation}
+%
+Now to the implementation of the algorithm. We start with a vector of
+data, $z$, as input and want to apply the transform,
+%
+\begin{eqnarray}
+x &=& W_N z \\
+  &=& T_{n_f} \dots T_2 T_1 z
+\end{eqnarray}
+%
+where $T_i = (P^{f_i}_{q_i} D^{f_i}_{q_i} \otimes I_{p_{i-1}}) (
+W_{f_i} \otimes I_{m_i})$.
+
+The outer structure of the implementation will be a loop over the
+$n_f$ factors, applying each matrix $T_i$ to the vector in turn to
+build up the complete transform.
+%
+\begin{algorithm}
+\FOR{$(i = 1 \dots n_f)$}
+        \STATE{$v \Leftarrow T_i v $}
+\ENDFOR
+\end{algorithm}
+%
+The order of the factors is not important. Now we examine the iteration
+$v \Leftarrow T_i v$, which we'll write as,
+%
+\begin{equation}
+v' = 
+(P^{f_i}_{q_i} D^{f_i}_{q_i} \otimes I_{p_{i-1}}) ~
+( W_{f_i} \otimes I_{m_i}) v
+\end{equation}
+%
+There are two Kronecker product matrices in this iteration. The
+rightmost matrix, which is the first to be applied, is a DFT of length
+$f_i$ applied to $N/f_i$ subsets of the data. We'll call this $t$,
+since it will be a temporary array,
+%
+\begin{equation}
+t = (W_{f_i} \otimes I_{m_i}) v
+\end{equation}
+%
+The second matrix applies a permutation and the exponential
+twiddle-factors. We'll call this $v'$, since it is the result of the
+full iteration on $v$,
+%
+\begin{equation}
+v' = (P^{f_i}_{q_i} D^{f_i}_{q_i} \otimes I_{p_{i-1}}) t 
+\end{equation}
+
+The effect of the matrix $(W_{f_i} \otimes I_{m_i})$ is best seen by
+an example. Suppose the factor is $f_i = 3$, and the length of the FFT
+is $N=6$, then the relevant Kronecker product is,
+%
+\begin{equation}
+t = (W_3 \otimes I_2) v 
+\end{equation}
+%
+which expands out to,
+%
+\begin{equation}
+\left(
+\begin{array}{c}
+t_0 \\
+t_1 \\
+t_2 \\
+t_3 \\
+t_4 \\
+t_5
+\end{array}
+\right)
+=
+\left(
+\begin{array}{cccccc}
+W_3(1,1) & 0 & W_3(1,2) & 0 & W_3(1,3) & 0 \\
+0 & W_3(1,1) & 0 & W_3(1,2) & 0 & W_3(1,3) \\
+W_3(2,1) & 0 & W_3(2,2) & 0 & W_3(2,3) & 0 \\
+0 & W_3(2,1) & 0 & W_3(2,2) & 0 & W_3(2,3) \\
+W_3(3,1) & 0 & W_3(3,2) & 0 & W_3(3,3) & 0 \\
+0 & W_3(3,1) & 0 & W_3(3,2) & 0 & W_3(3,3) 
+\end{array}
+\right)
+\left(
+\begin{array}{c}
+v_0 \\
+v_1 \\
+v_2 \\
+v_3 \\
+v_4 \\
+v_5
+\end{array}
+\right)
+\end{equation}
+%
+We can rearrange the components in a computationally convenient form,
+\begin{equation}
+\left(
+\begin{array}{c}
+t_0 \\
+t_2 \\
+t_4 \\
+t_1 \\
+t_3 \\
+t_5
+\end{array}
+\right)
+=
+\left(
+\begin{array}{cccccc}
+W_3(1,1) & W_3(1,2) & W_3(1,3) & 0 & 0 & 0 \\
+W_3(2,1) & W_3(2,2) & W_3(2,3) & 0 & 0 & 0 \\
+W_3(3,1) & W_3(3,2) & W_3(3,3) & 0 & 0 & 0 \\
+0 & 0 & 0 & W_3(1,1) & W_3(1,2) & W_3(1,3) \\
+0 & 0 & 0 & W_3(2,1) & W_3(2,2) & W_3(2,3) \\
+0 & 0 & 0 & W_3(3,1) & W_3(3,2) & W_3(3,3)
+\end{array}
+\right)
+\left(
+\begin{array}{c}
+v_0 \\
+v_2 \\
+v_4 \\
+v_1 \\
+v_3 \\
+v_5
+\end{array}
+\right)
+\end{equation}
+%
+which clearly shows that we just need to apply the $3\times 3$ DFT
+matrix $W_3$ twice, once to the sub-vector of elements $(v_0, v_2, v_4)$,
+and independently to the remaining sub-vector $(v_1, v_3, v_5)$.
+
+In the general case, if we index $t$ as $t_k = t(\lambda,\mu) =
+t_{\lambda m + \mu}$ then $\lambda = 0 \dots f-1$ is an index within
+each transform of length $f$ and $\mu = 0 \dots m-1$ labels the
+independent subsets of data. We can see this by showing the
+calculation with all indices present,
+%
+\begin{equation}
+t = (W_f \otimes I_m) z 
+\end{equation}
+%
+becomes,
+%
+\begin{eqnarray}
+t_{\lambda m + \mu} &=& \sum_{\lambda'=0}^{f-1} \sum_{\mu'=0}^{m-1} 
+        (W_f \otimes I_m)_{(\lambda m + \mu)(\lambda' m + \mu')}
+        z_{\lambda'm + \mu} \\
+&=& \sum_{\lambda'\mu'} (W_f)_{\lambda\lambda'} \delta_{\mu\mu'} 
+        z_{\lambda'm+\mu'} \\
+&=& \sum_{\lambda'} (W_f)_{\lambda\lambda'} z_{\lambda'm+\mu}
+\end{eqnarray}
+%
+The DFTs on the index $\lambda$ will be computed using
+special optimized modules for each $f$.
+
+To calculate the next stage,
+%
+\begin{equation}
+v'=(P^f_q D^f_q \otimes I_{p_{i-1}}) t
+\end{equation}
+%
+we note that the Kronecker product has the property of performing
+$p_{i-1}$ independent multiplications of $PD$ on $q_{i-1}$ different
+subsets of $t$. The index $\mu$ of $t(\lambda,\mu)$ which runs from 0
+to $m$ will include $q_i$ copies of each $PD$ operation because
+$m=p_{i-1}q$. i.e. we can split the index $\mu$ further into $\mu = a
+p_{i-1} + b$, where $a = 0 \dots q-1$ and $b=0 \dots p_{i-1}$,
+%
+\begin{eqnarray}
+\lambda m + \mu &=& \lambda m + a p_{i-1} + b \\
+        &=& (\lambda q + a) p_{i-1} + b.
+\end{eqnarray}
+%
+Now we can expand the second stage,
+%
+\begin{eqnarray}
+v'&=& (P^f_q D^f_q \otimes I_{p_{i-1}}) t \\
+v'_{\lambda m + \mu} &=& \sum_{\lambda' \mu'} 
+ (P^f_q D^f_q \otimes I_{p_{i-1}})_{(\lambda m + \mu)(\lambda' m + \mu')}
+        t_{\lambda' m + \mu'} \\
+v'_{(\lambda q + a) p_{i-1} + b} &=& \sum_{\lambda' a' b'}
+(
+P^f_q D^f_q \otimes I_{p_{i-1}}
+)_{((\lambda q+ a)p_{i-1} + b)((\lambda' q+ a')p_{i-1} + b')} 
+t_{(\lambda' q + a')p_{i-1} +b'} 
+\end{eqnarray}
+%
+The first step in removing redundant indices is to take advantage of
+the identity matrix $I$ and separate the subspaces of the Kronecker
+product,
+%
+\begin{equation}
+(
+P^f_q D^f_q \otimes I_{p_{i-1}}
+)_{((\lambda q+ a)p_{i-1} + b)((\lambda' q+ a')p_{i-1} + b')} 
+=
+(P^f_q D^f_q)_{(\lambda q + a)(\lambda' q + a')}
+\delta_{bb'}
+\end{equation}
+%
+This eliminates one sum, leaving us with,
+%
+\begin{equation}
+v'_{(\lambda q + a) p_{i-1} + b} 
+= 
+\sum_{\lambda' a' }
+(P^f_q D^f_q)_{(\lambda q + a)(\lambda' q + a')} t_{(\lambda'q+a')p_{i-1} + b}
+\end{equation}
+%
+We can insert the definition of $D^f_q$ to give,
+%
+\begin{equation}
+\phantom{v'_{(\lambda q + a) p_{i-1} + b}}
+= \sum_{\lambda'a'} (P^f_q)_{(\lambda q + a)(\lambda'q + a')} 
+\omega^{\lambda'a'}_{q_{i-1}} t_{(\lambda'q+a')p_{i-1}+b}
+\end{equation}
+%
+Using the definition of $P^f_q$, which exchanges an index of $\lambda
+q + a$ with $a f + \lambda$, we get a final result with no matrix
+multiplication,
+%
+\begin{equation}
+v'_{(a f + \lambda) p_{i-1} + b}
+= \omega^{\lambda a}_{q_{i-1}} t_{(\lambda q + a)p_{i-1} + b}
+\end{equation}
+%
+All we have to do is premultiply each element of the temporary vector
+$t$ by an exponential twiddle factor and store the result in another
+index location, according to the digit reversal permutation of $P$.
+
+Here is the algorithm to implement the mixed-radix FFT,
+%
+\begin{algorithm}
+\FOR{$i = 1 \dots n_f$}
+\FOR{$a = 0 \dots q-1$}
+\FOR{$b = 0 \dots p_{i-1} - 1$}
+\FOR{$\lambda = 0 \dots f-1$}
+\STATE{$t_\lambda \Leftarrow 
+       \sum_{\lambda'=0}^{f-1} W_f(\lambda,\lambda') v_{b+\lambda'm+ap_{i-1}}$}
+       \COMMENT{DFT matrix-multiply module}
+\ENDFOR
+\FOR{$\lambda = 0 \dots f-1$}
+\STATE{$v'_{(af+\lambda)p_{i-1}+b} 
+        \Leftarrow \omega^{\lambda a}_{q_{i-1}} t_\lambda$}
+\ENDFOR
+\ENDFOR
+\ENDFOR
+\STATE{$v \Leftarrow v'$}
+\ENDFOR
+\end{algorithm}
+%
+\subsection{Details of the implementation}
+%
+First the function {\tt gsl\_fft\_complex\_wavetable\_alloc} allocates
+$n$ elements of scratch space (to hold the vector $v'$ for each
+iteration) and $n$ elements for a trigonometric lookup table of
+twiddle factors.
+
+Then the length $n$ must be factorized. There is a general
+factorization function {\tt gsl\_fft\_factorize} which takes a list of
+preferred factors. It first factors out the preferred factors and then
+removes general remaining prime factors.
+
+The algorithm used to generate the trigonometric lookup table is
+%
+\begin{algorithm}
+\FOR {$a = 1 \dots n_f$}
+\FOR {$b = 1 \dots f_i - 1$}
+\FOR {$c = 1 \dots q_i$}
+\STATE $\mbox{trig[k++]} = \exp(- 2\pi i b c p_{a-1}/N)$
+\ENDFOR
+\ENDFOR
+\ENDFOR
+\end{algorithm}
+%
+Note that $\sum_{1}^{n_f} \sum_{0}^{f_i-1} \sum_{1}^{q_i} =
+\sum_{1}^{n_f} (f_i-1)q_i = n - 1$ so $n$ elements are always
+sufficient to store the lookup table. This is chosen because we need
+to compute $\omega_{q_i-1}^{\lambda a} t_\lambda$ in
+the FFT. In terms of the lookup table we can write this as,
+%
+\begin{eqnarray}
+\omega_{q_{i-1}}^{\lambda a} t_\lambda 
+&=&  \exp(-2\pi i \lambda a/q_{i-1}) t_\lambda \\
+&=&  \exp(-2\pi i \lambda a p_{i-1}/N) t_\lambda \\
+&=& \left\{
+    \begin{array}{ll}
+    t_\lambda & a=0 \\
+    \mbox{trig}[\mbox{twiddle[i]}+\lambda q+(a-1)] t_\lambda & a\not=0
+\end{array}
+\right.
+\end{eqnarray}
+%
+\begin{sloppypar}
+\noindent 
+The array {\tt twiddle[i]} maintains a set of pointers into {\tt trig}
+for the starting points for the outer loop.  The core of the
+implementation is {\tt gsl\_fft\_complex}. This function loops over
+the chosen factors of $N$, computing the iteration $v'=T_i v$ for each
+pass. When the DFT for a factor is implemented the iteration is
+handed-off to a dedicated small-$N$ module, such as {\tt
+gsl\_fft\_complex\_pass\_3} or {\tt
+gsl\_fft\_complex\_pass\_5}.  Unimplemented factors are handled
+by the general-$N$ routine {\tt gsl\_fft\_complex\_pass\_n}. The
+structure of one of the small-$N$ modules is a simple transcription of
+the basic algorithm given above.  Here is an example for {\tt
+gsl\_fft\_complex\_pass\_3}. For a pass with a factor of 3 we have to
+calculate the following expression,
+\end{sloppypar}%
+\begin{equation}
+v'_{(a f + \lambda) p_{i-1} + b} 
+= 
+\sum_{\lambda' = 0,1,2} 
+\omega^{\lambda a}_{q_{i-1}} W^{\lambda \lambda'}_{3} 
+v_{b + \lambda' m + a p_{i-1}}
+\end{equation}
+%
+for $b = 0 \dots p_{i-1} - 1$, $a = 0 \dots q_{i} - 1$ and $\lambda =
+0, 1, 2$.  This is implemented as,
+%
+\begin{algorithm}
+\FOR {$a = 0 \dots q-1$}
+\FOR {$b = 0 \dots p_{i-1} - 1$}
+\STATE {$
+        \left(
+        \begin{array}{c}
+        t_0 \\ t_1 \\ t_2
+        \end{array}
+        \right)
+        =
+        \left(
+        \begin{array}{ccc}
+        W^{0}_3 & W^{0}_3 & W^{0}_3 \\
+        W^{0}_3 & W^{1}_3 & W^{2}_3 \\
+        W^{0}_3 & W^{2}_3 & W^{4}_3
+        \end{array}
+        \right)
+        \left(
+        \begin{array}{l}
+        v_{b + a p_{i-1}} \\ 
+        v_{b + a p_{i-1} + m} \\ 
+        v_{b + a p_{i-1} +2m}
+        \end{array}
+        \right)$}
+        \STATE {$ v'_{a p_{i} + b}  = t_0$}
+        \STATE {$ v'_{a p_{i} + b + p_{i-1}} = \omega^{a}_{q_{i-1}} t_1$} 
+        \STATE {$ v'_{a p_{i} + b + 2 p_{i-1}} = \omega^{2a}_{q_{i-1}} t_2$}
+\ENDFOR
+\ENDFOR
+\end{algorithm}
+%
+In the code we use the variables {\tt from0}, {\tt from1}, {\tt from2}
+to index the input locations,
+%
+\begin{eqnarray}
+\mbox{\tt from0} &=& b + a p_{i-1} \\
+\mbox{\tt from1} &=& b + a p_{i-1} + m \\
+\mbox{\tt from2} &=& b + a p_{i-1} + 2m
+\end{eqnarray}
+%
+and the variables {\tt to0}, {\tt to1}, {\tt to2} to index the output
+locations in the scratch vector $v'$,
+%
+\begin{eqnarray}
+\mbox{\tt to0} &=& b + a p_{i} \\
+\mbox{\tt to1} &=& b + a p_{i} + p_{i-1} \\
+\mbox{\tt to2} &=& b + a p_{i} + 2 p_{i-1}
+\end{eqnarray}
+%
+The DFT matrix multiplication is computed using the optimized
+sub-transform modules given in the next section. The twiddle factors
+$\omega^a_{q_{i-1}}$ are taken out of the {\tt trig} array.
+
+To compute the inverse transform we go back to the definition of the
+fourier transform and note that the inverse matrix is just the complex
+conjugate of the forward matrix (with a factor of $1/N$),
+%
+\begin{equation}
+W^{-1}_N = W^*_N / N 
+\end{equation}
+%
+Therefore we can easily compute the inverse transform by conjugating
+all the complex elements of the DFT matrices and twiddle factors that
+we apply. (An alternative strategy is to conjugate the input data,
+take a forward transform, and then conjugate the output data).
+
+\section{Fast Sub-transform Modules}
+%
+To implement the mixed-radix FFT we still need to compute the
+small-$N$ DFTs for each factor. Fortunately many highly-optimized
+small-$N$ modules are available, following the work of Winograd who
+showed how to derive efficient small-$N$ sub-transforms by number
+theoretic techniques.
+
+The algorithms in this section all compute,
+%
+\begin{equation}
+x_a = \sum_{b=0}^{N-1} W_N^{ab} z_b
+\end{equation}
+%
+The sub-transforms given here are the ones recommended by Temperton
+and differ slightly from the canonical Winograd modules. According to
+Temperton~\cite{temperton83} they are slightly more robust against
+rounding errors and trade off some additions for multiplications.
+%
+For the $N=2$ DFT,
+%
+\begin{equation}
+\begin{array}{ll}
+x_0 = z_0 + z_1, &
+x_1 = z_0 - z_1. 
+\end{array}
+\end{equation}
+%
+For the $N=3$ DFT,
+%
+\begin{equation}
+\begin{array}{lll}
+t_1 = z_1 + z_2, &
+t_2 = z_0 - t_1/2, &
+t_3 = \sin(\pi/3) (z_1 - z_2), 
+\end{array}
+\end{equation}
+\begin{equation}
+\begin{array}{lll}
+x_0 = z_0 + t_1, &
+x_1 = t_2 + i t_3, &
+x_2 = t_2 - i t_3. 
+\end{array}
+\end{equation}
+%
+The $N=4$ transform involves only additions and subtractions,
+%
+\begin{equation}
+\begin{array}{llll}
+t_1 = z_0 + z_2, &
+t_2 = z_1 + z_3, &
+t_3 = z_0 - z_2, &
+t_4 = z_1 - z_3,
+\end{array}
+\end{equation}
+\begin{equation}
+\begin{array}{llll}
+x_0 = t_1 + t_2, &
+x_1 = t_3 + i t_4, &
+x_2 = t_1 - t_2, &
+x_3 = t_3 - i t_4.
+\end{array}
+\end{equation}
+%
+For the $N=5$ DFT,
+%
+\begin{equation}
+\begin{array}{llll}
+t_1 = z_1 + z_4, &
+t_2 = z_2 + z_3, &
+t_3 = z_1 - z_4, &
+t_4 = z_2 - z_3,
+\end{array}
+\end{equation}
+\begin{equation}
+\begin{array}{llll}
+t_5 = t_1 + t_2, &
+t_6 = (\sqrt{5}/4) (t_1 - t_2), &
+t_7 = z_0 - t_5/4, \\
+\end{array}
+\end{equation}
+\begin{equation}
+\begin{array}{llll}
+t_8 = t_7 + t_6, &
+t_9 = t_7 - t_6, \\
+\end{array}
+\end{equation}
+\begin{equation}
+\begin{array}{llll}
+t_{10} = \sin(2\pi/5) t_3 + \sin(2\pi/10) t_4, &
+t_{11} = \sin(2\pi/10) t_3 - \sin(2\pi/5) t_4,
+\end{array}
+\end{equation}
+\begin{equation}
+\begin{array}{llll}
+x_0 = z_0 + t_5,
+\end{array}
+\end{equation}
+\begin{equation}
+\begin{array}{llll}
+x_1 = t_8 + i t_{10}, &
+x_2 = t_9 + i t_{11},
+\end{array}
+\end{equation}
+\begin{equation}
+\begin{array}{llll}
+x_3 = t_9 - i t_{11}, &
+x_4 = t_8 - i t_{10}.
+\end{array}
+\end{equation}
+%
+The DFT matrix for $N=6$ can be written as a combination of $N=3$ and
+$N=2$ transforms with index permutations,
+%
+\begin{equation}
+\left(
+\begin{array}{c}
+x_0 \\
+x_4 \\
+x_2 \\
+\hline x_3 \\
+x_1 \\
+x_5
+\end{array}
+\right)
+= 
+\left(
+\begin{array}{ccc|ccc}
+  &   &  &  &   & \\
+  &W_3&  &  &W_3& \\
+  &   &  &  &   & \\
+\hline  &   &  &  &   & \\
+  &W_3&  &  &-W_3& \\
+  &   &  &  &   & 
+\end{array}
+\right)
+\left(
+\begin{array}{c}
+z_0 \\
+z_2 \\
+z_4 \\
+\hline z_3 \\
+z_5 \\
+z_1 
+\end{array}
+\right)
+\end{equation}
+%
+This simplification is an example of the Prime Factor Algorithm, which
+can be used because the factors 2 and 3 are mutually prime.  For more
+details consult one of the books on number theory for
+FFTs~\cite{elliott82,blahut}. We can take advantage of the simple
+indexing scheme of the PFA to write the $N=6$ DFT as,
+%
+\begin{equation}
+\begin{array}{lll}
+t_1 = z_2 + z_4, &
+t_2 = z_0 - t_1/2, &
+t_3 = \sin(\pi/3) (z_2 - z_4),
+\end{array}
+\end{equation}
+\begin{equation}
+\begin{array}{lll}
+t_4 = z_5 + z_1, &
+t_5 = z_3 - t_4/2, &
+t_6 = \sin(\pi/3) (z_5 - z_1), 
+\end{array}
+\end{equation}
+\begin{equation}
+\begin{array}{lll}
+t_7 = z_0 + t_1, &
+t_8 = t_2 + i t_3, &
+t_9 = t_2 - i t_3,
+\end{array}
+\end{equation}
+\begin{equation}
+\begin{array}{lll}
+t_{10} = z_3 + t_4, &
+t_{11} = t_5 + i t_6, &
+t_{12} = t_5 - i t_6, 
+\end{array}
+\end{equation}
+\begin{equation}
+\begin{array}{lll}
+x_0 = t_7 + t_{10}, &
+x_4 = t_8 + t_{11}, &
+x_2 = t_9 + t_{12}, 
+\end{array}
+\end{equation}
+\begin{equation}
+\begin{array}{lll}
+x_3 = t_7 - t_{10}, &
+x_1 = t_8 - t_{11}, &
+x_5 = t_9 - t_{12}.
+\end{array}
+\end{equation}
+
+For any remaining general factors we use Singleton's efficient method
+for computing a DFT~\cite{singleton}. Although it is an $O(N^2)$
+algorithm it does reduce the number of multiplications by a factor of
+4 compared with a naive evaluation of the DFT. If we look at the
+general stucture of a DFT matrix, shown schematically below,
+%
+\begin{equation}
+\left(
+\begin{array}{c}
+h_0 \\
+h_1 \\
+h_2 \\
+\vdots \\
+h_{N-2} \\
+h_{N-1}
+\end{array}
+\right)
+=
+\left(
+\begin{array}{cccccc}
+1 & 1 & 1 & \cdots & 1 & 1 \\
+1 & W & W & \cdots & W & W \\
+1 & W & W & \cdots & W & W \\
+\vdots & \vdots & \vdots & \cdots & \vdots & \vdots \\
+1 & W & W & \cdots & W & W \\
+1 & W & W & \cdots & W & W 
+\end{array}
+\right)
+\left(
+\begin{array}{c}
+g_0 \\
+g_1 \\
+g_2 \\
+\vdots \\
+g_{N-2} \\
+g_{N-1}
+\end{array}
+\right)
+\end{equation}
+%
+we see that the outer elements of the DFT matrix are all unity. We can
+remove these trivial multiplications but we will still be left with an
+$(N-1) \times (N-1)$ sub-matrix of complex entries, which would appear
+to require $(N-1)^2$ complex multiplications.  Singleton's method,
+uses symmetries of the DFT matrix to convert the complex
+multiplications to an equivalent number of real multiplications. We
+start with the definition of the DFT in component form,
+%
+\begin{equation}
+a_k + i b_k = \sum_{j=0} (x_j+iy_j)(\cos(2\pi jk/f) - i\sin(2\pi jk/f))
+\end{equation}
+%
+The zeroth component can be computed using only additions,
+%
+\begin{equation}
+a_0 + i b_0 = \sum_{j=0}^{(f-1)} x_j + i y_j
+\end{equation}
+%
+We can rewrite the remaining components as,
+%
+\begin{eqnarray}
+a_k + i b_k & =   x_0 + i y_0 & + 
+  \sum_{j=1}^{(f-1)/2} (x_j + x_{f-j}) \cos(2\pi jk/f)
+ + (y_j - y_{f-j}) \sin(2\pi jk/f) \\
+& & + i\sum_{j=1}^{(f-1)/2} (y_j + y_{f-j}) \cos(2\pi jk/f) 
+ - (x_j - x_{f-j}) \sin(2\pi jk/f)
+\end{eqnarray}
+%
+by using the following trigonometric identities,
+%
+\begin{eqnarray}
+ \cos(2\pi(f-j)k/f) &=& \phantom{-}\cos(2\pi jk/f) \\
+ \sin(2\pi(f-j)k/f) &=&  -\sin(2\pi jk/f)
+\end{eqnarray}
+%
+These remaining components can all be computed using four partial
+sums,
+%
+\begin{eqnarray}
+a_k + i b_k & = & (a^+_k - a^-_k) + i (b^+_k + b^-_k) \\
+a_{f-k} + i b_{f-k} & = & (a^+_k + a^-_k) + i (b^+_k - b^-_k)
+\end{eqnarray}
+%
+for $k = 1, 2, \dots, (f-1)/2$, where,
+%
+\begin{eqnarray}
+a^+_k &=& x_0 + \sum_{j=1}^{(f-1)/2} (x_j + x_{f-j}) \cos(2\pi jk/f) \\
+a^-_k &=& \phantom{x_0} - \sum_{j=1}^{(f-1)/2} (y_j - y_{f-j}) \sin(2\pi jk/f) \\
+b^+_k &=& y_0 + \sum_{j=1}^{(f-1)/2} (y_j + y_{f-j}) \cos(2\pi jk/f) \\
+b^-_k &=& \phantom{y_0} - \sum_{j=1}^{(f-1)/2} (x_j - x_{f-j}) \sin(2\pi jk/f)
+\end{eqnarray}
+%
+Note that the higher components $k'=f-k$ can be obtained directly
+without further computation once $a^+$, $a^-$, $b^+$ and $b^-$ are
+known. There are $4 \times (f-1)/2$ different sums, each involving
+$(f-1)/2$ real multiplications, giving a total of $(f-1)^2$ real
+multiplications instead of $(f-1)^2$ complex multiplications.
+
+To implement Singleton's method we make use of the input and output
+vectors $v$ and $v'$ as scratch space, copying data back and forth
+between them to obtain the final result.  First we use $v'$ to store
+the terms of the symmetrized and anti-symmetrized vectors of the form
+$x_j + x_{f-j}$ and $x_j - y_{f-j}$. Then we multiply these by the
+appropriate trigonometric factors to compute the partial sums $a^+$,
+$a^-$, $b^+$ and $b^-$, storing the results $a_k + i b_k$ and $a_{f-k}
++ i b_{f-k}$ back in $v$. Finally we multiply the DFT output by any
+necessary twiddle factors and place the results in $v'$.
+
+\section{FFTs for real data}
+%
+This section is based on the articles {\em Fast Mixed-Radix Real
+  Fourier Transforms} by Clive Temperton~\cite{temperton83real} and
+{\em Real-Valued Fast Fourier Transform Algorithms} by Sorensen,
+Jones, Heideman and Burrus~\cite{burrus87real}. The DFT of a real
+sequence has a special symmetry, called a {\em conjugate-complex} or
+{\em half-complex} symmetry,
+%
+\begin{equation}
+h(a) = h(N-a)^*
+\end{equation}
+%
+The element $h(0)$ is real, and when $N$ is even $h(N/2)$ is also
+real. It is straightforward to prove the symmetry,
+%
+\begin{eqnarray}
+h(a) &=& \sum W^{ab}_N g(b) \\
+h(N-a)^* &=& \sum W^{-(N-a)b}_N g(b)^*  \\
+         &=& \sum W^{-Nb}_N W^{ab}_N g(b) \qquad{(W^N_N=1)} \\
+         &=& \sum W^{ab}_N g(b)
+\end{eqnarray}
+%
+Real-valued data is very common in practice (perhaps more common that
+complex data) so it is worth having efficient FFT routines for real
+data. In principle an FFT for real data should need half the
+operations of an FFT on the equivalent complex data (where the
+imaginary parts are set to zero). There are two different strategies
+for computing FFTs of real-valued data:
+
+One strategy is to ``pack'' the real data (of length $N$) into a
+complex array (of length $N/2$) by index transformations. A complex
+FFT routine can then be used to compute the transform of that array.
+By further index transformations the result can actually by
+``unpacked'' to the FFT of the original real data. It is also possible
+to do two real FFTs simultaneously by packing one in the real part and
+the other in the imaginary part of the complex array.  These
+techniques have some disadvantages. The packing and unpacking
+procedures always add $O(N)$ operations, and packing a real array of
+length $N$ into a complex array of length $N/2$ is only possible if
+$N$ is even. In addition, if two unconnected datasets with very
+different magnitudes are packed together in the same FFT there could
+be ``cross-talk'' between them due to a loss of precision.
+
+A more straightforward strategy is to start with an FFT algorithm,
+such as the complex mixed-radix algorithm, and prune out all the
+operations involving the zero imaginary parts of the initial data. The
+FFT is linear so the imaginary part of the data can be decoupled from
+the real part. This procedure leads to a dedicated FFT for real-valued
+data which works for any length and does not perform any unnecessary
+operations. It also allows us to derive a corresponding inverse FFT
+routine which transforms a half-complex sequence back into real data.
+
+\subsection{Radix-2 FFTs for real data}
+%
+Before embarking on the full mixed-radix real FFT we'll start with the
+radix-2 case. It contains all the essential features of the
+general-$N$ algorithm. To make it easier to see the analogy between
+the two we will use the mixed-radix notation to describe the
+factors. The factors are all 2,
+%
+\begin{equation}
+f_1 = 2, f_2 = 2, \dots, f_{n_f} = 2
+\end{equation}
+%
+and the products $p_i$ are powers of 2,
+%
+\begin{eqnarray}
+p_0 & = & 1 \\
+p_1 & = & f_1 = 2 \\
+p_2 & = & f_1 f_2 = 4 \\
+\dots &=& \dots \\
+p_i & = & f_1 f_2 \dots f_i = 2^i 
+\end{eqnarray}
+%
+Using this notation we can rewrite the radix-2 decimation-in-time
+algorithm as,
+%
+\begin{algorithm}
+\STATE bit-reverse ordering of $g$
+\FOR {$i = 1 \dots n$}
+  \FOR {$a = 0 \dots p_{i-1} - 1$}
+    \FOR{$b = 0 \dots q_i - 1$}
+        \STATE{$
+                \left(
+                \begin{array}{c} 
+                g(b p_i + a) \\
+                g(b p_i + p_{i-1} + a)
+                \end{array}
+                \right)
+                =
+                \left(
+                \begin{array}{c}
+                g(b p_i + a) + W^a_{p_i} g(b p_i + p_{i-1} + a) \\
+                g(b p_i + a) - W^a_{p_i} g(b p_i + p_{i-1} + a)
+                \end{array}
+                \right) $}
+     \ENDFOR
+  \ENDFOR
+\ENDFOR
+\end{algorithm}
+%
+where we have used $p_i = 2 \Delta$, and factored $2 \Delta$ out of
+the original definition of $b$ ($b \to b p_i$).
+
+If we go back to the original recurrence relations we can see how to
+write the intermediate results in a way which make the
+real/half-complex symmetries explicit at each step. The first pass is
+just a set of FFTs of length-2 on real values,
+%
+\begin{equation}
+g_1([b_0 b_1 b_2 a_0]) = \sum_{b_3} W^{a_0 b_3}_2 g([b_0 b_1 b_2 b_3])
+\end{equation}
+%
+Using the symmetry $FFT(x)_k = FFT(x)^*_{N-k}$ we have the reality
+condition,
+%
+\begin{eqnarray}
+g_1([b_0 b_1 b_2 0]) &=& \mbox{real} \\
+g_1([b_0 b_1 b_2 1]) &=& \mbox{real'} 
+\end{eqnarray}
+%
+In the next pass we have a set of length-4 FFTs on the original data,
+%
+\begin{eqnarray}
+g_2([b_0 b_1 b_1 a_0]) 
+&=&
+\sum_{b_2}\sum_{b_3} 
+W^{[a_1 a_0]b_2}_4 W^{a_0 b_3}_2 
+g([b_0 b_1 b_2 b_3]) \\
+&=&
+\sum_{b_2}\sum_{b_3} 
+W^{[a_1 a_0][b_3 b_2]}_4
+g([b_0 b_1 b_2 b_3])
+\end{eqnarray}
+%
+This time symmetry gives us the following conditions on the
+transformed data,
+%
+\begin{eqnarray}
+g_2([b_0 b_1 0 0]) &=& \mbox{real} \\
+g_2([b_0 b_1 0 1]) &=& x + i y  \\
+g_2([b_0 b_1 1 0]) &=& \mbox{real'} \\
+g_2([b_0 b_1 1 1]) &=& x - i y 
+\end{eqnarray}
+%
+We can see a pattern emerging here: the $i$-th pass computes a set of
+independent length-$2^i$ FFTs on the original real data,
+%
+\begin{eqnarray}
+g_i ( b p_i  + a ) = \sum_{a' = 0}^{p_i-1} W_{p_i}^{aa'} g(b p_i + a') 
+\quad 
+\mbox{for $b = 0 \dots q_i - 1$}
+\end{eqnarray}
+%
+As a consequence the we can apply the symmetry for an FFT of real data
+to all the intermediate results -- not just the final result.
+In general after the $i$-th pass we will
+have the symmetry,
+%
+\begin{eqnarray}
+g_i(b p_i) &=& \mbox{real} \\
+g_i(b p_i + a) &=& g_i(b p_i + p_i - a)^* \qquad a = 1 \dots p_{i}/2 - 1  \\
+g_i(b p_i + p_{i}/2) &=& \mbox{real'} 
+\end{eqnarray}
+%
+In the next section we'll show that this is a general property of
+decimation-in-time algorithms. The same is not true for the
+decimation-in-frequency algorithm, which does not have any simple
+symmetries in the intermediate results.
+
+Since we can obtain the values of $g_i(b p_i + a)$ for $a > p_i/2$
+from the values for $a < p_i/2$ we can cut our computation and
+storage in half compared with the full-complex case.
+%
+We can easily rewrite the algorithm to show how the computation can be
+halved, simply by limiting all terms to involve only values for $a
+\leq p_{i}/2$. Whenever we encounter a term $g_i(b p_i + a)$ with $a >
+p_{i}/2$ we rewrite it in terms of its complex symmetry partner,
+$g_i(b p_i + a')^*$, where $a' = p_i - a$.  The butterfly computes two
+values for each value of $a$, $b p_i + a$ and $b p_i + p_{i-1} - a$,
+so we actually only need to compute from $a = 0$ to $p_{i-1}/2$.  This
+gives the following algorithm,
+%
+\begin{algorithm}
+\FOR {$a = 0 \dots p_{i-1}/2$}
+  \FOR{$b = 0 \dots q_i - 1$}
+        \STATE{$
+                \left(
+                \begin{array}{c} 
+                g(b p_i + a) \\
+                g(b p_i + p_{i-1} - a)^*
+                \end{array}
+                \right)
+                =
+                \left(
+                \begin{array}{c}
+                g(b p_{i} + a) + W^a_{p_i} g(b p_i + p_{i-1} + a) \\
+                g(b p_{i} + a) - W^a_{p_i} g(b p_i + p_{i-1} + a)
+                \end{array}
+                \right) $}
+     \ENDFOR
+  \ENDFOR
+\end{algorithm}
+%
+Although we have halved the number of operations we also need a
+storage arrangement which will halve the memory requirement. The
+algorithm above is still formulated in terms of a complex array $g$,
+but the input to our routine will naturally be an array of $N$ real
+values which we want to use in-place.
+
+Therefore we need a storage scheme which lays out the real and
+imaginary parts within the real array, in a natural way so that there
+is no need for complicated index calculations. In the radix-2
+algorithm we do not have any additional scratch space. The storage
+scheme has to be designed to accommodate the in-place calculation
+taking account of dual node pairs.
+
+Here is a scheme which takes these restrictions into account: On the
+$i$-th pass we store the real part of $g(b p_i + a)$ in location $b
+p_i + a$. We store the imaginary part in location $b p_i + p_i -
+a$. This is the redundant location which corresponds to the conjugate
+term $g(b p_i + a)^* = g(b p_i + p_i -a)$, so it is not needed.  When
+the results are purely real (as in the case $a = 0$ and $a = p_i/2$ we
+store only the real part and drop the zero imaginary part).
+
+This storage scheme has to work in-place, because the radix-2 routines
+should not use any scratch space. We will now check the in-place
+property for each butterfly.  A crucial point is that the scheme is
+pass-dependent. Namely, when we are computing the result for pass $i$
+we are reading the results of pass $i-1$, and we must access them
+using the scheme from the previous pass, i.e. we have to remember that
+the results from the previous pass were stored using $b p_{i-1} + a$,
+not $b p_i + a$, and the symmetry for these results will be $g_{i-1}(b
+p_{i-1} + a) = g_{i-1}(b p_{i-1} + p_{i-1} - a)^*$. To take this into
+account we'll write the right hand side of the iteration, $g_{i-1}$,
+in terms of $p_{i-1}$. For example, instead of $b p_i$, which occurs
+naturally in $g_i(b p_i + a)$ we will use $2 b p_{i-1}$, since $p_i =
+2 p_{i-1}$.
+
+Let's start with the butterfly for $a = 0$,
+%
+\begin{equation}
+\left(
+\begin{array}{c} 
+g(b p_i) \\
+g(b p_i + p_{i-1})^*
+\end{array}
+\right)
+=
+\left(
+\begin{array}{c}
+g(2 b p_{i-1}) + g((2 b + 1) p_{i-1}) \\
+g(2 b p_{i-1}) - g((2 b + 1) p_{i-1})
+\end{array}
+\right)
+\end{equation}
+% 
+By the symmetry $g_{i-1}(b p_{i-1} + a) = g_{i-1}(b p_{i-1} + p_{i-1}
+- a)^*$ all the inputs are purely real.  The input $g(2 b p_{i-1})$ is
+read from location $2 b p_{i-1}$ and $g((2 b + 1) p_{i-1})$ is read
+from the location $(2 b + 1) p_{i-1}$. Here is the full breakdown,
+%
+\begin{center}
+\renewcommand{\arraystretch}{1.5}
+\begin{tabular}{|l|lll|}
+\hline Term & & Location & \\
+\hline
+$g(2 b p_{i-1})$        
+        & real part & $2 b p_{i-1} $ &$= b p_i$ \\
+        & imag part & --- & \\
+$g((2 b+1) p_{i-1})$ 
+        & real part & $(2 b + 1) p_{i-1}  $&$= b p_i + p_{i-1} $ \\
+        & imag part & --- & \\
+\hline
+$g(b p_{i})$ 
+        & real part & $b p_i$ &\\
+        & imag part & --- & \\
+$g(b p_{i} + p_{i-1})$ 
+        & real part & $b p_i + p_{i-1}$& \\
+        & imag part & --- &  \\
+\hline
+\end{tabular}
+\end{center}
+%
+The conjugation of the output term $g(b p_i + p_{i-1})^*$ is
+irrelevant here since the results are purely real. The real results
+are stored in locations $b p_i$ and $b p_i + p_{i-1}$, which
+overwrites the inputs in-place.
+
+The general butterfly for $a = 1 \dots p_{i-1}/2 - 1$ is,
+%
+\begin{equation}
+\left(
+\begin{array}{c} 
+g(b p_i + a) \\
+g(b p_i + p_{i-1} - a)^*
+\end{array}
+\right)
+=
+\left(
+\begin{array}{c}
+g(2 b p_{i-1} + a) + W^a_{p_i} g((2 b + 1) p_{i-1} + a) \\
+g(2 b p_{i-1} + a) - W^a_{p_i} g((2 b + 1) p_{i-1} + a)
+\end{array}
+\right)
+\end{equation}
+%
+All the terms are complex. To store a conjugated term like $g(b' p_i +
+a')^*$ where $a > p_i/2$ we take the real part and store it in
+location $b' p_i + a'$ and then take imaginary part, negate it, and
+store the result in location $b' p_i + p_i - a'$.
+
+Here is the full breakdown of the inputs and outputs from the
+butterfly,
+%
+\begin{center}
+\renewcommand{\arraystretch}{1.5}
+\begin{tabular}{|l|lll|}
+\hline Term & & Location & \\
+\hline
+$g(2 b p_{i-1} + a)$    
+        & real part & $2 b p_{i-1} + a $ &$= b p_i + a$ \\
+        & imag part & $2 b p_{i-1} + p_{i-1} - a$ &$= b p_i + p_{i-1} - a$ \\
+$g((2 b+1) p_{i-1} + a)$ 
+        & real part & $(2 b+1) p_{i-1} + a $&$= b p_i + p_{i-1} + a $ \\
+        & imag part & $(2 b+1) p_{i-1} + p_{i-1} - a $&$= b p_i + p_i - a$\\
+\hline
+$g(b p_{i} + a)$ 
+        & real part & $b p_i + a$ &\\
+        & imag part & $b p_i + p_i - a$& \\
+$g(b p_{i} + p_{i-1} - a)$ 
+        & real part & $b p_i + p_{i-1} - a$& \\
+        & imag part & $b p_i + p_{i-1} + a$&  \\
+\hline
+\end{tabular}
+\end{center}
+%
+By comparing the input locations and output locations we can see
+that the calculation is done in place.
+
+The final butterfly for $a = p_{i-1}/2$ is,
+%
+\begin{equation}
+\left(
+\begin{array}{c} 
+g(b p_i + p_{i-1}/2) \\
+g(b p_i + p_{i-1} - p_{i-1}/2)^*
+\end{array}
+\right)
+=
+\left(
+\begin{array}{c}
+g(2 b p_{i-1} + p_{i-1}/2) - i g((2 b + 1) p_{i-1} + p_{i-1}/2) \\
+g(2 b p_{i-1} + p_{i-1}/2) + i g((2 b + 1) p_{i-1} + p_{i-1}/2)
+\end{array}
+\right)
+\end{equation}
+%
+where we have substituted for the twiddle factor, $W^a_{p_i} = -i$,
+%
+\begin{eqnarray}
+W^{p_{i-1}/2}_{p_i} &=& \exp(-2\pi i p_{i-1}/2 p_i) \\
+                    &=& \exp(-2\pi i /4) \\
+                    &=& -i 
+\end{eqnarray}
+%
+For this butterfly the second line is just the conjugate of the first,
+because $p_{i-1} - p_{i-1}/2 = p_{i-1}/2$. Therefore we only need to
+consider the first line. The breakdown of the inputs and outputs is,
+%
+\begin{center}
+\renewcommand{\arraystretch}{1.5}
+\begin{tabular}{|l|lll|}
+\hline Term & & Location & \\
+\hline
+$g(2 b p_{i-1} + p_{i-1}/2)$    
+        & real part & $2 b p_{i-1} + p_{i-1}/2 $ &$= b p_i + p_{i-1}/2$ \\
+        & imag part & --- & \\
+$g((2 b + 1) p_{i-1} + p_{i-1}/2)$ 
+        & real part & $(2 b + 1) p_{i-1} + p_{i-1}/2 $&$= b p_i + p_{i} - p_{i-1}/2 $ \\
+        & imag part & --- & \\
+\hline
+$g(b p_{i} + p_{i-1}/2)$ 
+        & real part & $b p_i + p_{i-1}/2$ &\\
+        & imag part & $b p_i + p_i - p_{i-1}/2$& \\
+\hline
+\end{tabular}
+\end{center}
+%
+By comparing the locations of the inputs and outputs with the
+operations in the butterfly we find that this computation is very
+simple: the effect of the butterfly is to negate the location $b p_i +
+p_i - p_{i-1}/2$ and leave other locations unchanged. This is clearly
+an in-place operation.
+
+Here is the radix-2 algorithm for real data, in full, with the cases
+of $a=0$, $a=1\dots p_{i-1}/2 - 1$ and $a = p_{i-1}/2$ in separate
+blocks,
+%
+\begin{algorithm}
+\STATE bit-reverse ordering of $g$
+\FOR {$i = 1 \dots n$}
+  \FOR{$b = 0 \dots q_i - 1$}
+  \STATE{$\left(
+          \begin{array}{c} 
+          g(b p_i) \\
+          g(b p_i + p_{i-1})
+          \end{array}
+          \right)
+          \Leftarrow
+          \left(
+          \begin{array}{c}
+          g(b p_{i}) + g(b p_{i} + p_{i-1}) \\
+          g(b p_{i}) - g(b p_{i} + p_{i-1})
+          \end{array}
+          \right)$}
+  \ENDFOR
+
+  \FOR {$a = 1 \dots p_{i-1}/2 - 1$}
+    \FOR{$b = 0 \dots q_i - 1$}
+        \STATE{$(\Real z_0, \Imag z_0) \Leftarrow 
+                (g(b p_i + a),  g(b p_i + p_{i-1} - a))$}
+        \STATE{$(\Real z_1, \Imag z_1) \Leftarrow 
+                (g(b p_i + p_{i-1} + a), g(b p_i + p_{i} - a))$}
+        \STATE{$t_0 \Leftarrow z_0 + W^a_{p_i} z_1$}
+        \STATE{$t_1 \Leftarrow z_0 - W^a_{p_i} z_1$}
+        \STATE{$(g(b p_{i} + a),g(b p_{i} + p_i - a) \Leftarrow 
+                (\Real t_0, \Imag t_0)$}
+        \STATE{$(g(b p_{i} + p_{i-1} - a), g(b p_{i} + p_{i-1} + a))
+                 \Leftarrow 
+                (\Real t_1, -\Imag t_1)$}
+     \ENDFOR
+  \ENDFOR
+
+  \FOR{$b = 0 \dots q_i - 1$}
+        \STATE{$g(b p_{i} - p_{i-1}/2) \Leftarrow -g(b p_{i} - p_{i-1}/2)$}
+  \ENDFOR
+
+\ENDFOR
+\end{algorithm}
+%
+We split the loop over $a$ into three parts, $a=0$, $a=1\dots
+p_{i-1}/2-1$ and $a = p_{i-1}/2$ for efficiency.  When $a=0$ we have
+$W^a_{p_i}=1$ so we can eliminate a complex multiplication within the
+loop over $b$. When $a=p_{i-1}/2$ we have $W^a_{p_i} = -i$ which does
+not require a full complex multiplication either.
+
+
+\subsubsection{Calculating the Inverse}
+%
+The inverse FFT of complex data was easy to calculate, simply by
+taking the complex conjugate of the DFT matrix. The input data and
+output data were both complex and did not have any special
+symmetry. For real data the inverse FFT is more complicated because
+the half-complex symmetry of the transformed data is 
+different from the purely real input data.
+
+We can compute an inverse by stepping backwards through the forward
+transform.  To simplify the inversion it's convenient to write the
+forward algorithm with the butterfly in matrix form,
+%
+\begin{algorithm}
+\FOR {$i = 1 \dots n$}
+  \FOR {$a = 0 \dots p_{i-1}/2$}
+    \FOR{$b = 0 \dots q_i - 1$}
+        \STATE{$
+                \left(
+                \begin{array}{c} 
+                g(b p_i + a) \\
+                g(b p_i + p_{i-1} + a)
+                \end{array}
+                \right)
+                =
+                \left(
+                \begin{array}{cc}
+                1 & W^a_{p_{i}} \\
+                1 & -W^a_{p_{i}}
+                \end{array}
+                \right)
+                \left(
+                \begin{array}{c}
+                g(2 b p_{i-1} + a) \\
+                g((2 b + 1) p_{i-1} + a)
+                \end{array}
+                \right) $}
+     \ENDFOR
+  \ENDFOR
+\ENDFOR
+\end{algorithm}
+%
+To invert the algorithm we run the iterations backwards and invert the
+matrix multiplication in the innermost loop,
+%
+\begin{algorithm}
+\FOR {$i = n \dots 1$}
+  \FOR {$a = 0 \dots p_{i-1}/2$}
+    \FOR{$b = 0 \dots q_i - 1$}
+        \STATE{$
+                \left(
+                \begin{array}{c}
+                g(2 b p_{i-1} + a) \\
+                g((2 b + 1) p_{i-1} + a)
+                \end{array}
+                \right)
+                =
+                \left(
+                \begin{array}{cc}
+                1 & W^a_{p_{i}} \\
+                1 & -W^a_{p_{i}}
+                \end{array}
+                \right)^{-1}
+                \left(
+                \begin{array}{c}
+                g(b p_i + a) \\
+                g(b p_i + p_{i-1} + a)
+                \end{array}
+                \right) $}
+     \ENDFOR
+  \ENDFOR
+\ENDFOR
+\end{algorithm}
+%
+There is no need to reverse the loops over $a$ and $b$ because the
+result is independent of their order. The inverse of the matrix that
+appears is,
+%
+\begin{equation}
+\left(
+\begin{array}{cc}
+1 & W^a_{p_{i}} \\
+1 & -W^a_{p_{i}}
+\end{array}
+\right)^{-1}
+=
+{1 \over 2}
+\left(
+\begin{array}{cc}
+1 & 1 \\
+W^{-a}_{p_{i}} & -W^{-a}_{p_{i}}
+\end{array}
+\right)
+\end{equation}
+%
+To save divisions we remove the factor of $1/2$ inside the loop. This
+computes the unnormalized inverse, and the normalized inverse can be
+retrieved by dividing the final result by $N = 2^n$.
+
+Here is the radix-2 half-complex to real inverse FFT algorithm, taking
+into account the radix-2 storage scheme,
+%
+\begin{algorithm}
+\FOR {$i = n \dots 1$}
+  \FOR{$b = 0 \dots q_i - 1$}
+  \STATE{$\left(
+          \begin{array}{c} 
+          g(b p_i) \\
+          g(b p_i + p_{i-1})
+          \end{array}
+          \right)
+          \Leftarrow
+          \left(
+          \begin{array}{c}
+          g(b p_{i}) + g(b p_{i} + p_{i-1}) \\
+          g(b p_{i}) - g(b p_{i} + p_{i-1})
+          \end{array}
+          \right)$}
+  \ENDFOR
+
+  \FOR {$a = 1 \dots p_{i-1}/2 - 1$}
+    \FOR{$b = 0 \dots q_i - 1$}
+        \STATE{$(\Real z_0, \Imag z_0)
+                \Leftarrow 
+                (g(b p_i + a), g(b p_i + p_{i} - a))$}
+        \STATE{$(\Real z_1, \Imag z_1) 
+                \Leftarrow 
+                (g(b p_i + p_{i-1} - a),  -g(b p_i + p_{i-1} + a))$}
+        \STATE{$t_0 \Leftarrow z_0 + z_1$}
+        \STATE{$t_1 \Leftarrow z_0 - z_1$}
+        \STATE{$(g(b p_{i} + a), g(b p_{i} + p_{i-1} - a))
+                 \Leftarrow 
+                (\Real t_0, \Imag t_0) $}
+        \STATE{$(g(b p_{i} + p_{i-1} + a),g(b p_{i} + p_{i} - a)) 
+                \Leftarrow 
+                (\Real(W^a_{p_i}t_1), \Imag(W^a_{p_i}t_1))$}
+     \ENDFOR
+  \ENDFOR
+
+  \FOR{$b = 0 \dots q_i - 1$}
+        \STATE{$g(b p_{i} + p_{i-1}/2) \Leftarrow 2 g(b p_{i} + p_{i-1}/2)$}
+        \STATE{$g(b p_{i} + p_{i-1} + p_{i-1}/2) \Leftarrow -2 g(b p_{i} + p_{i-1} + p_{i-1}/2)$}
+  \ENDFOR
+
+\ENDFOR
+\STATE bit-reverse ordering of $g$
+\end{algorithm}
+
+
+
+\subsection{Mixed-Radix FFTs for real data}
+%
+As discussed earlier the radix-2 decimation-in-time algorithm had the
+special property that its intermediate passes are interleaved fourier
+transforms of the original data, and this generalizes to the
+mixed-radix algorithm. The complex mixed-radix algorithm that we
+derived earlier was a decimation-in-frequency algorithm, but we can
+obtain a decimation-in-time version by taking the transpose of the
+decimation-in-frequency DFT matrix like this,
+%
+\begin{eqnarray}
+W_N &=& W_N^T  \\
+&=& (T_{n_f} \dots T_2 T_1)^T \\
+&=& T_1^T T_2^T \dots T_{n_f}^T
+\end{eqnarray}
+%
+with,
+%
+\begin{eqnarray}
+T_i^T &=& \left( (P^{f_i}_{q_i} D^{f_i}_{q_i} \otimes I_{p_{i-1}})
+        (W_{f_i} \otimes I_{m_i}) \right)^T \\
+        &=&     (W_{f_i} \otimes I_{m_i})
+                ( D^{f_i}_{q_i} (P^{f_i}_{q_i})^T \otimes I_{p_{i-1}}).
+\end{eqnarray}
+%
+We have used the fact that $W$, $D$ and $I$ are symmetric and that the
+permutation matrix $P$ obeys,
+%
+\begin{equation}
+(P^a_b)^T = P^b_a.
+\end{equation}
+%
+From the definitions of $D$ and $P$ we can derive the following identity,
+%
+\begin{equation}
+D^a_b P^b_a = P^b_a D^b_a.
+\end{equation}
+%
+This allows us to put the transpose into a simple form,
+%
+\begin{equation}
+T_i^T =         (W_{f_i} \otimes I_{m_i})
+                (P^{q_i}_{f_i} D^{q_i}_{f_i} \otimes I_{p_{i-1}}).
+\end{equation}
+%
+The transposed matrix, $T^T$ applies the digit-reversal $P$ before the
+DFT $W$, giving the required decimation-in-time algorithm.  The
+transpose reverses the order of the factors --- $T_{n_f}$ is applied
+first and $T_1$ is applied last. For convenience we can reverse the
+order of the factors, $f_1 \leftrightarrow f_{n_f}$, $f_2
+\leftrightarrow f_{n_f-1}$, \dots and make the corresponding
+substitution $p_{i-1} \leftrightarrow q_i$. These substitutions give
+us a decimation-in-time algorithm with the same ordering as the
+decimation-in-frequency algorithm,
+%
+\begin{equation}
+W_N = T_{n_f} \dots T_2 T_1
+\end{equation}
+%
+\begin{equation}
+T_i = (W_{f_i} \otimes I_{m_i}) 
+        (P^{p_{i-1}}_{f_i} D^{p_{i-1}}_{f_i} \otimes I_{q_i})
+\end{equation}
+%
+where $p_i$, $q_i$ and $m_i$ now have the same meanings as before,
+namely,
+%
+\begin{eqnarray}
+p_i &=& f_1 f_2 \dots f_i \quad (p_0 = 1)  \\
+q_i &=& N / p_i \\
+m_i &=& N / f_i 
+\end{eqnarray}
+%
+Now we would like to prove that the iteration for computing $x = W z =
+T_{n_f} \dots T_2 T_1 z$ has the special property interleaving
+property. If we write the result of each intermediate pass as
+$v^{(i)}$,
+%
+\begin{eqnarray}
+v^{(0)} &=& z \\
+v^{(1)} &=& T_1 v^{(0)} \\
+v^{(2)} &=& T_2 v^{(1)} \\
+\dots   &=& \dots \\
+v^{(i)} &=& T_i v^{(i-1)} 
+\end{eqnarray}
+%
+then we will show that the intermediate results $v^{(i)}$ on any pass
+can be written as,
+%
+\begin{equation}
+v^{(i)} = (W_{p_i} \otimes I_{q_i}) z
+\end{equation}
+%
+Each intermediate stage will be a set of $q_i$ interleaved fourier
+transforms, each of length $p_i$. We can prove this result by
+induction. First we assume that the result is true for $v^{(i-1)}$,
+%
+\begin{equation}
+v^{(i-1)} = (W_{p_{i-1}} \otimes I_{q_{i-1}}) z \qquad \mbox{(assumption)}
+\end{equation}
+%
+And then we examine the next iteration using this assumption,
+%
+\begin{eqnarray}
+v^{(i)} &=& T_i v^{(i-1)} \\
+        &=& T_i (W_{p_{i-1}} \otimes I_{q_{i-1}}) z \\
+        &=& (W_{f_i} \otimes I_{m_i}) 
+                (P^{p_{i-1}}_{f_i} D^{p_{i-1}}_{f_i} \otimes I_{q_i})
+                (W_{p_{i-1}} \otimes I_{q_{i-1}}) z \label{dit-induction}
+\end{eqnarray}
+%
+Using the relation $m_i = p_{i-1} q_i$, we can write $I_{m_i}$ as
+$I_{p_{i-1} q_i}$ and $I_{q_{i-1}}$ as $I_{f_i q_i}$. By combining these
+with the basic matrix identity,
+%
+\begin{equation}
+I_{ab} = I_a \otimes I_b
+\end{equation}
+%
+we can rewrite $v^{(i)}$ in the following form,
+%
+\begin{equation}
+v^{(i)} =  (((W_{f_i} \otimes I_{p_{i-1}}) 
+                (P^{p_{i-1}}_{f_i} D^{p_{i-1}}_{f_i})
+                (W_{p_{i-1}} \otimes I_{f_i})) \otimes I_{q_i}) z .
+\end{equation}
+%
+The first part of this matrix product is the two-factor expansion of
+$W_{ab}$, for $a = p_{i-1}$ and $b = f_i$,
+%
+\begin{equation}
+W_{p_{i-1} f_i} = ((W_{f_i} \otimes I_{p_{i-1}}) 
+                  (P^{p_{i-1}}_{f_i} D^{p_{i-1}}_{f_i})
+                  (W_{p_{i-1}} \otimes I_{f_i})).
+\end{equation}
+%
+If we substitute this result, remembering that $p_i = p_{i-1} f_i$, we
+obtain,
+%
+\begin{equation}
+v^{(i)} = (W_{p_i} \otimes I_{q_i}) z
+\end{equation}
+%
+which is the desired result. The case $i=1$ can be verified
+explicitly, and induction then shows that the result is true for all
+$i$.  As discussed for the radix-2 algorithm this result is important
+because if the initial data $z$ is real then each intermediate pass is
+a set of interleaved fourier transforms of $z$, having half-complex
+symmetries (appropriately applied in the subspaces of the Kronecker
+product). Consequently only $N$ real numbers are needed to store the
+intermediate and final results.
+
+\subsection{Implementation}
+%
+The implementation of the mixed-radix real FFT algorithm follows the
+same principles as the full complex transform. Some of the steps are
+applied in the opposite order because we are dealing with a decimation
+in time algorithm instead of a decimation in frequency algorithm, but
+the basic outer structure of the algorithm is the same. We want to
+apply the factorized version of the DFT matrix $W_N$ to the input
+vector $z$,
+%
+\begin{eqnarray}
+x &=& W_N z \\
+  &=& T_{n_f} \dots T_2 T_1 z
+\end{eqnarray}
+%
+We loop over the $n_f$ factors, applying each matrix $T_i$ to the
+vector in turn to build up the complete transform,
+%
+\begin{algorithm}
+\FOR{$(i = 1 \dots n_f)$}
+        \STATE{$v \Leftarrow T_i v $}
+\ENDFOR
+\end{algorithm}
+%
+In this case the definition of $T_i$ is different because we have
+taken the transpose,
+%
+\begin{equation}
+T_i = 
+  (W_{f_i} \otimes I_{m_i}) 
+  (P^{p_{i-1}}_{f_i} D^{p_{i-1}}_{f_i} \otimes I_{q_i}).
+\end{equation}
+%
+We'll define a temporary vector $t$ to denote the results of applying the
+rightmost matrix,
+%
+\begin{equation}
+t = (P^{p_{i-1}}_{f_i} D^{p_{i-1}}_{f_i} \otimes I_{q_i}) v
+\end{equation}
+%
+If we expand this out into individual components, as before, we find a
+similar simplification,
+%
+\begin{eqnarray}
+t_{aq+b} 
+&=&
+\sum_{a'b'} 
+(P^{p_{i-1}}_{f_i} D^{p_{i-1}}_{f_i} \otimes I_{q_i})_{(aq+b)(a'q+b')}
+v_{a'q+b'} \\
+&=&
+\sum_{a'} 
+(P^{p_{i-1}}_{f_i} D^{p_{i-1}}_{f_i})_{a a'}
+v_{a'q+b} 
+\end{eqnarray}
+%
+We have factorized the indices into the form $aq+b$, with $0 \leq a <
+p_{i}$ and $0 \leq b < q$.  Just as in the decimation in frequency
+algorithm we can split the index $a$ to remove the matrix
+multiplication completely. We have to write $a$ as $\mu f + \lambda$,
+where $0 \leq \mu < p_{i-1}$ and $0 \leq \lambda < f$,
+%
+\begin{eqnarray}
+t_{(\mu f + \lambda)q+b} 
+&=&
+\sum_{\mu'\lambda'} 
+(P^{p_{i-1}}_{f_i} D^{p_{i-1}}_{f_i})_{(\mu f + \lambda)(\mu' f + \lambda')}
+v_{(\mu' f + \lambda')q+b} 
+\\
+&=&
+\sum_{\mu'\lambda'} 
+(P^{p_{i-1}}_{f_i})_{(\mu f + \lambda)(\mu' f + \lambda')}
+\omega^{\mu'\lambda'}_{p_{i}}
+v_{(\mu' f + \lambda')q+b}
+\end{eqnarray}
+%
+The matrix $P^{p_{i-1}}_{f_i}$ exchanges an index of $(\mu f +
+\lambda) q + b$ with $(\lambda p_{i-1} + \mu) q + b$, giving a final
+result of,
+%
+\begin{eqnarray}
+t_{(\lambda p_{i-1} + \mu) q + b} 
+= 
+w^{\mu\lambda}_{p_i} v_{(\mu f + \lambda)q +b}
+\end{eqnarray}
+%
+To calculate the next stage,
+%
+\begin{equation}
+v' = (W_{f_i} \otimes I_{m_i}) t,
+\end{equation}
+%
+we temporarily rearrange the index of $t$ to separate the $m_{i}$
+independent DFTs in the Kronecker product,
+%
+\begin{equation}
+v'_{(\lambda p_{i-1} + \mu) q + b} 
+=
+\sum_{\lambda' \mu' b'}
+(W_{f_i} \otimes I_{m_i})_{
+((\lambda p_{i-1} + \mu) q + b)
+((\lambda' p_{i-1} + \mu') q + b')}
+t_{(\lambda' p_{i-1} + \mu') q + b'}
+\end{equation}
+%
+If we use the identity $m = p_{i-1} q$ to rewrite the index of $t$
+like this,
+%
+\begin{equation}
+t_{(\lambda p_{i-1} + \mu) q + b} = t_{\lambda m + (\mu q + b)}
+\end{equation}
+%
+we can split the Kronecker product,
+%
+\begin{eqnarray}
+v'_{(\lambda p_{i-1} + \mu) q + b}
+&=&
+\sum_{\lambda' \mu' b'}
+(W_{f_i} \otimes I_{m_i})_{
+((\lambda p_{i-1} + \mu) q + b)
+((\lambda' p_{i-1} + \mu') q + b')}
+t_{(\lambda' p_{i-1} + \mu') q + b'}\\
+&=&
+\sum_{\lambda'}
+(W_{f_i})_{\lambda \lambda'}
+t_{\lambda' m_i + (\mu q + b)} 
+\end{eqnarray}
+%
+If we switch back to the original form of the index in the last line we obtain,
+%
+\begin{eqnarray}
+\phantom{v'_{(\lambda p_{i-1} + \mu) q + b}}
+&=&
+\sum_{\lambda'}
+(W_{f_i})_{\lambda \lambda'}
+t_{(\lambda p_{i-1} + \mu) q + b}
+\end{eqnarray}
+%
+which allows us to substitute our previous result for $t$,
+%
+\begin{equation}
+v'_{(\lambda p_{i-1} + \mu) q + b} 
+= 
+\sum_{\lambda'}
+(W_{f_i})_{\lambda \lambda'}
+w^{\mu\lambda'}_{p_i} v_{(\mu f + \lambda')q + b}
+\end{equation}
+%
+This gives us the basic decimation-in-time mixed-radix algorithm for
+complex data which we will be able to specialize to real data,
+%
+\begin{algorithm}
+\FOR{$i = 1 \dots n_f$}
+\FOR{$\mu = 0 \dots p_{i-1} - 1$}
+\FOR{$b = 0 \dots q-1$}
+\FOR{$\lambda = 0 \dots f-1$}
+\STATE{$t_\lambda \Leftarrow 
+               \omega^{\mu\lambda'}_{p_{i}} v_{(\mu f + \lambda')q + b}$}
+\ENDFOR
+\FOR{$\lambda = 0 \dots f-1$}
+\STATE{$v'_{(\lambda p_{i-1} + \mu)q + b} =
+       \sum_{\lambda'=0}^{f-1}  W_f(\lambda,\lambda') t_{\lambda'}$}
+       \COMMENT{DFT matrix-multiply module}
+\ENDFOR
+\ENDFOR
+\ENDFOR
+\STATE{$v \Leftarrow v'$}
+\ENDFOR
+\end{algorithm}
+%
+We are now at the point where we can convert an algorithm formulated
+in terms of complex variables to one in terms of real variables by
+choosing a suitable storage scheme.  We will adopt the FFTPACK storage
+convention. FFTPACK uses a scheme where individual FFTs are
+contiguous, not interleaved, and real-imaginary pairs are stored in
+neighboring locations. This has better locality than was possible for
+the radix-2 case.
+
+The interleaving of the intermediate FFTs results from the Kronecker
+product, $W_p \otimes I_q$. The FFTs can be made contiguous if we
+reorder the Kronecker product on the intermediate passes,
+%
+\begin{equation}
+W_p \otimes I_q \Rightarrow I_q \otimes W_p
+\end{equation}
+%
+This can be implemented by a simple change in indexing.  On pass-$i$
+we store element $v_{a q_i + b}$ in location $v_{b p_i+a}$. We
+compensate for this change by making the same transposition when
+reading the data. Note that this only affects the indices of the
+intermediate passes.  On the zeroth iteration the transposition has no
+effect because $p_0 = 1$. Similarly there is no effect on the last
+iteration, which has $q_{n_f} = 1$. This is how the algorithm above
+looks after this index transformation,
+%
+\begin{algorithm}
+\FOR{$i = 1 \dots n_f$}
+\FOR{$\mu = 0 \dots p_{i-1} - 1$}
+\FOR{$b = 0 \dots q-1$}
+\FOR{$\lambda = 0 \dots f-1$}
+\STATE{$t_\lambda \Leftarrow 
+               \omega^{\mu\lambda'}_{p_{i}} v_{(\lambda'q + b)p_{i-1} + \mu}$}
+\ENDFOR
+\FOR{$\lambda = 0 \dots f-1$}
+\STATE{$v'_{b p + (\lambda p_{i-1} + \mu)} =
+       \sum_{\lambda'=0}^{f-1}  W_f(\lambda,\lambda') t_{\lambda'}$}
+       \COMMENT{DFT matrix-multiply module}
+\ENDFOR
+\ENDFOR
+\ENDFOR
+\STATE{$v \Leftarrow v'$}
+\ENDFOR
+\end{algorithm}
+%
+We transpose the input terms by writing the index in the form $a
+q_{i-1} + b$, to take account of the pass-dependence of the scheme,
+%
+\begin{equation}
+v_{(\mu f + \lambda')q + b} = v_{\mu q_{i-1} + \lambda'q + b}
+\end{equation}
+%
+We used the identity $q_{i-1} = f q$ to split the index. Note that in
+this form $\lambda'q + b$ runs from 0 to $q_{i-1} - 1$ as $b$ runs
+from 0 to $q-1$ and $\lambda'$ runs from 0 to $f-1$. The transposition
+for the input terms then gives,
+%
+\begin{equation}
+v_{\mu q_{i-1} + \lambda'q + b} \Rightarrow  v_{(\lambda'q + b) p_{i-1} + \mu}
+\end{equation}
+%
+In the FFTPACK scheme the intermediate output data have the same
+half-complex symmetry as the radix-2 example, namely,
+%
+\begin{equation}
+v^{(i)}_{b p + a} = v^{(i)*}_{b p + (p - a)}
+\end{equation}
+%
+and on the input data from the previous pass have the symmetry,
+%
+\begin{equation}
+v^{(i-1)}_{(\lambda' q + b) p_{i-1} + \mu} = v^{(i-1)*}_{(\lambda' q +
+b) p_{i-1} + (p_{i-1} - \mu)}
+\end{equation}
+%
+Using these symmetries we can halve the storage and computation
+requirements for each pass. Compared with the radix-2 algorithm we
+have more freedom because the computation does not have to be done in
+place. The storage scheme adopted by FFTPACK places elements
+sequentially with real and imaginary parts in neighboring
+locations. Imaginary parts which are known to be zero are not
+stored. Here are the full details of the scheme,
+%
+\begin{center}
+\renewcommand{\arraystretch}{1.5}
+\begin{tabular}{|l|lll|}
+\hline Term & & Location & \\
+\hline
+$g(b p_i)$        
+        & real part & $b p_{i} $ & \\
+        & imag part & --- & \\
+\hline
+$g(b p_i + a)$ 
+        & real part & $b p_{i} + 2a - 1 $& for $a = 1 \dots p_i/2 - 1$ \\
+        & imag part & $b p_{i} + 2a$ & \\
+\hline
+$g(b p_{i} + p_{i}/2)$ 
+        & real part & $b p_i + p_{i} - 1$ & if $p_i$ is even\\
+        & imag part & --- & \\
+\hline
+\end{tabular}
+\end{center}
+%
+The real element for $a=0$ is stored in location $bp$.  The real parts
+for $a = 1 \dots p/2 - 1$ are stored in locations $bp + 2a -1$ and the
+imaginary parts are stored in locations $b p + 2 a$.  When $p$ is even
+the term for $a = p/2$ is purely real and we store it in location $bp
++ p - 1$. The zero imaginary part is not stored.
+
+When we compute the basic iteration,
+%
+\begin{equation}
+v^{(i)}_{b p + (\lambda p_{i-1} + \mu)} = \sum_{\lambda'} 
+W^{\lambda \lambda'}_f
+\omega^{\mu\lambda'}_{p_i} v^{(i-1)}_{(\lambda' q + b)p_{i-1} + \mu}
+\end{equation}
+%
+we eliminate the redundant conjugate terms with $a > p_{i}/2$ as we
+did in the radix-2 algorithm. Whenever we need to store a term with $a
+> p_{i}/2$ we consider instead the corresponding conjugate term with
+$a' = p - a$. Similarly when reading data we replace terms with $\mu >
+p_{i-1}/2$ with the corresponding conjugate term for $\mu' = p_{i-1} -
+\mu$.
+
+Since the input data on each stage has half-complex symmetry we only
+need to consider the range $\mu=0 \dots p_{i-1}/2$. We can consider
+the best ways to implement the basic iteration for each pass, $\mu = 0
+\dots p_{i-1}/2$.
+
+On the first pass where $\mu=0$ we will be accessing elements which
+are the zeroth components of the independent transforms $W_{p_{i-1}}
+\otimes I_{q_{i-1}}$, and are purely real.
+%
+We can code the pass with $\mu=0$ as a special case for real input
+data, and conjugate-complex output. When $\mu=0$ the twiddle factors
+$\omega^{\mu\lambda'}_{p_i}$ are all unity, giving a further saving.
+We can obtain small-$N$ real-data DFT modules by removing the
+redundant operations from the complex modules.
+%
+For example the $N=3$ module was,
+%
+\begin{equation}
+\begin{array}{lll}
+t_1 = z_1 + z_2, &
+t_2 = z_0 - t_1/2, &
+t_3 = \sin(\pi/3) (z_1 - z_2), 
+\end{array}
+\end{equation}
+\begin{equation}
+\begin{array}{lll}
+x_0 = z_0 + t_1, &
+x_1 = t_2 + i t_3, &
+x_2 = t_2 - i t_3. 
+\end{array}
+\end{equation}
+%
+In the complex case all the operations were complex, for complex input
+data $z_0$, $z_1$, $z_2$. In the real case $z_0$, $z_1$ and $z_2$ are
+all real. Consequently $t_1$, $t_2$ and $t_3$ are also real, and the
+symmetry $x_1 = t_1 + i t_2 = x^*_2$ means that we do not have to
+compute $x_2$ once we have computed $x_1$.
+
+For subsequent passes $\mu = 1 \dots p_{i-1}/2 - 1$ the input data is
+complex and we have to compute full complex DFTs using the same
+modules as in the complex case. Note that the inputs are all of the
+form $v_{(\lambda q + b) p_{i-1} + \mu}$ with $\mu < p_{i-1}/2$ so we
+never need to use the symmetry to access the conjugate elements with
+$\mu > p_{i-1}/2$.
+
+If $p_{i-1}$ is even then we reach the halfway point $\mu=p_{i-1}/2$,
+which is another special case. The input data in this case is purely
+real because $\mu = p_{i-1} - \mu$ for $\mu = p_{i-1}/2$. We can code
+this as a special case, using real inputs and real-data DFT modules as
+we did for $\mu=0$. However, for $\mu = p_{i-1}/2$ the twiddle factors
+are not unity,
+%
+\begin{eqnarray}
+\omega^{\mu\lambda'}_{p_i} &=& \omega^{(p_{i-1}/2)\lambda'}_{p_i} \\
+&=& \exp(-i\pi\lambda'/f_i) 
+\end{eqnarray}
+%
+These twiddle factors introduce an additional phase which modifies the
+symmetry of the outputs. Instead of the conjugate-complex symmetry
+which applied for $\mu=0$ there is a shifted conjugate-complex
+symmetry,
+%
+\begin{equation}
+t_\lambda = t^*_{f-(\lambda+1)}
+\end{equation}
+%
+This is easily proved,
+%
+\begin{eqnarray}
+t_\lambda 
+&=& 
+\sum e^{-2\pi i \lambda\lambda'/f_i} e^{-i\pi \lambda'/f_i} r_{\lambda'} \\
+t_{f - (\lambda + 1)}
+&=& 
+\sum e^{-2\pi i (f-\lambda-1)\lambda'/f_i} e^{-i\pi \lambda'/f_i} r_{\lambda'} \\
+&=& 
+\sum e^{2\pi i \lambda\lambda'/f_i} e^{i\pi \lambda'/f_i} r_{\lambda'} \\
+&=& t^*_\lambda
+\end{eqnarray}
+%
+The symmetry of the output means that we only need to compute half of
+the output terms, the remaining terms being conjugates or zero
+imaginary parts. For example, when $f=4$ the outputs are $(x_0 + i
+y_0, x_1 + i y_1, x_1 - i y_1, x_0 - i y_0)$. For $f=5$ the outputs
+are $(x_0 + i y_0, x_1 + i y_1, x_2, x_1 - i y_1, x_0 - i y_0)$. By
+combining the twiddle factors and DFT matrix we can make a combined
+module which applies both at the same time. By starting from the
+complex DFT modules and bringing in twiddle factors we can derive
+optimized modules. Here are the modules given by Temperton for $z = W
+\Omega x$ where $x$ is real and $z$ has the shifted conjugate-complex
+symmetry. The matrix of twiddle factors, $\Omega$, is given by,
+%
+\begin{equation}
+\Omega = \mathrm{diag}(1, e^{-i\pi/f}, e^{-2\pi i/f}, \dots, e^{-i\pi(f-1)/f})
+\end{equation}
+%
+We write $z$ in terms of two real vectors $z = a + i b$.
+%
+For $N=2$,
+%
+\begin{equation}
+\begin{array}{ll}
+a_0 = x_0, & 
+b_0 = - x_1.
+\end{array}
+\end{equation}
+%
+For $N=3$,
+%
+\begin{equation}
+\begin{array}{l}
+t_1 = x_1 - x_2,
+\end{array}
+\end{equation}
+\begin{equation}
+\begin{array}{ll}
+a_0 = x_0 + t_1/2, & b_0 = x_0 - t_1,
+\end{array}
+\end{equation}
+\begin{equation}
+\begin{array}{l}
+a_1 = - \sin(\pi/3) (x_1 + x_2)
+\end{array}
+\end{equation}
+%
+For $N=4$,
+%
+\begin{equation}
+\begin{array}{ll}
+t_1 = (x_1 - x_3)/\sqrt{2}, & t_2 = (x_1 + x_3)/\sqrt{2},
+\end{array}
+\end{equation}
+\begin{equation}
+\begin{array}{ll}
+a_0 = x_0 + t_1, & b_0 = -x_2 - t_2,
+\end{array}
+\end{equation}
+\begin{equation}
+\begin{array}{ll}
+a_1 = x_0 - t_1, & b_1 = x_2 - t_2.
+\end{array}
+\end{equation}
+%
+For $N=5$,
+%
+\begin{equation}
+\begin{array}{ll}
+t_1 = x_1 - x_4, & t_2 = x_1 + x_4,
+\end{array}
+\end{equation}
+\begin{equation}
+\begin{array}{ll}
+t_3 = x_2 - x_3, & t_4 = x_2 + x_3,
+\end{array}
+\end{equation}
+\begin{equation}
+\begin{array}{ll}
+t_5 = t_1 - t_3, & t_6 = x_0 + t_5 / 4,
+\end{array}
+\end{equation}
+\begin{equation}
+\begin{array}{ll}
+t_7 = (\sqrt{5}/4)(t_1 + t_3) & 
+\end{array}
+\end{equation}
+\begin{equation}
+\begin{array}{ll}
+a_0 = t_6 + t_7, & b_0 = -\sin(2\pi/10) t_2 - \sin(2\pi/5) t_4, 
+\end{array}
+\end{equation}
+\begin{equation}
+\begin{array}{ll}
+a_1 = t_6 - t_7, & b_1 = -\sin(2\pi/5) t_2 + \sin(2\pi/10) t_4,
+\end{array}
+\end{equation}
+\begin{equation}
+\begin{array}{ll}
+a_2 = x_0 - t_5 &
+\end{array}
+\end{equation}
+%
+For $N=6$,
+%
+\begin{equation}
+\begin{array}{ll}
+t_1 = \sin(\pi/3)(x_5 - x_1), & t_2 = \sin(\pi/3) (x_2 + x_4),
+\end{array}
+\end{equation}
+\begin{equation}
+\begin{array}{ll}
+t_3 = x_2 - x_4, & t_4 = x_1 + x_5,
+\end{array}
+\end{equation}
+\begin{equation}
+\begin{array}{ll}
+t_5 = x_0 + t_3 / 2, & t_6 = -x_3 - t_4 / 2,
+\end{array}
+\end{equation}
+\begin{equation}
+\begin{array}{ll}
+a_0 = t_5 - t_1, & b_0 = t_6 - t_2,
+\end{array}
+\end{equation}
+\begin{equation}
+\begin{array}{ll}
+a_1 = x_0 - t_3, & b_1 = x_3 - t_4,
+\end{array}
+\end{equation}
+\begin{equation}
+\begin{array}{ll}
+a_2 = t_5 + t_1, & b_2 = t_6 + t_2
+\end{array}
+\end{equation}
+
+\section{Computing the mixed-radix inverse for real data}
+%
+To compute the inverse of the mixed-radix FFT on real data we step
+through the algorithm in reverse and invert each operation.
+
+This gives the following algorithm using FFTPACK indexing,
+%
+\begin{algorithm}
+\FOR{$i = n_f \dots 1$}
+\FOR{$\mu = 0 \dots p_{i-1} - 1$}
+\FOR{$b = 0 \dots q-1$}
+\FOR{$\lambda = 0 \dots f-1$}
+\STATE{$t_{\lambda'} =
+       \sum_{\lambda'=0}^{f-1}  W_f(\lambda,\lambda')
+       v_{b p + (\lambda p_{i-1} + \mu)}$}
+       \COMMENT{DFT matrix-multiply module}
+\ENDFOR
+\FOR{$\lambda = 0 \dots f-1$}
+\STATE{$v'_{(\lambda'q + b)p_{i-1} + \mu} \Leftarrow 
+               \omega^{-\mu\lambda'}_{p_{i}} t_\lambda$}
+\ENDFOR
+
+\ENDFOR
+\ENDFOR
+\STATE{$v \Leftarrow v'$}
+\ENDFOR
+\end{algorithm}
+%
+When $\mu=0$ we are applying an inverse DFT to half-complex data,
+giving a real result. The twiddle factors are all unity. We can code
+this as a special case, just as we did for the forward routine. We
+start with complex module and eliminate the redundant terms. In this
+case it is the final result which has the zero imaginary part, and we
+eliminate redundant terms by using the half-complex symmetry of the
+input data. 
+
+When $\mu=1 \dots p_{i-1}/2 - 1$ we have full complex transforms on
+complex data, so no simplification is possible.
+
+When $\mu = p_{i-1}/2$ (which occurs only when $p_{i-1}$ is even) we
+have a combination of twiddle factors and DFTs on data with the
+shifted half-complex symmetry which give a real result. We implement
+this as a special module, essentially by inverting the system of
+equations given for the forward case. We use the modules given by
+Temperton, appropriately modified for our version of the algorithm. He
+uses a slightly different convention which differs by factors of two
+for some terms (consult his paper for details~\cite{temperton83real}).
+
+For $N=2$,
+%
+\begin{equation}
+\begin{array}{ll}
+x_0 = 2 a_0, & x_1 = - 2 b_0 .
+\end{array}
+\end{equation}
+%
+For $N=3$,
+%
+\begin{equation}
+\begin{array}{ll}
+t_1 = a_0 - a_1, & t_2 = \sqrt{3} b_1, \\
+\end{array}
+\end{equation}
+\begin{equation}
+\begin{array}{lll}
+x_0 = 2 a_0 + a_1, & x_1 = t_1 - t_2, & x_2 = - t_1 - t_2 
+\end{array}
+\end{equation}
+%
+For $N=4$,
+\begin{equation}
+\begin{array}{ll}
+t_1 = \sqrt{2} (b_0 + b_1), & t_2 = \sqrt{2} (a_0 - a_1),
+\end{array}
+\end{equation}
+\begin{equation}
+\begin{array}{ll}
+x_0 = 2(a_0 + a_1), & x_1 = t_2 - t_1 , 
+\end{array}
+\end{equation}
+\begin{equation}
+\begin{array}{ll}
+x_2 = 2(b_1 - b_0), & x_3 = -(t_2 + t_1).
+\end{array}
+\end{equation}
+%
+For $N=5$,
+%
+\begin{equation}
+\begin{array}{ll}
+t_1 = 2 (a_0 + a_1), & t_2 = t_1 / 4 - a_2,
+\end{array}
+\end{equation}
+\begin{equation}
+\begin{array}{ll}
+t_3 = (\sqrt{5}/2) (a_0 - a_1), 
+\end{array}
+\end{equation}
+\begin{equation}
+\begin{array}{l}
+t_4 = 2(\sin(2\pi/10) b_0 + \sin(2\pi/5) b_1),
+\end{array}
+\end{equation}
+\begin{equation}
+\begin{array}{l}
+t_5 = 2(\sin(2\pi/10) b_0 - \sin(2\pi/5) b_1),
+\end{array}
+\end{equation}
+\begin{equation}
+\begin{array}{ll}
+t_6 = t_3 + t_2, & t_7 = t_3 - t_2,
+\end{array}
+\end{equation}
+\begin{equation}
+\begin{array}{ll}
+x_0 = t_1 + a_2, & x_1 = t_6 - t_4 ,
+\end{array}
+\end{equation}
+\begin{equation}
+\begin{array}{ll}
+x_2 = t_7 - t_5, & x_3 = - t_7 - t_5,
+\end{array}
+\end{equation}
+\begin{equation}
+\begin{array}{ll}
+x_4 = -t_6 - t_4.
+\end{array}
+\end{equation}
+
+\section{Conclusions}
+%
+We have described the basic algorithms for one-dimensional radix-2 and
+mixed-radix FFTs. It would be nice to have a pedagogical explanation
+of the split-radix FFT algorithm, which is faster than the simple
+radix-2 algorithm we used. We could also have a whole chapter on
+multidimensional FFTs.
+%
+%\section{Multidimensional FFTs}
+%\section{Testing FFTs, Numerical Analysis}
+
+%\nocite{*}
+\bibliographystyle{unsrt} 
+\bibliography{fftalgorithms} 
+
+\end{document}
+
+
+
+
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diff --git a/gsl-1.9/doc/fitting.texi b/gsl-1.9/doc/fitting.texi
new file mode 100644
index 0000000..f407e3b
--- /dev/null
+++ b/gsl-1.9/doc/fitting.texi
@@ -0,0 +1,501 @@
+@cindex fitting
+@cindex least squares fit
+@cindex regression, least squares
+@cindex weighted linear fits
+@cindex unweighted linear fits
+This chapter describes routines for performing least squares fits to
+experimental data using linear combinations of functions.  The data
+may be weighted or unweighted, i.e. with known or unknown errors.  For
+weighted data the functions compute the best fit parameters and their
+associated covariance matrix.  For unweighted data the covariance
+matrix is estimated from the scatter of the points, giving a
+variance-covariance matrix.
+
+The functions are divided into separate versions for simple one- or
+two-parameter regression and multiple-parameter fits.  The functions
+are declared in the header file @file{gsl_fit.h}.
+
+@menu
+* Fitting Overview::            
+* Linear regression::           
+* Linear fitting without a constant term::  
+* Multi-parameter fitting::     
+* Fitting Examples::            
+* Fitting References and Further Reading::  
+@end menu
+
+@node Fitting Overview
+@section Overview
+
+Least-squares fits are found by minimizing @math{\chi^2}
+(chi-squared), the weighted sum of squared residuals over @math{n}
+experimental datapoints @math{(x_i, y_i)} for the model @math{Y(c,x)},
+@tex
+\beforedisplay
+$$
+\chi^2 = \sum_i w_i (y_i - Y(c, x_i))^2
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+\chi^2 = \sum_i w_i (y_i - Y(c, x_i))^2
+@end example
+
+@end ifinfo
+@noindent
+The @math{p} parameters of the model are @c{$c = \{c_0, c_1, \dots\}$}
+@math{c = @{c_0, c_1, @dots{}@}}.  The
+weight factors @math{w_i} are given by @math{w_i = 1/\sigma_i^2},
+where @math{\sigma_i} is the experimental error on the data-point
+@math{y_i}.  The errors are assumed to be
+gaussian and uncorrelated. 
+For unweighted data the chi-squared sum is computed without any weight factors. 
+
+The fitting routines return the best-fit parameters @math{c} and their
+@math{p \times p} covariance matrix.  The covariance matrix measures the
+statistical errors on the best-fit parameters resulting from the 
+errors on the data, @math{\sigma_i}, and is defined
+@cindex covariance matrix, linear fits
+as @c{$C_{ab} = \langle \delta c_a \delta c_b \rangle$}
+@math{C_@{ab@} = <\delta c_a \delta c_b>} where @c{$\langle \, \rangle$}
+@math{< >} denotes an average over the gaussian error distributions of the underlying datapoints.
+
+The covariance matrix is calculated by error propagation from the data
+errors @math{\sigma_i}.  The change in a fitted parameter @math{\delta
+c_a} caused by a small change in the data @math{\delta y_i} is given
+by
+@tex
+\beforedisplay
+$$
+\delta c_a = \sum_i {\partial c_a \over \partial y_i} \delta y_i
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+\delta c_a = \sum_i (dc_a/dy_i) \delta y_i
+@end example
+
+@end ifinfo
+@noindent
+allowing the covariance matrix to be written in terms of the errors on the data,
+@tex
+\beforedisplay
+$$
+C_{ab} =  \sum_{i,j} {\partial c_a \over \partial y_i}
+                     {\partial c_b \over \partial y_j} 
+                     \langle \delta y_i \delta y_j \rangle
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+C_@{ab@} = \sum_@{i,j@} (dc_a/dy_i) (dc_b/dy_j) <\delta y_i \delta y_j>
+@end example
+
+@end ifinfo
+@noindent
+For uncorrelated data the fluctuations of the underlying datapoints satisfy
+@c{$\langle \delta y_i \delta y_j \rangle = \sigma_i^2 \delta_{ij}$}
+@math{<\delta y_i \delta y_j> = \sigma_i^2 \delta_@{ij@}}, giving a 
+corresponding parameter covariance matrix of
+@tex
+\beforedisplay
+$$
+C_{ab} = \sum_{i} {1 \over w_i} {\partial c_a \over \partial y_i} {\partial c_b \over \partial y_i} 
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+C_@{ab@} = \sum_i (1/w_i) (dc_a/dy_i) (dc_b/dy_i) 
+@end example
+
+@end ifinfo
+@noindent
+When computing the covariance matrix for unweighted data, i.e. data with unknown errors, 
+the weight factors @math{w_i} in this sum are replaced by the single estimate @math{w =
+1/\sigma^2}, where @math{\sigma^2} is the computed variance of the
+residuals about the best-fit model, @math{\sigma^2 = \sum (y_i - Y(c,x_i))^2 / (n-p)}.  
+This is referred to as the @dfn{variance-covariance matrix}.
+@cindex variance-covariance matrix, linear fits
+
+The standard deviations of the best-fit parameters are given by the
+square root of the corresponding diagonal elements of
+the covariance matrix, @c{$\sigma_{c_a} = \sqrt{C_{aa}}$}
+@math{\sigma_@{c_a@} = \sqrt@{C_@{aa@}@}}.
+
+@node   Linear regression
+@section Linear regression
+@cindex linear regression
+The functions described in this section can be used to perform
+least-squares fits to a straight line model, @math{Y(c,x) = c_0 + c_1 x}.
+
+@cindex covariance matrix, from linear regression
+@deftypefun int gsl_fit_linear (const double * @var{x}, const size_t @var{xstride}, const double * @var{y}, const size_t @var{ystride}, size_t @var{n}, double * @var{c0}, double * @var{c1}, double * @var{cov00}, double * @var{cov01}, double * @var{cov11}, double * @var{sumsq})
+This function computes the best-fit linear regression coefficients
+(@var{c0},@var{c1}) of the model @math{Y = c_0 + c_1 X} for the dataset
+(@var{x}, @var{y}), two vectors of length @var{n} with strides
+@var{xstride} and @var{ystride}.  The errors on @var{y} are assumed unknown so 
+the variance-covariance matrix for the
+parameters (@var{c0}, @var{c1}) is estimated from the scatter of the
+points around the best-fit line and returned via the parameters
+(@var{cov00}, @var{cov01}, @var{cov11}).   
+The sum of squares of the residuals from the best-fit line is returned
+in @var{sumsq}.
+@end deftypefun
+
+@deftypefun int gsl_fit_wlinear (const double * @var{x}, const size_t @var{xstride}, const double * @var{w}, const size_t @var{wstride}, const double * @var{y}, const size_t @var{ystride}, size_t @var{n}, double * @var{c0}, double * @var{c1}, double * @var{cov00}, double * @var{cov01}, double * @var{cov11}, double * @var{chisq})
+This function computes the best-fit linear regression coefficients
+(@var{c0},@var{c1}) of the model @math{Y = c_0 + c_1 X} for the weighted
+dataset (@var{x}, @var{y}), two vectors of length @var{n} with strides
+@var{xstride} and @var{ystride}.  The vector @var{w}, of length @var{n}
+and stride @var{wstride}, specifies the weight of each datapoint. The
+weight is the reciprocal of the variance for each datapoint in @var{y}.
+
+The covariance matrix for the parameters (@var{c0}, @var{c1}) is
+computed using the weights and returned via the parameters
+(@var{cov00}, @var{cov01}, @var{cov11}).  The weighted sum of squares
+of the residuals from the best-fit line, @math{\chi^2}, is returned in
+@var{chisq}.
+@end deftypefun
+
+@deftypefun int gsl_fit_linear_est (double @var{x}, double @var{c0}, double @var{c1}, double @var{c00}, double @var{c01}, double @var{c11}, double * @var{y}, double * @var{y_err})
+This function uses the best-fit linear regression coefficients
+@var{c0},@var{c1} and their covariance
+@var{cov00},@var{cov01},@var{cov11} to compute the fitted function
+@var{y} and its standard deviation @var{y_err} for the model @math{Y =
+c_0 + c_1 X} at the point @var{x}.
+@end deftypefun
+
+@node Linear fitting without a constant term
+@section Linear fitting without a constant term
+
+The functions described in this section can be used to perform
+least-squares fits to a straight line model without a constant term,
+@math{Y = c_1 X}.
+
+@deftypefun int gsl_fit_mul (const double * @var{x}, const size_t @var{xstride}, const double * @var{y}, const size_t @var{ystride}, size_t @var{n}, double * @var{c1}, double * @var{cov11}, double * @var{sumsq})
+This function computes the best-fit linear regression coefficient
+@var{c1} of the model @math{Y = c_1 X} for the datasets (@var{x},
+@var{y}), two vectors of length @var{n} with strides @var{xstride} and
+@var{ystride}.  The errors on @var{y} are assumed unknown so the 
+variance of the parameter @var{c1} is estimated from
+the scatter of the points around the best-fit line and returned via the
+parameter @var{cov11}.  The sum of squares of the residuals from the
+best-fit line is returned in @var{sumsq}.
+@end deftypefun
+
+@deftypefun int gsl_fit_wmul (const double * @var{x}, const size_t @var{xstride}, const double * @var{w}, const size_t @var{wstride}, const double * @var{y}, const size_t @var{ystride}, size_t @var{n}, double * @var{c1}, double * @var{cov11}, double * @var{sumsq})
+This function computes the best-fit linear regression coefficient
+@var{c1} of the model @math{Y = c_1 X} for the weighted datasets
+(@var{x}, @var{y}), two vectors of length @var{n} with strides
+@var{xstride} and @var{ystride}.  The vector @var{w}, of length @var{n}
+and stride @var{wstride}, specifies the weight of each datapoint. The
+weight is the reciprocal of the variance for each datapoint in @var{y}.
+
+The variance of the parameter @var{c1} is computed using the weights
+and returned via the parameter @var{cov11}.  The weighted sum of
+squares of the residuals from the best-fit line, @math{\chi^2}, is
+returned in @var{chisq}.
+@end deftypefun
+
+@deftypefun int gsl_fit_mul_est (double @var{x}, double @var{c1}, double @var{c11}, double * @var{y}, double * @var{y_err})
+This function uses the best-fit linear regression coefficient @var{c1}
+and its covariance @var{cov11} to compute the fitted function
+@var{y} and its standard deviation @var{y_err} for the model @math{Y =
+c_1 X} at the point @var{x}.
+@end deftypefun
+
+@node Multi-parameter fitting
+@section Multi-parameter fitting
+@cindex multi-parameter regression
+@cindex fits, multi-parameter linear
+The functions described in this section perform least-squares fits to a
+general linear model, @math{y = X c} where @math{y} is a vector of
+@math{n} observations, @math{X} is an @math{n} by @math{p} matrix of
+predictor variables, and the elements of the vector @math{c} are the @math{p} unknown best-fit parameters which are to be estimated.  The chi-squared value is given by @c{$\chi^2 = \sum_i w_i (y_i - \sum_j X_{ij} c_j)^2$}
+@math{\chi^2 = \sum_i w_i (y_i - \sum_j X_@{ij@} c_j)^2}.
+
+This formulation can be used for fits to any number of functions and/or
+variables by preparing the @math{n}-by-@math{p} matrix @math{X}
+appropriately.  For example, to fit to a @math{p}-th order polynomial in
+@var{x}, use the following matrix,
+@tex
+\beforedisplay
+$$
+X_{ij} = x_i^j
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+X_@{ij@} = x_i^j
+@end example
+
+@end ifinfo
+@noindent
+where the index @math{i} runs over the observations and the index
+@math{j} runs from 0 to @math{p-1}.
+
+To fit to a set of @math{p} sinusoidal functions with fixed frequencies
+@math{\omega_1}, @math{\omega_2}, @dots{}, @math{\omega_p}, use,
+@tex
+\beforedisplay
+$$
+X_{ij} = \sin(\omega_j x_i)
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+X_@{ij@} = sin(\omega_j x_i)
+@end example
+
+@end ifinfo
+@noindent
+To fit to @math{p} independent variables @math{x_1}, @math{x_2}, @dots{},
+@math{x_p}, use,
+@tex
+\beforedisplay
+$$
+X_{ij} = x_j(i)
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+X_@{ij@} = x_j(i)
+@end example
+
+@end ifinfo
+@noindent
+where @math{x_j(i)} is the @math{i}-th value of the predictor variable
+@math{x_j}.
+
+The functions described in this section are declared in the header file
+@file{gsl_multifit.h}.
+
+The solution of the general linear least-squares system requires an
+additional working space for intermediate results, such as the singular
+value decomposition of the matrix @math{X}.
+
+@deftypefun {gsl_multifit_linear_workspace *} gsl_multifit_linear_alloc (size_t @var{n}, size_t @var{p})
+This function allocates a workspace for fitting a model to @var{n}
+observations using @var{p} parameters.
+@end deftypefun
+
+@deftypefun void gsl_multifit_linear_free (gsl_multifit_linear_workspace * @var{work})
+This function frees the memory associated with the workspace @var{w}.
+@end deftypefun
+
+@deftypefun int gsl_multifit_linear (const gsl_matrix * @var{X}, const gsl_vector * @var{y}, gsl_vector * @var{c}, gsl_matrix * @var{cov}, double * @var{chisq}, gsl_multifit_linear_workspace * @var{work})
+@deftypefunx int gsl_multifit_linear_svd (const gsl_matrix * @var{X}, const gsl_vector * @var{y}, double @var{tol}, size_t * @var{rank}, gsl_vector * @var{c}, gsl_matrix * @var{cov}, double * @var{chisq}, gsl_multifit_linear_workspace * @var{work})
+These functions compute the best-fit parameters @var{c} of the model
+@math{y = X c} for the observations @var{y} and the matrix of predictor
+variables @var{X}.  The variance-covariance matrix of the model
+parameters @var{cov} is estimated from the scatter of the observations
+about the best-fit.  The sum of squares of the residuals from the
+best-fit, @math{\chi^2}, is returned in @var{chisq}. 
+
+The best-fit is found by singular value decomposition of the matrix
+@var{X} using the preallocated workspace provided in @var{work}. The
+modified Golub-Reinsch SVD algorithm is used, with column scaling to
+improve the accuracy of the singular values. Any components which have
+zero singular value (to machine precision) are discarded from the fit.
+In the second form of the function the components are discarded if the
+ratio of singular values @math{s_i/s_0} falls below the user-specified
+tolerance @var{tol}, and the effective rank is returned in @var{rank}.
+@end deftypefun
+
+@deftypefun int gsl_multifit_wlinear (const gsl_matrix * @var{X}, const gsl_vector * @var{w}, const gsl_vector * @var{y}, gsl_vector * @var{c}, gsl_matrix * @var{cov}, double * @var{chisq}, gsl_multifit_linear_workspace * @var{work})
+@deftypefunx int gsl_multifit_wlinear_svd (const gsl_matrix * @var{X}, const gsl_vector * @var{w}, const gsl_vector * @var{y}, double @var{tol}, size_t * @var{rank}, gsl_vector * @var{c}, gsl_matrix * @var{cov}, double * @var{chisq}, gsl_multifit_linear_workspace * @var{work})
+
+This function computes the best-fit parameters @var{c} of the weighted
+model @math{y = X c} for the observations @var{y} with weights @var{w}
+and the matrix of predictor variables @var{X}.  The covariance matrix of
+the model parameters @var{cov} is computed with the given weights.  The
+weighted sum of squares of the residuals from the best-fit,
+@math{\chi^2}, is returned in @var{chisq}.
+
+The best-fit is found by singular value decomposition of the matrix
+@var{X} using the preallocated workspace provided in @var{work}. Any
+components which have zero singular value (to machine precision) are
+discarded from the fit.  In the second form of the function the
+components are discarded if the ratio of singular values @math{s_i/s_0}
+falls below the user-specified tolerance @var{tol}, and the effective
+rank is returned in @var{rank}.
+@end deftypefun
+
+@deftypefun int gsl_multifit_linear_est (const gsl_vector * @var{x}, const gsl_vector * @var{c}, const gsl_matrix * @var{cov}, double * @var{y}, double * @var{y_err})
+This function uses the best-fit multilinear regression coefficients
+@var{c} and their covariance matrix
+@var{cov} to compute the fitted function value
+@var{y} and its standard deviation @var{y_err} for the model @math{y = x.c} 
+at the point @var{x}.
+@end deftypefun
+
+@node Fitting Examples
+@section Examples
+
+The following program computes a least squares straight-line fit to a
+simple dataset, and outputs the best-fit line and its
+associated one standard-deviation error bars.
+
+@example
+@verbatiminclude examples/fitting.c
+@end example
+
+@noindent
+The following commands extract the data from the output of the program
+and display it using the @sc{gnu} plotutils @code{graph} utility, 
+
+@example
+$ ./demo > tmp
+$ more tmp
+# best fit: Y = -106.6 + 0.06 X
+# covariance matrix:
+# [ 39602, -19.9
+#   -19.9, 0.01]
+# chisq = 0.8
+
+$ for n in data fit hi lo ; 
+   do 
+     grep "^$n" tmp | cut -d: -f2 > $n ; 
+   done
+$ graph -T X -X x -Y y -y 0 20 -m 0 -S 2 -Ie data 
+     -S 0 -I a -m 1 fit -m 2 hi -m 2 lo
+@end example
+
+@iftex
+@sp 1
+@center @image{fit-wlinear,3.0in}
+@end iftex
+
+The next program performs a quadratic fit @math{y = c_0 + c_1 x + c_2
+x^2} to a weighted dataset using the generalised linear fitting function
+@code{gsl_multifit_wlinear}.  The model matrix @math{X} for a quadratic
+fit is given by,
+@tex
+\beforedisplay
+$$
+X=\pmatrix{1&x_0&x_0^2\cr
+1&x_1&x_1^2\cr
+1&x_2&x_2^2\cr
+\dots&\dots&\dots\cr}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+X = [ 1   , x_0  , x_0^2 ;
+      1   , x_1  , x_1^2 ;
+      1   , x_2  , x_2^2 ;
+      ... , ...  , ...   ]
+@end example
+
+@end ifinfo
+@noindent
+where the column of ones corresponds to the constant term @math{c_0}.
+The two remaining columns corresponds to the terms @math{c_1 x} and
+@math{c_2 x^2}.
+
+The program reads @var{n} lines of data in the format (@var{x}, @var{y},
+@var{err}) where @var{err} is the error (standard deviation) in the
+value @var{y}.
+
+@example
+@verbatiminclude examples/fitting2.c
+@end example
+
+@noindent
+A suitable set of data for fitting can be generated using the following
+program.  It outputs a set of points with gaussian errors from the curve
+@math{y = e^x} in the region @math{0 < x < 2}.
+
+@example
+@verbatiminclude examples/fitting3.c
+@end example
+
+@noindent
+The data can be prepared by running the resulting executable program,
+
+@example
+$ ./generate > exp.dat
+$ more exp.dat
+0.1 0.97935 0.110517
+0.2 1.3359 0.12214
+0.3 1.52573 0.134986
+0.4 1.60318 0.149182
+0.5 1.81731 0.164872
+0.6 1.92475 0.182212
+....
+@end example
+
+@noindent
+To fit the data use the previous program, with the number of data points
+given as the first argument.  In this case there are 19 data points.
+
+@example
+$ ./fit 19 < exp.dat
+0.1 0.97935 +/- 0.110517
+0.2 1.3359 +/- 0.12214
+...
+# best fit: Y = 1.02318 + 0.956201 X + 0.876796 X^2
+# covariance matrix:
+[ +1.25612e-02, -3.64387e-02, +1.94389e-02  
+  -3.64387e-02, +1.42339e-01, -8.48761e-02  
+  +1.94389e-02, -8.48761e-02, +5.60243e-02 ]
+# chisq = 23.0987
+@end example
+
+@noindent
+The parameters of the quadratic fit match the coefficients of the
+expansion of @math{e^x}, taking into account the errors on the
+parameters and the @math{O(x^3)} difference between the exponential and
+quadratic functions for the larger values of @math{x}.  The errors on
+the parameters are given by the square-root of the corresponding
+diagonal elements of the covariance matrix.  The chi-squared per degree
+of freedom is 1.4, indicating a reasonable fit to the data.
+
+@iftex
+@sp 1
+@center @image{fit-wlinear2,3.0in}
+@end iftex
+
+@node Fitting References and Further Reading
+@section References and Further Reading
+
+A summary of formulas and techniques for least squares fitting can be
+found in the ``Statistics'' chapter of the Annual Review of Particle
+Physics prepared by the Particle Data Group,
+
+@itemize @asis
+@item
+@cite{Review of Particle Properties},
+R.M. Barnett et al., Physical Review D54, 1 (1996)
+@uref{http://pdg.lbl.gov/}
+@end itemize
+
+@noindent
+The Review of Particle Physics is available online at the website given
+above.
+
+@cindex NIST Statistical Reference Datasets
+@cindex Statistical Reference Datasets (StRD)
+The tests used to prepare these routines are based on the NIST
+Statistical Reference Datasets. The datasets and their documentation are
+available from NIST at the following website,
+
+@center @uref{http://www.nist.gov/itl/div898/strd/index.html}.
+
+
diff --git a/gsl-1.9/doc/freemanuals.texi b/gsl-1.9/doc/freemanuals.texi
new file mode 100644
index 0000000..0223f8c
--- /dev/null
+++ b/gsl-1.9/doc/freemanuals.texi
@@ -0,0 +1,99 @@
+@cindex free documentation
+
+@quotation
+@i{The following article was written by Richard Stallman, founder of the
+GNU Project.}
+@end quotation
+
+
+The biggest deficiency in the free software community today is not in
+the software---it is the lack of good free documentation that we can
+include with the free software.  Many of our most important
+programs do not come with free reference manuals and free introductory
+texts.  Documentation is an essential part of any software package;
+when an important free software package does not come with a free
+manual and a free tutorial, that is a major gap.  We have many such
+gaps today.
+
+Consider Perl, for instance.  The tutorial manuals that people
+normally use are non-free.  How did this come about?  Because the
+authors of those manuals published them with restrictive terms---no
+copying, no modification, source files not available---which exclude
+them from the free software world.
+
+That wasn't the first time this sort of thing happened, and it was far
+from the last.  Many times we have heard a GNU user eagerly describe a
+manual that he is writing, his intended contribution to the community,
+only to learn that he had ruined everything by signing a publication
+contract to make it non-free.
+
+Free documentation, like free software, is a matter of freedom, not
+price.  The problem with the non-free manual is not that publishers
+charge a price for printed copies---that in itself is fine.  (The Free
+Software Foundation sells printed copies of manuals, too.)  The
+problem is the restrictions on the use of the manual.  Free manuals
+are available in source code form, and give you permission to copy and
+modify.  Non-free manuals do not allow this.
+
+The criteria of freedom for a free manual are roughly the same as for
+free software.  Redistribution (including the normal kinds of
+commercial redistribution) must be permitted, so that the manual can
+accompany every copy of the program, both on-line and on paper.
+
+Permission for modification of the technical content is crucial too.
+When people modify the software, adding or changing features, if they
+are conscientious they will change the manual too---so they can
+provide accurate and clear documentation for the modified program.  A
+manual that leaves you no choice but to write a new manual to document
+a changed version of the program is not really available to our
+community.
+
+Some kinds of limits on the way modification is handled are
+acceptable.  For example, requirements to preserve the original
+author's copyright notice, the distribution terms, or the list of
+authors, are ok.  It is also no problem to require modified versions
+to include notice that they were modified.  Even entire sections that
+may not be deleted or changed are acceptable, as long as they deal
+with nontechnical topics (like this one).  These kinds of restrictions
+are acceptable because they don't obstruct the community's normal use
+of the manual.
+
+However, it must be possible to modify all the @emph{technical}
+content of the manual, and then distribute the result in all the usual
+media, through all the usual channels.  Otherwise, the restrictions
+obstruct the use of the manual, it is not free, and we need another
+manual to replace it.
+
+Please spread the word about this issue.  Our community continues to
+lose manuals to proprietary publishing.  If we spread the word that
+free software needs free reference manuals and free tutorials, perhaps
+the next person who wants to contribute by writing documentation will
+realize, before it is too late, that only free manuals contribute to
+the free software community.
+
+If you are writing documentation, please insist on publishing it under
+the GNU Free Documentation License or another free documentation
+license.  Remember that this decision requires your approval---you
+don't have to let the publisher decide.  Some commercial publishers
+will use a free license if you insist, but they will not propose the
+option; it is up to you to raise the issue and say firmly that this is
+what you want.  If the publisher you are dealing with refuses, please
+try other publishers.  If you're not sure whether a proposed license
+is free, write to @email{licensing@@gnu.org}.
+
+You can encourage commercial publishers to sell more free, copylefted
+manuals and tutorials by buying them, and particularly by buying
+copies from the publishers that paid for their writing or for major
+improvements.  Meanwhile, try to avoid buying non-free documentation
+at all.  Check the distribution terms of a manual before you buy it,
+and insist that whoever seeks your business must respect your freedom.
+Check the history of the book, and try reward the publishers that have
+paid or pay the authors to work on it.
+
+The Free Software Foundation maintains a list of free documentation
+published by other publishers:
+
+@itemize @asis
+@item
+@uref{http://www.fsf.org/doc/other-free-books.html}
+@end itemize
diff --git a/gsl-1.9/doc/gpl.texi b/gsl-1.9/doc/gpl.texi
new file mode 100644
index 0000000..28f738f
--- /dev/null
+++ b/gsl-1.9/doc/gpl.texi
@@ -0,0 +1,407 @@
+@center Version 2, June 1991
+@iftex
+@smallerfonts @rm
+@end iftex
+
+@c This file is intended to be included in another file.
+
+@display
+Copyright @copyright{} 1989, 1991 Free Software Foundation, Inc.
+59 Temple Place - Suite 330, Boston, MA  02111-1307, USA
+
+Everyone is permitted to copy and distribute verbatim copies of this
+license document, but changing it is not allowed.
+@end display
+
+@heading Preamble
+
+  The licenses for most software are designed to take away your
+freedom to share and change it.  By contrast, the GNU General Public
+License is intended to guarantee your freedom to share and change free
+software---to make sure the software is free for all its users.  This
+General Public License applies to most of the Free Software
+Foundation's software and to any other program whose authors commit to
+using it.  (Some other Free Software Foundation software is covered by
+the GNU Library General Public License instead.)  You can apply it to
+your programs, too.
+
+  When we speak of free software, we are referring to freedom, not
+price.  Our General Public Licenses are designed to make sure that you
+have the freedom to distribute copies of free software (and charge for
+this service if you wish), that you receive source code or can get it
+if you want it, that you can change the software or use pieces of it
+in new free programs; and that you know you can do these things.
+
+  To protect your rights, we need to make restrictions that forbid
+anyone to deny you these rights or to ask you to surrender the rights.
+These restrictions translate to certain responsibilities for you if you
+distribute copies of the software, or if you modify it.
+
+  For example, if you distribute copies of such a program, whether
+gratis or for a fee, you must give the recipients all the rights that
+you have.  You must make sure that they, too, receive or can get the
+source code.  And you must show them these terms so they know their
+rights.
+
+  We protect your rights with two steps: (1) copyright the software, and
+(2) offer you this license which gives you legal permission to copy,
+distribute and/or modify the software.
+
+  Also, for each author's protection and ours, we want to make certain
+that everyone understands that there is no warranty for this free
+software.  If the software is modified by someone else and passed on, we
+want its recipients to know that what they have is not the original, so
+that any problems introduced by others will not reflect on the original
+authors' reputations.
+
+  Finally, any free program is threatened constantly by software
+patents.  We wish to avoid the danger that redistributors of a free
+program will individually obtain patent licenses, in effect making the
+program proprietary.  To prevent this, we have made it clear that any
+patent must be licensed for everyone's free use or not licensed at all.
+
+  The precise terms and conditions for copying, distribution and
+modification follow.
+
+@iftex
+@heading TERMS AND CONDITIONS FOR COPYING, DISTRIBUTION AND MODIFICATION
+@end iftex
+@ifinfo
+@center TERMS AND CONDITIONS FOR COPYING, DISTRIBUTION AND MODIFICATION
+@end ifinfo
+
+@enumerate 0
+@item
+This License applies to any program or other work which contains
+a notice placed by the copyright holder saying it may be distributed
+under the terms of this General Public License.  The ``Program'', below,
+refers to any such program or work, and a ``work based on the Program''
+means either the Program or any derivative work under copyright law:
+that is to say, a work containing the Program or a portion of it,
+either verbatim or with modifications and/or translated into another
+language.  (Hereinafter, translation is included without limitation in
+the term ``modification''.)  Each licensee is addressed as ``you''.
+
+Activities other than copying, distribution and modification are not
+covered by this License; they are outside its scope.  The act of
+running the Program is not restricted, and the output from the Program
+is covered only if its contents constitute a work based on the
+Program (independent of having been made by running the Program).
+Whether that is true depends on what the Program does.
+
+@item
+You may copy and distribute verbatim copies of the Program's
+source code as you receive it, in any medium, provided that you
+conspicuously and appropriately publish on each copy an appropriate
+copyright notice and disclaimer of warranty; keep intact all the
+notices that refer to this License and to the absence of any warranty;
+and give any other recipients of the Program a copy of this License
+along with the Program.
+
+You may charge a fee for the physical act of transferring a copy, and
+you may at your option offer warranty protection in exchange for a fee.
+
+@item
+You may modify your copy or copies of the Program or any portion
+of it, thus forming a work based on the Program, and copy and
+distribute such modifications or work under the terms of Section 1
+above, provided that you also meet all of these conditions:
+
+@enumerate a
+@item
+You must cause the modified files to carry prominent notices
+stating that you changed the files and the date of any change.
+
+@item
+You must cause any work that you distribute or publish, that in
+whole or in part contains or is derived from the Program or any
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+parties under the terms of this License.
+
+@item
+If the modified program normally reads commands interactively
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+the Program is not required to print an announcement.)
+@end enumerate
+
+These requirements apply to the modified work as a whole.  If
+identifiable sections of that work are not derived from the Program,
+and can be reasonably considered independent and separate works in
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+sections when you distribute them as separate works.  But when you
+distribute the same sections as part of a whole which is a work based
+on the Program, the distribution of the whole must be on the terms of
+this License, whose permissions for other licensees extend to the
+entire whole, and thus to each and every part regardless of who wrote it.
+
+Thus, it is not the intent of this section to claim rights or contest
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+
+In addition, mere aggregation of another work not based on the Program
+with the Program (or with a work based on the Program) on a volume of
+a storage or distribution medium does not bring the other work under
+the scope of this License.
+
+@item
+You may copy and distribute the Program (or a work based on it,
+under Section 2) in object code or executable form under the terms of
+Sections 1 and 2 above provided that you also do one of the following:
+
+@enumerate a
+@item
+Accompany it with the complete corresponding machine-readable
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+
+@item
+Accompany it with a written offer, valid for at least three
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+cost of physically performing source distribution, a complete
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+distributed under the terms of Sections 1 and 2 above on a medium
+customarily used for software interchange; or,
+
+@item
+Accompany it with the information you received as to the offer
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+received the program in object code or executable form with such
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+@end enumerate
+
+The source code for a work means the preferred form of the work for
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+access to copy the source code from the same place counts as
+distribution of the source code, even though third parties are not
+compelled to copy the source along with the object code.
+
+@item
+You may not copy, modify, sublicense, or distribute the Program
+except as expressly provided under this License.  Any attempt
+otherwise to copy, modify, sublicense or distribute the Program is
+void, and will automatically terminate your rights under this License.
+However, parties who have received copies, or rights, from you under
+this License will not have their licenses terminated so long as such
+parties remain in full compliance.
+
+@item
+You are not required to accept this License, since you have not
+signed it.  However, nothing else grants you permission to modify or
+distribute the Program or its derivative works.  These actions are
+prohibited by law if you do not accept this License.  Therefore, by
+modifying or distributing the Program (or any work based on the
+Program), you indicate your acceptance of this License to do so, and
+all its terms and conditions for copying, distributing or modifying
+the Program or works based on it.
+
+@item
+Each time you redistribute the Program (or any work based on the
+Program), the recipient automatically receives a license from the
+original licensor to copy, distribute or modify the Program subject to
+these terms and conditions.  You may not impose any further
+restrictions on the recipients' exercise of the rights granted herein.
+You are not responsible for enforcing compliance by third parties to
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+
+@item
+If, as a consequence of a court judgment or allegation of patent
+infringement or for any other reason (not limited to patent issues),
+conditions are imposed on you (whether by court order, agreement or
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+may not distribute the Program at all.  For example, if a patent
+license would not permit royalty-free redistribution of the Program by
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+
+If any portion of this section is held invalid or unenforceable under
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+
+It is not the purpose of this section to induce you to infringe any
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+
+This section is intended to make thoroughly clear what is believed to
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+
+@item
+If the distribution and/or use of the Program is restricted in
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+original copyright holder who places the Program under this License
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+countries not thus excluded.  In such case, this License incorporates
+the limitation as if written in the body of this License.
+
+@item
+The Free Software Foundation may publish revised and/or new versions
+of the General Public License from time to time.  Such new versions will
+be similar in spirit to the present version, but may differ in detail to
+address new problems or concerns.
+
+Each version is given a distinguishing version number.  If the Program
+specifies a version number of this License which applies to it and ``any
+later version'', you have the option of following the terms and conditions
+either of that version or of any later version published by the Free
+Software Foundation.  If the Program does not specify a version number of
+this License, you may choose any version ever published by the Free Software
+Foundation.
+
+@item
+If you wish to incorporate parts of the Program into other free
+programs whose distribution conditions are different, write to the author
+to ask for permission.  For software which is copyrighted by the Free
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+make exceptions for this.  Our decision will be guided by the two goals
+of preserving the free status of all derivatives of our free software and
+of promoting the sharing and reuse of software generally.
+
+@iftex
+@heading NO WARRANTY
+@end iftex
+@ifinfo
+@center NO WARRANTY
+@end ifinfo
+
+@item
+BECAUSE THE PROGRAM IS LICENSED FREE OF CHARGE, THERE IS NO WARRANTY
+FOR THE PROGRAM, TO THE EXTENT PERMITTED BY APPLICABLE LAW.  EXCEPT WHEN
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+REPAIR OR CORRECTION.
+
+@item
+IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING
+WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MAY MODIFY AND/OR
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+OUT OF THE USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED
+TO LOSS OF DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY
+YOU OR THIRD PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER
+PROGRAMS), EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE
+POSSIBILITY OF SUCH DAMAGES.
+@end enumerate
+
+@iftex
+@heading END OF TERMS AND CONDITIONS
+@end iftex
+@ifinfo
+@center END OF TERMS AND CONDITIONS
+@end ifinfo
+
+@page
+@heading Appendix: How to Apply These Terms to Your New Programs
+
+  If you develop a new program, and you want it to be of the greatest
+possible use to the public, the best way to achieve this is to make it
+free software which everyone can redistribute and change under these terms.
+
+  To do so, attach the following notices to the program.  It is safest
+to attach them to the start of each source file to most effectively
+convey the exclusion of warranty; and each file should have at least
+the ``copyright'' line and a pointer to where the full notice is found.
+
+@smallexample
+@var{one line to give the program's name and a brief idea @*of what it does.}
+Copyright (C) @var{yyyy}  @var{name of author}
+
+This program is free software; you can redistribute it
+and/or modify it under the terms of the GNU General Public
+License as published by the Free Software Foundation; either
+version 2 of the License, or (at your option) any later
+version.
+
+This program is distributed in the hope that it will be
+useful, but WITHOUT ANY WARRANTY; without even the implied
+warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR
+PURPOSE.  See the GNU General Public License for more
+details.
+
+You should have received a copy of the GNU General Public
+License along with this program; if not, write to the Free
+Software Foundation, Inc., 59 Temple Place - Suite 330,
+Boston, MA 02111-1307, USA.
+@end smallexample
+
+Also add information on how to contact you by electronic and paper mail.
+
+If the program is interactive, make it output a short notice like this
+when it starts in an interactive mode:
+
+@smallexample
+Gnomovision version 69, Copyright (C) 19@var{yy} @var{name of author}
+Gnomovision comes with ABSOLUTELY NO WARRANTY; for details
+type `show w'.  This is free software, and you are welcome
+to redistribute it under certain conditions; type `show c'
+for details.
+@end smallexample
+
+The hypothetical commands @samp{show w} and @samp{show c} should show
+the appropriate parts of the General Public License.  Of course, the
+commands you use may be called something other than @samp{show w} and
+@samp{show c}; they could even be mouse-clicks or menu items---whatever
+suits your program.
+
+@need 1500
+You should also get your employer (if you work as a programmer) or your
+school, if any, to sign a ``copyright disclaimer'' for the program, if
+necessary.  Here is a sample; alter the names:
+
+@smallexample
+Yoyodyne, Inc., hereby disclaims all copyright interest in
+the program `Gnomovision' (which makes passes at compilers)
+written by James Hacker.
+
+@var{signature of Ty Coon}, 1 April 1989
+Ty Coon, President of Vice
+@end smallexample
+
+@iftex
+@smallerfonts @rm
+@end iftex
+This General Public License does not permit incorporating your program into
+proprietary programs.  If your program is a subroutine library, you may
+consider it more useful to permit linking proprietary applications with the
+library.  If this is what you want to do, use the GNU Library General
+Public License instead of this License.
+
+@iftex
+@*
+@textfonts @rm
+@end iftex
diff --git a/gsl-1.9/doc/gsl-config.1 b/gsl-1.9/doc/gsl-config.1
new file mode 100644
index 0000000..e8c1ebc
--- /dev/null
+++ b/gsl-1.9/doc/gsl-config.1
@@ -0,0 +1,42 @@
+.TH GSL 1 "22 May 2001"
+.SH NAME
+gsl-config - script to get version number and compiler flags of the installed GSL library
+.SH SYNOPSIS
+.B gsl-config
+[\-\-prefix]  [\-\-version] [\-\-libs] [\-\-libs\-without\-cblas] [\-\-cflags]
+.SH DESCRIPTION
+.PP
+\fIgsl-config\fP is a tool that is used to configure to determine
+the compiler and linker flags that should be used to compile
+and link programs that use \fIGSL\fP. It is also used internally
+to the .m4 macros for GNU autoconf that are included with \fIGSL\fP.
+.
+.SH OPTIONS
+.l
+\fIgsl-config\fP accepts the following options:
+.TP 8
+.B  \-\-version
+Print the currently installed version of \fIGSL\fP on the standard output.
+.TP 8
+.B  \-\-libs
+Print the linker flags that are necessary to link a \fIGSL\fP program, with cblas
+.TP 8
+.B  \-\-libs\-without\-cblas
+Print the linker flags that are necessary to link a \fIGSL\fP program, without cblas
+.TP 8
+.B  \-\-cflags
+Print the compiler flags that are necessary to compile a \fIGSL\fP program.
+.TP 8
+.B  \-\-prefix
+Show the GSL installation prefix.
+.SH SEE ALSO
+.BR gtk-config (1),
+.BR gnome-config (1)
+.SH COPYRIGHT
+Copyright \(co  2001 Christopher R. Gabriel
+
+Permission to use, copy, modify, and distribute this software and its
+documentation for any purpose and without fee is hereby granted,
+provided that the above copyright notice appear in all copies and that
+both that copyright notice and this permission notice appear in
+supporting documentation.
diff --git a/gsl-1.9/doc/gsl-design.texi b/gsl-1.9/doc/gsl-design.texi
new file mode 100644
index 0000000..7e513ef
--- /dev/null
+++ b/gsl-1.9/doc/gsl-design.texi
@@ -0,0 +1,1610 @@
+\input texinfo @c -*-texinfo-*-
+@c %**start of header
+@setfilename gsl-design.info
+@settitle GNU Scientific Library
+@finalout
+@c -@setchapternewpage odd
+@c %**end of header
+
+@dircategory Scientific software
+@direntry
+* GSL-design: (GSL-design).             GNU Scientific Library -- Design
+@end direntry
+
+@comment @include version-design.texi
+@set GSL @i{GNU Scientific Library}
+
+@ifinfo
+This file documents the @value{GSL}.
+
+Copyright (C) 1996, 1997, 1998, 1999, 2000, 2001, 2004 The GSL Project.
+
+Permission is granted to copy, distribute and/or modify this document
+under the terms of the GNU Free Documentation License, Version 1.2 or
+any later version published by the Free Software Foundation; with no
+Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts.  A
+copy of the license is included in the section entitled ``GNU Free
+Documentation License''.
+@end ifinfo
+
+@titlepage
+@title GNU Scientific Library -- Design document
+@comment @subtitle Edition @value{EDITION}, for gsl Version @value{VERSION}
+@comment @subtitle @value{UPDATED}
+@author Mark Galassi 
+Los Alamos National Laboratory
+
+@author James Theiler 
+Astrophysics and Radiation Measurements Group, Los Alamos National Laboratory
+
+@author Brian Gough 
+Network Theory Limited
+
+@page
+@vskip 0pt plus 1filll
+Copyright @copyright{} 1996,1997,1998,1999,2000,2001,2004 The GSL Project.
+
+Permission is granted to make and distribute verbatim copies of
+this manual provided the copyright notice and this permission notice
+are preserved on all copies.
+
+Permission is granted to copy and distribute modified versions of this
+manual under the conditions for verbatim copying, provided that the entire
+resulting derived work is distributed under the terms of a permission
+notice identical to this one.
+
+Permission is granted to copy and distribute translations of this manual
+into another language, under the above conditions for modified versions,
+except that this permission notice may be stated in a translation approved
+by the Foundation.
+@end titlepage
+
+@contents
+
+@node Top, Motivation, (dir), (dir)
+@top About GSL
+
+@ifinfo
+This file documents the design of @value{GSL}, a collection of numerical
+routines for scientific computing.
+
+More information about GSL can be found at the project homepage,
+@uref{http://www.gnu.org/software/gsl/}.
+@end ifinfo
+
+The @value{GSL} is a library of scientific subroutines.  It aims to
+provide a convenient interface to routines that do standard (and not so
+standard) tasks that arise in scientific research.  More than that, it
+also provides the source code.  Users are welcome to alter, adjust,
+modify, and improve the interfaces and/or implementations of whichever
+routines might be needed for a particular purpose.
+
+GSL is intended to provide a free equivalent to existing proprietary
+numerical libraries written in C or Fortran, such as NAG, IMSL's CNL,
+IBM's ESSL, and SGI's SCSL.
+
+The target platform is a low-end desktop workstation. The goal is to
+provide something which is generally useful, and the library is aimed at
+general users rather than specialists.
+
+@menu
+* Motivation::                  
+* Contributing::                
+* Design::                      
+* Bibliography::                
+* Copying::                     
+* GNU Free Documentation License::  
+@end menu
+
+@node Motivation, Contributing, Top, Top
+@chapter Motivation
+@cindex numerical analysis
+@cindex free software
+
+There is a need for scientists and engineers to have a numerical library
+that:
+@itemize @bullet
+@item
+is free (in the sense of freedom, not in the sense of gratis; see the
+GNU General Public License), so that people can use that library,
+redistribute it, modify it @dots{}
+@item
+is written in C using modern coding conventions, calling conventions,
+scoping @dots{}
+@item
+is clearly and pedagogically documented; preferably with TeXinfo, so as
+to allow online info, WWW and TeX output.
+@item
+uses top quality state-of-the-art algorithms.
+@item
+is portable and configurable using @emph{autoconf} and @emph{automake}.
+@item
+basically, is GNUlitically correct.
+@end itemize
+
+There are strengths and weaknesses with existing libraries:
+
+@emph{Netlib} (http://www.netlib.org/) is probably the most advanced set
+of numerical algorithms available on the net, maintained by AT&T.
+Unfortunately most of the software is written in Fortran, with strange
+calling conventions in many places.  It is also not very well collected,
+so it is a lot of work to get started with netlib.
+
+@emph{GAMS} (http://gams.nist.gov/) is an extremely well organized set
+of pointers to scientific software, but like netlib, the individual
+routines vary in their quality and their level of documentation.
+
+@emph{Numerical Recipes} (http://www.nr.com,
+http://cfata2.harvard.edu/nr/) is an excellent book: it explains the
+algorithms in a very clear way.  Unfortunately the authors released the
+source code under a license which allows you to use it, but prevents you
+from re-distributing it.  Thus Numerical Recipes is not @emph{free} in
+the sense of @emph{freedom}.  On top of that, the implementation suffers
+from @emph{fortranitis} and other
+limitations. [http://www.lysator.liu.se/c/num-recipes-in-c.html]
+
+@emph{SLATEC} is a large public domain collection of numerical routines
+in Fortran written under a Department of Energy program in the
+1970's. The routines are well tested and have a reasonable overall
+design (given the limitations of that era).  GSL should aim to be a
+modern version of SLATEC.
+
+@emph{NSWC} is the Naval Surface Warfare Center numerical library.  It
+is a large public-domain Fortran library, containing a lot of
+high-quality code.  Documentation for the library is hard to find, only
+a few photocopies of the printed manual are still in circulation.
+
+@emph{NAG} and @emph{IMSL} both sell high-quality libraries which are
+proprietary.  The NAG library is more advanced and has wider scope than
+IMSL. The IMSL library leans more towards ease-of-use and makes
+extensive use of variable length argument lists to emulate "default
+arguments".
+
+@emph{ESSL} and @emph{SCSL} are proprietary libraries from IBM and SGI.
+
+@emph{Forth Scientific Library} [see the URL
+http://www.taygeta.com/fsl/sciforth.html].  Mainly of interest to Forth
+users.
+
+@emph{Numerical Algorithms with C} G. Engeln-Mullges, F. Uhlig. A nice
+numerical library written in ANSI C with an accompanying
+textbook. Source code is available but the library is not free software.
+
+@emph{NUMAL} A C version of the NUMAL library has been written by
+H.T. Lau and is published as a book and disk with the title "A Numerical
+Library in C for Scientists and Engineers". Source code is available but
+the library is not free software.
+
+@emph{C Mathematical Function Handbook} by Louis Baker. A library of
+function approximations and methods corresponding to those in the
+"Handbook of Mathematical Functions" by Abramowitz and Stegun.  Source
+code is available but the library is not free software.
+
+@emph{CCMATH} by Daniel A. Atkinson. A C numerical library covering
+similar areas to GSL. The code is quite terse.  Earlier versions were
+under the GPL but unfortunately it has changed to the LGPL in recent
+versions.
+
+@emph{CEPHES} A useful collection of high-quality special functions
+written in C. Not GPL'ed.
+
+@emph{WNLIB} A small collection of numerical routines written in C by
+Will Naylor. Public domain.
+
+@emph{MESHACH} A comprehensive matrix-vector linear algebra library
+written in C. Freely available but not GPL'ed (non-commercial license).
+
+@emph{CERNLIB} is a large high-quality Fortran library developed at CERN
+over many years.  It was originally non-free software but has recently
+been released under the GPL.
+
+@emph{COLT} is a free numerical library in Java developed at CERN by
+Wolfgang Hoschek.  It is under a BSD-style license.
+
+The long-term goal will be to provide a framework to which the real
+numerical experts (or their graduate students) will contribute. 
+
+@node  Contributing, Design, Motivation, Top
+@chapter Contributing
+
+This design document was originally written in 1996.  As of 2004, GSL
+itself is essentially feature complete, the developers are not actively
+working on any major new functionality.
+
+The main emphasis is now on ensuring the stability of the existing
+functions, improving consistency, tidying up a few problem areas and
+fixing any bugs that are reported.  Potential contributors are
+encouraged to gain familiarity with the library by investigating and
+fixing known problems listed in the @file{BUGS} file in the CVS
+repository.
+
+Adding large amounts of new code is difficult because it leads to
+differences in the maturity of different parts of the library.  To
+maintain stability, any new functionality is encouraged as
+@dfn{packages}, built on top of GSL and maintained independently by the
+author, as in other free software projects (such as the Perl CPAN
+archive and TeX CTAN archive, etc).
+
+@menu
+* Packages::                    
+@end menu
+
+@node Packages,  , Contributing, Contributing
+@section Packages
+
+The design of GSL permits extensions to be used alongside the existing
+library easily by simple linking.  For example, additional random number
+generators can be provided in a separate library:
+
+@example
+$ tar xvfz rngextra-0.1.tar.gz
+$ cd rngextra-0.1
+$ ./configure; make; make check; make install
+$ ...
+$ gcc -Wall main.c -lrngextra -lgsl -lgslcblas -lm
+@end example
+
+The points below summarise the package design guidelines.  These are
+intended to ensure that packages are consistent with GSL itself, to make
+life easier for the end-user and make it possible to distribute popular
+well-tested packages as part of the core GSL in future.
+
+@itemize @bullet
+@item Follow the GSL and GNU coding standards described in this document
+
+This means using the standard GNU packaging tools, such as Automake,
+providing documentation in Texinfo format, and a test suite.  The test
+suite should run using @samp{make check}, and use the test functions
+provided in GSL to produce the output with @code{PASS:}/@code{FAIL:}
+lines.  It is not essential to use libtool since packages are likely to
+be small, a static library is sufficient and simpler to build.
+
+@item Use a new unique prefix for the package (do not use @samp{gsl_} -- this is reserved for internal use).
+
+For example, a package of additional random number generators might use
+the prefix @code{rngextra}.
+
+@example
+#include <rngextra.h>
+
+gsl_rng * r = gsl_rng_alloc (rngextra_lsfr32);
+@end example
+
+@item Use a meaningful version number which reflects the state of development
+
+Generally, @code{0.x} are alpha versions, which provide no guarantees.
+Following that, @code{0.9.x} are beta versions, which should be essentially
+complete, subject only to minor changes and bug fixes.  The first major
+release is @code{1.0}.  Any version number of @code{1.0} or higher
+should be suitable for production use with a well-defined API.
+
+The API must not change in a major release and should be
+backwards-compatible in its behavior (excluding actual bug-fixes), so
+that existing code do not have to be modified.  Note that the API
+includes all exported definitions, including data-structures defined
+with @code{struct}.  If you need to change the API in a package, it
+requires a new major release (e.g. @code{2.0}).
+
+@item Use the GNU General Public License (GPL)
+
+Follow the normal procedures of obtaining a copyright disclaimer if you
+would like to have the package considered for inclusion in GSL itself in
+the future (@pxref{Legal issues}).
+@end itemize
+
+Post announcements of your package releases to
+@email{gsl-discuss@@sources.redhat.com} so that information about them
+can be added to the GSL webpages.
+
+For security, sign your package with GPG (@code{gpg --detach-sign
+@var{file}}).
+
+An example package @samp{rngextra} containing two additional random
+number generators can be found at
+@url{http://www.network-theory.co.uk/download/rngextra/}.
+
+@node Design, Bibliography, Contributing, Top
+@chapter Design
+
+@menu
+* Language for implementation::  
+* Interface to other languages::  
+* What routines are implemented::  
+* What routines are not implemented::  
+* Design of  Numerical Libraries::  
+* Code Reuse::                  
+* Standards and conventions::   
+* Background and Preparation::  
+* Choice of Algorithms::        
+* Documentation::               
+* Namespace::                   
+* Header files::                
+* Target system::               
+* Function Names::              
+* Object-orientation::          
+* Comments::                    
+* Minimal structs::             
+* Algorithm decomposition::     
+* Memory allocation and ownership::  
+* Memory layout::               
+* Linear Algebra Levels::       
+* Error estimates::             
+* Exceptions and Error handling::  
+* Persistence::                 
+* Using Return Values::         
+* Variable Names::              
+* Datatype widths::             
+* size_t::                      
+* Arrays vs Pointers::          
+* Pointers::                    
+* Constness::                   
+* Pseudo-templates::            
+* Arbitrary Constants::         
+* Test suites::                 
+* Compilation::                 
+* Thread-safety::               
+* Legal issues::                
+* Non-UNIX portability::        
+* Compatibility with other libraries::  
+* Parallelism::                 
+* Precision::                   
+* Miscellaneous::               
+@end menu
+
+@node Language for implementation, Interface to other languages, Design, Design
+@section Language for implementation
+
+@strong{One language only (C)}
+
+Advantages: simpler, compiler available and quite universal.
+
+@node Interface to other languages, What routines are implemented, Language for implementation, Design
+@section Interface to other languages
+
+Wrapper packages are supplied as "extra" packages; not as part of the
+"core". They are maintained separately by independent contributors.
+
+Use standard tools to make wrappers: swig, g-wrap
+
+@node What routines are implemented, What routines are not implemented, Interface to other languages, Design
+@section What routines are implemented
+
+Anything which is in any of the existing libraries.  Obviously it makes
+sense to prioritize and write code for the most important areas first.
+
+@c @itemize @bullet
+@c @item Random number generators
+
+@c Includes both random number generators and routines to give various
+@c interesting distributions.
+
+@c @item Statistics
+
+@c @item Special Functions
+
+@c What I (jt) envision for this section is a collection of routines for
+@c reliable and accurate (but not necessarily fast or efficient) estimation
+@c of values for special functions, explicitly using Taylor series, asymptotic 
+@c expansions, continued fraction expansions, etc.  As well as these routines,
+@c fast approximations will also be provided, primarily based on Chebyschev
+@c polynomials and ratios of polynomials.  In this vision, the approximations
+@c will be the "standard" routines for the users, and the exact (so-called)
+@c routines will be used for verification of the approximations.  It may also
+@c be useful to provide various identity-checking routines as part of the
+@c verification suite.
+
+@c @item Curve fitting
+
+@c polynomial, special functions, spline
+
+@c @item Ordinary differential equations
+
+@c @item Partial differential equations
+
+@c @item Fourier Analysis
+
+@c @item Wavelets
+
+@c @item Matrix operations: linear equations
+
+@c @item Matrix operations: eigenvalues and spectral analysis
+
+@c @item Matrix operations: any others?
+
+@c @item Direct integration
+
+@c @item Monte carlo methods
+
+@c @item Simulated annealing
+
+@c @item Genetic algorithms
+
+@c We need to think about what kinds of algorithms are basic generally
+@c useful numerical algorithms, and which ones are special purpose
+@c research projects.  We should concentrate on supplying the former.
+
+@c @item Cellular automata
+
+@c @end itemize
+
+@node What routines are not implemented, Design of  Numerical Libraries, What routines are implemented, Design
+@section What routines are not implemented
+
+@itemize @bullet
+@item anything which already exists as a high-quality GPL'ed package.
+
+@item anything which is too big
+ -- i.e. an application in its own right rather than a subroutine
+
+For example, partial differential equation solvers are often huge and
+very specialized applications (since there are so many types of PDEs,
+types of solution, types of grid, etc).  This sort of thing should
+remain separate.  It is better to point people to the good applications
+which exist.
+
+@item anything which is independent and useful separately.
+
+Arguably functions for manipulating date and time, or financial
+functions might be included in a "scientific" library.  However, these
+sorts of modules could equally well be used independently in other
+programs, so it makes sense for them to be separate libraries.
+@end itemize
+
+@node  Design of  Numerical Libraries, Code Reuse, What routines are not implemented, Design
+@section  Design of  Numerical Libraries
+
+In writing a numerical library there is a unavoidable conflict between
+completeness and simplicity.  Completeness refers to the ability to
+perform operations on different objects so that the group is
+"closed". In mathematics objects can be combined and operated on in an
+infinite number of ways.  For example, I can take the derivative of a
+scalar field with respect to a vector and the derivative of a vector
+field wrt a scalar (along a path).
+
+There is a definite tendency to unconsciously try to reproduce all these
+possibilities in a numerical library, by adding new features one by
+one. After all, it is always easy enough to support just one more
+feature.... so why not?
+
+Looking at the big picture, no-one would start out by saying "I want to
+be able to represent every possible mathematical object and operation
+using C structs" -- this is a strategy which is doomed to fail.  There
+is a limited amount of complexity which can be represented in a
+programming language like C.  Attempts to reproduce the complexity of
+mathematics within such a language would just lead to a morass of
+unmaintainable code.  However, it's easy to go down that road if you
+don't think about it ahead of time.
+
+It is better to choose simplicity over completeness.  In designing new
+parts of the library keep modules independent where possible. If
+interdependencies between modules are introduced be sure about where you
+are going to draw the line.
+
+@node Code Reuse, Standards and conventions, Design of  Numerical Libraries, Design
+@section Code Reuse
+
+It is useful if people can grab a single source file and include it in
+their own programs without needing the whole library.  Try to allow
+standalone files like this whenever it is reasonable.  Obviously the
+user might need to define a few macros, such as GSL_ERROR, to compile
+the file but that is ok.  Examples where this can be done: grabbing a
+single random number generator.
+
+@node Standards and conventions, Background and Preparation, Code Reuse, Design
+@section Standards and conventions
+
+The people who kick off this project should set the coding standards and
+conventions.  In order of precedence the  standards that we follow are,
+
+@itemize @bullet
+@item We follow the GNU Coding Standards.
+@item We follow the conventions of the ANSI Standard C Library.
+@item We follow the conventions of the GNU C Library.
+@item We follow the conventions of the glib GTK support Library.
+@end itemize
+
+The references for these standards are the @cite{GNU Coding Standards}
+document, Harbison and Steele @cite{C: A Reference Manual}, the
+@cite{GNU C Library Manual} (version 2), and the Glib source code.
+
+For mathematical formulas, always follow the conventions in Abramowitz &
+Stegun, the @cite{Handbook of Mathematical Functions}, since it is the
+definitive reference and also in the public domain.
+
+If the project has a philosophy it is to "Think in C".  Since we are
+working in C we should only do what is natural in C, rather than trying
+to simulate features of other languages.  If there is something which is
+unnatural in C and has to be simulated then we avoid using it. If this
+means leaving something out of the library, or only offering a limited
+version then so be it.  It is not worthwhile making the library
+over-complicated.  There are numerical libraries in other languages, and
+if people need the features of those languages it would be sensible for
+them to use the corresponding libraries, rather than coercing a C
+library into doing that job.  
+
+It should be borne in mind at all time that C is a macro-assembler.  If
+you are in doubt about something being too complicated ask yourself the
+question "Would I try to write this in macro-assembler?" If the answer
+is obviously "No" then do not try to include it in GSL. [BJG]
+
+It will be useful to read the following papers,
+
+@itemize @asis
+@item 
+Kiem-Phong Vo, ``The Discipline and Method Architecture for Reusable
+Libraries'', Software - Practice & Experience, v.30, pp.107-128, 2000.
+@end itemize
+
+@noindent
+It is available from
+@uref{http://www.research.att.com/sw/tools/sfio/dm-spe.ps} or the earlier
+technical report Kiem-Phong Vo, "An Architecture for Reusable Libraries"
+@uref{http://citeseer.nj.nec.com/48973.html}.
+
+There are associated papers on Vmalloc, SFIO, and CDT which are also
+relevant to the design of portable C libraries.
+
+@itemize @asis
+@item
+Kiem-Phong Vo, ``Vmalloc: A General and Efficient Memory
+Allocator''. Software Practice & Experience, 26:1--18, 1996.
+
+@uref{http://www.research.att.com/sw/tools/vmalloc/vmalloc.ps}
+@item
+Kiem-Phong Vo. ``Cdt: A Container Data Type Library''. Soft. Prac. &
+Exp., 27:1177--1197, 1997
+
+@uref{http://www.research.att.com/sw/tools/cdt/cdt.ps}
+@item
+David G. Korn and Kiem-Phong Vo, ``Sfio: Safe/Fast String/File IO'',
+Proceedings of the Summer '91 Usenix Conference, pp.  235-256, 1991.
+
+@uref{http://citeseer.nj.nec.com/korn91sfio.html}
+@end itemize
+
+Source code should be indented according to the GNU Coding Standards,
+with spaces not tabs. For example, by using the @code{indent} command:
+
+@example
+indent -gnu -nut *.c *.h
+@end example
+
+@noindent
+The @code{-nut} option converts tabs into spaces.
+
+@node Background and Preparation, Choice of Algorithms, Standards and conventions, Design
+@section Background and Preparation
+
+Before implementing something be sure to research the subject
+thoroughly!  This will save a lot of time in the long-run.  The two most
+important steps are,
+
+@enumerate
+@item
+to determine whether there is already a free library (GPL or
+GPL-compatible) which does the job.  If so, there is no need to
+reimplement it.  Carry out a search on Netlib, GAMs, na-net,
+sci.math.num-analysis and the web in general. This should also provide
+you with a list of existing proprietary libraries which are relevant,
+keep a note of these for future reference in step 2.
+
+@item 
+make a comparative survey of existing implementations in the
+commercial/free libraries. Examine the typical APIs, methods of
+communication between program and subroutine, and classify them so that
+you are familiar with the key concepts or features that an
+implementation may or may not have, depending on the relevant tradeoffs
+chosen.  Be sure to review the documentation of existing libraries for
+useful references.
+
+@item
+read up on the subject and determine the state-of-the-art.  Find the
+latest review papers.  A search of the following journals should be
+undertaken.
+
+@itemize @asis
+@item ACM Transactions on Mathematical Software
+@item Numerische Mathematik
+@item Journal of Computation and Applied Mathematics
+@item Computer Physics Communications
+@item SIAM Journal of Numerical Analysis
+@item SIAM Journal of Scientific Computing
+@end itemize
+@end enumerate
+
+@noindent
+Keep in mind that GSL is not a research project.  Making a good
+implementation is difficult enough, without also needing to invent new
+algorithms.  We want to implement existing algorithms whenever
+possible. Making minor improvements is ok, but don't let it be a
+time-sink.
+
+@node Choice of Algorithms, Documentation, Background and Preparation, Design
+@section Choice of Algorithms
+
+Whenever possible choose algorithms which scale well and always remember
+to handle asymptotic cases.  This is particularly relevant for functions
+with integer arguments.  It is tempting to implement these using the
+simple @math{O(n)} algorithms used to define the functions, such as the
+many recurrence relations found in Abramowitz and Stegun.  While such
+methods might be acceptable for @math{n=O(10-100)} they will not be
+satisfactory for a user who needs to compute the same function for
+@math{n=1000000}.
+
+Similarly, do not make the implicit assumption that multivariate data
+has been scaled to have components of the same size or O(1).  Algorithms
+should take care of any necessary scaling or balancing internally, and
+use appropriate norms (e.g. |Dx| where D is a diagonal scaling matrix,
+rather than |x|).
+
+@node Documentation, Namespace, Choice of Algorithms, Design
+@section Documentation
+Documentation: the project leaders should give examples of how things
+are to be documented.  High quality documentation is absolutely
+mandatory, so documentation should introduce the topic, and give careful
+reference for the provided functions. The priority is to provide
+reference documentation for each function. It is not necessary to
+provide tutorial documentation.
+
+Use free software, such as GNU Plotutils, to produce the graphs in the
+manual.
+
+Some of the graphs have been made with gnuplot which is not truly free
+(or GNU) software, and some have been made with proprietary
+programs. These should be replaced with output from GNU plotutils.
+
+When citing references be sure to use the standard, definitive and
+best reference books in the field, rather than lesser known text-books
+or introductory books which happen to be available (e.g. from
+undergraduate studies).  For example, references concerning algorithms
+should be to Knuth, references concerning statistics should be to
+Kendall & Stuart, references concerning special functions should be to
+Abramowitz & Stegun (Handbook of Mathematical Functions AMS-55), etc.
+Whereever possible refer to Abramowitz & Stegun rather than other
+reference books because it is a public domain work, so it is
+inexpensive and freely redistributable.
+
+The standard references have a better chance of being available in an
+accessible library for the user.  If they are not available and the user
+decides to buy a copy in order to look up the reference then this also
+gives them the best quality book which should also cover the largest
+number of other references in the GSL Manual.  If many different books
+were to be referenced this would be an expensive and inefficient use of
+resources for a user who needs to look up the details of the algorithms.
+Reference books also stay in print much longer than text books, which
+are often out-of-print after a few years.
+
+Similarly, cite original papers wherever possible.  Be sure to keep
+copies of these for your own reference (e.g. when dealing with bug
+reports) or to pass on to future maintainers.
+
+If you need help in tracking down references, ask on the
+@code{gsl-discuss} mailing list.  There is a group of volunteers with
+access to good libraries who have offered to help GSL developers get
+copies of papers.
+
+@c [JT section: written by James Theiler
+
+@c And we furthermore promise to try as hard as possible to document
+@c the software: this will ideally involve discussion of why you might want
+@c to use it, what precisely it does, how precisely to invoke it, 
+@c how more-or-less it works, and where we learned about the algorithm,
+@c and (unless we wrote it from scratch) where we got the code.
+@c We do not plan to write this entire package from scratch, but to cannibalize
+@c existing mathematical freeware, just as we expect our own software to
+@c be cannibalized.]
+
+To write mathematics in the texinfo file you can use the @code{@@math}
+command with @emph{simple} TeX commands. These are automatically
+surrounded by @code{$...$} for math mode. For example,
+
+@example
+to calculate the coefficient @@math@{\alpha@} use the function...
+@end example
+
+@noindent
+will be correctly formatted in both online and TeX versions of the
+documentation.
+
+Note that you cannot use the special characters @{ and @}
+inside the @code{@@math} command because these conflict between TeX
+and Texinfo.  This is a problem if you want to write something like
+@code{\sqrt@{x+y@}}.  
+
+To work around it you can preceed the math command with a special
+macro @code{@@c} which contains the explicit TeX commands you want to
+use (no restrictions), and put an ASCII approximation into the
+@code{@@math} command (you can write @code{@@@{} and
+@code{@@@}} there for the left and right braces).  The explicit TeX
+commands are used in the TeX ouput and the argument of @code{@@math}
+in the plain info output.  
+
+Note that the @code{@@c@{@}} macro must go at the end of the
+preceeding line, because everything else after it is ignored---as far
+as texinfo is concerned it's actually a 'comment'. The comment
+command @@c has been modified to capture a TeX expression which is
+output by the next @@math command. For ordinary comments use the @@comment
+command.
+
+For example,
+
+@example
+this is a test @@c@{$\sqrt@{x+y@}$@}
+@@math@{\sqrt@@@{x+y@@@}@}
+@end example
+
+@noindent
+is equivalent to @code{this is a test $\sqrt@{x+y@}$} in plain TeX
+and @code{this is a test @@math@{\sqrt@@@{x+y@@@}@}} in Info.  
+
+It looks nicer if some of the more cryptic TeX commands are given
+a C-style ascii version, e.g.
+
+@example
+for @@c@{$x \ge y$@}
+@@math@{x >= y@}
+@end example
+
+@noindent
+will be appropriately displayed in both TeX and Info.
+
+
+@node Namespace, Header files, Documentation, Design
+@section Namespace
+
+Use @code{gsl_} as a prefix for all exported functions and variables.
+
+Use @code{GSL_} as a prefix for all exported macros.
+
+All exported header files should have a filename with the prefix @code{gsl_}.
+
+All installed libraries should have a name like libgslhistogram.a
+
+Any installed executables (utility programs etc) should have the prefix
+@code{gsl-} (with a hyphen, not an underscore).
+
+All function names, variables, etc should be in lower case.  Macros and
+preprocessor variables should be in upper case.
+
+Some common conventions in variable and function names:
+
+@table @code
+@item p1
+plus 1, e.g. function @code{log1p(x)} or a variable like @code{kp1}, @math{=k+1}.
+
+@item m1
+minus 1, e.g. function @code{expm1(x)} or a variable like @code{km1}, @math{=k-1}.
+@end table
+
+@node Header files, Target system, Namespace, Design
+@section Header files
+
+Installed header files should be idempotent, i.e. surround them by the
+preprocessor conditionals like the following,
+
+@example
+#ifndef __GSL_HISTOGRAM_H__
+#define __GSL_HISTOGRAM_H__
+...
+#endif /* __GSL_HISTOGRAM_H__ */
+@end example
+
+@node Target system, Function Names, Header files, Design
+@section Target system
+
+The target system is ANSI C, with a full Standard C Library, and IEEE
+arithmetic.
+
+@node Function Names, Object-orientation, Target system, Design
+@section Function Names
+
+Each module has a name, which prefixes any function names in that
+module, e.g. the module gsl_fft has function names like
+gsl_fft_init. The modules correspond to subdirectories of the library
+source tree.
+
+@node Object-orientation, Comments, Function Names, Design
+@section Object-orientation
+
+The algorithms should be object oriented, but only to the extent that is
+easy in portable ANSI C.  The use of casting or other tricks to simulate
+inheritance is not desirable, and the user should not have to be aware
+of anything like that.  This means many types of patterns are ruled
+out.  However, this is not considered a problem -- they are too
+complicated for the library.  
+
+Note: it is possible to define an abstract base class easily in C, using
+function pointers.  See the rng directory for an example. 
+
+When reimplementing public domain fortran code, please try to introduce
+the appropriate object concepts as structs, rather than translating the
+code literally in terms of arrays.  The structs can be useful just
+within the file, you don't need to export them to the user.
+
+For example, if a fortran program repeatedly uses a subroutine like,
+
+@example
+SUBROUTINE  RESIZE (X, K, ND, K1)
+@end example
+
+@noindent
+where X(K,D) represents a grid to be resized to X(K1,D) you can make
+this more readable by introducing a struct,
+
+@smallexample
+struct grid @{
+    int nd;  /* number of dimensions */
+    int k;   /* number of bins */
+    double * x;   /* partition of axes, array of size x[k][nd] */
+@}
+
+void
+resize_grid (struct grid * g, int k_new)
+@{
+...
+@}
+@end smallexample
+
+@noindent
+Similarly, if you have a frequently recurring code fragment within a
+single file you can define a static or static inline function for it.
+This is typesafe and saves writing out everything in full.
+
+
+@node Comments, Minimal structs, Object-orientation, Design
+@section Comments
+
+Follow the GNU Coding Standards.  A relevant quote is,
+
+``Please write complete sentences and capitalize the first word.  If a
+lower-case identifier comes at the beginning of a sentence, don't
+capitalize it!  Changing the spelling makes it a different identifier.
+If you don't like starting a sentence with a lower case letter, write
+the sentence differently (e.g., "The identifier lower-case is ...").''
+
+@node Minimal structs, Algorithm decomposition, Comments, Design
+@section Minimal structs
+
+We prefer to make structs which are @dfn{minimal}.  For example, if a
+certain type of problem can be solved by several classes of algorithm
+(e.g. with and without derivative information) it is better to make
+separate types of struct to handle those cases.  i.e. run time type
+identification is not desirable.
+
+@node Algorithm decomposition, Memory allocation and ownership, Minimal structs, Design
+@section Algorithm decomposition
+
+Iterative algorithms should be decomposed into an INITIALIZE, ITERATE,
+TEST form, so that the user can control the progress of the iteration
+and print out intermediate results.  This is better than using
+call-backs or using flags to control whether the function prints out
+intermediate results.  In fact, call-backs should not be used -- if they
+seem necessary then it's a sign that the algorithm should be broken down
+further into individual components so that the user has complete control
+over them.  
+
+For example, when solving a differential equation the user may need to
+be able to advance the solution by individual steps, while tracking a
+realtime process.  This is only possible if the algorithm is broken down
+into step-level components.  Higher level decompositions would not give
+sufficient flexibility.
+
+@node Memory allocation and ownership, Memory layout, Algorithm decomposition, Design
+@section Memory allocation and ownership
+
+Functions which allocate memory on the heap should end in _alloc
+(e.g. gsl_foo_alloc) and be deallocated by a corresponding _free function
+(gsl_foo_free).
+
+Be sure to free any memory allocated by your function if you have to
+return an error in a partially initialized object.
+
+Don't allocate memory 'temporarily' inside a function and then free it
+before the function returns.  This prevents the user from controlling
+memory allocation.  All memory should be allocated and freed through
+separate functions and passed around as a "workspace" argument.  This
+allows memory allocation to be factored out of tight loops.
+
+@node Memory layout, Linear Algebra Levels, Memory allocation and ownership, Design
+@section Memory layout
+
+We use flat blocks of memory to store matrices and vectors, not C-style
+pointer-to-pointer arrays.  The matrices are stored in row-major order
+-- i.e. the column index (second index) moves continuously through memory. 
+
+@node Linear Algebra Levels, Error estimates, Memory layout, Design
+@section Linear Algebra Levels
+
+Functions using linear algebra are divided into two levels:
+
+For purely "1d" functions we use the C-style arguments (double *,
+stride, size) so that it is simpler to use the functions in a normal C
+program, without needing to invoke all the gsl_vector machinery.
+
+The philosophy here is to minimize the learning curve. If someone only
+needs to use one function, like an fft, they can do so without having
+to learn about gsl_vector.  
+
+This leads to the question of why we don't do the same for matrices.
+In that case the argument list gets too long and confusing, with
+(size1, size2, tda) for each matrix and potential ambiguities over row
+vs column ordering. In this case, it makes sense to use gsl_vector and
+gsl_matrix, which take care of this for the user.
+
+So really the library has two levels -- a lower level based on C types
+for 1d operations, and a higher level based on gsl_matrix and
+gsl_vector for general linear algebra.
+
+Of course, it would be possible to define a vector version of the
+lower level functions too. So far we have not done that because it was
+not essential -- it could be done but it is easy enough to get by
+using the C arguments, by typing v->data, v->stride, v->size instead.
+A gsl_vector version of low-level functions would mainly be a
+convenience.
+
+Please use BLAS routines internally within the library whenever possible
+for efficiency.
+
+@node Error estimates, Exceptions and Error handling, Linear Algebra Levels, Design
+@section Error estimates
+
+In the special functions error bounds are given as twice the expected
+``gaussian'' error.  i.e. 2-sigma, so the result is inside the error
+98% of the time.  People expect the true value to be within +/- the
+quoted error (this wouldn't be the case 32% of the time for 1 sigma).
+Obviously the errors are not gaussian but a factor of two works well
+in practice.
+
+@node Exceptions and Error handling, Persistence, Error estimates, Design
+@section Exceptions and Error handling
+
+The basic error handling procedure is the return code (see gsl_errno.h
+for a list of allowed values).  Use the GSL_ERROR macro to mark an
+error.  The current definition of this macro is not ideal but it can be
+changed at compile time. 
+
+You should always use the GSL_ERROR macro to indicate an error, rather
+than just returning an error code. The macro allows the user to trap
+errors using the debugger (by setting a breakpoint on the function
+gsl_error).  
+
+The only circumstances where GSL_ERROR should not be used are where the
+return value is "indicative" rather than an error -- for example, the
+iterative routines use the return code to indicate the success or
+failure of an iteration. By the nature of an iterative algorithm
+"failure" (a return code of GSL_CONTINUE) is a normal occurrence and
+there is no need to use GSL_ERROR there.
+
+Be sure to free any memory allocated by your function if you return an
+error (in particular for errors in partially initialized objects).
+
+@node Persistence, Using Return Values, Exceptions and Error handling, Design
+@section Persistence
+
+If you make an object foo which uses blocks of memory (e.g. vector,
+matrix, histogram) you can provide functions for reading and writing
+those blocks,
+
+@smallexample
+int gsl_foo_fread (FILE * stream, gsl_foo * v);
+int gsl_foo_fwrite (FILE * stream, const gsl_foo * v);
+int gsl_foo_fscanf (FILE * stream, gsl_foo * v);
+int gsl_foo_fprintf (FILE * stream, const gsl_foo * v, const char *format);
+@end smallexample
+
+@noindent
+Only dump out the blocks of memory, not any associated parameters such
+as lengths.  The idea is for the user to build higher level input/output
+facilities using the functions the library provides.  The fprintf/fscanf
+versions should be portable between architectures, while the binary
+versions should be the "raw" version of the data. Use the functions
+
+@smallexample
+int gsl_block_fread (FILE * stream, gsl_block * b);
+int gsl_block_fwrite (FILE * stream, const gsl_block * b);
+int gsl_block_fscanf (FILE * stream, gsl_block * b);
+int gsl_block_fprintf (FILE * stream, const gsl_block * b, const char *format);
+@end smallexample
+
+@noindent
+or
+
+@smallexample
+int gsl_block_raw_fread (FILE * stream, double * b, size_t n, size_t stride);
+int gsl_block_raw_fwrite (FILE * stream, const double * b, size_t n, size_t stri
+de);
+int gsl_block_raw_fscanf (FILE * stream, double * b, size_t n, size_t stride);
+int gsl_block_raw_fprintf (FILE * stream, const double * b, size_t n, size_t str
+ide, const char *format);
+@end smallexample
+
+@noindent
+to do the actual reading and writing.
+
+@node Using Return Values, Variable Names, Persistence, Design
+@section Using Return Values
+
+Always assign a return value to a variable before using it.  This allows
+easier debugging of the function, and inspection and modification of the
+return value.  If the variable is only needed temporarily then enclose
+it in a suitable scope.
+
+For example, instead of writing,
+
+@example
+a = f(g(h(x,y)))
+@end example
+
+@noindent
+use temporary variables to store the intermediate values,
+@example
+@{
+  double u = h(x,y);
+  double v = g(u);
+  a = f(v);
+@}
+@end example
+
+@noindent
+These can then be inspected more easily in the debugger, and breakpoints
+can be placed more precisely.  The compiler will eliminate the temporary
+variables automatically when the program is compiled with optimization.
+
+@node Variable Names, Datatype widths, Using Return Values, Design
+@section Variable Names
+
+Try to follow existing conventions for variable names,
+
+@table @code
+@item dim
+number of dimensions
+@item w
+pointer to workspace 
+@item state
+pointer to state variable (use @code{s} if you need to save characters)
+@item result
+pointer to result (output variable)
+@item abserr
+absolute error 
+@item relerr
+relative error
+@item epsabs
+absolute tolerance 
+@item epsrel
+relative tolerance
+@item size
+the size of an array or vector e.g. double array[size] 
+@item stride
+the stride of a vector
+@item size1
+the number of rows in a matrix
+@item size2
+the number of columns in a matrix
+@item n
+general integer number, e.g. number of elements of array, in fft, etc
+@item r
+random number generator (gsl_rng)
+@end table
+
+@node Datatype widths, size_t, Variable Names, Design
+@section Datatype widths
+
+Be aware that in ANSI C the type @code{int} is only guaranteed to
+provide 16-bits. It may provide more, but is not guaranteed to.
+Therefore if you require 32 bits you must use @code{long int}, which
+will have 32 bits or more.  Of course, on many platforms the type
+@code{int} does have 32 bits instead of 16 bits but we have to code to
+the ANSI standard rather than a specific platform.
+
+@node size_t, Arrays vs Pointers, Datatype widths, Design
+@section size_t
+
+All objects (blocks of memory, etc) should be measured in terms of a
+@code{size_t} type.  Therefore any iterations (e.g. @code{for(i=0; i<N;
+i++)}) should also use an index of type @code{size_t}.  
+
+Don't mix @code{int} and @code{size_t}.  They are @emph{not}
+interchangeable.
+
+If you need to write a descending loop you have to be careful because
+@code{size_t} is unsigned, so instead of
+
+@example
+for (i = N - 1; i >= 0; i--) @{ ... @} /* DOESN'T WORK */
+@end example
+
+@noindent
+use something like
+
+@example
+for (i = N; i > 0 && i--;) @{ ... @}
+@end example
+
+@noindent
+to avoid problems with wrap-around at @code{i=0}.  
+
+If you really want to avoid confusion use a separate variable to invert
+the loop order,
+@example
+for (i = 0; i < N; i++) @{ j = N - i; ... @}
+@end example
+
+@node Arrays vs Pointers, Pointers, size_t, Design
+@section Arrays vs Pointers
+
+A function can be declared with either pointer arguments or array
+arguments.  The C standard considers these to be equivalent. However, it
+is useful to distinguish between the case of a pointer, representing a
+single object which is being modified, and an array which represents a
+set of objects with unit stride (that are modified or not depending on
+the presence of @code{const}).  For vectors, where the stride is not
+required to be unity, the pointer form is preferred.
+
+@smallexample
+/* real value, set on output */
+int foo (double * x);                           
+
+/* real vector, modified */
+int foo (double * x, size_t stride, size_t n);  
+
+/* constant real vector */
+int foo (const double * x, size_t stride, size_t n);  
+
+/* real array, modified */
+int bar (double x[], size_t n);                 
+
+/* real array, not modified */
+int baz (const double x[], size_t n);           
+@end smallexample
+
+@node  Pointers, Constness, Arrays vs Pointers, Design
+@section Pointers
+
+Avoid dereferencing pointers on the right-hand side of an expression where
+possible.  It's better to introduce a temporary variable.  This is
+easier for the compiler to optimise and also more readable since it
+avoids confusion between the use of @code{*} for multiplication and
+dereferencing.
+
+@example
+while (fabs (f) < 0.5)
+@{
+  *e = *e - 1;
+  f  *= 2;
+@}
+@end example
+
+@noindent
+is better written as,
+
+@example
+@{ 
+  int p = *e;
+
+  while (fabs(f) < 0.5)
+    @{
+     p--;
+     f *= 2;
+    @}
+
+  *e = p;
+@}
+@end example
+
+@node Constness, Pseudo-templates, Pointers, Design
+@section Constness
+
+Use @code{const} in function prototypes wherever an object pointed to by
+a pointer is constant (obviously).  For variables which are meaningfully
+constant within a function/scope use @code{const} also.  This prevents
+you from accidentally modifying a variable which should be constant
+(e.g. length of an array, etc).  It can also help the compiler do
+optimization.  These comments also apply to arguments passed by value
+which should be made @code{const} when that is meaningful.
+
+@node Pseudo-templates, Arbitrary Constants, Constness, Design
+@section Pseudo-templates
+
+There are some pseudo-template macros available in @file{templates_on.h}
+and @file{templates_off.h}.  See a directory link @file{block} for
+details on how to use them.  Use sparingly, they are a bit of a
+nightmare, but unavoidable in places.
+
+In particular, the convention is: templates are used for operations on
+"data" only (vectors, matrices, statistics, sorting).  This is intended
+to cover the case where the program must interface with an external
+data-source which produces a fixed type. e.g. a big array of char's
+produced by an 8-bit counter.
+
+All other functions can use double, for floating point, or the
+appropriate integer type for integers (e.g. unsigned long int for random
+numbers).  It is not the intention to provide a fully templated version
+of the library. 
+
+That would be "putting a quart into a pint pot". To summarize, almost
+everything should be in a "natural type" which is appropriate for
+typical usage, and templates are there to handle a few cases where it is
+unavoidable that other data-types will be encountered.
+
+For floating point work "double" is considered a "natural type".  This
+sort of idea is a part of the C language.
+
+@node  Arbitrary Constants, Test suites, Pseudo-templates, Design
+@section Arbitrary Constants
+
+Avoid arbitrary constants.
+
+For example, don't hard code "small" values like '1e-30', '1e-100' or
+@code{10*GSL_DBL_EPSILON} into the routines.  This is not appropriate
+for a general purpose library.
+
+Compute values accurately using IEEE arithmetic.  If errors are
+potentially significant then error terms should be estimated reliably
+and returned to the user, by analytically deriving an error propagation
+formula, not using guesswork.
+
+A careful consideration of the algorithm usually shows that arbitrary
+constants are unnecessary, and represent an important parameter which
+should be accessible to the user.
+
+For example, consider the following code:
+
+@example
+if (residual < 1e-30) @{
+   return 0.0;  /* residual is zero within round-off error */
+@}
+@end example
+
+@noindent
+This should be rewritten as,
+
+@example
+   return residual;
+@end example
+
+@noindent
+in order to allow the user to determine whether the residual is 
+significant or not.
+
+The only place where it is acceptable to use constants like
+@code{GSL_DBL_EPSILON} is in function approximations, (e.g. taylor
+series, asymptotic expansions, etc).  In these cases it is not an
+arbitrary constant, but an inherent part of the algorithm.
+
+@node Test suites, Compilation, Arbitrary Constants, Design
+@section Test suites
+
+The implementor of each module should provide a reasonable test suite
+for the routines.
+
+The test suite should be a program that uses the library and checks the
+result against known results, or invokes the library several times and
+does a statistical analysis on the results (for example in the case of
+random number generators).
+
+Ideally the one test program per directory should aim for 100% path
+coverage of the code.  Obviously it would be a lot of work to really
+achieve this, so prioritize testing on the critical parts and use
+inspection for the rest.  Test all the error conditions by explicitly
+provoking them, because we consider it a serious defect if the function
+does not return an error for an invalid parameter. N.B. Don't bother to
+test for null pointers -- it's sufficient for the library to segfault if
+the user provides an invalid pointer.
+
+The tests should be deterministic.  Use the @code{gsl_test} functions
+provided to perform separate tests for each feature with a separate
+output PASS/FAIL line, so that any failure can be uniquely identified.
+
+Use realistic test cases with 'high entropy'.  Tests on simple values
+such as 1 or 0 may not reveal bugs.  For example, a test using a value
+of @math{x=1} will not pick up a missing factor of @math{x} in the code.
+Similarly, a test using a value of @math{x=0} will not pick any missing
+terms involving @math{x} in the code.  Use values like @math{2.385} to
+avoid silent failures. 
+
+If your test uses multiple values make sure there are no simple
+relations between them that could allow bugs to be missed through silent
+cancellations.
+
+If you need some random floats to put in the test programs use @code{od -f
+/dev/random} as a source of inspiration.
+
+Don't use @code{sprintf} to create output strings in the tests.  It can
+cause hard to find bugs in the test programs themselves.  The functions
+@code{gsl_test_...}  support format string arguments so use these
+instead.
+
+@node Compilation, Thread-safety, Test suites, Design
+@section Compilation
+
+Make sure everything compiles cleanly.  Use the strict compilation
+options for extra checking.
+
+@smallexample
+make CFLAGS="-ansi -pedantic -Werror -W -Wall -Wtraditional -Wconversion 
+  -Wshadow -Wpointer-arith -Wcast-qual -Wcast-align -Wwrite-strings 
+  -Wstrict-prototypes -fshort-enums -fno-common -Wmissing-prototypes 
+  -Wnested-externs -Dinline= -g -O4"
+@end smallexample
+
+@noindent
+Also use @code{checkergcc} to check for memory problems on the stack and
+the heap.  It's the best memory checking tool.  If checkergcc isn't
+available then Electric Fence will check the heap, which is better than
+no checking.
+
+There is a new tool @code{valgrind} for checking memory access.  Test
+the code with this as well.
+
+Make sure that the library will also compile with C++ compilers
+(g++).  This should not be too much of a problem if you have been writing
+in ANSI C.
+
+@node Thread-safety, Legal issues, Compilation, Design
+@section Thread-safety
+
+The library should be usable in thread-safe programs.  All the functions
+should be thread-safe, in the sense that they shouldn't use static
+variables.
+
+We don't require everything to be completely thread safe, but anything
+that isn't should be obvious.  For example, some global variables are
+used to control the overall behavior of the library (range-checking
+on/off, function to call on fatal error, etc).  Since these are accessed
+directly by the user it is obvious to the multi-threaded programmer that
+they shouldn't be modified by different threads.
+
+There is no need to provide any explicit support for threads
+(e.g. locking mechanisms etc), just to avoid anything which would make
+it impossible for someone to call a GSL routine from a multithreaded
+program.
+
+
+@node Legal issues, Non-UNIX portability, Thread-safety, Design
+@section Legal issues
+
+@itemize @bullet
+@item
+Each contributor must make sure her code is under the GNU General Public
+License (GPL).   This means getting a disclaimer from your employer.
+@item
+We must clearly understand ownership of existing code and algorithms.
+@item
+Each contributor can retain ownership of their code, or sign it over to
+FSF as they prefer. 
+
+There is a standard disclaimer in the GPL (take a look at it).  The more
+specific you make your disclaimer the more likely it is that it will be
+accepted by an employer.  For example,
+
+@smallexample
+Yoyodyne, Inc., hereby disclaims all copyright interest in the software
+`GNU Scientific Library - Legendre Functions' (routines for computing
+legendre functions numerically in C) written by James Hacker.
+
+<signature of Ty Coon>, 1 April 1989
+Ty Coon, President of Vice
+@end smallexample
+
+@item 
+Obviously: don't use or translate non-free code. 
+
+In particular don't copy or translate code from @cite{Numerical Recipes}
+or @cite{ACM TOMS}.
+
+Numerical Recipes is under a strict license and is not free software.
+The publishers Cambridge University Press claim copyright on all aspects
+of the book and the code, including function names, variable names and
+ordering of mathematical subexpressions.  Routines in GSL should not
+refer to Numerical Recipes or be based on it in any way.  
+
+The ACM algorithms published in TOMS (Transactions on Mathematical
+Software) are not public domain, even though they are distributed on the
+internet -- the ACM uses a special non-commercial license which is not
+compatible with the GPL. The details of this license can be found on the
+cover page of ACM Transactions on Mathematical Software or on the ACM
+Website.
+
+Only use code which is explicitly under a free license: GPL or Public
+Domain.  If there is no license on the code then this does not mean it
+is public domain -- an explicit statement is required. If in doubt check
+with the author.
+
+@item
+I @strong{think} one can reference algorithms from classic books on
+numerical analysis (BJG: yes, provided the code is an independent
+implementation and not copied from any existing software.  For
+example, it would be ok to read the papers in ACM TOMS and make an
+independent implementation from their description).
+@end itemize
+
+@node Non-UNIX portability, Compatibility with other libraries, Legal issues, Design
+@section Non-UNIX portability
+
+There is good reason to make this library work on non-UNIX systems.  It
+is probably safe to ignore DOS and only worry about windows95/windowsNT
+portability (so filenames can be long, I think).
+
+On the other hand, nobody should be forced to use non-UNIX systems for
+development.
+
+The best solution is probably to issue guidelines for portability, like
+saying "don't use XYZ unless you absolutely have to".  Then the Windows
+people will be able to do their porting.
+
+@node Compatibility with other libraries, Parallelism, Non-UNIX portability, Design
+@section Compatibility with other libraries
+
+We do not regard compatibility with other numerical libraries as a
+priority.
+
+However, other libraries, such as Numerical Recipes, are widely used.
+If somebody writes the code to allow drop-in replacement of these
+libraries it would be useful to people.  If it is done, it would be as a
+separate wrapper that can be maintained and shipped separately.
+
+There is a separate issue of system libraries, such as BSD math library
+and functions like @code{expm1}, @code{log1p}, @code{hypot}.  The
+functions in this library are available on nearly every platform (but
+not all).  
+
+In this case, it is best to write code in terms of these native
+functions to take advantage of the vendor-supplied system library (for
+example log1p is a machine instruction on the Intel x86). The library
+also provides portable implementations e.g. @code{gsl_hypot} which are
+used as an automatic fall back via autoconf when necessary. See the
+usage of @code{hypot} in @file{gsl/complex/math.c}, the implementation
+of @code{gsl_hypot} and the corresponding parts of files
+@file{configure.in} and @file{config.h.in} as an example.
+
+@node Parallelism, Precision, Compatibility with other libraries, Design
+@section Parallelism
+
+We don't intend to provide support for parallelism within the library
+itself. A parallel library would require a completely different design
+and would carry overhead that other applications do not need.
+
+@node Precision, Miscellaneous, Parallelism, Design
+@section Precision
+
+For algorithms which use cutoffs or other precision-related terms please
+express these in terms of @code{GSL_DBL_EPSILON} and @code{GSL_DBL_MIN}, or powers or
+combinations of these.  This makes it easier to port the routines to
+different precisions.
+
+@node Miscellaneous,  , Precision, Design
+@section Miscellaneous
+
+Don't use the letter @code{l} as a variable name --- it is difficult to
+distinguish from the number @code{1}. (This seems to be a favorite in
+old Fortran programs).
+
+Final tip: one perfect routine is better than any number of routines
+containing errors.
+
+@node Bibliography, Copying, Design, Top
+@chapter Bibliography 
+
+@section General numerics
+
+@itemize
+
+@item
+@cite{Numerical Computation} (2 Volumes) by C.W. Ueberhuber,
+Springer 1997,  ISBN 3540620583  (Vol 1) and  ISBN 3540620575  (Vol 2).
+
+@item
+@cite{Accuracy and Stability of Numerical Algorithms} by  N.J. Higham, 
+SIAM,  ISBN 0898715210.
+
+@item
+@cite{Sources and Development of Mathematical Software} edited by W.R. Cowell,
+Prentice Hall, ISBN 0138235015.
+
+@item
+@cite{A Survey of Numerical Mathematics (2 vols)} by D.M. Young and R.T. Gregory,
+ ISBN 0486656918, ISBN 0486656926.
+
+@item
+@cite{Methods and Programs for Mathematical Functions} by Stephen L. Moshier,
+Hard to find (ISBN 13578980X or 0135789982, possibly others).
+
+@item
+@cite{Numerical Methods That Work} by Forman S. Acton,
+ ISBN 0883854503.
+
+@item
+@cite{Real Computing Made Real: Preventing Errors in Scientific and Engineering Calculations} by Forman S. Acton,
+ ISBN 0486442217. 
+@end itemize
+
+@section Reference
+
+@itemize
+@item
+@cite{Handbook of Mathematical Functions} edited by Abramowitz & Stegun,
+Dover,  ISBN 0486612724.
+
+@item
+@cite{The Art of Computer Programming} (3rd Edition, 3 Volumes) by D. Knuth,
+Addison Wesley,  ISBN 0201485419.
+@end itemize
+
+@section Subject specific
+
+@itemize
+@item
+@cite{Matrix Computations} (3rd Ed) by G.H. Golub, C.F. Van Loan,
+Johns Hopkins University Press 1996,  ISBN 0801854148.
+
+@item
+@cite{LAPACK Users' Guide} (3rd Edition),
+SIAM 1999,  ISBN 0898714478.
+
+@item
+@cite{Treatise on the Theory of Bessel Functions 2ND Edition} by G N Watson,
+ ISBN 0521483913.
+
+@item
+@cite{Higher Transcendental Functions satisfying nonhomogenous linear differential equations} by A W Babister,
+ ISBN 1114401773.
+
+@end itemize
+
+@node Copying, GNU Free Documentation License, Bibliography, Top
+@unnumbered Copying
+
+   The subroutines and source code in the @value{GSL} package are "free";
+this means that everyone is free to use them and free to redistribute
+them on a free basis.  The @value{GSL}-related programs are not in the
+public domain; they are copyrighted and there are restrictions on their
+distribution, but these restrictions are designed to permit everything
+that a good cooperating citizen would want to do.  What is not allowed
+is to try to prevent others from further sharing any version of these
+programs that they might get from you.
+
+   Specifically, we want to make sure that you have the right to give
+away copies of the programs that relate to @value{GSL}, that you receive
+source code or else can get it if you want it, that you can change these
+programs or use pieces of them in new free programs, and that you know
+you can do these things.
+
+   To make sure that everyone has such rights, we have to forbid you to
+deprive anyone else of these rights.  For example, if you distribute
+copies of the @value{GSL}-related code, you must give the recipients all
+the rights that you have.  You must make sure that they, too, receive or
+can get the source code.  And you must tell them their rights.
+
+   Also, for our own protection, we must make certain that everyone
+finds out that there is no warranty for the programs that relate to
+@value{GSL}.  If these programs are modified by someone else and passed
+on, we want their recipients to know that what they have is not what we
+distributed, so that any problems introduced by others will not reflect
+on our reputation.
+
+   The precise conditions of the licenses for the programs currently
+being distributed that relate to @value{GSL} are found in the General
+Public Licenses that accompany them.
+
+@node GNU Free Documentation License,  , Copying, Top
+@unnumbered GNU Free Documentation License
+@include fdl.texi
+
+@c @printindex cp
+
+@c @node Function Index
+@c @unnumbered Function Index
+
+@c @printindex fn
+
+@c @node Variable Index
+@c @unnumbered Variable Index
+
+@c @printindex vr
+
+@c @node Type Index
+@c @unnumbered Type Index
+
+@c @printindex tp
+
+@bye
diff --git a/gsl-1.9/doc/gsl-histogram.1 b/gsl-1.9/doc/gsl-histogram.1
new file mode 100644
index 0000000..66f9b5a
--- /dev/null
+++ b/gsl-1.9/doc/gsl-histogram.1
@@ -0,0 +1,42 @@
+.\" Man page contributed by Dirk Eddelbuettel <edd@debian.org>
+.\" and released under the GNU General Public License
+.TH GSL-HISTOGRAM 1 "" GNU
+.SH NAME
+gsl-histogram - compute histogram of data on stdin
+.SH SYNOPSYS
+.B gsl-histogram xmin xmax [n]
+.SH DESCRIPTION
+.B gsl-histogram 
+is a demonstration program for the GNU Scientific Library.
+It takes three arguments, specifying the upper and lower bounds of the
+histogram and the number of bins.  It then reads numbers from `stdin',
+one line at a time, and adds them to the histogram.  When there is no
+more data to read it prints out the accumulated histogram using
+gsl_histogram_fprintf.  If n is unspecified then bins of integer width
+are used.
+.SH EXAMPLE
+Here is an example.  We generate 10000 random samples from a Cauchy
+distribution with a width of 30 and histogram them over the range -100 to
+100, using 200 bins.
+ 
+     gsl-randist 0 10000 cauchy 30 | gsl-histogram -100 100 200 > histogram.dat
+ 
+A plot of the resulting histogram will show the familiar shape of the
+Cauchy distribution with fluctuations caused by the finite sample
+size.
+
+     awk '{print $1, $3 ; print $2, $3}' histogram.dat | graph -T X
+
+.SH SEE ALSO
+.BR gsl(3) ,
+.BR gsl-randist(1) .
+
+.SH AUTHOR
+.B gsl-histogram 
+was written by Brian Gough.
+Copyright 1996-2000; for copying conditions see the GNU General
+Public Licence. 
+
+This manual page was added by the Dirk Eddelbuettel
+<edd@debian.org>, the Debian GNU/Linux maintainer for
+.BR GSL .
diff --git a/gsl-1.9/doc/gsl-randist.1 b/gsl-1.9/doc/gsl-randist.1
new file mode 100644
index 0000000..4a06169
--- /dev/null
+++ b/gsl-1.9/doc/gsl-randist.1
@@ -0,0 +1,38 @@
+.\" Man page contributed by Dirk Eddelbuettel <edd@debian.org>
+.\" and released under the GNU General Public License
+.TH GSL-RANDIST 1 "" GNU
+.SH NAME
+gsl-randist - generate random samples from various distributions
+.SH SYNOPSYS
+.B gsl-randist seed n DIST param1 param2 [..]
+.SH DESCRIPTION
+.B gsl-randist 
+is a demonstration program for the GNU Scientific Library.
+It generates n random samples from the distribution DIST using the distribution
+parameters param1, param2, ...
+.SH EXAMPLE
+Here is an example.  We generate 10000 random samples from a Cauchy
+distribution with a width of 30 and histogram them over the range -100 to
+100, using 200 bins.
+ 
+     gsl-randist 0 10000 cauchy 30 | gsl-histogram -100 100 200 > histogram.dat
+ 
+A plot of the resulting histogram will show the familiar shape of the
+Cauchy distribution with fluctuations caused by the finite sample
+size.
+
+     awk '{print $1, $3 ; print $2, $3}' histogram.dat | graph -T X
+
+.SH SEE ALSO
+.BR gsl(3) ,
+.BR gsl-histogram(1) .
+
+.SH AUTHOR
+.B gsl-randist 
+was written by James Theiler and Brian Gough.
+Copyright 1996-2000; for copying conditions see the GNU General
+Public Licence. 
+
+This manual page was added by the Dirk Eddelbuettel
+<edd@debian.org>, the Debian GNU/Linux maintainer for
+.BR GSL .
diff --git a/gsl-1.9/doc/gsl-ref.info b/gsl-1.9/doc/gsl-ref.info
new file mode 100644
index 0000000..0423ca6
--- /dev/null
+++ b/gsl-1.9/doc/gsl-ref.info
@@ -0,0 +1,596 @@
+This is gsl-ref.info, produced by makeinfo version 4.8 from
+gsl-ref.texi.
+
+INFO-DIR-SECTION Scientific software
+START-INFO-DIR-ENTRY
+* gsl-ref: (gsl-ref).                   GNU Scientific Library - Reference
+END-INFO-DIR-ENTRY
+
+   Copyright (C) 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004,
+2005, 2006, 2007 The GSL Team.
+
+   Permission is granted to copy, distribute and/or modify this document
+under the terms of the GNU Free Documentation License, Version 1.2 or
+any later version published by the Free Software Foundation; with the
+Invariant Sections being "GNU General Public License" and "Free Software
+Needs Free Documentation", the Front-Cover text being "A GNU Manual",
+and with the Back-Cover Text being (a) (see below).  A copy of the
+license is included in the section entitled "GNU Free Documentation
+License".
+
+   (a) The Back-Cover Text is: "You have freedom to copy and modify this
+GNU Manual, like GNU software."
+
+
+Indirect:
+gsl-ref.info-1: 939
+gsl-ref.info-2: 298842
+gsl-ref.info-3: 596860
+gsl-ref.info-4: 895867
+gsl-ref.info-5: 1072596
+gsl-ref.info-6: 1311271
+
+Tag Table:
+(Indirect)
+Node: Top939
+Node: Introduction3646
+Node: Routines available in GSL4298
+Node: GSL is Free Software6213
+Node: Obtaining GSL8605
+Node: No Warranty9746
+Node: Reporting Bugs10252
+Node: Further Information11136
+Node: Conventions used in this manual12149
+Node: Using the library12913
+Node: An Example Program13499
+Node: Compiling and Linking14203
+Node: Linking programs with the library15273
+Node: Linking with an alternative BLAS library16418
+Node: Shared Libraries17348
+Node: ANSI C Compliance18764
+Node: Inline functions19857
+Node: Long double20900
+Node: Portability functions22298
+Node: Alternative optimized functions23695
+Node: Support for different numeric types25153
+Node: Compatibility with C++28216
+Node: Aliasing of arrays28788
+Node: Thread-safety29539
+Node: Deprecated Functions30626
+Node: Code Reuse31251
+Node: Error Handling31895
+Node: Error Reporting32645
+Node: Error Codes34537
+Node: Error Handlers36393
+Node: Using GSL error reporting in your own functions40028
+Node: Error Reporting Examples41986
+Node: Mathematical Functions43190
+Node: Mathematical Constants43951
+Node: Infinities and Not-a-number45081
+Node: Elementary Functions46149
+Node: Small integer powers48404
+Node: Testing the Sign of Numbers49966
+Node: Testing for Odd and Even Numbers50397
+Node: Maximum and Minimum functions50953
+Node: Approximate Comparison of Floating Point Numbers53035
+Node: Complex Numbers54241
+Ref: Complex Numbers-Footnote-155549
+Node: Complex numbers55613
+Node: Properties of complex numbers57416
+Node: Complex arithmetic operators58403
+Node: Elementary Complex Functions61149
+Node: Complex Trigonometric Functions62969
+Node: Inverse Complex Trigonometric Functions64190
+Node: Complex Hyperbolic Functions66796
+Node: Inverse Complex Hyperbolic Functions68092
+Node: Complex Number References and Further Reading70080
+Node: Polynomials71476
+Node: Polynomial Evaluation72243
+Node: Divided Difference Representation of Polynomials72703
+Node: Quadratic Equations74372
+Node: Cubic Equations76312
+Node: General Polynomial Equations77683
+Node: Roots of Polynomials Examples80028
+Node: Roots of Polynomials References and Further Reading81417
+Node: Special Functions82362
+Node: Special Function Usage84356
+Node: The gsl_sf_result struct85539
+Node: Special Function Modes86803
+Node: Airy Functions and Derivatives87739
+Node: Airy Functions88435
+Node: Derivatives of Airy Functions89795
+Node: Zeros of Airy Functions91307
+Node: Zeros of Derivatives of Airy Functions92026
+Node: Bessel Functions92785
+Node: Regular Cylindrical Bessel Functions93977
+Node: Irregular Cylindrical Bessel Functions95301
+Node: Regular Modified Cylindrical Bessel Functions96757
+Node: Irregular Modified Cylindrical Bessel Functions99584
+Node: Regular Spherical Bessel Functions102552
+Node: Irregular Spherical Bessel Functions104764
+Node: Regular Modified Spherical Bessel Functions106451
+Node: Irregular Modified Spherical Bessel Functions108454
+Node: Regular Bessel Function - Fractional Order110517
+Node: Irregular Bessel Functions - Fractional Order111515
+Node: Regular Modified Bessel Functions - Fractional Order112097
+Node: Irregular Modified Bessel Functions - Fractional Order113022
+Node: Zeros of Regular Bessel Functions114258
+Node: Clausen Functions115361
+Node: Coulomb Functions115965
+Node: Normalized Hydrogenic Bound States116414
+Node: Coulomb Wave Functions117527
+Node: Coulomb Wave Function Normalization Constant120937
+Node: Coupling Coefficients121690
+Node: 3-j Symbols122413
+Node: 6-j Symbols123017
+Node: 9-j Symbols123641
+Node: Dawson Function124347
+Node: Debye Functions124914
+Node: Dilogarithm126688
+Node: Real Argument126983
+Node: Complex Argument127552
+Node: Elementary Operations128021
+Node: Elliptic Integrals128845
+Node: Definition of Legendre Forms129417
+Node: Definition of Carlson Forms130237
+Node: Legendre Form of Complete Elliptic Integrals130934
+Node: Legendre Form of Incomplete Elliptic Integrals132502
+Node: Carlson Forms134690
+Node: Elliptic Functions (Jacobi)136237
+Node: Error Functions136818
+Node: Error Function137269
+Node: Complementary Error Function137642
+Node: Log Complementary Error Function138102
+Node: Probability functions138554
+Node: Exponential Functions139807
+Node: Exponential Function140190
+Node: Relative Exponential Functions141413
+Node: Exponentiation With Error Estimate143084
+Node: Exponential Integrals144293
+Node: Exponential Integral144775
+Node: Ei(x)145425
+Node: Hyperbolic Integrals145873
+Node: Ei_3(x)146529
+Node: Trigonometric Integrals146921
+Node: Arctangent Integral147520
+Node: Fermi-Dirac Function147920
+Node: Complete Fermi-Dirac Integrals148290
+Node: Incomplete Fermi-Dirac Integrals150751
+Node: Gamma and Beta Functions151394
+Node: Gamma Functions151830
+Node: Factorials154951
+Node: Pochhammer Symbol157504
+Node: Incomplete Gamma Functions159016
+Node: Beta Functions160265
+Node: Incomplete Beta Function161017
+Node: Gegenbauer Functions161548
+Node: Hypergeometric Functions163198
+Node: Laguerre Functions167465
+Node: Lambert W Functions169028
+Node: Legendre Functions and Spherical Harmonics170066
+Node: Legendre Polynomials170655
+Node: Associated Legendre Polynomials and Spherical Harmonics172735
+Node: Conical Functions175536
+Node: Radial Functions for Hyperbolic Space177659
+Node: Logarithm and Related Functions179836
+Node: Mathieu Functions181460
+Node: Mathieu Function Workspace182809
+Node: Mathieu Function Characteristic Values183622
+Node: Angular Mathieu Functions184690
+Node: Radial Mathieu Functions185722
+Node: Power Function186960
+Node: Psi (Digamma) Function187887
+Node: Digamma Function188407
+Node: Trigamma Function189237
+Node: Polygamma Function189814
+Node: Synchrotron Functions190198
+Node: Transport Functions190961
+Node: Trigonometric Functions192108
+Node: Circular Trigonometric Functions192731
+Node: Trigonometric Functions for Complex Arguments193740
+Node: Hyperbolic Trigonometric Functions194774
+Node: Conversion Functions195377
+Node: Restriction Functions196160
+Node: Trigonometric Functions With Error Estimates197092
+Node: Zeta Functions198002
+Node: Riemann Zeta Function198460
+Node: Riemann Zeta Function Minus One199139
+Node: Hurwitz Zeta Function199910
+Node: Eta Function200415
+Node: Special Functions Examples200974
+Node: Special Functions References and Further Reading202668
+Node: Vectors and Matrices203775
+Node: Data types204534
+Node: Blocks205737
+Node: Block allocation206653
+Node: Reading and writing blocks208137
+Node: Example programs for blocks210201
+Node: Vectors210828
+Node: Vector allocation212714
+Node: Accessing vector elements214332
+Node: Initializing vector elements216574
+Node: Reading and writing vectors217268
+Node: Vector views219358
+Node: Copying vectors226800
+Node: Exchanging elements227660
+Node: Vector operations228218
+Node: Finding maximum and minimum elements of vectors229732
+Node: Vector properties231257
+Node: Example programs for vectors231880
+Node: Matrices234169
+Node: Matrix allocation237027
+Node: Accessing matrix elements238694
+Node: Initializing matrix elements240402
+Node: Reading and writing matrices241198
+Node: Matrix views243301
+Node: Creating row and column views250441
+Node: Copying matrices253612
+Node: Copying rows and columns254208
+Node: Exchanging rows and columns255885
+Node: Matrix operations257363
+Node: Finding maximum and minimum elements of matrices259003
+Node: Matrix properties260907
+Node: Example programs for matrices261655
+Node: Vector and Matrix References and Further Reading265726
+Node: Permutations266212
+Node: The Permutation struct267500
+Node: Permutation allocation268004
+Node: Accessing permutation elements269438
+Node: Permutation properties270201
+Node: Permutation functions270900
+Node: Applying Permutations272157
+Node: Reading and writing permutations274082
+Node: Permutations in cyclic form276264
+Node: Permutation Examples279689
+Node: Permutation References and Further Reading282232
+Node: Combinations282906
+Node: The Combination struct283751
+Node: Combination allocation284303
+Node: Accessing combination elements286088
+Node: Combination properties286667
+Node: Combination functions287553
+Node: Reading and writing combinations288493
+Node: Combination Examples290693
+Node: Combination References and Further Reading292001
+Node: Sorting292405
+Node: Sorting objects293355
+Node: Sorting vectors296266
+Node: Selecting the k smallest or largest elements298842
+Node: Computing the rank302176
+Node: Sorting Examples303377
+Node: Sorting References and Further Reading305021
+Node: BLAS Support305567
+Node: GSL BLAS Interface308542
+Node: Level 1 GSL BLAS Interface309029
+Node: Level 2 GSL BLAS Interface315772
+Node: Level 3 GSL BLAS Interface325066
+Node: BLAS Examples336202
+Node: BLAS References and Further Reading337623
+Node: Linear Algebra338982
+Node: LU Decomposition340164
+Node: QR Decomposition345372
+Node: QR Decomposition with Column Pivoting351174
+Node: Singular Value Decomposition355584
+Node: Cholesky Decomposition359174
+Node: Tridiagonal Decomposition of Real Symmetric Matrices361181
+Node: Tridiagonal Decomposition of Hermitian Matrices363029
+Node: Hessenberg Decomposition of Real Matrices364984
+Node: Bidiagonalization367329
+Node: Householder Transformations369739
+Node: Householder solver for linear systems371587
+Node: Tridiagonal Systems372492
+Node: Balancing375145
+Node: Linear Algebra Examples375946
+Node: Linear Algebra References and Further Reading377934
+Node: Eigensystems379424
+Node: Real Symmetric Matrices380603
+Node: Complex Hermitian Matrices382835
+Node: Real Nonsymmetric Matrices385343
+Node: Sorting Eigenvalues and Eigenvectors390537
+Node: Eigenvalue and Eigenvector Examples392404
+Node: Eigenvalue and Eigenvector References397790
+Node: Fast Fourier Transforms398498
+Node: Mathematical Definitions399707
+Node: Overview of complex data FFTs402112
+Node: Radix-2 FFT routines for complex data405055
+Node: Mixed-radix FFT routines for complex data409267
+Node: Overview of real data FFTs418596
+Node: Radix-2 FFT routines for real data421030
+Node: Mixed-radix FFT routines for real data424037
+Node: FFT References and Further Reading434617
+Node: Numerical Integration437486
+Node: Numerical Integration Introduction439031
+Node: Integrands without weight functions440965
+Node: Integrands with weight functions441794
+Node: Integrands with singular weight functions442505
+Node: QNG non-adaptive Gauss-Kronrod integration443423
+Node: QAG adaptive integration444694
+Node: QAGS adaptive integration with singularities447325
+Node: QAGP adaptive integration with known singular points449125
+Node: QAGI adaptive integration on infinite intervals450439
+Node: QAWC adaptive integration for Cauchy principal values452732
+Node: QAWS adaptive integration for singular functions453907
+Node: QAWO adaptive integration for oscillatory functions457129
+Node: QAWF adaptive integration for Fourier integrals460908
+Node: Numerical integration error codes463533
+Node: Numerical integration examples464301
+Node: Numerical integration References and Further Reading466346
+Node: Random Number Generation467075
+Node: General comments on random numbers468577
+Node: The Random Number Generator Interface470527
+Node: Random number generator initialization471885
+Node: Sampling from a random number generator473901
+Node: Auxiliary random number generator functions477066
+Node: Random number environment variables479386
+Node: Copying random number generator state481912
+Node: Reading and writing random number generator state482887
+Node: Random number generator algorithms484299
+Node: Unix random number generators494108
+Node: Other random number generators497826
+Node: Random Number Generator Performance506459
+Node: Random Number Generator Examples507777
+Node: Random Number References and Further Reading509333
+Node: Random Number Acknowledgements510630
+Node: Quasi-Random Sequences511116
+Node: Quasi-random number generator initialization512223
+Node: Sampling from a quasi-random number generator513238
+Node: Auxiliary quasi-random number generator functions513855
+Node: Saving and resorting quasi-random number generator state514801
+Node: Quasi-random number generator algorithms515609
+Node: Quasi-random number generator examples516384
+Node: Quasi-random number references517369
+Node: Random Number Distributions517895
+Node: Random Number Distribution Introduction521094
+Node: The Gaussian Distribution522889
+Node: The Gaussian Tail Distribution525551
+Node: The Bivariate Gaussian Distribution527220
+Node: The Exponential Distribution528543
+Node: The Laplace Distribution529682
+Node: The Exponential Power Distribution530779
+Node: The Cauchy Distribution532047
+Node: The Rayleigh Distribution533296
+Node: The Rayleigh Tail Distribution534468
+Node: The Landau Distribution535340
+Node: The Levy alpha-Stable Distributions536295
+Node: The Levy skew alpha-Stable Distribution537353
+Node: The Gamma Distribution538965
+Node: The Flat (Uniform) Distribution540622
+Node: The Lognormal Distribution541774
+Node: The Chi-squared Distribution543122
+Node: The F-distribution544526
+Node: The t-distribution546162
+Node: The Beta Distribution547556
+Node: The Logistic Distribution548715
+Node: The Pareto Distribution549844
+Node: Spherical Vector Distributions551014
+Node: The Weibull Distribution553848
+Node: The Type-1 Gumbel Distribution555042
+Node: The Type-2 Gumbel Distribution556279
+Node: The Dirichlet Distribution557510
+Node: General Discrete Distributions559178
+Node: The Poisson Distribution563043
+Node: The Bernoulli Distribution564051
+Node: The Binomial Distribution564802
+Node: The Multinomial Distribution566007
+Node: The Negative Binomial Distribution567776
+Node: The Pascal Distribution569141
+Node: The Geometric Distribution570299
+Node: The Hypergeometric Distribution571547
+Node: The Logarithmic Distribution573200
+Node: Shuffling and Sampling573991
+Node: Random Number Distribution Examples576810
+Node: Random Number Distribution References and Further Reading580006
+Node: Statistics582072
+Node: Mean and standard deviation and variance583351
+Node: Absolute deviation586396
+Node: Higher moments (skewness and kurtosis)587688
+Node: Autocorrelation589821
+Node: Covariance590635
+Node: Weighted Samples591616
+Node: Maximum and Minimum values596860
+Node: Median and Percentiles599599
+Node: Example statistical programs602014
+Node: Statistics References and Further Reading604673
+Node: Histograms605881
+Node: The histogram struct607641
+Node: Histogram allocation609444
+Node: Copying Histograms612405
+Node: Updating and accessing histogram elements613081
+Node: Searching histogram ranges616348
+Node: Histogram Statistics617347
+Node: Histogram Operations619207
+Node: Reading and writing histograms621279
+Node: Resampling from histograms624321
+Node: The histogram probability distribution struct625119
+Node: Example programs for histograms628150
+Node: Two dimensional histograms630213
+Node: The 2D histogram struct630934
+Node: 2D Histogram allocation632742
+Node: Copying 2D Histograms634813
+Node: Updating and accessing 2D histogram elements635518
+Node: Searching 2D histogram ranges639173
+Node: 2D Histogram Statistics640192
+Node: 2D Histogram Operations643051
+Node: Reading and writing 2D histograms645225
+Node: Resampling from 2D histograms648852
+Node: Example programs for 2D histograms651871
+Node: N-tuples653699
+Node: The ntuple struct654955
+Node: Creating ntuples655433
+Node: Opening an existing ntuple file656100
+Node: Writing ntuples656728
+Node: Reading ntuples657189
+Node: Closing an ntuple file657520
+Node: Histogramming ntuple values657860
+Node: Example ntuple programs659866
+Node: Ntuple References and Further Reading663195
+Node: Monte Carlo Integration663516
+Node: Monte Carlo Interface664761
+Node: PLAIN Monte Carlo667384
+Node: MISER669822
+Node: VEGAS675660
+Node: Monte Carlo Examples683995
+Node: Monte Carlo Integration References and Further Reading689938
+Node: Simulated Annealing690720
+Node: Simulated Annealing algorithm691927
+Node: Simulated Annealing functions693081
+Node: Examples with Simulated Annealing697607
+Node: Trivial example698161
+Node: Traveling Salesman Problem700812
+Node: Simulated Annealing References and Further Reading704125
+Node: Ordinary Differential Equations704536
+Node: Defining the ODE System705357
+Node: Stepping Functions707549
+Node: Adaptive Step-size Control711728
+Node: Evolution717186
+Node: ODE Example programs719864
+Node: ODE References and Further Reading724934
+Node: Interpolation725573
+Node: Introduction to Interpolation726586
+Node: Interpolation Functions727027
+Node: Interpolation Types728203
+Node: Index Look-up and Acceleration730813
+Node: Evaluation of Interpolating Functions732443
+Node: Higher-level Interface734827
+Node: Interpolation Example programs736866
+Node: Interpolation References and Further Reading740113
+Node: Numerical Differentiation740686
+Node: Numerical Differentiation functions741275
+Node: Numerical Differentiation Examples744133
+Node: Numerical Differentiation References745546
+Node: Chebyshev Approximations746097
+Node: Chebyshev Definitions747113
+Node: Creation and Calculation of Chebyshev Series747903
+Node: Chebyshev Series Evaluation748875
+Node: Derivatives and Integrals750259
+Node: Chebyshev Approximation examples751501
+Node: Chebyshev Approximation References and Further Reading752997
+Node: Series Acceleration753446
+Node: Acceleration functions754211
+Node: Acceleration functions without error estimation756536
+Node: Example of accelerating a series759146
+Node: Series Acceleration References761492
+Node: Wavelet Transforms762380
+Node: DWT Definitions762921
+Node: DWT Initialization763873
+Node: DWT Transform Functions766452
+Node: DWT in one dimension766987
+Node: DWT in two dimension769006
+Node: DWT Examples773607
+Node: DWT References775424
+Node: Discrete Hankel Transforms777587
+Node: Discrete Hankel Transform Definition778055
+Node: Discrete Hankel Transform Functions780194
+Node: Discrete Hankel Transform References781725
+Node: One dimensional Root-Finding782129
+Node: Root Finding Overview783389
+Node: Root Finding Caveats785247
+Node: Initializing the Solver787014
+Node: Providing the function to solve789647
+Node: Search Bounds and Guesses793413
+Node: Root Finding Iteration794274
+Node: Search Stopping Parameters796123
+Node: Root Bracketing Algorithms798637
+Node: Root Finding Algorithms using Derivatives801935
+Node: Root Finding Examples805438
+Node: Root Finding References and Further Reading812726
+Node: One dimensional Minimization813365
+Node: Minimization Overview814667
+Node: Minimization Caveats816373
+Node: Initializing the Minimizer817710
+Node: Providing the function to minimize819951
+Node: Minimization Iteration820429
+Node: Minimization Stopping Parameters822565
+Node: Minimization Algorithms824174
+Node: Minimization Examples826613
+Node: Minimization References and Further Reading829557
+Node: Multidimensional Root-Finding830013
+Node: Overview of Multidimensional Root Finding831502
+Node: Initializing the Multidimensional Solver833689
+Node: Providing the multidimensional system of equations to solve836887
+Node: Iteration of the multidimensional solver841792
+Node: Search Stopping Parameters for the multidimensional solver843977
+Node: Algorithms using Derivatives845714
+Node: Algorithms without Derivatives850460
+Node: Example programs for Multidimensional Root finding853548
+Node: References and Further Reading for Multidimensional Root Finding862161
+Node: Multidimensional Minimization863400
+Node: Multimin Overview864692
+Node: Multimin Caveats866664
+Node: Initializing the Multidimensional Minimizer867413
+Node: Providing a function to minimize870574
+Node: Multimin Iteration874239
+Node: Multimin Stopping Criteria876230
+Node: Multimin Algorithms877782
+Node: Multimin Examples882741
+Node: Multimin References and Further Reading889358
+Node: Least-Squares Fitting890221
+Node: Fitting Overview891220
+Node: Linear regression893436
+Node: Linear fitting without a constant term895867
+Node: Multi-parameter fitting898058
+Node: Fitting Examples902963
+Node: Fitting References and Further Reading909788
+Node: Nonlinear Least-Squares Fitting910610
+Node: Overview of Nonlinear Least-Squares Fitting912060
+Node: Initializing the Nonlinear Least-Squares Solver913527
+Node: Providing the Function to be Minimized916380
+Node: Iteration of the Minimization Algorithm919317
+Node: Search Stopping Parameters for Minimization Algorithms921167
+Node: Minimization Algorithms using Derivatives923262
+Node: Minimization Algorithms without Derivatives926510
+Node: Computing the covariance matrix of best fit parameters926902
+Node: Example programs for Nonlinear Least-Squares Fitting928936
+Node: References and Further Reading for Nonlinear Least-Squares Fitting936664
+Node: Basis Splines937400
+Node: Overview of B-splines937975
+Node: Initializing the B-splines solver939260
+Node: Constructing the knots vector940003
+Node: Evaluation of B-spline basis functions940790
+Node: Example programs for B-splines941523
+Node: References and Further Reading945220
+Node: Physical Constants945734
+Node: Fundamental Constants947144
+Node: Astronomy and Astrophysics948283
+Node: Atomic and Nuclear Physics948948
+Node: Measurement of Time950585
+Node: Imperial Units951015
+Node: Speed and Nautical Units951457
+Node: Printers Units951961
+Node: Volume Area and Length952284
+Node: Mass and Weight952970
+Node: Thermal Energy and Power953789
+Node: Pressure954212
+Node: Viscosity954825
+Node: Light and Illumination955101
+Node: Radioactivity955693
+Node: Force and Energy956028
+Node: Prefixes956432
+Node: Physical Constant Examples957175
+Node: Physical Constant References and Further Reading958964
+Node: IEEE floating-point arithmetic959675
+Node: Representation of floating point numbers960261
+Node: Setting up your IEEE environment964742
+Node: IEEE References and Further Reading971351
+Node: Debugging Numerical Programs972506
+Node: Using gdb972990
+Node: Examining floating point registers976333
+Node: Handling floating point exceptions977618
+Node: GCC warning options for numerical programs979030
+Node: Debugging References983223
+Node: Contributors to GSL983935
+Node: Autoconf Macros988118
+Node: GSL CBLAS Library992103
+Node: Level 1 CBLAS Functions992630
+Node: Level 2 CBLAS Functions997932
+Node: Level 3 CBLAS Functions1014602
+Node: GSL CBLAS Examples1024264
+Node: Free Software Needs Free Documentation1025826
+Node: GNU General Public License1030895
+Node: GNU Free Documentation License1050165
+Node: Function Index1072596
+Node: Variable Index1306453
+Node: Type Index1307625
+Node: Concept Index1311271
+
+End Tag Table
diff --git a/gsl-1.9/doc/gsl-ref.info-1 b/gsl-1.9/doc/gsl-ref.info-1
new file mode 100644
index 0000000..2d4e483
--- /dev/null
+++ b/gsl-1.9/doc/gsl-ref.info-1
@@ -0,0 +1,7392 @@
+This is gsl-ref.info, produced by makeinfo version 4.8 from
+gsl-ref.texi.
+
+INFO-DIR-SECTION Scientific software
+START-INFO-DIR-ENTRY
+* gsl-ref: (gsl-ref).                   GNU Scientific Library - Reference
+END-INFO-DIR-ENTRY
+
+   Copyright (C) 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004,
+2005, 2006, 2007 The GSL Team.
+
+   Permission is granted to copy, distribute and/or modify this document
+under the terms of the GNU Free Documentation License, Version 1.2 or
+any later version published by the Free Software Foundation; with the
+Invariant Sections being "GNU General Public License" and "Free Software
+Needs Free Documentation", the Front-Cover text being "A GNU Manual",
+and with the Back-Cover Text being (a) (see below).  A copy of the
+license is included in the section entitled "GNU Free Documentation
+License".
+
+   (a) The Back-Cover Text is: "You have freedom to copy and modify this
+GNU Manual, like GNU software."
+
+
+File: gsl-ref.info,  Node: Top,  Next: Introduction,  Prev: (dir),  Up: (dir)
+
+GSL
+***
+
+This file documents the GNU Scientific Library (GSL), a collection of
+numerical routines for scientific computing.  It corresponds to release
+1.9 of the library.  Please report any errors in this manual to
+<bug-gsl@gnu.org>.
+
+   More information about GSL can be found at the project homepage,
+`http://www.gnu.org/software/gsl/'.
+
+   Printed copies of this manual can be purchased from Network Theory
+Ltd at `http://www.network-theory.co.uk/gsl/manual/'. The money raised
+from sales of the manual helps support the development of GSL.
+
+   A Japanese translation of this manual is available from the GSL
+project homepage thanks to Daisuke Tominaga.
+
+   Copyright (C) 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004,
+2005, 2006, 2007 The GSL Team.
+
+   Permission is granted to copy, distribute and/or modify this document
+under the terms of the GNU Free Documentation License, Version 1.2 or
+any later version published by the Free Software Foundation; with the
+Invariant Sections being "GNU General Public License" and "Free Software
+Needs Free Documentation", the Front-Cover text being "A GNU Manual",
+and with the Back-Cover Text being (a) (see below).  A copy of the
+license is included in the section entitled "GNU Free Documentation
+License".
+
+   (a) The Back-Cover Text is: "You have freedom to copy and modify this
+GNU Manual, like GNU software."
+
+
+* Menu:
+
+* Introduction::
+* Using the library::
+* Error Handling::
+* Mathematical Functions::
+* Complex Numbers::
+* Polynomials::
+* Special Functions::
+* Vectors and Matrices::
+* Permutations::
+* Combinations::
+* Sorting::
+* BLAS Support::
+* Linear Algebra::
+* Eigensystems::
+* Fast Fourier Transforms::
+* Numerical Integration::
+* Random Number Generation::
+* Quasi-Random Sequences::
+* Random Number Distributions::
+* Statistics::
+* Histograms::
+* N-tuples::
+* Monte Carlo Integration::
+* Simulated Annealing::
+* Ordinary Differential Equations::
+* Interpolation::
+* Numerical Differentiation::
+* Chebyshev Approximations::
+* Series Acceleration::
+* Wavelet Transforms::
+* Discrete Hankel Transforms::
+* One dimensional Root-Finding::
+* One dimensional Minimization::
+* Multidimensional Root-Finding::
+* Multidimensional Minimization::
+* Least-Squares Fitting::
+* Nonlinear Least-Squares Fitting::
+* Basis Splines::
+* Physical Constants::
+* IEEE floating-point arithmetic::
+* Debugging Numerical Programs::
+* Contributors to GSL::
+* Autoconf Macros::
+* GSL CBLAS Library::
+* Free Software Needs Free Documentation::
+* GNU General Public License::
+* GNU Free Documentation License::
+* Function Index::
+* Variable Index::
+* Type Index::
+* Concept Index::
+
+
+File: gsl-ref.info,  Node: Introduction,  Next: Using the library,  Prev: Top,  Up: Top
+
+1 Introduction
+**************
+
+The GNU Scientific Library (GSL) is a collection of routines for
+numerical computing.  The routines have been written from scratch in C,
+and present a modern Applications Programming Interface (API) for C
+programmers, allowing wrappers to be written for very high level
+languages.  The source code is distributed under the GNU General Public
+License.
+
+* Menu:
+
+* Routines available in GSL::
+* GSL is Free Software::
+* Obtaining GSL::
+* No Warranty::
+* Reporting Bugs::
+* Further Information::
+* Conventions used in this manual::
+
+
+File: gsl-ref.info,  Node: Routines available in GSL,  Next: GSL is Free Software,  Up: Introduction
+
+1.1 Routines available in GSL
+=============================
+
+The library covers a wide range of topics in numerical computing.
+Routines are available for the following areas,
+
+     Complex Numbers                  Roots of Polynomials             
+     Special Functions                Vectors and Matrices             
+     Permutations                     Combinations                     
+     Sorting                          BLAS Support                     
+     Linear Algebra                   CBLAS Library                    
+     Fast Fourier Transforms          Eigensystems                     
+     Random Numbers                   Quadrature                       
+     Random Distributions             Quasi-Random Sequences           
+     Histograms                       Statistics                       
+     Monte Carlo Integration          N-Tuples                         
+     Differential Equations           Simulated Annealing              
+     Numerical Differentiation        Interpolation                    
+     Series Acceleration              Chebyshev Approximations         
+     Root-Finding                     Discrete Hankel Transforms       
+     Least-Squares Fitting            Minimization                     
+     IEEE Floating-Point              Physical Constants               
+     Wavelets                                                          
+
+The use of these routines is described in this manual.  Each chapter
+provides detailed definitions of the functions, followed by example
+programs and references to the articles on which the algorithms are
+based.
+
+   Where possible the routines have been based on reliable public-domain
+packages such as FFTPACK and QUADPACK, which the developers of GSL have
+reimplemented in C with modern coding conventions.
+
+
+File: gsl-ref.info,  Node: GSL is Free Software,  Next: Obtaining GSL,  Prev: Routines available in GSL,  Up: Introduction
+
+1.2 GSL is Free Software
+========================
+
+The subroutines in the GNU Scientific Library are "free software"; this
+means that everyone is free to use them, and to redistribute them in
+other free programs.  The library is not in the public domain; it is
+copyrighted and there are conditions on its distribution.  These
+conditions are designed to permit everything that a good cooperating
+citizen would want to do.  What is not allowed is to try to prevent
+others from further sharing any version of the software that they might
+get from you.
+
+   Specifically, we want to make sure that you have the right to share
+copies of programs that you are given which use the GNU Scientific
+Library, that you receive their source code or else can get it if you
+want it, that you can change these programs or use pieces of them in new
+free programs, and that you know you can do these things.
+
+   To make sure that everyone has such rights, we have to forbid you to
+deprive anyone else of these rights.  For example, if you distribute
+copies of any code which uses the GNU Scientific Library, you must give
+the recipients all the rights that you have received.  You must make
+sure that they, too, receive or can get the source code, both to the
+library and the code which uses it.  And you must tell them their
+rights.  This means that the library should not be redistributed in
+proprietary programs.
+
+   Also, for our own protection, we must make certain that everyone
+finds out that there is no warranty for the GNU Scientific Library.  If
+these programs are modified by someone else and passed on, we want their
+recipients to know that what they have is not what we distributed, so
+that any problems introduced by others will not reflect on our
+reputation.
+
+   The precise conditions for the distribution of software related to
+the GNU Scientific Library are found in the GNU General Public License
+(*note GNU General Public License::).  Further information about this
+license is available from the GNU Project webpage `Frequently Asked
+Questions about the GNU GPL',
+
+     `http://www.gnu.org/copyleft/gpl-faq.html'
+
+The Free Software Foundation also operates a license consulting service
+for commercial users (contact details available from
+`http://www.fsf.org/').
+
+
+File: gsl-ref.info,  Node: Obtaining GSL,  Next: No Warranty,  Prev: GSL is Free Software,  Up: Introduction
+
+1.3 Obtaining GSL
+=================
+
+The source code for the library can be obtained in different ways, by
+copying it from a friend, purchasing it on CDROM or downloading it from
+the internet. A list of public ftp servers which carry the source code
+can be found on the GNU website,
+
+     `http://www.gnu.org/software/gsl/'
+
+The preferred platform for the library is a GNU system, which allows it
+to take advantage of additional features in the GNU C compiler and GNU C
+library.  However, the library is fully portable and should compile on
+most systems with a C compiler.  Precompiled versions of the library
+can be purchased from commercial redistributors listed on the website
+above.
+
+   Announcements of new releases, updates and other relevant events are
+made on the `info-gsl@gnu.org' mailing list.  To subscribe to this
+low-volume list, send an email of the following form:
+
+     To: info-gsl-request@gnu.org
+     Subject: subscribe
+
+You will receive a response asking you to reply in order to confirm
+your subscription.
+
+
+File: gsl-ref.info,  Node: No Warranty,  Next: Reporting Bugs,  Prev: Obtaining GSL,  Up: Introduction
+
+1.4 No Warranty
+===============
+
+The software described in this manual has no warranty, it is provided
+"as is".  It is your responsibility to validate the behavior of the
+routines and their accuracy using the source code provided, or to
+purchase support and warranties from commercial redistributors.  Consult
+the GNU General Public license for further details (*note GNU General
+Public License::).
+
+
+File: gsl-ref.info,  Node: Reporting Bugs,  Next: Further Information,  Prev: No Warranty,  Up: Introduction
+
+1.5 Reporting Bugs
+==================
+
+A list of known bugs can be found in the `BUGS' file included in the
+GSL distribution.  Details of compilation problems can be found in the
+`INSTALL' file.
+
+   If you find a bug which is not listed in these files, please report
+it to <bug-gsl@gnu.org>.
+
+   All bug reports should include:
+
+   * The version number of GSL
+
+   * The hardware and operating system
+
+   * The compiler used, including version number and compilation options
+
+   * A description of the bug behavior
+
+   * A short program which exercises the bug
+
+It is useful if you can check whether the same problem occurs when the
+library is compiled without optimization.  Thank you.
+
+   Any errors or omissions in this manual can also be reported to the
+same address.
+
+
+File: gsl-ref.info,  Node: Further Information,  Next: Conventions used in this manual,  Prev: Reporting Bugs,  Up: Introduction
+
+1.6 Further Information
+=======================
+
+Additional information, including online copies of this manual, links to
+related projects, and mailing list archives are available from the
+website mentioned above.
+
+   Any questions about the use and installation of the library can be
+asked on the mailing list `help-gsl@gnu.org'.  To subscribe to this
+list, send an email of the following form:
+
+     To: help-gsl-request@gnu.org
+     Subject: subscribe
+
+This mailing list can be used to ask questions not covered by this
+manual, and to contact the developers of the library.
+
+   If you would like to refer to the GNU Scientific Library in a journal
+article, the recommended way is to cite this reference manual, e.g. `M.
+Galassi et al, GNU Scientific Library Reference Manual (2nd Ed.), ISBN
+0954161734'.
+
+   If you want to give a url, use "`http://www.gnu.org/software/gsl/'".
+
+
+File: gsl-ref.info,  Node: Conventions used in this manual,  Prev: Further Information,  Up: Introduction
+
+1.7 Conventions used in this manual
+===================================
+
+This manual contains many examples which can be typed at the keyboard.
+A command entered at the terminal is shown like this,
+
+     $ command
+
+The first character on the line is the terminal prompt, and should not
+be typed.  The dollar sign `$' is used as the standard prompt in this
+manual, although some systems may use a different character.
+
+   The examples assume the use of the GNU operating system.  There may
+be minor differences in the output on other systems.  The commands for
+setting environment variables use the Bourne shell syntax of the
+standard GNU shell (`bash').
+
+
+File: gsl-ref.info,  Node: Using the library,  Next: Error Handling,  Prev: Introduction,  Up: Top
+
+2 Using the library
+*******************
+
+This chapter describes how to compile programs that use GSL, and
+introduces its conventions.
+
+* Menu:
+
+* An Example Program::
+* Compiling and Linking::
+* Shared Libraries::
+* ANSI C Compliance::
+* Inline functions::
+* Long double::
+* Portability functions::
+* Alternative optimized functions::
+* Support for different numeric types::
+* Compatibility with C++::
+* Aliasing of arrays::
+* Thread-safety::
+* Deprecated Functions::
+* Code Reuse::
+
+
+File: gsl-ref.info,  Node: An Example Program,  Next: Compiling and Linking,  Up: Using the library
+
+2.1 An Example Program
+======================
+
+The following short program demonstrates the use of the library by
+computing the value of the Bessel function J_0(x) for x=5,
+
+     #include <stdio.h>
+     #include <gsl/gsl_sf_bessel.h>
+
+     int
+     main (void)
+     {
+       double x = 5.0;
+       double y = gsl_sf_bessel_J0 (x);
+       printf ("J0(%g) = %.18e\n", x, y);
+       return 0;
+     }
+
+The output is shown below, and should be correct to double-precision
+accuracy,
+
+     J0(5) = -1.775967713143382920e-01
+
+The steps needed to compile this program are described in the following
+sections.
+
+
+File: gsl-ref.info,  Node: Compiling and Linking,  Next: Shared Libraries,  Prev: An Example Program,  Up: Using the library
+
+2.2 Compiling and Linking
+=========================
+
+The library header files are installed in their own `gsl' directory.
+You should write any preprocessor include statements with a `gsl/'
+directory prefix thus,
+
+     #include <gsl/gsl_math.h>
+
+If the directory is not installed on the standard search path of your
+compiler you will also need to provide its location to the preprocessor
+as a command line flag.  The default location of the `gsl' directory is
+`/usr/local/include/gsl'.  A typical compilation command for a source
+file `example.c' with the GNU C compiler `gcc' is,
+
+     $ gcc -Wall -I/usr/local/include -c example.c
+
+This results in an object file `example.o'.   The default include path
+for `gcc' searches `/usr/local/include' automatically so the `-I'
+option can actually be omitted when GSL is installed in its default
+location.
+
+* Menu:
+
+* Linking programs with the library::
+* Linking with an alternative BLAS library::
+
+
+File: gsl-ref.info,  Node: Linking programs with the library,  Next: Linking with an alternative BLAS library,  Up: Compiling and Linking
+
+2.2.1 Linking programs with the library
+---------------------------------------
+
+The library is installed as a single file, `libgsl.a'.  A shared
+version of the library `libgsl.so' is also installed on systems that
+support shared libraries.  The default location of these files is
+`/usr/local/lib'.  If this directory is not on the standard search path
+of your linker you will also need to provide its location as a command
+line flag.
+
+   To link against the library you need to specify both the main
+library and a supporting CBLAS library, which provides standard basic
+linear algebra subroutines.  A suitable CBLAS implementation is
+provided in the library `libgslcblas.a' if your system does not provide
+one.  The following example shows how to link an application with the
+library,
+
+     $ gcc -L/usr/local/lib example.o -lgsl -lgslcblas -lm
+
+The default library path for `gcc' searches `/usr/local/lib'
+automatically so the `-L' option can be omitted when GSL is installed
+in its default location.
+
+
+File: gsl-ref.info,  Node: Linking with an alternative BLAS library,  Prev: Linking programs with the library,  Up: Compiling and Linking
+
+2.2.2 Linking with an alternative BLAS library
+----------------------------------------------
+
+The following command line shows how you would link the same application
+with an alternative CBLAS library called `libcblas',
+
+     $ gcc example.o -lgsl -lcblas -lm
+
+For the best performance an optimized platform-specific CBLAS library
+should be used for `-lcblas'.  The library must conform to the CBLAS
+standard.  The ATLAS package provides a portable high-performance BLAS
+library with a CBLAS interface.  It is free software and should be
+installed for any work requiring fast vector and matrix operations.
+The following command line will link with the ATLAS library and its
+CBLAS interface,
+
+     $ gcc example.o -lgsl -lcblas -latlas -lm
+
+For more information see *Note BLAS Support::.
+
+
+File: gsl-ref.info,  Node: Shared Libraries,  Next: ANSI C Compliance,  Prev: Compiling and Linking,  Up: Using the library
+
+2.3 Shared Libraries
+====================
+
+To run a program linked with the shared version of the library the
+operating system must be able to locate the corresponding `.so' file at
+runtime.  If the library cannot be found, the following error will
+occur:
+
+     $ ./a.out
+     ./a.out: error while loading shared libraries:
+     libgsl.so.0: cannot open shared object file: No such
+     file or directory
+
+To avoid this error, define the shell variable `LD_LIBRARY_PATH' to
+include the directory where the library is installed.
+
+   For example, in the Bourne shell (`/bin/sh' or `/bin/bash'), the
+library search path can be set with the following commands:
+
+     $ LD_LIBRARY_PATH=/usr/local/lib:$LD_LIBRARY_PATH
+     $ export LD_LIBRARY_PATH
+     $ ./example
+
+In the C-shell (`/bin/csh' or `/bin/tcsh') the equivalent command is,
+
+     % setenv LD_LIBRARY_PATH /usr/local/lib:$LD_LIBRARY_PATH
+
+The standard prompt for the C-shell in the example above is the percent
+character `%', and should not be typed as part of the command.
+
+   To save retyping these commands each session they should be placed
+in an individual or system-wide login file.
+
+   To compile a statically linked version of the program, use the
+`-static' flag in `gcc',
+
+     $ gcc -static example.o -lgsl -lgslcblas -lm
+
+
+File: gsl-ref.info,  Node: ANSI C Compliance,  Next: Inline functions,  Prev: Shared Libraries,  Up: Using the library
+
+2.4 ANSI C Compliance
+=====================
+
+The library is written in ANSI C and is intended to conform to the ANSI
+C standard (C89).  It should be portable to any system with a working
+ANSI C compiler.
+
+   The library does not rely on any non-ANSI extensions in the
+interface it exports to the user.  Programs you write using GSL can be
+ANSI compliant.  Extensions which can be used in a way compatible with
+pure ANSI C are supported, however, via conditional compilation.  This
+allows the library to take advantage of compiler extensions on those
+platforms which support them.
+
+   When an ANSI C feature is known to be broken on a particular system
+the library will exclude any related functions at compile-time.  This
+should make it impossible to link a program that would use these
+functions and give incorrect results.
+
+   To avoid namespace conflicts all exported function names and
+variables have the prefix `gsl_', while exported macros have the prefix
+`GSL_'.
+
+
+File: gsl-ref.info,  Node: Inline functions,  Next: Long double,  Prev: ANSI C Compliance,  Up: Using the library
+
+2.5 Inline functions
+====================
+
+The `inline' keyword is not part of the original ANSI C standard (C89)
+and the library does not export any inline function definitions by
+default. However, the library provides optional inline versions of
+performance-critical functions by conditional compilation.  The inline
+versions of these functions can be included by defining the macro
+`HAVE_INLINE' when compiling an application,
+
+     $ gcc -Wall -c -DHAVE_INLINE example.c
+
+If you use `autoconf' this macro can be defined automatically.  If you
+do not define the macro `HAVE_INLINE' then the slower non-inlined
+versions of the functions will be used instead.
+
+   Note that the actual usage of the inline keyword is `extern inline',
+which eliminates unnecessary function definitions in GCC.  If the form
+`extern inline' causes problems with other compilers a stricter
+autoconf test can be used, see *Note Autoconf Macros::.
+
+
+File: gsl-ref.info,  Node: Long double,  Next: Portability functions,  Prev: Inline functions,  Up: Using the library
+
+2.6 Long double
+===============
+
+The extended numerical type `long double' is part of the ANSI C
+standard and should be available in every modern compiler.  However, the
+precision of `long double' is platform dependent, and this should be
+considered when using it.  The IEEE standard only specifies the minimum
+precision of extended precision numbers, while the precision of
+`double' is the same on all platforms.
+
+   In some system libraries the `stdio.h' formatted input/output
+functions `printf' and `scanf' are not implemented correctly for `long
+double'.  Undefined or incorrect results are avoided by testing these
+functions during the `configure' stage of library compilation and
+eliminating certain GSL functions which depend on them if necessary.
+The corresponding line in the `configure' output looks like this,
+
+     checking whether printf works with long double... no
+
+Consequently when `long double' formatted input/output does not work on
+a given system it should be impossible to link a program which uses GSL
+functions dependent on this.
+
+   If it is necessary to work on a system which does not support
+formatted `long double' input/output then the options are to use binary
+formats or to convert `long double' results into `double' for reading
+and writing.
+
+
+File: gsl-ref.info,  Node: Portability functions,  Next: Alternative optimized functions,  Prev: Long double,  Up: Using the library
+
+2.7 Portability functions
+=========================
+
+To help in writing portable applications GSL provides some
+implementations of functions that are found in other libraries, such as
+the BSD math library.  You can write your application to use the native
+versions of these functions, and substitute the GSL versions via a
+preprocessor macro if they are unavailable on another platform.
+
+   For example, after determining whether the BSD function `hypot' is
+available you can include the following macro definitions in a file
+`config.h' with your application,
+
+     /* Substitute gsl_hypot for missing system hypot */
+
+     #ifndef HAVE_HYPOT
+     #define hypot gsl_hypot
+     #endif
+
+The application source files can then use the include command `#include
+<config.h>' to replace each occurrence of `hypot' by `gsl_hypot' when
+`hypot' is not available.  This substitution can be made automatically
+if you use `autoconf', see *Note Autoconf Macros::.
+
+   In most circumstances the best strategy is to use the native
+versions of these functions when available, and fall back to GSL
+versions otherwise, since this allows your application to take
+advantage of any platform-specific optimizations in the system library.
+This is the strategy used within GSL itself.
+
+
+File: gsl-ref.info,  Node: Alternative optimized functions,  Next: Support for different numeric types,  Prev: Portability functions,  Up: Using the library
+
+2.8 Alternative optimized functions
+===================================
+
+The main implementation of some functions in the library will not be
+optimal on all architectures.  For example, there are several ways to
+compute a Gaussian random variate and their relative speeds are
+platform-dependent.  In cases like this the library provides alternative
+implementations of these functions with the same interface.  If you
+write your application using calls to the standard implementation you
+can select an alternative version later via a preprocessor definition.
+It is also possible to introduce your own optimized functions this way
+while retaining portability.  The following lines demonstrate the use of
+a platform-dependent choice of methods for sampling from the Gaussian
+distribution,
+
+     #ifdef SPARC
+     #define gsl_ran_gaussian gsl_ran_gaussian_ratio_method
+     #endif
+     #ifdef INTEL
+     #define gsl_ran_gaussian my_gaussian
+     #endif
+
+These lines would be placed in the configuration header file `config.h'
+of the application, which should then be included by all the source
+files.  Note that the alternative implementations will not produce
+bit-for-bit identical results, and in the case of random number
+distributions will produce an entirely different stream of random
+variates.
+
+
+File: gsl-ref.info,  Node: Support for different numeric types,  Next: Compatibility with C++,  Prev: Alternative optimized functions,  Up: Using the library
+
+2.9 Support for different numeric types
+=======================================
+
+Many functions in the library are defined for different numeric types.
+This feature is implemented by varying the name of the function with a
+type-related modifier--a primitive form of C++ templates.  The modifier
+is inserted into the function name after the initial module prefix.
+The following table shows the function names defined for all the
+numeric types of an imaginary module `gsl_foo' with function `fn',
+
+     gsl_foo_fn               double
+     gsl_foo_long_double_fn   long double
+     gsl_foo_float_fn         float
+     gsl_foo_long_fn          long
+     gsl_foo_ulong_fn         unsigned long
+     gsl_foo_int_fn           int
+     gsl_foo_uint_fn          unsigned int
+     gsl_foo_short_fn         short
+     gsl_foo_ushort_fn        unsigned short
+     gsl_foo_char_fn          char
+     gsl_foo_uchar_fn         unsigned char
+
+The normal numeric precision `double' is considered the default and
+does not require a suffix.  For example, the function `gsl_stats_mean'
+computes the mean of double precision numbers, while the function
+`gsl_stats_int_mean' computes the mean of integers.
+
+   A corresponding scheme is used for library defined types, such as
+`gsl_vector' and `gsl_matrix'.  In this case the modifier is appended
+to the type name.  For example, if a module defines a new
+type-dependent struct or typedef `gsl_foo' it is modified for other
+types in the following way,
+
+     gsl_foo                  double
+     gsl_foo_long_double      long double
+     gsl_foo_float            float
+     gsl_foo_long             long
+     gsl_foo_ulong            unsigned long
+     gsl_foo_int              int
+     gsl_foo_uint             unsigned int
+     gsl_foo_short            short
+     gsl_foo_ushort           unsigned short
+     gsl_foo_char             char
+     gsl_foo_uchar            unsigned char
+
+When a module contains type-dependent definitions the library provides
+individual header files for each type.  The filenames are modified as
+shown in the below.  For convenience the default header includes the
+definitions for all the types.  To include only the double precision
+header file, or any other specific type, use its individual filename.
+
+     #include <gsl/gsl_foo.h>               All types
+     #include <gsl/gsl_foo_double.h>        double
+     #include <gsl/gsl_foo_long_double.h>   long double
+     #include <gsl/gsl_foo_float.h>         float
+     #include <gsl/gsl_foo_long.h>          long
+     #include <gsl/gsl_foo_ulong.h>         unsigned long
+     #include <gsl/gsl_foo_int.h>           int
+     #include <gsl/gsl_foo_uint.h>          unsigned int
+     #include <gsl/gsl_foo_short.h>         short
+     #include <gsl/gsl_foo_ushort.h>        unsigned short
+     #include <gsl/gsl_foo_char.h>          char
+     #include <gsl/gsl_foo_uchar.h>         unsigned char
+
+
+File: gsl-ref.info,  Node: Compatibility with C++,  Next: Aliasing of arrays,  Prev: Support for different numeric types,  Up: Using the library
+
+2.10 Compatibility with C++
+===========================
+
+The library header files automatically define functions to have `extern
+"C"' linkage when included in C++ programs.  This allows the functions
+to be called directly from C++.
+
+   To use C++ exception handling within user-defined functions passed to
+the library as parameters, the library must be built with the
+additional `CFLAGS' compilation option `-fexceptions'.
+
+
+File: gsl-ref.info,  Node: Aliasing of arrays,  Next: Thread-safety,  Prev: Compatibility with C++,  Up: Using the library
+
+2.11 Aliasing of arrays
+=======================
+
+The library assumes that arrays, vectors and matrices passed as
+modifiable arguments are not aliased and do not overlap with each other.
+This removes the need for the library to handle overlapping memory
+regions as a special case, and allows additional optimizations to be
+used.  If overlapping memory regions are passed as modifiable arguments
+then the results of such functions will be undefined.  If the arguments
+will not be modified (for example, if a function prototype declares them
+as `const' arguments) then overlapping or aliased memory regions can be
+safely used.
+
+
+File: gsl-ref.info,  Node: Thread-safety,  Next: Deprecated Functions,  Prev: Aliasing of arrays,  Up: Using the library
+
+2.12 Thread-safety
+==================
+
+The library can be used in multi-threaded programs.  All the functions
+are thread-safe, in the sense that they do not use static variables.
+Memory is always associated with objects and not with functions.  For
+functions which use "workspace" objects as temporary storage the
+workspaces should be allocated on a per-thread basis.  For functions
+which use "table" objects as read-only memory the tables can be used by
+multiple threads simultaneously.  Table arguments are always declared
+`const' in function prototypes, to indicate that they may be safely
+accessed by different threads.
+
+   There are a small number of static global variables which are used to
+control the overall behavior of the library (e.g. whether to use
+range-checking, the function to call on fatal error, etc).  These
+variables are set directly by the user, so they should be initialized
+once at program startup and not modified by different threads.
+
+
+File: gsl-ref.info,  Node: Deprecated Functions,  Next: Code Reuse,  Prev: Thread-safety,  Up: Using the library
+
+2.13 Deprecated Functions
+=========================
+
+From time to time, it may be necessary for the definitions of some
+functions to be altered or removed from the library.  In these
+circumstances the functions will first be declared "deprecated" and
+then removed from subsequent versions of the library.  Functions that
+are deprecated can be disabled in the current release by setting the
+preprocessor definition `GSL_DISABLE_DEPRECATED'.  This allows existing
+code to be tested for forwards compatibility.
+
+
+File: gsl-ref.info,  Node: Code Reuse,  Prev: Deprecated Functions,  Up: Using the library
+
+2.14 Code Reuse
+===============
+
+Where possible the routines in the library have been written to avoid
+dependencies between modules and files.  This should make it possible to
+extract individual functions for use in your own applications, without
+needing to have the whole library installed.  You may need to define
+certain macros such as `GSL_ERROR' and remove some `#include'
+statements in order to compile the files as standalone units. Reuse of
+the library code in this way is encouraged, subject to the terms of the
+GNU General Public License.
+
+
+File: gsl-ref.info,  Node: Error Handling,  Next: Mathematical Functions,  Prev: Using the library,  Up: Top
+
+3 Error Handling
+****************
+
+This chapter describes the way that GSL functions report and handle
+errors.  By examining the status information returned by every function
+you can determine whether it succeeded or failed, and if it failed you
+can find out what the precise cause of failure was.  You can also define
+your own error handling functions to modify the default behavior of the
+library.
+
+   The functions described in this section are declared in the header
+file `gsl_errno.h'.
+
+* Menu:
+
+* Error Reporting::
+* Error Codes::
+* Error Handlers::
+* Using GSL error reporting in your own functions::
+* Error Reporting Examples::
+
+
+File: gsl-ref.info,  Node: Error Reporting,  Next: Error Codes,  Up: Error Handling
+
+3.1 Error Reporting
+===================
+
+The library follows the thread-safe error reporting conventions of the
+POSIX Threads library.  Functions return a non-zero error code to
+indicate an error and `0' to indicate success.
+
+     int status = gsl_function (...)
+
+     if (status) { /* an error occurred */
+       .....
+       /* status value specifies the type of error */
+     }
+
+   The routines report an error whenever they cannot perform the task
+requested of them.  For example, a root-finding function would return a
+non-zero error code if could not converge to the requested accuracy, or
+exceeded a limit on the number of iterations.  Situations like this are
+a normal occurrence when using any mathematical library and you should
+check the return status of the functions that you call.
+
+   Whenever a routine reports an error the return value specifies the
+type of error.  The return value is analogous to the value of the
+variable `errno' in the C library.  The caller can examine the return
+code and decide what action to take, including ignoring the error if it
+is not considered serious.
+
+   In addition to reporting errors by return codes the library also has
+an error handler function `gsl_error'.  This function is called by
+other library functions when they report an error, just before they
+return to the caller.  The default behavior of the error handler is to
+print a message and abort the program,
+
+     gsl: file.c:67: ERROR: invalid argument supplied by user
+     Default GSL error handler invoked.
+     Aborted
+
+   The purpose of the `gsl_error' handler is to provide a function
+where a breakpoint can be set that will catch library errors when
+running under the debugger.  It is not intended for use in production
+programs, which should handle any errors using the return codes.
+
+
+File: gsl-ref.info,  Node: Error Codes,  Next: Error Handlers,  Prev: Error Reporting,  Up: Error Handling
+
+3.2 Error Codes
+===============
+
+The error code numbers returned by library functions are defined in the
+file `gsl_errno.h'.  They all have the prefix `GSL_' and expand to
+non-zero constant integer values. Error codes above 1024 are reserved
+for applications, and are not used by the library.  Many of the error
+codes use the same base name as the corresponding error code in the C
+library.  Here are some of the most common error codes,
+
+ -- Macro: int GSL_EDOM
+     Domain error; used by mathematical functions when an argument
+     value does not fall into the domain over which the function is
+     defined (like EDOM in the C library)
+
+ -- Macro: int GSL_ERANGE
+     Range error; used by mathematical functions when the result value
+     is not representable because of overflow or underflow (like ERANGE
+     in the C library)
+
+ -- Macro: int GSL_ENOMEM
+     No memory available.  The system cannot allocate more virtual
+     memory because its capacity is full (like ENOMEM in the C
+     library).  This error is reported when a GSL routine encounters
+     problems when trying to allocate memory with `malloc'.
+
+ -- Macro: int GSL_EINVAL
+     Invalid argument.  This is used to indicate various kinds of
+     problems with passing the wrong argument to a library function
+     (like EINVAL in the C library).
+
+   The error codes can be converted into an error message using the
+function `gsl_strerror'.
+
+ -- Function: const char * gsl_strerror (const int GSL_ERRNO)
+     This function returns a pointer to a string describing the error
+     code GSL_ERRNO. For example,
+
+          printf ("error: %s\n", gsl_strerror (status));
+
+     would print an error message like `error: output range error' for a
+     status value of `GSL_ERANGE'.
+
+
+File: gsl-ref.info,  Node: Error Handlers,  Next: Using GSL error reporting in your own functions,  Prev: Error Codes,  Up: Error Handling
+
+3.3 Error Handlers
+==================
+
+The default behavior of the GSL error handler is to print a short
+message and call `abort'.  When this default is in use programs will
+stop with a core-dump whenever a library routine reports an error.
+This is intended as a fail-safe default for programs which do not check
+the return status of library routines (we don't encourage you to write
+programs this way).
+
+   If you turn off the default error handler it is your responsibility
+to check the return values of routines and handle them yourself.  You
+can also customize the error behavior by providing a new error handler.
+For example, an alternative error handler could log all errors to a
+file, ignore certain error conditions (such as underflows), or start the
+debugger and attach it to the current process when an error occurs.
+
+   All GSL error handlers have the type `gsl_error_handler_t', which is
+defined in `gsl_errno.h',
+
+ -- Data Type: gsl_error_handler_t
+     This is the type of GSL error handler functions.  An error handler
+     will be passed four arguments which specify the reason for the
+     error (a string), the name of the source file in which it occurred
+     (also a string), the line number in that file (an integer) and the
+     error number (an integer).  The source file and line number are
+     set at compile time using the `__FILE__' and `__LINE__' directives
+     in the preprocessor.  An error handler function returns type
+     `void'.  Error handler functions should be defined like this,
+
+          void handler (const char * reason,
+                        const char * file,
+                        int line,
+                        int gsl_errno)
+
+To request the use of your own error handler you need to call the
+function `gsl_set_error_handler' which is also declared in
+`gsl_errno.h',
+
+ -- Function: gsl_error_handler_t * gsl_set_error_handler
+          (gsl_error_handler_t NEW_HANDLER)
+     This function sets a new error handler, NEW_HANDLER, for the GSL
+     library routines.  The previous handler is returned (so that you
+     can restore it later).  Note that the pointer to a user defined
+     error handler function is stored in a static variable, so there
+     can be only one error handler per program.  This function should
+     be not be used in multi-threaded programs except to set up a
+     program-wide error handler from a master thread.  The following
+     example shows how to set and restore a new error handler,
+
+          /* save original handler, install new handler */
+          old_handler = gsl_set_error_handler (&my_handler);
+
+          /* code uses new handler */
+          .....
+
+          /* restore original handler */
+          gsl_set_error_handler (old_handler);
+
+     To use the default behavior (`abort' on error) set the error
+     handler to `NULL',
+
+          old_handler = gsl_set_error_handler (NULL);
+
+ -- Function: gsl_error_handler_t * gsl_set_error_handler_off ()
+     This function turns off the error handler by defining an error
+     handler which does nothing. This will cause the program to
+     continue after any error, so the return values from any library
+     routines must be checked.  This is the recommended behavior for
+     production programs.  The previous handler is returned (so that
+     you can restore it later).
+
+   The error behavior can be changed for specific applications by
+recompiling the library with a customized definition of the `GSL_ERROR'
+macro in the file `gsl_errno.h'.
+
+
+File: gsl-ref.info,  Node: Using GSL error reporting in your own functions,  Next: Error Reporting Examples,  Prev: Error Handlers,  Up: Error Handling
+
+3.4 Using GSL error reporting in your own functions
+===================================================
+
+If you are writing numerical functions in a program which also uses GSL
+code you may find it convenient to adopt the same error reporting
+conventions as in the library.
+
+   To report an error you need to call the function `gsl_error' with a
+string describing the error and then return an appropriate error code
+from `gsl_errno.h', or a special value, such as `NaN'.  For convenience
+the file `gsl_errno.h' defines two macros which carry out these steps:
+
+ -- Macro: GSL_ERROR (REASON, GSL_ERRNO)
+     This macro reports an error using the GSL conventions and returns a
+     status value of `gsl_errno'.  It expands to the following code
+     fragment,
+
+          gsl_error (reason, __FILE__, __LINE__, gsl_errno);
+          return gsl_errno;
+
+     The macro definition in `gsl_errno.h' actually wraps the code in a
+     `do { ... } while (0)' block to prevent possible parsing problems.
+
+   Here is an example of how the macro could be used to report that a
+routine did not achieve a requested tolerance.  To report the error the
+routine needs to return the error code `GSL_ETOL'.
+
+     if (residual > tolerance)
+       {
+         GSL_ERROR("residual exceeds tolerance", GSL_ETOL);
+       }
+
+ -- Macro: GSL_ERROR_VAL (REASON, GSL_ERRNO, VALUE)
+     This macro is the same as `GSL_ERROR' but returns a user-defined
+     value of VALUE instead of an error code.  It can be used for
+     mathematical functions that return a floating point value.
+
+   The following example shows how to return a `NaN' at a mathematical
+singularity using the `GSL_ERROR_VAL' macro,
+
+     if (x == 0)
+       {
+         GSL_ERROR_VAL("argument lies on singularity",
+                       GSL_ERANGE, GSL_NAN);
+       }
+
+
+File: gsl-ref.info,  Node: Error Reporting Examples,  Prev: Using GSL error reporting in your own functions,  Up: Error Handling
+
+3.5 Examples
+============
+
+Here is an example of some code which checks the return value of a
+function where an error might be reported,
+
+     #include <stdio.h>
+     #include <gsl/gsl_errno.h>
+     #include <gsl/gsl_fft_complex.h>
+
+     ...
+       int status;
+       size_t n = 37;
+
+       gsl_set_error_handler_off();
+
+       status = gsl_fft_complex_radix2_forward (data, n);
+
+       if (status) {
+         if (status == GSL_EINVAL) {
+            fprintf (stderr, "invalid argument, n=%d\n", n);
+         } else {
+            fprintf (stderr, "failed, gsl_errno=%d\n",
+                             status);
+         }
+         exit (-1);
+       }
+     ...
+
+The function `gsl_fft_complex_radix2' only accepts integer lengths
+which are a power of two.  If the variable `n' is not a power of two
+then the call to the library function will return `GSL_EINVAL',
+indicating that the length argument is invalid.  The function call to
+`gsl_set_error_handler_off' stops the default error handler from
+aborting the program.  The `else' clause catches any other possible
+errors.
+
+
+File: gsl-ref.info,  Node: Mathematical Functions,  Next: Complex Numbers,  Prev: Error Handling,  Up: Top
+
+4 Mathematical Functions
+************************
+
+This chapter describes basic mathematical functions.  Some of these
+functions are present in system libraries, but the alternative versions
+given here can be used as a substitute when the system functions are not
+available.
+
+   The functions and macros described in this chapter are defined in the
+header file `gsl_math.h'.
+
+* Menu:
+
+* Mathematical Constants::
+* Infinities and Not-a-number::
+* Elementary Functions::
+* Small integer powers::
+* Testing the Sign of Numbers::
+* Testing for Odd and Even Numbers::
+* Maximum and Minimum functions::
+* Approximate Comparison of Floating Point Numbers::
+
+
+File: gsl-ref.info,  Node: Mathematical Constants,  Next: Infinities and Not-a-number,  Up: Mathematical Functions
+
+4.1 Mathematical Constants
+==========================
+
+The library ensures that the standard BSD mathematical constants are
+defined. For reference, here is a list of the constants:
+
+`M_E'
+     The base of exponentials, e
+
+`M_LOG2E'
+     The base-2 logarithm of e, \log_2 (e)
+
+`M_LOG10E'
+     The base-10 logarithm of e, \log_10 (e)
+
+`M_SQRT2'
+     The square root of two, \sqrt 2
+
+`M_SQRT1_2'
+     The square root of one-half, \sqrt{1/2}
+
+`M_SQRT3'
+     The square root of three, \sqrt 3
+
+`M_PI'
+     The constant pi, \pi
+
+`M_PI_2'
+     Pi divided by two, \pi/2
+
+`M_PI_4'
+     Pi divided by four, \pi/4
+
+`M_SQRTPI'
+     The square root of pi, \sqrt\pi
+
+`M_2_SQRTPI'
+     Two divided by the square root of pi, 2/\sqrt\pi
+
+`M_1_PI'
+     The reciprocal of pi, 1/\pi
+
+`M_2_PI'
+     Twice the reciprocal of pi, 2/\pi
+
+`M_LN10'
+     The natural logarithm of ten, \ln(10)
+
+`M_LN2'
+     The natural logarithm of two, \ln(2)
+
+`M_LNPI'
+     The natural logarithm of pi, \ln(\pi)
+
+`M_EULER'
+     Euler's constant, \gamma
+
+
+
+File: gsl-ref.info,  Node: Infinities and Not-a-number,  Next: Elementary Functions,  Prev: Mathematical Constants,  Up: Mathematical Functions
+
+4.2 Infinities and Not-a-number
+===============================
+
+ -- Macro: GSL_POSINF
+     This macro contains the IEEE representation of positive infinity,
+     +\infty. It is computed from the expression `+1.0/0.0'.
+
+ -- Macro: GSL_NEGINF
+     This macro contains the IEEE representation of negative infinity,
+     -\infty. It is computed from the expression `-1.0/0.0'.
+
+ -- Macro: GSL_NAN
+     This macro contains the IEEE representation of the Not-a-Number
+     symbol, `NaN'. It is computed from the ratio `0.0/0.0'.
+
+ -- Function: int gsl_isnan (const double X)
+     This function returns 1 if X is not-a-number.
+
+ -- Function: int gsl_isinf (const double X)
+     This function returns +1 if X is positive infinity, -1 if X is
+     negative infinity and 0 otherwise.
+
+ -- Function: int gsl_finite (const double X)
+     This function returns 1 if X is a real number, and 0 if it is
+     infinite or not-a-number.
+
+
+File: gsl-ref.info,  Node: Elementary Functions,  Next: Small integer powers,  Prev: Infinities and Not-a-number,  Up: Mathematical Functions
+
+4.3 Elementary Functions
+========================
+
+The following routines provide portable implementations of functions
+found in the BSD math library.  When native versions are not available
+the functions described here can be used instead.  The substitution can
+be made automatically if you use `autoconf' to compile your application
+(*note Portability functions::).
+
+ -- Function: double gsl_log1p (const double X)
+     This function computes the value of \log(1+x) in a way that is
+     accurate for small X. It provides an alternative to the BSD math
+     function `log1p(x)'.
+
+ -- Function: double gsl_expm1 (const double X)
+     This function computes the value of \exp(x)-1 in a way that is
+     accurate for small X. It provides an alternative to the BSD math
+     function `expm1(x)'.
+
+ -- Function: double gsl_hypot (const double X, const double Y)
+     This function computes the value of \sqrt{x^2 + y^2} in a way that
+     avoids overflow. It provides an alternative to the BSD math
+     function `hypot(x,y)'.
+
+ -- Function: double gsl_acosh (const double X)
+     This function computes the value of \arccosh(x). It provides an
+     alternative to the standard math function `acosh(x)'.
+
+ -- Function: double gsl_asinh (const double X)
+     This function computes the value of \arcsinh(x). It provides an
+     alternative to the standard math function `asinh(x)'.
+
+ -- Function: double gsl_atanh (const double X)
+     This function computes the value of \arctanh(x). It provides an
+     alternative to the standard math function `atanh(x)'.
+
+ -- Function: double gsl_ldexp (double X, int E)
+     This function computes the value of x * 2^e. It provides an
+     alternative to the standard math function `ldexp(x,e)'.
+
+ -- Function: double gsl_frexp (double X, int * E)
+     This function splits the number x into its normalized fraction f
+     and exponent e, such that x = f * 2^e and 0.5 <= f < 1. The
+     function returns f and stores the exponent in e. If x is zero,
+     both f and e are set to zero. This function provides an
+     alternative to the standard math function `frexp(x, e)'.
+
+
+File: gsl-ref.info,  Node: Small integer powers,  Next: Testing the Sign of Numbers,  Prev: Elementary Functions,  Up: Mathematical Functions
+
+4.4 Small integer powers
+========================
+
+A common complaint about the standard C library is its lack of a
+function for calculating (small) integer powers.  GSL provides some
+simple functions to fill this gap.  For reasons of efficiency, these
+functions do not check for overflow or underflow conditions.
+
+ -- Function: double gsl_pow_int (double X, int N)
+     This routine computes the power x^n for integer N.  The power is
+     computed efficiently--for example, x^8 is computed as ((x^2)^2)^2,
+     requiring only 3 multiplications.  A version of this function
+     which also computes the numerical error in the result is available
+     as `gsl_sf_pow_int_e'.
+
+ -- Function: double gsl_pow_2 (const double X)
+ -- Function: double gsl_pow_3 (const double X)
+ -- Function: double gsl_pow_4 (const double X)
+ -- Function: double gsl_pow_5 (const double X)
+ -- Function: double gsl_pow_6 (const double X)
+ -- Function: double gsl_pow_7 (const double X)
+ -- Function: double gsl_pow_8 (const double X)
+ -- Function: double gsl_pow_9 (const double X)
+     These functions can be used to compute small integer powers x^2,
+     x^3, etc. efficiently. The functions will be inlined when possible
+     so that use of these functions should be as efficient as
+     explicitly writing the corresponding product expression.
+
+     #include <gsl/gsl_math.h>
+     double y = gsl_pow_4 (3.141)  /* compute 3.141**4 */
+
+
+File: gsl-ref.info,  Node: Testing the Sign of Numbers,  Next: Testing for Odd and Even Numbers,  Prev: Small integer powers,  Up: Mathematical Functions
+
+4.5 Testing the Sign of Numbers
+===============================
+
+ -- Macro: GSL_SIGN (x)
+     This macro returns the sign of X. It is defined as `((x) >= 0 ? 1
+     : -1)'. Note that with this definition the sign of zero is positive
+     (regardless of its IEEE sign bit).
+
+
+File: gsl-ref.info,  Node: Testing for Odd and Even Numbers,  Next: Maximum and Minimum functions,  Prev: Testing the Sign of Numbers,  Up: Mathematical Functions
+
+4.6 Testing for Odd and Even Numbers
+====================================
+
+ -- Macro: GSL_IS_ODD (n)
+     This macro evaluates to 1 if N is odd and 0 if N is even. The
+     argument N must be of integer type.
+
+ -- Macro: GSL_IS_EVEN (n)
+     This macro is the opposite of `GSL_IS_ODD(n)'. It evaluates to 1 if
+     N is even and 0 if N is odd. The argument N must be of integer
+     type.
+
+
+File: gsl-ref.info,  Node: Maximum and Minimum functions,  Next: Approximate Comparison of Floating Point Numbers,  Prev: Testing for Odd and Even Numbers,  Up: Mathematical Functions
+
+4.7 Maximum and Minimum functions
+=================================
+
+ -- Macro: GSL_MAX (a, b)
+     This macro returns the maximum of A and B. It is defined as `((a)
+     > (b) ? (a):(b))'.
+
+ -- Macro: GSL_MIN (a, b)
+     This macro returns the minimum of A and B. It is defined as `((a)
+     < (b) ? (a):(b))'.
+
+ -- Function: extern inline double GSL_MAX_DBL (double A, double B)
+     This function returns the maximum of the double precision numbers
+     A and B using an inline function. The use of a function allows for
+     type checking of the arguments as an extra safety feature. On
+     platforms where inline functions are not available the macro
+     `GSL_MAX' will be automatically substituted.
+
+ -- Function: extern inline double GSL_MIN_DBL (double A, double B)
+     This function returns the minimum of the double precision numbers
+     A and B using an inline function. The use of a function allows for
+     type checking of the arguments as an extra safety feature. On
+     platforms where inline functions are not available the macro
+     `GSL_MIN' will be automatically substituted.
+
+ -- Function: extern inline int GSL_MAX_INT (int A, int B)
+ -- Function: extern inline int GSL_MIN_INT (int A, int B)
+     These functions return the maximum or minimum of the integers A
+     and B using an inline function.  On platforms where inline
+     functions are not available the macros `GSL_MAX' or `GSL_MIN' will
+     be automatically substituted.
+
+ -- Function: extern inline long double GSL_MAX_LDBL (long double A,
+          long double B)
+ -- Function: extern inline long double GSL_MIN_LDBL (long double A,
+          long double B)
+     These functions return the maximum or minimum of the long doubles A
+     and B using an inline function.  On platforms where inline
+     functions are not available the macros `GSL_MAX' or `GSL_MIN' will
+     be automatically substituted.
+
+
+File: gsl-ref.info,  Node: Approximate Comparison of Floating Point Numbers,  Prev: Maximum and Minimum functions,  Up: Mathematical Functions
+
+4.8 Approximate Comparison of Floating Point Numbers
+====================================================
+
+It is sometimes useful to be able to compare two floating point numbers
+approximately, to allow for rounding and truncation errors.  The
+following function implements the approximate floating-point comparison
+algorithm proposed by D.E. Knuth in Section 4.2.2 of `Seminumerical
+Algorithms' (3rd edition).
+
+ -- Function: int gsl_fcmp (double X, double Y, double EPSILON)
+     This function determines whether x and y are approximately equal
+     to a relative accuracy EPSILON.
+
+     The relative accuracy is measured using an interval of size 2
+     \delta, where \delta = 2^k \epsilon and k is the maximum base-2
+     exponent of x and y as computed by the function `frexp'.
+
+     If x and y lie within this interval, they are considered
+     approximately equal and the function returns 0. Otherwise if x <
+     y, the function returns -1, or if x > y, the function returns +1.
+
+     The implementation is based on the package `fcmp' by T.C. Belding.
+
+
+File: gsl-ref.info,  Node: Complex Numbers,  Next: Polynomials,  Prev: Mathematical Functions,  Up: Top
+
+5 Complex Numbers
+*****************
+
+The functions described in this chapter provide support for complex
+numbers.  The algorithms take care to avoid unnecessary intermediate
+underflows and overflows, allowing the functions to be evaluated over
+as much of the complex plane as possible.
+
+   For multiple-valued functions the branch cuts have been chosen to
+follow the conventions of Abramowitz and Stegun in the `Handbook of
+Mathematical Functions'. The functions return principal values which are
+the same as those in GNU Calc, which in turn are the same as those in
+`Common Lisp, The Language (Second Edition)'(1) and the HP-28/48 series
+of calculators.
+
+   The complex types are defined in the header file `gsl_complex.h',
+while the corresponding complex functions and arithmetic operations are
+defined in `gsl_complex_math.h'.
+
+* Menu:
+
+* Complex numbers::
+* Properties of complex numbers::
+* Complex arithmetic operators::
+* Elementary Complex Functions::
+* Complex Trigonometric Functions::
+* Inverse Complex Trigonometric Functions::
+* Complex Hyperbolic Functions::
+* Inverse Complex Hyperbolic Functions::
+* Complex Number References and Further Reading::
+
+   ---------- Footnotes ----------
+
+   (1) Note that the first edition uses different definitions.
+
+
+File: gsl-ref.info,  Node: Complex numbers,  Next: Properties of complex numbers,  Up: Complex Numbers
+
+5.1 Complex numbers
+===================
+
+Complex numbers are represented using the type `gsl_complex'. The
+internal representation of this type may vary across platforms and
+should not be accessed directly. The functions and macros described
+below allow complex numbers to be manipulated in a portable way.
+
+   For reference, the default form of the `gsl_complex' type is given
+by the following struct,
+
+     typedef struct
+     {
+       double dat[2];
+     } gsl_complex;
+
+The real and imaginary part are stored in contiguous elements of a two
+element array. This eliminates any padding between the real and
+imaginary parts, `dat[0]' and `dat[1]', allowing the struct to be
+mapped correctly onto packed complex arrays.
+
+ -- Function: gsl_complex gsl_complex_rect (double X, double Y)
+     This function uses the rectangular cartesian components (X,Y) to
+     return the complex number z = x + i y.
+
+ -- Function: gsl_complex gsl_complex_polar (double R, double THETA)
+     This function returns the complex number z = r \exp(i \theta) = r
+     (\cos(\theta) + i \sin(\theta)) from the polar representation
+     (R,THETA).
+
+ -- Macro: GSL_REAL (Z)
+ -- Macro: GSL_IMAG (Z)
+     These macros return the real and imaginary parts of the complex
+     number Z.
+
+ -- Macro: GSL_SET_COMPLEX (ZP, X, Y)
+     This macro uses the cartesian components (X,Y) to set the real and
+     imaginary parts of the complex number pointed to by ZP.  For
+     example,
+
+          GSL_SET_COMPLEX(&z, 3, 4)
+
+     sets Z to be 3 + 4i.
+
+ -- Macro: GSL_SET_REAL (ZP,X)
+ -- Macro: GSL_SET_IMAG (ZP,Y)
+     These macros allow the real and imaginary parts of the complex
+     number pointed to by ZP to be set independently.
+
+
+File: gsl-ref.info,  Node: Properties of complex numbers,  Next: Complex arithmetic operators,  Prev: Complex numbers,  Up: Complex Numbers
+
+5.2 Properties of complex numbers
+=================================
+
+ -- Function: double gsl_complex_arg (gsl_complex Z)
+     This function returns the argument of the complex number Z,
+     \arg(z), where -\pi < \arg(z) <= \pi.
+
+ -- Function: double gsl_complex_abs (gsl_complex Z)
+     This function returns the magnitude of the complex number Z, |z|.
+
+ -- Function: double gsl_complex_abs2 (gsl_complex Z)
+     This function returns the squared magnitude of the complex number
+     Z, |z|^2.
+
+ -- Function: double gsl_complex_logabs (gsl_complex Z)
+     This function returns the natural logarithm of the magnitude of the
+     complex number Z, \log|z|.  It allows an accurate evaluation of
+     \log|z| when |z| is close to one. The direct evaluation of
+     `log(gsl_complex_abs(z))' would lead to a loss of precision in
+     this case.
+
+
+File: gsl-ref.info,  Node: Complex arithmetic operators,  Next: Elementary Complex Functions,  Prev: Properties of complex numbers,  Up: Complex Numbers
+
+5.3 Complex arithmetic operators
+================================
+
+ -- Function: gsl_complex gsl_complex_add (gsl_complex A, gsl_complex B)
+     This function returns the sum of the complex numbers A and B,
+     z=a+b.
+
+ -- Function: gsl_complex gsl_complex_sub (gsl_complex A, gsl_complex B)
+     This function returns the difference of the complex numbers A and
+     B, z=a-b.
+
+ -- Function: gsl_complex gsl_complex_mul (gsl_complex A, gsl_complex B)
+     This function returns the product of the complex numbers A and B,
+     z=ab.
+
+ -- Function: gsl_complex gsl_complex_div (gsl_complex A, gsl_complex B)
+     This function returns the quotient of the complex numbers A and B,
+     z=a/b.
+
+ -- Function: gsl_complex gsl_complex_add_real (gsl_complex A, double X)
+     This function returns the sum of the complex number A and the real
+     number X, z=a+x.
+
+ -- Function: gsl_complex gsl_complex_sub_real (gsl_complex A, double X)
+     This function returns the difference of the complex number A and
+     the real number X, z=a-x.
+
+ -- Function: gsl_complex gsl_complex_mul_real (gsl_complex A, double X)
+     This function returns the product of the complex number A and the
+     real number X, z=ax.
+
+ -- Function: gsl_complex gsl_complex_div_real (gsl_complex A, double X)
+     This function returns the quotient of the complex number A and the
+     real number X, z=a/x.
+
+ -- Function: gsl_complex gsl_complex_add_imag (gsl_complex A, double Y)
+     This function returns the sum of the complex number A and the
+     imaginary number iY, z=a+iy.
+
+ -- Function: gsl_complex gsl_complex_sub_imag (gsl_complex A, double Y)
+     This function returns the difference of the complex number A and
+     the imaginary number iY, z=a-iy.
+
+ -- Function: gsl_complex gsl_complex_mul_imag (gsl_complex A, double Y)
+     This function returns the product of the complex number A and the
+     imaginary number iY, z=a*(iy).
+
+ -- Function: gsl_complex gsl_complex_div_imag (gsl_complex A, double Y)
+     This function returns the quotient of the complex number A and the
+     imaginary number iY, z=a/(iy).
+
+ -- Function: gsl_complex gsl_complex_conjugate (gsl_complex Z)
+     This function returns the complex conjugate of the complex number
+     Z, z^* = x - i y.
+
+ -- Function: gsl_complex gsl_complex_inverse (gsl_complex Z)
+     This function returns the inverse, or reciprocal, of the complex
+     number Z, 1/z = (x - i y)/(x^2 + y^2).
+
+ -- Function: gsl_complex gsl_complex_negative (gsl_complex Z)
+     This function returns the negative of the complex number Z, -z =
+     (-x) + i(-y).
+
+
+File: gsl-ref.info,  Node: Elementary Complex Functions,  Next: Complex Trigonometric Functions,  Prev: Complex arithmetic operators,  Up: Complex Numbers
+
+5.4 Elementary Complex Functions
+================================
+
+ -- Function: gsl_complex gsl_complex_sqrt (gsl_complex Z)
+     This function returns the square root of the complex number Z,
+     \sqrt z. The branch cut is the negative real axis. The result
+     always lies in the right half of the complex plane.
+
+ -- Function: gsl_complex gsl_complex_sqrt_real (double X)
+     This function returns the complex square root of the real number
+     X, where X may be negative.
+
+ -- Function: gsl_complex gsl_complex_pow (gsl_complex Z, gsl_complex A)
+     The function returns the complex number Z raised to the complex
+     power A, z^a. This is computed as \exp(\log(z)*a) using complex
+     logarithms and complex exponentials.
+
+ -- Function: gsl_complex gsl_complex_pow_real (gsl_complex Z, double X)
+     This function returns the complex number Z raised to the real
+     power X, z^x.
+
+ -- Function: gsl_complex gsl_complex_exp (gsl_complex Z)
+     This function returns the complex exponential of the complex number
+     Z, \exp(z).
+
+ -- Function: gsl_complex gsl_complex_log (gsl_complex Z)
+     This function returns the complex natural logarithm (base e) of
+     the complex number Z, \log(z).  The branch cut is the negative
+     real axis.
+
+ -- Function: gsl_complex gsl_complex_log10 (gsl_complex Z)
+     This function returns the complex base-10 logarithm of the complex
+     number Z, \log_10 (z).
+
+ -- Function: gsl_complex gsl_complex_log_b (gsl_complex Z, gsl_complex
+          B)
+     This function returns the complex base-B logarithm of the complex
+     number Z, \log_b(z). This quantity is computed as the ratio
+     \log(z)/\log(b).
+
+
+File: gsl-ref.info,  Node: Complex Trigonometric Functions,  Next: Inverse Complex Trigonometric Functions,  Prev: Elementary Complex Functions,  Up: Complex Numbers
+
+5.5 Complex Trigonometric Functions
+===================================
+
+ -- Function: gsl_complex gsl_complex_sin (gsl_complex Z)
+     This function returns the complex sine of the complex number Z,
+     \sin(z) = (\exp(iz) - \exp(-iz))/(2i).
+
+ -- Function: gsl_complex gsl_complex_cos (gsl_complex Z)
+     This function returns the complex cosine of the complex number Z,
+     \cos(z) = (\exp(iz) + \exp(-iz))/2.
+
+ -- Function: gsl_complex gsl_complex_tan (gsl_complex Z)
+     This function returns the complex tangent of the complex number Z,
+     \tan(z) = \sin(z)/\cos(z).
+
+ -- Function: gsl_complex gsl_complex_sec (gsl_complex Z)
+     This function returns the complex secant of the complex number Z,
+     \sec(z) = 1/\cos(z).
+
+ -- Function: gsl_complex gsl_complex_csc (gsl_complex Z)
+     This function returns the complex cosecant of the complex number Z,
+     \csc(z) = 1/\sin(z).
+
+ -- Function: gsl_complex gsl_complex_cot (gsl_complex Z)
+     This function returns the complex cotangent of the complex number
+     Z, \cot(z) = 1/\tan(z).
+
+
+File: gsl-ref.info,  Node: Inverse Complex Trigonometric Functions,  Next: Complex Hyperbolic Functions,  Prev: Complex Trigonometric Functions,  Up: Complex Numbers
+
+5.6 Inverse Complex Trigonometric Functions
+===========================================
+
+ -- Function: gsl_complex gsl_complex_arcsin (gsl_complex Z)
+     This function returns the complex arcsine of the complex number Z,
+     \arcsin(z). The branch cuts are on the real axis, less than -1 and
+     greater than 1.
+
+ -- Function: gsl_complex gsl_complex_arcsin_real (double Z)
+     This function returns the complex arcsine of the real number Z,
+     \arcsin(z). For z between -1 and 1, the function returns a real
+     value in the range [-\pi/2,\pi/2]. For z less than -1 the result
+     has a real part of -\pi/2 and a positive imaginary part.  For z
+     greater than 1 the result has a real part of \pi/2 and a negative
+     imaginary part.
+
+ -- Function: gsl_complex gsl_complex_arccos (gsl_complex Z)
+     This function returns the complex arccosine of the complex number
+     Z, \arccos(z). The branch cuts are on the real axis, less than -1
+     and greater than 1.
+
+ -- Function: gsl_complex gsl_complex_arccos_real (double Z)
+     This function returns the complex arccosine of the real number Z,
+     \arccos(z). For z between -1 and 1, the function returns a real
+     value in the range [0,\pi]. For z less than -1 the result has a
+     real part of \pi and a negative imaginary part.  For z greater
+     than 1 the result is purely imaginary and positive.
+
+ -- Function: gsl_complex gsl_complex_arctan (gsl_complex Z)
+     This function returns the complex arctangent of the complex number
+     Z, \arctan(z). The branch cuts are on the imaginary axis, below -i
+     and above i.
+
+ -- Function: gsl_complex gsl_complex_arcsec (gsl_complex Z)
+     This function returns the complex arcsecant of the complex number
+     Z, \arcsec(z) = \arccos(1/z).
+
+ -- Function: gsl_complex gsl_complex_arcsec_real (double Z)
+     This function returns the complex arcsecant of the real number Z,
+     \arcsec(z) = \arccos(1/z).
+
+ -- Function: gsl_complex gsl_complex_arccsc (gsl_complex Z)
+     This function returns the complex arccosecant of the complex
+     number Z, \arccsc(z) = \arcsin(1/z).
+
+ -- Function: gsl_complex gsl_complex_arccsc_real (double Z)
+     This function returns the complex arccosecant of the real number Z,
+     \arccsc(z) = \arcsin(1/z).
+
+ -- Function: gsl_complex gsl_complex_arccot (gsl_complex Z)
+     This function returns the complex arccotangent of the complex
+     number Z, \arccot(z) = \arctan(1/z).
+
+
+File: gsl-ref.info,  Node: Complex Hyperbolic Functions,  Next: Inverse Complex Hyperbolic Functions,  Prev: Inverse Complex Trigonometric Functions,  Up: Complex Numbers
+
+5.7 Complex Hyperbolic Functions
+================================
+
+ -- Function: gsl_complex gsl_complex_sinh (gsl_complex Z)
+     This function returns the complex hyperbolic sine of the complex
+     number Z, \sinh(z) = (\exp(z) - \exp(-z))/2.
+
+ -- Function: gsl_complex gsl_complex_cosh (gsl_complex Z)
+     This function returns the complex hyperbolic cosine of the complex
+     number Z, \cosh(z) = (\exp(z) + \exp(-z))/2.
+
+ -- Function: gsl_complex gsl_complex_tanh (gsl_complex Z)
+     This function returns the complex hyperbolic tangent of the
+     complex number Z, \tanh(z) = \sinh(z)/\cosh(z).
+
+ -- Function: gsl_complex gsl_complex_sech (gsl_complex Z)
+     This function returns the complex hyperbolic secant of the complex
+     number Z, \sech(z) = 1/\cosh(z).
+
+ -- Function: gsl_complex gsl_complex_csch (gsl_complex Z)
+     This function returns the complex hyperbolic cosecant of the
+     complex number Z, \csch(z) = 1/\sinh(z).
+
+ -- Function: gsl_complex gsl_complex_coth (gsl_complex Z)
+     This function returns the complex hyperbolic cotangent of the
+     complex number Z, \coth(z) = 1/\tanh(z).
+
+
+File: gsl-ref.info,  Node: Inverse Complex Hyperbolic Functions,  Next: Complex Number References and Further Reading,  Prev: Complex Hyperbolic Functions,  Up: Complex Numbers
+
+5.8 Inverse Complex Hyperbolic Functions
+========================================
+
+ -- Function: gsl_complex gsl_complex_arcsinh (gsl_complex Z)
+     This function returns the complex hyperbolic arcsine of the
+     complex number Z, \arcsinh(z).  The branch cuts are on the
+     imaginary axis, below -i and above i.
+
+ -- Function: gsl_complex gsl_complex_arccosh (gsl_complex Z)
+     This function returns the complex hyperbolic arccosine of the
+     complex number Z, \arccosh(z).  The branch cut is on the real
+     axis, less than 1.  Note that in this case we use the negative
+     square root in formula 4.6.21 of Abramowitz & Stegun giving
+     \arccosh(z)=\log(z-\sqrt{z^2-1}).
+
+ -- Function: gsl_complex gsl_complex_arccosh_real (double Z)
+     This function returns the complex hyperbolic arccosine of the real
+     number Z, \arccosh(z).
+
+ -- Function: gsl_complex gsl_complex_arctanh (gsl_complex Z)
+     This function returns the complex hyperbolic arctangent of the
+     complex number Z, \arctanh(z).  The branch cuts are on the real
+     axis, less than -1 and greater than 1.
+
+ -- Function: gsl_complex gsl_complex_arctanh_real (double Z)
+     This function returns the complex hyperbolic arctangent of the real
+     number Z, \arctanh(z).
+
+ -- Function: gsl_complex gsl_complex_arcsech (gsl_complex Z)
+     This function returns the complex hyperbolic arcsecant of the
+     complex number Z, \arcsech(z) = \arccosh(1/z).
+
+ -- Function: gsl_complex gsl_complex_arccsch (gsl_complex Z)
+     This function returns the complex hyperbolic arccosecant of the
+     complex number Z, \arccsch(z) = \arcsin(1/z).
+
+ -- Function: gsl_complex gsl_complex_arccoth (gsl_complex Z)
+     This function returns the complex hyperbolic arccotangent of the
+     complex number Z, \arccoth(z) = \arctanh(1/z).
+
+
+File: gsl-ref.info,  Node: Complex Number References and Further Reading,  Prev: Inverse Complex Hyperbolic Functions,  Up: Complex Numbers
+
+5.9 References and Further Reading
+==================================
+
+The implementations of the elementary and trigonometric functions are
+based on the following papers,
+
+     T. E. Hull, Thomas F. Fairgrieve, Ping Tak Peter Tang,
+     "Implementing Complex Elementary Functions Using Exception
+     Handling", `ACM Transactions on Mathematical Software', Volume 20
+     (1994), pp 215-244, Corrigenda, p553
+
+     T. E. Hull, Thomas F. Fairgrieve, Ping Tak Peter Tang,
+     "Implementing the complex arcsin and arccosine functions using
+     exception handling", `ACM Transactions on Mathematical Software',
+     Volume 23 (1997) pp 299-335
+
+The general formulas and details of branch cuts can be found in the
+following books,
+
+     Abramowitz and Stegun, `Handbook of Mathematical Functions',
+     "Circular Functions in Terms of Real and Imaginary Parts", Formulas
+     4.3.55-58, "Inverse Circular Functions in Terms of Real and
+     Imaginary Parts", Formulas 4.4.37-39, "Hyperbolic Functions in
+     Terms of Real and Imaginary Parts", Formulas 4.5.49-52, "Inverse
+     Hyperbolic Functions--relation to Inverse Circular Functions",
+     Formulas 4.6.14-19.
+
+     Dave Gillespie, `Calc Manual', Free Software Foundation, ISBN
+     1-882114-18-3
+
+
+File: gsl-ref.info,  Node: Polynomials,  Next: Special Functions,  Prev: Complex Numbers,  Up: Top
+
+6 Polynomials
+*************
+
+This chapter describes functions for evaluating and solving polynomials.
+There are routines for finding real and complex roots of quadratic and
+cubic equations using analytic methods.  An iterative polynomial solver
+is also available for finding the roots of general polynomials with real
+coefficients (of any order).  The functions are declared in the header
+file `gsl_poly.h'.
+
+* Menu:
+
+* Polynomial Evaluation::
+* Divided Difference Representation of Polynomials::
+* Quadratic Equations::
+* Cubic Equations::
+* General Polynomial Equations::
+* Roots of Polynomials Examples::
+* Roots of Polynomials References and Further Reading::
+
+
+File: gsl-ref.info,  Node: Polynomial Evaluation,  Next: Divided Difference Representation of Polynomials,  Up: Polynomials
+
+6.1 Polynomial Evaluation
+=========================
+
+ -- Function: double gsl_poly_eval (const double C[], const int LEN,
+          const double X)
+     This function evaluates the polynomial c[0] + c[1] x + c[2] x^2 +
+     \dots + c[len-1] x^{len-1} using Horner's method for stability.
+     The function is inlined when possible.
+
+
+File: gsl-ref.info,  Node: Divided Difference Representation of Polynomials,  Next: Quadratic Equations,  Prev: Polynomial Evaluation,  Up: Polynomials
+
+6.2 Divided Difference Representation of Polynomials
+====================================================
+
+The functions described here manipulate polynomials stored in Newton's
+divided-difference representation.  The use of divided-differences is
+described in Abramowitz & Stegun sections 25.1.4 and 25.2.26.
+
+ -- Function: int gsl_poly_dd_init (double DD[], const double XA[],
+          const double YA[], size_t SIZE)
+     This function computes a divided-difference representation of the
+     interpolating polynomial for the points (XA, YA) stored in the
+     arrays XA and YA of length SIZE.  On output the
+     divided-differences of (XA,YA) are stored in the array DD, also of
+     length SIZE.
+
+ -- Function: double gsl_poly_dd_eval (const double DD[], const double
+          XA[], const size_t SIZE, const double X)
+     This function evaluates the polynomial stored in
+     divided-difference form in the arrays DD and XA of length SIZE at
+     the point X.
+
+ -- Function: int gsl_poly_dd_taylor (double C[], double XP, const
+          double DD[], const double XA[], size_t SIZE, double W[])
+     This function converts the divided-difference representation of a
+     polynomial to a Taylor expansion.  The divided-difference
+     representation is supplied in the arrays DD and XA of length SIZE.
+     On output the Taylor coefficients of the polynomial expanded about
+     the point XP are stored in the array C also of length SIZE.  A
+     workspace of length SIZE must be provided in the array W.
+
+
+File: gsl-ref.info,  Node: Quadratic Equations,  Next: Cubic Equations,  Prev: Divided Difference Representation of Polynomials,  Up: Polynomials
+
+6.3 Quadratic Equations
+=======================
+
+ -- Function: int gsl_poly_solve_quadratic (double A, double B, double
+          C, double * X0, double * X1)
+     This function finds the real roots of the quadratic equation,
+
+          a x^2 + b x + c = 0
+
+     The number of real roots (either zero, one or two) is returned, and
+     their locations are stored in X0 and X1.  If no real roots are
+     found then X0 and X1 are not modified.  If one real root is found
+     (i.e. if a=0) then it is stored in X0.  When two real roots are
+     found they are stored in X0 and X1 in ascending order.  The case
+     of coincident roots is not considered special.  For example
+     (x-1)^2=0 will have two roots, which happen to have exactly equal
+     values.
+
+     The number of roots found depends on the sign of the discriminant
+     b^2 - 4 a c.  This will be subject to rounding and cancellation
+     errors when computed in double precision, and will also be subject
+     to errors if the coefficients of the polynomial are inexact.
+     These errors may cause a discrete change in the number of roots.
+     However, for polynomials with small integer coefficients the
+     discriminant can always be computed exactly.
+
+
+ -- Function: int gsl_poly_complex_solve_quadratic (double A, double B,
+          double C, gsl_complex * Z0, gsl_complex * Z1)
+     This function finds the complex roots of the quadratic equation,
+
+          a z^2 + b z + c = 0
+
+     The number of complex roots is returned (either one or two) and the
+     locations of the roots are stored in Z0 and Z1.  The roots are
+     returned in ascending order, sorted first by their real components
+     and then by their imaginary components.  If only one real root is
+     found (i.e. if a=0) then it is stored in Z0.
+
+
+
+File: gsl-ref.info,  Node: Cubic Equations,  Next: General Polynomial Equations,  Prev: Quadratic Equations,  Up: Polynomials
+
+6.4 Cubic Equations
+===================
+
+ -- Function: int gsl_poly_solve_cubic (double A, double B, double C,
+          double * X0, double * X1, double * X2)
+     This function finds the real roots of the cubic equation,
+
+          x^3 + a x^2 + b x + c = 0
+
+     with a leading coefficient of unity.  The number of real roots
+     (either one or three) is returned, and their locations are stored
+     in X0, X1 and X2.  If one real root is found then only X0 is
+     modified.  When three real roots are found they are stored in X0,
+     X1 and X2 in ascending order.  The case of coincident roots is not
+     considered special.  For example, the equation (x-1)^3=0 will have
+     three roots with exactly equal values.
+
+
+ -- Function: int gsl_poly_complex_solve_cubic (double A, double B,
+          double C, gsl_complex * Z0, gsl_complex * Z1, gsl_complex *
+          Z2)
+     This function finds the complex roots of the cubic equation,
+
+          z^3 + a z^2 + b z + c = 0
+
+     The number of complex roots is returned (always three) and the
+     locations of the roots are stored in Z0, Z1 and Z2.  The roots are
+     returned in ascending order, sorted first by their real components
+     and then by their imaginary components.
+
+
+
+File: gsl-ref.info,  Node: General Polynomial Equations,  Next: Roots of Polynomials Examples,  Prev: Cubic Equations,  Up: Polynomials
+
+6.5 General Polynomial Equations
+================================
+
+The roots of polynomial equations cannot be found analytically beyond
+the special cases of the quadratic, cubic and quartic equation.  The
+algorithm described in this section uses an iterative method to find the
+approximate locations of roots of higher order polynomials.
+
+ -- Function: gsl_poly_complex_workspace *
+gsl_poly_complex_workspace_alloc (size_t N)
+     This function allocates space for a `gsl_poly_complex_workspace'
+     struct and a workspace suitable for solving a polynomial with N
+     coefficients using the routine `gsl_poly_complex_solve'.
+
+     The function returns a pointer to the newly allocated
+     `gsl_poly_complex_workspace' if no errors were detected, and a null
+     pointer in the case of error.
+
+ -- Function: void gsl_poly_complex_workspace_free
+          (gsl_poly_complex_workspace * W)
+     This function frees all the memory associated with the workspace W.
+
+ -- Function: int gsl_poly_complex_solve (const double * A, size_t N,
+          gsl_poly_complex_workspace * W, gsl_complex_packed_ptr Z)
+     This function computes the roots of the general polynomial P(x) =
+     a_0 + a_1 x + a_2 x^2 + ... + a_{n-1} x^{n-1} using balanced-QR
+     reduction of the companion matrix.  The parameter N specifies the
+     length of the coefficient array.  The coefficient of the highest
+     order term must be non-zero.  The function requires a workspace W
+     of the appropriate size.  The n-1 roots are returned in the packed
+     complex array Z of length 2(n-1), alternating real and imaginary
+     parts.
+
+     The function returns `GSL_SUCCESS' if all the roots are found. If
+     the QR reduction does not converge, the error handler is invoked
+     with an error code of `GSL_EFAILED'.  Note that due to finite
+     precision, roots of higher multiplicity are returned as a cluster
+     of simple roots with reduced accuracy.  The solution of
+     polynomials with higher-order roots requires specialized
+     algorithms that take the multiplicity structure into account (see
+     e.g. Z. Zeng, Algorithm 835, ACM Transactions on Mathematical
+     Software, Volume 30, Issue 2 (2004), pp 218-236).
+
+
+File: gsl-ref.info,  Node: Roots of Polynomials Examples,  Next: Roots of Polynomials References and Further Reading,  Prev: General Polynomial Equations,  Up: Polynomials
+
+6.6 Examples
+============
+
+To demonstrate the use of the general polynomial solver we will take the
+polynomial P(x) = x^5 - 1 which has the following roots,
+
+     1, e^{2\pi i /5}, e^{4\pi i /5}, e^{6\pi i /5}, e^{8\pi i /5}
+
+The following program will find these roots.
+
+     #include <stdio.h>
+     #include <gsl/gsl_poly.h>
+
+     int
+     main (void)
+     {
+       int i;
+       /* coefficients of P(x) =  -1 + x^5  */
+       double a[6] = { -1, 0, 0, 0, 0, 1 };
+       double z[10];
+
+       gsl_poly_complex_workspace * w
+           = gsl_poly_complex_workspace_alloc (6);
+
+       gsl_poly_complex_solve (a, 6, w, z);
+
+       gsl_poly_complex_workspace_free (w);
+
+       for (i = 0; i < 5; i++)
+         {
+           printf ("z%d = %+.18f %+.18f\n",
+                   i, z[2*i], z[2*i+1]);
+         }
+
+       return 0;
+     }
+
+The output of the program is,
+
+     $ ./a.out
+     z0 = -0.809016994374947451 +0.587785252292473137
+     z1 = -0.809016994374947451 -0.587785252292473137
+     z2 = +0.309016994374947451 +0.951056516295153642
+     z3 = +0.309016994374947451 -0.951056516295153642
+     z4 = +1.000000000000000000 +0.000000000000000000
+
+which agrees with the analytic result, z_n = \exp(2 \pi n i/5).
+
+
+File: gsl-ref.info,  Node: Roots of Polynomials References and Further Reading,  Prev: Roots of Polynomials Examples,  Up: Polynomials
+
+6.7 References and Further Reading
+==================================
+
+The balanced-QR method and its error analysis are described in the
+following papers,
+
+     R.S. Martin, G. Peters and J.H. Wilkinson, "The QR Algorithm for
+     Real Hessenberg Matrices", `Numerische Mathematik', 14 (1970),
+     219-231.
+
+     B.N. Parlett and C. Reinsch, "Balancing a Matrix for Calculation of
+     Eigenvalues and Eigenvectors", `Numerische Mathematik', 13 (1969),
+     293-304.
+
+     A. Edelman and H. Murakami, "Polynomial roots from companion matrix
+     eigenvalues", `Mathematics of Computation', Vol. 64, No. 210
+     (1995), 763-776.
+
+The formulas for divided differences are given in Abramowitz and Stegun,
+
+     Abramowitz and Stegun, `Handbook of Mathematical Functions',
+     Sections 25.1.4 and 25.2.26.
+
+
+File: gsl-ref.info,  Node: Special Functions,  Next: Vectors and Matrices,  Prev: Polynomials,  Up: Top
+
+7 Special Functions
+*******************
+
+This chapter describes the GSL special function library.  The library
+includes routines for calculating the values of Airy functions, Bessel
+functions, Clausen functions, Coulomb wave functions, Coupling
+coefficients, the Dawson function, Debye functions, Dilogarithms,
+Elliptic integrals, Jacobi elliptic functions, Error functions,
+Exponential integrals, Fermi-Dirac functions, Gamma functions,
+Gegenbauer functions, Hypergeometric functions, Laguerre functions,
+Legendre functions and Spherical Harmonics, the Psi (Digamma) Function,
+Synchrotron functions, Transport functions, Trigonometric functions and
+Zeta functions.  Each routine also computes an estimate of the numerical
+error in the calculated value of the function.
+
+   The functions in this chapter are declared in individual header
+files, such as `gsl_sf_airy.h', `gsl_sf_bessel.h', etc.  The complete
+set of header files can be included using the file `gsl_sf.h'.
+
+* Menu:
+
+* Special Function Usage::
+* The gsl_sf_result struct::
+* Special Function Modes::
+* Airy Functions and Derivatives::
+* Bessel Functions::
+* Clausen Functions::
+* Coulomb Functions::
+* Coupling Coefficients::
+* Dawson Function::
+* Debye Functions::
+* Dilogarithm::
+* Elementary Operations::
+* Elliptic Integrals::
+* Elliptic Functions (Jacobi)::
+* Error Functions::
+* Exponential Functions::
+* Exponential Integrals::
+* Fermi-Dirac Function::
+* Gamma and Beta Functions::
+* Gegenbauer Functions::
+* Hypergeometric Functions::
+* Laguerre Functions::
+* Lambert W Functions::
+* Legendre Functions and Spherical Harmonics::
+* Logarithm and Related Functions::
+* Mathieu Functions::
+* Power Function::
+* Psi (Digamma) Function::
+* Synchrotron Functions::
+* Transport Functions::
+* Trigonometric Functions::
+* Zeta Functions::
+* Special Functions Examples::
+* Special Functions References and Further Reading::
+
+
+File: gsl-ref.info,  Node: Special Function Usage,  Next: The gsl_sf_result struct,  Up: Special Functions
+
+7.1 Usage
+=========
+
+The special functions are available in two calling conventions, a
+"natural form" which returns the numerical value of the function and an
+"error-handling form" which returns an error code.  The two types of
+function provide alternative ways of accessing the same underlying code.
+
+   The "natural form" returns only the value of the function and can be
+used directly in mathematical expressions.  For example, the following
+function call will compute the value of the Bessel function J_0(x),
+
+     double y = gsl_sf_bessel_J0 (x);
+
+There is no way to access an error code or to estimate the error using
+this method.  To allow access to this information the alternative
+error-handling form stores the value and error in a modifiable argument,
+
+     gsl_sf_result result;
+     int status = gsl_sf_bessel_J0_e (x, &result);
+
+The error-handling functions have the suffix `_e'. The returned status
+value indicates error conditions such as overflow, underflow or loss of
+precision.  If there are no errors the error-handling functions return
+`GSL_SUCCESS'.
+
+
+File: gsl-ref.info,  Node: The gsl_sf_result struct,  Next: Special Function Modes,  Prev: Special Function Usage,  Up: Special Functions
+
+7.2 The gsl_sf_result struct
+============================
+
+The error handling form of the special functions always calculate an
+error estimate along with the value of the result.  Therefore,
+structures are provided for amalgamating a value and error estimate.
+These structures are declared in the header file `gsl_sf_result.h'.
+
+   The `gsl_sf_result' struct contains value and error fields.
+
+     typedef struct
+     {
+       double val;
+       double err;
+     } gsl_sf_result;
+
+The field VAL contains the value and the field ERR contains an estimate
+of the absolute error in the value.
+
+   In some cases, an overflow or underflow can be detected and handled
+by a function.  In this case, it may be possible to return a scaling
+exponent as well as an error/value pair in order to save the result from
+exceeding the dynamic range of the built-in types.  The
+`gsl_sf_result_e10' struct contains value and error fields as well as
+an exponent field such that the actual result is obtained as `result *
+10^(e10)'.
+
+     typedef struct
+     {
+       double val;
+       double err;
+       int    e10;
+     } gsl_sf_result_e10;
+
+
+File: gsl-ref.info,  Node: Special Function Modes,  Next: Airy Functions and Derivatives,  Prev: The gsl_sf_result struct,  Up: Special Functions
+
+7.3 Modes
+=========
+
+The goal of the library is to achieve double precision accuracy wherever
+possible.  However the cost of evaluating some special functions to
+double precision can be significant, particularly where very high order
+terms are required.  In these cases a `mode' argument allows the
+accuracy of the function to be reduced in order to improve performance.
+The following precision levels are available for the mode argument,
+
+`GSL_PREC_DOUBLE'
+     Double-precision, a relative accuracy of approximately 2 * 10^-16.
+
+`GSL_PREC_SINGLE'
+     Single-precision, a relative accuracy of approximately 10^-7.
+
+`GSL_PREC_APPROX'
+     Approximate values, a relative accuracy of approximately 5 * 10^-4.
+
+The approximate mode provides the fastest evaluation at the lowest
+accuracy.
+
+
+File: gsl-ref.info,  Node: Airy Functions and Derivatives,  Next: Bessel Functions,  Prev: Special Function Modes,  Up: Special Functions
+
+7.4 Airy Functions and Derivatives
+==================================
+
+The Airy functions Ai(x) and Bi(x) are defined by the integral
+representations,
+
+     Ai(x) = (1/\pi) \int_0^\infty \cos((1/3) t^3 + xt) dt
+     Bi(x) = (1/\pi) \int_0^\infty (e^(-(1/3) t^3) + \sin((1/3) t^3 + xt)) dt
+
+For further information see Abramowitz & Stegun, Section 10.4. The Airy
+functions are defined in the header file `gsl_sf_airy.h'.
+
+* Menu:
+
+* Airy Functions::
+* Derivatives of Airy Functions::
+* Zeros of Airy Functions::
+* Zeros of Derivatives of Airy Functions::
+
+
+File: gsl-ref.info,  Node: Airy Functions,  Next: Derivatives of Airy Functions,  Up: Airy Functions and Derivatives
+
+7.4.1 Airy Functions
+--------------------
+
+ -- Function: double gsl_sf_airy_Ai (double X, gsl_mode_t MODE)
+ -- Function: int gsl_sf_airy_Ai_e (double X, gsl_mode_t MODE,
+          gsl_sf_result * RESULT)
+     These routines compute the Airy function Ai(x) with an accuracy
+     specified by MODE.
+
+ -- Function: double gsl_sf_airy_Bi (double X, gsl_mode_t MODE)
+ -- Function: int gsl_sf_airy_Bi_e (double X, gsl_mode_t MODE,
+          gsl_sf_result * RESULT)
+     These routines compute the Airy function Bi(x) with an accuracy
+     specified by MODE.
+
+ -- Function: double gsl_sf_airy_Ai_scaled (double X, gsl_mode_t MODE)
+ -- Function: int gsl_sf_airy_Ai_scaled_e (double X, gsl_mode_t MODE,
+          gsl_sf_result * RESULT)
+     These routines compute a scaled version of the Airy function
+     S_A(x) Ai(x).  For x>0 the scaling factor S_A(x) is \exp(+(2/3)
+     x^(3/2)), and is 1 for x<0.
+
+ -- Function: double gsl_sf_airy_Bi_scaled (double X, gsl_mode_t MODE)
+ -- Function: int gsl_sf_airy_Bi_scaled_e (double X, gsl_mode_t MODE,
+          gsl_sf_result * RESULT)
+     These routines compute a scaled version of the Airy function
+     S_B(x) Bi(x).  For x>0 the scaling factor S_B(x) is exp(-(2/3)
+     x^(3/2)), and is 1 for x<0.
+
+
+File: gsl-ref.info,  Node: Derivatives of Airy Functions,  Next: Zeros of Airy Functions,  Prev: Airy Functions,  Up: Airy Functions and Derivatives
+
+7.4.2 Derivatives of Airy Functions
+-----------------------------------
+
+ -- Function: double gsl_sf_airy_Ai_deriv (double X, gsl_mode_t MODE)
+ -- Function: int gsl_sf_airy_Ai_deriv_e (double X, gsl_mode_t MODE,
+          gsl_sf_result * RESULT)
+     These routines compute the Airy function derivative Ai'(x) with an
+     accuracy specified by MODE.
+
+ -- Function: double gsl_sf_airy_Bi_deriv (double X, gsl_mode_t MODE)
+ -- Function: int gsl_sf_airy_Bi_deriv_e (double X, gsl_mode_t MODE,
+          gsl_sf_result * RESULT)
+     These routines compute the Airy function derivative Bi'(x) with an
+     accuracy specified by MODE.
+
+ -- Function: double gsl_sf_airy_Ai_deriv_scaled (double X, gsl_mode_t
+          MODE)
+ -- Function: int gsl_sf_airy_Ai_deriv_scaled_e (double X, gsl_mode_t
+          MODE, gsl_sf_result * RESULT)
+     These routines compute the scaled Airy function derivative S_A(x)
+     Ai'(x).  For x>0 the scaling factor S_A(x) is \exp(+(2/3)
+     x^(3/2)), and is 1 for x<0.
+
+ -- Function: double gsl_sf_airy_Bi_deriv_scaled (double X, gsl_mode_t
+          MODE)
+ -- Function: int gsl_sf_airy_Bi_deriv_scaled_e (double X, gsl_mode_t
+          MODE, gsl_sf_result * RESULT)
+     These routines compute the scaled Airy function derivative S_B(x)
+     Bi'(x).  For x>0 the scaling factor S_B(x) is exp(-(2/3) x^(3/2)),
+     and is 1 for x<0.
+
+
+File: gsl-ref.info,  Node: Zeros of Airy Functions,  Next: Zeros of Derivatives of Airy Functions,  Prev: Derivatives of Airy Functions,  Up: Airy Functions and Derivatives
+
+7.4.3 Zeros of Airy Functions
+-----------------------------
+
+ -- Function: double gsl_sf_airy_zero_Ai (unsigned int S)
+ -- Function: int gsl_sf_airy_zero_Ai_e (unsigned int S, gsl_sf_result
+          * RESULT)
+     These routines compute the location of the S-th zero of the Airy
+     function Ai(x).
+
+ -- Function: double gsl_sf_airy_zero_Bi (unsigned int S)
+ -- Function: int gsl_sf_airy_zero_Bi_e (unsigned int S, gsl_sf_result
+          * RESULT)
+     These routines compute the location of the S-th zero of the Airy
+     function Bi(x).
+
+
+File: gsl-ref.info,  Node: Zeros of Derivatives of Airy Functions,  Prev: Zeros of Airy Functions,  Up: Airy Functions and Derivatives
+
+7.4.4 Zeros of Derivatives of Airy Functions
+--------------------------------------------
+
+ -- Function: double gsl_sf_airy_zero_Ai_deriv (unsigned int S)
+ -- Function: int gsl_sf_airy_zero_Ai_deriv_e (unsigned int S,
+          gsl_sf_result * RESULT)
+     These routines compute the location of the S-th zero of the Airy
+     function derivative Ai'(x).
+
+ -- Function: double gsl_sf_airy_zero_Bi_deriv (unsigned int S)
+ -- Function: int gsl_sf_airy_zero_Bi_deriv_e (unsigned int S,
+          gsl_sf_result * RESULT)
+     These routines compute the location of the S-th zero of the Airy
+     function derivative Bi'(x).
+
+
+File: gsl-ref.info,  Node: Bessel Functions,  Next: Clausen Functions,  Prev: Airy Functions and Derivatives,  Up: Special Functions
+
+7.5 Bessel Functions
+====================
+
+The routines described in this section compute the Cylindrical Bessel
+functions J_n(x), Y_n(x), Modified cylindrical Bessel functions I_n(x),
+K_n(x), Spherical Bessel functions j_l(x), y_l(x), and Modified
+Spherical Bessel functions i_l(x), k_l(x).  For more information see
+Abramowitz & Stegun, Chapters 9 and 10.  The Bessel functions are
+defined in the header file `gsl_sf_bessel.h'.
+
+* Menu:
+
+* Regular Cylindrical Bessel Functions::
+* Irregular Cylindrical Bessel Functions::
+* Regular Modified Cylindrical Bessel Functions::
+* Irregular Modified Cylindrical Bessel Functions::
+* Regular Spherical Bessel Functions::
+* Irregular Spherical Bessel Functions::
+* Regular Modified Spherical Bessel Functions::
+* Irregular Modified Spherical Bessel Functions::
+* Regular Bessel Function - Fractional Order::
+* Irregular Bessel Functions - Fractional Order::
+* Regular Modified Bessel Functions - Fractional Order::
+* Irregular Modified Bessel Functions - Fractional Order::
+* Zeros of Regular Bessel Functions::
+
+
+File: gsl-ref.info,  Node: Regular Cylindrical Bessel Functions,  Next: Irregular Cylindrical Bessel Functions,  Up: Bessel Functions
+
+7.5.1 Regular Cylindrical Bessel Functions
+------------------------------------------
+
+ -- Function: double gsl_sf_bessel_J0 (double X)
+ -- Function: int gsl_sf_bessel_J0_e (double X, gsl_sf_result * RESULT)
+     These routines compute the regular cylindrical Bessel function of
+     zeroth order, J_0(x).
+
+ -- Function: double gsl_sf_bessel_J1 (double X)
+ -- Function: int gsl_sf_bessel_J1_e (double X, gsl_sf_result * RESULT)
+     These routines compute the regular cylindrical Bessel function of
+     first order, J_1(x).
+
+ -- Function: double gsl_sf_bessel_Jn (int N, double X)
+ -- Function: int gsl_sf_bessel_Jn_e (int N, double X, gsl_sf_result *
+          RESULT)
+     These routines compute the regular cylindrical Bessel function of
+     order N, J_n(x).
+
+ -- Function: int gsl_sf_bessel_Jn_array (int NMIN, int NMAX, double X,
+          double RESULT_ARRAY[])
+     This routine computes the values of the regular cylindrical Bessel
+     functions J_n(x) for n from NMIN to NMAX inclusive, storing the
+     results in the array RESULT_ARRAY.  The values are computed using
+     recurrence relations for efficiency, and therefore may differ
+     slightly from the exact values.
+
+
+File: gsl-ref.info,  Node: Irregular Cylindrical Bessel Functions,  Next: Regular Modified Cylindrical Bessel Functions,  Prev: Regular Cylindrical Bessel Functions,  Up: Bessel Functions
+
+7.5.2 Irregular Cylindrical Bessel Functions
+--------------------------------------------
+
+ -- Function: double gsl_sf_bessel_Y0 (double X)
+ -- Function: int gsl_sf_bessel_Y0_e (double X, gsl_sf_result * RESULT)
+     These routines compute the irregular cylindrical Bessel function
+     of zeroth order, Y_0(x), for x>0.
+
+ -- Function: double gsl_sf_bessel_Y1 (double X)
+ -- Function: int gsl_sf_bessel_Y1_e (double X, gsl_sf_result * RESULT)
+     These routines compute the irregular cylindrical Bessel function
+     of first order, Y_1(x), for x>0.
+
+ -- Function: double gsl_sf_bessel_Yn (int N,double X)
+ -- Function: int gsl_sf_bessel_Yn_e (int N,double X, gsl_sf_result *
+          RESULT)
+     These routines compute the irregular cylindrical Bessel function of
+     order N, Y_n(x), for x>0.
+
+ -- Function: int gsl_sf_bessel_Yn_array (int NMIN, int NMAX, double X,
+          double RESULT_ARRAY[])
+     This routine computes the values of the irregular cylindrical
+     Bessel functions Y_n(x) for n from NMIN to NMAX inclusive, storing
+     the results in the array RESULT_ARRAY.  The domain of the function
+     is x>0.  The values are computed using recurrence relations for
+     efficiency, and therefore may differ slightly from the exact
+     values.
+
+
+File: gsl-ref.info,  Node: Regular Modified Cylindrical Bessel Functions,  Next: Irregular Modified Cylindrical Bessel Functions,  Prev: Irregular Cylindrical Bessel Functions,  Up: Bessel Functions
+
+7.5.3 Regular Modified Cylindrical Bessel Functions
+---------------------------------------------------
+
+ -- Function: double gsl_sf_bessel_I0 (double X)
+ -- Function: int gsl_sf_bessel_I0_e (double X, gsl_sf_result * RESULT)
+     These routines compute the regular modified cylindrical Bessel
+     function of zeroth order, I_0(x).
+
+ -- Function: double gsl_sf_bessel_I1 (double X)
+ -- Function: int gsl_sf_bessel_I1_e (double X, gsl_sf_result * RESULT)
+     These routines compute the regular modified cylindrical Bessel
+     function of first order, I_1(x).
+
+ -- Function: double gsl_sf_bessel_In (int N, double X)
+ -- Function: int gsl_sf_bessel_In_e (int N, double X, gsl_sf_result *
+          RESULT)
+     These routines compute the regular modified cylindrical Bessel
+     function of order N, I_n(x).
+
+ -- Function: int gsl_sf_bessel_In_array (int NMIN, int NMAX, double X,
+          double RESULT_ARRAY[])
+     This routine computes the values of the regular modified
+     cylindrical Bessel functions I_n(x) for n from NMIN to NMAX
+     inclusive, storing the results in the array RESULT_ARRAY.  The
+     start of the range NMIN must be positive or zero.  The values are
+     computed using recurrence relations for efficiency, and therefore
+     may differ slightly from the exact values.
+
+ -- Function: double gsl_sf_bessel_I0_scaled (double X)
+ -- Function: int gsl_sf_bessel_I0_scaled_e (double X, gsl_sf_result *
+          RESULT)
+     These routines compute the scaled regular modified cylindrical
+     Bessel function of zeroth order \exp(-|x|) I_0(x).
+
+ -- Function: double gsl_sf_bessel_I1_scaled (double X)
+ -- Function: int gsl_sf_bessel_I1_scaled_e (double X, gsl_sf_result *
+          RESULT)
+     These routines compute the scaled regular modified cylindrical
+     Bessel function of first order \exp(-|x|) I_1(x).
+
+ -- Function: double gsl_sf_bessel_In_scaled (int N, double X)
+ -- Function: int gsl_sf_bessel_In_scaled_e (int N, double X,
+          gsl_sf_result * RESULT)
+     These routines compute the scaled regular modified cylindrical
+     Bessel function of order N, \exp(-|x|) I_n(x)
+
+ -- Function: int gsl_sf_bessel_In_scaled_array (int NMIN, int NMAX,
+          double X, double RESULT_ARRAY[])
+     This routine computes the values of the scaled regular cylindrical
+     Bessel functions \exp(-|x|) I_n(x) for n from NMIN to NMAX
+     inclusive, storing the results in the array RESULT_ARRAY. The
+     start of the range NMIN must be positive or zero.  The values are
+     computed using recurrence relations for efficiency, and therefore
+     may differ slightly from the exact values.
+
+
+File: gsl-ref.info,  Node: Irregular Modified Cylindrical Bessel Functions,  Next: Regular Spherical Bessel Functions,  Prev: Regular Modified Cylindrical Bessel Functions,  Up: Bessel Functions
+
+7.5.4 Irregular Modified Cylindrical Bessel Functions
+-----------------------------------------------------
+
+ -- Function: double gsl_sf_bessel_K0 (double X)
+ -- Function: int gsl_sf_bessel_K0_e (double X, gsl_sf_result * RESULT)
+     These routines compute the irregular modified cylindrical Bessel
+     function of zeroth order, K_0(x), for x > 0.
+
+ -- Function: double gsl_sf_bessel_K1 (double X)
+ -- Function: int gsl_sf_bessel_K1_e (double X, gsl_sf_result * RESULT)
+     These routines compute the irregular modified cylindrical Bessel
+     function of first order, K_1(x), for x > 0.
+
+ -- Function: double gsl_sf_bessel_Kn (int N, double X)
+ -- Function: int gsl_sf_bessel_Kn_e (int N, double X, gsl_sf_result *
+          RESULT)
+     These routines compute the irregular modified cylindrical Bessel
+     function of order N, K_n(x), for x > 0.
+
+ -- Function: int gsl_sf_bessel_Kn_array (int NMIN, int NMAX, double X,
+          double RESULT_ARRAY[])
+     This routine computes the values of the irregular modified
+     cylindrical Bessel functions K_n(x) for n from NMIN to NMAX
+     inclusive, storing the results in the array RESULT_ARRAY. The
+     start of the range NMIN must be positive or zero. The domain of
+     the function is x>0. The values are computed using recurrence
+     relations for efficiency, and therefore may differ slightly from
+     the exact values.
+
+ -- Function: double gsl_sf_bessel_K0_scaled (double X)
+ -- Function: int gsl_sf_bessel_K0_scaled_e (double X, gsl_sf_result *
+          RESULT)
+     These routines compute the scaled irregular modified cylindrical
+     Bessel function of zeroth order \exp(x) K_0(x) for x>0.
+
+ -- Function: double gsl_sf_bessel_K1_scaled (double X)
+ -- Function: int gsl_sf_bessel_K1_scaled_e (double X, gsl_sf_result *
+          RESULT)
+     These routines compute the scaled irregular modified cylindrical
+     Bessel function of first order \exp(x) K_1(x) for x>0.
+
+ -- Function: double gsl_sf_bessel_Kn_scaled (int N, double X)
+ -- Function: int gsl_sf_bessel_Kn_scaled_e (int N, double X,
+          gsl_sf_result * RESULT)
+     These routines compute the scaled irregular modified cylindrical
+     Bessel function of order N, \exp(x) K_n(x), for x>0.
+
+ -- Function: int gsl_sf_bessel_Kn_scaled_array (int NMIN, int NMAX,
+          double X, double RESULT_ARRAY[])
+     This routine computes the values of the scaled irregular
+     cylindrical Bessel functions \exp(x) K_n(x) for n from NMIN to
+     NMAX inclusive, storing the results in the array RESULT_ARRAY. The
+     start of the range NMIN must be positive or zero.  The domain of
+     the function is x>0. The values are computed using recurrence
+     relations for efficiency, and therefore may differ slightly from
+     the exact values.
+
+
+File: gsl-ref.info,  Node: Regular Spherical Bessel Functions,  Next: Irregular Spherical Bessel Functions,  Prev: Irregular Modified Cylindrical Bessel Functions,  Up: Bessel Functions
+
+7.5.5 Regular Spherical Bessel Functions
+----------------------------------------
+
+ -- Function: double gsl_sf_bessel_j0 (double X)
+ -- Function: int gsl_sf_bessel_j0_e (double X, gsl_sf_result * RESULT)
+     These routines compute the regular spherical Bessel function of
+     zeroth order, j_0(x) = \sin(x)/x.
+
+ -- Function: double gsl_sf_bessel_j1 (double X)
+ -- Function: int gsl_sf_bessel_j1_e (double X, gsl_sf_result * RESULT)
+     These routines compute the regular spherical Bessel function of
+     first order, j_1(x) = (\sin(x)/x - \cos(x))/x.
+
+ -- Function: double gsl_sf_bessel_j2 (double X)
+ -- Function: int gsl_sf_bessel_j2_e (double X, gsl_sf_result * RESULT)
+     These routines compute the regular spherical Bessel function of
+     second order, j_2(x) = ((3/x^2 - 1)\sin(x) - 3\cos(x)/x)/x.
+
+ -- Function: double gsl_sf_bessel_jl (int L, double X)
+ -- Function: int gsl_sf_bessel_jl_e (int L, double X, gsl_sf_result *
+          RESULT)
+     These routines compute the regular spherical Bessel function of
+     order L, j_l(x), for l >= 0 and x >= 0.
+
+ -- Function: int gsl_sf_bessel_jl_array (int LMAX, double X, double
+          RESULT_ARRAY[])
+     This routine computes the values of the regular spherical Bessel
+     functions j_l(x) for l from 0 to LMAX inclusive  for lmax >= 0 and
+     x >= 0, storing the results in the array RESULT_ARRAY.  The values
+     are computed using recurrence relations for efficiency, and
+     therefore may differ slightly from the exact values.
+
+ -- Function: int gsl_sf_bessel_jl_steed_array (int LMAX, double X,
+          double * JL_X_ARRAY)
+     This routine uses Steed's method to compute the values of the
+     regular spherical Bessel functions j_l(x) for l from 0 to LMAX
+     inclusive for lmax >= 0 and x >= 0, storing the results in the
+     array RESULT_ARRAY.  The Steed/Barnett algorithm is described in
+     `Comp. Phys. Comm.' 21, 297 (1981).  Steed's method is more stable
+     than the recurrence used in the other functions but is also slower.
+
+
+File: gsl-ref.info,  Node: Irregular Spherical Bessel Functions,  Next: Regular Modified Spherical Bessel Functions,  Prev: Regular Spherical Bessel Functions,  Up: Bessel Functions
+
+7.5.6 Irregular Spherical Bessel Functions
+------------------------------------------
+
+ -- Function: double gsl_sf_bessel_y0 (double X)
+ -- Function: int gsl_sf_bessel_y0_e (double X, gsl_sf_result * RESULT)
+     These routines compute the irregular spherical Bessel function of
+     zeroth order, y_0(x) = -\cos(x)/x.
+
+ -- Function: double gsl_sf_bessel_y1 (double X)
+ -- Function: int gsl_sf_bessel_y1_e (double X, gsl_sf_result * RESULT)
+     These routines compute the irregular spherical Bessel function of
+     first order, y_1(x) = -(\cos(x)/x + \sin(x))/x.
+
+ -- Function: double gsl_sf_bessel_y2 (double X)
+ -- Function: int gsl_sf_bessel_y2_e (double X, gsl_sf_result * RESULT)
+     These routines compute the irregular spherical Bessel function of
+     second order, y_2(x) = (-3/x^3 + 1/x)\cos(x) - (3/x^2)\sin(x).
+
+ -- Function: double gsl_sf_bessel_yl (int L, double X)
+ -- Function: int gsl_sf_bessel_yl_e (int L, double X, gsl_sf_result *
+          RESULT)
+     These routines compute the irregular spherical Bessel function of
+     order L, y_l(x), for l >= 0.
+
+ -- Function: int gsl_sf_bessel_yl_array (int LMAX, double X, double
+          RESULT_ARRAY[])
+     This routine computes the values of the irregular spherical Bessel
+     functions y_l(x) for l from 0 to LMAX inclusive  for lmax >= 0,
+     storing the results in the array RESULT_ARRAY.  The values are
+     computed using recurrence relations for efficiency, and therefore
+     may differ slightly from the exact values.
+
+
+File: gsl-ref.info,  Node: Regular Modified Spherical Bessel Functions,  Next: Irregular Modified Spherical Bessel Functions,  Prev: Irregular Spherical Bessel Functions,  Up: Bessel Functions
+
+7.5.7 Regular Modified Spherical Bessel Functions
+-------------------------------------------------
+
+The regular modified spherical Bessel functions i_l(x) are related to
+the modified Bessel functions of fractional order, i_l(x) =
+\sqrt{\pi/(2x)} I_{l+1/2}(x)
+
+ -- Function: double gsl_sf_bessel_i0_scaled (double X)
+ -- Function: int gsl_sf_bessel_i0_scaled_e (double X, gsl_sf_result *
+          RESULT)
+     These routines compute the scaled regular modified spherical Bessel
+     function of zeroth order, \exp(-|x|) i_0(x).
+
+ -- Function: double gsl_sf_bessel_i1_scaled (double X)
+ -- Function: int gsl_sf_bessel_i1_scaled_e (double X, gsl_sf_result *
+          RESULT)
+     These routines compute the scaled regular modified spherical Bessel
+     function of first order, \exp(-|x|) i_1(x).
+
+ -- Function: double gsl_sf_bessel_i2_scaled (double X)
+ -- Function: int gsl_sf_bessel_i2_scaled_e (double X, gsl_sf_result *
+          RESULT)
+     These routines compute the scaled regular modified spherical Bessel
+     function of second order,  \exp(-|x|) i_2(x)
+
+ -- Function: double gsl_sf_bessel_il_scaled (int L, double X)
+ -- Function: int gsl_sf_bessel_il_scaled_e (int L, double X,
+          gsl_sf_result * RESULT)
+     These routines compute the scaled regular modified spherical Bessel
+     function of order L,  \exp(-|x|) i_l(x)
+
+ -- Function: int gsl_sf_bessel_il_scaled_array (int LMAX, double X,
+          double RESULT_ARRAY[])
+     This routine computes the values of the scaled regular modified
+     cylindrical Bessel functions \exp(-|x|) i_l(x) for l from 0 to
+     LMAX inclusive for lmax >= 0, storing the results in the array
+     RESULT_ARRAY.  The values are computed using recurrence relations
+     for efficiency, and therefore may differ slightly from the exact
+     values.
+
+
+File: gsl-ref.info,  Node: Irregular Modified Spherical Bessel Functions,  Next: Regular Bessel Function - Fractional Order,  Prev: Regular Modified Spherical Bessel Functions,  Up: Bessel Functions
+
+7.5.8 Irregular Modified Spherical Bessel Functions
+---------------------------------------------------
+
+The irregular modified spherical Bessel functions k_l(x) are related to
+the irregular modified Bessel functions of fractional order, k_l(x) =
+\sqrt{\pi/(2x)} K_{l+1/2}(x).
+
+ -- Function: double gsl_sf_bessel_k0_scaled (double X)
+ -- Function: int gsl_sf_bessel_k0_scaled_e (double X, gsl_sf_result *
+          RESULT)
+     These routines compute the scaled irregular modified spherical
+     Bessel function of zeroth order, \exp(x) k_0(x), for x>0.
+
+ -- Function: double gsl_sf_bessel_k1_scaled (double X)
+ -- Function: int gsl_sf_bessel_k1_scaled_e (double X, gsl_sf_result *
+          RESULT)
+     These routines compute the scaled irregular modified spherical
+     Bessel function of first order, \exp(x) k_1(x), for x>0.
+
+ -- Function: double gsl_sf_bessel_k2_scaled (double X)
+ -- Function: int gsl_sf_bessel_k2_scaled_e (double X, gsl_sf_result *
+          RESULT)
+     These routines compute the scaled irregular modified spherical
+     Bessel function of second order, \exp(x) k_2(x), for x>0.
+
+ -- Function: double gsl_sf_bessel_kl_scaled (int L, double X)
+ -- Function: int gsl_sf_bessel_kl_scaled_e (int L, double X,
+          gsl_sf_result * RESULT)
+     These routines compute the scaled irregular modified spherical
+     Bessel function of order L, \exp(x) k_l(x), for x>0.
+
+ -- Function: int gsl_sf_bessel_kl_scaled_array (int LMAX, double X,
+          double RESULT_ARRAY[])
+     This routine computes the values of the scaled irregular modified
+     spherical Bessel functions \exp(x) k_l(x) for l from 0 to LMAX
+     inclusive for lmax >= 0 and x>0, storing the results in the array
+     RESULT_ARRAY.  The values are computed using recurrence relations
+     for efficiency, and therefore may differ slightly from the exact
+     values.
+
+
+File: gsl-ref.info,  Node: Regular Bessel Function - Fractional Order,  Next: Irregular Bessel Functions - Fractional Order,  Prev: Irregular Modified Spherical Bessel Functions,  Up: Bessel Functions
+
+7.5.9 Regular Bessel Function--Fractional Order
+-----------------------------------------------
+
+ -- Function: double gsl_sf_bessel_Jnu (double NU, double X)
+ -- Function: int gsl_sf_bessel_Jnu_e (double NU, double X,
+          gsl_sf_result * RESULT)
+     These routines compute the regular cylindrical Bessel function of
+     fractional order \nu, J_\nu(x).
+
+ -- Function: int gsl_sf_bessel_sequence_Jnu_e (double NU, gsl_mode_t
+          MODE, size_t SIZE, double V[])
+     This function computes the regular cylindrical Bessel function of
+     fractional order \nu, J_\nu(x), evaluated at a series of x values.
+     The array V of length SIZE contains the x values.  They are
+     assumed to be strictly ordered and positive.  The array is
+     over-written with the values of J_\nu(x_i).
+
+
+File: gsl-ref.info,  Node: Irregular Bessel Functions - Fractional Order,  Next: Regular Modified Bessel Functions - Fractional Order,  Prev: Regular Bessel Function - Fractional Order,  Up: Bessel Functions
+
+7.5.10 Irregular Bessel Functions--Fractional Order
+---------------------------------------------------
+
+ -- Function: double gsl_sf_bessel_Ynu (double NU, double X)
+ -- Function: int gsl_sf_bessel_Ynu_e (double NU, double X,
+          gsl_sf_result * RESULT)
+     These routines compute the irregular cylindrical Bessel function of
+     fractional order \nu, Y_\nu(x).
+
+
+File: gsl-ref.info,  Node: Regular Modified Bessel Functions - Fractional Order,  Next: Irregular Modified Bessel Functions - Fractional Order,  Prev: Irregular Bessel Functions - Fractional Order,  Up: Bessel Functions
+
+7.5.11 Regular Modified Bessel Functions--Fractional Order
+----------------------------------------------------------
+
+ -- Function: double gsl_sf_bessel_Inu (double NU, double X)
+ -- Function: int gsl_sf_bessel_Inu_e (double NU, double X,
+          gsl_sf_result * RESULT)
+     These routines compute the regular modified Bessel function of
+     fractional order \nu, I_\nu(x) for x>0, \nu>0.
+
+ -- Function: double gsl_sf_bessel_Inu_scaled (double NU, double X)
+ -- Function: int gsl_sf_bessel_Inu_scaled_e (double NU, double X,
+          gsl_sf_result * RESULT)
+     These routines compute the scaled regular modified Bessel function
+     of fractional order \nu, \exp(-|x|)I_\nu(x) for x>0, \nu>0.
+
+
+File: gsl-ref.info,  Node: Irregular Modified Bessel Functions - Fractional Order,  Next: Zeros of Regular Bessel Functions,  Prev: Regular Modified Bessel Functions - Fractional Order,  Up: Bessel Functions
+
+7.5.12 Irregular Modified Bessel Functions--Fractional Order
+------------------------------------------------------------
+
+ -- Function: double gsl_sf_bessel_Knu (double NU, double X)
+ -- Function: int gsl_sf_bessel_Knu_e (double NU, double X,
+          gsl_sf_result * RESULT)
+     These routines compute the irregular modified Bessel function of
+     fractional order \nu, K_\nu(x) for x>0, \nu>0.
+
+ -- Function: double gsl_sf_bessel_lnKnu (double NU, double X)
+ -- Function: int gsl_sf_bessel_lnKnu_e (double NU, double X,
+          gsl_sf_result * RESULT)
+     These routines compute the logarithm of the irregular modified
+     Bessel function of fractional order \nu, \ln(K_\nu(x)) for x>0,
+     \nu>0.
+
+ -- Function: double gsl_sf_bessel_Knu_scaled (double NU, double X)
+ -- Function: int gsl_sf_bessel_Knu_scaled_e (double NU, double X,
+          gsl_sf_result * RESULT)
+     These routines compute the scaled irregular modified Bessel
+     function of fractional order \nu, \exp(+|x|) K_\nu(x) for x>0,
+     \nu>0.
+
+
+File: gsl-ref.info,  Node: Zeros of Regular Bessel Functions,  Prev: Irregular Modified Bessel Functions - Fractional Order,  Up: Bessel Functions
+
+7.5.13 Zeros of Regular Bessel Functions
+----------------------------------------
+
+ -- Function: double gsl_sf_bessel_zero_J0 (unsigned int S)
+ -- Function: int gsl_sf_bessel_zero_J0_e (unsigned int S,
+          gsl_sf_result * RESULT)
+     These routines compute the location of the S-th positive zero of
+     the Bessel function J_0(x).
+
+ -- Function: double gsl_sf_bessel_zero_J1 (unsigned int S)
+ -- Function: int gsl_sf_bessel_zero_J1_e (unsigned int S,
+          gsl_sf_result * RESULT)
+     These routines compute the location of the S-th positive zero of
+     the Bessel function J_1(x).
+
+ -- Function: double gsl_sf_bessel_zero_Jnu (double NU, unsigned int S)
+ -- Function: int gsl_sf_bessel_zero_Jnu_e (double NU, unsigned int S,
+          gsl_sf_result * RESULT)
+     These routines compute the location of the S-th positive zero of
+     the Bessel function J_\nu(x).  The current implementation does not
+     support negative values of NU.
+
+
+File: gsl-ref.info,  Node: Clausen Functions,  Next: Coulomb Functions,  Prev: Bessel Functions,  Up: Special Functions
+
+7.6 Clausen Functions
+=====================
+
+The Clausen function is defined by the following integral,
+
+     Cl_2(x) = - \int_0^x dt \log(2 \sin(t/2))
+
+It is related to the dilogarithm by Cl_2(\theta) = \Im
+Li_2(\exp(i\theta)).  The Clausen functions are declared in the header
+file `gsl_sf_clausen.h'.
+
+ -- Function: double gsl_sf_clausen (double X)
+ -- Function: int gsl_sf_clausen_e (double X, gsl_sf_result * RESULT)
+     These routines compute the Clausen integral Cl_2(x).
+
+
+File: gsl-ref.info,  Node: Coulomb Functions,  Next: Coupling Coefficients,  Prev: Clausen Functions,  Up: Special Functions
+
+7.7 Coulomb Functions
+=====================
+
+The prototypes of the Coulomb functions are declared in the header file
+`gsl_sf_coulomb.h'.  Both bound state and scattering solutions are
+available.
+
+* Menu:
+
+* Normalized Hydrogenic Bound States::
+* Coulomb Wave Functions::
+* Coulomb Wave Function Normalization Constant::
+
+
+File: gsl-ref.info,  Node: Normalized Hydrogenic Bound States,  Next: Coulomb Wave Functions,  Up: Coulomb Functions
+
+7.7.1 Normalized Hydrogenic Bound States
+----------------------------------------
+
+ -- Function: double gsl_sf_hydrogenicR_1 (double Z, double R)
+ -- Function: int gsl_sf_hydrogenicR_1_e (double Z, double R,
+          gsl_sf_result * RESULT)
+     These routines compute the lowest-order normalized hydrogenic bound
+     state radial wavefunction R_1 := 2Z \sqrt{Z} \exp(-Z r).
+
+ -- Function: double gsl_sf_hydrogenicR (int N, int L, double Z, double
+          R)
+ -- Function: int gsl_sf_hydrogenicR_e (int N, int L, double Z, double
+          R, gsl_sf_result * RESULT)
+     These routines compute the N-th normalized hydrogenic bound state
+     radial wavefunction,
+
+          R_n := 2 (Z^{3/2}/n^2) \sqrt{(n-l-1)!/(n+l)!} \exp(-Z r/n) (2Zr/n)^l
+                    L^{2l+1}_{n-l-1}(2Zr/n).
+
+     where L^a_b(x) is the generalized Laguerre polynomial (*note
+     Laguerre Functions::).  The normalization is chosen such that the
+     wavefunction \psi is given by \psi(n,l,r) = R_n Y_{lm}.
+
+
+File: gsl-ref.info,  Node: Coulomb Wave Functions,  Next: Coulomb Wave Function Normalization Constant,  Prev: Normalized Hydrogenic Bound States,  Up: Coulomb Functions
+
+7.7.2 Coulomb Wave Functions
+----------------------------
+
+The Coulomb wave functions F_L(\eta,x), G_L(\eta,x) are described in
+Abramowitz & Stegun, Chapter 14.  Because there can be a large dynamic
+range of values for these functions, overflows are handled gracefully.
+If an overflow occurs, `GSL_EOVRFLW' is signalled and exponent(s) are
+returned through the modifiable parameters EXP_F, EXP_G. The full
+solution can be reconstructed from the following relations,
+
+     F_L(eta,x)  =  fc[k_L] * exp(exp_F)
+     G_L(eta,x)  =  gc[k_L] * exp(exp_G)
+
+     F_L'(eta,x) = fcp[k_L] * exp(exp_F)
+     G_L'(eta,x) = gcp[k_L] * exp(exp_G)
+
+
+ -- Function: int gsl_sf_coulomb_wave_FG_e (double ETA, double X,
+          double L_F, int K, gsl_sf_result * F, gsl_sf_result * FP,
+          gsl_sf_result * G, gsl_sf_result * GP, double * EXP_F, double
+          * EXP_G)
+     This function computes the Coulomb wave functions F_L(\eta,x),
+     G_{L-k}(\eta,x) and their derivatives F'_L(\eta,x),
+     G'_{L-k}(\eta,x) with respect to x.  The parameters are restricted
+     to L, L-k > -1/2, x > 0 and integer k.  Note that L itself is not
+     restricted to being an integer. The results are stored in the
+     parameters F, G for the function values and FP, GP for the
+     derivative values.  If an overflow occurs, `GSL_EOVRFLW' is
+     returned and scaling exponents are stored in the modifiable
+     parameters EXP_F, EXP_G.
+
+ -- Function: int gsl_sf_coulomb_wave_F_array (double L_MIN, int KMAX,
+          double ETA, double X, double FC_ARRAY[], double * F_EXPONENT)
+     This function computes the Coulomb wave function F_L(\eta,x) for L
+     = Lmin \dots Lmin + kmax, storing the results in FC_ARRAY.  In the
+     case of overflow the exponent is stored in F_EXPONENT.
+
+ -- Function: int gsl_sf_coulomb_wave_FG_array (double L_MIN, int KMAX,
+          double ETA, double X, double FC_ARRAY[], double GC_ARRAY[],
+          double * F_EXPONENT, double * G_EXPONENT)
+     This function computes the functions F_L(\eta,x), G_L(\eta,x) for
+     L = Lmin \dots Lmin + kmax storing the results in FC_ARRAY and
+     GC_ARRAY.  In the case of overflow the exponents are stored in
+     F_EXPONENT and G_EXPONENT.
+
+ -- Function: int gsl_sf_coulomb_wave_FGp_array (double L_MIN, int
+          KMAX, double ETA, double X, double FC_ARRAY[], double
+          FCP_ARRAY[], double GC_ARRAY[], double GCP_ARRAY[], double *
+          F_EXPONENT, double * G_EXPONENT)
+     This function computes the functions F_L(\eta,x), G_L(\eta,x) and
+     their derivatives F'_L(\eta,x), G'_L(\eta,x) for L = Lmin \dots
+     Lmin + kmax storing the results in FC_ARRAY, GC_ARRAY, FCP_ARRAY
+     and GCP_ARRAY.  In the case of overflow the exponents are stored
+     in F_EXPONENT and G_EXPONENT.
+
+ -- Function: int gsl_sf_coulomb_wave_sphF_array (double L_MIN, int
+          KMAX, double ETA, double X, double FC_ARRAY[], double
+          F_EXPONENT[])
+     This function computes the Coulomb wave function divided by the
+     argument F_L(\eta, x)/x for L = Lmin \dots Lmin + kmax, storing the
+     results in FC_ARRAY.  In the case of overflow the exponent is
+     stored in F_EXPONENT. This function reduces to spherical Bessel
+     functions in the limit \eta \to 0.
+
+
+File: gsl-ref.info,  Node: Coulomb Wave Function Normalization Constant,  Prev: Coulomb Wave Functions,  Up: Coulomb Functions
+
+7.7.3 Coulomb Wave Function Normalization Constant
+--------------------------------------------------
+
+The Coulomb wave function normalization constant is defined in
+Abramowitz 14.1.7.
+
+ -- Function: int gsl_sf_coulomb_CL_e (double L, double ETA,
+          gsl_sf_result * RESULT)
+     This function computes the Coulomb wave function normalization
+     constant C_L(\eta) for L > -1.
+
+ -- Function: int gsl_sf_coulomb_CL_array (double LMIN, int KMAX,
+          double ETA, double CL[])
+     This function computes the Coulomb wave function normalization
+     constant C_L(\eta) for L = Lmin \dots Lmin + kmax, Lmin > -1.
+
+
+File: gsl-ref.info,  Node: Coupling Coefficients,  Next: Dawson Function,  Prev: Coulomb Functions,  Up: Special Functions
+
+7.8 Coupling Coefficients
+=========================
+
+The Wigner 3-j, 6-j and 9-j symbols give the coupling coefficients for
+combined angular momentum vectors.  Since the arguments of the standard
+coupling coefficient functions are integer or half-integer, the
+arguments of the following functions are, by convention, integers equal
+to twice the actual spin value.  For information on the 3-j coefficients
+see Abramowitz & Stegun, Section 27.9.  The functions described in this
+section are declared in the header file `gsl_sf_coupling.h'.
+
+* Menu:
+
+* 3-j Symbols::
+* 6-j Symbols::
+* 9-j Symbols::
+
+
+File: gsl-ref.info,  Node: 3-j Symbols,  Next: 6-j Symbols,  Up: Coupling Coefficients
+
+7.8.1 3-j Symbols
+-----------------
+
+ -- Function: double gsl_sf_coupling_3j (int TWO_JA, int TWO_JB, int
+          TWO_JC, int TWO_MA, int TWO_MB, int TWO_MC)
+ -- Function: int gsl_sf_coupling_3j_e (int TWO_JA, int TWO_JB, int
+          TWO_JC, int TWO_MA, int TWO_MB, int TWO_MC, gsl_sf_result *
+          RESULT)
+     These routines compute the Wigner 3-j coefficient,
+
+          (ja jb jc
+           ma mb mc)
+
+     where the arguments are given in half-integer units, ja =
+     TWO_JA/2, ma = TWO_MA/2, etc.
+
+
+File: gsl-ref.info,  Node: 6-j Symbols,  Next: 9-j Symbols,  Prev: 3-j Symbols,  Up: Coupling Coefficients
+
+7.8.2 6-j Symbols
+-----------------
+
+ -- Function: double gsl_sf_coupling_6j (int TWO_JA, int TWO_JB, int
+          TWO_JC, int TWO_JD, int TWO_JE, int TWO_JF)
+ -- Function: int gsl_sf_coupling_6j_e (int TWO_JA, int TWO_JB, int
+          TWO_JC, int TWO_JD, int TWO_JE, int TWO_JF, gsl_sf_result *
+          RESULT)
+     These routines compute the Wigner 6-j coefficient,
+
+          {ja jb jc
+           jd je jf}
+
+     where the arguments are given in half-integer units, ja =
+     TWO_JA/2, ma = TWO_MA/2, etc.
+
+
+File: gsl-ref.info,  Node: 9-j Symbols,  Prev: 6-j Symbols,  Up: Coupling Coefficients
+
+7.8.3 9-j Symbols
+-----------------
+
+ -- Function: double gsl_sf_coupling_9j (int TWO_JA, int TWO_JB, int
+          TWO_JC, int TWO_JD, int TWO_JE, int TWO_JF, int TWO_JG, int
+          TWO_JH, int TWO_JI)
+ -- Function: int gsl_sf_coupling_9j_e (int TWO_JA, int TWO_JB, int
+          TWO_JC, int TWO_JD, int TWO_JE, int TWO_JF, int TWO_JG, int
+          TWO_JH, int TWO_JI, gsl_sf_result * RESULT)
+     These routines compute the Wigner 9-j coefficient,
+
+          {ja jb jc
+           jd je jf
+           jg jh ji}
+
+     where the arguments are given in half-integer units, ja =
+     TWO_JA/2, ma = TWO_MA/2, etc.
+
+
+File: gsl-ref.info,  Node: Dawson Function,  Next: Debye Functions,  Prev: Coupling Coefficients,  Up: Special Functions
+
+7.9 Dawson Function
+===================
+
+The Dawson integral is defined by \exp(-x^2) \int_0^x dt \exp(t^2).  A
+table of Dawson's integral can be found in Abramowitz & Stegun, Table
+7.5.  The Dawson functions are declared in the header file
+`gsl_sf_dawson.h'.
+
+ -- Function: double gsl_sf_dawson (double X)
+ -- Function: int gsl_sf_dawson_e (double X, gsl_sf_result * RESULT)
+     These routines compute the value of Dawson's integral for X.
+
+
+File: gsl-ref.info,  Node: Debye Functions,  Next: Dilogarithm,  Prev: Dawson Function,  Up: Special Functions
+
+7.10 Debye Functions
+====================
+
+The Debye functions D_n(x) are defined by the following integral,
+
+     D_n(x) = n/x^n \int_0^x dt (t^n/(e^t - 1))
+
+For further information see Abramowitz & Stegun, Section 27.1.  The
+Debye functions are declared in the header file `gsl_sf_debye.h'.
+
+ -- Function: double gsl_sf_debye_1 (double X)
+ -- Function: int gsl_sf_debye_1_e (double X, gsl_sf_result * RESULT)
+     These routines compute the first-order Debye function D_1(x) =
+     (1/x) \int_0^x dt (t/(e^t - 1)).
+
+ -- Function: double gsl_sf_debye_2 (double X)
+ -- Function: int gsl_sf_debye_2_e (double X, gsl_sf_result * RESULT)
+     These routines compute the second-order Debye function D_2(x) =
+     (2/x^2) \int_0^x dt (t^2/(e^t - 1)).
+
+ -- Function: double gsl_sf_debye_3 (double X)
+ -- Function: int gsl_sf_debye_3_e (double X, gsl_sf_result * RESULT)
+     These routines compute the third-order Debye function D_3(x) =
+     (3/x^3) \int_0^x dt (t^3/(e^t - 1)).
+
+ -- Function: double gsl_sf_debye_4 (double X)
+ -- Function: int gsl_sf_debye_4_e (double X, gsl_sf_result * RESULT)
+     These routines compute the fourth-order Debye function D_4(x) =
+     (4/x^4) \int_0^x dt (t^4/(e^t - 1)).
+
+ -- Function: double gsl_sf_debye_5 (double X)
+ -- Function: int gsl_sf_debye_5_e (double X, gsl_sf_result * RESULT)
+     These routines compute the fifth-order Debye function D_5(x) =
+     (5/x^5) \int_0^x dt (t^5/(e^t - 1)).
+
+ -- Function: double gsl_sf_debye_6 (double X)
+ -- Function: int gsl_sf_debye_6_e (double X, gsl_sf_result * RESULT)
+     These routines compute the sixth-order Debye function D_6(x) =
+     (6/x^6) \int_0^x dt (t^6/(e^t - 1)).
+
+
+File: gsl-ref.info,  Node: Dilogarithm,  Next: Elementary Operations,  Prev: Debye Functions,  Up: Special Functions
+
+7.11 Dilogarithm
+================
+
+The functions described in this section are declared in the header file
+`gsl_sf_dilog.h'.
+
+* Menu:
+
+* Real Argument::
+* Complex Argument::
+
+
+File: gsl-ref.info,  Node: Real Argument,  Next: Complex Argument,  Up: Dilogarithm
+
+7.11.1 Real Argument
+--------------------
+
+ -- Function: double gsl_sf_dilog (double X)
+ -- Function: int gsl_sf_dilog_e (double X, gsl_sf_result * RESULT)
+     These routines compute the dilogarithm for a real argument. In
+     Lewin's notation this is Li_2(x), the real part of the dilogarithm
+     of a real x.  It is defined by the integral representation Li_2(x)
+     = - \Re \int_0^x ds \log(1-s) / s.  Note that \Im(Li_2(x)) = 0 for
+     x <= 1, and -\pi\log(x) for x > 1.
+
+
+
+File: gsl-ref.info,  Node: Complex Argument,  Prev: Real Argument,  Up: Dilogarithm
+
+7.11.2 Complex Argument
+-----------------------
+
+ -- Function: int gsl_sf_complex_dilog_e (double R, double THETA,
+          gsl_sf_result * RESULT_RE, gsl_sf_result * RESULT_IM)
+     This function computes the full complex-valued dilogarithm for the
+     complex argument z = r \exp(i \theta). The real and imaginary
+     parts of the result are returned in RESULT_RE, RESULT_IM.
+
+
+File: gsl-ref.info,  Node: Elementary Operations,  Next: Elliptic Integrals,  Prev: Dilogarithm,  Up: Special Functions
+
+7.12 Elementary Operations
+==========================
+
+The following functions allow for the propagation of errors when
+combining quantities by multiplication.  The functions are declared in
+the header file `gsl_sf_elementary.h'.
+
+ -- Function: int gsl_sf_multiply_e (double X, double Y, gsl_sf_result
+          * RESULT)
+     This function multiplies X and Y storing the product and its
+     associated error in RESULT.
+
+ -- Function: int gsl_sf_multiply_err_e (double X, double DX, double Y,
+          double DY, gsl_sf_result * RESULT)
+     This function multiplies X and Y with associated absolute errors
+     DX and DY.  The product xy +/- xy \sqrt((dx/x)^2 +(dy/y)^2) is
+     stored in RESULT.
+
+
+File: gsl-ref.info,  Node: Elliptic Integrals,  Next: Elliptic Functions (Jacobi),  Prev: Elementary Operations,  Up: Special Functions
+
+7.13 Elliptic Integrals
+=======================
+
+The functions described in this section are declared in the header file
+`gsl_sf_ellint.h'.  Further information about the elliptic integrals
+can be found in Abramowitz & Stegun, Chapter 17.
+
+* Menu:
+
+* Definition of Legendre Forms::
+* Definition of Carlson Forms::
+* Legendre Form of Complete Elliptic Integrals::
+* Legendre Form of Incomplete Elliptic Integrals::
+* Carlson Forms::
+
+
+File: gsl-ref.info,  Node: Definition of Legendre Forms,  Next: Definition of Carlson Forms,  Up: Elliptic Integrals
+
+7.13.1 Definition of Legendre Forms
+-----------------------------------
+
+The Legendre forms of elliptic integrals F(\phi,k), E(\phi,k) and
+\Pi(\phi,k,n) are defined by,
+
+       F(\phi,k) = \int_0^\phi dt 1/\sqrt((1 - k^2 \sin^2(t)))
+
+       E(\phi,k) = \int_0^\phi dt   \sqrt((1 - k^2 \sin^2(t)))
+
+     Pi(\phi,k,n) = \int_0^\phi dt 1/((1 + n \sin^2(t))\sqrt(1 - k^2 \sin^2(t)))
+
+The complete Legendre forms are denoted by K(k) = F(\pi/2, k) and E(k)
+= E(\pi/2, k).
+
+   The notation used here is based on Carlson, `Numerische Mathematik'
+33 (1979) 1 and differs slightly from that used by Abramowitz & Stegun,
+where the functions are given in terms of the parameter m = k^2 and n
+is replaced by -n.
+
+
+File: gsl-ref.info,  Node: Definition of Carlson Forms,  Next: Legendre Form of Complete Elliptic Integrals,  Prev: Definition of Legendre Forms,  Up: Elliptic Integrals
+
+7.13.2 Definition of Carlson Forms
+----------------------------------
+
+The Carlson symmetric forms of elliptical integrals RC(x,y), RD(x,y,z),
+RF(x,y,z) and RJ(x,y,z,p) are defined by,
+
+         RC(x,y) = 1/2 \int_0^\infty dt (t+x)^(-1/2) (t+y)^(-1)
+
+       RD(x,y,z) = 3/2 \int_0^\infty dt (t+x)^(-1/2) (t+y)^(-1/2) (t+z)^(-3/2)
+
+       RF(x,y,z) = 1/2 \int_0^\infty dt (t+x)^(-1/2) (t+y)^(-1/2) (t+z)^(-1/2)
+
+     RJ(x,y,z,p) = 3/2 \int_0^\infty dt
+                      (t+x)^(-1/2) (t+y)^(-1/2) (t+z)^(-1/2) (t+p)^(-1)
+
+
+File: gsl-ref.info,  Node: Legendre Form of Complete Elliptic Integrals,  Next: Legendre Form of Incomplete Elliptic Integrals,  Prev: Definition of Carlson Forms,  Up: Elliptic Integrals
+
+7.13.3 Legendre Form of Complete Elliptic Integrals
+---------------------------------------------------
+
+ -- Function: double gsl_sf_ellint_Kcomp (double K, gsl_mode_t MODE)
+ -- Function: int gsl_sf_ellint_Kcomp_e (double K, gsl_mode_t MODE,
+          gsl_sf_result * RESULT)
+     These routines compute the complete elliptic integral K(k) to the
+     accuracy specified by the mode variable MODE.  Note that
+     Abramowitz & Stegun define this function in terms of the parameter
+     m = k^2.
+
+ -- Function: double gsl_sf_ellint_Ecomp (double K, gsl_mode_t MODE)
+ -- Function: int gsl_sf_ellint_Ecomp_e (double K, gsl_mode_t MODE,
+          gsl_sf_result * RESULT)
+     These routines compute the complete elliptic integral E(k) to the
+     accuracy specified by the mode variable MODE.  Note that
+     Abramowitz & Stegun define this function in terms of the parameter
+     m = k^2.
+
+ -- Function: double gsl_sf_ellint_Pcomp (double K, double N,
+          gsl_mode_t MODE)
+ -- Function: int gsl_sf_ellint_Pcomp_e (double K, double N, gsl_mode_t
+          MODE, gsl_sf_result * RESULT)
+     These routines compute the complete elliptic integral \Pi(k,n) to
+     the accuracy specified by the mode variable MODE.  Note that
+     Abramowitz & Stegun define this function in terms of the
+     parameters m = k^2 and \sin^2(\alpha) = k^2, with the change of
+     sign n \to -n.
+
+
+File: gsl-ref.info,  Node: Legendre Form of Incomplete Elliptic Integrals,  Next: Carlson Forms,  Prev: Legendre Form of Complete Elliptic Integrals,  Up: Elliptic Integrals
+
+7.13.4 Legendre Form of Incomplete Elliptic Integrals
+-----------------------------------------------------
+
+ -- Function: double gsl_sf_ellint_F (double PHI, double K, gsl_mode_t
+          MODE)
+ -- Function: int gsl_sf_ellint_F_e (double PHI, double K, gsl_mode_t
+          MODE, gsl_sf_result * RESULT)
+     These routines compute the incomplete elliptic integral F(\phi,k)
+     to the accuracy specified by the mode variable MODE.  Note that
+     Abramowitz & Stegun define this function in terms of the parameter
+     m = k^2.
+
+ -- Function: double gsl_sf_ellint_E (double PHI, double K, gsl_mode_t
+          MODE)
+ -- Function: int gsl_sf_ellint_E_e (double PHI, double K, gsl_mode_t
+          MODE, gsl_sf_result * RESULT)
+     These routines compute the incomplete elliptic integral E(\phi,k)
+     to the accuracy specified by the mode variable MODE.  Note that
+     Abramowitz & Stegun define this function in terms of the parameter
+     m = k^2.
+
+ -- Function: double gsl_sf_ellint_P (double PHI, double K, double N,
+          gsl_mode_t MODE)
+ -- Function: int gsl_sf_ellint_P_e (double PHI, double K, double N,
+          gsl_mode_t MODE, gsl_sf_result * RESULT)
+     These routines compute the incomplete elliptic integral
+     \Pi(\phi,k,n) to the accuracy specified by the mode variable MODE.
+     Note that Abramowitz & Stegun define this function in terms of the
+     parameters m = k^2 and \sin^2(\alpha) = k^2, with the change of
+     sign n \to -n.
+
+ -- Function: double gsl_sf_ellint_D (double PHI, double K, double N,
+          gsl_mode_t MODE)
+ -- Function: int gsl_sf_ellint_D_e (double PHI, double K, double N,
+          gsl_mode_t MODE, gsl_sf_result * RESULT)
+     These functions compute the incomplete elliptic integral D(\phi,k)
+     which is defined through the Carlson form RD(x,y,z) by the
+     following relation,
+
+          D(\phi,k,n) = (1/3)(\sin(\phi))^3 RD (1-\sin^2(\phi), 1-k^2 \sin^2(\phi), 1).
+     The argument N is not used and will be removed in a future release.
+
+
+
+File: gsl-ref.info,  Node: Carlson Forms,  Prev: Legendre Form of Incomplete Elliptic Integrals,  Up: Elliptic Integrals
+
+7.13.5 Carlson Forms
+--------------------
+
+ -- Function: double gsl_sf_ellint_RC (double X, double Y, gsl_mode_t
+          MODE)
+ -- Function: int gsl_sf_ellint_RC_e (double X, double Y, gsl_mode_t
+          MODE, gsl_sf_result * RESULT)
+     These routines compute the incomplete elliptic integral RC(x,y) to
+     the accuracy specified by the mode variable MODE.
+
+ -- Function: double gsl_sf_ellint_RD (double X, double Y, double Z,
+          gsl_mode_t MODE)
+ -- Function: int gsl_sf_ellint_RD_e (double X, double Y, double Z,
+          gsl_mode_t MODE, gsl_sf_result * RESULT)
+     These routines compute the incomplete elliptic integral RD(x,y,z)
+     to the accuracy specified by the mode variable MODE.
+
+ -- Function: double gsl_sf_ellint_RF (double X, double Y, double Z,
+          gsl_mode_t MODE)
+ -- Function: int gsl_sf_ellint_RF_e (double X, double Y, double Z,
+          gsl_mode_t MODE, gsl_sf_result * RESULT)
+     These routines compute the incomplete elliptic integral RF(x,y,z)
+     to the accuracy specified by the mode variable MODE.
+
+ -- Function: double gsl_sf_ellint_RJ (double X, double Y, double Z,
+          double P, gsl_mode_t MODE)
+ -- Function: int gsl_sf_ellint_RJ_e (double X, double Y, double Z,
+          double P, gsl_mode_t MODE, gsl_sf_result * RESULT)
+     These routines compute the incomplete elliptic integral RJ(x,y,z,p)
+     to the accuracy specified by the mode variable MODE.
+
+
+File: gsl-ref.info,  Node: Elliptic Functions (Jacobi),  Next: Error Functions,  Prev: Elliptic Integrals,  Up: Special Functions
+
+7.14 Elliptic Functions (Jacobi)
+================================
+
+The Jacobian Elliptic functions are defined in Abramowitz & Stegun,
+Chapter 16.  The functions are declared in the header file
+`gsl_sf_elljac.h'.
+
+ -- Function: int gsl_sf_elljac_e (double U, double M, double * SN,
+          double * CN, double * DN)
+     This function computes the Jacobian elliptic functions sn(u|m),
+     cn(u|m), dn(u|m) by descending Landen transformations.
+
+
+File: gsl-ref.info,  Node: Error Functions,  Next: Exponential Functions,  Prev: Elliptic Functions (Jacobi),  Up: Special Functions
+
+7.15 Error Functions
+====================
+
+The error function is described in Abramowitz & Stegun, Chapter 7.  The
+functions in this section are declared in the header file
+`gsl_sf_erf.h'.
+
+* Menu:
+
+* Error Function::
+* Complementary Error Function::
+* Log Complementary Error Function::
+* Probability functions::
+
+
+File: gsl-ref.info,  Node: Error Function,  Next: Complementary Error Function,  Up: Error Functions
+
+7.15.1 Error Function
+---------------------
+
+ -- Function: double gsl_sf_erf (double X)
+ -- Function: int gsl_sf_erf_e (double X, gsl_sf_result * RESULT)
+     These routines compute the error function erf(x), where erf(x) =
+     (2/\sqrt(\pi)) \int_0^x dt \exp(-t^2).
+
+
+File: gsl-ref.info,  Node: Complementary Error Function,  Next: Log Complementary Error Function,  Prev: Error Function,  Up: Error Functions
+
+7.15.2 Complementary Error Function
+-----------------------------------
+
+ -- Function: double gsl_sf_erfc (double X)
+ -- Function: int gsl_sf_erfc_e (double X, gsl_sf_result * RESULT)
+     These routines compute the complementary error function erfc(x) =
+     1 - erf(x) = (2/\sqrt(\pi)) \int_x^\infty \exp(-t^2).
+
+
+File: gsl-ref.info,  Node: Log Complementary Error Function,  Next: Probability functions,  Prev: Complementary Error Function,  Up: Error Functions
+
+7.15.3 Log Complementary Error Function
+---------------------------------------
+
+ -- Function: double gsl_sf_log_erfc (double X)
+ -- Function: int gsl_sf_log_erfc_e (double X, gsl_sf_result * RESULT)
+     These routines compute the logarithm of the complementary error
+     function \log(\erfc(x)).
+
+
+File: gsl-ref.info,  Node: Probability functions,  Prev: Log Complementary Error Function,  Up: Error Functions
+
+7.15.4 Probability functions
+----------------------------
+
+The probability functions for the Normal or Gaussian distribution are
+described in Abramowitz & Stegun, Section 26.2.
+
+ -- Function: double gsl_sf_erf_Z (double X)
+ -- Function: int gsl_sf_erf_Z_e (double X, gsl_sf_result * RESULT)
+     These routines compute the Gaussian probability density function
+     Z(x) = (1/\sqrt{2\pi}) \exp(-x^2/2).
+
+ -- Function: double gsl_sf_erf_Q (double X)
+ -- Function: int gsl_sf_erf_Q_e (double X, gsl_sf_result * RESULT)
+     These routines compute the upper tail of the Gaussian probability
+     function Q(x) = (1/\sqrt{2\pi}) \int_x^\infty dt \exp(-t^2/2).
+
+   The "hazard function" for the normal distribution, also known as the
+inverse Mill's ratio, is defined as,
+
+     h(x) = Z(x)/Q(x) = \sqrt{2/\pi} \exp(-x^2 / 2) / \erfc(x/\sqrt 2)
+
+It decreases rapidly as x approaches -\infty and asymptotes to h(x)
+\sim x as x approaches +\infty.
+
+ -- Function: double gsl_sf_hazard (double X)
+ -- Function: int gsl_sf_hazard_e (double X, gsl_sf_result * RESULT)
+     These routines compute the hazard function for the normal
+     distribution.
+
+
+File: gsl-ref.info,  Node: Exponential Functions,  Next: Exponential Integrals,  Prev: Error Functions,  Up: Special Functions
+
+7.16 Exponential Functions
+==========================
+
+The functions described in this section are declared in the header file
+`gsl_sf_exp.h'.
+
+* Menu:
+
+* Exponential Function::
+* Relative Exponential Functions::
+* Exponentiation With Error Estimate::
+
+
+File: gsl-ref.info,  Node: Exponential Function,  Next: Relative Exponential Functions,  Up: Exponential Functions
+
+7.16.1 Exponential Function
+---------------------------
+
+ -- Function: double gsl_sf_exp (double X)
+ -- Function: int gsl_sf_exp_e (double X, gsl_sf_result * RESULT)
+     These routines provide an exponential function \exp(x) using GSL
+     semantics and error checking.
+
+ -- Function: int gsl_sf_exp_e10_e (double X, gsl_sf_result_e10 *
+          RESULT)
+     This function computes the exponential \exp(x) using the
+     `gsl_sf_result_e10' type to return a result with extended range.
+     This function may be useful if the value of \exp(x) would overflow
+     the  numeric range of `double'.
+
+ -- Function: double gsl_sf_exp_mult (double X, double Y)
+ -- Function: int gsl_sf_exp_mult_e (double X, double Y, gsl_sf_result
+          * RESULT)
+     These routines exponentiate X and multiply by the factor Y to
+     return the product y \exp(x).
+
+ -- Function: int gsl_sf_exp_mult_e10_e (const double X, const double
+          Y, gsl_sf_result_e10 * RESULT)
+     This function computes the product y \exp(x) using the
+     `gsl_sf_result_e10' type to return a result with extended numeric
+     range.
+
+
+File: gsl-ref.info,  Node: Relative Exponential Functions,  Next: Exponentiation With Error Estimate,  Prev: Exponential Function,  Up: Exponential Functions
+
+7.16.2 Relative Exponential Functions
+-------------------------------------
+
+ -- Function: double gsl_sf_expm1 (double X)
+ -- Function: int gsl_sf_expm1_e (double X, gsl_sf_result * RESULT)
+     These routines compute the quantity \exp(x)-1 using an algorithm
+     that is accurate for small x.
+
+ -- Function: double gsl_sf_exprel (double X)
+ -- Function: int gsl_sf_exprel_e (double X, gsl_sf_result * RESULT)
+     These routines compute the quantity (\exp(x)-1)/x using an
+     algorithm that is accurate for small x.  For small x the algorithm
+     is based on the expansion (\exp(x)-1)/x = 1 + x/2 + x^2/(2*3) +
+     x^3/(2*3*4) + \dots.
+
+ -- Function: double gsl_sf_exprel_2 (double X)
+ -- Function: int gsl_sf_exprel_2_e (double X, gsl_sf_result * RESULT)
+     These routines compute the quantity 2(\exp(x)-1-x)/x^2 using an
+     algorithm that is accurate for small x.  For small x the algorithm
+     is based on the expansion 2(\exp(x)-1-x)/x^2 = 1 + x/3 + x^2/(3*4)
+     + x^3/(3*4*5) + \dots.
+
+ -- Function: double gsl_sf_exprel_n (int N, double X)
+ -- Function: int gsl_sf_exprel_n_e (int N, double X, gsl_sf_result *
+          RESULT)
+     These routines compute the N-relative exponential, which is the
+     N-th generalization of the functions `gsl_sf_exprel' and
+     `gsl_sf_exprel2'.  The N-relative exponential is given by,
+
+          exprel_N(x) = N!/x^N (\exp(x) - \sum_{k=0}^{N-1} x^k/k!)
+                      = 1 + x/(N+1) + x^2/((N+1)(N+2)) + ...
+                      = 1F1 (1,1+N,x)
+
+
+File: gsl-ref.info,  Node: Exponentiation With Error Estimate,  Prev: Relative Exponential Functions,  Up: Exponential Functions
+
+7.16.3 Exponentiation With Error Estimate
+-----------------------------------------
+
+ -- Function: int gsl_sf_exp_err_e (double X, double DX, gsl_sf_result
+          * RESULT)
+     This function exponentiates X with an associated absolute error DX.
+
+ -- Function: int gsl_sf_exp_err_e10_e (double X, double DX,
+          gsl_sf_result_e10 * RESULT)
+     This function exponentiates a quantity X with an associated
+     absolute error DX using the `gsl_sf_result_e10' type to return a
+     result with extended range.
+
+ -- Function: int gsl_sf_exp_mult_err_e (double X, double DX, double Y,
+          double DY, gsl_sf_result * RESULT)
+     This routine computes the product y \exp(x) for the quantities X,
+     Y with associated absolute errors DX, DY.
+
+ -- Function: int gsl_sf_exp_mult_err_e10_e (double X, double DX,
+          double Y, double DY, gsl_sf_result_e10 * RESULT)
+     This routine computes the product y \exp(x) for the quantities X,
+     Y with associated absolute errors DX, DY using the
+     `gsl_sf_result_e10' type to return a result with extended range.
+
+
+File: gsl-ref.info,  Node: Exponential Integrals,  Next: Fermi-Dirac Function,  Prev: Exponential Functions,  Up: Special Functions
+
+7.17 Exponential Integrals
+==========================
+
+Information on the exponential integrals can be found in Abramowitz &
+Stegun, Chapter 5.  These functions are declared in the header file
+`gsl_sf_expint.h'.
+
+* Menu:
+
+* Exponential Integral::
+* Ei(x)::
+* Hyperbolic Integrals::
+* Ei_3(x)::
+* Trigonometric Integrals::
+* Arctangent Integral::
+
+
+File: gsl-ref.info,  Node: Exponential Integral,  Next: Ei(x),  Up: Exponential Integrals
+
+7.17.1 Exponential Integral
+---------------------------
+
+ -- Function: double gsl_sf_expint_E1 (double X)
+ -- Function: int gsl_sf_expint_E1_e (double X, gsl_sf_result * RESULT)
+     These routines compute the exponential integral E_1(x),
+
+          E_1(x) := \Re \int_1^\infty dt \exp(-xt)/t.
+
+
+
+ -- Function: double gsl_sf_expint_E2 (double X)
+ -- Function: int gsl_sf_expint_E2_e (double X, gsl_sf_result * RESULT)
+     These routines compute the second-order exponential integral
+     E_2(x),
+
+          E_2(x) := \Re \int_1^\infty dt \exp(-xt)/t^2.
+
+
+
+
+File: gsl-ref.info,  Node: Ei(x),  Next: Hyperbolic Integrals,  Prev: Exponential Integral,  Up: Exponential Integrals
+
+7.17.2 Ei(x)
+------------
+
+ -- Function: double gsl_sf_expint_Ei (double X)
+ -- Function: int gsl_sf_expint_Ei_e (double X, gsl_sf_result * RESULT)
+     These routines compute the exponential integral Ei(x),
+
+          Ei(x) := - PV(\int_{-x}^\infty dt \exp(-t)/t)
+
+     where PV denotes the principal value of the integral.
+
+
+File: gsl-ref.info,  Node: Hyperbolic Integrals,  Next: Ei_3(x),  Prev: Ei(x),  Up: Exponential Integrals
+
+7.17.3 Hyperbolic Integrals
+---------------------------
+
+ -- Function: double gsl_sf_Shi (double X)
+ -- Function: int gsl_sf_Shi_e (double X, gsl_sf_result * RESULT)
+     These routines compute the integral Shi(x) = \int_0^x dt
+     \sinh(t)/t.
+
+ -- Function: double gsl_sf_Chi (double X)
+ -- Function: int gsl_sf_Chi_e (double X, gsl_sf_result * RESULT)
+     These routines compute the integral  Chi(x) := \Re[ \gamma_E +
+     \log(x) + \int_0^x dt (\cosh[t]-1)/t] , where \gamma_E is the
+     Euler constant (available as the macro `M_EULER').
+
+
+File: gsl-ref.info,  Node: Ei_3(x),  Next: Trigonometric Integrals,  Prev: Hyperbolic Integrals,  Up: Exponential Integrals
+
+7.17.4 Ei_3(x)
+--------------
+
+ -- Function: double gsl_sf_expint_3 (double X)
+ -- Function: int gsl_sf_expint_3_e (double X, gsl_sf_result * RESULT)
+     These routines compute the third-order exponential integral
+     Ei_3(x) = \int_0^xdt \exp(-t^3) for x >= 0.
+
+
+File: gsl-ref.info,  Node: Trigonometric Integrals,  Next: Arctangent Integral,  Prev: Ei_3(x),  Up: Exponential Integrals
+
+7.17.5 Trigonometric Integrals
+------------------------------
+
+ -- Function: double gsl_sf_Si (const double X)
+ -- Function: int gsl_sf_Si_e (double X, gsl_sf_result * RESULT)
+     These routines compute the Sine integral Si(x) = \int_0^x dt
+     \sin(t)/t.
+
+ -- Function: double gsl_sf_Ci (const double X)
+ -- Function: int gsl_sf_Ci_e (double X, gsl_sf_result * RESULT)
+     These routines compute the Cosine integral Ci(x) = -\int_x^\infty
+     dt \cos(t)/t for x > 0.
+
+
+File: gsl-ref.info,  Node: Arctangent Integral,  Prev: Trigonometric Integrals,  Up: Exponential Integrals
+
+7.17.6 Arctangent Integral
+--------------------------
+
+ -- Function: double gsl_sf_atanint (double X)
+ -- Function: int gsl_sf_atanint_e (double X, gsl_sf_result * RESULT)
+     These routines compute the Arctangent integral, which is defined
+     as AtanInt(x) = \int_0^x dt \arctan(t)/t.
+
+
+File: gsl-ref.info,  Node: Fermi-Dirac Function,  Next: Gamma and Beta Functions,  Prev: Exponential Integrals,  Up: Special Functions
+
+7.18 Fermi-Dirac Function
+=========================
+
+The functions described in this section are declared in the header file
+`gsl_sf_fermi_dirac.h'.
+
+* Menu:
+
+* Complete Fermi-Dirac Integrals::
+* Incomplete Fermi-Dirac Integrals::
+
+
+File: gsl-ref.info,  Node: Complete Fermi-Dirac Integrals,  Next: Incomplete Fermi-Dirac Integrals,  Up: Fermi-Dirac Function
+
+7.18.1 Complete Fermi-Dirac Integrals
+-------------------------------------
+
+The complete Fermi-Dirac integral F_j(x) is given by,
+
+     F_j(x)   := (1/r\Gamma(j+1)) \int_0^\infty dt (t^j / (\exp(t-x) + 1))
+
+ -- Function: double gsl_sf_fermi_dirac_m1 (double X)
+ -- Function: int gsl_sf_fermi_dirac_m1_e (double X, gsl_sf_result *
+          RESULT)
+     These routines compute the complete Fermi-Dirac integral with an
+     index of -1.  This integral is given by F_{-1}(x) = e^x / (1 +
+     e^x).
+
+ -- Function: double gsl_sf_fermi_dirac_0 (double X)
+ -- Function: int gsl_sf_fermi_dirac_0_e (double X, gsl_sf_result *
+          RESULT)
+     These routines compute the complete Fermi-Dirac integral with an
+     index of 0.  This integral is given by F_0(x) = \ln(1 + e^x).
+
+ -- Function: double gsl_sf_fermi_dirac_1 (double X)
+ -- Function: int gsl_sf_fermi_dirac_1_e (double X, gsl_sf_result *
+          RESULT)
+     These routines compute the complete Fermi-Dirac integral with an
+     index of 1, F_1(x) = \int_0^\infty dt (t /(\exp(t-x)+1)).
+
+ -- Function: double gsl_sf_fermi_dirac_2 (double X)
+ -- Function: int gsl_sf_fermi_dirac_2_e (double X, gsl_sf_result *
+          RESULT)
+     These routines compute the complete Fermi-Dirac integral with an
+     index of 2, F_2(x) = (1/2) \int_0^\infty dt (t^2 /(\exp(t-x)+1)).
+
+ -- Function: double gsl_sf_fermi_dirac_int (int J, double X)
+ -- Function: int gsl_sf_fermi_dirac_int_e (int J, double X,
+          gsl_sf_result * RESULT)
+     These routines compute the complete Fermi-Dirac integral with an
+     integer index of j, F_j(x) = (1/\Gamma(j+1)) \int_0^\infty dt (t^j
+     /(\exp(t-x)+1)).
+
+ -- Function: double gsl_sf_fermi_dirac_mhalf (double X)
+ -- Function: int gsl_sf_fermi_dirac_mhalf_e (double X, gsl_sf_result *
+          RESULT)
+     These routines compute the complete Fermi-Dirac integral
+     F_{-1/2}(x).
+
+ -- Function: double gsl_sf_fermi_dirac_half (double X)
+ -- Function: int gsl_sf_fermi_dirac_half_e (double X, gsl_sf_result *
+          RESULT)
+     These routines compute the complete Fermi-Dirac integral
+     F_{1/2}(x).
+
+ -- Function: double gsl_sf_fermi_dirac_3half (double X)
+ -- Function: int gsl_sf_fermi_dirac_3half_e (double X, gsl_sf_result *
+          RESULT)
+     These routines compute the complete Fermi-Dirac integral
+     F_{3/2}(x).
+
+
+File: gsl-ref.info,  Node: Incomplete Fermi-Dirac Integrals,  Prev: Complete Fermi-Dirac Integrals,  Up: Fermi-Dirac Function
+
+7.18.2 Incomplete Fermi-Dirac Integrals
+---------------------------------------
+
+The incomplete Fermi-Dirac integral F_j(x,b) is given by,
+
+     F_j(x,b)   := (1/\Gamma(j+1)) \int_b^\infty dt (t^j / (\Exp(t-x) + 1))
+
+ -- Function: double gsl_sf_fermi_dirac_inc_0 (double X, double B)
+ -- Function: int gsl_sf_fermi_dirac_inc_0_e (double X, double B,
+          gsl_sf_result * RESULT)
+     These routines compute the incomplete Fermi-Dirac integral with an
+     index of zero, F_0(x,b) = \ln(1 + e^{b-x}) - (b-x).
+
+
+File: gsl-ref.info,  Node: Gamma and Beta Functions,  Next: Gegenbauer Functions,  Prev: Fermi-Dirac Function,  Up: Special Functions
+
+7.19 Gamma and Beta Functions
+=============================
+
+The  functions described in this section are declared in the header
+file `gsl_sf_gamma.h'.
+
+* Menu:
+
+* Gamma Functions::
+* Factorials::
+* Pochhammer Symbol::
+* Incomplete Gamma Functions::
+* Beta Functions::
+* Incomplete Beta Function::
+
+
+File: gsl-ref.info,  Node: Gamma Functions,  Next: Factorials,  Up: Gamma and Beta Functions
+
+7.19.1 Gamma Functions
+----------------------
+
+The Gamma function is defined by the following integral,
+
+     \Gamma(x) = \int_0^\infty dt  t^{x-1} \exp(-t)
+
+It is related to the factorial function by \Gamma(n)=(n-1)!  for
+positive integer n.  Further information on the Gamma function can be
+found in Abramowitz & Stegun, Chapter 6.  The functions described in
+this section are declared in the header file `gsl_sf_gamma.h'.
+
+ -- Function: double gsl_sf_gamma (double X)
+ -- Function: int gsl_sf_gamma_e (double X, gsl_sf_result * RESULT)
+     These routines compute the Gamma function \Gamma(x), subject to x
+     not being a negative integer or zero.  The function is computed
+     using the real Lanczos method. The maximum value of x such that
+     \Gamma(x) is not considered an overflow is given by the macro
+     `GSL_SF_GAMMA_XMAX' and is 171.0.
+
+ -- Function: double gsl_sf_lngamma (double X)
+ -- Function: int gsl_sf_lngamma_e (double X, gsl_sf_result * RESULT)
+     These routines compute the logarithm of the Gamma function,
+     \log(\Gamma(x)), subject to x not being a negative integer or
+     zero.  For x<0 the real part of \log(\Gamma(x)) is returned, which
+     is equivalent to \log(|\Gamma(x)|).  The function is computed
+     using the real Lanczos method.
+
+ -- Function: int gsl_sf_lngamma_sgn_e (double X, gsl_sf_result *
+          RESULT_LG, double * SGN)
+     This routine computes the sign of the gamma function and the
+     logarithm of its magnitude, subject to x not being a negative
+     integer or zero.  The function is computed using the real Lanczos
+     method.  The value of the gamma function can be reconstructed
+     using the relation \Gamma(x) = sgn * \exp(resultlg).
+
+ -- Function: double gsl_sf_gammastar (double X)
+ -- Function: int gsl_sf_gammastar_e (double X, gsl_sf_result * RESULT)
+     These routines compute the regulated Gamma Function \Gamma^*(x)
+     for x > 0. The regulated gamma function is given by,
+
+          \Gamma^*(x) = \Gamma(x)/(\sqrt{2\pi} x^{(x-1/2)} \exp(-x))
+                      = (1 + (1/12x) + ...)  for x \to \infty
+     and is a useful suggestion of Temme.
+
+ -- Function: double gsl_sf_gammainv (double X)
+ -- Function: int gsl_sf_gammainv_e (double X, gsl_sf_result * RESULT)
+     These routines compute the reciprocal of the gamma function,
+     1/\Gamma(x) using the real Lanczos method.
+
+ -- Function: int gsl_sf_lngamma_complex_e (double ZR, double ZI,
+          gsl_sf_result * LNR, gsl_sf_result * ARG)
+     This routine computes \log(\Gamma(z)) for complex z=z_r+i z_i and
+     z not a negative integer or zero, using the complex Lanczos
+     method.  The returned parameters are lnr = \log|\Gamma(z)| and arg
+     = \arg(\Gamma(z)) in (-\pi,\pi].  Note that the phase part (ARG)
+     is not well-determined when |z| is very large, due to inevitable
+     roundoff in restricting to (-\pi,\pi].  This will result in a
+     `GSL_ELOSS' error when it occurs.  The absolute value part (LNR),
+     however, never suffers from loss of precision.
+
+
+File: gsl-ref.info,  Node: Factorials,  Next: Pochhammer Symbol,  Prev: Gamma Functions,  Up: Gamma and Beta Functions
+
+7.19.2 Factorials
+-----------------
+
+Although factorials can be computed from the Gamma function, using the
+relation n! = \Gamma(n+1) for non-negative integer n, it is usually
+more efficient to call the functions in this section, particularly for
+small values of n, whose factorial values are maintained in hardcoded
+tables.
+
+ -- Function: double gsl_sf_fact (unsigned int N)
+ -- Function: int gsl_sf_fact_e (unsigned int N, gsl_sf_result * RESULT)
+     These routines compute the factorial n!.  The factorial is related
+     to the Gamma function by n! = \Gamma(n+1).  The maximum value of n
+     such that n! is not considered an overflow is given by the macro
+     `GSL_SF_FACT_NMAX' and is 170.
+
+ -- Function: double gsl_sf_doublefact (unsigned int N)
+ -- Function: int gsl_sf_doublefact_e (unsigned int N, gsl_sf_result *
+          RESULT)
+     These routines compute the double factorial n!! = n(n-2)(n-4)
+     \dots.  The maximum value of n such that n!! is not considered an
+     overflow is given by the macro `GSL_SF_DOUBLEFACT_NMAX' and is 297.
+
+ -- Function: double gsl_sf_lnfact (unsigned int N)
+ -- Function: int gsl_sf_lnfact_e (unsigned int N, gsl_sf_result *
+          RESULT)
+     These routines compute the logarithm of the factorial of N,
+     \log(n!).  The algorithm is faster than computing \ln(\Gamma(n+1))
+     via `gsl_sf_lngamma' for n < 170, but defers for larger N.
+
+ -- Function: double gsl_sf_lndoublefact (unsigned int N)
+ -- Function: int gsl_sf_lndoublefact_e (unsigned int N, gsl_sf_result
+          * RESULT)
+     These routines compute the logarithm of the double factorial of N,
+     \log(n!!).
+
+ -- Function: double gsl_sf_choose (unsigned int N, unsigned int M)
+ -- Function: int gsl_sf_choose_e (unsigned int N, unsigned int M,
+          gsl_sf_result * RESULT)
+     These routines compute the combinatorial factor `n choose m' =
+     n!/(m!(n-m)!)
+
+ -- Function: double gsl_sf_lnchoose (unsigned int N, unsigned int M)
+ -- Function: int gsl_sf_lnchoose_e (unsigned int N, unsigned int M,
+          gsl_sf_result * RESULT)
+     These routines compute the logarithm of `n choose m'.  This is
+     equivalent to the sum \log(n!) - \log(m!) - \log((n-m)!).
+
+ -- Function: double gsl_sf_taylorcoeff (int N, double X)
+ -- Function: int gsl_sf_taylorcoeff_e (int N, double X, gsl_sf_result
+          * RESULT)
+     These routines compute the Taylor coefficient x^n / n! for x >= 0,
+     n >= 0.
+
+
+File: gsl-ref.info,  Node: Pochhammer Symbol,  Next: Incomplete Gamma Functions,  Prev: Factorials,  Up: Gamma and Beta Functions
+
+7.19.3 Pochhammer Symbol
+------------------------
+
+ -- Function: double gsl_sf_poch (double A, double X)
+ -- Function: int gsl_sf_poch_e (double A, double X, gsl_sf_result *
+          RESULT)
+     These routines compute the Pochhammer symbol (a)_x = \Gamma(a +
+     x)/\Gamma(a), subject to a and a+x not being negative integers or
+     zero. The Pochhammer symbol is also known as the Apell symbol and
+     sometimes written as (a,x).
+
+ -- Function: double gsl_sf_lnpoch (double A, double X)
+ -- Function: int gsl_sf_lnpoch_e (double A, double X, gsl_sf_result *
+          RESULT)
+     These routines compute the logarithm of the Pochhammer symbol,
+     \log((a)_x) = \log(\Gamma(a + x)/\Gamma(a)) for a > 0, a+x > 0.
+
+ -- Function: int gsl_sf_lnpoch_sgn_e (double A, double X,
+          gsl_sf_result * RESULT, double * SGN)
+     These routines compute the sign of the Pochhammer symbol and the
+     logarithm of its magnitude.  The computed parameters are result =
+     \log(|(a)_x|) and sgn = \sgn((a)_x) where (a)_x = \Gamma(a +
+     x)/\Gamma(a), subject to a, a+x not being negative integers or
+     zero.
+
+ -- Function: double gsl_sf_pochrel (double A, double X)
+ -- Function: int gsl_sf_pochrel_e (double A, double X, gsl_sf_result *
+          RESULT)
+     These routines compute the relative Pochhammer symbol ((a)_x -
+     1)/x where (a)_x = \Gamma(a + x)/\Gamma(a).
+
+
+File: gsl-ref.info,  Node: Incomplete Gamma Functions,  Next: Beta Functions,  Prev: Pochhammer Symbol,  Up: Gamma and Beta Functions
+
+7.19.4 Incomplete Gamma Functions
+---------------------------------
+
+ -- Function: double gsl_sf_gamma_inc (double A, double X)
+ -- Function: int gsl_sf_gamma_inc_e (double A, double X, gsl_sf_result
+          * RESULT)
+     These functions compute the unnormalized incomplete Gamma Function
+     \Gamma(a,x) = \int_x^\infty dt t^{a-1} \exp(-t) for a real and x
+     >= 0.
+
+ -- Function: double gsl_sf_gamma_inc_Q (double A, double X)
+ -- Function: int gsl_sf_gamma_inc_Q_e (double A, double X,
+          gsl_sf_result * RESULT)
+     These routines compute the normalized incomplete Gamma Function
+     Q(a,x) = 1/\Gamma(a) \int_x^\infty dt t^{a-1} \exp(-t) for a > 0,
+     x >= 0.
+
+ -- Function: double gsl_sf_gamma_inc_P (double A, double X)
+ -- Function: int gsl_sf_gamma_inc_P_e (double A, double X,
+          gsl_sf_result * RESULT)
+     These routines compute the complementary normalized incomplete
+     Gamma Function P(a,x) = 1 - Q(a,x) = 1/\Gamma(a) \int_0^x dt
+     t^{a-1} \exp(-t) for a > 0, x >= 0.
+
+     Note that Abramowitz & Stegun call P(a,x) the incomplete gamma
+     function (section 6.5).
+
+
+File: gsl-ref.info,  Node: Beta Functions,  Next: Incomplete Beta Function,  Prev: Incomplete Gamma Functions,  Up: Gamma and Beta Functions
+
+7.19.5 Beta Functions
+---------------------
+
+ -- Function: double gsl_sf_beta (double A, double B)
+ -- Function: int gsl_sf_beta_e (double A, double B, gsl_sf_result *
+          RESULT)
+     These routines compute the Beta Function, B(a,b) =
+     \Gamma(a)\Gamma(b)/\Gamma(a+b) subject to a and b not being
+     negative integers.
+
+ -- Function: double gsl_sf_lnbeta (double A, double B)
+ -- Function: int gsl_sf_lnbeta_e (double A, double B, gsl_sf_result *
+          RESULT)
+     These routines compute the logarithm of the Beta Function,
+     \log(B(a,b)) subject to a and b not being negative integers.
+
+
+File: gsl-ref.info,  Node: Incomplete Beta Function,  Prev: Beta Functions,  Up: Gamma and Beta Functions
+
+7.19.6 Incomplete Beta Function
+-------------------------------
+
+ -- Function: double gsl_sf_beta_inc (double A, double B, double X)
+ -- Function: int gsl_sf_beta_inc_e (double A, double B, double X,
+          gsl_sf_result * RESULT)
+     These routines compute the normalized incomplete Beta function
+     I_x(a,b)=B_x(a,b)/B(a,b) where B_x(a,b) = \int_0^x t^{a-1}
+     (1-t)^{b-1} dt for a > 0, b > 0, and 0 <= x <= 1.
+
+
+File: gsl-ref.info,  Node: Gegenbauer Functions,  Next: Hypergeometric Functions,  Prev: Gamma and Beta Functions,  Up: Special Functions
+
+7.20 Gegenbauer Functions
+=========================
+
+The Gegenbauer polynomials are defined in Abramowitz & Stegun, Chapter
+22, where they are known as Ultraspherical polynomials.  The functions
+described in this section are declared in the header file
+`gsl_sf_gegenbauer.h'.
+
+ -- Function: double gsl_sf_gegenpoly_1 (double LAMBDA, double X)
+ -- Function: double gsl_sf_gegenpoly_2 (double LAMBDA, double X)
+ -- Function: double gsl_sf_gegenpoly_3 (double LAMBDA, double X)
+ -- Function: int gsl_sf_gegenpoly_1_e (double LAMBDA, double X,
+          gsl_sf_result * RESULT)
+ -- Function: int gsl_sf_gegenpoly_2_e (double LAMBDA, double X,
+          gsl_sf_result * RESULT)
+ -- Function: int gsl_sf_gegenpoly_3_e (double LAMBDA, double X,
+          gsl_sf_result * RESULT)
+     These functions evaluate the Gegenbauer polynomials
+     C^{(\lambda)}_n(x) using explicit representations for n =1, 2, 3.
+
+ -- Function: double gsl_sf_gegenpoly_n (int N, double LAMBDA, double X)
+ -- Function: int gsl_sf_gegenpoly_n_e (int N, double LAMBDA, double X,
+          gsl_sf_result * RESULT)
+     These functions evaluate the Gegenbauer polynomial
+     C^{(\lambda)}_n(x) for a specific value of N, LAMBDA, X subject to
+     \lambda > -1/2, n >= 0.
+
+ -- Function: int gsl_sf_gegenpoly_array (int NMAX, double LAMBDA,
+          double X, double RESULT_ARRAY[])
+     This function computes an array of Gegenbauer polynomials
+     C^{(\lambda)}_n(x) for n = 0, 1, 2, \dots, nmax, subject to
+     \lambda > -1/2, nmax >= 0.
+
+
+File: gsl-ref.info,  Node: Hypergeometric Functions,  Next: Laguerre Functions,  Prev: Gegenbauer Functions,  Up: Special Functions
+
+7.21 Hypergeometric Functions
+=============================
+
+Hypergeometric functions are described in Abramowitz & Stegun, Chapters
+13 and 15.  These functions are declared in the header file
+`gsl_sf_hyperg.h'.
+
+ -- Function: double gsl_sf_hyperg_0F1 (double C, double X)
+ -- Function: int gsl_sf_hyperg_0F1_e (double C, double X,
+          gsl_sf_result * RESULT)
+     These routines compute the hypergeometric function 0F1(c,x).
+
+ -- Function: double gsl_sf_hyperg_1F1_int (int M, int N, double X)
+ -- Function: int gsl_sf_hyperg_1F1_int_e (int M, int N, double X,
+          gsl_sf_result * RESULT)
+     These routines compute the confluent hypergeometric function
+     1F1(m,n,x) = M(m,n,x) for integer parameters M, N.
+
+ -- Function: double gsl_sf_hyperg_1F1 (double A, double B, double X)
+ -- Function: int gsl_sf_hyperg_1F1_e (double A, double B, double X,
+          gsl_sf_result * RESULT)
+     These routines compute the confluent hypergeometric function
+     1F1(a,b,x) = M(a,b,x) for general parameters A, B.
+
+ -- Function: double gsl_sf_hyperg_U_int (int M, int N, double X)
+ -- Function: int gsl_sf_hyperg_U_int_e (int M, int N, double X,
+          gsl_sf_result * RESULT)
+     These routines compute the confluent hypergeometric function
+     U(m,n,x) for integer parameters M, N.
+
+ -- Function: int gsl_sf_hyperg_U_int_e10_e (int M, int N, double X,
+          gsl_sf_result_e10 * RESULT)
+     This routine computes the confluent hypergeometric function
+     U(m,n,x) for integer parameters M, N using the `gsl_sf_result_e10'
+     type to return a result with extended range.
+
+ -- Function: double gsl_sf_hyperg_U (double A, double B, double X)
+ -- Function: int gsl_sf_hyperg_U_e (double A, double B, double X,
+          gsl_sf_result * RESULT)
+     These routines compute the confluent hypergeometric function
+     U(a,b,x).
+
+ -- Function: int gsl_sf_hyperg_U_e10_e (double A, double B, double X,
+          gsl_sf_result_e10 * RESULT)
+     This routine computes the confluent hypergeometric function
+     U(a,b,x) using the `gsl_sf_result_e10' type to return a result
+     with extended range.
+
+ -- Function: double gsl_sf_hyperg_2F1 (double A, double B, double C,
+          double X)
+ -- Function: int gsl_sf_hyperg_2F1_e (double A, double B, double C,
+          double X, gsl_sf_result * RESULT)
+     These routines compute the Gauss hypergeometric function
+     2F1(a,b,c,x) for |x| < 1.
+
+     If the arguments (a,b,c,x) are too close to a singularity then the
+     function can return the error code `GSL_EMAXITER' when the series
+     approximation converges too slowly.  This occurs in the region of
+     x=1, c - a - b = m for integer m.
+
+ -- Function: double gsl_sf_hyperg_2F1_conj (double AR, double AI,
+          double C, double X)
+ -- Function: int gsl_sf_hyperg_2F1_conj_e (double AR, double AI,
+          double C, double X, gsl_sf_result * RESULT)
+     These routines compute the Gauss hypergeometric function 2F1(a_R +
+     i a_I, a_R - i a_I, c, x) with complex parameters for |x| < 1.
+     exceptions:
+
+ -- Function: double gsl_sf_hyperg_2F1_renorm (double A, double B,
+          double C, double X)
+ -- Function: int gsl_sf_hyperg_2F1_renorm_e (double A, double B,
+          double C, double X, gsl_sf_result * RESULT)
+     These routines compute the renormalized Gauss hypergeometric
+     function 2F1(a,b,c,x) / \Gamma(c) for |x| < 1.
+
+ -- Function: double gsl_sf_hyperg_2F1_conj_renorm (double AR, double
+          AI, double C, double X)
+ -- Function: int gsl_sf_hyperg_2F1_conj_renorm_e (double AR, double
+          AI, double C, double X, gsl_sf_result * RESULT)
+     These routines compute the renormalized Gauss hypergeometric
+     function 2F1(a_R + i a_I, a_R - i a_I, c, x) / \Gamma(c) for |x| <
+     1.
+
+ -- Function: double gsl_sf_hyperg_2F0 (double A, double B, double X)
+ -- Function: int gsl_sf_hyperg_2F0_e (double A, double B, double X,
+          gsl_sf_result * RESULT)
+     These routines compute the hypergeometric function 2F0(a,b,x).
+     The series representation is a divergent hypergeometric series.
+     However, for x < 0 we have 2F0(a,b,x) = (-1/x)^a U(a,1+a-b,-1/x)
+
+
+File: gsl-ref.info,  Node: Laguerre Functions,  Next: Lambert W Functions,  Prev: Hypergeometric Functions,  Up: Special Functions
+
+7.22 Laguerre Functions
+=======================
+
+The generalized Laguerre polynomials are defined in terms of confluent
+hypergeometric functions as L^a_n(x) = ((a+1)_n / n!) 1F1(-n,a+1,x),
+and are sometimes referred to as the associated Laguerre polynomials.
+They are related to the plain Laguerre polynomials L_n(x) by L^0_n(x) =
+L_n(x) and L^k_n(x) = (-1)^k (d^k/dx^k) L_(n+k)(x).  For more
+information see Abramowitz & Stegun, Chapter 22.
+
+   The functions described in this section are declared in the header
+file `gsl_sf_laguerre.h'.
+
+ -- Function: double gsl_sf_laguerre_1 (double A, double X)
+ -- Function: double gsl_sf_laguerre_2 (double A, double X)
+ -- Function: double gsl_sf_laguerre_3 (double A, double X)
+ -- Function: int gsl_sf_laguerre_1_e (double A, double X,
+          gsl_sf_result * RESULT)
+ -- Function: int gsl_sf_laguerre_2_e (double A, double X,
+          gsl_sf_result * RESULT)
+ -- Function: int gsl_sf_laguerre_3_e (double A, double X,
+          gsl_sf_result * RESULT)
+     These routines evaluate the generalized Laguerre polynomials
+     L^a_1(x), L^a_2(x), L^a_3(x) using explicit representations.
+
+ -- Function: double gsl_sf_laguerre_n (const int N, const double A,
+          const double X)
+ -- Function: int gsl_sf_laguerre_n_e (int N, double A, double X,
+          gsl_sf_result * RESULT)
+     These routines evaluate the generalized Laguerre polynomials
+     L^a_n(x) for a > -1, n >= 0.
+
+
+
+File: gsl-ref.info,  Node: Lambert W Functions,  Next: Legendre Functions and Spherical Harmonics,  Prev: Laguerre Functions,  Up: Special Functions
+
+7.23 Lambert W Functions
+========================
+
+Lambert's W functions, W(x), are defined to be solutions of the
+equation W(x) \exp(W(x)) = x. This function has multiple branches for x
+< 0; however, it has only two real-valued branches. We define W_0(x) to
+be the principal branch, where W > -1 for x < 0, and W_{-1}(x) to be
+the other real branch, where W < -1 for x < 0.  The Lambert functions
+are declared in the header file `gsl_sf_lambert.h'.
+
+ -- Function: double gsl_sf_lambert_W0 (double X)
+ -- Function: int gsl_sf_lambert_W0_e (double X, gsl_sf_result * RESULT)
+     These compute the principal branch of the Lambert W function,
+     W_0(x).
+
+ -- Function: double gsl_sf_lambert_Wm1 (double X)
+ -- Function: int gsl_sf_lambert_Wm1_e (double X, gsl_sf_result *
+          RESULT)
+     These compute the secondary real-valued branch of the Lambert W
+     function, W_{-1}(x).
+
+
+File: gsl-ref.info,  Node: Legendre Functions and Spherical Harmonics,  Next: Logarithm and Related Functions,  Prev: Lambert W Functions,  Up: Special Functions
+
+7.24 Legendre Functions and Spherical Harmonics
+===============================================
+
+The Legendre Functions and Legendre Polynomials are described in
+Abramowitz & Stegun, Chapter 8.  These functions are declared in the
+header file `gsl_sf_legendre.h'.
+
+* Menu:
+
+* Legendre Polynomials::
+* Associated Legendre Polynomials and Spherical Harmonics::
+* Conical Functions::
+* Radial Functions for Hyperbolic Space::
+
+
+File: gsl-ref.info,  Node: Legendre Polynomials,  Next: Associated Legendre Polynomials and Spherical Harmonics,  Up: Legendre Functions and Spherical Harmonics
+
+7.24.1 Legendre Polynomials
+---------------------------
+
+ -- Function: double gsl_sf_legendre_P1 (double X)
+ -- Function: double gsl_sf_legendre_P2 (double X)
+ -- Function: double gsl_sf_legendre_P3 (double X)
+ -- Function: int gsl_sf_legendre_P1_e (double X, gsl_sf_result *
+          RESULT)
+ -- Function: int gsl_sf_legendre_P2_e (double X, gsl_sf_result *
+          RESULT)
+ -- Function: int gsl_sf_legendre_P3_e (double X, gsl_sf_result *
+          RESULT)
+     These functions evaluate the Legendre polynomials P_l(x) using
+     explicit representations for l=1, 2, 3.
+
+ -- Function: double gsl_sf_legendre_Pl (int L, double X)
+ -- Function: int gsl_sf_legendre_Pl_e (int L, double X, gsl_sf_result
+          * RESULT)
+     These functions evaluate the Legendre polynomial P_l(x) for a
+     specific value of L, X subject to l >= 0, |x| <= 1
+
+ -- Function: int gsl_sf_legendre_Pl_array (int LMAX, double X, double
+          RESULT_ARRAY[])
+ -- Function: int gsl_sf_legendre_Pl_deriv_array (int LMAX, double X,
+          double RESULT_ARRAY[], double RESULT_DERIV_ARRAY[])
+     These functions compute an array of Legendre polynomials P_l(x),
+     and optionally their derivatives dP_l(x)/dx, for l = 0, \dots,
+     lmax, |x| <= 1
+
+ -- Function: double gsl_sf_legendre_Q0 (double X)
+ -- Function: int gsl_sf_legendre_Q0_e (double X, gsl_sf_result *
+          RESULT)
+     These routines compute the Legendre function Q_0(x) for x > -1, x
+     != 1.
+
+ -- Function: double gsl_sf_legendre_Q1 (double X)
+ -- Function: int gsl_sf_legendre_Q1_e (double X, gsl_sf_result *
+          RESULT)
+     These routines compute the Legendre function Q_1(x) for x > -1, x
+     != 1.
+
+ -- Function: double gsl_sf_legendre_Ql (int L, double X)
+ -- Function: int gsl_sf_legendre_Ql_e (int L, double X, gsl_sf_result
+          * RESULT)
+     These routines compute the Legendre function Q_l(x) for x > -1, x
+     != 1 and l >= 0.
+
+
+File: gsl-ref.info,  Node: Associated Legendre Polynomials and Spherical Harmonics,  Next: Conical Functions,  Prev: Legendre Polynomials,  Up: Legendre Functions and Spherical Harmonics
+
+7.24.2 Associated Legendre Polynomials and Spherical Harmonics
+--------------------------------------------------------------
+
+The following functions compute the associated Legendre Polynomials
+P_l^m(x).  Note that this function grows combinatorially with l and can
+overflow for l larger than about 150.  There is no trouble for small m,
+but overflow occurs when m and l are both large.  Rather than allow
+overflows, these functions refuse to calculate P_l^m(x) and return
+`GSL_EOVRFLW' when they can sense that l and m are too big.
+
+   If you want to calculate a spherical harmonic, then _do not_ use
+these functions.  Instead use `gsl_sf_legendre_sphPlm' below, which
+uses a similar recursion, but with the normalized functions.
+
+ -- Function: double gsl_sf_legendre_Plm (int L, int M, double X)
+ -- Function: int gsl_sf_legendre_Plm_e (int L, int M, double X,
+          gsl_sf_result * RESULT)
+     These routines compute the associated Legendre polynomial P_l^m(x)
+     for m >= 0, l >= m, |x| <= 1.
+
+ -- Function: int gsl_sf_legendre_Plm_array (int LMAX, int M, double X,
+          double RESULT_ARRAY[])
+ -- Function: int gsl_sf_legendre_Plm_deriv_array (int LMAX, int M,
+          double X, double RESULT_ARRAY[], double RESULT_DERIV_ARRAY[])
+     These functions compute an array of Legendre polynomials P_l^m(x),
+     and optionally their derivatives dP_l^m(x)/dx, for m >= 0, l =
+     |m|, ..., lmax, |x| <= 1.
+
+ -- Function: double gsl_sf_legendre_sphPlm (int L, int M, double X)
+ -- Function: int gsl_sf_legendre_sphPlm_e (int L, int M, double X,
+          gsl_sf_result * RESULT)
+     These routines compute the normalized associated Legendre
+     polynomial $\sqrt{(2l+1)/(4\pi)} \sqrt{(l-m)!/(l+m)!} P_l^m(x)$
+     suitable for use in spherical harmonics.  The parameters must
+     satisfy m >= 0, l >= m, |x| <= 1. Theses routines avoid the
+     overflows that occur for the standard normalization of P_l^m(x).
+
+ -- Function: int gsl_sf_legendre_sphPlm_array (int LMAX, int M, double
+          X, double RESULT_ARRAY[])
+ -- Function: int gsl_sf_legendre_sphPlm_deriv_array (int LMAX, int M,
+          double X, double RESULT_ARRAY[], double RESULT_DERIV_ARRAY[])
+     These functions compute an array of normalized associated Legendre
+     functions $\sqrt{(2l+1)/(4\pi)} \sqrt{(l-m)!/(l+m)!} P_l^m(x)$,
+     and optionally their derivatives, for m >= 0, l = |m|, ..., lmax,
+     |x| <= 1.0
+
+ -- Function: int gsl_sf_legendre_array_size (const int LMAX, const int
+          M)
+     This function returns the size of RESULT_ARRAY[] needed for the
+     array versions of P_l^m(x), LMAX - M + 1.
+
+
+File: gsl-ref.info,  Node: Conical Functions,  Next: Radial Functions for Hyperbolic Space,  Prev: Associated Legendre Polynomials and Spherical Harmonics,  Up: Legendre Functions and Spherical Harmonics
+
+7.24.3 Conical Functions
+------------------------
+
+The Conical Functions P^\mu_{-(1/2)+i\lambda}(x) and
+Q^\mu_{-(1/2)+i\lambda} are described in Abramowitz & Stegun, Section
+8.12.
+
+ -- Function: double gsl_sf_conicalP_half (double LAMBDA, double X)
+ -- Function: int gsl_sf_conicalP_half_e (double LAMBDA, double X,
+          gsl_sf_result * RESULT)
+     These routines compute the irregular Spherical Conical Function
+     P^{1/2}_{-1/2 + i \lambda}(x) for x > -1.
+
+ -- Function: double gsl_sf_conicalP_mhalf (double LAMBDA, double X)
+ -- Function: int gsl_sf_conicalP_mhalf_e (double LAMBDA, double X,
+          gsl_sf_result * RESULT)
+     These routines compute the regular Spherical Conical Function
+     P^{-1/2}_{-1/2 + i \lambda}(x) for x > -1.
+
+ -- Function: double gsl_sf_conicalP_0 (double LAMBDA, double X)
+ -- Function: int gsl_sf_conicalP_0_e (double LAMBDA, double X,
+          gsl_sf_result * RESULT)
+     These routines compute the conical function P^0_{-1/2 + i
+     \lambda}(x) for x > -1.
+
+ -- Function: double gsl_sf_conicalP_1 (double LAMBDA, double X)
+ -- Function: int gsl_sf_conicalP_1_e (double LAMBDA, double X,
+          gsl_sf_result * RESULT)
+     These routines compute the conical function P^1_{-1/2 + i
+     \lambda}(x) for x > -1.
+
+ -- Function: double gsl_sf_conicalP_sph_reg (int L, double LAMBDA,
+          double X)
+ -- Function: int gsl_sf_conicalP_sph_reg_e (int L, double LAMBDA,
+          double X, gsl_sf_result * RESULT)
+     These routines compute the Regular Spherical Conical Function
+     P^{-1/2-l}_{-1/2 + i \lambda}(x) for x > -1, l >= -1.
+
+ -- Function: double gsl_sf_conicalP_cyl_reg (int M, double LAMBDA,
+          double X)
+ -- Function: int gsl_sf_conicalP_cyl_reg_e (int M, double LAMBDA,
+          double X, gsl_sf_result * RESULT)
+     These routines compute the Regular Cylindrical Conical Function
+     P^{-m}_{-1/2 + i \lambda}(x) for x > -1, m >= -1.
+
+
+File: gsl-ref.info,  Node: Radial Functions for Hyperbolic Space,  Prev: Conical Functions,  Up: Legendre Functions and Spherical Harmonics
+
+7.24.4 Radial Functions for Hyperbolic Space
+--------------------------------------------
+
+The following spherical functions are specializations of Legendre
+functions which give the regular eigenfunctions of the Laplacian on a
+3-dimensional hyperbolic space H3d.  Of particular interest is the flat
+limit, \lambda \to \infty, \eta \to 0, \lambda\eta fixed.
+
+ -- Function: double gsl_sf_legendre_H3d_0 (double LAMBDA, double ETA)
+ -- Function: int gsl_sf_legendre_H3d_0_e (double LAMBDA, double ETA,
+          gsl_sf_result * RESULT)
+     These routines compute the zeroth radial eigenfunction of the
+     Laplacian on the 3-dimensional hyperbolic space,
+     L^{H3d}_0(\lambda,\eta) := \sin(\lambda\eta)/(\lambda\sinh(\eta))
+     for \eta >= 0.  In the flat limit this takes the form
+     L^{H3d}_0(\lambda,\eta) = j_0(\lambda\eta).
+
+ -- Function: double gsl_sf_legendre_H3d_1 (double LAMBDA, double ETA)
+ -- Function: int gsl_sf_legendre_H3d_1_e (double LAMBDA, double ETA,
+          gsl_sf_result * RESULT)
+     These routines compute the first radial eigenfunction of the
+     Laplacian on the 3-dimensional hyperbolic space,
+     L^{H3d}_1(\lambda,\eta) := 1/\sqrt{\lambda^2 + 1} \sin(\lambda
+     \eta)/(\lambda \sinh(\eta)) (\coth(\eta) - \lambda
+     \cot(\lambda\eta)) for \eta >= 0.  In the flat limit this takes
+     the form L^{H3d}_1(\lambda,\eta) = j_1(\lambda\eta).
+
+ -- Function: double gsl_sf_legendre_H3d (int L, double LAMBDA, double
+          ETA)
+ -- Function: int gsl_sf_legendre_H3d_e (int L, double LAMBDA, double
+          ETA, gsl_sf_result * RESULT)
+     These routines compute the L-th radial eigenfunction of the
+     Laplacian on the 3-dimensional hyperbolic space \eta >= 0, l >= 0.
+     In the flat limit this takes the form L^{H3d}_l(\lambda,\eta) =
+     j_l(\lambda\eta).
+
+ -- Function: int gsl_sf_legendre_H3d_array (int LMAX, double LAMBDA,
+          double ETA, double RESULT_ARRAY[])
+     This function computes an array of radial eigenfunctions
+     L^{H3d}_l(\lambda, \eta) for 0 <= l <= lmax.
+
+
+File: gsl-ref.info,  Node: Logarithm and Related Functions,  Next: Mathieu Functions,  Prev: Legendre Functions and Spherical Harmonics,  Up: Special Functions
+
+7.25 Logarithm and Related Functions
+====================================
+
+Information on the properties of the Logarithm function can be found in
+Abramowitz & Stegun, Chapter 4.  The functions described in this section
+are declared in the header file `gsl_sf_log.h'.
+
+ -- Function: double gsl_sf_log (double X)
+ -- Function: int gsl_sf_log_e (double X, gsl_sf_result * RESULT)
+     These routines compute the logarithm of X, \log(x), for x > 0.
+
+ -- Function: double gsl_sf_log_abs (double X)
+ -- Function: int gsl_sf_log_abs_e (double X, gsl_sf_result * RESULT)
+     These routines compute the logarithm of the magnitude of X,
+     \log(|x|), for x \ne 0.
+
+ -- Function: int gsl_sf_complex_log_e (double ZR, double ZI,
+          gsl_sf_result * LNR, gsl_sf_result * THETA)
+     This routine computes the complex logarithm of z = z_r + i z_i.
+     The results are returned as LNR, THETA such that \exp(lnr + i
+     \theta) = z_r + i z_i, where \theta lies in the range [-\pi,\pi].
+
+ -- Function: double gsl_sf_log_1plusx (double X)
+ -- Function: int gsl_sf_log_1plusx_e (double X, gsl_sf_result * RESULT)
+     These routines compute \log(1 + x) for x > -1 using an algorithm
+     that is accurate for small x.
+
+ -- Function: double gsl_sf_log_1plusx_mx (double X)
+ -- Function: int gsl_sf_log_1plusx_mx_e (double X, gsl_sf_result *
+          RESULT)
+     These routines compute \log(1 + x) - x for x > -1 using an
+     algorithm that is accurate for small x.
+
+
+File: gsl-ref.info,  Node: Mathieu Functions,  Next: Power Function,  Prev: Logarithm and Related Functions,  Up: Special Functions
+
+7.26 Mathieu Functions
+======================
+
+The routines described in this section compute the angular and radial
+Mathieu functions, and their characteristic values.  Mathieu functions
+are the solutions of the following two differential equations:
+
+     d^2y/dv^2 + (a - 2q\cos 2v)y = 0
+     d^2f/du^2 - (a - 2q\cosh 2u)f = 0
+
+The angular Mathieu functions ce_r(x,q), se_r(x,q) are the even and odd
+periodic solutions of the first equation, which is known as Mathieu's
+equation. These exist only for the discrete sequence of  characteristic
+values a=a_r(q) (even-periodic) and a=b_r(q) (odd-periodic).
+
+   The radial Mathieu functions Mc^{(j)}_{r}(z,q), Ms^{(j)}_{r}(z,q)
+are the solutions of the second equation, which is referred to as
+Mathieu's modified equation.  The radial Mathieu functions of the
+first, second, third and fourth kind are denoted by the parameter j,
+which takes the value 1, 2, 3 or 4.
+
+   For more information on the Mathieu functions, see Abramowitz and
+Stegun, Chapter 20.  These functions are defined in the header file
+`gsl_sf_mathieu.h'.
+
+* Menu:
+
+* Mathieu Function Workspace::
+* Mathieu Function Characteristic Values::
+* Angular Mathieu Functions::
+* Radial Mathieu Functions::
+
+
+File: gsl-ref.info,  Node: Mathieu Function Workspace,  Next: Mathieu Function Characteristic Values,  Up: Mathieu Functions
+
+7.26.1 Mathieu Function Workspace
+---------------------------------
+
+The Mathieu functions can be computed for a single order or for
+multiple orders, using array-based routines.  The array-based routines
+require a preallocated workspace.
+
+ -- Function: gsl_sf_mathieu_workspace * gsl_sf_mathieu_alloc (size_t
+          N, double QMAX)
+     This function returns a workspace for the array versions of the
+     Mathieu routines.  The arguments N and QMAX specify the maximum
+     order and q-value of Mathieu functions which can be computed with
+     this workspace.
+
+
+ -- Function: void gsl_sf_mathieu_free (gsl_sf_mathieu_workspace *WORK)
+     This function frees the workspace WORK.
+
+
+File: gsl-ref.info,  Node: Mathieu Function Characteristic Values,  Next: Angular Mathieu Functions,  Prev: Mathieu Function Workspace,  Up: Mathieu Functions
+
+7.26.2 Mathieu Function Characteristic Values
+---------------------------------------------
+
+ -- Function: int gsl_sf_mathieu_a (int N, double Q, gsl_sf_result
+          *RESULT)
+ -- Function: int gsl_sf_mathieu_b (int N, double Q, gsl_sf_result
+          *RESULT)
+     These routines compute the characteristic values a_n(q), b_n(q) of
+     the Mathieu functions ce_n(q,x) and se_n(q,x), respectively.
+
+ -- Function: int gsl_sf_mathieu_a_array (int ORDER_MIN, int ORDER_MAX,
+          double Q, gsl_sf_mathieu_workspace *WORK, double
+          RESULT_ARRAY[])
+ -- Function: int gsl_sf_mathieu_b_array (int ORDER_MIN, int ORDER_MAX,
+          double Q, gsl_sf_mathieu_workspace *WORK, double
+          RESULT_ARRAY[])
+     These routines compute a series of Mathieu characteristic values
+     a_n(q), b_n(q) for n from ORDER_MIN to ORDER_MAX inclusive,
+     storing the results in the array RESULT_ARRAY.
+
+
+File: gsl-ref.info,  Node: Angular Mathieu Functions,  Next: Radial Mathieu Functions,  Prev: Mathieu Function Characteristic Values,  Up: Mathieu Functions
+
+7.26.3 Angular Mathieu Functions
+--------------------------------
+
+ -- Function: int gsl_sf_mathieu_ce (int N, double Q, double X,
+          gsl_sf_result *RESULT)
+ -- Function: int gsl_sf_mathieu_se (int N, double Q, double X,
+          gsl_sf_result *RESULT)
+     These routines compute the angular Mathieu functions ce_n(q,x) and
+     se_n(q,x), respectively.
+
+ -- Function: int gsl_sf_mathieu_ce_array (int NMIN, int NMAX, double
+          Q, double X, gsl_sf_mathieu_workspace *WORK, double
+          RESULT_ARRAY[])
+ -- Function: int gsl_sf_mathieu_se_array (int NMIN, int NMAX, double
+          Q, double X, gsl_sf_mathieu_workspace *WORK, double
+          RESULT_ARRAY[])
+     These routines compute a series of the angular Mathieu functions
+     ce_n(q,x) and se_n(q,x) of order n from NMIN to NMAX inclusive,
+     storing the results in the array RESULT_ARRAY.
+
+
+File: gsl-ref.info,  Node: Radial Mathieu Functions,  Prev: Angular Mathieu Functions,  Up: Mathieu Functions
+
+7.26.4 Radial Mathieu Functions
+-------------------------------
+
+ -- Function: int gsl_sf_mathieu_Mc (int J, int N, double Q, double X,
+          gsl_sf_result *RESULT)
+ -- Function: int gsl_sf_mathieu_Ms (int J, int N, double Q, double X,
+          gsl_sf_result *RESULT)
+     These routines compute the radial J-th kind Mathieu functions
+     Mc_n^{(j)}(q,x) and Ms_n^{(j)}(q,x) of order N.
+
+     The allowed values of J are 1 and 2.  The functions for j = 3,4
+     can be computed as M_n^{(3)} = M_n^{(1)} + iM_n^{(2)} and
+     M_n^{(4)} = M_n^{(1)} - iM_n^{(2)}, where M_n^{(j)} = Mc_n^{(j)} or
+     Ms_n^{(j)}.
+
+ -- Function: int gsl_sf_mathieu_Mc_array (int J, int NMIN, int NMAX,
+          double Q, double X, gsl_sf_mathieu_workspace *WORK, double
+          RESULT_ARRAY[])
+ -- Function: int gsl_sf_mathieu_Ms_array (int J, int NMIN, int NMAX,
+          double Q, double X, gsl_sf_mathieu_workspace *WORK, double
+          RESULT_ARRAY[])
+     These routines compute a series of the radial Mathieu functions of
+     kind J, with order from NMIN to NMAX inclusive, storing the
+     results in the array RESULT_ARRAY.
+
+
+File: gsl-ref.info,  Node: Power Function,  Next: Psi (Digamma) Function,  Prev: Mathieu Functions,  Up: Special Functions
+
+7.27 Power Function
+===================
+
+The following functions are equivalent to the function `gsl_pow_int'
+(*note Small integer powers::) with an error estimate.  These functions
+are declared in the header file `gsl_sf_pow_int.h'.
+
+ -- Function: double gsl_sf_pow_int (double X, int N)
+ -- Function: int gsl_sf_pow_int_e (double X, int N, gsl_sf_result *
+          RESULT)
+     These routines compute the power x^n for integer N.  The power is
+     computed using the minimum number of multiplications. For example,
+     x^8 is computed as ((x^2)^2)^2, requiring only 3 multiplications.
+     For reasons of efficiency, these functions do not check for
+     overflow or underflow conditions.
+
+     #include <gsl/gsl_sf_pow_int.h>
+     /* compute 3.0**12 */
+     double y = gsl_sf_pow_int(3.0, 12);
+
+
+File: gsl-ref.info,  Node: Psi (Digamma) Function,  Next: Synchrotron Functions,  Prev: Power Function,  Up: Special Functions
+
+7.28 Psi (Digamma) Function
+===========================
+
+The polygamma functions of order n are defined by
+
+     \psi^{(n)}(x) = (d/dx)^n \psi(x) = (d/dx)^{n+1} \log(\Gamma(x))
+
+where \psi(x) = \Gamma'(x)/\Gamma(x) is known as the digamma function.
+These functions are declared in the header file `gsl_sf_psi.h'.
+
+* Menu:
+
+* Digamma Function::
+* Trigamma Function::
+* Polygamma Function::
+
+
+File: gsl-ref.info,  Node: Digamma Function,  Next: Trigamma Function,  Up: Psi (Digamma) Function
+
+7.28.1 Digamma Function
+-----------------------
+
+ -- Function: double gsl_sf_psi_int (int N)
+ -- Function: int gsl_sf_psi_int_e (int N, gsl_sf_result * RESULT)
+     These routines compute the digamma function \psi(n) for positive
+     integer N.  The digamma function is also called the Psi function.
+
+ -- Function: double gsl_sf_psi (double X)
+ -- Function: int gsl_sf_psi_e (double X, gsl_sf_result * RESULT)
+     These routines compute the digamma function \psi(x) for general x,
+     x \ne 0.
+
+ -- Function: double gsl_sf_psi_1piy (double Y)
+ -- Function: int gsl_sf_psi_1piy_e (double Y, gsl_sf_result * RESULT)
+     These routines compute the real part of the digamma function on
+     the line 1+i y, \Re[\psi(1 + i y)].
+
+
+File: gsl-ref.info,  Node: Trigamma Function,  Next: Polygamma Function,  Prev: Digamma Function,  Up: Psi (Digamma) Function
+
+7.28.2 Trigamma Function
+------------------------
+
+ -- Function: double gsl_sf_psi_1_int (int N)
+ -- Function: int gsl_sf_psi_1_int_e (int N, gsl_sf_result * RESULT)
+     These routines compute the Trigamma function \psi'(n) for positive
+     integer n.
+
+ -- Function: double gsl_sf_psi_1 (double X)
+ -- Function: int gsl_sf_psi_1_e (double X, gsl_sf_result * RESULT)
+     These routines compute the Trigamma function \psi'(x) for general
+     x.
+
+
+File: gsl-ref.info,  Node: Polygamma Function,  Prev: Trigamma Function,  Up: Psi (Digamma) Function
+
+7.28.3 Polygamma Function
+-------------------------
+
+ -- Function: double gsl_sf_psi_n (int N, double X)
+ -- Function: int gsl_sf_psi_n_e (int N, double X, gsl_sf_result *
+          RESULT)
+     These routines compute the polygamma function \psi^{(n)}(x) for n
+     >= 0, x > 0.
+
+
+File: gsl-ref.info,  Node: Synchrotron Functions,  Next: Transport Functions,  Prev: Psi (Digamma) Function,  Up: Special Functions
+
+7.29 Synchrotron Functions
+==========================
+
+The functions described in this section are declared in the header file
+`gsl_sf_synchrotron.h'.
+
+ -- Function: double gsl_sf_synchrotron_1 (double X)
+ -- Function: int gsl_sf_synchrotron_1_e (double X, gsl_sf_result *
+          RESULT)
+     These routines compute the first synchrotron function x
+     \int_x^\infty dt K_{5/3}(t) for x >= 0.
+
+ -- Function: double gsl_sf_synchrotron_2 (double X)
+ -- Function: int gsl_sf_synchrotron_2_e (double X, gsl_sf_result *
+          RESULT)
+     These routines compute the second synchrotron function x
+     K_{2/3}(x) for x >= 0.
+
+
+File: gsl-ref.info,  Node: Transport Functions,  Next: Trigonometric Functions,  Prev: Synchrotron Functions,  Up: Special Functions
+
+7.30 Transport Functions
+========================
+
+The transport functions J(n,x) are defined by the integral
+representations J(n,x) := \int_0^x dt t^n e^t /(e^t - 1)^2.  They are
+declared in the header file `gsl_sf_transport.h'.
+
+ -- Function: double gsl_sf_transport_2 (double X)
+ -- Function: int gsl_sf_transport_2_e (double X, gsl_sf_result *
+          RESULT)
+     These routines compute the transport function J(2,x).
+
+ -- Function: double gsl_sf_transport_3 (double X)
+ -- Function: int gsl_sf_transport_3_e (double X, gsl_sf_result *
+          RESULT)
+     These routines compute the transport function J(3,x).
+
+ -- Function: double gsl_sf_transport_4 (double X)
+ -- Function: int gsl_sf_transport_4_e (double X, gsl_sf_result *
+          RESULT)
+     These routines compute the transport function J(4,x).
+
+ -- Function: double gsl_sf_transport_5 (double X)
+ -- Function: int gsl_sf_transport_5_e (double X, gsl_sf_result *
+          RESULT)
+     These routines compute the transport function J(5,x).
+
+
+File: gsl-ref.info,  Node: Trigonometric Functions,  Next: Zeta Functions,  Prev: Transport Functions,  Up: Special Functions
+
+7.31 Trigonometric Functions
+============================
+
+The library includes its own trigonometric functions in order to provide
+consistency across platforms and reliable error estimates.  These
+functions are declared in the header file `gsl_sf_trig.h'.
+
+* Menu:
+
+* Circular Trigonometric Functions::
+* Trigonometric Functions for Complex Arguments::
+* Hyperbolic Trigonometric Functions::
+* Conversion Functions::
+* Restriction Functions::
+* Trigonometric Functions With Error Estimates::
+
+
+File: gsl-ref.info,  Node: Circular Trigonometric Functions,  Next: Trigonometric Functions for Complex Arguments,  Up: Trigonometric Functions
+
+7.31.1 Circular Trigonometric Functions
+---------------------------------------
+
+ -- Function: double gsl_sf_sin (double X)
+ -- Function: int gsl_sf_sin_e (double X, gsl_sf_result * RESULT)
+     These routines compute the sine function \sin(x).
+
+ -- Function: double gsl_sf_cos (double X)
+ -- Function: int gsl_sf_cos_e (double X, gsl_sf_result * RESULT)
+     These routines compute the cosine function \cos(x).
+
+ -- Function: double gsl_sf_hypot (double X, double Y)
+ -- Function: int gsl_sf_hypot_e (double X, double Y, gsl_sf_result *
+          RESULT)
+     These routines compute the hypotenuse function \sqrt{x^2 + y^2}
+     avoiding overflow and underflow.
+
+ -- Function: double gsl_sf_sinc (double X)
+ -- Function: int gsl_sf_sinc_e (double X, gsl_sf_result * RESULT)
+     These routines compute \sinc(x) = \sin(\pi x) / (\pi x) for any
+     value of X.
+
+
+File: gsl-ref.info,  Node: Trigonometric Functions for Complex Arguments,  Next: Hyperbolic Trigonometric Functions,  Prev: Circular Trigonometric Functions,  Up: Trigonometric Functions
+
+7.31.2 Trigonometric Functions for Complex Arguments
+----------------------------------------------------
+
+ -- Function: int gsl_sf_complex_sin_e (double ZR, double ZI,
+          gsl_sf_result * SZR, gsl_sf_result * SZI)
+     This function computes the complex sine, \sin(z_r + i z_i) storing
+     the real and imaginary parts in SZR, SZI.
+
+ -- Function: int gsl_sf_complex_cos_e (double ZR, double ZI,
+          gsl_sf_result * CZR, gsl_sf_result * CZI)
+     This function computes the complex cosine, \cos(z_r + i z_i)
+     storing the real and imaginary parts in SZR, SZI.
+
+ -- Function: int gsl_sf_complex_logsin_e (double ZR, double ZI,
+          gsl_sf_result * LSZR, gsl_sf_result * LSZI)
+     This function computes the logarithm of the complex sine,
+     \log(\sin(z_r + i z_i)) storing the real and imaginary parts in
+     SZR, SZI.
+
+
+File: gsl-ref.info,  Node: Hyperbolic Trigonometric Functions,  Next: Conversion Functions,  Prev: Trigonometric Functions for Complex Arguments,  Up: Trigonometric Functions
+
+7.31.3 Hyperbolic Trigonometric Functions
+-----------------------------------------
+
+ -- Function: double gsl_sf_lnsinh (double X)
+ -- Function: int gsl_sf_lnsinh_e (double X, gsl_sf_result * RESULT)
+     These routines compute \log(\sinh(x)) for x > 0.
+
+ -- Function: double gsl_sf_lncosh (double X)
+ -- Function: int gsl_sf_lncosh_e (double X, gsl_sf_result * RESULT)
+     These routines compute \log(\cosh(x)) for any X.
+
+
+File: gsl-ref.info,  Node: Conversion Functions,  Next: Restriction Functions,  Prev: Hyperbolic Trigonometric Functions,  Up: Trigonometric Functions
+
+7.31.4 Conversion Functions
+---------------------------
+
+ -- Function: int gsl_sf_polar_to_rect (double R, double THETA,
+          gsl_sf_result * X, gsl_sf_result * Y);
+     This function converts the polar coordinates (R,THETA) to
+     rectilinear coordinates (X,Y), x = r\cos(\theta), y =
+     r\sin(\theta).
+
+ -- Function: int gsl_sf_rect_to_polar (double X, double Y,
+          gsl_sf_result * R, gsl_sf_result * THETA)
+     This function converts the rectilinear coordinates (X,Y) to polar
+     coordinates (R,THETA), such that x = r\cos(\theta), y =
+     r\sin(\theta).  The argument THETA lies in the range [-\pi, \pi].
+
+
+File: gsl-ref.info,  Node: Restriction Functions,  Next: Trigonometric Functions With Error Estimates,  Prev: Conversion Functions,  Up: Trigonometric Functions
+
+7.31.5 Restriction Functions
+----------------------------
+
+ -- Function: double gsl_sf_angle_restrict_symm (double THETA)
+ -- Function: int gsl_sf_angle_restrict_symm_e (double * THETA)
+     These routines force the angle THETA to lie in the range
+     (-\pi,\pi].
+
+     Note that the mathematical value of \pi is slightly greater than
+     `M_PI', so the machine numbers `M_PI' and `-M_PI' are included in
+     the range.
+
+ -- Function: double gsl_sf_angle_restrict_pos (double THETA)
+ -- Function: int gsl_sf_angle_restrict_pos_e (double * THETA)
+     These routines force the angle THETA to lie in the range [0, 2\pi).
+
+     Note that the mathematical value of 2\pi is slightly greater than
+     `2*M_PI', so the machine number `2*M_PI' is included in the range.
+
+
+
+File: gsl-ref.info,  Node: Trigonometric Functions With Error Estimates,  Prev: Restriction Functions,  Up: Trigonometric Functions
+
+7.31.6 Trigonometric Functions With Error Estimates
+---------------------------------------------------
+
+ -- Function: int gsl_sf_sin_err_e (double X, double DX, gsl_sf_result
+          * RESULT)
+     This routine computes the sine of an angle X with an associated
+     absolute error DX, \sin(x \pm dx).  Note that this function is
+     provided in the error-handling form only since its purpose is to
+     compute the propagated error.
+
+ -- Function: int gsl_sf_cos_err_e (double X, double DX, gsl_sf_result
+          * RESULT)
+     This routine computes the cosine of an angle X with an associated
+     absolute error DX, \cos(x \pm dx).  Note that this function is
+     provided in the error-handling form only since its purpose is to
+     compute the propagated error.
+
+
+File: gsl-ref.info,  Node: Zeta Functions,  Next: Special Functions Examples,  Prev: Trigonometric Functions,  Up: Special Functions
+
+7.32 Zeta Functions
+===================
+
+The Riemann zeta function is defined in Abramowitz & Stegun, Section
+23.2.  The functions described in this section are declared in the
+header file `gsl_sf_zeta.h'.
+
+* Menu:
+
+* Riemann Zeta Function::
+* Riemann Zeta Function Minus One::
+* Hurwitz Zeta Function::
+* Eta Function::
+
+
+File: gsl-ref.info,  Node: Riemann Zeta Function,  Next: Riemann Zeta Function Minus One,  Up: Zeta Functions
+
+7.32.1 Riemann Zeta Function
+----------------------------
+
+The Riemann zeta function is defined by the infinite sum \zeta(s) =
+\sum_{k=1}^\infty k^{-s}.
+
+ -- Function: double gsl_sf_zeta_int (int N)
+ -- Function: int gsl_sf_zeta_int_e (int N, gsl_sf_result * RESULT)
+     These routines compute the Riemann zeta function \zeta(n) for
+     integer N, n \ne 1.
+
+ -- Function: double gsl_sf_zeta (double S)
+ -- Function: int gsl_sf_zeta_e (double S, gsl_sf_result * RESULT)
+     These routines compute the Riemann zeta function \zeta(s) for
+     arbitrary S, s \ne 1.
+
+
+File: gsl-ref.info,  Node: Riemann Zeta Function Minus One,  Next: Hurwitz Zeta Function,  Prev: Riemann Zeta Function,  Up: Zeta Functions
+
+7.32.2 Riemann Zeta Function Minus One
+--------------------------------------
+
+For large positive argument, the Riemann zeta function approaches one.
+In this region the fractional part is interesting, and therefore we
+need a function to evaluate it explicitly.
+
+ -- Function: double gsl_sf_zetam1_int (int N)
+ -- Function: int gsl_sf_zetam1_int_e (int N, gsl_sf_result * RESULT)
+     These routines compute \zeta(n) - 1 for integer N, n \ne 1.
+
+ -- Function: double gsl_sf_zetam1 (double S)
+ -- Function: int gsl_sf_zetam1_e (double S, gsl_sf_result * RESULT)
+     These routines compute \zeta(s) - 1 for arbitrary S, s \ne 1.
+
+
+File: gsl-ref.info,  Node: Hurwitz Zeta Function,  Next: Eta Function,  Prev: Riemann Zeta Function Minus One,  Up: Zeta Functions
+
+7.32.3 Hurwitz Zeta Function
+----------------------------
+
+The Hurwitz zeta function is defined by \zeta(s,q) = \sum_0^\infty
+(k+q)^{-s}.
+
+ -- Function: double gsl_sf_hzeta (double S, double Q)
+ -- Function: int gsl_sf_hzeta_e (double S, double Q, gsl_sf_result *
+          RESULT)
+     These routines compute the Hurwitz zeta function \zeta(s,q) for s
+     > 1, q > 0.
+
+
+File: gsl-ref.info,  Node: Eta Function,  Prev: Hurwitz Zeta Function,  Up: Zeta Functions
+
+7.32.4 Eta Function
+-------------------
+
+The eta function is defined by \eta(s) = (1-2^{1-s}) \zeta(s).
+
+ -- Function: double gsl_sf_eta_int (int N)
+ -- Function: int gsl_sf_eta_int_e (int N, gsl_sf_result * RESULT)
+     These routines compute the eta function \eta(n) for integer N.
+
+ -- Function: double gsl_sf_eta (double S)
+ -- Function: int gsl_sf_eta_e (double S, gsl_sf_result * RESULT)
+     These routines compute the eta function \eta(s) for arbitrary S.
+
+
+File: gsl-ref.info,  Node: Special Functions Examples,  Next: Special Functions References and Further Reading,  Prev: Zeta Functions,  Up: Special Functions
+
+7.33 Examples
+=============
+
+The following example demonstrates the use of the error handling form of
+the special functions, in this case to compute the Bessel function
+J_0(5.0),
+
+     #include <stdio.h>
+     #include <gsl/gsl_errno.h>
+     #include <gsl/gsl_sf_bessel.h>
+
+     int
+     main (void)
+     {
+       double x = 5.0;
+       gsl_sf_result result;
+
+       double expected = -0.17759677131433830434739701;
+
+       int status = gsl_sf_bessel_J0_e (x, &result);
+
+       printf ("status  = %s\n", gsl_strerror(status));
+       printf ("J0(5.0) = %.18f\n"
+               "      +/- % .18f\n",
+               result.val, result.err);
+       printf ("exact   = %.18f\n", expected);
+       return status;
+     }
+
+Here are the results of running the program,
+
+     $ ./a.out
+     status  = success
+     J0(5.0) = -0.177596771314338292
+           +/-  0.000000000000000193
+     exact   = -0.177596771314338292
+
+The next program computes the same quantity using the natural form of
+the function. In this case the error term RESULT.ERR and return status
+are not accessible.
+
+     #include <stdio.h>
+     #include <gsl/gsl_sf_bessel.h>
+
+     int
+     main (void)
+     {
+       double x = 5.0;
+       double expected = -0.17759677131433830434739701;
+
+       double y = gsl_sf_bessel_J0 (x);
+
+       printf ("J0(5.0) = %.18f\n", y);
+       printf ("exact   = %.18f\n", expected);
+       return 0;
+     }
+
+The results of the function are the same,
+
+     $ ./a.out
+     J0(5.0) = -0.177596771314338292
+     exact   = -0.177596771314338292
+
+
+File: gsl-ref.info,  Node: Special Functions References and Further Reading,  Prev: Special Functions Examples,  Up: Special Functions
+
+7.34 References and Further Reading
+===================================
+
+The library follows the conventions of `Abramowitz & Stegun' where
+possible,
+     Abramowitz & Stegun (eds.), `Handbook of Mathematical Functions'
+
+The following papers contain information on the algorithms used to
+compute the special functions, 
+     MISCFUN: A software package to compute uncommon special functions.
+     `ACM Trans. Math. Soft.', vol. 22, 1996, 288-301
+
+     G.N. Watson, A Treatise on the Theory of Bessel Functions, 2nd
+     Edition (Cambridge University Press, 1944).
+
+     G. Nemeth, Mathematical Approximations of Special Functions, Nova
+     Science Publishers, ISBN 1-56072-052-2
+
+     B.C. Carlson, Special Functions of Applied Mathematics (1977)
+
+     W.J. Thompson, Atlas for Computing Mathematical Functions, John
+     Wiley & Sons, New York (1997).
+
+     Y.Y. Luke, Algorithms for the Computation of Mathematical
+     Functions, Academic Press, New York (1977).
+
+
+
+File: gsl-ref.info,  Node: Vectors and Matrices,  Next: Permutations,  Prev: Special Functions,  Up: Top
+
+8 Vectors and Matrices
+**********************
+
+The functions described in this chapter provide a simple vector and
+matrix interface to ordinary C arrays. The memory management of these
+arrays is implemented using a single underlying type, known as a block.
+By writing your functions in terms of vectors and matrices you can pass
+a single structure containing both data and dimensions as an argument
+without needing additional function parameters.  The structures are
+compatible with the vector and matrix formats used by BLAS routines.
+
+* Menu:
+
+* Data types::
+* Blocks::
+* Vectors::
+* Matrices::
+* Vector and Matrix References and Further Reading::
+
+
+File: gsl-ref.info,  Node: Data types,  Next: Blocks,  Up: Vectors and Matrices
+
+8.1 Data types
+==============
+
+All the functions are available for each of the standard data-types.
+The versions for `double' have the prefix `gsl_block', `gsl_vector' and
+`gsl_matrix'.  Similarly the versions for single-precision `float'
+arrays have the prefix `gsl_block_float', `gsl_vector_float' and
+`gsl_matrix_float'.  The full list of available types is given below,
+
+     gsl_block                       double
+     gsl_block_float                 float
+     gsl_block_long_double           long double
+     gsl_block_int                   int
+     gsl_block_uint                  unsigned int
+     gsl_block_long                  long
+     gsl_block_ulong                 unsigned long
+     gsl_block_short                 short
+     gsl_block_ushort                unsigned short
+     gsl_block_char                  char
+     gsl_block_uchar                 unsigned char
+     gsl_block_complex               complex double
+     gsl_block_complex_float         complex float
+     gsl_block_complex_long_double   complex long double
+
+Corresponding types exist for the `gsl_vector' and `gsl_matrix'
+functions.
+
+
+File: gsl-ref.info,  Node: Blocks,  Next: Vectors,  Prev: Data types,  Up: Vectors and Matrices
+
+8.2 Blocks
+==========
+
+For consistency all memory is allocated through a `gsl_block'
+structure.  The structure contains two components, the size of an area
+of memory and a pointer to the memory.  The `gsl_block' structure looks
+like this,
+
+     typedef struct
+     {
+       size_t size;
+       double * data;
+     } gsl_block;
+
+Vectors and matrices are made by "slicing" an underlying block. A slice
+is a set of elements formed from an initial offset and a combination of
+indices and step-sizes. In the case of a matrix the step-size for the
+column index represents the row-length.  The step-size for a vector is
+known as the "stride".
+
+   The functions for allocating and deallocating blocks are defined in
+`gsl_block.h'
+
+* Menu:
+
+* Block allocation::
+* Reading and writing blocks::
+* Example programs for blocks::
+
+
+File: gsl-ref.info,  Node: Block allocation,  Next: Reading and writing blocks,  Up: Blocks
+
+8.2.1 Block allocation
+----------------------
+
+The functions for allocating memory to a block follow the style of
+`malloc' and `free'.  In addition they also perform their own error
+checking.  If there is insufficient memory available to allocate a
+block then the functions call the GSL error handler (with an error
+number of `GSL_ENOMEM') in addition to returning a null pointer.  Thus
+if you use the library error handler to abort your program then it
+isn't necessary to check every `alloc'.
+
+ -- Function: gsl_block * gsl_block_alloc (size_t N)
+     This function allocates memory for a block of N double-precision
+     elements, returning a pointer to the block struct.  The block is
+     not initialized and so the values of its elements are undefined.
+     Use the function `gsl_block_calloc' if you want to ensure that all
+     the elements are initialized to zero.
+
+     A null pointer is returned if insufficient memory is available to
+     create the block.
+
+ -- Function: gsl_block * gsl_block_calloc (size_t N)
+     This function allocates memory for a block and initializes all the
+     elements of the block to zero.
+
+ -- Function: void gsl_block_free (gsl_block * B)
+     This function frees the memory used by a block B previously
+     allocated with `gsl_block_alloc' or `gsl_block_calloc'.  The block
+     B must be a valid block object (a null pointer is not allowed).
+
+
+File: gsl-ref.info,  Node: Reading and writing blocks,  Next: Example programs for blocks,  Prev: Block allocation,  Up: Blocks
+
+8.2.2 Reading and writing blocks
+--------------------------------
+
+The library provides functions for reading and writing blocks to a file
+as binary data or formatted text.
+
+ -- Function: int gsl_block_fwrite (FILE * STREAM, const gsl_block * B)
+     This function writes the elements of the block B to the stream
+     STREAM in binary format.  The return value is 0 for success and
+     `GSL_EFAILED' if there was a problem writing to the file.  Since
+     the data is written in the native binary format it may not be
+     portable between different architectures.
+
+ -- Function: int gsl_block_fread (FILE * STREAM, gsl_block * B)
+     This function reads into the block B from the open stream STREAM
+     in binary format.  The block B must be preallocated with the
+     correct length since the function uses the size of B to determine
+     how many bytes to read.  The return value is 0 for success and
+     `GSL_EFAILED' if there was a problem reading from the file.  The
+     data is assumed to have been written in the native binary format
+     on the same architecture.
+
+ -- Function: int gsl_block_fprintf (FILE * STREAM, const gsl_block *
+          B, const char * FORMAT)
+     This function writes the elements of the block B line-by-line to
+     the stream STREAM using the format specifier FORMAT, which should
+     be one of the `%g', `%e' or `%f' formats for floating point
+     numbers and `%d' for integers.  The function returns 0 for success
+     and `GSL_EFAILED' if there was a problem writing to the file.
+
+ -- Function: int gsl_block_fscanf (FILE * STREAM, gsl_block * B)
+     This function reads formatted data from the stream STREAM into the
+     block B.  The block B must be preallocated with the correct length
+     since the function uses the size of B to determine how many
+     numbers to read.  The function returns 0 for success and
+     `GSL_EFAILED' if there was a problem reading from the file.
+
+
+File: gsl-ref.info,  Node: Example programs for blocks,  Prev: Reading and writing blocks,  Up: Blocks
+
+8.2.3 Example programs for blocks
+---------------------------------
+
+The following program shows how to allocate a block,
+
+     #include <stdio.h>
+     #include <gsl/gsl_block.h>
+
+     int
+     main (void)
+     {
+       gsl_block * b = gsl_block_alloc (100);
+
+       printf ("length of block = %u\n", b->size);
+       printf ("block data address = %#x\n", b->data);
+
+       gsl_block_free (b);
+       return 0;
+     }
+
+Here is the output from the program,
+
+     length of block = 100
+     block data address = 0x804b0d8
+
+
+File: gsl-ref.info,  Node: Vectors,  Next: Matrices,  Prev: Blocks,  Up: Vectors and Matrices
+
+8.3 Vectors
+===========
+
+Vectors are defined by a `gsl_vector' structure which describes a slice
+of a block.  Different vectors can be created which point to the same
+block.  A vector slice is a set of equally-spaced elements of an area
+of memory.
+
+   The `gsl_vector' structure contains five components, the "size", the
+"stride", a pointer to the memory where the elements are stored, DATA,
+a pointer to the block owned by the vector, BLOCK, if any, and an
+ownership flag, OWNER.  The structure is very simple and looks like
+this,
+
+     typedef struct
+     {
+       size_t size;
+       size_t stride;
+       double * data;
+       gsl_block * block;
+       int owner;
+     } gsl_vector;
+
+The SIZE is simply the number of vector elements.  The range of valid
+indices runs from 0 to `size-1'.  The STRIDE is the step-size from one
+element to the next in physical memory, measured in units of the
+appropriate datatype.  The pointer DATA gives the location of the first
+element of the vector in memory.  The pointer BLOCK stores the location
+of the memory block in which the vector elements are located (if any).
+If the vector owns this block then the OWNER field is set to one and
+the block will be deallocated when the vector is freed.  If the vector
+points to a block owned by another object then the OWNER field is zero
+and any underlying block will not be deallocated with the vector.
+
+   The functions for allocating and accessing vectors are defined in
+`gsl_vector.h'
+
+* Menu:
+
+* Vector allocation::
+* Accessing vector elements::
+* Initializing vector elements::
+* Reading and writing vectors::
+* Vector views::
+* Copying vectors::
+* Exchanging elements::
+* Vector operations::
+* Finding maximum and minimum elements of vectors::
+* Vector properties::
+* Example programs for vectors::
+
+
+File: gsl-ref.info,  Node: Vector allocation,  Next: Accessing vector elements,  Up: Vectors
+
+8.3.1 Vector allocation
+-----------------------
+
+The functions for allocating memory to a vector follow the style of
+`malloc' and `free'.  In addition they also perform their own error
+checking.  If there is insufficient memory available to allocate a
+vector then the functions call the GSL error handler (with an error
+number of `GSL_ENOMEM') in addition to returning a null pointer.  Thus
+if you use the library error handler to abort your program then it
+isn't necessary to check every `alloc'.
+
+ -- Function: gsl_vector * gsl_vector_alloc (size_t N)
+     This function creates a vector of length N, returning a pointer to
+     a newly initialized vector struct. A new block is allocated for the
+     elements of the vector, and stored in the BLOCK component of the
+     vector struct.  The block is "owned" by the vector, and will be
+     deallocated when the vector is deallocated.
+
+ -- Function: gsl_vector * gsl_vector_calloc (size_t N)
+     This function allocates memory for a vector of length N and
+     initializes all the elements of the vector to zero.
+
+ -- Function: void gsl_vector_free (gsl_vector * V)
+     This function frees a previously allocated vector V.  If the
+     vector was created using `gsl_vector_alloc' then the block
+     underlying the vector will also be deallocated.  If the vector has
+     been created from another object then the memory is still owned by
+     that object and will not be deallocated.  The vector V must be a
+     valid vector object (a null pointer is not allowed).
+
+
+File: gsl-ref.info,  Node: Accessing vector elements,  Next: Initializing vector elements,  Prev: Vector allocation,  Up: Vectors
+
+8.3.2 Accessing vector elements
+-------------------------------
+
+Unlike FORTRAN compilers, C compilers do not usually provide support
+for range checking of vectors and matrices.  Range checking is
+available in the GNU C Compiler bounds-checking extension, but it is not
+part of the default installation of GCC.  The functions
+`gsl_vector_get' and `gsl_vector_set' can perform portable range
+checking for you and report an error if you attempt to access elements
+outside the allowed range.
+
+   The functions for accessing the elements of a vector or matrix are
+defined in `gsl_vector.h' and declared `extern inline' to eliminate
+function-call overhead.  You must compile your program with the macro
+`HAVE_INLINE' defined to use these functions.
+
+   If necessary you can turn off range checking completely without
+modifying any source files by recompiling your program with the
+preprocessor definition `GSL_RANGE_CHECK_OFF'.  Provided your compiler
+supports inline functions the effect of turning off range checking is
+to replace calls to `gsl_vector_get(v,i)' by `v->data[i*v->stride]' and
+calls to `gsl_vector_set(v,i,x)' by `v->data[i*v->stride]=x'.  Thus
+there should be no performance penalty for using the range checking
+functions when range checking is turned off.
+
+ -- Function: double gsl_vector_get (const gsl_vector * V, size_t I)
+     This function returns the I-th element of a vector V.  If I lies
+     outside the allowed range of 0 to N-1 then the error handler is
+     invoked and 0 is returned.
+
+ -- Function: void gsl_vector_set (gsl_vector * V, size_t I, double X)
+     This function sets the value of the I-th element of a vector V to
+     X.  If I lies outside the allowed range of 0 to N-1 then the error
+     handler is invoked.
+
+ -- Function: double * gsl_vector_ptr (gsl_vector * V, size_t I)
+ -- Function: const double * gsl_vector_const_ptr (const gsl_vector *
+          V, size_t I)
+     These functions return a pointer to the I-th element of a vector
+     V.  If I lies outside the allowed range of 0 to N-1 then the error
+     handler is invoked and a null pointer is returned.
+
+
+File: gsl-ref.info,  Node: Initializing vector elements,  Next: Reading and writing vectors,  Prev: Accessing vector elements,  Up: Vectors
+
+8.3.3 Initializing vector elements
+----------------------------------
+
+ -- Function: void gsl_vector_set_all (gsl_vector * V, double X)
+     This function sets all the elements of the vector V to the value X.
+
+ -- Function: void gsl_vector_set_zero (gsl_vector * V)
+     This function sets all the elements of the vector V to zero.
+
+ -- Function: int gsl_vector_set_basis (gsl_vector * V, size_t I)
+     This function makes a basis vector by setting all the elements of
+     the vector V to zero except for the I-th element which is set to
+     one.
+
+
+File: gsl-ref.info,  Node: Reading and writing vectors,  Next: Vector views,  Prev: Initializing vector elements,  Up: Vectors
+
+8.3.4 Reading and writing vectors
+---------------------------------
+
+The library provides functions for reading and writing vectors to a file
+as binary data or formatted text.
+
+ -- Function: int gsl_vector_fwrite (FILE * STREAM, const gsl_vector *
+          V)
+     This function writes the elements of the vector V to the stream
+     STREAM in binary format.  The return value is 0 for success and
+     `GSL_EFAILED' if there was a problem writing to the file.  Since
+     the data is written in the native binary format it may not be
+     portable between different architectures.
+
+ -- Function: int gsl_vector_fread (FILE * STREAM, gsl_vector * V)
+     This function reads into the vector V from the open stream STREAM
+     in binary format.  The vector V must be preallocated with the
+     correct length since the function uses the size of V to determine
+     how many bytes to read.  The return value is 0 for success and
+     `GSL_EFAILED' if there was a problem reading from the file.  The
+     data is assumed to have been written in the native binary format
+     on the same architecture.
+
+ -- Function: int gsl_vector_fprintf (FILE * STREAM, const gsl_vector *
+          V, const char * FORMAT)
+     This function writes the elements of the vector V line-by-line to
+     the stream STREAM using the format specifier FORMAT, which should
+     be one of the `%g', `%e' or `%f' formats for floating point
+     numbers and `%d' for integers.  The function returns 0 for success
+     and `GSL_EFAILED' if there was a problem writing to the file.
+
+ -- Function: int gsl_vector_fscanf (FILE * STREAM, gsl_vector * V)
+     This function reads formatted data from the stream STREAM into the
+     vector V.  The vector V must be preallocated with the correct
+     length since the function uses the size of V to determine how many
+     numbers to read.  The function returns 0 for success and
+     `GSL_EFAILED' if there was a problem reading from the file.
+
+
+File: gsl-ref.info,  Node: Vector views,  Next: Copying vectors,  Prev: Reading and writing vectors,  Up: Vectors
+
+8.3.5 Vector views
+------------------
+
+In addition to creating vectors from slices of blocks it is also
+possible to slice vectors and create vector views.  For example, a
+subvector of another vector can be described with a view, or two views
+can be made which provide access to the even and odd elements of a
+vector.
+
+   A vector view is a temporary object, stored on the stack, which can
+be used to operate on a subset of vector elements.  Vector views can be
+defined for both constant and non-constant vectors, using separate types
+that preserve constness.  A vector view has the type `gsl_vector_view'
+and a constant vector view has the type `gsl_vector_const_view'.  In
+both cases the elements of the view can be accessed as a `gsl_vector'
+using the `vector' component of the view object.  A pointer to a vector
+of type `gsl_vector *' or `const gsl_vector *' can be obtained by
+taking the address of this component with the `&' operator.
+
+   When using this pointer it is important to ensure that the view
+itself remains in scope--the simplest way to do so is by always writing
+the pointer as `&'VIEW`.vector', and never storing this value in
+another variable.
+
+ -- Function: gsl_vector_view gsl_vector_subvector (gsl_vector * V,
+          size_t OFFSET, size_t N)
+ -- Function: gsl_vector_const_view gsl_vector_const_subvector (const
+          gsl_vector * V, size_t OFFSET, size_t N)
+     These functions return a vector view of a subvector of another
+     vector V.  The start of the new vector is offset by OFFSET elements
+     from the start of the original vector.  The new vector has N
+     elements.  Mathematically, the I-th element of the new vector V'
+     is given by,
+
+          v'(i) = v->data[(offset + i)*v->stride]
+
+     where the index I runs from 0 to `n-1'.
+
+     The `data' pointer of the returned vector struct is set to null if
+     the combined parameters (OFFSET,N) overrun the end of the original
+     vector.
+
+     The new vector is only a view of the block underlying the original
+     vector, V.  The block containing the elements of V is not owned by
+     the new vector.  When the view goes out of scope the original
+     vector V and its block will continue to exist.  The original
+     memory can only be deallocated by freeing the original vector.  Of
+     course, the original vector should not be deallocated while the
+     view is still in use.
+
+     The function `gsl_vector_const_subvector' is equivalent to
+     `gsl_vector_subvector' but can be used for vectors which are
+     declared `const'.
+
+ -- Function: gsl_vector_view gsl_vector_subvector_with_stride
+          (gsl_vector * V, size_t OFFSET, size_t STRIDE, size_t N)
+ -- Function: gsl_vector_const_view
+gsl_vector_const_subvector_with_stride (const gsl_vector * V, size_t
+          OFFSET, size_t STRIDE, size_t N)
+     These functions return a vector view of a subvector of another
+     vector V with an additional stride argument. The subvector is
+     formed in the same way as for `gsl_vector_subvector' but the new
+     vector has N elements with a step-size of STRIDE from one element
+     to the next in the original vector.  Mathematically, the I-th
+     element of the new vector V' is given by,
+
+          v'(i) = v->data[(offset + i*stride)*v->stride]
+
+     where the index I runs from 0 to `n-1'.
+
+     Note that subvector views give direct access to the underlying
+     elements of the original vector. For example, the following code
+     will zero the even elements of the vector `v' of length `n', while
+     leaving the odd elements untouched,
+
+          gsl_vector_view v_even
+            = gsl_vector_subvector_with_stride (v, 0, 2, n/2);
+          gsl_vector_set_zero (&v_even.vector);
+
+     A vector view can be passed to any subroutine which takes a vector
+     argument just as a directly allocated vector would be, using
+     `&'VIEW`.vector'.  For example, the following code computes the
+     norm of the odd elements of `v' using the BLAS routine DNRM2,
+
+          gsl_vector_view v_odd
+            = gsl_vector_subvector_with_stride (v, 1, 2, n/2);
+          double r = gsl_blas_dnrm2 (&v_odd.vector);
+
+     The function `gsl_vector_const_subvector_with_stride' is equivalent
+     to `gsl_vector_subvector_with_stride' but can be used for vectors
+     which are declared `const'.
+
+ -- Function: gsl_vector_view gsl_vector_complex_real
+          (gsl_vector_complex * V)
+ -- Function: gsl_vector_const_view gsl_vector_complex_const_real
+          (const gsl_vector_complex * V)
+     These functions return a vector view of the real parts of the
+     complex vector V.
+
+     The function `gsl_vector_complex_const_real' is equivalent to
+     `gsl_vector_complex_real' but can be used for vectors which are
+     declared `const'.
+
+ -- Function: gsl_vector_view gsl_vector_complex_imag
+          (gsl_vector_complex * V)
+ -- Function: gsl_vector_const_view gsl_vector_complex_const_imag
+          (const gsl_vector_complex * V)
+     These functions return a vector view of the imaginary parts of the
+     complex vector V.
+
+     The function `gsl_vector_complex_const_imag' is equivalent to
+     `gsl_vector_complex_imag' but can be used for vectors which are
+     declared `const'.
+
+ -- Function: gsl_vector_view gsl_vector_view_array (double * BASE,
+          size_t N)
+ -- Function: gsl_vector_const_view gsl_vector_const_view_array (const
+          double * BASE, size_t N)
+     These functions return a vector view of an array.  The start of
+     the new vector is given by BASE and has N elements.
+     Mathematically, the I-th element of the new vector V' is given by,
+
+          v'(i) = base[i]
+
+     where the index I runs from 0 to `n-1'.
+
+     The array containing the elements of V is not owned by the new
+     vector view.  When the view goes out of scope the original array
+     will continue to exist.  The original memory can only be
+     deallocated by freeing the original pointer BASE.  Of course, the
+     original array should not be deallocated while the view is still
+     in use.
+
+     The function `gsl_vector_const_view_array' is equivalent to
+     `gsl_vector_view_array' but can be used for arrays which are
+     declared `const'.
+
+ -- Function: gsl_vector_view gsl_vector_view_array_with_stride (double
+          * BASE, size_t STRIDE, size_t N)
+ -- Function: gsl_vector_const_view
+gsl_vector_const_view_array_with_stride (const double * BASE, size_t
+          STRIDE, size_t N)
+     These functions return a vector view of an array BASE with an
+     additional stride argument. The subvector is formed in the same
+     way as for `gsl_vector_view_array' but the new vector has N
+     elements with a step-size of STRIDE from one element to the next
+     in the original array.  Mathematically, the I-th element of the new
+     vector V' is given by,
+
+          v'(i) = base[i*stride]
+
+     where the index I runs from 0 to `n-1'.
+
+     Note that the view gives direct access to the underlying elements
+     of the original array.  A vector view can be passed to any
+     subroutine which takes a vector argument just as a directly
+     allocated vector would be, using `&'VIEW`.vector'.
+
+     The function `gsl_vector_const_view_array_with_stride' is
+     equivalent to `gsl_vector_view_array_with_stride' but can be used
+     for arrays which are declared `const'.
+
+
+File: gsl-ref.info,  Node: Copying vectors,  Next: Exchanging elements,  Prev: Vector views,  Up: Vectors
+
+8.3.6 Copying vectors
+---------------------
+
+Common operations on vectors such as addition and multiplication are
+available in the BLAS part of the library (*note BLAS Support::).
+However, it is useful to have a small number of utility functions which
+do not require the full BLAS code.  The following functions fall into
+this category.
+
+ -- Function: int gsl_vector_memcpy (gsl_vector * DEST, const
+          gsl_vector * SRC)
+     This function copies the elements of the vector SRC into the
+     vector DEST.  The two vectors must have the same length.
+
+ -- Function: int gsl_vector_swap (gsl_vector * V, gsl_vector * W)
+     This function exchanges the elements of the vectors V and W by
+     copying.  The two vectors must have the same length.
+
+
+File: gsl-ref.info,  Node: Exchanging elements,  Next: Vector operations,  Prev: Copying vectors,  Up: Vectors
+
+8.3.7 Exchanging elements
+-------------------------
+
+The following function can be used to exchange, or permute, the elements
+of a vector.
+
+ -- Function: int gsl_vector_swap_elements (gsl_vector * V, size_t I,
+          size_t J)
+     This function exchanges the I-th and J-th elements of the vector V
+     in-place.
+
+ -- Function: int gsl_vector_reverse (gsl_vector * V)
+     This function reverses the order of the elements of the vector V.
+
+
+File: gsl-ref.info,  Node: Vector operations,  Next: Finding maximum and minimum elements of vectors,  Prev: Exchanging elements,  Up: Vectors
+
+8.3.8 Vector operations
+-----------------------
+
+The following operations are only defined for real vectors.
+
+ -- Function: int gsl_vector_add (gsl_vector * A, const gsl_vector * B)
+     This function adds the elements of vector B to the elements of
+     vector A, a'_i = a_i + b_i. The two vectors must have the same
+     length.
+
+ -- Function: int gsl_vector_sub (gsl_vector * A, const gsl_vector * B)
+     This function subtracts the elements of vector B from the elements
+     of vector A, a'_i = a_i - b_i. The two vectors must have the same
+     length.
+
+ -- Function: int gsl_vector_mul (gsl_vector * A, const gsl_vector * B)
+     This function multiplies the elements of vector A by the elements
+     of vector B, a'_i = a_i * b_i. The two vectors must have the same
+     length.
+
+ -- Function: int gsl_vector_div (gsl_vector * A, const gsl_vector * B)
+     This function divides the elements of vector A by the elements of
+     vector B, a'_i = a_i / b_i. The two vectors must have the same
+     length.
+
+ -- Function: int gsl_vector_scale (gsl_vector * A, const double X)
+     This function multiplies the elements of vector A by the constant
+     factor X, a'_i = x a_i.
+
+ -- Function: int gsl_vector_add_constant (gsl_vector * A, const double
+          X)
+     This function adds the constant value X to the elements of the
+     vector A, a'_i = a_i + x.
+
+
+File: gsl-ref.info,  Node: Finding maximum and minimum elements of vectors,  Next: Vector properties,  Prev: Vector operations,  Up: Vectors
+
+8.3.9 Finding maximum and minimum elements of vectors
+-----------------------------------------------------
+
+ -- Function: double gsl_vector_max (const gsl_vector * V)
+     This function returns the maximum value in the vector V.
+
+ -- Function: double gsl_vector_min (const gsl_vector * V)
+     This function returns the minimum value in the vector V.
+
+ -- Function: void gsl_vector_minmax (const gsl_vector * V, double *
+          MIN_OUT, double * MAX_OUT)
+     This function returns the minimum and maximum values in the vector
+     V, storing them in MIN_OUT and MAX_OUT.
+
+ -- Function: size_t gsl_vector_max_index (const gsl_vector * V)
+     This function returns the index of the maximum value in the vector
+     V.  When there are several equal maximum elements then the lowest
+     index is returned.
+
+ -- Function: size_t gsl_vector_min_index (const gsl_vector * V)
+     This function returns the index of the minimum value in the vector
+     V.  When there are several equal minimum elements then the lowest
+     index is returned.
+
+ -- Function: void gsl_vector_minmax_index (const gsl_vector * V,
+          size_t * IMIN, size_t * IMAX)
+     This function returns the indices of the minimum and maximum
+     values in the vector V, storing them in IMIN and IMAX. When there
+     are several equal minimum or maximum elements then the lowest
+     indices are returned.
+
+
+File: gsl-ref.info,  Node: Vector properties,  Next: Example programs for vectors,  Prev: Finding maximum and minimum elements of vectors,  Up: Vectors
+
+8.3.10 Vector properties
+------------------------
+
+ -- Function: int gsl_vector_isnull (const gsl_vector * V)
+ -- Function: int gsl_vector_ispos (const gsl_vector * V)
+ -- Function: int gsl_vector_isneg (const gsl_vector * V)
+     These functions return 1 if all the elements of the vector V are
+     zero, strictly positive, or strictly negative respectively, and 0
+     otherwise.  To test for a non-negative vector, use the expression
+     `!gsl_vector_isneg(v)'.
+
+
+File: gsl-ref.info,  Node: Example programs for vectors,  Prev: Vector properties,  Up: Vectors
+
+8.3.11 Example programs for vectors
+-----------------------------------
+
+This program shows how to allocate, initialize and read from a vector
+using the functions `gsl_vector_alloc', `gsl_vector_set' and
+`gsl_vector_get'.
+
+     #include <stdio.h>
+     #include <gsl/gsl_vector.h>
+
+     int
+     main (void)
+     {
+       int i;
+       gsl_vector * v = gsl_vector_alloc (3);
+
+       for (i = 0; i < 3; i++)
+         {
+           gsl_vector_set (v, i, 1.23 + i);
+         }
+
+       for (i = 0; i < 100; i++) /* OUT OF RANGE ERROR */
+         {
+           printf ("v_%d = %g\n", i, gsl_vector_get (v, i));
+         }
+
+       gsl_vector_free (v);
+       return 0;
+     }
+
+Here is the output from the program.  The final loop attempts to read
+outside the range of the vector `v', and the error is trapped by the
+range-checking code in `gsl_vector_get'.
+
+     $ ./a.out
+     v_0 = 1.23
+     v_1 = 2.23
+     v_2 = 3.23
+     gsl: vector_source.c:12: ERROR: index out of range
+     Default GSL error handler invoked.
+     Aborted (core dumped)
+
+The next program shows how to write a vector to a file.
+
+     #include <stdio.h>
+     #include <gsl/gsl_vector.h>
+
+     int
+     main (void)
+     {
+       int i;
+       gsl_vector * v = gsl_vector_alloc (100);
+
+       for (i = 0; i < 100; i++)
+         {
+           gsl_vector_set (v, i, 1.23 + i);
+         }
+
+       {
+          FILE * f = fopen ("test.dat", "w");
+          gsl_vector_fprintf (f, v, "%.5g");
+          fclose (f);
+       }
+
+       gsl_vector_free (v);
+       return 0;
+     }
+
+After running this program the file `test.dat' should contain the
+elements of `v', written using the format specifier `%.5g'.  The vector
+could then be read back in using the function `gsl_vector_fscanf (f,
+v)' as follows:
+
+     #include <stdio.h>
+     #include <gsl/gsl_vector.h>
+
+     int
+     main (void)
+     {
+       int i;
+       gsl_vector * v = gsl_vector_alloc (10);
+
+       {
+          FILE * f = fopen ("test.dat", "r");
+          gsl_vector_fscanf (f, v);
+          fclose (f);
+       }
+
+       for (i = 0; i < 10; i++)
+         {
+           printf ("%g\n", gsl_vector_get(v, i));
+         }
+
+       gsl_vector_free (v);
+       return 0;
+     }
+
+
+File: gsl-ref.info,  Node: Matrices,  Next: Vector and Matrix References and Further Reading,  Prev: Vectors,  Up: Vectors and Matrices
+
+8.4 Matrices
+============
+
+Matrices are defined by a `gsl_matrix' structure which describes a
+generalized slice of a block.  Like a vector it represents a set of
+elements in an area of memory, but uses two indices instead of one.
+
+   The `gsl_matrix' structure contains six components, the two
+dimensions of the matrix, a physical dimension, a pointer to the memory
+where the elements of the matrix are stored, DATA, a pointer to the
+block owned by the matrix BLOCK, if any, and an ownership flag, OWNER.
+The physical dimension determines the memory layout and can differ from
+the matrix dimension to allow the use of submatrices.  The `gsl_matrix'
+structure is very simple and looks like this,
+
+     typedef struct
+     {
+       size_t size1;
+       size_t size2;
+       size_t tda;
+       double * data;
+       gsl_block * block;
+       int owner;
+     } gsl_matrix;
+
+Matrices are stored in row-major order, meaning that each row of
+elements forms a contiguous block in memory.  This is the standard
+"C-language ordering" of two-dimensional arrays. Note that FORTRAN
+stores arrays in column-major order. The number of rows is SIZE1.  The
+range of valid row indices runs from 0 to `size1-1'.  Similarly SIZE2
+is the number of columns.  The range of valid column indices runs from
+0 to `size2-1'.  The physical row dimension TDA, or "trailing
+dimension", specifies the size of a row of the matrix as laid out in
+memory.
+
+   For example, in the following matrix SIZE1 is 3, SIZE2 is 4, and TDA
+is 8.  The physical memory layout of the matrix begins in the top left
+hand-corner and proceeds from left to right along each row in turn.
+
+     00 01 02 03 XX XX XX XX
+     10 11 12 13 XX XX XX XX
+     20 21 22 23 XX XX XX XX
+
+Each unused memory location is represented by "`XX'".  The pointer DATA
+gives the location of the first element of the matrix in memory.  The
+pointer BLOCK stores the location of the memory block in which the
+elements of the matrix are located (if any).  If the matrix owns this
+block then the OWNER field is set to one and the block will be
+deallocated when the matrix is freed.  If the matrix is only a slice of
+a block owned by another object then the OWNER field is zero and any
+underlying block will not be freed.
+
+   The functions for allocating and accessing matrices are defined in
+`gsl_matrix.h'
+
+* Menu:
+
+* Matrix allocation::
+* Accessing matrix elements::
+* Initializing matrix elements::
+* Reading and writing matrices::
+* Matrix views::
+* Creating row and column views::
+* Copying matrices::
+* Copying rows and columns::
+* Exchanging rows and columns::
+* Matrix operations::
+* Finding maximum and minimum elements of matrices::
+* Matrix properties::
+* Example programs for matrices::
+
+
+File: gsl-ref.info,  Node: Matrix allocation,  Next: Accessing matrix elements,  Up: Matrices
+
+8.4.1 Matrix allocation
+-----------------------
+
+The functions for allocating memory to a matrix follow the style of
+`malloc' and `free'.  They also perform their own error checking.  If
+there is insufficient memory available to allocate a vector then the
+functions call the GSL error handler (with an error number of
+`GSL_ENOMEM') in addition to returning a null pointer.  Thus if you use
+the library error handler to abort your program then it isn't necessary
+to check every `alloc'.
+
+ -- Function: gsl_matrix * gsl_matrix_alloc (size_t N1, size_t N2)
+     This function creates a matrix of size N1 rows by N2 columns,
+     returning a pointer to a newly initialized matrix struct. A new
+     block is allocated for the elements of the matrix, and stored in
+     the BLOCK component of the matrix struct.  The block is "owned" by
+     the matrix, and will be deallocated when the matrix is deallocated.
+
+ -- Function: gsl_matrix * gsl_matrix_calloc (size_t N1, size_t N2)
+     This function allocates memory for a matrix of size N1 rows by N2
+     columns and initializes all the elements of the matrix to zero.
+
+ -- Function: void gsl_matrix_free (gsl_matrix * M)
+     This function frees a previously allocated matrix M.  If the
+     matrix was created using `gsl_matrix_alloc' then the block
+     underlying the matrix will also be deallocated.  If the matrix has
+     been created from another object then the memory is still owned by
+     that object and will not be deallocated.  The matrix M must be a
+     valid matrix object (a null pointer is not allowed).
+
+
+File: gsl-ref.info,  Node: Accessing matrix elements,  Next: Initializing matrix elements,  Prev: Matrix allocation,  Up: Matrices
+
+8.4.2 Accessing matrix elements
+-------------------------------
+
+The functions for accessing the elements of a matrix use the same range
+checking system as vectors.  You can turn off range checking by
+recompiling your program with the preprocessor definition
+`GSL_RANGE_CHECK_OFF'.
+
+   The elements of the matrix are stored in "C-order", where the second
+index moves continuously through memory.  More precisely, the element
+accessed by the function `gsl_matrix_get(m,i,j)' and
+`gsl_matrix_set(m,i,j,x)' is
+
+     m->data[i * m->tda + j]
+
+where TDA is the physical row-length of the matrix.
+
+ -- Function: double gsl_matrix_get (const gsl_matrix * M, size_t I,
+          size_t J)
+     This function returns the (i,j)-th element of a matrix M.  If I or
+     J lie outside the allowed range of 0 to N1-1 and 0 to N2-1 then
+     the error handler is invoked and 0 is returned.
+
+ -- Function: void gsl_matrix_set (gsl_matrix * M, size_t I, size_t J,
+          double X)
+     This function sets the value of the (i,j)-th element of a matrix M
+     to X.  If I or J lies outside the allowed range of 0 to N1-1 and 0
+     to N2-1 then the error handler is invoked.
+
+ -- Function: double * gsl_matrix_ptr (gsl_matrix * M, size_t I, size_t
+          J)
+ -- Function: const double * gsl_matrix_const_ptr (const gsl_matrix *
+          M, size_t I, size_t J)
+     These functions return a pointer to the (i,j)-th element of a
+     matrix M.  If I or J lie outside the allowed range of 0 to N1-1
+     and 0 to N2-1 then the error handler is invoked and a null pointer
+     is returned.
+
+
+File: gsl-ref.info,  Node: Initializing matrix elements,  Next: Reading and writing matrices,  Prev: Accessing matrix elements,  Up: Matrices
+
+8.4.3 Initializing matrix elements
+----------------------------------
+
+ -- Function: void gsl_matrix_set_all (gsl_matrix * M, double X)
+     This function sets all the elements of the matrix M to the value X.
+
+ -- Function: void gsl_matrix_set_zero (gsl_matrix * M)
+     This function sets all the elements of the matrix M to zero.
+
+ -- Function: void gsl_matrix_set_identity (gsl_matrix * M)
+     This function sets the elements of the matrix M to the
+     corresponding elements of the identity matrix, m(i,j) =
+     \delta(i,j), i.e. a unit diagonal with all off-diagonal elements
+     zero.  This applies to both square and rectangular matrices.
+
+
+File: gsl-ref.info,  Node: Reading and writing matrices,  Next: Matrix views,  Prev: Initializing matrix elements,  Up: Matrices
+
+8.4.4 Reading and writing matrices
+----------------------------------
+
+The library provides functions for reading and writing matrices to a
+file as binary data or formatted text.
+
+ -- Function: int gsl_matrix_fwrite (FILE * STREAM, const gsl_matrix *
+          M)
+     This function writes the elements of the matrix M to the stream
+     STREAM in binary format.  The return value is 0 for success and
+     `GSL_EFAILED' if there was a problem writing to the file.  Since
+     the data is written in the native binary format it may not be
+     portable between different architectures.
+
+ -- Function: int gsl_matrix_fread (FILE * STREAM, gsl_matrix * M)
+     This function reads into the matrix M from the open stream STREAM
+     in binary format.  The matrix M must be preallocated with the
+     correct dimensions since the function uses the size of M to
+     determine how many bytes to read.  The return value is 0 for
+     success and `GSL_EFAILED' if there was a problem reading from the
+     file.  The data is assumed to have been written in the native
+     binary format on the same architecture.
+
+ -- Function: int gsl_matrix_fprintf (FILE * STREAM, const gsl_matrix *
+          M, const char * FORMAT)
+     This function writes the elements of the matrix M line-by-line to
+     the stream STREAM using the format specifier FORMAT, which should
+     be one of the `%g', `%e' or `%f' formats for floating point
+     numbers and `%d' for integers.  The function returns 0 for success
+     and `GSL_EFAILED' if there was a problem writing to the file.
+
+ -- Function: int gsl_matrix_fscanf (FILE * STREAM, gsl_matrix * M)
+     This function reads formatted data from the stream STREAM into the
+     matrix M.  The matrix M must be preallocated with the correct
+     dimensions since the function uses the size of M to determine how
+     many numbers to read.  The function returns 0 for success and
+     `GSL_EFAILED' if there was a problem reading from the file.
+
+
+File: gsl-ref.info,  Node: Matrix views,  Next: Creating row and column views,  Prev: Reading and writing matrices,  Up: Matrices
+
+8.4.5 Matrix views
+------------------
+
+A matrix view is a temporary object, stored on the stack, which can be
+used to operate on a subset of matrix elements.  Matrix views can be
+defined for both constant and non-constant matrices using separate types
+that preserve constness.  A matrix view has the type `gsl_matrix_view'
+and a constant matrix view has the type `gsl_matrix_const_view'.  In
+both cases the elements of the view can by accessed using the `matrix'
+component of the view object.  A pointer `gsl_matrix *' or `const
+gsl_matrix *' can be obtained by taking the address of the `matrix'
+component with the `&' operator.  In addition to matrix views it is
+also possible to create vector views of a matrix, such as row or column
+views.
+
+ -- Function: gsl_matrix_view gsl_matrix_submatrix (gsl_matrix * M,
+          size_t K1, size_t K2, size_t N1, size_t N2)
+ -- Function: gsl_matrix_const_view gsl_matrix_const_submatrix (const
+          gsl_matrix * M, size_t K1, size_t K2, size_t N1, size_t N2)
+     These functions return a matrix view of a submatrix of the matrix
+     M.  The upper-left element of the submatrix is the element (K1,K2)
+     of the original matrix.  The submatrix has N1 rows and N2 columns.
+     The physical number of columns in memory given by TDA is
+     unchanged.  Mathematically, the (i,j)-th element of the new matrix
+     is given by,
+
+          m'(i,j) = m->data[(k1*m->tda + k2) + i*m->tda + j]
+
+     where the index I runs from 0 to `n1-1' and the index J runs from
+     0 to `n2-1'.
+
+     The `data' pointer of the returned matrix struct is set to null if
+     the combined parameters (I,J,N1,N2,TDA) overrun the ends of the
+     original matrix.
+
+     The new matrix view is only a view of the block underlying the
+     existing matrix, M.  The block containing the elements of M is not
+     owned by the new matrix view.  When the view goes out of scope the
+     original matrix M and its block will continue to exist.  The
+     original memory can only be deallocated by freeing the original
+     matrix.  Of course, the original matrix should not be deallocated
+     while the view is still in use.
+
+     The function `gsl_matrix_const_submatrix' is equivalent to
+     `gsl_matrix_submatrix' but can be used for matrices which are
+     declared `const'.
+
+ -- Function: gsl_matrix_view gsl_matrix_view_array (double * BASE,
+          size_t N1, size_t N2)
+ -- Function: gsl_matrix_const_view gsl_matrix_const_view_array (const
+          double * BASE, size_t N1, size_t N2)
+     These functions return a matrix view of the array BASE.  The
+     matrix has N1 rows and N2 columns.  The physical number of columns
+     in memory is also given by N2.  Mathematically, the (i,j)-th
+     element of the new matrix is given by,
+
+          m'(i,j) = base[i*n2 + j]
+
+     where the index I runs from 0 to `n1-1' and the index J runs from
+     0 to `n2-1'.
+
+     The new matrix is only a view of the array BASE.  When the view
+     goes out of scope the original array BASE will continue to exist.
+     The original memory can only be deallocated by freeing the original
+     array.  Of course, the original array should not be deallocated
+     while the view is still in use.
+
+     The function `gsl_matrix_const_view_array' is equivalent to
+     `gsl_matrix_view_array' but can be used for matrices which are
+     declared `const'.
+
+ -- Function: gsl_matrix_view gsl_matrix_view_array_with_tda (double *
+          BASE, size_t N1, size_t N2, size_t TDA)
+ -- Function: gsl_matrix_const_view
+          gsl_matrix_const_view_array_with_tda (const double * BASE,
+          size_t N1, size_t N2, size_t TDA)
+     These functions return a matrix view of the array BASE with a
+     physical number of columns TDA which may differ from the
+     corresponding dimension of the matrix.  The matrix has N1 rows and
+     N2 columns, and the physical number of columns in memory is given
+     by TDA.  Mathematically, the (i,j)-th element of the new matrix is
+     given by,
+
+          m'(i,j) = base[i*tda + j]
+
+     where the index I runs from 0 to `n1-1' and the index J runs from
+     0 to `n2-1'.
+
+     The new matrix is only a view of the array BASE.  When the view
+     goes out of scope the original array BASE will continue to exist.
+     The original memory can only be deallocated by freeing the original
+     array.  Of course, the original array should not be deallocated
+     while the view is still in use.
+
+     The function `gsl_matrix_const_view_array_with_tda' is equivalent
+     to `gsl_matrix_view_array_with_tda' but can be used for matrices
+     which are declared `const'.
+
+ -- Function: gsl_matrix_view gsl_matrix_view_vector (gsl_vector * V,
+          size_t N1, size_t N2)
+ -- Function: gsl_matrix_const_view gsl_matrix_const_view_vector (const
+          gsl_vector * V, size_t N1, size_t N2)
+     These functions return a matrix view of the vector V.  The matrix
+     has N1 rows and N2 columns. The vector must have unit stride. The
+     physical number of columns in memory is also given by N2.
+     Mathematically, the (i,j)-th element of the new matrix is given by,
+
+          m'(i,j) = v->data[i*n2 + j]
+
+     where the index I runs from 0 to `n1-1' and the index J runs from
+     0 to `n2-1'.
+
+     The new matrix is only a view of the vector V.  When the view goes
+     out of scope the original vector V will continue to exist.  The
+     original memory can only be deallocated by freeing the original
+     vector.  Of course, the original vector should not be deallocated
+     while the view is still in use.
+
+     The function `gsl_matrix_const_view_vector' is equivalent to
+     `gsl_matrix_view_vector' but can be used for matrices which are
+     declared `const'.
+
+ -- Function: gsl_matrix_view gsl_matrix_view_vector_with_tda
+          (gsl_vector * V, size_t N1, size_t N2, size_t TDA)
+ -- Function: gsl_matrix_const_view
+gsl_matrix_const_view_vector_with_tda (const gsl_vector * V, size_t N1,
+          size_t N2, size_t TDA)
+     These functions return a matrix view of the vector V with a
+     physical number of columns TDA which may differ from the
+     corresponding matrix dimension.  The vector must have unit stride.
+     The matrix has N1 rows and N2 columns, and the physical number of
+     columns in memory is given by TDA.  Mathematically, the (i,j)-th
+     element of the new matrix is given by,
+
+          m'(i,j) = v->data[i*tda + j]
+
+     where the index I runs from 0 to `n1-1' and the index J runs from
+     0 to `n2-1'.
+
+     The new matrix is only a view of the vector V.  When the view goes
+     out of scope the original vector V will continue to exist.  The
+     original memory can only be deallocated by freeing the original
+     vector.  Of course, the original vector should not be deallocated
+     while the view is still in use.
+
+     The function `gsl_matrix_const_view_vector_with_tda' is equivalent
+     to `gsl_matrix_view_vector_with_tda' but can be used for matrices
+     which are declared `const'.
+
+
+File: gsl-ref.info,  Node: Creating row and column views,  Next: Copying matrices,  Prev: Matrix views,  Up: Matrices
+
+8.4.6 Creating row and column views
+-----------------------------------
+
+In general there are two ways to access an object, by reference or by
+copying.  The functions described in this section create vector views
+which allow access to a row or column of a matrix by reference.
+Modifying elements of the view is equivalent to modifying the matrix,
+since both the vector view and the matrix point to the same memory
+block.
+
+ -- Function: gsl_vector_view gsl_matrix_row (gsl_matrix * M, size_t I)
+ -- Function: gsl_vector_const_view gsl_matrix_const_row (const
+          gsl_matrix * M, size_t I)
+     These functions return a vector view of the I-th row of the matrix
+     M.  The `data' pointer of the new vector is set to null if I is
+     out of range.
+
+     The function `gsl_vector_const_row' is equivalent to
+     `gsl_matrix_row' but can be used for matrices which are declared
+     `const'.
+
+ -- Function: gsl_vector_view gsl_matrix_column (gsl_matrix * M, size_t
+          J)
+ -- Function: gsl_vector_const_view gsl_matrix_const_column (const
+          gsl_matrix * M, size_t J)
+     These functions return a vector view of the J-th column of the
+     matrix M.  The `data' pointer of the new vector is set to null if
+     J is out of range.
+
+     The function `gsl_vector_const_column' is equivalent to
+     `gsl_matrix_column' but can be used for matrices which are declared
+     `const'.
+
+ -- Function: gsl_vector_view gsl_matrix_diagonal (gsl_matrix * M)
+ -- Function: gsl_vector_const_view gsl_matrix_const_diagonal (const
+          gsl_matrix * M)
+     These functions returns a vector view of the diagonal of the matrix
+     M. The matrix M is not required to be square. For a rectangular
+     matrix the length of the diagonal is the same as the smaller
+     dimension of the matrix.
+
+     The function `gsl_matrix_const_diagonal' is equivalent to
+     `gsl_matrix_diagonal' but can be used for matrices which are
+     declared `const'.
+
+ -- Function: gsl_vector_view gsl_matrix_subdiagonal (gsl_matrix * M,
+          size_t K)
+ -- Function: gsl_vector_const_view gsl_matrix_const_subdiagonal (const
+          gsl_matrix * M, size_t K)
+     These functions return a vector view of the K-th subdiagonal of
+     the matrix M. The matrix M is not required to be square.  The
+     diagonal of the matrix corresponds to k = 0.
+
+     The function `gsl_matrix_const_subdiagonal' is equivalent to
+     `gsl_matrix_subdiagonal' but can be used for matrices which are
+     declared `const'.
+
+ -- Function: gsl_vector_view gsl_matrix_superdiagonal (gsl_matrix * M,
+          size_t K)
+ -- Function: gsl_vector_const_view gsl_matrix_const_superdiagonal
+          (const gsl_matrix * M, size_t K)
+     These functions return a vector view of the K-th superdiagonal of
+     the matrix M. The matrix M is not required to be square. The
+     diagonal of the matrix corresponds to k = 0.
+
+     The function `gsl_matrix_const_superdiagonal' is equivalent to
+     `gsl_matrix_superdiagonal' but can be used for matrices which are
+     declared `const'.
+
+
+File: gsl-ref.info,  Node: Copying matrices,  Next: Copying rows and columns,  Prev: Creating row and column views,  Up: Matrices
+
+8.4.7 Copying matrices
+----------------------
+
+ -- Function: int gsl_matrix_memcpy (gsl_matrix * DEST, const
+          gsl_matrix * SRC)
+     This function copies the elements of the matrix SRC into the
+     matrix DEST.  The two matrices must have the same size.
+
+ -- Function: int gsl_matrix_swap (gsl_matrix * M1, gsl_matrix * M2)
+     This function exchanges the elements of the matrices M1 and M2 by
+     copying.  The two matrices must have the same size.
+
+
+File: gsl-ref.info,  Node: Copying rows and columns,  Next: Exchanging rows and columns,  Prev: Copying matrices,  Up: Matrices
+
+8.4.8 Copying rows and columns
+------------------------------
+
+The functions described in this section copy a row or column of a matrix
+into a vector.  This allows the elements of the vector and the matrix to
+be modified independently.  Note that if the matrix and the vector point
+to overlapping regions of memory then the result will be undefined.  The
+same effect can be achieved with more generality using
+`gsl_vector_memcpy' with vector views of rows and columns.
+
+ -- Function: int gsl_matrix_get_row (gsl_vector * V, const gsl_matrix
+          * M, size_t I)
+     This function copies the elements of the I-th row of the matrix M
+     into the vector V.  The length of the vector must be the same as
+     the length of the row.
+
+ -- Function: int gsl_matrix_get_col (gsl_vector * V, const gsl_matrix
+          * M, size_t J)
+     This function copies the elements of the J-th column of the matrix
+     M into the vector V.  The length of the vector must be the same as
+     the length of the column.
+
+ -- Function: int gsl_matrix_set_row (gsl_matrix * M, size_t I, const
+          gsl_vector * V)
+     This function copies the elements of the vector V into the I-th
+     row of the matrix M.  The length of the vector must be the same as
+     the length of the row.
+
+ -- Function: int gsl_matrix_set_col (gsl_matrix * M, size_t J, const
+          gsl_vector * V)
+     This function copies the elements of the vector V into the J-th
+     column of the matrix M.  The length of the vector must be the same
+     as the length of the column.
+
+
+File: gsl-ref.info,  Node: Exchanging rows and columns,  Next: Matrix operations,  Prev: Copying rows and columns,  Up: Matrices
+
+8.4.9 Exchanging rows and columns
+---------------------------------
+
+The following functions can be used to exchange the rows and columns of
+a matrix.
+
+ -- Function: int gsl_matrix_swap_rows (gsl_matrix * M, size_t I,
+          size_t J)
+     This function exchanges the I-th and J-th rows of the matrix M
+     in-place.
+
+ -- Function: int gsl_matrix_swap_columns (gsl_matrix * M, size_t I,
+          size_t J)
+     This function exchanges the I-th and J-th columns of the matrix M
+     in-place.
+
+ -- Function: int gsl_matrix_swap_rowcol (gsl_matrix * M, size_t I,
+          size_t J)
+     This function exchanges the I-th row and J-th column of the matrix
+     M in-place.  The matrix must be square for this operation to be
+     possible.
+
+ -- Function: int gsl_matrix_transpose_memcpy (gsl_matrix * DEST, const
+          gsl_matrix * SRC)
+     This function makes the matrix DEST the transpose of the matrix
+     SRC by copying the elements of SRC into DEST.  This function works
+     for all matrices provided that the dimensions of the matrix DEST
+     match the transposed dimensions of the matrix SRC.
+
+ -- Function: int gsl_matrix_transpose (gsl_matrix * M)
+     This function replaces the matrix M by its transpose by copying
+     the elements of the matrix in-place.  The matrix must be square
+     for this operation to be possible.
+
+
+File: gsl-ref.info,  Node: Matrix operations,  Next: Finding maximum and minimum elements of matrices,  Prev: Exchanging rows and columns,  Up: Matrices
+
+8.4.10 Matrix operations
+------------------------
+
+The following operations are defined for real and complex matrices.
+
+ -- Function: int gsl_matrix_add (gsl_matrix * A, const gsl_matrix * B)
+     This function adds the elements of matrix B to the elements of
+     matrix A, a'(i,j) = a(i,j) + b(i,j). The two matrices must have the
+     same dimensions.
+
+ -- Function: int gsl_matrix_sub (gsl_matrix * A, const gsl_matrix * B)
+     This function subtracts the elements of matrix B from the elements
+     of matrix A, a'(i,j) = a(i,j) - b(i,j). The two matrices must have
+     the same dimensions.
+
+ -- Function: int gsl_matrix_mul_elements (gsl_matrix * A, const
+          gsl_matrix * B)
+     This function multiplies the elements of matrix A by the elements
+     of matrix B, a'(i,j) = a(i,j) * b(i,j). The two matrices must have
+     the same dimensions.
+
+ -- Function: int gsl_matrix_div_elements (gsl_matrix * A, const
+          gsl_matrix * B)
+     This function divides the elements of matrix A by the elements of
+     matrix B, a'(i,j) = a(i,j) / b(i,j). The two matrices must have the
+     same dimensions.
+
+ -- Function: int gsl_matrix_scale (gsl_matrix * A, const double X)
+     This function multiplies the elements of matrix A by the constant
+     factor X, a'(i,j) = x a(i,j).
+
+ -- Function: int gsl_matrix_add_constant (gsl_matrix * A, const double
+          X)
+     This function adds the constant value X to the elements of the
+     matrix A, a'(i,j) = a(i,j) + x.
+
+
+File: gsl-ref.info,  Node: Finding maximum and minimum elements of matrices,  Next: Matrix properties,  Prev: Matrix operations,  Up: Matrices
+
+8.4.11 Finding maximum and minimum elements of matrices
+-------------------------------------------------------
+
+The following operations are only defined for real matrices.
+
+ -- Function: double gsl_matrix_max (const gsl_matrix * M)
+     This function returns the maximum value in the matrix M.
+
+ -- Function: double gsl_matrix_min (const gsl_matrix * M)
+     This function returns the minimum value in the matrix M.
+
+ -- Function: void gsl_matrix_minmax (const gsl_matrix * M, double *
+          MIN_OUT, double * MAX_OUT)
+     This function returns the minimum and maximum values in the matrix
+     M, storing them in MIN_OUT and MAX_OUT.
+
+ -- Function: void gsl_matrix_max_index (const gsl_matrix * M, size_t *
+          IMAX, size_t * JMAX)
+     This function returns the indices of the maximum value in the
+     matrix M, storing them in IMAX and JMAX.  When there are several
+     equal maximum elements then the first element found is returned,
+     searching in row-major order.
+
+ -- Function: void gsl_matrix_min_index (const gsl_matrix * M, size_t *
+          IMIN, size_t * JMIN)
+     This function returns the indices of the minimum value in the
+     matrix M, storing them in IMIN and JMIN.  When there are several
+     equal minimum elements then the first element found is returned,
+     searching in row-major order.
+
+ -- Function: void gsl_matrix_minmax_index (const gsl_matrix * M,
+          size_t * IMIN, size_t * JMIN, size_t * IMAX, size_t * JMAX)
+     This function returns the indices of the minimum and maximum
+     values in the matrix M, storing them in (IMIN,JMIN) and
+     (IMAX,JMAX). When there are several equal minimum or maximum
+     elements then the first elements found are returned, searching in
+     row-major order.
+
+
+File: gsl-ref.info,  Node: Matrix properties,  Next: Example programs for matrices,  Prev: Finding maximum and minimum elements of matrices,  Up: Matrices
+
+8.4.12 Matrix properties
+------------------------
+
+ -- Function: int gsl_matrix_isnull (const gsl_matrix * M)
+ -- Function: int gsl_matrix_ispos (const gsl_matrix * M)
+ -- Function: int gsl_matrix_isneg (const gsl_matrix * M)
+     These functions return 1 if all the elements of the matrix M are
+     zero, strictly positive, or strictly negative respectively, and 0
+     otherwise. To test for a non-negative matrix, use the expression
+     `!gsl_matrix_isneg(m)'.  To test whether a matrix is
+     positive-definite, use the Cholesky decomposition (*note Cholesky
+     Decomposition::).
+
+
+File: gsl-ref.info,  Node: Example programs for matrices,  Prev: Matrix properties,  Up: Matrices
+
+8.4.13 Example programs for matrices
+------------------------------------
+
+The program below shows how to allocate, initialize and read from a
+matrix using the functions `gsl_matrix_alloc', `gsl_matrix_set' and
+`gsl_matrix_get'.
+
+     #include <stdio.h>
+     #include <gsl/gsl_matrix.h>
+
+     int
+     main (void)
+     {
+       int i, j;
+       gsl_matrix * m = gsl_matrix_alloc (10, 3);
+
+       for (i = 0; i < 10; i++)
+         for (j = 0; j < 3; j++)
+           gsl_matrix_set (m, i, j, 0.23 + 100*i + j);
+
+       for (i = 0; i < 100; i++)  /* OUT OF RANGE ERROR */
+         for (j = 0; j < 3; j++)
+           printf ("m(%d,%d) = %g\n", i, j,
+                   gsl_matrix_get (m, i, j));
+
+       gsl_matrix_free (m);
+
+       return 0;
+     }
+
+Here is the output from the program.  The final loop attempts to read
+outside the range of the matrix `m', and the error is trapped by the
+range-checking code in `gsl_matrix_get'.
+
+     $ ./a.out
+     m(0,0) = 0.23
+     m(0,1) = 1.23
+     m(0,2) = 2.23
+     m(1,0) = 100.23
+     m(1,1) = 101.23
+     m(1,2) = 102.23
+     ...
+     m(9,2) = 902.23
+     gsl: matrix_source.c:13: ERROR: first index out of range
+     Default GSL error handler invoked.
+     Aborted (core dumped)
+
+The next program shows how to write a matrix to a file.
+
+     #include <stdio.h>
+     #include <gsl/gsl_matrix.h>
+
+     int
+     main (void)
+     {
+       int i, j, k = 0;
+       gsl_matrix * m = gsl_matrix_alloc (100, 100);
+       gsl_matrix * a = gsl_matrix_alloc (100, 100);
+
+       for (i = 0; i < 100; i++)
+         for (j = 0; j < 100; j++)
+           gsl_matrix_set (m, i, j, 0.23 + i + j);
+
+       {
+          FILE * f = fopen ("test.dat", "wb");
+          gsl_matrix_fwrite (f, m);
+          fclose (f);
+       }
+
+       {
+          FILE * f = fopen ("test.dat", "rb");
+          gsl_matrix_fread (f, a);
+          fclose (f);
+       }
+
+       for (i = 0; i < 100; i++)
+         for (j = 0; j < 100; j++)
+           {
+             double mij = gsl_matrix_get (m, i, j);
+             double aij = gsl_matrix_get (a, i, j);
+             if (mij != aij) k++;
+           }
+
+       gsl_matrix_free (m);
+       gsl_matrix_free (a);
+
+       printf ("differences = %d (should be zero)\n", k);
+       return (k > 0);
+     }
+
+After running this program the file `test.dat' should contain the
+elements of `m', written in binary format.  The matrix which is read
+back in using the function `gsl_matrix_fread' should be exactly equal
+to the original matrix.
+
+   The following program demonstrates the use of vector views.  The
+program computes the column norms of a matrix.
+
+     #include <math.h>
+     #include <stdio.h>
+     #include <gsl/gsl_matrix.h>
+     #include <gsl/gsl_blas.h>
+
+     int
+     main (void)
+     {
+       size_t i,j;
+
+       gsl_matrix *m = gsl_matrix_alloc (10, 10);
+
+       for (i = 0; i < 10; i++)
+         for (j = 0; j < 10; j++)
+           gsl_matrix_set (m, i, j, sin (i) + cos (j));
+
+       for (j = 0; j < 10; j++)
+         {
+           gsl_vector_view column = gsl_matrix_column (m, j);
+           double d;
+
+           d = gsl_blas_dnrm2 (&column.vector);
+
+           printf ("matrix column %d, norm = %g\n", j, d);
+         }
+
+       gsl_matrix_free (m);
+
+       return 0;
+     }
+
+Here is the output of the program,
+
+     $ ./a.out
+     matrix column 0, norm = 4.31461
+     matrix column 1, norm = 3.1205
+     matrix column 2, norm = 2.19316
+     matrix column 3, norm = 3.26114
+     matrix column 4, norm = 2.53416
+     matrix column 5, norm = 2.57281
+     matrix column 6, norm = 4.20469
+     matrix column 7, norm = 3.65202
+     matrix column 8, norm = 2.08524
+     matrix column 9, norm = 3.07313
+
+The results can be confirmed using GNU OCTAVE,
+
+     $ octave
+     GNU Octave, version 2.0.16.92
+     octave> m = sin(0:9)' * ones(1,10)
+                    + ones(10,1) * cos(0:9);
+     octave> sqrt(sum(m.^2))
+     ans =
+       4.3146  3.1205  2.1932  3.2611  2.5342  2.5728
+       4.2047  3.6520  2.0852  3.0731
+
+
+File: gsl-ref.info,  Node: Vector and Matrix References and Further Reading,  Prev: Matrices,  Up: Vectors and Matrices
+
+8.5 References and Further Reading
+==================================
+
+The block, vector and matrix objects in GSL follow the `valarray' model
+of C++.  A description of this model can be found in the following
+reference,
+
+     B. Stroustrup, `The C++ Programming Language' (3rd Ed), Section
+     22.4 Vector Arithmetic.  Addison-Wesley 1997, ISBN 0-201-88954-4.
+
+
+File: gsl-ref.info,  Node: Permutations,  Next: Combinations,  Prev: Vectors and Matrices,  Up: Top
+
+9 Permutations
+**************
+
+This chapter describes functions for creating and manipulating
+permutations. A permutation p is represented by an array of n integers
+in the range 0 to n-1, where each value p_i occurs once and only once.
+The application of a permutation p to a vector v yields a new vector v'
+where v'_i = v_{p_i}.  For example, the array (0,1,3,2) represents a
+permutation which exchanges the last two elements of a four element
+vector.  The corresponding identity permutation is (0,1,2,3).
+
+   Note that the permutations produced by the linear algebra routines
+correspond to the exchange of matrix columns, and so should be
+considered as applying to row-vectors in the form v' = v P rather than
+column-vectors, when permuting the elements of a vector.
+
+   The functions described in this chapter are defined in the header
+file `gsl_permutation.h'.
+
+* Menu:
+
+* The Permutation struct::
+* Permutation allocation::
+* Accessing permutation elements::
+* Permutation properties::
+* Permutation functions::
+* Applying Permutations::
+* Reading and writing permutations::
+* Permutations in cyclic form::
+* Permutation Examples::
+* Permutation References and Further Reading::
+
+
+File: gsl-ref.info,  Node: The Permutation struct,  Next: Permutation allocation,  Up: Permutations
+
+9.1 The Permutation struct
+==========================
+
+A permutation is defined by a structure containing two components, the
+size of the permutation and a pointer to the permutation array.  The
+elements of the permutation array are all of type `size_t'.  The
+`gsl_permutation' structure looks like this,
+
+     typedef struct
+     {
+       size_t size;
+       size_t * data;
+     } gsl_permutation;
+
+
+
+File: gsl-ref.info,  Node: Permutation allocation,  Next: Accessing permutation elements,  Prev: The Permutation struct,  Up: Permutations
+
+9.2 Permutation allocation
+==========================
+
+ -- Function: gsl_permutation * gsl_permutation_alloc (size_t N)
+     This function allocates memory for a new permutation of size N.
+     The permutation is not initialized and its elements are undefined.
+     Use the function `gsl_permutation_calloc' if you want to create a
+     permutation which is initialized to the identity. A null pointer is
+     returned if insufficient memory is available to create the
+     permutation.
+
+ -- Function: gsl_permutation * gsl_permutation_calloc (size_t N)
+     This function allocates memory for a new permutation of size N and
+     initializes it to the identity. A null pointer is returned if
+     insufficient memory is available to create the permutation.
+
+ -- Function: void gsl_permutation_init (gsl_permutation * P)
+     This function initializes the permutation P to the identity, i.e.
+     (0,1,2,...,n-1).
+
+ -- Function: void gsl_permutation_free (gsl_permutation * P)
+     This function frees all the memory used by the permutation P.
+
+ -- Function: int gsl_permutation_memcpy (gsl_permutation * DEST, const
+          gsl_permutation * SRC)
+     This function copies the elements of the permutation SRC into the
+     permutation DEST.  The two permutations must have the same size.
+
+
+File: gsl-ref.info,  Node: Accessing permutation elements,  Next: Permutation properties,  Prev: Permutation allocation,  Up: Permutations
+
+9.3 Accessing permutation elements
+==================================
+
+The following functions can be used to access and manipulate
+permutations.
+
+ -- Function: size_t gsl_permutation_get (const gsl_permutation * P,
+          const size_t I)
+     This function returns the value of the I-th element of the
+     permutation P.  If I lies outside the allowed range of 0 to N-1
+     then the error handler is invoked and 0 is returned.
+
+ -- Function: int gsl_permutation_swap (gsl_permutation * P, const
+          size_t I, const size_t J)
+     This function exchanges the I-th and J-th elements of the
+     permutation P.
+
+
+File: gsl-ref.info,  Node: Permutation properties,  Next: Permutation functions,  Prev: Accessing permutation elements,  Up: Permutations
+
+9.4 Permutation properties
+==========================
+
+ -- Function: size_t gsl_permutation_size (const gsl_permutation * P)
+     This function returns the size of the permutation P.
+
+ -- Function: size_t * gsl_permutation_data (const gsl_permutation * P)
+     This function returns a pointer to the array of elements in the
+     permutation P.
+
+ -- Function: int gsl_permutation_valid (gsl_permutation * P)
+     This function checks that the permutation P is valid.  The N
+     elements should contain each of the numbers 0 to N-1 once and only
+     once.
+
+
+File: gsl-ref.info,  Node: Permutation functions,  Next: Applying Permutations,  Prev: Permutation properties,  Up: Permutations
+
+9.5 Permutation functions
+=========================
+
+ -- Function: void gsl_permutation_reverse (gsl_permutation * P)
+     This function reverses the elements of the permutation P.
+
+ -- Function: int gsl_permutation_inverse (gsl_permutation * INV, const
+          gsl_permutation * P)
+     This function computes the inverse of the permutation P, storing
+     the result in INV.
+
+ -- Function: int gsl_permutation_next (gsl_permutation * P)
+     This function advances the permutation P to the next permutation
+     in lexicographic order and returns `GSL_SUCCESS'.  If no further
+     permutations are available it returns `GSL_FAILURE' and leaves P
+     unmodified.  Starting with the identity permutation and repeatedly
+     applying this function will iterate through all possible
+     permutations of a given order.
+
+ -- Function: int gsl_permutation_prev (gsl_permutation * P)
+     This function steps backwards from the permutation P to the
+     previous permutation in lexicographic order, returning
+     `GSL_SUCCESS'.  If no previous permutation is available it returns
+     `GSL_FAILURE' and leaves P unmodified.
+
+
+File: gsl-ref.info,  Node: Applying Permutations,  Next: Reading and writing permutations,  Prev: Permutation functions,  Up: Permutations
+
+9.6 Applying Permutations
+=========================
+
+ -- Function: int gsl_permute (const size_t * P, double * DATA, size_t
+          STRIDE, size_t N)
+     This function applies the permutation P to the array DATA of size
+     N with stride STRIDE.
+
+ -- Function: int gsl_permute_inverse (const size_t * P, double * DATA,
+          size_t STRIDE, size_t N)
+     This function applies the inverse of the permutation P to the
+     array DATA of size N with stride STRIDE.
+
+ -- Function: int gsl_permute_vector (const gsl_permutation * P,
+          gsl_vector * V)
+     This function applies the permutation P to the elements of the
+     vector V, considered as a row-vector acted on by a permutation
+     matrix from the right, v' = v P.  The j-th column of the
+     permutation matrix P is given by the p_j-th column of the identity
+     matrix. The permutation P and the vector V must have the same
+     length.
+
+ -- Function: int gsl_permute_vector_inverse (const gsl_permutation *
+          P, gsl_vector * V)
+     This function applies the inverse of the permutation P to the
+     elements of the vector V, considered as a row-vector acted on by
+     an inverse permutation matrix from the right, v' = v P^T.  Note
+     that for permutation matrices the inverse is the same as the
+     transpose.  The j-th column of the permutation matrix P is given by
+     the p_j-th column of the identity matrix. The permutation P and
+     the vector V must have the same length.
+
+ -- Function: int gsl_permutation_mul (gsl_permutation * P, const
+          gsl_permutation * PA, const gsl_permutation * PB)
+     This function combines the two permutations PA and PB into a
+     single permutation P, where p = pa . pb. The permutation P is
+     equivalent to applying pb first and then PA.
+
+
+File: gsl-ref.info,  Node: Reading and writing permutations,  Next: Permutations in cyclic form,  Prev: Applying Permutations,  Up: Permutations
+
+9.7 Reading and writing permutations
+====================================
+
+The library provides functions for reading and writing permutations to a
+file as binary data or formatted text.
+
+ -- Function: int gsl_permutation_fwrite (FILE * STREAM, const
+          gsl_permutation * P)
+     This function writes the elements of the permutation P to the
+     stream STREAM in binary format.  The function returns
+     `GSL_EFAILED' if there was a problem writing to the file.  Since
+     the data is written in the native binary format it may not be
+     portable between different architectures.
+
+ -- Function: int gsl_permutation_fread (FILE * STREAM, gsl_permutation
+          * P)
+     This function reads into the permutation P from the open stream
+     STREAM in binary format.  The permutation P must be preallocated
+     with the correct length since the function uses the size of P to
+     determine how many bytes to read.  The function returns
+     `GSL_EFAILED' if there was a problem reading from the file.  The
+     data is assumed to have been written in the native binary format
+     on the same architecture.
+
+ -- Function: int gsl_permutation_fprintf (FILE * STREAM, const
+          gsl_permutation * P, const char * FORMAT)
+     This function writes the elements of the permutation P
+     line-by-line to the stream STREAM using the format specifier
+     FORMAT, which should be suitable for a type of SIZE_T.  On a GNU
+     system the type modifier `Z' represents `size_t', so `"%Zu\n"' is
+     a suitable format.  The function returns `GSL_EFAILED' if there
+     was a problem writing to the file.
+
+ -- Function: int gsl_permutation_fscanf (FILE * STREAM,
+          gsl_permutation * P)
+     This function reads formatted data from the stream STREAM into the
+     permutation P.  The permutation P must be preallocated with the
+     correct length since the function uses the size of P to determine
+     how many numbers to read.  The function returns `GSL_EFAILED' if
+     there was a problem reading from the file.
+
+
+File: gsl-ref.info,  Node: Permutations in cyclic form,  Next: Permutation Examples,  Prev: Reading and writing permutations,  Up: Permutations
+
+9.8 Permutations in cyclic form
+===============================
+
+A permutation can be represented in both "linear" and "cyclic"
+notations.  The functions described in this section convert between the
+two forms.  The linear notation is an index mapping, and has already
+been described above.  The cyclic notation expresses a permutation as a
+series of circular rearrangements of groups of elements, or "cycles".
+
+   For example, under the cycle (1 2 3), 1 is replaced by 2, 2 is
+replaced by 3 and 3 is replaced by 1 in a circular fashion. Cycles of
+different sets of elements can be combined independently, for example
+(1 2 3) (4 5) combines the cycle (1 2 3) with the cycle (4 5), which is
+an exchange of elements 4 and 5.  A cycle of length one represents an
+element which is unchanged by the permutation and is referred to as a
+"singleton".
+
+   It can be shown that every permutation can be decomposed into
+combinations of cycles.  The decomposition is not unique, but can always
+be rearranged into a standard "canonical form" by a reordering of
+elements.  The library uses the canonical form defined in Knuth's `Art
+of Computer Programming' (Vol 1, 3rd Ed, 1997) Section 1.3.3, p.178.
+
+   The procedure for obtaining the canonical form given by Knuth is,
+
+  1. Write all singleton cycles explicitly
+
+  2. Within each cycle, put the smallest number first
+
+  3. Order the cycles in decreasing order of the first number in the
+     cycle.
+
+For example, the linear representation (2 4 3 0 1) is represented as (1
+4) (0 2 3) in canonical form. The permutation corresponds to an
+exchange of elements 1 and 4, and rotation of elements 0, 2 and 3.
+
+   The important property of the canonical form is that it can be
+reconstructed from the contents of each cycle without the brackets. In
+addition, by removing the brackets it can be considered as a linear
+representation of a different permutation. In the example given above
+the permutation (2 4 3 0 1) would become (1 4 0 2 3).  This mapping has
+many applications in the theory of permutations.
+
+ -- Function: int gsl_permutation_linear_to_canonical (gsl_permutation
+          * Q, const gsl_permutation * P)
+     This function computes the canonical form of the permutation P and
+     stores it in the output argument Q.
+
+ -- Function: int gsl_permutation_canonical_to_linear (gsl_permutation
+          * P, const gsl_permutation * Q)
+     This function converts a permutation Q in canonical form back into
+     linear form storing it in the output argument P.
+
+ -- Function: size_t gsl_permutation_inversions (const gsl_permutation
+          * P)
+     This function counts the number of inversions in the permutation
+     P.  An inversion is any pair of elements that are not in order.
+     For example, the permutation 2031 has three inversions,
+     corresponding to the pairs (2,0) (2,1) and (3,1).  The identity
+     permutation has no inversions.
+
+ -- Function: size_t gsl_permutation_linear_cycles (const
+          gsl_permutation * P)
+     This function counts the number of cycles in the permutation P,
+     given in linear form.
+
+ -- Function: size_t gsl_permutation_canonical_cycles (const
+          gsl_permutation * Q)
+     This function counts the number of cycles in the permutation Q,
+     given in canonical form.
+
+
+File: gsl-ref.info,  Node: Permutation Examples,  Next: Permutation References and Further Reading,  Prev: Permutations in cyclic form,  Up: Permutations
+
+9.9 Examples
+============
+
+The example program below creates a random permutation (by shuffling the
+elements of the identity) and finds its inverse.
+
+     #include <stdio.h>
+     #include <gsl/gsl_rng.h>
+     #include <gsl/gsl_randist.h>
+     #include <gsl/gsl_permutation.h>
+
+     int
+     main (void)
+     {
+       const size_t N = 10;
+       const gsl_rng_type * T;
+       gsl_rng * r;
+
+       gsl_permutation * p = gsl_permutation_alloc (N);
+       gsl_permutation * q = gsl_permutation_alloc (N);
+
+       gsl_rng_env_setup();
+       T = gsl_rng_default;
+       r = gsl_rng_alloc (T);
+
+       printf ("initial permutation:");
+       gsl_permutation_init (p);
+       gsl_permutation_fprintf (stdout, p, " %u");
+       printf ("\n");
+
+       printf (" random permutation:");
+       gsl_ran_shuffle (r, p->data, N, sizeof(size_t));
+       gsl_permutation_fprintf (stdout, p, " %u");
+       printf ("\n");
+
+       printf ("inverse permutation:");
+       gsl_permutation_inverse (q, p);
+       gsl_permutation_fprintf (stdout, q, " %u");
+       printf ("\n");
+
+       gsl_permutation_free (p);
+       gsl_permutation_free (q);
+       gsl_rng_free (r);
+
+       return 0;
+     }
+
+Here is the output from the program,
+
+     $ ./a.out
+     initial permutation: 0 1 2 3 4 5 6 7 8 9
+      random permutation: 1 3 5 2 7 6 0 4 9 8
+     inverse permutation: 6 0 3 1 7 2 5 4 9 8
+
+The random permutation `p[i]' and its inverse `q[i]' are related
+through the identity `p[q[i]] = i', which can be verified from the
+output.
+
+   The next example program steps forwards through all possible third
+order permutations, starting from the identity,
+
+     #include <stdio.h>
+     #include <gsl/gsl_permutation.h>
+
+     int
+     main (void)
+     {
+       gsl_permutation * p = gsl_permutation_alloc (3);
+
+       gsl_permutation_init (p);
+
+       do
+        {
+           gsl_permutation_fprintf (stdout, p, " %u");
+           printf ("\n");
+        }
+       while (gsl_permutation_next(p) == GSL_SUCCESS);
+
+       gsl_permutation_free (p);
+
+       return 0;
+     }
+
+Here is the output from the program,
+
+     $ ./a.out
+      0 1 2
+      0 2 1
+      1 0 2
+      1 2 0
+      2 0 1
+      2 1 0
+
+The permutations are generated in lexicographic order.  To reverse the
+sequence, begin with the final permutation (which is the reverse of the
+identity) and replace `gsl_permutation_next' with
+`gsl_permutation_prev'.
+
+
+File: gsl-ref.info,  Node: Permutation References and Further Reading,  Prev: Permutation Examples,  Up: Permutations
+
+9.10 References and Further Reading
+===================================
+
+The subject of permutations is covered extensively in Knuth's `Sorting
+and Searching',
+
+     Donald E. Knuth, `The Art of Computer Programming: Sorting and
+     Searching' (Vol 3, 3rd Ed, 1997), Addison-Wesley, ISBN 0201896850.
+
+For the definition of the "canonical form" see,
+
+     Donald E. Knuth, `The Art of Computer Programming: Fundamental
+     Algorithms' (Vol 1, 3rd Ed, 1997), Addison-Wesley, ISBN 0201896850.
+     Section 1.3.3, `An Unusual Correspondence', p.178-179.
+
+
+File: gsl-ref.info,  Node: Combinations,  Next: Sorting,  Prev: Permutations,  Up: Top
+
+10 Combinations
+***************
+
+This chapter describes functions for creating and manipulating
+combinations. A combination c is represented by an array of k integers
+in the range 0 to n-1, where each value c_i occurs at most once.  The
+combination c corresponds to indices of k elements chosen from an n
+element vector.  Combinations are useful for iterating over all
+k-element subsets of a set.
+
+   The functions described in this chapter are defined in the header
+file `gsl_combination.h'.
+
+* Menu:
+
+* The Combination struct::
+* Combination allocation::
+* Accessing combination elements::
+* Combination properties::
+* Combination functions::
+* Reading and writing combinations::
+* Combination Examples::
+* Combination References and Further Reading::
+
+
+File: gsl-ref.info,  Node: The Combination struct,  Next: Combination allocation,  Up: Combinations
+
+10.1 The Combination struct
+===========================
+
+A combination is defined by a structure containing three components, the
+values of n and k, and a pointer to the combination array.  The
+elements of the combination array are all of type `size_t', and are
+stored in increasing order.  The `gsl_combination' structure looks like
+this,
+
+     typedef struct
+     {
+       size_t n;
+       size_t k;
+       size_t *data;
+     } gsl_combination;
+
+
+
+File: gsl-ref.info,  Node: Combination allocation,  Next: Accessing combination elements,  Prev: The Combination struct,  Up: Combinations
+
+10.2 Combination allocation
+===========================
+
+ -- Function: gsl_combination * gsl_combination_alloc (size_t N, size_t
+          K)
+     This function allocates memory for a new combination with
+     parameters N, K.  The combination is not initialized and its
+     elements are undefined.  Use the function `gsl_combination_calloc'
+     if you want to create a combination which is initialized to the
+     lexicographically first combination. A null pointer is returned if
+     insufficient memory is available to create the combination.
+
+ -- Function: gsl_combination * gsl_combination_calloc (size_t N,
+          size_t K)
+     This function allocates memory for a new combination with
+     parameters N, K and initializes it to the lexicographically first
+     combination. A null pointer is returned if insufficient memory is
+     available to create the combination.
+
+ -- Function: void gsl_combination_init_first (gsl_combination * C)
+     This function initializes the combination C to the
+     lexicographically first combination, i.e.  (0,1,2,...,k-1).
+
+ -- Function: void gsl_combination_init_last (gsl_combination * C)
+     This function initializes the combination C to the
+     lexicographically last combination, i.e.  (n-k,n-k+1,...,n-1).
+
+ -- Function: void gsl_combination_free (gsl_combination * C)
+     This function frees all the memory used by the combination C.
+
+ -- Function: int gsl_combination_memcpy (gsl_combination * DEST, const
+          gsl_combination * SRC)
+     This function copies the elements of the combination SRC into the
+     combination DEST.  The two combinations must have the same size.
+
+
+File: gsl-ref.info,  Node: Accessing combination elements,  Next: Combination properties,  Prev: Combination allocation,  Up: Combinations
+
+10.3 Accessing combination elements
+===================================
+
+The following function can be used to access the elements of a
+combination.
+
+ -- Function: size_t gsl_combination_get (const gsl_combination * C,
+          const size_t I)
+     This function returns the value of the I-th element of the
+     combination C.  If I lies outside the allowed range of 0 to K-1
+     then the error handler is invoked and 0 is returned.
+
+
+File: gsl-ref.info,  Node: Combination properties,  Next: Combination functions,  Prev: Accessing combination elements,  Up: Combinations
+
+10.4 Combination properties
+===========================
+
+ -- Function: size_t gsl_combination_n (const gsl_combination * C)
+     This function returns the range (n) of the combination C.
+
+ -- Function: size_t gsl_combination_k (const gsl_combination * C)
+     This function returns the number of elements (k) in the
+     combination C.
+
+ -- Function: size_t * gsl_combination_data (const gsl_combination * C)
+     This function returns a pointer to the array of elements in the
+     combination C.
+
+ -- Function: int gsl_combination_valid (gsl_combination * C)
+     This function checks that the combination C is valid.  The K
+     elements should lie in the range 0 to N-1, with each value
+     occurring once at most and in increasing order.
+
+
+File: gsl-ref.info,  Node: Combination functions,  Next: Reading and writing combinations,  Prev: Combination properties,  Up: Combinations
+
+10.5 Combination functions
+==========================
+
+ -- Function: int gsl_combination_next (gsl_combination * C)
+     This function advances the combination C to the next combination
+     in lexicographic order and returns `GSL_SUCCESS'.  If no further
+     combinations are available it returns `GSL_FAILURE' and leaves C
+     unmodified.  Starting with the first combination and repeatedly
+     applying this function will iterate through all possible
+     combinations of a given order.
+
+ -- Function: int gsl_combination_prev (gsl_combination * C)
+     This function steps backwards from the combination C to the
+     previous combination in lexicographic order, returning
+     `GSL_SUCCESS'.  If no previous combination is available it returns
+     `GSL_FAILURE' and leaves C unmodified.
+
+
+File: gsl-ref.info,  Node: Reading and writing combinations,  Next: Combination Examples,  Prev: Combination functions,  Up: Combinations
+
+10.6 Reading and writing combinations
+=====================================
+
+The library provides functions for reading and writing combinations to a
+file as binary data or formatted text.
+
+ -- Function: int gsl_combination_fwrite (FILE * STREAM, const
+          gsl_combination * C)
+     This function writes the elements of the combination C to the
+     stream STREAM in binary format.  The function returns
+     `GSL_EFAILED' if there was a problem writing to the file.  Since
+     the data is written in the native binary format it may not be
+     portable between different architectures.
+
+ -- Function: int gsl_combination_fread (FILE * STREAM, gsl_combination
+          * C)
+     This function reads elements from the open stream STREAM into the
+     combination C in binary format.  The combination C must be
+     preallocated with correct values of n and k since the function
+     uses the size of C to determine how many bytes to read.  The
+     function returns `GSL_EFAILED' if there was a problem reading from
+     the file.  The data is assumed to have been written in the native
+     binary format on the same architecture.
+
+ -- Function: int gsl_combination_fprintf (FILE * STREAM, const
+          gsl_combination * C, const char * FORMAT)
+     This function writes the elements of the combination C
+     line-by-line to the stream STREAM using the format specifier
+     FORMAT, which should be suitable for a type of SIZE_T.  On a GNU
+     system the type modifier `Z' represents `size_t', so `"%Zu\n"' is
+     a suitable format.  The function returns `GSL_EFAILED' if there
+     was a problem writing to the file.
+
+ -- Function: int gsl_combination_fscanf (FILE * STREAM,
+          gsl_combination * C)
+     This function reads formatted data from the stream STREAM into the
+     combination C.  The combination C must be preallocated with
+     correct values of n and k since the function uses the size of C to
+     determine how many numbers to read.  The function returns
+     `GSL_EFAILED' if there was a problem reading from the file.
+
+
+File: gsl-ref.info,  Node: Combination Examples,  Next: Combination References and Further Reading,  Prev: Reading and writing combinations,  Up: Combinations
+
+10.7 Examples
+=============
+
+The example program below prints all subsets of the set {0,1,2,3}
+ordered by size.  Subsets of the same size are ordered
+lexicographically.
+
+     #include <stdio.h>
+     #include <gsl/gsl_combination.h>
+
+     int
+     main (void)
+     {
+       gsl_combination * c;
+       size_t i;
+
+       printf ("All subsets of {0,1,2,3} by size:\n") ;
+       for (i = 0; i <= 4; i++)
+         {
+           c = gsl_combination_calloc (4, i);
+           do
+             {
+               printf ("{");
+               gsl_combination_fprintf (stdout, c, " %u");
+               printf (" }\n");
+             }
+           while (gsl_combination_next (c) == GSL_SUCCESS);
+           gsl_combination_free (c);
+         }
+
+       return 0;
+     }
+
+Here is the output from the program,
+
+     $ ./a.out
+     All subsets of {0,1,2,3} by size:
+     { }
+     { 0 }
+     { 1 }
+     { 2 }
+     { 3 }
+     { 0 1 }
+     { 0 2 }
+     { 0 3 }
+     { 1 2 }
+     { 1 3 }
+     { 2 3 }
+     { 0 1 2 }
+     { 0 1 3 }
+     { 0 2 3 }
+     { 1 2 3 }
+     { 0 1 2 3 }
+
+All 16 subsets are generated, and the subsets of each size are sorted
+lexicographically.
+
+
+File: gsl-ref.info,  Node: Combination References and Further Reading,  Prev: Combination Examples,  Up: Combinations
+
+10.8 References and Further Reading
+===================================
+
+Further information on combinations can be found in,
+
+     Donald L. Kreher, Douglas R. Stinson, `Combinatorial Algorithms:
+     Generation, Enumeration and Search', 1998, CRC Press LLC, ISBN
+     084933988X
+
+
+
+File: gsl-ref.info,  Node: Sorting,  Next: BLAS Support,  Prev: Combinations,  Up: Top
+
+11 Sorting
+**********
+
+This chapter describes functions for sorting data, both directly and
+indirectly (using an index).  All the functions use the "heapsort"
+algorithm.  Heapsort is an O(N \log N) algorithm which operates
+in-place and does not require any additional storage.  It also provides
+consistent performance, the running time for its worst-case (ordered
+data) being not significantly longer than the average and best cases.
+Note that the heapsort algorithm does not preserve the relative ordering
+of equal elements--it is an "unstable" sort.  However the resulting
+order of equal elements will be consistent across different platforms
+when using these functions.
+
+* Menu:
+
+* Sorting objects::
+* Sorting vectors::
+* Selecting the k smallest or largest elements::
+* Computing the rank::
+* Sorting Examples::
+* Sorting References and Further Reading::
+
+
+File: gsl-ref.info,  Node: Sorting objects,  Next: Sorting vectors,  Up: Sorting
+
+11.1 Sorting objects
+====================
+
+The following function provides a simple alternative to the standard
+library function `qsort'.  It is intended for systems lacking `qsort',
+not as a replacement for it.  The function `qsort' should be used
+whenever possible, as it will be faster and can provide stable ordering
+of equal elements.  Documentation for `qsort' is available in the `GNU
+C Library Reference Manual'.
+
+   The functions described in this section are defined in the header
+file `gsl_heapsort.h'.
+
+ -- Function: void gsl_heapsort (void * ARRAY, size_t COUNT, size_t
+          SIZE, gsl_comparison_fn_t COMPARE)
+     This function sorts the COUNT elements of the array ARRAY, each of
+     size SIZE, into ascending order using the comparison function
+     COMPARE.  The type of the comparison function is defined by,
+
+          int (*gsl_comparison_fn_t) (const void * a,
+                                      const void * b)
+
+     A comparison function should return a negative integer if the first
+     argument is less than the second argument, `0' if the two arguments
+     are equal and a positive integer if the first argument is greater
+     than the second argument.
+
+     For example, the following function can be used to sort doubles
+     into ascending numerical order.
+
+          int
+          compare_doubles (const double * a,
+                           const double * b)
+          {
+              if (*a > *b)
+                 return 1;
+              else if (*a < *b)
+                 return -1;
+              else
+                 return 0;
+          }
+
+     The appropriate function call to perform the sort is,
+
+          gsl_heapsort (array, count, sizeof(double),
+                        compare_doubles);
+
+     Note that unlike `qsort' the heapsort algorithm cannot be made into
+     a stable sort by pointer arithmetic.  The trick of comparing
+     pointers for equal elements in the comparison function does not
+     work for the heapsort algorithm.  The heapsort algorithm performs
+     an internal rearrangement of the data which destroys its initial
+     ordering.
+
+ -- Function: int gsl_heapsort_index (size_t * P, const void * ARRAY,
+          size_t COUNT, size_t SIZE, gsl_comparison_fn_t COMPARE)
+     This function indirectly sorts the COUNT elements of the array
+     ARRAY, each of size SIZE, into ascending order using the
+     comparison function COMPARE.  The resulting permutation is stored
+     in P, an array of length N.  The elements of P give the index of
+     the array element which would have been stored in that position if
+     the array had been sorted in place.  The first element of P gives
+     the index of the least element in ARRAY, and the last element of P
+     gives the index of the greatest element in ARRAY.  The array
+     itself is not changed.
+
+
+File: gsl-ref.info,  Node: Sorting vectors,  Next: Selecting the k smallest or largest elements,  Prev: Sorting objects,  Up: Sorting
+
+11.2 Sorting vectors
+====================
+
+The following functions will sort the elements of an array or vector,
+either directly or indirectly.  They are defined for all real and
+integer types using the normal suffix rules.  For example, the `float'
+versions of the array functions are `gsl_sort_float' and
+`gsl_sort_float_index'.  The corresponding vector functions are
+`gsl_sort_vector_float' and `gsl_sort_vector_float_index'.  The
+prototypes are available in the header files `gsl_sort_float.h'
+`gsl_sort_vector_float.h'.  The complete set of prototypes can be
+included using the header files `gsl_sort.h' and `gsl_sort_vector.h'.
+
+   There are no functions for sorting complex arrays or vectors, since
+the ordering of complex numbers is not uniquely defined.  To sort a
+complex vector by magnitude compute a real vector containing the
+magnitudes of the complex elements, and sort this vector indirectly.
+The resulting index gives the appropriate ordering of the original
+complex vector.
+
+ -- Function: void gsl_sort (double * DATA, size_t STRIDE, size_t N)
+     This function sorts the N elements of the array DATA with stride
+     STRIDE into ascending numerical order.
+
+ -- Function: void gsl_sort_vector (gsl_vector * V)
+     This function sorts the elements of the vector V into ascending
+     numerical order.
+
+ -- Function: void gsl_sort_index (size_t * P, const double * DATA,
+          size_t STRIDE, size_t N)
+     This function indirectly sorts the N elements of the array DATA
+     with stride STRIDE into ascending order, storing the resulting
+     permutation in P.  The array P must be allocated with a sufficient
+     length to store the N elements of the permutation.  The elements
+     of P give the index of the array element which would have been
+     stored in that position if the array had been sorted in place.
+     The array DATA is not changed.
+
+ -- Function: int gsl_sort_vector_index (gsl_permutation * P, const
+          gsl_vector * V)
+     This function indirectly sorts the elements of the vector V into
+     ascending order, storing the resulting permutation in P.  The
+     elements of P give the index of the vector element which would
+     have been stored in that position if the vector had been sorted in
+     place.  The first element of P gives the index of the least element
+     in V, and the last element of P gives the index of the greatest
+     element in V.  The vector V is not changed.
+
diff --git a/gsl-1.9/doc/gsl-ref.info-2 b/gsl-1.9/doc/gsl-ref.info-2
new file mode 100644
index 0000000..3fd15e9
--- /dev/null
+++ b/gsl-1.9/doc/gsl-ref.info-2
@@ -0,0 +1,7071 @@
+This is gsl-ref.info, produced by makeinfo version 4.8 from
+gsl-ref.texi.
+
+INFO-DIR-SECTION Scientific software
+START-INFO-DIR-ENTRY
+* gsl-ref: (gsl-ref).                   GNU Scientific Library - Reference
+END-INFO-DIR-ENTRY
+
+   Copyright (C) 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004,
+2005, 2006, 2007 The GSL Team.
+
+   Permission is granted to copy, distribute and/or modify this document
+under the terms of the GNU Free Documentation License, Version 1.2 or
+any later version published by the Free Software Foundation; with the
+Invariant Sections being "GNU General Public License" and "Free Software
+Needs Free Documentation", the Front-Cover text being "A GNU Manual",
+and with the Back-Cover Text being (a) (see below).  A copy of the
+license is included in the section entitled "GNU Free Documentation
+License".
+
+   (a) The Back-Cover Text is: "You have freedom to copy and modify this
+GNU Manual, like GNU software."
+
+
+File: gsl-ref.info,  Node: Selecting the k smallest or largest elements,  Next: Computing the rank,  Prev: Sorting vectors,  Up: Sorting
+
+11.3 Selecting the k smallest or largest elements
+=================================================
+
+The functions described in this section select the k smallest or
+largest elements of a data set of size N.  The routines use an O(kN)
+direct insertion algorithm which is suited to subsets that are small
+compared with the total size of the dataset. For example, the routines
+are useful for selecting the 10 largest values from one million data
+points, but not for selecting the largest 100,000 values.  If the
+subset is a significant part of the total dataset it may be faster to
+sort all the elements of the dataset directly with an O(N \log N)
+algorithm and obtain the smallest or largest values that way.
+
+ -- Function: int gsl_sort_smallest (double * DEST, size_t K, const
+          double * SRC, size_t STRIDE, size_t N)
+     This function copies the K smallest elements of the array SRC, of
+     size N and stride STRIDE, in ascending numerical order into the
+     array DEST.  The size K of the subset must be less than or equal
+     to N.  The data SRC is not modified by this operation.
+
+ -- Function: int gsl_sort_largest (double * DEST, size_t K, const
+          double * SRC, size_t STRIDE, size_t N)
+     This function copies the K largest elements of the array SRC, of
+     size N and stride STRIDE, in descending numerical order into the
+     array DEST. K must be less than or equal to N. The data SRC is not
+     modified by this operation.
+
+ -- Function: int gsl_sort_vector_smallest (double * DEST, size_t K,
+          const gsl_vector * V)
+ -- Function: int gsl_sort_vector_largest (double * DEST, size_t K,
+          const gsl_vector * V)
+     These functions copy the K smallest or largest elements of the
+     vector V into the array DEST. K must be less than or equal to the
+     length of the vector V.
+
+   The following functions find the indices of the k smallest or
+largest elements of a dataset,
+
+ -- Function: int gsl_sort_smallest_index (size_t * P, size_t K, const
+          double * SRC, size_t STRIDE, size_t N)
+     This function stores the indices of the K smallest elements of the
+     array SRC, of size N and stride STRIDE, in the array P.  The
+     indices are chosen so that the corresponding data is in ascending
+     numerical order.  K must be less than or equal to N. The data SRC
+     is not modified by this operation.
+
+ -- Function: int gsl_sort_largest_index (size_t * P, size_t K, const
+          double * SRC, size_t STRIDE, size_t N)
+     This function stores the indices of the K largest elements of the
+     array SRC, of size N and stride STRIDE, in the array P.  The
+     indices are chosen so that the corresponding data is in descending
+     numerical order.  K must be less than or equal to N. The data SRC
+     is not modified by this operation.
+
+ -- Function: int gsl_sort_vector_smallest_index (size_t * P, size_t K,
+          const gsl_vector * V)
+ -- Function: int gsl_sort_vector_largest_index (size_t * P, size_t K,
+          const gsl_vector * V)
+     These functions store the indices of the K smallest or largest
+     elements of the vector V in the array P. K must be less than or
+     equal to the length of the vector V.
+
+
+File: gsl-ref.info,  Node: Computing the rank,  Next: Sorting Examples,  Prev: Selecting the k smallest or largest elements,  Up: Sorting
+
+11.4 Computing the rank
+=======================
+
+The "rank" of an element is its order in the sorted data.  The rank is
+the inverse of the index permutation, P.  It can be computed using the
+following algorithm,
+
+     for (i = 0; i < p->size; i++)
+     {
+         size_t pi = p->data[i];
+         rank->data[pi] = i;
+     }
+
+This can be computed directly from the function
+`gsl_permutation_inverse(rank,p)'.
+
+   The following function will print the rank of each element of the
+vector V,
+
+     void
+     print_rank (gsl_vector * v)
+     {
+       size_t i;
+       size_t n = v->size;
+       gsl_permutation * perm = gsl_permutation_alloc(n);
+       gsl_permutation * rank = gsl_permutation_alloc(n);
+
+       gsl_sort_vector_index (perm, v);
+       gsl_permutation_inverse (rank, perm);
+
+       for (i = 0; i < n; i++)
+        {
+         double vi = gsl_vector_get(v, i);
+         printf ("element = %d, value = %g, rank = %d\n",
+                  i, vi, rank->data[i]);
+        }
+
+       gsl_permutation_free (perm);
+       gsl_permutation_free (rank);
+     }
+
+
+File: gsl-ref.info,  Node: Sorting Examples,  Next: Sorting References and Further Reading,  Prev: Computing the rank,  Up: Sorting
+
+11.5 Examples
+=============
+
+The following example shows how to use the permutation P to print the
+elements of the vector V in ascending order,
+
+     gsl_sort_vector_index (p, v);
+
+     for (i = 0; i < v->size; i++)
+     {
+         double vpi = gsl_vector_get (v, p->data[i]);
+         printf ("order = %d, value = %g\n", i, vpi);
+     }
+
+The next example uses the function `gsl_sort_smallest' to select the 5
+smallest numbers from 100000 uniform random variates stored in an array,
+
+     #include <gsl/gsl_rng.h>
+     #include <gsl/gsl_sort_double.h>
+
+     int
+     main (void)
+     {
+       const gsl_rng_type * T;
+       gsl_rng * r;
+
+       size_t i, k = 5, N = 100000;
+
+       double * x = malloc (N * sizeof(double));
+       double * small = malloc (k * sizeof(double));
+
+       gsl_rng_env_setup();
+
+       T = gsl_rng_default;
+       r = gsl_rng_alloc (T);
+
+       for (i = 0; i < N; i++)
+         {
+           x[i] = gsl_rng_uniform(r);
+         }
+
+       gsl_sort_smallest (small, k, x, 1, N);
+
+       printf ("%d smallest values from %d\n", k, N);
+
+       for (i = 0; i < k; i++)
+         {
+           printf ("%d: %.18f\n", i, small[i]);
+         }
+
+       free (x);
+       free (small);
+       gsl_rng_free (r);
+       return 0;
+     }
+   The output lists the 5 smallest values, in ascending order,
+
+     $ ./a.out
+     5 smallest values from 100000
+     0: 0.000003489200025797
+     1: 0.000008199829608202
+     2: 0.000008953968062997
+     3: 0.000010712770745158
+     4: 0.000033531803637743
+
+
+File: gsl-ref.info,  Node: Sorting References and Further Reading,  Prev: Sorting Examples,  Up: Sorting
+
+11.6 References and Further Reading
+===================================
+
+The subject of sorting is covered extensively in Knuth's `Sorting and
+Searching',
+
+     Donald E. Knuth, `The Art of Computer Programming: Sorting and
+     Searching' (Vol 3, 3rd Ed, 1997), Addison-Wesley, ISBN 0201896850.
+
+The Heapsort algorithm is described in the following book,
+
+     Robert Sedgewick, `Algorithms in C', Addison-Wesley, ISBN
+     0201514257.
+
+
+File: gsl-ref.info,  Node: BLAS Support,  Next: Linear Algebra,  Prev: Sorting,  Up: Top
+
+12 BLAS Support
+***************
+
+The Basic Linear Algebra Subprograms (BLAS) define a set of fundamental
+operations on vectors and matrices which can be used to create optimized
+higher-level linear algebra functionality.
+
+   The library provides a low-level layer which corresponds directly to
+the C-language BLAS standard, referred to here as "CBLAS", and a
+higher-level interface for operations on GSL vectors and matrices.
+Users who are interested in simple operations on GSL vector and matrix
+objects should use the high-level layer, which is declared in the file
+`gsl_blas.h'.  This should satisfy the needs of most users.  Note that
+GSL matrices are implemented using dense-storage so the interface only
+includes the corresponding dense-storage BLAS functions.  The full BLAS
+functionality for band-format and packed-format matrices is available
+through the low-level CBLAS interface.
+
+   The interface for the `gsl_cblas' layer is specified in the file
+`gsl_cblas.h'.  This interface corresponds to the BLAS Technical
+Forum's draft standard for the C interface to legacy BLAS
+implementations. Users who have access to other conforming CBLAS
+implementations can use these in place of the version provided by the
+library.  Note that users who have only a Fortran BLAS library can use
+a CBLAS conformant wrapper to convert it into a CBLAS library.  A
+reference CBLAS wrapper for legacy Fortran implementations exists as
+part of the draft CBLAS standard and can be obtained from Netlib.  The
+complete set of CBLAS functions is listed in an appendix (*note GSL
+CBLAS Library::).
+
+   There are three levels of BLAS operations,
+
+Level 1
+     Vector operations, e.g. y = \alpha x + y
+
+Level 2
+     Matrix-vector operations, e.g. y = \alpha A x + \beta y
+
+Level 3
+     Matrix-matrix operations, e.g. C = \alpha A B + C
+
+Each routine has a name which specifies the operation, the type of
+matrices involved and their precisions.  Some of the most common
+operations and their names are given below,
+
+DOT
+     scalar product, x^T y
+
+AXPY
+     vector sum, \alpha x + y
+
+MV
+     matrix-vector product, A x
+
+SV
+     matrix-vector solve, inv(A) x
+
+MM
+     matrix-matrix product, A B
+
+SM
+     matrix-matrix solve, inv(A) B
+
+The types of matrices are,
+
+GE
+     general
+
+GB
+     general band
+
+SY
+     symmetric
+
+SB
+     symmetric band
+
+SP
+     symmetric packed
+
+HE
+     hermitian
+
+HB
+     hermitian band
+
+HP
+     hermitian packed
+
+TR
+     triangular
+
+TB
+     triangular band
+
+TP
+     triangular packed
+
+Each operation is defined for four precisions,
+
+S
+     single real
+
+D
+     double real
+
+C
+     single complex
+
+Z
+     double complex
+
+Thus, for example, the name SGEMM stands for "single-precision general
+matrix-matrix multiply" and ZGEMM stands for "double-precision complex
+matrix-matrix multiply".
+
+* Menu:
+
+* GSL BLAS Interface::
+* BLAS Examples::
+* BLAS References and Further Reading::
+
+
+File: gsl-ref.info,  Node: GSL BLAS Interface,  Next: BLAS Examples,  Up: BLAS Support
+
+12.1 GSL BLAS Interface
+=======================
+
+GSL provides dense vector and matrix objects, based on the relevant
+built-in types.  The library provides an interface to the BLAS
+operations which apply to these objects.  The interface to this
+functionality is given in the file `gsl_blas.h'.
+
+* Menu:
+
+* Level 1 GSL BLAS Interface::
+* Level 2 GSL BLAS Interface::
+* Level 3 GSL BLAS Interface::
+
+
+File: gsl-ref.info,  Node: Level 1 GSL BLAS Interface,  Next: Level 2 GSL BLAS Interface,  Up: GSL BLAS Interface
+
+12.1.1 Level 1
+--------------
+
+ -- Function: int gsl_blas_sdsdot (float ALPHA, const gsl_vector_float
+          * X, const gsl_vector_float * Y, float * RESULT)
+     This function computes the sum \alpha + x^T y for the vectors X
+     and Y, returning the result in RESULT.
+
+ -- Function: int gsl_blas_sdot (const gsl_vector_float * X, const
+          gsl_vector_float * Y, float * RESULT)
+ -- Function: int gsl_blas_dsdot (const gsl_vector_float * X, const
+          gsl_vector_float * Y, double * RESULT)
+ -- Function: int gsl_blas_ddot (const gsl_vector * X, const gsl_vector
+          * Y, double * RESULT)
+     These functions compute the scalar product x^T y for the vectors X
+     and Y, returning the result in RESULT.
+
+ -- Function: int gsl_blas_cdotu (const gsl_vector_complex_float * X,
+          const gsl_vector_complex_float * Y, gsl_complex_float * DOTU)
+ -- Function: int gsl_blas_zdotu (const gsl_vector_complex * X, const
+          gsl_vector_complex * Y, gsl_complex * DOTU)
+     These functions compute the complex scalar product x^T y for the
+     vectors X and Y, returning the result in RESULT
+
+ -- Function: int gsl_blas_cdotc (const gsl_vector_complex_float * X,
+          const gsl_vector_complex_float * Y, gsl_complex_float * DOTC)
+ -- Function: int gsl_blas_zdotc (const gsl_vector_complex * X, const
+          gsl_vector_complex * Y, gsl_complex * DOTC)
+     These functions compute the complex conjugate scalar product x^H y
+     for the vectors X and Y, returning the result in RESULT
+
+ -- Function: float gsl_blas_snrm2 (const gsl_vector_float * X)
+ -- Function: double gsl_blas_dnrm2 (const gsl_vector * X)
+     These functions compute the Euclidean norm ||x||_2 = \sqrt {\sum
+     x_i^2} of the vector X.
+
+ -- Function: float gsl_blas_scnrm2 (const gsl_vector_complex_float * X)
+ -- Function: double gsl_blas_dznrm2 (const gsl_vector_complex * X)
+     These functions compute the Euclidean norm of the complex vector X,
+
+          ||x||_2 = \sqrt {\sum (\Re(x_i)^2 + \Im(x_i)^2)}.
+
+ -- Function: float gsl_blas_sasum (const gsl_vector_float * X)
+ -- Function: double gsl_blas_dasum (const gsl_vector * X)
+     These functions compute the absolute sum \sum |x_i| of the
+     elements of the vector X.
+
+ -- Function: float gsl_blas_scasum (const gsl_vector_complex_float * X)
+ -- Function: double gsl_blas_dzasum (const gsl_vector_complex * X)
+     These functions compute the sum of the magnitudes of the real and
+     imaginary parts of the complex vector X, \sum |\Re(x_i)| +
+     |\Im(x_i)|.
+
+ -- Function: CBLAS_INDEX_t gsl_blas_isamax (const gsl_vector_float * X)
+ -- Function: CBLAS_INDEX_t gsl_blas_idamax (const gsl_vector * X)
+ -- Function: CBLAS_INDEX_t gsl_blas_icamax (const
+          gsl_vector_complex_float * X)
+ -- Function: CBLAS_INDEX_t gsl_blas_izamax (const gsl_vector_complex *
+          X)
+     These functions return the index of the largest element of the
+     vector X. The largest element is determined by its absolute
+     magnitude for real vectors and by the sum of the magnitudes of the
+     real and imaginary parts |\Re(x_i)| + |\Im(x_i)| for complex
+     vectors.  If the largest value occurs several times then the index
+     of the first occurrence is returned.
+
+ -- Function: int gsl_blas_sswap (gsl_vector_float * X,
+          gsl_vector_float * Y)
+ -- Function: int gsl_blas_dswap (gsl_vector * X, gsl_vector * Y)
+ -- Function: int gsl_blas_cswap (gsl_vector_complex_float * X,
+          gsl_vector_complex_float * Y)
+ -- Function: int gsl_blas_zswap (gsl_vector_complex * X,
+          gsl_vector_complex * Y)
+     These functions exchange the elements of the vectors X and Y.
+
+ -- Function: int gsl_blas_scopy (const gsl_vector_float * X,
+          gsl_vector_float * Y)
+ -- Function: int gsl_blas_dcopy (const gsl_vector * X, gsl_vector * Y)
+ -- Function: int gsl_blas_ccopy (const gsl_vector_complex_float * X,
+          gsl_vector_complex_float * Y)
+ -- Function: int gsl_blas_zcopy (const gsl_vector_complex * X,
+          gsl_vector_complex * Y)
+     These functions copy the elements of the vector X into the vector
+     Y.
+
+ -- Function: int gsl_blas_saxpy (float ALPHA, const gsl_vector_float *
+          X, gsl_vector_float * Y)
+ -- Function: int gsl_blas_daxpy (double ALPHA, const gsl_vector * X,
+          gsl_vector * Y)
+ -- Function: int gsl_blas_caxpy (const gsl_complex_float ALPHA, const
+          gsl_vector_complex_float * X, gsl_vector_complex_float * Y)
+ -- Function: int gsl_blas_zaxpy (const gsl_complex ALPHA, const
+          gsl_vector_complex * X, gsl_vector_complex * Y)
+     These functions compute the sum y = \alpha x + y for the vectors X
+     and Y.
+
+ -- Function: void gsl_blas_sscal (float ALPHA, gsl_vector_float * X)
+ -- Function: void gsl_blas_dscal (double ALPHA, gsl_vector * X)
+ -- Function: void gsl_blas_cscal (const gsl_complex_float ALPHA,
+          gsl_vector_complex_float * X)
+ -- Function: void gsl_blas_zscal (const gsl_complex ALPHA,
+          gsl_vector_complex * X)
+ -- Function: void gsl_blas_csscal (float ALPHA,
+          gsl_vector_complex_float * X)
+ -- Function: void gsl_blas_zdscal (double ALPHA, gsl_vector_complex *
+          X)
+     These functions rescale the vector X by the multiplicative factor
+     ALPHA.
+
+ -- Function: int gsl_blas_srotg (float A[], float B[], float C[],
+          float S[])
+ -- Function: int gsl_blas_drotg (double A[], double B[], double C[],
+          double S[])
+     These functions compute a Givens rotation (c,s) which zeroes the
+     vector (a,b),
+
+          [  c  s ] [ a ] = [ r ]
+          [ -s  c ] [ b ]   [ 0 ]
+
+     The variables A and B are overwritten by the routine.
+
+ -- Function: int gsl_blas_srot (gsl_vector_float * X, gsl_vector_float
+          * Y, float C, float S)
+ -- Function: int gsl_blas_drot (gsl_vector * X, gsl_vector * Y, const
+          double C, const double S)
+     These functions apply a Givens rotation (x', y') = (c x + s y, -s
+     x + c y) to the vectors X, Y.
+
+ -- Function: int gsl_blas_srotmg (float D1[], float D2[], float B1[],
+          float B2, float P[])
+ -- Function: int gsl_blas_drotmg (double D1[], double D2[], double
+          B1[], double B2, double P[])
+     These functions compute a modified Givens transformation.  The
+     modified Givens transformation is defined in the original Level-1
+     BLAS specification, given in the references.
+
+ -- Function: int gsl_blas_srotm (gsl_vector_float * X,
+          gsl_vector_float * Y, const float P[])
+ -- Function: int gsl_blas_drotm (gsl_vector * X, gsl_vector * Y, const
+          double P[])
+     These functions apply a modified Givens transformation.
+
+
+File: gsl-ref.info,  Node: Level 2 GSL BLAS Interface,  Next: Level 3 GSL BLAS Interface,  Prev: Level 1 GSL BLAS Interface,  Up: GSL BLAS Interface
+
+12.1.2 Level 2
+--------------
+
+ -- Function: int gsl_blas_sgemv (CBLAS_TRANSPOSE_t TRANSA, float
+          ALPHA, const gsl_matrix_float * A, const gsl_vector_float *
+          X, float BETA, gsl_vector_float * Y)
+ -- Function: int gsl_blas_dgemv (CBLAS_TRANSPOSE_t TRANSA, double
+          ALPHA, const gsl_matrix * A, const gsl_vector * X, double
+          BETA, gsl_vector * Y)
+ -- Function: int gsl_blas_cgemv (CBLAS_TRANSPOSE_t TRANSA, const
+          gsl_complex_float ALPHA, const gsl_matrix_complex_float * A,
+          const gsl_vector_complex_float * X, const gsl_complex_float
+          BETA, gsl_vector_complex_float * Y)
+ -- Function: int gsl_blas_zgemv (CBLAS_TRANSPOSE_t TRANSA, const
+          gsl_complex ALPHA, const gsl_matrix_complex * A, const
+          gsl_vector_complex * X, const gsl_complex BETA,
+          gsl_vector_complex * Y)
+     These functions compute the matrix-vector product and sum y =
+     \alpha op(A) x + \beta y, where op(A) = A, A^T, A^H for TRANSA =
+     `CblasNoTrans', `CblasTrans', `CblasConjTrans'.
+
+ -- Function: int gsl_blas_strmv (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
+          TRANSA, CBLAS_DIAG_t DIAG, const gsl_matrix_float * A,
+          gsl_vector_float * X)
+ -- Function: int gsl_blas_dtrmv (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
+          TRANSA, CBLAS_DIAG_t DIAG, const gsl_matrix * A, gsl_vector *
+          X)
+ -- Function: int gsl_blas_ctrmv (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
+          TRANSA, CBLAS_DIAG_t DIAG, const gsl_matrix_complex_float *
+          A, gsl_vector_complex_float * X)
+ -- Function: int gsl_blas_ztrmv (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
+          TRANSA, CBLAS_DIAG_t DIAG, const gsl_matrix_complex * A,
+          gsl_vector_complex * X)
+     These functions compute the matrix-vector product x = op(A) x for
+     the triangular matrix A, where op(A) = A, A^T, A^H for TRANSA =
+     `CblasNoTrans', `CblasTrans', `CblasConjTrans'.  When UPLO is
+     `CblasUpper' then the upper triangle of A is used, and when UPLO
+     is `CblasLower' then the lower triangle of A is used.  If DIAG is
+     `CblasNonUnit' then the diagonal of the matrix is used, but if
+     DIAG is `CblasUnit' then the diagonal elements of the matrix A are
+     taken as unity and are not referenced.
+
+ -- Function: int gsl_blas_strsv (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
+          TRANSA, CBLAS_DIAG_t DIAG, const gsl_matrix_float * A,
+          gsl_vector_float * X)
+ -- Function: int gsl_blas_dtrsv (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
+          TRANSA, CBLAS_DIAG_t DIAG, const gsl_matrix * A, gsl_vector *
+          X)
+ -- Function: int gsl_blas_ctrsv (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
+          TRANSA, CBLAS_DIAG_t DIAG, const gsl_matrix_complex_float *
+          A, gsl_vector_complex_float * X)
+ -- Function: int gsl_blas_ztrsv (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
+          TRANSA, CBLAS_DIAG_t DIAG, const gsl_matrix_complex * A,
+          gsl_vector_complex * X)
+     These functions compute inv(op(A)) x for X, where op(A) = A, A^T,
+     A^H for TRANSA = `CblasNoTrans', `CblasTrans', `CblasConjTrans'.
+     When UPLO is `CblasUpper' then the upper triangle of A is used,
+     and when UPLO is `CblasLower' then the lower triangle of A is
+     used.  If DIAG is `CblasNonUnit' then the diagonal of the matrix
+     is used, but if DIAG is `CblasUnit' then the diagonal elements of
+     the matrix A are taken as unity and are not referenced.
+
+ -- Function: int gsl_blas_ssymv (CBLAS_UPLO_t UPLO, float ALPHA, const
+          gsl_matrix_float * A, const gsl_vector_float * X, float BETA,
+          gsl_vector_float * Y)
+ -- Function: int gsl_blas_dsymv (CBLAS_UPLO_t UPLO, double ALPHA,
+          const gsl_matrix * A, const gsl_vector * X, double BETA,
+          gsl_vector * Y)
+     These functions compute the matrix-vector product and sum y =
+     \alpha A x + \beta y for the symmetric matrix A.  Since the matrix
+     A is symmetric only its upper half or lower half need to be
+     stored.  When UPLO is `CblasUpper' then the upper triangle and
+     diagonal of A are used, and when UPLO is `CblasLower' then the
+     lower triangle and diagonal of A are used.
+
+ -- Function: int gsl_blas_chemv (CBLAS_UPLO_t UPLO, const
+          gsl_complex_float ALPHA, const gsl_matrix_complex_float * A,
+          const gsl_vector_complex_float * X, const gsl_complex_float
+          BETA, gsl_vector_complex_float * Y)
+ -- Function: int gsl_blas_zhemv (CBLAS_UPLO_t UPLO, const gsl_complex
+          ALPHA, const gsl_matrix_complex * A, const gsl_vector_complex
+          * X, const gsl_complex BETA, gsl_vector_complex * Y)
+     These functions compute the matrix-vector product and sum y =
+     \alpha A x + \beta y for the hermitian matrix A.  Since the matrix
+     A is hermitian only its upper half or lower half need to be
+     stored.  When UPLO is `CblasUpper' then the upper triangle and
+     diagonal of A are used, and when UPLO is `CblasLower' then the
+     lower triangle and diagonal of A are used.  The imaginary elements
+     of the diagonal are automatically assumed to be zero and are not
+     referenced.
+
+ -- Function: int gsl_blas_sger (float ALPHA, const gsl_vector_float *
+          X, const gsl_vector_float * Y, gsl_matrix_float * A)
+ -- Function: int gsl_blas_dger (double ALPHA, const gsl_vector * X,
+          const gsl_vector * Y, gsl_matrix * A)
+ -- Function: int gsl_blas_cgeru (const gsl_complex_float ALPHA, const
+          gsl_vector_complex_float * X, const gsl_vector_complex_float
+          * Y, gsl_matrix_complex_float * A)
+ -- Function: int gsl_blas_zgeru (const gsl_complex ALPHA, const
+          gsl_vector_complex * X, const gsl_vector_complex * Y,
+          gsl_matrix_complex * A)
+     These functions compute the rank-1 update A = \alpha x y^T + A of
+     the matrix A.
+
+ -- Function: int gsl_blas_cgerc (const gsl_complex_float ALPHA, const
+          gsl_vector_complex_float * X, const gsl_vector_complex_float
+          * Y, gsl_matrix_complex_float * A)
+ -- Function: int gsl_blas_zgerc (const gsl_complex ALPHA, const
+          gsl_vector_complex * X, const gsl_vector_complex * Y,
+          gsl_matrix_complex * A)
+     These functions compute the conjugate rank-1 update A = \alpha x
+     y^H + A of the matrix A.
+
+ -- Function: int gsl_blas_ssyr (CBLAS_UPLO_t UPLO, float ALPHA, const
+          gsl_vector_float * X, gsl_matrix_float * A)
+ -- Function: int gsl_blas_dsyr (CBLAS_UPLO_t UPLO, double ALPHA, const
+          gsl_vector * X, gsl_matrix * A)
+     These functions compute the symmetric rank-1 update A = \alpha x
+     x^T + A of the symmetric matrix A.  Since the matrix A is
+     symmetric only its upper half or lower half need to be stored.
+     When UPLO is `CblasUpper' then the upper triangle and diagonal of
+     A are used, and when UPLO is `CblasLower' then the lower triangle
+     and diagonal of A are used.
+
+ -- Function: int gsl_blas_cher (CBLAS_UPLO_t UPLO, float ALPHA, const
+          gsl_vector_complex_float * X, gsl_matrix_complex_float * A)
+ -- Function: int gsl_blas_zher (CBLAS_UPLO_t UPLO, double ALPHA, const
+          gsl_vector_complex * X, gsl_matrix_complex * A)
+     These functions compute the hermitian rank-1 update A = \alpha x
+     x^H + A of the hermitian matrix A.  Since the matrix A is
+     hermitian only its upper half or lower half need to be stored.
+     When UPLO is `CblasUpper' then the upper triangle and diagonal of
+     A are used, and when UPLO is `CblasLower' then the lower triangle
+     and diagonal of A are used.  The imaginary elements of the
+     diagonal are automatically set to zero.
+
+ -- Function: int gsl_blas_ssyr2 (CBLAS_UPLO_t UPLO, float ALPHA, const
+          gsl_vector_float * X, const gsl_vector_float * Y,
+          gsl_matrix_float * A)
+ -- Function: int gsl_blas_dsyr2 (CBLAS_UPLO_t UPLO, double ALPHA,
+          const gsl_vector * X, const gsl_vector * Y, gsl_matrix * A)
+     These functions compute the symmetric rank-2 update A = \alpha x
+     y^T + \alpha y x^T + A of the symmetric matrix A.  Since the
+     matrix A is symmetric only its upper half or lower half need to be
+     stored.  When UPLO is `CblasUpper' then the upper triangle and
+     diagonal of A are used, and when UPLO is `CblasLower' then the
+     lower triangle and diagonal of A are used.
+
+ -- Function: int gsl_blas_cher2 (CBLAS_UPLO_t UPLO, const
+          gsl_complex_float ALPHA, const gsl_vector_complex_float * X,
+          const gsl_vector_complex_float * Y, gsl_matrix_complex_float
+          * A)
+ -- Function: int gsl_blas_zher2 (CBLAS_UPLO_t UPLO, const gsl_complex
+          ALPHA, const gsl_vector_complex * X, const gsl_vector_complex
+          * Y, gsl_matrix_complex * A)
+     These functions compute the hermitian rank-2 update A = \alpha x
+     y^H + \alpha^* y x^H A of the hermitian matrix A.  Since the
+     matrix A is hermitian only its upper half or lower half need to be
+     stored.  When UPLO is `CblasUpper' then the upper triangle and
+     diagonal of A are used, and when UPLO is `CblasLower' then the
+     lower triangle and diagonal of A are used.  The imaginary elements
+     of the diagonal are automatically set to zero.
+
+
+File: gsl-ref.info,  Node: Level 3 GSL BLAS Interface,  Prev: Level 2 GSL BLAS Interface,  Up: GSL BLAS Interface
+
+12.1.3 Level 3
+--------------
+
+ -- Function: int gsl_blas_sgemm (CBLAS_TRANSPOSE_t TRANSA,
+          CBLAS_TRANSPOSE_t TRANSB, float ALPHA, const gsl_matrix_float
+          * A, const gsl_matrix_float * B, float BETA, gsl_matrix_float
+          * C)
+ -- Function: int gsl_blas_dgemm (CBLAS_TRANSPOSE_t TRANSA,
+          CBLAS_TRANSPOSE_t TRANSB, double ALPHA, const gsl_matrix * A,
+          const gsl_matrix * B, double BETA, gsl_matrix * C)
+ -- Function: int gsl_blas_cgemm (CBLAS_TRANSPOSE_t TRANSA,
+          CBLAS_TRANSPOSE_t TRANSB, const gsl_complex_float ALPHA,
+          const gsl_matrix_complex_float * A, const
+          gsl_matrix_complex_float * B, const gsl_complex_float BETA,
+          gsl_matrix_complex_float * C)
+ -- Function: int gsl_blas_zgemm (CBLAS_TRANSPOSE_t TRANSA,
+          CBLAS_TRANSPOSE_t TRANSB, const gsl_complex ALPHA, const
+          gsl_matrix_complex * A, const gsl_matrix_complex * B, const
+          gsl_complex BETA, gsl_matrix_complex * C)
+     These functions compute the matrix-matrix product and sum C =
+     \alpha op(A) op(B) + \beta C where op(A) = A, A^T, A^H for TRANSA
+     = `CblasNoTrans', `CblasTrans', `CblasConjTrans' and similarly for
+     the parameter TRANSB.
+
+ -- Function: int gsl_blas_ssymm (CBLAS_SIDE_t SIDE, CBLAS_UPLO_t UPLO,
+          float ALPHA, const gsl_matrix_float * A, const
+          gsl_matrix_float * B, float BETA, gsl_matrix_float * C)
+ -- Function: int gsl_blas_dsymm (CBLAS_SIDE_t SIDE, CBLAS_UPLO_t UPLO,
+          double ALPHA, const gsl_matrix * A, const gsl_matrix * B,
+          double BETA, gsl_matrix * C)
+ -- Function: int gsl_blas_csymm (CBLAS_SIDE_t SIDE, CBLAS_UPLO_t UPLO,
+          const gsl_complex_float ALPHA, const gsl_matrix_complex_float
+          * A, const gsl_matrix_complex_float * B, const
+          gsl_complex_float BETA, gsl_matrix_complex_float * C)
+ -- Function: int gsl_blas_zsymm (CBLAS_SIDE_t SIDE, CBLAS_UPLO_t UPLO,
+          const gsl_complex ALPHA, const gsl_matrix_complex * A, const
+          gsl_matrix_complex * B, const gsl_complex BETA,
+          gsl_matrix_complex * C)
+     These functions compute the matrix-matrix product and sum C =
+     \alpha A B + \beta C for SIDE is `CblasLeft' and C = \alpha B A +
+     \beta C for SIDE is `CblasRight', where the matrix A is symmetric.
+     When UPLO is `CblasUpper' then the upper triangle and diagonal of
+     A are used, and when UPLO is `CblasLower' then the lower triangle
+     and diagonal of A are used.
+
+ -- Function: int gsl_blas_chemm (CBLAS_SIDE_t SIDE, CBLAS_UPLO_t UPLO,
+          const gsl_complex_float ALPHA, const gsl_matrix_complex_float
+          * A, const gsl_matrix_complex_float * B, const
+          gsl_complex_float BETA, gsl_matrix_complex_float * C)
+ -- Function: int gsl_blas_zhemm (CBLAS_SIDE_t SIDE, CBLAS_UPLO_t UPLO,
+          const gsl_complex ALPHA, const gsl_matrix_complex * A, const
+          gsl_matrix_complex * B, const gsl_complex BETA,
+          gsl_matrix_complex * C)
+     These functions compute the matrix-matrix product and sum C =
+     \alpha A B + \beta C for SIDE is `CblasLeft' and C = \alpha B A +
+     \beta C for SIDE is `CblasRight', where the matrix A is hermitian.
+     When UPLO is `CblasUpper' then the upper triangle and diagonal of
+     A are used, and when UPLO is `CblasLower' then the lower triangle
+     and diagonal of A are used.  The imaginary elements of the
+     diagonal are automatically set to zero.
+
+ -- Function: int gsl_blas_strmm (CBLAS_SIDE_t SIDE, CBLAS_UPLO_t UPLO,
+          CBLAS_TRANSPOSE_t TRANSA, CBLAS_DIAG_t DIAG, float ALPHA,
+          const gsl_matrix_float * A, gsl_matrix_float * B)
+ -- Function: int gsl_blas_dtrmm (CBLAS_SIDE_t SIDE, CBLAS_UPLO_t UPLO,
+          CBLAS_TRANSPOSE_t TRANSA, CBLAS_DIAG_t DIAG, double ALPHA,
+          const gsl_matrix * A, gsl_matrix * B)
+ -- Function: int gsl_blas_ctrmm (CBLAS_SIDE_t SIDE, CBLAS_UPLO_t UPLO,
+          CBLAS_TRANSPOSE_t TRANSA, CBLAS_DIAG_t DIAG, const
+          gsl_complex_float ALPHA, const gsl_matrix_complex_float * A,
+          gsl_matrix_complex_float * B)
+ -- Function: int gsl_blas_ztrmm (CBLAS_SIDE_t SIDE, CBLAS_UPLO_t UPLO,
+          CBLAS_TRANSPOSE_t TRANSA, CBLAS_DIAG_t DIAG, const
+          gsl_complex ALPHA, const gsl_matrix_complex * A,
+          gsl_matrix_complex * B)
+     These functions compute the matrix-matrix product B = \alpha op(A)
+     B for SIDE is `CblasLeft' and B = \alpha B op(A) for SIDE is
+     `CblasRight'.  The matrix A is triangular and op(A) = A, A^T, A^H
+     for TRANSA = `CblasNoTrans', `CblasTrans', `CblasConjTrans'. When
+     UPLO is `CblasUpper' then the upper triangle of A is used, and
+     when UPLO is `CblasLower' then the lower triangle of A is used.
+     If DIAG is `CblasNonUnit' then the diagonal of A is used, but if
+     DIAG is `CblasUnit' then the diagonal elements of the matrix A are
+     taken as unity and are not referenced.
+
+ -- Function: int gsl_blas_strsm (CBLAS_SIDE_t SIDE, CBLAS_UPLO_t UPLO,
+          CBLAS_TRANSPOSE_t TRANSA, CBLAS_DIAG_t DIAG, float ALPHA,
+          const gsl_matrix_float * A, gsl_matrix_float * B)
+ -- Function: int gsl_blas_dtrsm (CBLAS_SIDE_t SIDE, CBLAS_UPLO_t UPLO,
+          CBLAS_TRANSPOSE_t TRANSA, CBLAS_DIAG_t DIAG, double ALPHA,
+          const gsl_matrix * A, gsl_matrix * B)
+ -- Function: int gsl_blas_ctrsm (CBLAS_SIDE_t SIDE, CBLAS_UPLO_t UPLO,
+          CBLAS_TRANSPOSE_t TRANSA, CBLAS_DIAG_t DIAG, const
+          gsl_complex_float ALPHA, const gsl_matrix_complex_float * A,
+          gsl_matrix_complex_float * B)
+ -- Function: int gsl_blas_ztrsm (CBLAS_SIDE_t SIDE, CBLAS_UPLO_t UPLO,
+          CBLAS_TRANSPOSE_t TRANSA, CBLAS_DIAG_t DIAG, const
+          gsl_complex ALPHA, const gsl_matrix_complex * A,
+          gsl_matrix_complex * B)
+     These functions compute the inverse-matrix matrix product B =
+     \alpha op(inv(A))B for SIDE is `CblasLeft' and B = \alpha B
+     op(inv(A)) for SIDE is `CblasRight'.  The matrix A is triangular
+     and op(A) = A, A^T, A^H for TRANSA = `CblasNoTrans', `CblasTrans',
+     `CblasConjTrans'. When UPLO is `CblasUpper' then the upper
+     triangle of A is used, and when UPLO is `CblasLower' then the
+     lower triangle of A is used.  If DIAG is `CblasNonUnit' then the
+     diagonal of A is used, but if DIAG is `CblasUnit' then the
+     diagonal elements of the matrix A are taken as unity and are not
+     referenced.
+
+ -- Function: int gsl_blas_ssyrk (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
+          TRANS, float ALPHA, const gsl_matrix_float * A, float BETA,
+          gsl_matrix_float * C)
+ -- Function: int gsl_blas_dsyrk (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
+          TRANS, double ALPHA, const gsl_matrix * A, double BETA,
+          gsl_matrix * C)
+ -- Function: int gsl_blas_csyrk (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
+          TRANS, const gsl_complex_float ALPHA, const
+          gsl_matrix_complex_float * A, const gsl_complex_float BETA,
+          gsl_matrix_complex_float * C)
+ -- Function: int gsl_blas_zsyrk (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
+          TRANS, const gsl_complex ALPHA, const gsl_matrix_complex * A,
+          const gsl_complex BETA, gsl_matrix_complex * C)
+     These functions compute a rank-k update of the symmetric matrix C,
+     C = \alpha A A^T + \beta C when TRANS is `CblasNoTrans' and C =
+     \alpha A^T A + \beta C when TRANS is `CblasTrans'.  Since the
+     matrix C is symmetric only its upper half or lower half need to be
+     stored.  When UPLO is `CblasUpper' then the upper triangle and
+     diagonal of C are used, and when UPLO is `CblasLower' then the
+     lower triangle and diagonal of C are used.
+
+ -- Function: int gsl_blas_cherk (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
+          TRANS, float ALPHA, const gsl_matrix_complex_float * A, float
+          BETA, gsl_matrix_complex_float * C)
+ -- Function: int gsl_blas_zherk (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
+          TRANS, double ALPHA, const gsl_matrix_complex * A, double
+          BETA, gsl_matrix_complex * C)
+     These functions compute a rank-k update of the hermitian matrix C,
+     C = \alpha A A^H + \beta C when TRANS is `CblasNoTrans' and C =
+     \alpha A^H A + \beta C when TRANS is `CblasTrans'.  Since the
+     matrix C is hermitian only its upper half or lower half need to be
+     stored.  When UPLO is `CblasUpper' then the upper triangle and
+     diagonal of C are used, and when UPLO is `CblasLower' then the
+     lower triangle and diagonal of C are used.  The imaginary elements
+     of the diagonal are automatically set to zero.
+
+ -- Function: int gsl_blas_ssyr2k (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
+          TRANS, float ALPHA, const gsl_matrix_float * A, const
+          gsl_matrix_float * B, float BETA, gsl_matrix_float * C)
+ -- Function: int gsl_blas_dsyr2k (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
+          TRANS, double ALPHA, const gsl_matrix * A, const gsl_matrix *
+          B, double BETA, gsl_matrix * C)
+ -- Function: int gsl_blas_csyr2k (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
+          TRANS, const gsl_complex_float ALPHA, const
+          gsl_matrix_complex_float * A, const gsl_matrix_complex_float
+          * B, const gsl_complex_float BETA, gsl_matrix_complex_float *
+          C)
+ -- Function: int gsl_blas_zsyr2k (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
+          TRANS, const gsl_complex ALPHA, const gsl_matrix_complex * A,
+          const gsl_matrix_complex * B, const gsl_complex BETA,
+          gsl_matrix_complex * C)
+     These functions compute a rank-2k update of the symmetric matrix C,
+     C = \alpha A B^T + \alpha B A^T + \beta C when TRANS is
+     `CblasNoTrans' and C = \alpha A^T B + \alpha B^T A + \beta C when
+     TRANS is `CblasTrans'.  Since the matrix C is symmetric only its
+     upper half or lower half need to be stored.  When UPLO is
+     `CblasUpper' then the upper triangle and diagonal of C are used,
+     and when UPLO is `CblasLower' then the lower triangle and diagonal
+     of C are used.
+
+ -- Function: int gsl_blas_cher2k (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
+          TRANS, const gsl_complex_float ALPHA, const
+          gsl_matrix_complex_float * A, const gsl_matrix_complex_float
+          * B, float BETA, gsl_matrix_complex_float * C)
+ -- Function: int gsl_blas_zher2k (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
+          TRANS, const gsl_complex ALPHA, const gsl_matrix_complex * A,
+          const gsl_matrix_complex * B, double BETA, gsl_matrix_complex
+          * C)
+     These functions compute a rank-2k update of the hermitian matrix C,
+     C = \alpha A B^H + \alpha^* B A^H + \beta C when TRANS is
+     `CblasNoTrans' and C = \alpha A^H B + \alpha^* B^H A + \beta C when
+     TRANS is `CblasConjTrans'.  Since the matrix C is hermitian only
+     its upper half or lower half need to be stored.  When UPLO is
+     `CblasUpper' then the upper triangle and diagonal of C are used,
+     and when UPLO is `CblasLower' then the lower triangle and diagonal
+     of C are used.  The imaginary elements of the diagonal are
+     automatically set to zero.
+
+
+File: gsl-ref.info,  Node: BLAS Examples,  Next: BLAS References and Further Reading,  Prev: GSL BLAS Interface,  Up: BLAS Support
+
+12.2 Examples
+=============
+
+The following program computes the product of two matrices using the
+Level-3 BLAS function DGEMM,
+
+     [ 0.11 0.12 0.13 ]  [ 1011 1012 ]     [ 367.76 368.12 ]
+     [ 0.21 0.22 0.23 ]  [ 1021 1022 ]  =  [ 674.06 674.72 ]
+                         [ 1031 1032 ]
+
+The matrices are stored in row major order, according to the C
+convention for arrays.
+
+     #include <stdio.h>
+     #include <gsl/gsl_blas.h>
+
+     int
+     main (void)
+     {
+       double a[] = { 0.11, 0.12, 0.13,
+                      0.21, 0.22, 0.23 };
+
+       double b[] = { 1011, 1012,
+                      1021, 1022,
+                      1031, 1032 };
+
+       double c[] = { 0.00, 0.00,
+                      0.00, 0.00 };
+
+       gsl_matrix_view A = gsl_matrix_view_array(a, 2, 3);
+       gsl_matrix_view B = gsl_matrix_view_array(b, 3, 2);
+       gsl_matrix_view C = gsl_matrix_view_array(c, 2, 2);
+
+       /* Compute C = A B */
+
+       gsl_blas_dgemm (CblasNoTrans, CblasNoTrans,
+                       1.0, &A.matrix, &B.matrix,
+                       0.0, &C.matrix);
+
+       printf ("[ %g, %g\n", c[0], c[1]);
+       printf ("  %g, %g ]\n", c[2], c[3]);
+
+       return 0;
+     }
+
+Here is the output from the program,
+
+     $ ./a.out
+     [ 367.76, 368.12
+       674.06, 674.72 ]
+
+
+File: gsl-ref.info,  Node: BLAS References and Further Reading,  Prev: BLAS Examples,  Up: BLAS Support
+
+12.3 References and Further Reading
+===================================
+
+Information on the BLAS standards, including both the legacy and draft
+interface standards, is available online from the BLAS Homepage and
+BLAS Technical Forum web-site.
+
+     `BLAS Homepage'
+     `http://www.netlib.org/blas/'
+
+     `BLAS Technical Forum'
+     `http://www.netlib.org/cgi-bin/checkout/blast/blast.pl'
+
+The following papers contain the specifications for Level 1, Level 2 and
+Level 3 BLAS.
+
+     C. Lawson, R. Hanson, D. Kincaid, F. Krogh, "Basic Linear Algebra
+     Subprograms for Fortran Usage", `ACM Transactions on Mathematical
+     Software', Vol. 5 (1979), Pages 308-325.
+
+     J.J. Dongarra, J. DuCroz, S. Hammarling, R. Hanson, "An Extended
+     Set of Fortran Basic Linear Algebra Subprograms", `ACM
+     Transactions on Mathematical Software', Vol. 14, No. 1 (1988),
+     Pages 1-32.
+
+     J.J. Dongarra, I. Duff, J. DuCroz, S. Hammarling, "A Set of Level
+     3 Basic Linear Algebra Subprograms", `ACM Transactions on
+     Mathematical Software', Vol. 16 (1990), Pages 1-28.
+
+Postscript versions of the latter two papers are available from
+`http://www.netlib.org/blas/'. A CBLAS wrapper for Fortran BLAS
+libraries is available from the same location.
+
+
+File: gsl-ref.info,  Node: Linear Algebra,  Next: Eigensystems,  Prev: BLAS Support,  Up: Top
+
+13 Linear Algebra
+*****************
+
+This chapter describes functions for solving linear systems.  The
+library provides linear algebra operations which operate directly on
+the `gsl_vector' and `gsl_matrix' objects.  These routines use the
+standard algorithms from Golub & Van Loan's `Matrix Computations'.
+
+   When dealing with very large systems the routines found in LAPACK
+should be considered.  These support specialized data representations
+and other optimizations.
+
+   The functions described in this chapter are declared in the header
+file `gsl_linalg.h'.
+
+* Menu:
+
+* LU Decomposition::
+* QR Decomposition::
+* QR Decomposition with Column Pivoting::
+* Singular Value Decomposition::
+* Cholesky Decomposition::
+* Tridiagonal Decomposition of Real Symmetric Matrices::
+* Tridiagonal Decomposition of Hermitian Matrices::
+* Hessenberg Decomposition of Real Matrices::
+* Bidiagonalization::
+* Householder Transformations::
+* Householder solver for linear systems::
+* Tridiagonal Systems::
+* Balancing::
+* Linear Algebra Examples::
+* Linear Algebra References and Further Reading::
+
+
+File: gsl-ref.info,  Node: LU Decomposition,  Next: QR Decomposition,  Up: Linear Algebra
+
+13.1 LU Decomposition
+=====================
+
+A general square matrix A has an LU decomposition into upper and lower
+triangular matrices,
+
+     P A = L U
+
+where P is a permutation matrix, L is unit lower triangular matrix and
+U is upper triangular matrix. For square matrices this decomposition
+can be used to convert the linear system A x = b into a pair of
+triangular systems (L y = P b, U x = y), which can be solved by forward
+and back-substitution.  Note that the LU decomposition is valid for
+singular matrices.
+
+ -- Function: int gsl_linalg_LU_decomp (gsl_matrix * A, gsl_permutation
+          * P, int * SIGNUM)
+ -- Function: int gsl_linalg_complex_LU_decomp (gsl_matrix_complex * A,
+          gsl_permutation * P, int * SIGNUM)
+     These functions factorize the square matrix A into the LU
+     decomposition PA = LU.  On output the diagonal and upper
+     triangular part of the input matrix A contain the matrix U. The
+     lower triangular part of the input matrix (excluding the diagonal)
+     contains L.  The diagonal elements of L are unity, and are not
+     stored.
+
+     The permutation matrix P is encoded in the permutation P. The j-th
+     column of the matrix P is given by the k-th column of the identity
+     matrix, where k = p_j the j-th element of the permutation vector.
+     The sign of the permutation is given by SIGNUM. It has the value
+     (-1)^n, where n is the number of interchanges in the permutation.
+
+     The algorithm used in the decomposition is Gaussian Elimination
+     with partial pivoting (Golub & Van Loan, `Matrix Computations',
+     Algorithm 3.4.1).
+
+ -- Function: int gsl_linalg_LU_solve (const gsl_matrix * LU, const
+          gsl_permutation * P, const gsl_vector * B, gsl_vector * X)
+ -- Function: int gsl_linalg_complex_LU_solve (const gsl_matrix_complex
+          * LU, const gsl_permutation * P, const gsl_vector_complex *
+          B, gsl_vector_complex * X)
+     These functions solve the square system A x = b using the LU
+     decomposition of A into (LU, P) given by `gsl_linalg_LU_decomp' or
+     `gsl_linalg_complex_LU_decomp'.
+
+ -- Function: int gsl_linalg_LU_svx (const gsl_matrix * LU, const
+          gsl_permutation * P, gsl_vector * X)
+ -- Function: int gsl_linalg_complex_LU_svx (const gsl_matrix_complex *
+          LU, const gsl_permutation * P, gsl_vector_complex * X)
+     These functions solve the square system A x = b in-place using the
+     LU decomposition of A into (LU,P). On input X should contain the
+     right-hand side b, which is replaced by the solution on output.
+
+ -- Function: int gsl_linalg_LU_refine (const gsl_matrix * A, const
+          gsl_matrix * LU, const gsl_permutation * P, const gsl_vector
+          * B, gsl_vector * X, gsl_vector * RESIDUAL)
+ -- Function: int gsl_linalg_complex_LU_refine (const
+          gsl_matrix_complex * A, const gsl_matrix_complex * LU, const
+          gsl_permutation * P, const gsl_vector_complex * B,
+          gsl_vector_complex * X, gsl_vector_complex * RESIDUAL)
+     These functions apply an iterative improvement to X, the solution
+     of A x = b, using the LU decomposition of A into (LU,P). The
+     initial residual r = A x - b is also computed and stored in
+     RESIDUAL.
+
+ -- Function: int gsl_linalg_LU_invert (const gsl_matrix * LU, const
+          gsl_permutation * P, gsl_matrix * INVERSE)
+ -- Function: int gsl_linalg_complex_LU_invert (const
+          gsl_matrix_complex * LU, const gsl_permutation * P,
+          gsl_matrix_complex * INVERSE)
+     These functions compute the inverse of a matrix A from its LU
+     decomposition (LU,P), storing the result in the matrix INVERSE.
+     The inverse is computed by solving the system A x = b for each
+     column of the identity matrix.  It is preferable to avoid direct
+     use of the inverse whenever possible, as the linear solver
+     functions can obtain the same result more efficiently and reliably
+     (consult any introductory textbook on numerical linear algebra for
+     details).
+
+ -- Function: double gsl_linalg_LU_det (gsl_matrix * LU, int SIGNUM)
+ -- Function: gsl_complex gsl_linalg_complex_LU_det (gsl_matrix_complex
+          * LU, int SIGNUM)
+     These functions compute the determinant of a matrix A from its LU
+     decomposition, LU. The determinant is computed as the product of
+     the diagonal elements of U and the sign of the row permutation
+     SIGNUM.
+
+ -- Function: double gsl_linalg_LU_lndet (gsl_matrix * LU)
+ -- Function: double gsl_linalg_complex_LU_lndet (gsl_matrix_complex *
+          LU)
+     These functions compute the logarithm of the absolute value of the
+     determinant of a matrix A, \ln|\det(A)|, from its LU
+     decomposition, LU.  This function may be useful if the direct
+     computation of the determinant would overflow or underflow.
+
+ -- Function: int gsl_linalg_LU_sgndet (gsl_matrix * LU, int SIGNUM)
+ -- Function: gsl_complex gsl_linalg_complex_LU_sgndet
+          (gsl_matrix_complex * LU, int SIGNUM)
+     These functions compute the sign or phase factor of the
+     determinant of a matrix A, \det(A)/|\det(A)|, from its LU
+     decomposition, LU.
+
+
+File: gsl-ref.info,  Node: QR Decomposition,  Next: QR Decomposition with Column Pivoting,  Prev: LU Decomposition,  Up: Linear Algebra
+
+13.2 QR Decomposition
+=====================
+
+A general rectangular M-by-N matrix A has a QR decomposition into the
+product of an orthogonal M-by-M square matrix Q (where Q^T Q = I) and
+an M-by-N right-triangular matrix R,
+
+     A = Q R
+
+This decomposition can be used to convert the linear system A x = b
+into the triangular system R x = Q^T b, which can be solved by
+back-substitution. Another use of the QR decomposition is to compute an
+orthonormal basis for a set of vectors. The first N columns of Q form
+an orthonormal basis for the range of A, ran(A), when A has full column
+rank.
+
+ -- Function: int gsl_linalg_QR_decomp (gsl_matrix * A, gsl_vector *
+          TAU)
+     This function factorizes the M-by-N matrix A into the QR
+     decomposition A = Q R.  On output the diagonal and upper
+     triangular part of the input matrix contain the matrix R. The
+     vector TAU and the columns of the lower triangular part of the
+     matrix A contain the Householder coefficients and Householder
+     vectors which encode the orthogonal matrix Q.  The vector TAU must
+     be of length k=\min(M,N). The matrix Q is related to these
+     components by, Q = Q_k ... Q_2 Q_1 where Q_i = I - \tau_i v_i
+     v_i^T and v_i is the Householder vector v_i =
+     (0,...,1,A(i+1,i),A(i+2,i),...,A(m,i)). This is the same storage
+     scheme as used by LAPACK.
+
+     The algorithm used to perform the decomposition is Householder QR
+     (Golub & Van Loan, `Matrix Computations', Algorithm 5.2.1).
+
+ -- Function: int gsl_linalg_QR_solve (const gsl_matrix * QR, const
+          gsl_vector * TAU, const gsl_vector * B, gsl_vector * X)
+     This function solves the square system A x = b using the QR
+     decomposition of A into (QR, TAU) given by `gsl_linalg_QR_decomp'.
+     The least-squares solution for rectangular systems can be found
+     using `gsl_linalg_QR_lssolve'.
+
+ -- Function: int gsl_linalg_QR_svx (const gsl_matrix * QR, const
+          gsl_vector * TAU, gsl_vector * X)
+     This function solves the square system A x = b in-place using the
+     QR decomposition of A into (QR,TAU) given by
+     `gsl_linalg_QR_decomp'. On input X should contain the right-hand
+     side b, which is replaced by the solution on output.
+
+ -- Function: int gsl_linalg_QR_lssolve (const gsl_matrix * QR, const
+          gsl_vector * TAU, const gsl_vector * B, gsl_vector * X,
+          gsl_vector * RESIDUAL)
+     This function finds the least squares solution to the
+     overdetermined system A x = b where the matrix A has more rows than
+     columns.  The least squares solution minimizes the Euclidean norm
+     of the residual, ||Ax - b||.The routine uses the QR decomposition
+     of A into (QR, TAU) given by `gsl_linalg_QR_decomp'.  The solution
+     is returned in X.  The residual is computed as a by-product and
+     stored in RESIDUAL.
+
+ -- Function: int gsl_linalg_QR_QTvec (const gsl_matrix * QR, const
+          gsl_vector * TAU, gsl_vector * V)
+     This function applies the matrix Q^T encoded in the decomposition
+     (QR,TAU) to the vector V, storing the result Q^T v in V.  The
+     matrix multiplication is carried out directly using the encoding
+     of the Householder vectors without needing to form the full matrix
+     Q^T.
+
+ -- Function: int gsl_linalg_QR_Qvec (const gsl_matrix * QR, const
+          gsl_vector * TAU, gsl_vector * V)
+     This function applies the matrix Q encoded in the decomposition
+     (QR,TAU) to the vector V, storing the result Q v in V.  The matrix
+     multiplication is carried out directly using the encoding of the
+     Householder vectors without needing to form the full matrix Q.
+
+ -- Function: int gsl_linalg_QR_Rsolve (const gsl_matrix * QR, const
+          gsl_vector * B, gsl_vector * X)
+     This function solves the triangular system R x = b for X. It may
+     be useful if the product b' = Q^T b has already been computed
+     using `gsl_linalg_QR_QTvec'.
+
+ -- Function: int gsl_linalg_QR_Rsvx (const gsl_matrix * QR, gsl_vector
+          * X)
+     This function solves the triangular system R x = b for X in-place.
+     On input X should contain the right-hand side b and is replaced by
+     the solution on output. This function may be useful if the product
+     b' = Q^T b has already been computed using `gsl_linalg_QR_QTvec'.
+
+ -- Function: int gsl_linalg_QR_unpack (const gsl_matrix * QR, const
+          gsl_vector * TAU, gsl_matrix * Q, gsl_matrix * R)
+     This function unpacks the encoded QR decomposition (QR,TAU) into
+     the matrices Q and R, where Q is M-by-M and R is M-by-N.
+
+ -- Function: int gsl_linalg_QR_QRsolve (gsl_matrix * Q, gsl_matrix *
+          R, const gsl_vector * B, gsl_vector * X)
+     This function solves the system R x = Q^T b for X. It can be used
+     when the QR decomposition of a matrix is available in unpacked
+     form as (Q, R).
+
+ -- Function: int gsl_linalg_QR_update (gsl_matrix * Q, gsl_matrix * R,
+          gsl_vector * W, const gsl_vector * V)
+     This function performs a rank-1 update w v^T of the QR
+     decomposition (Q, R). The update is given by Q'R' = Q R + w v^T
+     where the output matrices Q' and R' are also orthogonal and right
+     triangular. Note that W is destroyed by the update.
+
+ -- Function: int gsl_linalg_R_solve (const gsl_matrix * R, const
+          gsl_vector * B, gsl_vector * X)
+     This function solves the triangular system R x = b for the N-by-N
+     matrix R.
+
+ -- Function: int gsl_linalg_R_svx (const gsl_matrix * R, gsl_vector *
+          X)
+     This function solves the triangular system R x = b in-place. On
+     input X should contain the right-hand side b, which is replaced by
+     the solution on output.
+
+
+File: gsl-ref.info,  Node: QR Decomposition with Column Pivoting,  Next: Singular Value Decomposition,  Prev: QR Decomposition,  Up: Linear Algebra
+
+13.3 QR Decomposition with Column Pivoting
+==========================================
+
+The QR decomposition can be extended to the rank deficient case by
+introducing a column permutation P,
+
+     A P = Q R
+
+The first r columns of Q form an orthonormal basis for the range of A
+for a matrix with column rank r.  This decomposition can also be used
+to convert the linear system A x = b into the triangular system R y =
+Q^T b, x = P y, which can be solved by back-substitution and
+permutation.  We denote the QR decomposition with column pivoting by
+QRP^T since A = Q R P^T.
+
+ -- Function: int gsl_linalg_QRPT_decomp (gsl_matrix * A, gsl_vector *
+          TAU, gsl_permutation * P, int * SIGNUM, gsl_vector * NORM)
+     This function factorizes the M-by-N matrix A into the QRP^T
+     decomposition A = Q R P^T.  On output the diagonal and upper
+     triangular part of the input matrix contain the matrix R. The
+     permutation matrix P is stored in the permutation P.  The sign of
+     the permutation is given by SIGNUM. It has the value (-1)^n, where
+     n is the number of interchanges in the permutation. The vector TAU
+     and the columns of the lower triangular part of the matrix A
+     contain the Householder coefficients and vectors which encode the
+     orthogonal matrix Q.  The vector TAU must be of length
+     k=\min(M,N). The matrix Q is related to these components by, Q =
+     Q_k ... Q_2 Q_1 where Q_i = I - \tau_i v_i v_i^T and v_i is the
+     Householder vector v_i = (0,...,1,A(i+1,i),A(i+2,i),...,A(m,i)).
+     This is the same storage scheme as used by LAPACK.  The vector
+     NORM is a workspace of length N used for column pivoting.
+
+     The algorithm used to perform the decomposition is Householder QR
+     with column pivoting (Golub & Van Loan, `Matrix Computations',
+     Algorithm 5.4.1).
+
+ -- Function: int gsl_linalg_QRPT_decomp2 (const gsl_matrix * A,
+          gsl_matrix * Q, gsl_matrix * R, gsl_vector * TAU,
+          gsl_permutation * P, int * SIGNUM, gsl_vector * NORM)
+     This function factorizes the matrix A into the decomposition A = Q
+     R P^T without modifying A itself and storing the output in the
+     separate matrices Q and R.
+
+ -- Function: int gsl_linalg_QRPT_solve (const gsl_matrix * QR, const
+          gsl_vector * TAU, const gsl_permutation * P, const gsl_vector
+          * B, gsl_vector * X)
+     This function solves the square system A x = b using the QRP^T
+     decomposition of A into (QR, TAU, P) given by
+     `gsl_linalg_QRPT_decomp'.
+
+ -- Function: int gsl_linalg_QRPT_svx (const gsl_matrix * QR, const
+          gsl_vector * TAU, const gsl_permutation * P, gsl_vector * X)
+     This function solves the square system A x = b in-place using the
+     QRP^T decomposition of A into (QR,TAU,P). On input X should
+     contain the right-hand side b, which is replaced by the solution
+     on output.
+
+ -- Function: int gsl_linalg_QRPT_QRsolve (const gsl_matrix * Q, const
+          gsl_matrix * R, const gsl_permutation * P, const gsl_vector *
+          B, gsl_vector * X)
+     This function solves the square system R P^T x = Q^T b for X. It
+     can be used when the QR decomposition of a matrix is available in
+     unpacked form as (Q, R).
+
+ -- Function: int gsl_linalg_QRPT_update (gsl_matrix * Q, gsl_matrix *
+          R, const gsl_permutation * P, gsl_vector * U, const
+          gsl_vector * V)
+     This function performs a rank-1 update w v^T of the QRP^T
+     decomposition (Q, R, P). The update is given by Q'R' = Q R + w v^T
+     where the output matrices Q' and R' are also orthogonal and right
+     triangular. Note that W is destroyed by the update. The
+     permutation P is not changed.
+
+ -- Function: int gsl_linalg_QRPT_Rsolve (const gsl_matrix * QR, const
+          gsl_permutation * P, const gsl_vector * B, gsl_vector * X)
+     This function solves the triangular system R P^T x = b for the
+     N-by-N matrix R contained in QR.
+
+ -- Function: int gsl_linalg_QRPT_Rsvx (const gsl_matrix * QR, const
+          gsl_permutation * P, gsl_vector * X)
+     This function solves the triangular system R P^T x = b in-place
+     for the N-by-N matrix R contained in QR. On input X should contain
+     the right-hand side b, which is replaced by the solution on output.
+
+
+File: gsl-ref.info,  Node: Singular Value Decomposition,  Next: Cholesky Decomposition,  Prev: QR Decomposition with Column Pivoting,  Up: Linear Algebra
+
+13.4 Singular Value Decomposition
+=================================
+
+A general rectangular M-by-N matrix A has a singular value
+decomposition (SVD) into the product of an M-by-N orthogonal matrix U,
+an N-by-N diagonal matrix of singular values S and the transpose of an
+N-by-N orthogonal square matrix V,
+
+     A = U S V^T
+
+The singular values \sigma_i = S_{ii} are all non-negative and are
+generally chosen to form a non-increasing sequence \sigma_1 >= \sigma_2
+>= ... >= \sigma_N >= 0.
+
+   The singular value decomposition of a matrix has many practical uses.
+The condition number of the matrix is given by the ratio of the largest
+singular value to the smallest singular value. The presence of a zero
+singular value indicates that the matrix is singular. The number of
+non-zero singular values indicates the rank of the matrix.  In practice
+singular value decomposition of a rank-deficient matrix will not produce
+exact zeroes for singular values, due to finite numerical precision.
+Small singular values should be edited by choosing a suitable tolerance.
+
+   For a rank-deficient matrix, the null space of A is given by the
+columns of V corresponding to the zero singular values.  Similarly, the
+range of A is given by columns of U corresponding to the non-zero
+singular values.
+
+ -- Function: int gsl_linalg_SV_decomp (gsl_matrix * A, gsl_matrix * V,
+          gsl_vector * S, gsl_vector * WORK)
+     This function factorizes the M-by-N matrix A into the singular
+     value decomposition A = U S V^T for M >= N.  On output the matrix
+     A is replaced by U. The diagonal elements of the singular value
+     matrix S are stored in the vector S. The singular values are
+     non-negative and form a non-increasing sequence from S_1 to S_N.
+     The matrix V contains the elements of V in untransposed form. To
+     form the product U S V^T it is necessary to take the transpose of
+     V.  A workspace of length N is required in WORK.
+
+     This routine uses the Golub-Reinsch SVD algorithm.
+
+ -- Function: int gsl_linalg_SV_decomp_mod (gsl_matrix * A, gsl_matrix
+          * X, gsl_matrix * V, gsl_vector * S, gsl_vector * WORK)
+     This function computes the SVD using the modified Golub-Reinsch
+     algorithm, which is faster for M>>N.  It requires the vector WORK
+     of length N and the N-by-N matrix X as additional working space.
+
+ -- Function: int gsl_linalg_SV_decomp_jacobi (gsl_matrix * A,
+          gsl_matrix * V, gsl_vector * S)
+     This function computes the SVD of the M-by-N matrix A using
+     one-sided Jacobi orthogonalization for M >= N.  The Jacobi method
+     can compute singular values to higher relative accuracy than
+     Golub-Reinsch algorithms (see references for details).
+
+ -- Function: int gsl_linalg_SV_solve (gsl_matrix * U, gsl_matrix * V,
+          gsl_vector * S, const gsl_vector * B, gsl_vector * X)
+     This function solves the system A x = b using the singular value
+     decomposition (U, S, V) of A given by `gsl_linalg_SV_decomp'.
+
+     Only non-zero singular values are used in computing the solution.
+     The parts of the solution corresponding to singular values of zero
+     are ignored.  Other singular values can be edited out by setting
+     them to zero before calling this function.
+
+     In the over-determined case where A has more rows than columns the
+     system is solved in the least squares sense, returning the solution
+     X which minimizes ||A x - b||_2.
+
+
+File: gsl-ref.info,  Node: Cholesky Decomposition,  Next: Tridiagonal Decomposition of Real Symmetric Matrices,  Prev: Singular Value Decomposition,  Up: Linear Algebra
+
+13.5 Cholesky Decomposition
+===========================
+
+A symmetric, positive definite square matrix A has a Cholesky
+decomposition into a product of a lower triangular matrix L and its
+transpose L^T,
+
+     A = L L^T
+
+This is sometimes referred to as taking the square-root of a matrix. The
+Cholesky decomposition can only be carried out when all the eigenvalues
+of the matrix are positive.  This decomposition can be used to convert
+the linear system A x = b into a pair of triangular systems (L y = b,
+L^T x = y), which can be solved by forward and back-substitution.
+
+ -- Function: int gsl_linalg_cholesky_decomp (gsl_matrix * A)
+     This function factorizes the positive-definite symmetric square
+     matrix A into the Cholesky decomposition A = L L^T. On input only
+     the diagonal and lower-triangular part of the matrix A are needed.
+     On output the diagonal and lower triangular part of the input
+     matrix A contain the matrix L. The upper triangular part of the
+     input matrix contains L^T, the diagonal terms being identical for
+     both L and L^T.  If the matrix is not positive-definite then the
+     decomposition will fail, returning the error code `GSL_EDOM'.
+
+ -- Function: int gsl_linalg_cholesky_solve (const gsl_matrix *
+          CHOLESKY, const gsl_vector * B, gsl_vector * X)
+     This function solves the system A x = b using the Cholesky
+     decomposition of A into the matrix CHOLESKY given by
+     `gsl_linalg_cholesky_decomp'.
+
+ -- Function: int gsl_linalg_cholesky_svx (const gsl_matrix * CHOLESKY,
+          gsl_vector * X)
+     This function solves the system A x = b in-place using the
+     Cholesky decomposition of A into the matrix CHOLESKY given by
+     `gsl_linalg_cholesky_decomp'. On input X should contain the
+     right-hand side b, which is replaced by the solution on output.
+
+
+File: gsl-ref.info,  Node: Tridiagonal Decomposition of Real Symmetric Matrices,  Next: Tridiagonal Decomposition of Hermitian Matrices,  Prev: Cholesky Decomposition,  Up: Linear Algebra
+
+13.6 Tridiagonal Decomposition of Real Symmetric Matrices
+=========================================================
+
+A symmetric matrix A can be factorized by similarity transformations
+into the form,
+
+     A = Q T Q^T
+
+where Q is an orthogonal matrix and T is a symmetric tridiagonal matrix.
+
+ -- Function: int gsl_linalg_symmtd_decomp (gsl_matrix * A, gsl_vector
+          * TAU)
+     This function factorizes the symmetric square matrix A into the
+     symmetric tridiagonal decomposition Q T Q^T.  On output the
+     diagonal and subdiagonal part of the input matrix A contain the
+     tridiagonal matrix T.  The remaining lower triangular part of the
+     input matrix contains the Householder vectors which, together with
+     the Householder coefficients TAU, encode the orthogonal matrix Q.
+     This storage scheme is the same as used by LAPACK.  The upper
+     triangular part of A is not referenced.
+
+ -- Function: int gsl_linalg_symmtd_unpack (const gsl_matrix * A, const
+          gsl_vector * TAU, gsl_matrix * Q, gsl_vector * DIAG,
+          gsl_vector * SUBDIAG)
+     This function unpacks the encoded symmetric tridiagonal
+     decomposition (A, TAU) obtained from `gsl_linalg_symmtd_decomp'
+     into the orthogonal matrix Q, the vector of diagonal elements DIAG
+     and the vector of subdiagonal elements SUBDIAG.
+
+ -- Function: int gsl_linalg_symmtd_unpack_T (const gsl_matrix * A,
+          gsl_vector * DIAG, gsl_vector * SUBDIAG)
+     This function unpacks the diagonal and subdiagonal of the encoded
+     symmetric tridiagonal decomposition (A, TAU) obtained from
+     `gsl_linalg_symmtd_decomp' into the vectors DIAG and SUBDIAG.
+
+
+File: gsl-ref.info,  Node: Tridiagonal Decomposition of Hermitian Matrices,  Next: Hessenberg Decomposition of Real Matrices,  Prev: Tridiagonal Decomposition of Real Symmetric Matrices,  Up: Linear Algebra
+
+13.7 Tridiagonal Decomposition of Hermitian Matrices
+====================================================
+
+A hermitian matrix A can be factorized by similarity transformations
+into the form,
+
+     A = U T U^T
+
+where U is a unitary matrix and T is a real symmetric tridiagonal
+matrix.
+
+ -- Function: int gsl_linalg_hermtd_decomp (gsl_matrix_complex * A,
+          gsl_vector_complex * TAU)
+     This function factorizes the hermitian matrix A into the symmetric
+     tridiagonal decomposition U T U^T.  On output the real parts of
+     the diagonal and subdiagonal part of the input matrix A contain
+     the tridiagonal matrix T.  The remaining lower triangular part of
+     the input matrix contains the Householder vectors which, together
+     with the Householder coefficients TAU, encode the orthogonal matrix
+     Q. This storage scheme is the same as used by LAPACK.  The upper
+     triangular part of A and imaginary parts of the diagonal are not
+     referenced.
+
+ -- Function: int gsl_linalg_hermtd_unpack (const gsl_matrix_complex *
+          A, const gsl_vector_complex * TAU, gsl_matrix_complex * Q,
+          gsl_vector * DIAG, gsl_vector * SUBDIAG)
+     This function unpacks the encoded tridiagonal decomposition (A,
+     TAU) obtained from `gsl_linalg_hermtd_decomp' into the unitary
+     matrix U, the real vector of diagonal elements DIAG and the real
+     vector of subdiagonal elements SUBDIAG.
+
+ -- Function: int gsl_linalg_hermtd_unpack_T (const gsl_matrix_complex
+          * A, gsl_vector * DIAG, gsl_vector * SUBDIAG)
+     This function unpacks the diagonal and subdiagonal of the encoded
+     tridiagonal decomposition (A, TAU) obtained from the
+     `gsl_linalg_hermtd_decomp' into the real vectors DIAG and SUBDIAG.
+
+
+File: gsl-ref.info,  Node: Hessenberg Decomposition of Real Matrices,  Next: Bidiagonalization,  Prev: Tridiagonal Decomposition of Hermitian Matrices,  Up: Linear Algebra
+
+13.8 Hessenberg Decomposition of Real Matrices
+==============================================
+
+A general matrix A can be decomposed by orthogonal similarity
+transformations into the form
+
+     A = U H U^T
+
+   where U is orthogonal and H is an upper Hessenberg matrix, meaning
+that it has zeros below the first subdiagonal. The Hessenberg reduction
+is the first step in the Schur decomposition for the nonsymmetric
+eigenvalue problem, but has applications in other areas as well.
+
+ -- Function: int gsl_linalg_hessenberg (gsl_matrix * A, gsl_vector *
+          TAU)
+     This function computes the Hessenberg decomposition of the matrix
+     A by applying the similarity transformation H = U^T A U.  On
+     output, H is stored in the upper portion of A. The information
+     required to construct the matrix U is stored in the lower
+     triangular portion of A. U is a product of N - 2 Householder
+     matrices. The Householder vectors are stored in the lower portion
+     of A (below the subdiagonal) and the Householder coefficients are
+     stored in the vector TAU.  TAU must be of length N.
+
+ -- Function: int gsl_linalg_hessenberg_unpack (gsl_matrix * H,
+          gsl_vector * TAU, gsl_matrix * U)
+     This function constructs the orthogonal matrix U from the
+     information stored in the Hessenberg matrix H along with the
+     vector TAU. H and TAU are outputs from `gsl_linalg_hessenberg'.
+
+ -- Function: int gsl_linalg_hessenberg_unpack_accum (gsl_matrix * H,
+          gsl_vector * TAU, gsl_matrix * V)
+     This function is similar to `gsl_linalg_hessenberg_unpack', except
+     it accumulates the matrix U into V, so that V' = VU.  The matrix V
+     must be initialized prior to calling this function.  Setting V to
+     the identity matrix provides the same result as
+     `gsl_linalg_hessenberg_unpack'. If H is order N, then V must have
+     N columns but may have any number of rows.
+
+ -- Function: void gsl_linalg_hessenberg_set_zero (gsl_matrix * H)
+     This function sets the lower triangular portion of H, below the
+     subdiagonal, to zero. It is useful for clearing out the
+     Householder vectors after calling `gsl_linalg_hessenberg'.
+
+
+File: gsl-ref.info,  Node: Bidiagonalization,  Next: Householder Transformations,  Prev: Hessenberg Decomposition of Real Matrices,  Up: Linear Algebra
+
+13.9 Bidiagonalization
+======================
+
+A general matrix A can be factorized by similarity transformations into
+the form,
+
+     A = U B V^T
+
+where U and V are orthogonal matrices and B is a N-by-N bidiagonal
+matrix with non-zero entries only on the diagonal and superdiagonal.
+The size of U is M-by-N and the size of V is N-by-N.
+
+ -- Function: int gsl_linalg_bidiag_decomp (gsl_matrix * A, gsl_vector
+          * TAU_U, gsl_vector * TAU_V)
+     This function factorizes the M-by-N matrix A into bidiagonal form
+     U B V^T.  The diagonal and superdiagonal of the matrix B are
+     stored in the diagonal and superdiagonal of A.  The orthogonal
+     matrices U and V are stored as compressed Householder vectors in
+     the remaining elements of A.  The Householder coefficients are
+     stored in the vectors TAU_U and TAU_V.  The length of TAU_U must
+     equal the number of elements in the diagonal of A and the length
+     of TAU_V should be one element shorter.
+
+ -- Function: int gsl_linalg_bidiag_unpack (const gsl_matrix * A, const
+          gsl_vector * TAU_U, gsl_matrix * U, const gsl_vector * TAU_V,
+          gsl_matrix * V, gsl_vector * DIAG, gsl_vector * SUPERDIAG)
+     This function unpacks the bidiagonal decomposition of A given by
+     `gsl_linalg_bidiag_decomp', (A, TAU_U, TAU_V) into the separate
+     orthogonal matrices U, V and the diagonal vector DIAG and
+     superdiagonal SUPERDIAG.  Note that U is stored as a compact
+     M-by-N orthogonal matrix satisfying U^T U = I for efficiency.
+
+ -- Function: int gsl_linalg_bidiag_unpack2 (gsl_matrix * A, gsl_vector
+          * TAU_U, gsl_vector * TAU_V, gsl_matrix * V)
+     This function unpacks the bidiagonal decomposition of A given by
+     `gsl_linalg_bidiag_decomp', (A, TAU_U, TAU_V) into the separate
+     orthogonal matrices U, V and the diagonal vector DIAG and
+     superdiagonal SUPERDIAG.  The matrix U is stored in-place in A.
+
+ -- Function: int gsl_linalg_bidiag_unpack_B (const gsl_matrix * A,
+          gsl_vector * DIAG, gsl_vector * SUPERDIAG)
+     This function unpacks the diagonal and superdiagonal of the
+     bidiagonal decomposition of A given by `gsl_linalg_bidiag_decomp',
+     into the diagonal vector DIAG and superdiagonal vector SUPERDIAG.
+
+
+File: gsl-ref.info,  Node: Householder Transformations,  Next: Householder solver for linear systems,  Prev: Bidiagonalization,  Up: Linear Algebra
+
+13.10 Householder Transformations
+=================================
+
+A Householder transformation is a rank-1 modification of the identity
+matrix which can be used to zero out selected elements of a vector.  A
+Householder matrix P takes the form,
+
+     P = I - \tau v v^T
+
+where v is a vector (called the "Householder vector") and \tau = 2/(v^T
+v).  The functions described in this section use the rank-1 structure
+of the Householder matrix to create and apply Householder
+transformations efficiently.
+
+ -- Function: double gsl_linalg_householder_transform (gsl_vector * V)
+     This function prepares a Householder transformation P = I - \tau v
+     v^T which can be used to zero all the elements of the input vector
+     except the first.  On output the transformation is stored in the
+     vector V and the scalar \tau is returned.
+
+ -- Function: int gsl_linalg_householder_hm (double tau, const
+          gsl_vector * v, gsl_matrix * A)
+     This function applies the Householder matrix P defined by the
+     scalar TAU and the vector V to the left-hand side of the matrix A.
+     On output the result P A is stored in A.
+
+ -- Function: int gsl_linalg_householder_mh (double tau, const
+          gsl_vector * v, gsl_matrix * A)
+     This function applies the Householder matrix P defined by the
+     scalar TAU and the vector V to the right-hand side of the matrix
+     A. On output the result A P is stored in A.
+
+ -- Function: int gsl_linalg_householder_hv (double tau, const
+          gsl_vector * v, gsl_vector * w)
+     This function applies the Householder transformation P defined by
+     the scalar TAU and the vector V to the vector W.  On output the
+     result P w is stored in W.
+
+
+File: gsl-ref.info,  Node: Householder solver for linear systems,  Next: Tridiagonal Systems,  Prev: Householder Transformations,  Up: Linear Algebra
+
+13.11 Householder solver for linear systems
+===========================================
+
+ -- Function: int gsl_linalg_HH_solve (gsl_matrix * A, const gsl_vector
+          * B, gsl_vector * X)
+     This function solves the system A x = b directly using Householder
+     transformations. On output the solution is stored in X and B is
+     not modified. The matrix A is destroyed by the Householder
+     transformations.
+
+ -- Function: int gsl_linalg_HH_svx (gsl_matrix * A, gsl_vector * X)
+     This function solves the system A x = b in-place using Householder
+     transformations.  On input X should contain the right-hand side b,
+     which is replaced by the solution on output.  The matrix A is
+     destroyed by the Householder transformations.
+
+
+File: gsl-ref.info,  Node: Tridiagonal Systems,  Next: Balancing,  Prev: Householder solver for linear systems,  Up: Linear Algebra
+
+13.12 Tridiagonal Systems
+=========================
+
+ -- Function: int gsl_linalg_solve_tridiag (const gsl_vector * DIAG,
+          const gsl_vector * E, const gsl_vector * F, const gsl_vector
+          * B, gsl_vector * X)
+     This function solves the general N-by-N system A x = b where A is
+     tridiagonal (N >= 2). The super-diagonal and sub-diagonal vectors
+     E and F must be one element shorter than the diagonal vector DIAG.
+     The form of A for the 4-by-4 case is shown below,
+
+          A = ( d_0 e_0  0   0  )
+              ( f_0 d_1 e_1  0  )
+              (  0  f_1 d_2 e_2 )
+              (  0   0  f_2 d_3 )
+
+ -- Function: int gsl_linalg_solve_symm_tridiag (const gsl_vector *
+          DIAG, const gsl_vector * E, const gsl_vector * B, gsl_vector
+          * X)
+     This function solves the general N-by-N system A x = b where A is
+     symmetric tridiagonal (N >= 2).  The off-diagonal vector E must be
+     one element shorter than the diagonal vector DIAG.  The form of A
+     for the 4-by-4 case is shown below,
+
+          A = ( d_0 e_0  0   0  )
+              ( e_0 d_1 e_1  0  )
+              (  0  e_1 d_2 e_2 )
+              (  0   0  e_2 d_3 )
+     The current implementation uses a variant of Cholesky decomposition
+     which can cause division by zero if the matrix is not positive
+     definite.
+
+ -- Function: int gsl_linalg_solve_cyc_tridiag (const gsl_vector *
+          DIAG, const gsl_vector * E, const gsl_vector * F, const
+          gsl_vector * B, gsl_vector * X)
+     This function solves the general N-by-N system A x = b where A is
+     cyclic tridiagonal (N >= 3).  The cyclic super-diagonal and
+     sub-diagonal vectors E and F must have the same number of elements
+     as the diagonal vector DIAG.  The form of A for the 4-by-4 case is
+     shown below,
+
+          A = ( d_0 e_0  0  f_3 )
+              ( f_0 d_1 e_1  0  )
+              (  0  f_1 d_2 e_2 )
+              ( e_3  0  f_2 d_3 )
+
+ -- Function: int gsl_linalg_solve_symm_cyc_tridiag (const gsl_vector *
+          DIAG, const gsl_vector * E, const gsl_vector * B, gsl_vector
+          * X)
+     This function solves the general N-by-N system A x = b where A is
+     symmetric cyclic tridiagonal (N >= 3).  The cyclic off-diagonal
+     vector E must have the same number of elements as the diagonal
+     vector DIAG.  The form of A for the 4-by-4 case is shown below,
+
+          A = ( d_0 e_0  0  e_3 )
+              ( e_0 d_1 e_1  0  )
+              (  0  e_1 d_2 e_2 )
+              ( e_3  0  e_2 d_3 )
+
+
+File: gsl-ref.info,  Node: Balancing,  Next: Linear Algebra Examples,  Prev: Tridiagonal Systems,  Up: Linear Algebra
+
+13.13 Balancing
+===============
+
+The process of balancing a matrix applies similarity transformations to
+make the rows and columns have comparable norms. This is useful, for
+example, to reduce roundoff errors in the solution of eigenvalue
+problems. Balancing a matrix A consists of replacing A with a similar
+matrix
+
+     A' = D^(-1) A D
+
+   where D is a diagonal matrix whose entries are powers of the
+floating point radix.
+
+ -- Function: int gsl_linalg_balance_matrix (gsl_matrix * A, gsl_vector
+          * D)
+     This function replaces the matrix A with its balanced counterpart
+     and stores the diagonal elements of the similarity transformation
+     into the vector D.
+
+
+File: gsl-ref.info,  Node: Linear Algebra Examples,  Next: Linear Algebra References and Further Reading,  Prev: Balancing,  Up: Linear Algebra
+
+13.14 Examples
+==============
+
+The following program solves the linear system A x = b. The system to
+be solved is,
+
+     [ 0.18 0.60 0.57 0.96 ] [x0]   [1.0]
+     [ 0.41 0.24 0.99 0.58 ] [x1] = [2.0]
+     [ 0.14 0.30 0.97 0.66 ] [x2]   [3.0]
+     [ 0.51 0.13 0.19 0.85 ] [x3]   [4.0]
+
+and the solution is found using LU decomposition of the matrix A.
+
+     #include <stdio.h>
+     #include <gsl/gsl_linalg.h>
+
+     int
+     main (void)
+     {
+       double a_data[] = { 0.18, 0.60, 0.57, 0.96,
+                           0.41, 0.24, 0.99, 0.58,
+                           0.14, 0.30, 0.97, 0.66,
+                           0.51, 0.13, 0.19, 0.85 };
+
+       double b_data[] = { 1.0, 2.0, 3.0, 4.0 };
+
+       gsl_matrix_view m
+         = gsl_matrix_view_array (a_data, 4, 4);
+
+       gsl_vector_view b
+         = gsl_vector_view_array (b_data, 4);
+
+       gsl_vector *x = gsl_vector_alloc (4);
+
+       int s;
+
+       gsl_permutation * p = gsl_permutation_alloc (4);
+
+       gsl_linalg_LU_decomp (&m.matrix, p, &s);
+
+       gsl_linalg_LU_solve (&m.matrix, p, &b.vector, x);
+
+       printf ("x = \n");
+       gsl_vector_fprintf (stdout, x, "%g");
+
+       gsl_permutation_free (p);
+       gsl_vector_free (x);
+       return 0;
+     }
+
+Here is the output from the program,
+
+     x = -4.05205
+     -12.6056
+     1.66091
+     8.69377
+
+This can be verified by multiplying the solution x by the original
+matrix A using GNU OCTAVE,
+
+     octave> A = [ 0.18, 0.60, 0.57, 0.96;
+                   0.41, 0.24, 0.99, 0.58;
+                   0.14, 0.30, 0.97, 0.66;
+                   0.51, 0.13, 0.19, 0.85 ];
+
+     octave> x = [ -4.05205; -12.6056; 1.66091; 8.69377];
+
+     octave> A * x
+     ans =
+       1.0000
+       2.0000
+       3.0000
+       4.0000
+
+This reproduces the original right-hand side vector, b, in accordance
+with the equation A x = b.
+
+
+File: gsl-ref.info,  Node: Linear Algebra References and Further Reading,  Prev: Linear Algebra Examples,  Up: Linear Algebra
+
+13.15 References and Further Reading
+====================================
+
+Further information on the algorithms described in this section can be
+found in the following book,
+
+     G. H. Golub, C. F. Van Loan, `Matrix Computations' (3rd Ed, 1996),
+     Johns Hopkins University Press, ISBN 0-8018-5414-8.
+
+The LAPACK library is described in the following manual,
+
+     `LAPACK Users' Guide' (Third Edition, 1999), Published by SIAM,
+     ISBN 0-89871-447-8.
+
+     `http://www.netlib.org/lapack'
+
+The LAPACK source code can be found at the website above, along with an
+online copy of the users guide.
+
+The Modified Golub-Reinsch algorithm is described in the following
+paper,
+
+     T.F. Chan, "An Improved Algorithm for Computing the Singular Value
+     Decomposition", `ACM Transactions on Mathematical Software', 8
+     (1982), pp 72-83.
+
+The Jacobi algorithm for singular value decomposition is described in
+the following papers,
+
+     J.C. Nash, "A one-sided transformation method for the singular
+     value decomposition and algebraic eigenproblem", `Computer
+     Journal', Volume 18, Number 1 (1973), p 74-76
+
+     James Demmel, Kresimir Veselic, "Jacobi's Method is more accurate
+     than QR", `Lapack Working Note 15' (LAWN-15), October 1989.
+     Available from netlib, `http://www.netlib.org/lapack/' in the
+     `lawns' or `lawnspdf' directories.
+
+
+File: gsl-ref.info,  Node: Eigensystems,  Next: Fast Fourier Transforms,  Prev: Linear Algebra,  Up: Top
+
+14 Eigensystems
+***************
+
+This chapter describes functions for computing eigenvalues and
+eigenvectors of matrices.  There are routines for real symmetric, real
+nonsymmetric, and complex hermitian matrices. Eigenvalues can be
+computed with or without eigenvectors. The hermitian matrix algorithms
+used are symmetric bidiagonalization followed by QR reduction. The
+nonsymmetric algorithm is the Francis QR double-shift.
+
+   These routines are intended for "small" systems where simple
+algorithms are acceptable.  Anyone interested in finding eigenvalues
+and eigenvectors of large matrices will want to use the sophisticated
+routines found in LAPACK. The Fortran version of LAPACK is recommended
+as the standard package for large-scale linear algebra.
+
+   The functions described in this chapter are declared in the header
+file `gsl_eigen.h'.
+
+* Menu:
+
+* Real Symmetric Matrices::
+* Complex Hermitian Matrices::
+* Real Nonsymmetric Matrices::
+* Sorting Eigenvalues and Eigenvectors::
+* Eigenvalue and Eigenvector Examples::
+* Eigenvalue and Eigenvector References::
+
+
+File: gsl-ref.info,  Node: Real Symmetric Matrices,  Next: Complex Hermitian Matrices,  Up: Eigensystems
+
+14.1 Real Symmetric Matrices
+============================
+
+ -- Function: gsl_eigen_symm_workspace * gsl_eigen_symm_alloc (const
+          size_t N)
+     This function allocates a workspace for computing eigenvalues of
+     N-by-N real symmetric matrices.  The size of the workspace is
+     O(2n).
+
+ -- Function: void gsl_eigen_symm_free (gsl_eigen_symm_workspace * W)
+     This function frees the memory associated with the workspace W.
+
+ -- Function: int gsl_eigen_symm (gsl_matrix * A, gsl_vector * EVAL,
+          gsl_eigen_symm_workspace * W)
+     This function computes the eigenvalues of the real symmetric matrix
+     A.  Additional workspace of the appropriate size must be provided
+     in W.  The diagonal and lower triangular part of A are destroyed
+     during the computation, but the strict upper triangular part is
+     not referenced.  The eigenvalues are stored in the vector EVAL and
+     are unordered.
+
+ -- Function: gsl_eigen_symmv_workspace * gsl_eigen_symmv_alloc (const
+          size_t N)
+     This function allocates a workspace for computing eigenvalues and
+     eigenvectors of N-by-N real symmetric matrices.  The size of the
+     workspace is O(4n).
+
+ -- Function: void gsl_eigen_symmv_free (gsl_eigen_symmv_workspace * W)
+     This function frees the memory associated with the workspace W.
+
+ -- Function: int gsl_eigen_symmv (gsl_matrix * A, gsl_vector * EVAL,
+          gsl_matrix * EVEC, gsl_eigen_symmv_workspace * W)
+     This function computes the eigenvalues and eigenvectors of the real
+     symmetric matrix A.  Additional workspace of the appropriate size
+     must be provided in W.  The diagonal and lower triangular part of
+     A are destroyed during the computation, but the strict upper
+     triangular part is not referenced.  The eigenvalues are stored in
+     the vector EVAL and are unordered.  The corresponding eigenvectors
+     are stored in the columns of the matrix EVEC.  For example, the
+     eigenvector in the first column corresponds to the first
+     eigenvalue.  The eigenvectors are guaranteed to be mutually
+     orthogonal and normalised to unit magnitude.
+
+
+File: gsl-ref.info,  Node: Complex Hermitian Matrices,  Next: Real Nonsymmetric Matrices,  Prev: Real Symmetric Matrices,  Up: Eigensystems
+
+14.2 Complex Hermitian Matrices
+===============================
+
+ -- Function: gsl_eigen_herm_workspace * gsl_eigen_herm_alloc (const
+          size_t N)
+     This function allocates a workspace for computing eigenvalues of
+     N-by-N complex hermitian matrices.  The size of the workspace is
+     O(3n).
+
+ -- Function: void gsl_eigen_herm_free (gsl_eigen_herm_workspace * W)
+     This function frees the memory associated with the workspace W.
+
+ -- Function: int gsl_eigen_herm (gsl_matrix_complex * A, gsl_vector *
+          EVAL, gsl_eigen_herm_workspace * W)
+     This function computes the eigenvalues of the complex hermitian
+     matrix A.  Additional workspace of the appropriate size must be
+     provided in W.  The diagonal and lower triangular part of A are
+     destroyed during the computation, but the strict upper triangular
+     part is not referenced.  The imaginary parts of the diagonal are
+     assumed to be zero and are not referenced. The eigenvalues are
+     stored in the vector EVAL and are unordered.
+
+ -- Function: gsl_eigen_hermv_workspace * gsl_eigen_hermv_alloc (const
+          size_t N)
+     This function allocates a workspace for computing eigenvalues and
+     eigenvectors of N-by-N complex hermitian matrices.  The size of
+     the workspace is O(5n).
+
+ -- Function: void gsl_eigen_hermv_free (gsl_eigen_hermv_workspace * W)
+     This function frees the memory associated with the workspace W.
+
+ -- Function: int gsl_eigen_hermv (gsl_matrix_complex * A, gsl_vector *
+          EVAL, gsl_matrix_complex * EVEC, gsl_eigen_hermv_workspace *
+          W)
+     This function computes the eigenvalues and eigenvectors of the
+     complex hermitian matrix A.  Additional workspace of the
+     appropriate size must be provided in W.  The diagonal and lower
+     triangular part of A are destroyed during the computation, but the
+     strict upper triangular part is not referenced. The imaginary
+     parts of the diagonal are assumed to be zero and are not
+     referenced.  The eigenvalues are stored in the vector EVAL and are
+     unordered.  The corresponding complex eigenvectors are stored in
+     the columns of the matrix EVEC.  For example, the eigenvector in
+     the first column corresponds to the first eigenvalue.  The
+     eigenvectors are guaranteed to be mutually orthogonal and
+     normalised to unit magnitude.
+
+
+File: gsl-ref.info,  Node: Real Nonsymmetric Matrices,  Next: Sorting Eigenvalues and Eigenvectors,  Prev: Complex Hermitian Matrices,  Up: Eigensystems
+
+14.3 Real Nonsymmetric Matrices
+===============================
+
+The solution of the real nonsymmetric eigensystem problem for a matrix
+A involves computing the Schur decomposition
+
+     A = Z T Z^T
+
+   where Z is an orthogonal matrix of Schur vectors and T, the Schur
+form, is quasi upper triangular with diagonal 1-by-1 blocks which are
+real eigenvalues of A, and diagonal 2-by-2 blocks whose eigenvalues are
+complex conjugate eigenvalues of A. The algorithm used is the double
+shift Francis method.
+
+ -- Function: gsl_eigen_nonsymm_workspace * gsl_eigen_nonsymm_alloc
+          (const size_t N)
+     This function allocates a workspace for computing eigenvalues of
+     N-by-N real nonsymmetric matrices. The size of the workspace is
+     O(2n).
+
+ -- Function: void gsl_eigen_nonsymm_free (gsl_eigen_nonsymm_workspace
+          * W)
+     This function frees the memory associated with the workspace W.
+
+ -- Function: void gsl_eigen_nonsymm_params (const int COMPUTE_T, const
+          int BALANCE, gsl_eigen_nonsymm_workspace * W)
+     This function sets some parameters which determine how the
+     eigenvalue problem is solved in subsequent calls to
+     `gsl_eigen_nonsymm'.
+
+     If COMPUTE_T is set to 1, the full Schur form T will be computed
+     by `gsl_eigen_nonsymm'. If it is set to 0, T will not be computed
+     (this is the default setting). Computing the full Schur form T
+     requires approximately 1.5-2 times the number of flops.
+
+     If BALANCE is set to 1, a balancing transformation is applied to
+     the matrix prior to computing eigenvalues. This transformation is
+     designed to make the rows and columns of the matrix have comparable
+     norms, and can result in more accurate eigenvalues for matrices
+     whose entries vary widely in magnitude. See *Note Balancing:: for
+     more information. Note that the balancing transformation does not
+     preserve the orthogonality of the Schur vectors, so if you wish to
+     compute the Schur vectors with `gsl_eigen_nonsymm_Z' you will
+     obtain the Schur vectors of the balanced matrix instead of the
+     original matrix. The relationship will be
+
+          T = Q^t D^(-1) A D Q
+
+     where Q is the matrix of Schur vectors for the balanced matrix, and
+     D is the balancing transformation. Then `gsl_eigen_nonsymm_Z' will
+     compute a matrix Z which satisfies
+
+          T = Z^(-1) A Z
+
+     with Z = D Q. Note that Z will not be orthogonal. For this reason,
+     balancing is not performed by default.
+
+ -- Function: int gsl_eigen_nonsymm (gsl_matrix * A, gsl_vector_complex
+          * EVAL, gsl_eigen_nonsymm_workspace * W)
+     This function computes the eigenvalues of the real nonsymmetric
+     matrix A and stores them in the vector EVAL. If T is desired, it
+     is stored in the upper portion of A on output.  Otherwise, on
+     output, the diagonal of A will contain the 1-by-1 real eigenvalues
+     and 2-by-2 complex conjugate eigenvalue systems, and the rest of A
+     is destroyed. In rare cases, this function will fail to find all
+     eigenvalues. If this happens, an error code is returned and the
+     number of converged eigenvalues is stored in `w->n_evals'.  The
+     converged eigenvalues are stored in the beginning of EVAL.
+
+ -- Function: int gsl_eigen_nonsymm_Z (gsl_matrix * A,
+          gsl_vector_complex * EVAL, gsl_matrix * Z,
+          gsl_eigen_nonsymm_workspace * W)
+     This function is identical to `gsl_eigen_nonsymm' except it also
+     computes the Schur vectors and stores them into Z.
+
+ -- Function: gsl_eigen_nonsymmv_workspace * gsl_eigen_nonsymmv_alloc
+          (const size_t N)
+     This function allocates a workspace for computing eigenvalues and
+     eigenvectors of N-by-N real nonsymmetric matrices. The size of the
+     workspace is O(5n).
+
+ -- Function: void gsl_eigen_nonsymmv_free
+          (gsl_eigen_nonsymmv_workspace * W)
+     This function frees the memory associated with the workspace W.
+
+ -- Function: int gsl_eigen_nonsymmv (gsl_matrix * A,
+          gsl_vector_complex * EVAL, gsl_matrix_complex * EVEC,
+          gsl_eigen_nonsymmv_workspace * W)
+     This function computes eigenvalues and right eigenvectors of the
+     N-by-N real nonsymmetric matrix A. It first calls
+     `gsl_eigen_nonsymm' to compute the eigenvalues, Schur form T, and
+     Schur vectors. Then it finds eigenvectors of T and backtransforms
+     them using the Schur vectors. The Schur vectors are destroyed in
+     the process, but can be saved by using `gsl_eigen_nonsymmv_Z'. The
+     computed eigenvectors are normalized to have Euclidean norm 1. On
+     output, the upper portion of A contains the Schur form T. If
+     `gsl_eigen_nonsymm' fails, no eigenvectors are computed, and an
+     error code is returned.
+
+ -- Function: int gsl_eigen_nonsymmv_Z (gsl_matrix * A,
+          gsl_vector_complex * EVAL, gsl_matrix_complex * EVEC,
+          gsl_matrix * Z, gsl_eigen_nonsymmv_workspace * W)
+     This function is identical to `gsl_eigen_nonsymmv' except it also
+     saves the Schur vectors into Z.
+
+
+File: gsl-ref.info,  Node: Sorting Eigenvalues and Eigenvectors,  Next: Eigenvalue and Eigenvector Examples,  Prev: Real Nonsymmetric Matrices,  Up: Eigensystems
+
+14.4 Sorting Eigenvalues and Eigenvectors
+=========================================
+
+ -- Function: int gsl_eigen_symmv_sort (gsl_vector * EVAL, gsl_matrix *
+          EVEC, gsl_eigen_sort_t SORT_TYPE)
+     This function simultaneously sorts the eigenvalues stored in the
+     vector EVAL and the corresponding real eigenvectors stored in the
+     columns of the matrix EVEC into ascending or descending order
+     according to the value of the parameter SORT_TYPE,
+
+    `GSL_EIGEN_SORT_VAL_ASC'
+          ascending order in numerical value
+
+    `GSL_EIGEN_SORT_VAL_DESC'
+          descending order in numerical value
+
+    `GSL_EIGEN_SORT_ABS_ASC'
+          ascending order in magnitude
+
+    `GSL_EIGEN_SORT_ABS_DESC'
+          descending order in magnitude
+
+
+ -- Function: int gsl_eigen_hermv_sort (gsl_vector * EVAL,
+          gsl_matrix_complex * EVEC, gsl_eigen_sort_t SORT_TYPE)
+     This function simultaneously sorts the eigenvalues stored in the
+     vector EVAL and the corresponding complex eigenvectors stored in
+     the columns of the matrix EVEC into ascending or descending order
+     according to the value of the parameter SORT_TYPE as shown above.
+
+ -- Function: int gsl_eigen_nonsymmv_sort (gsl_vector_complex * EVAL,
+          gsl_matrix_complex * EVEC, gsl_eigen_sort_t SORT_TYPE)
+     This function simultaneously sorts the eigenvalues stored in the
+     vector EVAL and the corresponding complex eigenvectors stored in
+     the columns of the matrix EVEC into ascending or descending order
+     according to the value of the parameter SORT_TYPE as shown above.
+     Only GSL_EIGEN_SORT_ABS_ASC and GSL_EIGEN_SORT_ABS_DESC are
+     supported due to the eigenvalues being complex.
+
+
+File: gsl-ref.info,  Node: Eigenvalue and Eigenvector Examples,  Next: Eigenvalue and Eigenvector References,  Prev: Sorting Eigenvalues and Eigenvectors,  Up: Eigensystems
+
+14.5 Examples
+=============
+
+The following program computes the eigenvalues and eigenvectors of the
+4-th order Hilbert matrix, H(i,j) = 1/(i + j + 1).
+
+     #include <stdio.h>
+     #include <gsl/gsl_math.h>
+     #include <gsl/gsl_eigen.h>
+
+     int
+     main (void)
+     {
+       double data[] = { 1.0  , 1/2.0, 1/3.0, 1/4.0,
+                         1/2.0, 1/3.0, 1/4.0, 1/5.0,
+                         1/3.0, 1/4.0, 1/5.0, 1/6.0,
+                         1/4.0, 1/5.0, 1/6.0, 1/7.0 };
+
+       gsl_matrix_view m
+         = gsl_matrix_view_array (data, 4, 4);
+
+       gsl_vector *eval = gsl_vector_alloc (4);
+       gsl_matrix *evec = gsl_matrix_alloc (4, 4);
+
+       gsl_eigen_symmv_workspace * w =
+         gsl_eigen_symmv_alloc (4);
+
+       gsl_eigen_symmv (&m.matrix, eval, evec, w);
+
+       gsl_eigen_symmv_free (w);
+
+       gsl_eigen_symmv_sort (eval, evec,
+                             GSL_EIGEN_SORT_ABS_ASC);
+
+       {
+         int i;
+
+         for (i = 0; i < 4; i++)
+           {
+             double eval_i
+                = gsl_vector_get (eval, i);
+             gsl_vector_view evec_i
+                = gsl_matrix_column (evec, i);
+
+             printf ("eigenvalue = %g\n", eval_i);
+             printf ("eigenvector = \n");
+             gsl_vector_fprintf (stdout,
+                                 &evec_i.vector, "%g");
+           }
+       }
+
+       gsl_vector_free (eval);
+       gsl_matrix_free (evec);
+
+       return 0;
+     }
+
+Here is the beginning of the output from the program,
+
+     $ ./a.out
+     eigenvalue = 9.67023e-05
+     eigenvector =
+     -0.0291933
+     0.328712
+     -0.791411
+     0.514553
+     ...
+
+This can be compared with the corresponding output from GNU OCTAVE,
+
+     octave> [v,d] = eig(hilb(4));
+     octave> diag(d)
+     ans =
+
+        9.6702e-05
+        6.7383e-03
+        1.6914e-01
+        1.5002e+00
+
+     octave> v
+     v =
+
+        0.029193   0.179186  -0.582076   0.792608
+       -0.328712  -0.741918   0.370502   0.451923
+        0.791411   0.100228   0.509579   0.322416
+       -0.514553   0.638283   0.514048   0.252161
+
+Note that the eigenvectors can differ by a change of sign, since the
+sign of an eigenvector is arbitrary.
+
+   The following program illustrates the use of the nonsymmetric
+eigensolver, by computing the eigenvalues and eigenvectors of the
+Vandermonde matrix V(x;i,j) = x_i^n - j with x = (-1,-2,3,4).
+
+     #include <stdio.h>
+     #include <gsl/gsl_math.h>
+     #include <gsl/gsl_eigen.h>
+
+     int
+     main (void)
+     {
+       double data[] = { -1.0, 1.0, -1.0, 1.0,
+                         -8.0, 4.0, -2.0, 1.0,
+                         27.0, 9.0, 3.0, 1.0,
+                         64.0, 16.0, 4.0, 1.0 };
+
+       gsl_matrix_view m
+         = gsl_matrix_view_array (data, 4, 4);
+
+       gsl_vector_complex *eval = gsl_vector_complex_alloc (4);
+       gsl_matrix_complex *evec = gsl_matrix_complex_alloc (4, 4);
+
+       gsl_eigen_nonsymmv_workspace * w =
+         gsl_eigen_nonsymmv_alloc (4);
+
+       gsl_eigen_nonsymmv (&m.matrix, eval, evec, w);
+
+       gsl_eigen_nonsymmv_free (w);
+
+       gsl_eigen_nonsymmv_sort (eval, evec,
+                                GSL_EIGEN_SORT_ABS_DESC);
+
+       {
+         int i, j;
+
+         for (i = 0; i < 4; i++)
+           {
+             gsl_complex eval_i
+                = gsl_vector_complex_get (eval, i);
+             gsl_vector_complex_view evec_i
+                = gsl_matrix_complex_column (evec, i);
+
+             printf ("eigenvalue = %g + %gi\n",
+                     GSL_REAL(eval_i), GSL_IMAG(eval_i));
+             printf ("eigenvector = \n");
+             for (j = 0; j < 4; ++j)
+               {
+                 gsl_complex z = gsl_vector_complex_get(&evec_i.vector, j);
+                 printf("%g + %gi\n", GSL_REAL(z), GSL_IMAG(z));
+               }
+           }
+       }
+
+       gsl_vector_complex_free(eval);
+       gsl_matrix_complex_free(evec);
+
+       return 0;
+     }
+
+Here is the beginning of the output from the program,
+
+     $ ./a.out
+     eigenvalue = -6.41391 + 0i
+     eigenvector =
+     -0.0998822 + 0i
+     -0.111251 + 0i
+     0.292501 + 0i
+     0.944505 + 0i
+     eigenvalue = 5.54555 + 3.08545i
+     eigenvector =
+     -0.043487 + -0.0076308i
+     0.0642377 + -0.142127i
+     -0.515253 + 0.0405118i
+     -0.840592 + -0.00148565i
+     ...
+
+This can be compared with the corresponding output from GNU OCTAVE,
+
+     octave> [v,d] = eig(vander([-1 -2 3 4]));
+     octave> diag(d)
+     ans =
+
+       -6.4139 + 0.0000i
+        5.5456 + 3.0854i
+        5.5456 - 3.0854i
+        2.3228 + 0.0000i
+
+     octave> v
+     v =
+
+      Columns 1 through 3:
+
+       -0.09988 + 0.00000i  -0.04350 - 0.00755i  -0.04350 + 0.00755i
+       -0.11125 + 0.00000i   0.06399 - 0.14224i   0.06399 + 0.14224i
+        0.29250 + 0.00000i  -0.51518 + 0.04142i  -0.51518 - 0.04142i
+        0.94451 + 0.00000i  -0.84059 + 0.00000i  -0.84059 - 0.00000i
+
+      Column 4:
+
+       -0.14493 + 0.00000i
+        0.35660 + 0.00000i
+        0.91937 + 0.00000i
+        0.08118 + 0.00000i
+   Note that the eigenvectors corresponding to the eigenvalue 5.54555 +
+3.08545i are slightly different. This is because they differ by the
+multiplicative constant 0.9999984 + 0.0017674i which has magnitude 1.
+
+
+File: gsl-ref.info,  Node: Eigenvalue and Eigenvector References,  Prev: Eigenvalue and Eigenvector Examples,  Up: Eigensystems
+
+14.6 References and Further Reading
+===================================
+
+Further information on the algorithms described in this section can be
+found in the following book,
+
+     G. H. Golub, C. F. Van Loan, `Matrix Computations' (3rd Ed, 1996),
+     Johns Hopkins University Press, ISBN 0-8018-5414-8.
+
+The LAPACK library is described in,
+
+     `LAPACK Users' Guide' (Third Edition, 1999), Published by SIAM,
+     ISBN 0-89871-447-8.
+
+     `http://www.netlib.org/lapack'
+
+The LAPACK source code can be found at the website above along with an
+online copy of the users guide.
+
+
+File: gsl-ref.info,  Node: Fast Fourier Transforms,  Next: Numerical Integration,  Prev: Eigensystems,  Up: Top
+
+15 Fast Fourier Transforms (FFTs)
+*********************************
+
+This chapter describes functions for performing Fast Fourier Transforms
+(FFTs).  The library includes radix-2 routines (for lengths which are a
+power of two) and mixed-radix routines (which work for any length).  For
+efficiency there are separate versions of the routines for real data and
+for complex data.  The mixed-radix routines are a reimplementation of
+the FFTPACK library of Paul Swarztrauber.  Fortran code for FFTPACK is
+available on Netlib (FFTPACK also includes some routines for sine and
+cosine transforms but these are currently not available in GSL).  For
+details and derivations of the underlying algorithms consult the
+document `GSL FFT Algorithms' (*note FFT References and Further
+Reading::)
+
+* Menu:
+
+* Mathematical Definitions::
+* Overview of complex data FFTs::
+* Radix-2 FFT routines for complex data::
+* Mixed-radix FFT routines for complex data::
+* Overview of real data FFTs::
+* Radix-2 FFT routines for real data::
+* Mixed-radix FFT routines for real data::
+* FFT References and Further Reading::
+
+
+File: gsl-ref.info,  Node: Mathematical Definitions,  Next: Overview of complex data FFTs,  Up: Fast Fourier Transforms
+
+15.1 Mathematical Definitions
+=============================
+
+Fast Fourier Transforms are efficient algorithms for calculating the
+discrete fourier transform (DFT),
+
+     x_j = \sum_{k=0}^{N-1} z_k \exp(-2\pi i j k / N)
+
+   The DFT usually arises as an approximation to the continuous fourier
+transform when functions are sampled at discrete intervals in space or
+time.  The naive evaluation of the discrete fourier transform is a
+matrix-vector multiplication W\vec{z}. A general matrix-vector
+multiplication takes O(N^2) operations for N data-points.  Fast fourier
+transform algorithms use a divide-and-conquer strategy to factorize the
+matrix W into smaller sub-matrices, corresponding to the integer
+factors of the length N.  If N can be factorized into a product of
+integers f_1 f_2 ... f_n then the DFT can be computed in O(N \sum f_i)
+operations.  For a radix-2 FFT this gives an operation count of O(N
+\log_2 N).
+
+   All the FFT functions offer three types of transform: forwards,
+inverse and backwards, based on the same mathematical definitions.  The
+definition of the "forward fourier transform", x = FFT(z), is,
+
+     x_j = \sum_{k=0}^{N-1} z_k \exp(-2\pi i j k / N)
+
+and the definition of the "inverse fourier transform", x = IFFT(z), is,
+
+     z_j = {1 \over N} \sum_{k=0}^{N-1} x_k \exp(2\pi i j k / N).
+
+The factor of 1/N makes this a true inverse.  For example, a call to
+`gsl_fft_complex_forward' followed by a call to
+`gsl_fft_complex_inverse' should return the original data (within
+numerical errors).
+
+   In general there are two possible choices for the sign of the
+exponential in the transform/ inverse-transform pair. GSL follows the
+same convention as FFTPACK, using a negative exponential for the forward
+transform.  The advantage of this convention is that the inverse
+transform recreates the original function with simple fourier
+synthesis.  Numerical Recipes uses the opposite convention, a positive
+exponential in the forward transform.
+
+   The "backwards FFT" is simply our terminology for an unscaled
+version of the inverse FFT,
+
+     z^{backwards}_j = \sum_{k=0}^{N-1} x_k \exp(2\pi i j k / N).
+
+When the overall scale of the result is unimportant it is often
+convenient to use the backwards FFT instead of the inverse to save
+unnecessary divisions.
+
+
+File: gsl-ref.info,  Node: Overview of complex data FFTs,  Next: Radix-2 FFT routines for complex data,  Prev: Mathematical Definitions,  Up: Fast Fourier Transforms
+
+15.2 Overview of complex data FFTs
+==================================
+
+The inputs and outputs for the complex FFT routines are "packed arrays"
+of floating point numbers.  In a packed array the real and imaginary
+parts of each complex number are placed in alternate neighboring
+elements.  For example, the following definition of a packed array of
+length 6,
+
+     double x[3*2];
+     gsl_complex_packed_array data = x;
+
+can be used to hold an array of three complex numbers, `z[3]', in the
+following way,
+
+     data[0] = Re(z[0])
+     data[1] = Im(z[0])
+     data[2] = Re(z[1])
+     data[3] = Im(z[1])
+     data[4] = Re(z[2])
+     data[5] = Im(z[2])
+
+The array indices for the data have the same ordering as those in the
+definition of the DFT--i.e. there are no index transformations or
+permutations of the data.
+
+   A "stride" parameter allows the user to perform transforms on the
+elements `z[stride*i]' instead of `z[i]'.  A stride greater than 1 can
+be used to take an in-place FFT of the column of a matrix. A stride of
+1 accesses the array without any additional spacing between elements.
+
+   To perform an FFT on a vector argument, such as `gsl_vector_complex
+* v', use the following definitions (or their equivalents) when calling
+the functions described in this chapter:
+
+     gsl_complex_packed_array data = v->data;
+     size_t stride = v->stride;
+     size_t n = v->size;
+
+   For physical applications it is important to remember that the index
+appearing in the DFT does not correspond directly to a physical
+frequency.  If the time-step of the DFT is \Delta then the
+frequency-domain includes both positive and negative frequencies,
+ranging from -1/(2\Delta) through 0 to +1/(2\Delta).  The positive
+frequencies are stored from the beginning of the array up to the
+middle, and the negative frequencies are stored backwards from the end
+of the array.
+
+   Here is a table which shows the layout of the array DATA, and the
+correspondence between the time-domain data z, and the frequency-domain
+data x.
+
+     index    z               x = FFT(z)
+
+     0        z(t = 0)        x(f = 0)
+     1        z(t = 1)        x(f = 1/(N Delta))
+     2        z(t = 2)        x(f = 2/(N Delta))
+     .        ........        ..................
+     N/2      z(t = N/2)      x(f = +1/(2 Delta),
+                                    -1/(2 Delta))
+     .        ........        ..................
+     N-3      z(t = N-3)      x(f = -3/(N Delta))
+     N-2      z(t = N-2)      x(f = -2/(N Delta))
+     N-1      z(t = N-1)      x(f = -1/(N Delta))
+
+When N is even the location N/2 contains the most positive and negative
+frequencies (+1/(2 \Delta), -1/(2 \Delta)) which are equivalent.  If N
+is odd then general structure of the table above still applies, but N/2
+does not appear.
+
+
+File: gsl-ref.info,  Node: Radix-2 FFT routines for complex data,  Next: Mixed-radix FFT routines for complex data,  Prev: Overview of complex data FFTs,  Up: Fast Fourier Transforms
+
+15.3 Radix-2 FFT routines for complex data
+==========================================
+
+The radix-2 algorithms described in this section are simple and compact,
+although not necessarily the most efficient.  They use the Cooley-Tukey
+algorithm to compute in-place complex FFTs for lengths which are a power
+of 2--no additional storage is required.  The corresponding
+self-sorting mixed-radix routines offer better performance at the
+expense of requiring additional working space.
+
+   All the functions described in this section are declared in the
+header file `gsl_fft_complex.h'.
+
+ -- Function: int gsl_fft_complex_radix2_forward
+          (gsl_complex_packed_array DATA, size_t STRIDE, size_t N)
+ -- Function: int gsl_fft_complex_radix2_transform
+          (gsl_complex_packed_array DATA, size_t STRIDE, size_t N,
+          gsl_fft_direction SIGN)
+ -- Function: int gsl_fft_complex_radix2_backward
+          (gsl_complex_packed_array DATA, size_t STRIDE, size_t N)
+ -- Function: int gsl_fft_complex_radix2_inverse
+          (gsl_complex_packed_array DATA, size_t STRIDE, size_t N)
+     These functions compute forward, backward and inverse FFTs of
+     length N with stride STRIDE, on the packed complex array DATA
+     using an in-place radix-2 decimation-in-time algorithm.  The
+     length of the transform N is restricted to powers of two.  For the
+     `transform' version of the function the SIGN argument can be
+     either `forward' (-1) or `backward' (+1).
+
+     The functions return a value of `GSL_SUCCESS' if no errors were
+     detected, or `GSL_EDOM' if the length of the data N is not a power
+     of two.
+
+ -- Function: int gsl_fft_complex_radix2_dif_forward
+          (gsl_complex_packed_array DATA, size_t STRIDE, size_t N)
+ -- Function: int gsl_fft_complex_radix2_dif_transform
+          (gsl_complex_packed_array DATA, size_t STRIDE, size_t N,
+          gsl_fft_direction SIGN)
+ -- Function: int gsl_fft_complex_radix2_dif_backward
+          (gsl_complex_packed_array DATA, size_t STRIDE, size_t N)
+ -- Function: int gsl_fft_complex_radix2_dif_inverse
+          (gsl_complex_packed_array DATA, size_t STRIDE, size_t N)
+     These are decimation-in-frequency versions of the radix-2 FFT
+     functions.
+
+
+   Here is an example program which computes the FFT of a short pulse
+in a sample of length 128.  To make the resulting fourier transform
+real the pulse is defined for equal positive and negative times (-10
+... 10), where the negative times wrap around the end of the array.
+
+     #include <stdio.h>
+     #include <math.h>
+     #include <gsl/gsl_errno.h>
+     #include <gsl/gsl_fft_complex.h>
+
+     #define REAL(z,i) ((z)[2*(i)])
+     #define IMAG(z,i) ((z)[2*(i)+1])
+
+     int
+     main (void)
+     {
+       int i; double data[2*128];
+
+       for (i = 0; i < 128; i++)
+         {
+            REAL(data,i) = 0.0; IMAG(data,i) = 0.0;
+         }
+
+       REAL(data,0) = 1.0;
+
+       for (i = 1; i <= 10; i++)
+         {
+            REAL(data,i) = REAL(data,128-i) = 1.0;
+         }
+
+       for (i = 0; i < 128; i++)
+         {
+           printf ("%d %e %e\n", i,
+                   REAL(data,i), IMAG(data,i));
+         }
+       printf ("\n");
+
+       gsl_fft_complex_radix2_forward (data, 1, 128);
+
+       for (i = 0; i < 128; i++)
+         {
+           printf ("%d %e %e\n", i,
+                   REAL(data,i)/sqrt(128),
+                   IMAG(data,i)/sqrt(128));
+         }
+
+       return 0;
+     }
+
+Note that we have assumed that the program is using the default error
+handler (which calls `abort' for any errors).  If you are not using a
+safe error handler you would need to check the return status of
+`gsl_fft_complex_radix2_forward'.
+
+   The transformed data is rescaled by 1/\sqrt N so that it fits on the
+same plot as the input.  Only the real part is shown, by the choice of
+the input data the imaginary part is zero.  Allowing for the
+wrap-around of negative times at t=128, and working in units of k/N,
+the DFT approximates the continuum fourier transform, giving a
+modulated sine function.
+
+
+File: gsl-ref.info,  Node: Mixed-radix FFT routines for complex data,  Next: Overview of real data FFTs,  Prev: Radix-2 FFT routines for complex data,  Up: Fast Fourier Transforms
+
+15.4 Mixed-radix FFT routines for complex data
+==============================================
+
+This section describes mixed-radix FFT algorithms for complex data.  The
+mixed-radix functions work for FFTs of any length.  They are a
+reimplementation of Paul Swarztrauber's Fortran FFTPACK library.  The
+theory is explained in the review article `Self-sorting Mixed-radix
+FFTs' by Clive Temperton.  The routines here use the same indexing
+scheme and basic algorithms as FFTPACK.
+
+   The mixed-radix algorithm is based on sub-transform modules--highly
+optimized small length FFTs which are combined to create larger FFTs.
+There are efficient modules for factors of 2, 3, 4, 5, 6 and 7.  The
+modules for the composite factors of 4 and 6 are faster than combining
+the modules for 2*2 and 2*3.
+
+   For factors which are not implemented as modules there is a
+fall-back to a general length-n module which uses Singleton's method for
+efficiently computing a DFT. This module is O(n^2), and slower than a
+dedicated module would be but works for any length n.  Of course,
+lengths which use the general length-n module will still be factorized
+as much as possible.  For example, a length of 143 will be factorized
+into 11*13.  Large prime factors are the worst case scenario, e.g. as
+found in n=2*3*99991, and should be avoided because their O(n^2)
+scaling will dominate the run-time (consult the document `GSL FFT
+Algorithms' included in the GSL distribution if you encounter this
+problem).
+
+   The mixed-radix initialization function
+`gsl_fft_complex_wavetable_alloc' returns the list of factors chosen by
+the library for a given length N.  It can be used to check how well the
+length has been factorized, and estimate the run-time.  To a first
+approximation the run-time scales as N \sum f_i, where the f_i are the
+factors of N.  For programs under user control you may wish to issue a
+warning that the transform will be slow when the length is poorly
+factorized.  If you frequently encounter data lengths which cannot be
+factorized using the existing small-prime modules consult `GSL FFT
+Algorithms' for details on adding support for other factors.
+
+   All the functions described in this section are declared in the
+header file `gsl_fft_complex.h'.
+
+ -- Function: gsl_fft_complex_wavetable *
+gsl_fft_complex_wavetable_alloc (size_t N)
+     This function prepares a trigonometric lookup table for a complex
+     FFT of length N. The function returns a pointer to the newly
+     allocated `gsl_fft_complex_wavetable' if no errors were detected,
+     and a null pointer in the case of error.  The length N is
+     factorized into a product of subtransforms, and the factors and
+     their trigonometric coefficients are stored in the wavetable. The
+     trigonometric coefficients are computed using direct calls to
+     `sin' and `cos', for accuracy.  Recursion relations could be used
+     to compute the lookup table faster, but if an application performs
+     many FFTs of the same length then this computation is a one-off
+     overhead which does not affect the final throughput.
+
+     The wavetable structure can be used repeatedly for any transform
+     of the same length.  The table is not modified by calls to any of
+     the other FFT functions.  The same wavetable can be used for both
+     forward and backward (or inverse) transforms of a given length.
+
+ -- Function: void gsl_fft_complex_wavetable_free
+          (gsl_fft_complex_wavetable * WAVETABLE)
+     This function frees the memory associated with the wavetable
+     WAVETABLE.  The wavetable can be freed if no further FFTs of the
+     same length will be needed.
+
+These functions operate on a `gsl_fft_complex_wavetable' structure
+which contains internal parameters for the FFT.  It is not necessary to
+set any of the components directly but it can sometimes be useful to
+examine them.  For example, the chosen factorization of the FFT length
+is given and can be used to provide an estimate of the run-time or
+numerical error. The wavetable structure is declared in the header file
+`gsl_fft_complex.h'.
+
+ -- Data Type: gsl_fft_complex_wavetable
+     This is a structure that holds the factorization and trigonometric
+     lookup tables for the mixed radix fft algorithm.  It has the
+     following components:
+
+    `size_t n'
+          This is the number of complex data points
+
+    `size_t nf'
+          This is the number of factors that the length `n' was
+          decomposed into.
+
+    `size_t factor[64]'
+          This is the array of factors.  Only the first `nf' elements
+          are used.
+
+    `gsl_complex * trig'
+          This is a pointer to a preallocated trigonometric lookup
+          table of `n' complex elements.
+
+    `gsl_complex * twiddle[64]'
+          This is an array of pointers into `trig', giving the twiddle
+          factors for each pass.
+
+The mixed radix algorithms require additional working space to hold the
+intermediate steps of the transform.
+
+ -- Function: gsl_fft_complex_workspace *
+gsl_fft_complex_workspace_alloc (size_t N)
+     This function allocates a workspace for a complex transform of
+     length N.
+
+ -- Function: void gsl_fft_complex_workspace_free
+          (gsl_fft_complex_workspace * WORKSPACE)
+     This function frees the memory associated with the workspace
+     WORKSPACE. The workspace can be freed if no further FFTs of the
+     same length will be needed.
+
+The following functions compute the transform,
+
+ -- Function: int gsl_fft_complex_forward (gsl_complex_packed_array
+          DATA, size_t STRIDE, size_t N, const
+          gsl_fft_complex_wavetable * WAVETABLE,
+          gsl_fft_complex_workspace * WORK)
+ -- Function: int gsl_fft_complex_transform (gsl_complex_packed_array
+          DATA, size_t STRIDE, size_t N, const
+          gsl_fft_complex_wavetable * WAVETABLE,
+          gsl_fft_complex_workspace * WORK, gsl_fft_direction SIGN)
+ -- Function: int gsl_fft_complex_backward (gsl_complex_packed_array
+          DATA, size_t STRIDE, size_t N, const
+          gsl_fft_complex_wavetable * WAVETABLE,
+          gsl_fft_complex_workspace * WORK)
+ -- Function: int gsl_fft_complex_inverse (gsl_complex_packed_array
+          DATA, size_t STRIDE, size_t N, const
+          gsl_fft_complex_wavetable * WAVETABLE,
+          gsl_fft_complex_workspace * WORK)
+     These functions compute forward, backward and inverse FFTs of
+     length N with stride STRIDE, on the packed complex array DATA,
+     using a mixed radix decimation-in-frequency algorithm.  There is
+     no restriction on the length N.  Efficient modules are provided
+     for subtransforms of length 2, 3, 4, 5, 6 and 7.  Any remaining
+     factors are computed with a slow, O(n^2), general-n module. The
+     caller must supply a WAVETABLE containing the trigonometric lookup
+     tables and a workspace WORK.  For the `transform' version of the
+     function the SIGN argument can be either `forward' (-1) or
+     `backward' (+1).
+
+     The functions return a value of `0' if no errors were detected. The
+     following `gsl_errno' conditions are defined for these functions:
+
+    `GSL_EDOM'
+          The length of the data N is not a positive integer (i.e. N is
+          zero).
+
+    `GSL_EINVAL'
+          The length of the data N and the length used to compute the
+          given WAVETABLE do not match.
+
+   Here is an example program which computes the FFT of a short pulse
+in a sample of length 630 (=2*3*3*5*7) using the mixed-radix algorithm.
+
+     #include <stdio.h>
+     #include <math.h>
+     #include <gsl/gsl_errno.h>
+     #include <gsl/gsl_fft_complex.h>
+
+     #define REAL(z,i) ((z)[2*(i)])
+     #define IMAG(z,i) ((z)[2*(i)+1])
+
+     int
+     main (void)
+     {
+       int i;
+       const int n = 630;
+       double data[2*n];
+
+       gsl_fft_complex_wavetable * wavetable;
+       gsl_fft_complex_workspace * workspace;
+
+       for (i = 0; i < n; i++)
+         {
+           REAL(data,i) = 0.0;
+           IMAG(data,i) = 0.0;
+         }
+
+       data[0] = 1.0;
+
+       for (i = 1; i <= 10; i++)
+         {
+           REAL(data,i) = REAL(data,n-i) = 1.0;
+         }
+
+       for (i = 0; i < n; i++)
+         {
+           printf ("%d: %e %e\n", i, REAL(data,i),
+                                     IMAG(data,i));
+         }
+       printf ("\n");
+
+       wavetable = gsl_fft_complex_wavetable_alloc (n);
+       workspace = gsl_fft_complex_workspace_alloc (n);
+
+       for (i = 0; i < wavetable->nf; i++)
+         {
+            printf ("# factor %d: %d\n", i,
+                    wavetable->factor[i]);
+         }
+
+       gsl_fft_complex_forward (data, 1, n,
+                                wavetable, workspace);
+
+       for (i = 0; i < n; i++)
+         {
+           printf ("%d: %e %e\n", i, REAL(data,i),
+                                     IMAG(data,i));
+         }
+
+       gsl_fft_complex_wavetable_free (wavetable);
+       gsl_fft_complex_workspace_free (workspace);
+       return 0;
+     }
+
+Note that we have assumed that the program is using the default `gsl'
+error handler (which calls `abort' for any errors).  If you are not
+using a safe error handler you would need to check the return status of
+all the `gsl' routines.
+
+
+File: gsl-ref.info,  Node: Overview of real data FFTs,  Next: Radix-2 FFT routines for real data,  Prev: Mixed-radix FFT routines for complex data,  Up: Fast Fourier Transforms
+
+15.5 Overview of real data FFTs
+===============================
+
+The functions for real data are similar to those for complex data.
+However, there is an important difference between forward and inverse
+transforms.  The fourier transform of a real sequence is not real.  It
+is a complex sequence with a special symmetry:
+
+     z_k = z_{N-k}^*
+
+A sequence with this symmetry is called "conjugate-complex" or
+"half-complex".  This different structure requires different storage
+layouts for the forward transform (from real to half-complex) and
+inverse transform (from half-complex back to real).  As a consequence
+the routines are divided into two sets: functions in `gsl_fft_real'
+which operate on real sequences and functions in `gsl_fft_halfcomplex'
+which operate on half-complex sequences.
+
+   Functions in `gsl_fft_real' compute the frequency coefficients of a
+real sequence.  The half-complex coefficients c of a real sequence x
+are given by fourier analysis,
+
+     c_k = \sum_{j=0}^{N-1} x_j \exp(-2 \pi i j k /N)
+
+Functions in `gsl_fft_halfcomplex' compute inverse or backwards
+transforms.  They reconstruct real sequences by fourier synthesis from
+their half-complex frequency coefficients, c,
+
+     x_j = {1 \over N} \sum_{k=0}^{N-1} c_k \exp(2 \pi i j k /N)
+
+The symmetry of the half-complex sequence implies that only half of the
+complex numbers in the output need to be stored.  The remaining half can
+be reconstructed using the half-complex symmetry condition. This works
+for all lengths, even and odd--when the length is even the middle value
+where k=N/2 is also real.  Thus only N real numbers are required to
+store the half-complex sequence, and the transform of a real sequence
+can be stored in the same size array as the original data.
+
+   The precise storage arrangements depend on the algorithm, and are
+different for radix-2 and mixed-radix routines.  The radix-2 function
+operates in-place, which constrains the locations where each element can
+be stored.  The restriction forces real and imaginary parts to be stored
+far apart.  The mixed-radix algorithm does not have this restriction,
+and it stores the real and imaginary parts of a given term in
+neighboring locations (which is desirable for better locality of memory
+accesses).
+
+
+File: gsl-ref.info,  Node: Radix-2 FFT routines for real data,  Next: Mixed-radix FFT routines for real data,  Prev: Overview of real data FFTs,  Up: Fast Fourier Transforms
+
+15.6 Radix-2 FFT routines for real data
+=======================================
+
+This section describes radix-2 FFT algorithms for real data.  They use
+the Cooley-Tukey algorithm to compute in-place FFTs for lengths which
+are a power of 2.
+
+   The radix-2 FFT functions for real data are declared in the header
+files `gsl_fft_real.h'
+
+ -- Function: int gsl_fft_real_radix2_transform (double DATA[], size_t
+          STRIDE, size_t N)
+     This function computes an in-place radix-2 FFT of length N and
+     stride STRIDE on the real array DATA.  The output is a
+     half-complex sequence, which is stored in-place.  The arrangement
+     of the half-complex terms uses the following scheme: for k < N/2
+     the real part of the k-th term is stored in location k, and the
+     corresponding imaginary part is stored in location N-k.  Terms
+     with k > N/2 can be reconstructed using the symmetry z_k =
+     z^*_{N-k}.  The terms for k=0 and k=N/2 are both purely real, and
+     count as a special case.  Their real parts are stored in locations
+     0 and N/2 respectively, while their imaginary parts which are zero
+     are not stored.
+
+     The following table shows the correspondence between the output
+     DATA and the equivalent results obtained by considering the input
+     data as a complex sequence with zero imaginary part,
+
+          complex[0].real    =    data[0]
+          complex[0].imag    =    0
+          complex[1].real    =    data[1]
+          complex[1].imag    =    data[N-1]
+          ...............         ................
+          complex[k].real    =    data[k]
+          complex[k].imag    =    data[N-k]
+          ...............         ................
+          complex[N/2].real  =    data[N/2]
+          complex[N/2].imag  =    0
+          ...............         ................
+          complex[k'].real   =    data[k]        k' = N - k
+          complex[k'].imag   =   -data[N-k]
+          ...............         ................
+          complex[N-1].real  =    data[1]
+          complex[N-1].imag  =   -data[N-1]
+     Note that the output data can be converted into the full complex
+     sequence using the function `gsl_fft_halfcomplex_unpack' described
+     in the next section.
+
+   The radix-2 FFT functions for halfcomplex data are declared in the
+header file `gsl_fft_halfcomplex.h'.
+
+ -- Function: int gsl_fft_halfcomplex_radix2_inverse (double DATA[],
+          size_t STRIDE, size_t N)
+ -- Function: int gsl_fft_halfcomplex_radix2_backward (double DATA[],
+          size_t STRIDE, size_t N)
+     These functions compute the inverse or backwards in-place radix-2
+     FFT of length N and stride STRIDE on the half-complex sequence
+     DATA stored according the output scheme used by
+     `gsl_fft_real_radix2'.  The result is a real array stored in
+     natural order.
+
+
+
+File: gsl-ref.info,  Node: Mixed-radix FFT routines for real data,  Next: FFT References and Further Reading,  Prev: Radix-2 FFT routines for real data,  Up: Fast Fourier Transforms
+
+15.7 Mixed-radix FFT routines for real data
+===========================================
+
+This section describes mixed-radix FFT algorithms for real data.  The
+mixed-radix functions work for FFTs of any length.  They are a
+reimplementation of the real-FFT routines in the Fortran FFTPACK library
+by Paul Swarztrauber.  The theory behind the algorithm is explained in
+the article `Fast Mixed-Radix Real Fourier Transforms' by Clive
+Temperton.  The routines here use the same indexing scheme and basic
+algorithms as FFTPACK.
+
+   The functions use the FFTPACK storage convention for half-complex
+sequences.  In this convention the half-complex transform of a real
+sequence is stored with frequencies in increasing order, starting at
+zero, with the real and imaginary parts of each frequency in neighboring
+locations.  When a value is known to be real the imaginary part is not
+stored.  The imaginary part of the zero-frequency component is never
+stored.  It is known to be zero (since the zero frequency component is
+simply the sum of the input data (all real)).  For a sequence of even
+length the imaginary part of the frequency n/2 is not stored either,
+since the symmetry z_k = z_{N-k}^* implies that this is purely real too.
+
+   The storage scheme is best shown by some examples.  The table below
+shows the output for an odd-length sequence, n=5.  The two columns give
+the correspondence between the 5 values in the half-complex sequence
+returned by `gsl_fft_real_transform', HALFCOMPLEX[] and the values
+COMPLEX[] that would be returned if the same real input sequence were
+passed to `gsl_fft_complex_backward' as a complex sequence (with
+imaginary parts set to `0'),
+
+     complex[0].real  =  halfcomplex[0]
+     complex[0].imag  =  0
+     complex[1].real  =  halfcomplex[1]
+     complex[1].imag  =  halfcomplex[2]
+     complex[2].real  =  halfcomplex[3]
+     complex[2].imag  =  halfcomplex[4]
+     complex[3].real  =  halfcomplex[3]
+     complex[3].imag  = -halfcomplex[4]
+     complex[4].real  =  halfcomplex[1]
+     complex[4].imag  = -halfcomplex[2]
+
+The upper elements of the COMPLEX array, `complex[3]' and `complex[4]'
+are filled in using the symmetry condition.  The imaginary part of the
+zero-frequency term `complex[0].imag' is known to be zero by the
+symmetry.
+
+   The next table shows the output for an even-length sequence, n=6 In
+the even case there are two values which are purely real,
+
+     complex[0].real  =  halfcomplex[0]
+     complex[0].imag  =  0
+     complex[1].real  =  halfcomplex[1]
+     complex[1].imag  =  halfcomplex[2]
+     complex[2].real  =  halfcomplex[3]
+     complex[2].imag  =  halfcomplex[4]
+     complex[3].real  =  halfcomplex[5]
+     complex[3].imag  =  0
+     complex[4].real  =  halfcomplex[3]
+     complex[4].imag  = -halfcomplex[4]
+     complex[5].real  =  halfcomplex[1]
+     complex[5].imag  = -halfcomplex[2]
+
+The upper elements of the COMPLEX array, `complex[4]' and `complex[5]'
+are filled in using the symmetry condition.  Both `complex[0].imag' and
+`complex[3].imag' are known to be zero.
+
+   All these functions are declared in the header files
+`gsl_fft_real.h' and `gsl_fft_halfcomplex.h'.
+
+ -- Function: gsl_fft_real_wavetable * gsl_fft_real_wavetable_alloc
+          (size_t N)
+ -- Function: gsl_fft_halfcomplex_wavetable *
+gsl_fft_halfcomplex_wavetable_alloc (size_t N)
+     These functions prepare trigonometric lookup tables for an FFT of
+     size n real elements.  The functions return a pointer to the newly
+     allocated struct if no errors were detected, and a null pointer in
+     the case of error.  The length N is factorized into a product of
+     subtransforms, and the factors and their trigonometric
+     coefficients are stored in the wavetable. The trigonometric
+     coefficients are computed using direct calls to `sin' and `cos',
+     for accuracy.  Recursion relations could be used to compute the
+     lookup table faster, but if an application performs many FFTs of
+     the same length then computing the wavetable is a one-off overhead
+     which does not affect the final throughput.
+
+     The wavetable structure can be used repeatedly for any transform
+     of the same length.  The table is not modified by calls to any of
+     the other FFT functions.  The appropriate type of wavetable must
+     be used for forward real or inverse half-complex transforms.
+
+ -- Function: void gsl_fft_real_wavetable_free (gsl_fft_real_wavetable
+          * WAVETABLE)
+ -- Function: void gsl_fft_halfcomplex_wavetable_free
+          (gsl_fft_halfcomplex_wavetable * WAVETABLE)
+     These functions free the memory associated with the wavetable
+     WAVETABLE. The wavetable can be freed if no further FFTs of the
+     same length will be needed.
+
+The mixed radix algorithms require additional working space to hold the
+intermediate steps of the transform,
+
+ -- Function: gsl_fft_real_workspace * gsl_fft_real_workspace_alloc
+          (size_t N)
+     This function allocates a workspace for a real transform of length
+     N.  The same workspace can be used for both forward real and
+     inverse halfcomplex transforms.
+
+ -- Function: void gsl_fft_real_workspace_free (gsl_fft_real_workspace
+          * WORKSPACE)
+     This function frees the memory associated with the workspace
+     WORKSPACE. The workspace can be freed if no further FFTs of the
+     same length will be needed.
+
+The following functions compute the transforms of real and half-complex
+data,
+
+ -- Function: int gsl_fft_real_transform (double DATA[], size_t STRIDE,
+          size_t N, const gsl_fft_real_wavetable * WAVETABLE,
+          gsl_fft_real_workspace * WORK)
+ -- Function: int gsl_fft_halfcomplex_transform (double DATA[], size_t
+          STRIDE, size_t N, const gsl_fft_halfcomplex_wavetable *
+          WAVETABLE, gsl_fft_real_workspace * WORK)
+     These functions compute the FFT of DATA, a real or half-complex
+     array of length N, using a mixed radix decimation-in-frequency
+     algorithm.  For `gsl_fft_real_transform' DATA is an array of
+     time-ordered real data.  For `gsl_fft_halfcomplex_transform' DATA
+     contains fourier coefficients in the half-complex ordering
+     described above.  There is no restriction on the length N.
+     Efficient modules are provided for subtransforms of length 2, 3, 4
+     and 5.  Any remaining factors are computed with a slow, O(n^2),
+     general-n module.  The caller must supply a WAVETABLE containing
+     trigonometric lookup tables and a workspace WORK.
+
+ -- Function: int gsl_fft_real_unpack (const double REAL_COEFFICIENT[],
+          gsl_complex_packed_array COMPLEX_COEFFICIENT, size_t STRIDE,
+          size_t N)
+     This function converts a single real array, REAL_COEFFICIENT into
+     an equivalent complex array, COMPLEX_COEFFICIENT, (with imaginary
+     part set to zero), suitable for `gsl_fft_complex' routines.  The
+     algorithm for the conversion is simply,
+
+          for (i = 0; i < n; i++)
+            {
+              complex_coefficient[i].real
+                = real_coefficient[i];
+              complex_coefficient[i].imag
+                = 0.0;
+            }
+
+ -- Function: int gsl_fft_halfcomplex_unpack (const double
+          HALFCOMPLEX_COEFFICIENT[], gsl_complex_packed_array
+          COMPLEX_COEFFICIENT, size_t STRIDE, size_t N)
+     This function converts HALFCOMPLEX_COEFFICIENT, an array of
+     half-complex coefficients as returned by `gsl_fft_real_transform',
+     into an ordinary complex array, COMPLEX_COEFFICIENT.  It fills in
+     the complex array using the symmetry z_k = z_{N-k}^* to
+     reconstruct the redundant elements.  The algorithm for the
+     conversion is,
+
+          complex_coefficient[0].real
+            = halfcomplex_coefficient[0];
+          complex_coefficient[0].imag
+            = 0.0;
+
+          for (i = 1; i < n - i; i++)
+            {
+              double hc_real
+                = halfcomplex_coefficient[2 * i - 1];
+              double hc_imag
+                = halfcomplex_coefficient[2 * i];
+              complex_coefficient[i].real = hc_real;
+              complex_coefficient[i].imag = hc_imag;
+              complex_coefficient[n - i].real = hc_real;
+              complex_coefficient[n - i].imag = -hc_imag;
+            }
+
+          if (i == n - i)
+            {
+              complex_coefficient[i].real
+                = halfcomplex_coefficient[n - 1];
+              complex_coefficient[i].imag
+                = 0.0;
+            }
+
+   Here is an example program using `gsl_fft_real_transform' and
+`gsl_fft_halfcomplex_inverse'.  It generates a real signal in the shape
+of a square pulse.  The pulse is fourier transformed to frequency
+space, and all but the lowest ten frequency components are removed from
+the array of fourier coefficients returned by `gsl_fft_real_transform'.
+
+   The remaining fourier coefficients are transformed back to the
+time-domain, to give a filtered version of the square pulse.  Since
+fourier coefficients are stored using the half-complex symmetry both
+positive and negative frequencies are removed and the final filtered
+signal is also real.
+
+     #include <stdio.h>
+     #include <math.h>
+     #include <gsl/gsl_errno.h>
+     #include <gsl/gsl_fft_real.h>
+     #include <gsl/gsl_fft_halfcomplex.h>
+
+     int
+     main (void)
+     {
+       int i, n = 100;
+       double data[n];
+
+       gsl_fft_real_wavetable * real;
+       gsl_fft_halfcomplex_wavetable * hc;
+       gsl_fft_real_workspace * work;
+
+       for (i = 0; i < n; i++)
+         {
+           data[i] = 0.0;
+         }
+
+       for (i = n / 3; i < 2 * n / 3; i++)
+         {
+           data[i] = 1.0;
+         }
+
+       for (i = 0; i < n; i++)
+         {
+           printf ("%d: %e\n", i, data[i]);
+         }
+       printf ("\n");
+
+       work = gsl_fft_real_workspace_alloc (n);
+       real = gsl_fft_real_wavetable_alloc (n);
+
+       gsl_fft_real_transform (data, 1, n,
+                               real, work);
+
+       gsl_fft_real_wavetable_free (real);
+
+       for (i = 11; i < n; i++)
+         {
+           data[i] = 0;
+         }
+
+       hc = gsl_fft_halfcomplex_wavetable_alloc (n);
+
+       gsl_fft_halfcomplex_inverse (data, 1, n,
+                                    hc, work);
+       gsl_fft_halfcomplex_wavetable_free (hc);
+
+       for (i = 0; i < n; i++)
+         {
+           printf ("%d: %e\n", i, data[i]);
+         }
+
+       gsl_fft_real_workspace_free (work);
+       return 0;
+     }
+
+
+File: gsl-ref.info,  Node: FFT References and Further Reading,  Prev: Mixed-radix FFT routines for real data,  Up: Fast Fourier Transforms
+
+15.8 References and Further Reading
+===================================
+
+A good starting point for learning more about the FFT is the review
+article `Fast Fourier Transforms: A Tutorial Review and A State of the
+Art' by Duhamel and Vetterli,
+
+     P. Duhamel and M. Vetterli.  Fast fourier transforms: A tutorial
+     review and a state of the art.  `Signal Processing', 19:259-299,
+     1990.
+
+To find out about the algorithms used in the GSL routines you may want
+to consult the document `GSL FFT Algorithms' (it is included in GSL, as
+`doc/fftalgorithms.tex').  This has general information on FFTs and
+explicit derivations of the implementation for each routine.  There are
+also references to the relevant literature.  For convenience some of
+the more important references are reproduced below.
+
+There are several introductory books on the FFT with example programs,
+such as `The Fast Fourier Transform' by Brigham and `DFT/FFT and
+Convolution Algorithms' by Burrus and Parks,
+
+     E. Oran Brigham.  `The Fast Fourier Transform'.  Prentice Hall,
+     1974.
+
+     C. S. Burrus and T. W. Parks.  `DFT/FFT and Convolution
+     Algorithms'.  Wiley, 1984.
+
+Both these introductory books cover the radix-2 FFT in some detail.
+The mixed-radix algorithm at the heart of the FFTPACK routines is
+reviewed in Clive Temperton's paper,
+
+     Clive Temperton.  Self-sorting mixed-radix fast fourier transforms.
+     `Journal of Computational Physics', 52(1):1-23, 1983.
+
+The derivation of FFTs for real-valued data is explained in the
+following two articles,
+
+     Henrik V. Sorenson, Douglas L. Jones, Michael T. Heideman, and C.
+     Sidney Burrus.  Real-valued fast fourier transform algorithms.
+     `IEEE Transactions on Acoustics, Speech, and Signal Processing',
+     ASSP-35(6):849-863, 1987.
+
+     Clive Temperton.  Fast mixed-radix real fourier transforms.
+     `Journal of Computational Physics', 52:340-350, 1983.
+
+In 1979 the IEEE published a compendium of carefully-reviewed Fortran
+FFT programs in `Programs for Digital Signal Processing'.  It is a
+useful reference for implementations of many different FFT algorithms,
+
+     Digital Signal Processing Committee and IEEE Acoustics, Speech,
+     and Signal Processing Committee, editors.  `Programs for Digital
+     Signal Processing'.  IEEE Press, 1979.
+
+For large-scale FFT work we recommend the use of the dedicated FFTW
+library by Frigo and Johnson.  The FFTW library is self-optimizing--it
+automatically tunes itself for each hardware platform in order to
+achieve maximum performance.  It is available under the GNU GPL.
+
+     FFTW Website, `http://www.fftw.org/'
+
+The source code for FFTPACK is available from Netlib,
+
+     FFTPACK, `http://www.netlib.org/fftpack/'
+
+
+File: gsl-ref.info,  Node: Numerical Integration,  Next: Random Number Generation,  Prev: Fast Fourier Transforms,  Up: Top
+
+16 Numerical Integration
+************************
+
+This chapter describes routines for performing numerical integration
+(quadrature) of a function in one dimension.  There are routines for
+adaptive and non-adaptive integration of general functions, with
+specialised routines for specific cases.  These include integration over
+infinite and semi-infinite ranges, singular integrals, including
+logarithmic singularities, computation of Cauchy principal values and
+oscillatory integrals.  The library reimplements the algorithms used in
+QUADPACK, a numerical integration package written by Piessens,
+Doncker-Kapenga, Uberhuber and Kahaner.  Fortran code for QUADPACK is
+available on Netlib.
+
+   The functions described in this chapter are declared in the header
+file `gsl_integration.h'.
+
+* Menu:
+
+* Numerical Integration Introduction::
+* QNG non-adaptive Gauss-Kronrod integration::
+* QAG adaptive integration::
+* QAGS adaptive integration with singularities::
+* QAGP adaptive integration with known singular points::
+* QAGI adaptive integration on infinite intervals::
+* QAWC adaptive integration for Cauchy principal values::
+* QAWS adaptive integration for singular functions::
+* QAWO adaptive integration for oscillatory functions::
+* QAWF adaptive integration for Fourier integrals::
+* Numerical integration error codes::
+* Numerical integration examples::
+* Numerical integration References and Further Reading::
+
+
+File: gsl-ref.info,  Node: Numerical Integration Introduction,  Next: QNG non-adaptive Gauss-Kronrod integration,  Up: Numerical Integration
+
+16.1 Introduction
+=================
+
+Each algorithm computes an approximation to a definite integral of the
+form,
+
+     I = \int_a^b f(x) w(x) dx
+
+where w(x) is a weight function (for general integrands w(x)=1).  The
+user provides absolute and relative error bounds (epsabs, epsrel) which
+specify the following accuracy requirement,
+
+     |RESULT - I|  <= max(epsabs, epsrel |I|)
+
+where RESULT is the numerical approximation obtained by the algorithm.
+The algorithms attempt to estimate the absolute error ABSERR = |RESULT
+- I| in such a way that the following inequality holds,
+
+     |RESULT - I| <= ABSERR <= max(epsabs, epsrel |I|)
+
+The routines will fail to converge if the error bounds are too
+stringent, but always return the best approximation obtained up to that
+stage.
+
+   The algorithms in QUADPACK use a naming convention based on the
+following letters,
+
+     `Q' - quadrature routine
+
+     `N' - non-adaptive integrator
+     `A' - adaptive integrator
+
+     `G' - general integrand (user-defined)
+     `W' - weight function with integrand
+
+     `S' - singularities can be more readily integrated
+     `P' - points of special difficulty can be supplied
+     `I' - infinite range of integration
+     `O' - oscillatory weight function, cos or sin
+     `F' - Fourier integral
+     `C' - Cauchy principal value
+
+The algorithms are built on pairs of quadrature rules, a higher order
+rule and a lower order rule.  The higher order rule is used to compute
+the best approximation to an integral over a small range.  The
+difference between the results of the higher order rule and the lower
+order rule gives an estimate of the error in the approximation.
+
+* Menu:
+
+* Integrands without weight functions::
+* Integrands with weight functions::
+* Integrands with singular weight functions::
+
+
+File: gsl-ref.info,  Node: Integrands without weight functions,  Next: Integrands with weight functions,  Up: Numerical Integration Introduction
+
+16.1.1 Integrands without weight functions
+------------------------------------------
+
+The algorithms for general functions (without a weight function) are
+based on Gauss-Kronrod rules.
+
+   A Gauss-Kronrod rule begins with a classical Gaussian quadrature
+rule of order m.  This is extended with additional points between each
+of the abscissae to give a higher order Kronrod rule of order 2m+1.
+The Kronrod rule is efficient because it reuses existing function
+evaluations from the Gaussian rule.
+
+   The higher order Kronrod rule is used as the best approximation to
+the integral, and the difference between the two rules is used as an
+estimate of the error in the approximation.
+
+
+File: gsl-ref.info,  Node: Integrands with weight functions,  Next: Integrands with singular weight functions,  Prev: Integrands without weight functions,  Up: Numerical Integration Introduction
+
+16.1.2 Integrands with weight functions
+---------------------------------------
+
+For integrands with weight functions the algorithms use Clenshaw-Curtis
+quadrature rules.
+
+   A Clenshaw-Curtis rule begins with an n-th order Chebyshev
+polynomial approximation to the integrand.  This polynomial can be
+integrated exactly to give an approximation to the integral of the
+original function.  The Chebyshev expansion can be extended to higher
+orders to improve the approximation and provide an estimate of the
+error.
+
+
+File: gsl-ref.info,  Node: Integrands with singular weight functions,  Prev: Integrands with weight functions,  Up: Numerical Integration Introduction
+
+16.1.3 Integrands with singular weight functions
+------------------------------------------------
+
+The presence of singularities (or other behavior) in the integrand can
+cause slow convergence in the Chebyshev approximation.  The modified
+Clenshaw-Curtis rules used in QUADPACK separate out several common
+weight functions which cause slow convergence.
+
+   These weight functions are integrated analytically against the
+Chebyshev polynomials to precompute "modified Chebyshev moments".
+Combining the moments with the Chebyshev approximation to the function
+gives the desired integral.  The use of analytic integration for the
+singular part of the function allows exact cancellations and
+substantially improves the overall convergence behavior of the
+integration.
+
+
+File: gsl-ref.info,  Node: QNG non-adaptive Gauss-Kronrod integration,  Next: QAG adaptive integration,  Prev: Numerical Integration Introduction,  Up: Numerical Integration
+
+16.2 QNG non-adaptive Gauss-Kronrod integration
+===============================================
+
+The QNG algorithm is a non-adaptive procedure which uses fixed
+Gauss-Kronrod abscissae to sample the integrand at a maximum of 87
+points.  It is provided for fast integration of smooth functions.
+
+ -- Function: int gsl_integration_qng (const gsl_function * F, double
+          A, double B, double EPSABS, double EPSREL, double * RESULT,
+          double * ABSERR, size_t * NEVAL)
+     This function applies the Gauss-Kronrod 10-point, 21-point,
+     43-point and 87-point integration rules in succession until an
+     estimate of the integral of f over (a,b) is achieved within the
+     desired absolute and relative error limits, EPSABS and EPSREL.  The
+     function returns the final approximation, RESULT, an estimate of
+     the absolute error, ABSERR and the number of function evaluations
+     used, NEVAL.  The Gauss-Kronrod rules are designed in such a way
+     that each rule uses all the results of its predecessors, in order
+     to minimize the total number of function evaluations.
+
+
+File: gsl-ref.info,  Node: QAG adaptive integration,  Next: QAGS adaptive integration with singularities,  Prev: QNG non-adaptive Gauss-Kronrod integration,  Up: Numerical Integration
+
+16.3 QAG adaptive integration
+=============================
+
+The QAG algorithm is a simple adaptive integration procedure.  The
+integration region is divided into subintervals, and on each iteration
+the subinterval with the largest estimated error is bisected.  This
+reduces the overall error rapidly, as the subintervals become
+concentrated around local difficulties in the integrand.  These
+subintervals are managed by a `gsl_integration_workspace' struct, which
+handles the memory for the subinterval ranges, results and error
+estimates.
+
+ -- Function: gsl_integration_workspace *
+gsl_integration_workspace_alloc (size_t N)
+     This function allocates a workspace sufficient to hold N double
+     precision intervals, their integration results and error estimates.
+
+ -- Function: void gsl_integration_workspace_free
+          (gsl_integration_workspace * W)
+     This function frees the memory associated with the workspace W.
+
+ -- Function: int gsl_integration_qag (const gsl_function * F, double
+          A, double B, double EPSABS, double EPSREL, size_t LIMIT, int
+          KEY, gsl_integration_workspace * WORKSPACE, double * RESULT,
+          double * ABSERR)
+     This function applies an integration rule adaptively until an
+     estimate of the integral of f over (a,b) is achieved within the
+     desired absolute and relative error limits, EPSABS and EPSREL.
+     The function returns the final approximation, RESULT, and an
+     estimate of the absolute error, ABSERR.  The integration rule is
+     determined by the value of KEY, which should be chosen from the
+     following symbolic names,
+
+          GSL_INTEG_GAUSS15  (key = 1)
+          GSL_INTEG_GAUSS21  (key = 2)
+          GSL_INTEG_GAUSS31  (key = 3)
+          GSL_INTEG_GAUSS41  (key = 4)
+          GSL_INTEG_GAUSS51  (key = 5)
+          GSL_INTEG_GAUSS61  (key = 6)
+
+     corresponding to the 15, 21, 31, 41, 51 and 61 point Gauss-Kronrod
+     rules.  The higher-order rules give better accuracy for smooth
+     functions, while lower-order rules save time when the function
+     contains local difficulties, such as discontinuities.
+
+     On each iteration the adaptive integration strategy bisects the
+     interval with the largest error estimate.  The subintervals and
+     their results are stored in the memory provided by WORKSPACE.  The
+     maximum number of subintervals is given by LIMIT, which may not
+     exceed the allocated size of the workspace.
+
+
+File: gsl-ref.info,  Node: QAGS adaptive integration with singularities,  Next: QAGP adaptive integration with known singular points,  Prev: QAG adaptive integration,  Up: Numerical Integration
+
+16.4 QAGS adaptive integration with singularities
+=================================================
+
+The presence of an integrable singularity in the integration region
+causes an adaptive routine to concentrate new subintervals around the
+singularity.  As the subintervals decrease in size the successive
+approximations to the integral converge in a limiting fashion.  This
+approach to the limit can be accelerated using an extrapolation
+procedure.  The QAGS algorithm combines adaptive bisection with the Wynn
+epsilon-algorithm to speed up the integration of many types of
+integrable singularities.
+
+ -- Function: int gsl_integration_qags (const gsl_function * F, double
+          A, double B, double EPSABS, double EPSREL, size_t LIMIT,
+          gsl_integration_workspace * WORKSPACE, double * RESULT,
+          double * ABSERR)
+     This function applies the Gauss-Kronrod 21-point integration rule
+     adaptively until an estimate of the integral of f over (a,b) is
+     achieved within the desired absolute and relative error limits,
+     EPSABS and EPSREL.  The results are extrapolated using the
+     epsilon-algorithm, which accelerates the convergence of the
+     integral in the presence of discontinuities and integrable
+     singularities.  The function returns the final approximation from
+     the extrapolation, RESULT, and an estimate of the absolute error,
+     ABSERR.  The subintervals and their results are stored in the
+     memory provided by WORKSPACE.  The maximum number of subintervals
+     is given by LIMIT, which may not exceed the allocated size of the
+     workspace.
+
+
+
+File: gsl-ref.info,  Node: QAGP adaptive integration with known singular points,  Next: QAGI adaptive integration on infinite intervals,  Prev: QAGS adaptive integration with singularities,  Up: Numerical Integration
+
+16.5 QAGP adaptive integration with known singular points
+=========================================================
+
+ -- Function: int gsl_integration_qagp (const gsl_function * F, double
+          * PTS, size_t NPTS, double EPSABS, double EPSREL, size_t
+          LIMIT, gsl_integration_workspace * WORKSPACE, double *
+          RESULT, double * ABSERR)
+     This function applies the adaptive integration algorithm QAGS
+     taking account of the user-supplied locations of singular points.
+     The array PTS of length NPTS should contain the endpoints of the
+     integration ranges defined by the integration region and locations
+     of the singularities.  For example, to integrate over the region
+     (a,b) with break-points at x_1, x_2, x_3 (where a < x_1 < x_2 <
+     x_3 < b) the following PTS array should be used
+
+          pts[0] = a
+          pts[1] = x_1
+          pts[2] = x_2
+          pts[3] = x_3
+          pts[4] = b
+
+     with NPTS = 5.
+
+     If you know the locations of the singular points in the integration
+     region then this routine will be faster than `QAGS'.
+
+
+
+File: gsl-ref.info,  Node: QAGI adaptive integration on infinite intervals,  Next: QAWC adaptive integration for Cauchy principal values,  Prev: QAGP adaptive integration with known singular points,  Up: Numerical Integration
+
+16.6 QAGI adaptive integration on infinite intervals
+====================================================
+
+ -- Function: int gsl_integration_qagi (gsl_function * F, double
+          EPSABS, double EPSREL, size_t LIMIT,
+          gsl_integration_workspace * WORKSPACE, double * RESULT,
+          double * ABSERR)
+     This function computes the integral of the function F over the
+     infinite interval (-\infty,+\infty).  The integral is mapped onto
+     the semi-open interval (0,1] using the transformation x = (1-t)/t,
+
+          \int_{-\infty}^{+\infty} dx f(x) =
+               \int_0^1 dt (f((1-t)/t) + f((-1+t)/t))/t^2.
+
+     It is then integrated using the QAGS algorithm.  The normal
+     21-point Gauss-Kronrod rule of QAGS is replaced by a 15-point
+     rule, because the transformation can generate an integrable
+     singularity at the origin.  In this case a lower-order rule is
+     more efficient.
+
+ -- Function: int gsl_integration_qagiu (gsl_function * F, double A,
+          double EPSABS, double EPSREL, size_t LIMIT,
+          gsl_integration_workspace * WORKSPACE, double * RESULT,
+          double * ABSERR)
+     This function computes the integral of the function F over the
+     semi-infinite interval (a,+\infty).  The integral is mapped onto
+     the semi-open interval (0,1] using the transformation x = a +
+     (1-t)/t,
+
+          \int_{a}^{+\infty} dx f(x) =
+               \int_0^1 dt f(a + (1-t)/t)/t^2
+
+     and then integrated using the QAGS algorithm.
+
+ -- Function: int gsl_integration_qagil (gsl_function * F, double B,
+          double EPSABS, double EPSREL, size_t LIMIT,
+          gsl_integration_workspace * WORKSPACE, double * RESULT,
+          double * ABSERR)
+     This function computes the integral of the function F over the
+     semi-infinite interval (-\infty,b).  The integral is mapped onto
+     the semi-open interval (0,1] using the transformation x = b -
+     (1-t)/t,
+
+          \int_{+\infty}^{b} dx f(x) =
+               \int_0^1 dt f(b - (1-t)/t)/t^2
+
+     and then integrated using the QAGS algorithm.
+
+
+File: gsl-ref.info,  Node: QAWC adaptive integration for Cauchy principal values,  Next: QAWS adaptive integration for singular functions,  Prev: QAGI adaptive integration on infinite intervals,  Up: Numerical Integration
+
+16.7 QAWC adaptive integration for Cauchy principal values
+==========================================================
+
+ -- Function: int gsl_integration_qawc (gsl_function * F, double A,
+          double B, double C, double EPSABS, double EPSREL, size_t
+          LIMIT, gsl_integration_workspace * WORKSPACE, double *
+          RESULT, double * ABSERR)
+     This function computes the Cauchy principal value of the integral
+     of f over (a,b), with a singularity at C,
+
+          I = \int_a^b dx f(x) / (x - c)
+
+     The adaptive bisection algorithm of QAG is used, with
+     modifications to ensure that subdivisions do not occur at the
+     singular point x = c.  When a subinterval contains the point x = c
+     or is close to it then a special 25-point modified Clenshaw-Curtis
+     rule is used to control the singularity.  Further away from the
+     singularity the algorithm uses an ordinary 15-point Gauss-Kronrod
+     integration rule.
+
+
+
+File: gsl-ref.info,  Node: QAWS adaptive integration for singular functions,  Next: QAWO adaptive integration for oscillatory functions,  Prev: QAWC adaptive integration for Cauchy principal values,  Up: Numerical Integration
+
+16.8 QAWS adaptive integration for singular functions
+=====================================================
+
+The QAWS algorithm is designed for integrands with algebraic-logarithmic
+singularities at the end-points of an integration region.  In order to
+work efficiently the algorithm requires a precomputed table of
+Chebyshev moments.
+
+ -- Function: gsl_integration_qaws_table *
+gsl_integration_qaws_table_alloc (double ALPHA, double BETA, int MU,
+          int NU)
+     This function allocates space for a `gsl_integration_qaws_table'
+     struct describing a singular weight function W(x) with the
+     parameters (\alpha, \beta, \mu, \nu),
+
+          W(x) = (x-a)^alpha (b-x)^beta log^mu (x-a) log^nu (b-x)
+
+     where \alpha > -1, \beta > -1, and \mu = 0, 1, \nu = 0, 1.  The
+     weight function can take four different forms depending on the
+     values of \mu and \nu,
+
+          W(x) = (x-a)^alpha (b-x)^beta                   (mu = 0, nu = 0)
+          W(x) = (x-a)^alpha (b-x)^beta log(x-a)          (mu = 1, nu = 0)
+          W(x) = (x-a)^alpha (b-x)^beta log(b-x)          (mu = 0, nu = 1)
+          W(x) = (x-a)^alpha (b-x)^beta log(x-a) log(b-x) (mu = 1, nu = 1)
+
+     The singular points (a,b) do not have to be specified until the
+     integral is computed, where they are the endpoints of the
+     integration range.
+
+     The function returns a pointer to the newly allocated table
+     `gsl_integration_qaws_table' if no errors were detected, and 0 in
+     the case of error.
+
+ -- Function: int gsl_integration_qaws_table_set
+          (gsl_integration_qaws_table * T, double ALPHA, double BETA,
+          int MU, int NU)
+     This function modifies the parameters (\alpha, \beta, \mu, \nu) of
+     an existing `gsl_integration_qaws_table' struct T.
+
+ -- Function: void gsl_integration_qaws_table_free
+          (gsl_integration_qaws_table * T)
+     This function frees all the memory associated with the
+     `gsl_integration_qaws_table' struct T.
+
+ -- Function: int gsl_integration_qaws (gsl_function * F, const double
+          A, const double B, gsl_integration_qaws_table * T, const
+          double EPSABS, const double EPSREL, const size_t LIMIT,
+          gsl_integration_workspace * WORKSPACE, double * RESULT,
+          double * ABSERR)
+     This function computes the integral of the function f(x) over the
+     interval (a,b) with the singular weight function (x-a)^\alpha
+     (b-x)^\beta \log^\mu (x-a) \log^\nu (b-x).  The parameters of the
+     weight function (\alpha, \beta, \mu, \nu) are taken from the table
+     T.  The integral is,
+
+          I = \int_a^b dx f(x) (x-a)^alpha (b-x)^beta log^mu (x-a) log^nu (b-x).
+
+     The adaptive bisection algorithm of QAG is used.  When a
+     subinterval contains one of the endpoints then a special 25-point
+     modified Clenshaw-Curtis rule is used to control the
+     singularities.  For subintervals which do not include the
+     endpoints an ordinary 15-point Gauss-Kronrod integration rule is
+     used.
+
+
+
+File: gsl-ref.info,  Node: QAWO adaptive integration for oscillatory functions,  Next: QAWF adaptive integration for Fourier integrals,  Prev: QAWS adaptive integration for singular functions,  Up: Numerical Integration
+
+16.9 QAWO adaptive integration for oscillatory functions
+========================================================
+
+The QAWO algorithm is designed for integrands with an oscillatory
+factor, \sin(\omega x) or \cos(\omega x).  In order to work efficiently
+the algorithm requires a table of Chebyshev moments which must be
+pre-computed with calls to the functions below.
+
+ -- Function: gsl_integration_qawo_table *
+gsl_integration_qawo_table_alloc (double OMEGA, double L, enum
+          gsl_integration_qawo_enum SINE, size_t N)
+     This function allocates space for a `gsl_integration_qawo_table'
+     struct and its associated workspace describing a sine or cosine
+     weight function W(x) with the parameters (\omega, L),
+
+          W(x) = sin(omega x)
+          W(x) = cos(omega x)
+
+     The parameter L must be the length of the interval over which the
+     function will be integrated L = b - a.  The choice of sine or
+     cosine is made with the parameter SINE which should be chosen from
+     one of the two following symbolic values:
+
+          GSL_INTEG_COSINE
+          GSL_INTEG_SINE
+
+     The `gsl_integration_qawo_table' is a table of the trigonometric
+     coefficients required in the integration process.  The parameter N
+     determines the number of levels of coefficients that are computed.
+     Each level corresponds to one bisection of the interval L, so that
+     N levels are sufficient for subintervals down to the length L/2^n.
+     The integration routine `gsl_integration_qawo' returns the error
+     `GSL_ETABLE' if the number of levels is insufficient for the
+     requested accuracy.
+
+
+ -- Function: int gsl_integration_qawo_table_set
+          (gsl_integration_qawo_table * T, double OMEGA, double L, enum
+          gsl_integration_qawo_enum SINE)
+     This function changes the parameters OMEGA, L and SINE of the
+     existing workspace T.
+
+ -- Function: int gsl_integration_qawo_table_set_length
+          (gsl_integration_qawo_table * T, double L)
+     This function allows the length parameter L of the workspace T to
+     be changed.
+
+ -- Function: void gsl_integration_qawo_table_free
+          (gsl_integration_qawo_table * T)
+     This function frees all the memory associated with the workspace T.
+
+ -- Function: int gsl_integration_qawo (gsl_function * F, const double
+          A, const double EPSABS, const double EPSREL, const size_t
+          LIMIT, gsl_integration_workspace * WORKSPACE,
+          gsl_integration_qawo_table * WF, double * RESULT, double *
+          ABSERR)
+     This function uses an adaptive algorithm to compute the integral of
+     f over (a,b) with the weight function \sin(\omega x) or
+     \cos(\omega x) defined by the table WF,
+
+          I = \int_a^b dx f(x) sin(omega x)
+          I = \int_a^b dx f(x) cos(omega x)
+
+     The results are extrapolated using the epsilon-algorithm to
+     accelerate the convergence of the integral.  The function returns
+     the final approximation from the extrapolation, RESULT, and an
+     estimate of the absolute error, ABSERR.  The subintervals and
+     their results are stored in the memory provided by WORKSPACE.  The
+     maximum number of subintervals is given by LIMIT, which may not
+     exceed the allocated size of the workspace.
+
+     Those subintervals with "large" widths d where d\omega > 4 are
+     computed using a 25-point Clenshaw-Curtis integration rule, which
+     handles the oscillatory behavior.  Subintervals with a "small"
+     widths where d\omega < 4 are computed using a 15-point
+     Gauss-Kronrod integration.
+
+
+
+File: gsl-ref.info,  Node: QAWF adaptive integration for Fourier integrals,  Next: Numerical integration error codes,  Prev: QAWO adaptive integration for oscillatory functions,  Up: Numerical Integration
+
+16.10 QAWF adaptive integration for Fourier integrals
+=====================================================
+
+ -- Function: int gsl_integration_qawf (gsl_function * F, const double
+          A, const double EPSABS, const size_t LIMIT,
+          gsl_integration_workspace * WORKSPACE,
+          gsl_integration_workspace * CYCLE_WORKSPACE,
+          gsl_integration_qawo_table * WF, double * RESULT, double *
+          ABSERR)
+     This function attempts to compute a Fourier integral of the
+     function F over the semi-infinite interval [a,+\infty).
+
+          I = \int_a^{+\infty} dx f(x) sin(omega x)
+          I = \int_a^{+\infty} dx f(x) cos(omega x)
+
+     The parameter \omega and choice of \sin or \cos is taken from the
+     table WF (the length L can take any value, since it is overridden
+     by this function to a value appropriate for the fourier
+     integration).  The integral is computed using the QAWO algorithm
+     over each of the subintervals,
+
+          C_1 = [a, a + c]
+          C_2 = [a + c, a + 2 c]
+          ... = ...
+          C_k = [a + (k-1) c, a + k c]
+
+     where c = (2 floor(|\omega|) + 1) \pi/|\omega|.  The width c is
+     chosen to cover an odd number of periods so that the contributions
+     from the intervals alternate in sign and are monotonically
+     decreasing when F is positive and monotonically decreasing.  The
+     sum of this sequence of contributions is accelerated using the
+     epsilon-algorithm.
+
+     This function works to an overall absolute tolerance of ABSERR.
+     The following strategy is used: on each interval C_k the algorithm
+     tries to achieve the tolerance
+
+          TOL_k = u_k abserr
+
+     where u_k = (1 - p)p^{k-1} and p = 9/10.  The sum of the geometric
+     series of contributions from each interval gives an overall
+     tolerance of ABSERR.
+
+     If the integration of a subinterval leads to difficulties then the
+     accuracy requirement for subsequent intervals is relaxed,
+
+          TOL_k = u_k max(abserr, max_{i<k}{E_i})
+
+     where E_k is the estimated error on the interval C_k.
+
+     The subintervals and their results are stored in the memory
+     provided by WORKSPACE.  The maximum number of subintervals is
+     given by LIMIT, which may not exceed the allocated size of the
+     workspace.  The integration over each subinterval uses the memory
+     provided by CYCLE_WORKSPACE as workspace for the QAWO algorithm.
+
+
+
+File: gsl-ref.info,  Node: Numerical integration error codes,  Next: Numerical integration examples,  Prev: QAWF adaptive integration for Fourier integrals,  Up: Numerical Integration
+
+16.11 Error codes
+=================
+
+In addition to the standard error codes for invalid arguments the
+functions can return the following values,
+
+`GSL_EMAXITER'
+     the maximum number of subdivisions was exceeded.
+
+`GSL_EROUND'
+     cannot reach tolerance because of roundoff error, or roundoff
+     error was detected in the extrapolation table.
+
+`GSL_ESING'
+     a non-integrable singularity or other bad integrand behavior was
+     found in the integration interval.
+
+`GSL_EDIVERGE'
+     the integral is divergent, or too slowly convergent to be
+     integrated numerically.
+
+
+File: gsl-ref.info,  Node: Numerical integration examples,  Next: Numerical integration References and Further Reading,  Prev: Numerical integration error codes,  Up: Numerical Integration
+
+16.12 Examples
+==============
+
+The integrator `QAGS' will handle a large class of definite integrals.
+For example, consider the following integral, which has a
+algebraic-logarithmic singularity at the origin,
+
+     \int_0^1 x^{-1/2} log(x) dx = -4
+
+The program below computes this integral to a relative accuracy bound of
+`1e-7'.
+
+     #include <stdio.h>
+     #include <math.h>
+     #include <gsl/gsl_integration.h>
+
+     double f (double x, void * params) {
+       double alpha = *(double *) params;
+       double f = log(alpha*x) / sqrt(x);
+       return f;
+     }
+
+     int
+     main (void)
+     {
+       gsl_integration_workspace * w
+         = gsl_integration_workspace_alloc (1000);
+
+       double result, error;
+       double expected = -4.0;
+       double alpha = 1.0;
+
+       gsl_function F;
+       F.function = &f;
+       F.params = &alpha;
+
+       gsl_integration_qags (&F, 0, 1, 0, 1e-7, 1000,
+                             w, &result, &error);
+
+       printf ("result          = % .18f\n", result);
+       printf ("exact result    = % .18f\n", expected);
+       printf ("estimated error = % .18f\n", error);
+       printf ("actual error    = % .18f\n", result - expected);
+       printf ("intervals =  %d\n", w->size);
+
+       gsl_integration_workspace_free (w);
+
+       return 0;
+     }
+
+The results below show that the desired accuracy is achieved after 8
+subdivisions.
+
+     $ ./a.out
+     result          = -3.999999999999973799
+     exact result    = -4.000000000000000000
+     estimated error =  0.000000000000246025
+     actual error    =  0.000000000000026201
+     intervals =  8
+
+In fact, the extrapolation procedure used by `QAGS' produces an
+accuracy of almost twice as many digits.  The error estimate returned by
+the extrapolation procedure is larger than the actual error, giving a
+margin of safety of one order of magnitude.
+
+
+File: gsl-ref.info,  Node: Numerical integration References and Further Reading,  Prev: Numerical integration examples,  Up: Numerical Integration
+
+16.13 References and Further Reading
+====================================
+
+The following book is the definitive reference for QUADPACK, and was
+written by the original authors.  It provides descriptions of the
+algorithms, program listings, test programs and examples.  It also
+includes useful advice on numerical integration and many references to
+the numerical integration literature used in developing QUADPACK.
+
+     R. Piessens, E. de Doncker-Kapenga, C.W. Uberhuber, D.K. Kahaner.
+     `QUADPACK A subroutine package for automatic integration' Springer
+     Verlag, 1983.
+
+
+
+File: gsl-ref.info,  Node: Random Number Generation,  Next: Quasi-Random Sequences,  Prev: Numerical Integration,  Up: Top
+
+17 Random Number Generation
+***************************
+
+The library provides a large collection of random number generators
+which can be accessed through a uniform interface.  Environment
+variables allow you to select different generators and seeds at runtime,
+so that you can easily switch between generators without needing to
+recompile your program.  Each instance of a generator keeps track of its
+own state, allowing the generators to be used in multi-threaded
+programs.  Additional functions are available for transforming uniform
+random numbers into samples from continuous or discrete probability
+distributions such as the Gaussian, log-normal or Poisson distributions.
+
+   These functions are declared in the header file `gsl_rng.h'.
+
+* Menu:
+
+* General comments on random numbers::
+* The Random Number Generator Interface::
+* Random number generator initialization::
+* Sampling from a random number generator::
+* Auxiliary random number generator functions::
+* Random number environment variables::
+* Copying random number generator state::
+* Reading and writing random number generator state::
+* Random number generator algorithms::
+* Unix random number generators::
+* Other random number generators::
+* Random Number Generator Performance::
+* Random Number Generator Examples::
+* Random Number References and Further Reading::
+* Random Number Acknowledgements::
+
+
+File: gsl-ref.info,  Node: General comments on random numbers,  Next: The Random Number Generator Interface,  Up: Random Number Generation
+
+17.1 General comments on random numbers
+=======================================
+
+In 1988, Park and Miller wrote a paper entitled "Random number
+generators: good ones are hard to find." [Commun. ACM, 31, 1192-1201].
+Fortunately, some excellent random number generators are available,
+though poor ones are still in common use.  You may be happy with the
+system-supplied random number generator on your computer, but you should
+be aware that as computers get faster, requirements on random number
+generators increase.  Nowadays, a simulation that calls a random number
+generator millions of times can often finish before you can make it down
+the hall to the coffee machine and back.
+
+   A very nice review of random number generators was written by Pierre
+L'Ecuyer, as Chapter 4 of the book: Handbook on Simulation, Jerry Banks,
+ed. (Wiley, 1997).  The chapter is available in postscript from
+L'Ecuyer's ftp site (see references).  Knuth's volume on Seminumerical
+Algorithms (originally published in 1968) devotes 170 pages to random
+number generators, and has recently been updated in its 3rd edition
+(1997).  It is brilliant, a classic.  If you don't own it, you should
+stop reading right now, run to the nearest bookstore, and buy it.
+
+   A good random number generator will satisfy both theoretical and
+statistical properties.  Theoretical properties are often hard to obtain
+(they require real math!), but one prefers a random number generator
+with a long period, low serial correlation, and a tendency _not_ to
+"fall mainly on the planes."  Statistical tests are performed with
+numerical simulations.  Generally, a random number generator is used to
+estimate some quantity for which the theory of probability provides an
+exact answer.  Comparison to this exact answer provides a measure of
+"randomness".
+
+
+File: gsl-ref.info,  Node: The Random Number Generator Interface,  Next: Random number generator initialization,  Prev: General comments on random numbers,  Up: Random Number Generation
+
+17.2 The Random Number Generator Interface
+==========================================
+
+It is important to remember that a random number generator is not a
+"real" function like sine or cosine.  Unlike real functions, successive
+calls to a random number generator yield different return values.  Of
+course that is just what you want for a random number generator, but to
+achieve this effect, the generator must keep track of some kind of
+"state" variable.  Sometimes this state is just an integer (sometimes
+just the value of the previously generated random number), but often it
+is more complicated than that and may involve a whole array of numbers,
+possibly with some indices thrown in.  To use the random number
+generators, you do not need to know the details of what comprises the
+state, and besides that varies from algorithm to algorithm.
+
+   The random number generator library uses two special structs,
+`gsl_rng_type' which holds static information about each type of
+generator and `gsl_rng' which describes an instance of a generator
+created from a given `gsl_rng_type'.
+
+   The functions described in this section are declared in the header
+file `gsl_rng.h'.
+
+
+File: gsl-ref.info,  Node: Random number generator initialization,  Next: Sampling from a random number generator,  Prev: The Random Number Generator Interface,  Up: Random Number Generation
+
+17.3 Random number generator initialization
+===========================================
+
+ -- Function: gsl_rng * gsl_rng_alloc (const gsl_rng_type * T)
+     This function returns a pointer to a newly-created instance of a
+     random number generator of type T.  For example, the following
+     code creates an instance of the Tausworthe generator,
+
+          gsl_rng * r = gsl_rng_alloc (gsl_rng_taus);
+
+     If there is insufficient memory to create the generator then the
+     function returns a null pointer and the error handler is invoked
+     with an error code of `GSL_ENOMEM'.
+
+     The generator is automatically initialized with the default seed,
+     `gsl_rng_default_seed'.  This is zero by default but can be changed
+     either directly or by using the environment variable `GSL_RNG_SEED'
+     (*note Random number environment variables::).
+
+     The details of the available generator types are described later
+     in this chapter.
+
+ -- Function: void gsl_rng_set (const gsl_rng * R, unsigned long int S)
+     This function initializes (or `seeds') the random number
+     generator.  If the generator is seeded with the same value of S on
+     two different runs, the same stream of random numbers will be
+     generated by successive calls to the routines below.  If different
+     values of S are supplied, then the generated streams of random
+     numbers should be completely different.  If the seed S is zero
+     then the standard seed from the original implementation is used
+     instead.  For example, the original Fortran source code for the
+     `ranlux' generator used a seed of 314159265, and so choosing S
+     equal to zero reproduces this when using `gsl_rng_ranlux'.
+
+ -- Function: void gsl_rng_free (gsl_rng * R)
+     This function frees all the memory associated with the generator R.
+
+
+File: gsl-ref.info,  Node: Sampling from a random number generator,  Next: Auxiliary random number generator functions,  Prev: Random number generator initialization,  Up: Random Number Generation
+
+17.4 Sampling from a random number generator
+============================================
+
+The following functions return uniformly distributed random numbers,
+either as integers or double precision floating point numbers.  To
+obtain non-uniform distributions *note Random Number Distributions::.
+
+ -- Function: unsigned long int gsl_rng_get (const gsl_rng * R)
+     This function returns a random integer from the generator R.  The
+     minimum and maximum values depend on the algorithm used, but all
+     integers in the range [MIN,MAX] are equally likely.  The values of
+     MIN and MAX can determined using the auxiliary functions
+     `gsl_rng_max (r)' and `gsl_rng_min (r)'.
+
+ -- Function: double gsl_rng_uniform (const gsl_rng * R)
+     This function returns a double precision floating point number
+     uniformly distributed in the range [0,1).  The range includes 0.0
+     but excludes 1.0.  The value is typically obtained by dividing the
+     result of `gsl_rng_get(r)' by `gsl_rng_max(r) + 1.0' in double
+     precision.  Some generators compute this ratio internally so that
+     they can provide floating point numbers with more than 32 bits of
+     randomness (the maximum number of bits that can be portably
+     represented in a single `unsigned long int').
+
+ -- Function: double gsl_rng_uniform_pos (const gsl_rng * R)
+     This function returns a positive double precision floating point
+     number uniformly distributed in the range (0,1), excluding both
+     0.0 and 1.0.  The number is obtained by sampling the generator
+     with the algorithm of `gsl_rng_uniform' until a non-zero value is
+     obtained.  You can use this function if you need to avoid a
+     singularity at 0.0.
+
+ -- Function: unsigned long int gsl_rng_uniform_int (const gsl_rng * R,
+          unsigned long int N)
+     This function returns a random integer from 0 to n-1 inclusive by
+     scaling down and/or discarding samples from the generator R.  All
+     integers in the range [0,n-1] are produced with equal probability.
+     For generators with a non-zero minimum value an offset is applied
+     so that zero is returned with the correct probability.
+
+     Note that this function is designed for sampling from ranges
+     smaller than the range of the underlying generator.  The parameter
+     N must be less than or equal to the range of the generator R.  If
+     N is larger than the range of the generator then the function
+     calls the error handler with an error code of `GSL_EINVAL' and
+     returns zero.
+
+     In particular, this function is not intended for generating the
+     full range of unsigned integer values [0,2^32-1]. Instead choose a
+     generator with the maximal integer range and zero mimimum value,
+     such as `gsl_rng_ranlxd1', `gsl_rng_mt19937' or `gsl_rng_taus',
+     and sample it directly using `gsl_rng_get'.  The range of each
+     generator can be found using the auxiliary functions described in
+     the next section.
+
+
+File: gsl-ref.info,  Node: Auxiliary random number generator functions,  Next: Random number environment variables,  Prev: Sampling from a random number generator,  Up: Random Number Generation
+
+17.5 Auxiliary random number generator functions
+================================================
+
+The following functions provide information about an existing
+generator.  You should use them in preference to hard-coding the
+generator parameters into your own code.
+
+ -- Function: const char * gsl_rng_name (const gsl_rng * R)
+     This function returns a pointer to the name of the generator.  For
+     example,
+
+          printf ("r is a '%s' generator\n",
+                  gsl_rng_name (r));
+
+     would print something like `r is a 'taus' generator'.
+
+ -- Function: unsigned long int gsl_rng_max (const gsl_rng * R)
+     `gsl_rng_max' returns the largest value that `gsl_rng_get' can
+     return.
+
+ -- Function: unsigned long int gsl_rng_min (const gsl_rng * R)
+     `gsl_rng_min' returns the smallest value that `gsl_rng_get' can
+     return.  Usually this value is zero.  There are some generators
+     with algorithms that cannot return zero, and for these generators
+     the minimum value is 1.
+
+ -- Function: void * gsl_rng_state (const gsl_rng * R)
+ -- Function: size_t gsl_rng_size (const gsl_rng * R)
+     These functions return a pointer to the state of generator R and
+     its size.  You can use this information to access the state
+     directly.  For example, the following code will write the state of
+     a generator to a stream,
+
+          void * state = gsl_rng_state (r);
+          size_t n = gsl_rng_size (r);
+          fwrite (state, n, 1, stream);
+
+ -- Function: const gsl_rng_type ** gsl_rng_types_setup (void)
+     This function returns a pointer to an array of all the available
+     generator types, terminated by a null pointer. The function should
+     be called once at the start of the program, if needed.  The
+     following code fragment shows how to iterate over the array of
+     generator types to print the names of the available algorithms,
+
+          const gsl_rng_type **t, **t0;
+
+          t0 = gsl_rng_types_setup ();
+
+          printf ("Available generators:\n");
+
+          for (t = t0; *t != 0; t++)
+            {
+              printf ("%s\n", (*t)->name);
+            }
+
+
+File: gsl-ref.info,  Node: Random number environment variables,  Next: Copying random number generator state,  Prev: Auxiliary random number generator functions,  Up: Random Number Generation
+
+17.6 Random number environment variables
+========================================
+
+The library allows you to choose a default generator and seed from the
+environment variables `GSL_RNG_TYPE' and `GSL_RNG_SEED' and the
+function `gsl_rng_env_setup'.  This makes it easy try out different
+generators and seeds without having to recompile your program.
+
+ -- Function: const gsl_rng_type * gsl_rng_env_setup (void)
+     This function reads the environment variables `GSL_RNG_TYPE' and
+     `GSL_RNG_SEED' and uses their values to set the corresponding
+     library variables `gsl_rng_default' and `gsl_rng_default_seed'.
+     These global variables are defined as follows,
+
+          extern const gsl_rng_type *gsl_rng_default
+          extern unsigned long int gsl_rng_default_seed
+
+     The environment variable `GSL_RNG_TYPE' should be the name of a
+     generator, such as `taus' or `mt19937'.  The environment variable
+     `GSL_RNG_SEED' should contain the desired seed value.  It is
+     converted to an `unsigned long int' using the C library function
+     `strtoul'.
+
+     If you don't specify a generator for `GSL_RNG_TYPE' then
+     `gsl_rng_mt19937' is used as the default.  The initial value of
+     `gsl_rng_default_seed' is zero.
+
+
+Here is a short program which shows how to create a global generator
+using the environment variables `GSL_RNG_TYPE' and `GSL_RNG_SEED',
+
+     #include <stdio.h>
+     #include <gsl/gsl_rng.h>
+
+     gsl_rng * r;  /* global generator */
+
+     int
+     main (void)
+     {
+       const gsl_rng_type * T;
+
+       gsl_rng_env_setup();
+
+       T = gsl_rng_default;
+       r = gsl_rng_alloc (T);
+
+       printf ("generator type: %s\n", gsl_rng_name (r));
+       printf ("seed = %lu\n", gsl_rng_default_seed);
+       printf ("first value = %lu\n", gsl_rng_get (r));
+
+       gsl_rng_free (r);
+       return 0;
+     }
+
+Running the program without any environment variables uses the initial
+defaults, an `mt19937' generator with a seed of 0,
+
+     $ ./a.out
+     generator type: mt19937
+     seed = 0
+     first value = 4293858116
+
+By setting the two variables on the command line we can change the
+default generator and the seed,
+
+     $ GSL_RNG_TYPE="taus" GSL_RNG_SEED=123 ./a.out
+     GSL_RNG_TYPE=taus
+     GSL_RNG_SEED=123
+     generator type: taus
+     seed = 123
+     first value = 2720986350
+
+
+File: gsl-ref.info,  Node: Copying random number generator state,  Next: Reading and writing random number generator state,  Prev: Random number environment variables,  Up: Random Number Generation
+
+17.7 Copying random number generator state
+==========================================
+
+The above methods do not expose the random number `state' which changes
+from call to call.  It is often useful to be able to save and restore
+the state.  To permit these practices, a few somewhat more advanced
+functions are supplied.  These include:
+
+ -- Function: int gsl_rng_memcpy (gsl_rng * DEST, const gsl_rng * SRC)
+     This function copies the random number generator SRC into the
+     pre-existing generator DEST, making DEST into an exact copy of
+     SRC.  The two generators must be of the same type.
+
+ -- Function: gsl_rng * gsl_rng_clone (const gsl_rng * R)
+     This function returns a pointer to a newly created generator which
+     is an exact copy of the generator R.
+
+
+File: gsl-ref.info,  Node: Reading and writing random number generator state,  Next: Random number generator algorithms,  Prev: Copying random number generator state,  Up: Random Number Generation
+
+17.8 Reading and writing random number generator state
+======================================================
+
+The library provides functions for reading and writing the random
+number state to a file as binary data or formatted text.
+
+ -- Function: int gsl_rng_fwrite (FILE * STREAM, const gsl_rng * R)
+     This function writes the random number state of the random number
+     generator R to the stream STREAM in binary format.  The return
+     value is 0 for success and `GSL_EFAILED' if there was a problem
+     writing to the file.  Since the data is written in the native
+     binary format it may not be portable between different
+     architectures.
+
+ -- Function: int gsl_rng_fread (FILE * STREAM, gsl_rng * R)
+     This function reads the random number state into the random number
+     generator R from the open stream STREAM in binary format.  The
+     random number generator R must be preinitialized with the correct
+     random number generator type since type information is not saved.
+     The return value is 0 for success and `GSL_EFAILED' if there was a
+     problem reading from the file.  The data is assumed to have been
+     written in the native binary format on the same architecture.
+
+
+File: gsl-ref.info,  Node: Random number generator algorithms,  Next: Unix random number generators,  Prev: Reading and writing random number generator state,  Up: Random Number Generation
+
+17.9 Random number generator algorithms
+=======================================
+
+The functions described above make no reference to the actual algorithm
+used.  This is deliberate so that you can switch algorithms without
+having to change any of your application source code.  The library
+provides a large number of generators of different types, including
+simulation quality generators, generators provided for compatibility
+with other libraries and historical generators from the past.
+
+   The following generators are recommended for use in simulation.  They
+have extremely long periods, low correlation and pass most statistical
+tests.  For the most reliable source of uncorrelated numbers, the
+second-generation RANLUX generators have the strongest proof of
+randomness.
+
+ -- Generator: gsl_rng_mt19937
+     The MT19937 generator of Makoto Matsumoto and Takuji Nishimura is a
+     variant of the twisted generalized feedback shift-register
+     algorithm, and is known as the "Mersenne Twister" generator.  It
+     has a Mersenne prime period of 2^19937 - 1 (about 10^6000) and is
+     equi-distributed in 623 dimensions.  It has passed the DIEHARD
+     statistical tests.  It uses 624 words of state per generator and is
+     comparable in speed to the other generators.  The original
+     generator used a default seed of 4357 and choosing S equal to zero
+     in `gsl_rng_set' reproduces this.  Later versions switched to 5489
+     as the default seed, you can choose this explicitly via
+     `gsl_rng_set' instead if you require it.
+
+     For more information see,
+          Makoto Matsumoto and Takuji Nishimura, "Mersenne Twister: A
+          623-dimensionally equidistributed uniform pseudorandom number
+          generator". `ACM Transactions on Modeling and Computer
+          Simulation', Vol. 8, No. 1 (Jan. 1998), Pages 3-30
+
+     The generator `gsl_rng_mt19937' uses the second revision of the
+     seeding procedure published by the two authors above in 2002.  The
+     original seeding procedures could cause spurious artifacts for
+     some seed values. They are still available through the alternative
+     generators `gsl_rng_mt19937_1999' and `gsl_rng_mt19937_1998'.
+
+ -- Generator: gsl_rng_ranlxs0
+ -- Generator: gsl_rng_ranlxs1
+ -- Generator: gsl_rng_ranlxs2
+     The generator `ranlxs0' is a second-generation version of the
+     RANLUX algorithm of Lu"scher, which produces "luxury random
+     numbers".  This generator provides single precision output (24
+     bits) at three luxury levels `ranlxs0', `ranlxs1' and `ranlxs2',
+     in increasing order of strength.  It uses double-precision
+     floating point arithmetic internally and can be significantly
+     faster than the integer version of `ranlux', particularly on
+     64-bit architectures.  The period of the generator is about
+     10^171.  The algorithm has mathematically proven properties and
+     can provide truly decorrelated numbers at a known level of
+     randomness.  The higher luxury levels provide increased
+     decorrelation between samples as an additional safety margin.
+
+ -- Generator: gsl_rng_ranlxd1
+ -- Generator: gsl_rng_ranlxd2
+     These generators produce double precision output (48 bits) from the
+     RANLXS generator.  The library provides two luxury levels
+     `ranlxd1' and `ranlxd2', in increasing order of strength.
+
+ -- Generator: gsl_rng_ranlux
+ -- Generator: gsl_rng_ranlux389
+     The `ranlux' generator is an implementation of the original
+     algorithm developed by Lu"scher.  It uses a
+     lagged-fibonacci-with-skipping algorithm to produce "luxury random
+     numbers".  It is a 24-bit generator, originally designed for
+     single-precision IEEE floating point numbers.  This implementation
+     is based on integer arithmetic, while the second-generation
+     versions RANLXS and RANLXD described above provide floating-point
+     implementations which will be faster on many platforms.  The
+     period of the generator is about 10^171.  The algorithm has
+     mathematically proven properties and it can provide truly
+     decorrelated numbers at a known level of randomness.  The default
+     level of decorrelation recommended by Lu"scher is provided by
+     `gsl_rng_ranlux', while `gsl_rng_ranlux389' gives the highest
+     level of randomness, with all 24 bits decorrelated.  Both types of
+     generator use 24 words of state per generator.
+
+     For more information see,
+          M. Lu"scher, "A portable high-quality random number generator
+          for lattice field theory calculations", `Computer Physics
+          Communications', 79 (1994) 100-110.
+
+          F. James, "RANLUX: A Fortran implementation of the
+          high-quality pseudo-random number generator of Lu"scher",
+          `Computer Physics Communications', 79 (1994) 111-114
+
+ -- Generator: gsl_rng_cmrg
+     This is a combined multiple recursive generator by L'Ecuyer.  Its
+     sequence is,
+
+          z_n = (x_n - y_n) mod m_1
+
+     where the two underlying generators x_n and y_n are,
+
+          x_n = (a_1 x_{n-1} + a_2 x_{n-2} + a_3 x_{n-3}) mod m_1
+          y_n = (b_1 y_{n-1} + b_2 y_{n-2} + b_3 y_{n-3}) mod m_2
+
+     with coefficients a_1 = 0, a_2 = 63308, a_3 = -183326, b_1 = 86098,
+     b_2 = 0, b_3 = -539608, and moduli m_1 = 2^31 - 1 = 2147483647 and
+     m_2 = 2145483479.
+
+     The period of this generator is lcm(m_1^3-1, m_2^3-1), which is
+     approximately 2^185 (about 10^56).  It uses 6 words of state per
+     generator.  For more information see,
+
+          P. L'Ecuyer, "Combined Multiple Recursive Random Number
+          Generators", `Operations Research', 44, 5 (1996), 816-822.
+
+ -- Generator: gsl_rng_mrg
+     This is a fifth-order multiple recursive generator by L'Ecuyer,
+     Blouin and Coutre.  Its sequence is,
+
+          x_n = (a_1 x_{n-1} + a_5 x_{n-5}) mod m
+
+     with a_1 = 107374182, a_2 = a_3 = a_4 = 0, a_5 = 104480 and m =
+     2^31 - 1.
+
+     The period of this generator is about 10^46.  It uses 5 words of
+     state per generator.  More information can be found in the
+     following paper,
+          P. L'Ecuyer, F. Blouin, and R. Coutre, "A search for good
+          multiple recursive random number generators", `ACM
+          Transactions on Modeling and Computer Simulation' 3, 87-98
+          (1993).
+
+ -- Generator: gsl_rng_taus
+ -- Generator: gsl_rng_taus2
+     This is a maximally equidistributed combined Tausworthe generator
+     by L'Ecuyer.  The sequence is,
+
+          x_n = (s1_n ^^ s2_n ^^ s3_n)
+
+     where,
+
+          s1_{n+1} = (((s1_n&4294967294)<<12)^^(((s1_n<<13)^^s1_n)>>19))
+          s2_{n+1} = (((s2_n&4294967288)<< 4)^^(((s2_n<< 2)^^s2_n)>>25))
+          s3_{n+1} = (((s3_n&4294967280)<<17)^^(((s3_n<< 3)^^s3_n)>>11))
+
+     computed modulo 2^32.  In the formulas above ^^ denotes
+     "exclusive-or".  Note that the algorithm relies on the properties
+     of 32-bit unsigned integers and has been implemented using a
+     bitmask of `0xFFFFFFFF' to make it work on 64 bit machines.
+
+     The period of this generator is 2^88 (about 10^26).  It uses 3
+     words of state per generator.  For more information see,
+
+          P. L'Ecuyer, "Maximally Equidistributed Combined Tausworthe
+          Generators", `Mathematics of Computation', 65, 213 (1996),
+          203-213.
+
+     The generator `gsl_rng_taus2' uses the same algorithm as
+     `gsl_rng_taus' but with an improved seeding procedure described in
+     the paper,
+
+          P. L'Ecuyer, "Tables of Maximally Equidistributed Combined
+          LFSR Generators", `Mathematics of Computation', 68, 225
+          (1999), 261-269
+
+     The generator `gsl_rng_taus2' should now be used in preference to
+     `gsl_rng_taus'.
+
+ -- Generator: gsl_rng_gfsr4
+     The `gfsr4' generator is like a lagged-fibonacci generator, and
+     produces each number as an `xor''d sum of four previous values.
+
+          r_n = r_{n-A} ^^ r_{n-B} ^^ r_{n-C} ^^ r_{n-D}
+
+     Ziff (ref below) notes that "it is now widely known" that two-tap
+     registers (such as R250, which is described below) have serious
+     flaws, the most obvious one being the three-point correlation that
+     comes from the definition of the generator.  Nice mathematical
+     properties can be derived for GFSR's, and numerics bears out the
+     claim that 4-tap GFSR's with appropriately chosen offsets are as
+     random as can be measured, using the author's test.
+
+     This implementation uses the values suggested the example on p392
+     of Ziff's article: A=471, B=1586, C=6988, D=9689.
+
+     If the offsets are appropriately chosen (such as the one ones in
+     this implementation), then the sequence is said to be maximal;
+     that means that the period is 2^D - 1, where D is the longest lag.
+     (It is one less than 2^D because it is not permitted to have all
+     zeros in the `ra[]' array.)  For this implementation with D=9689
+     that works out to about 10^2917.
+
+     Note that the implementation of this generator using a 32-bit
+     integer amounts to 32 parallel implementations of one-bit
+     generators.  One consequence of this is that the period of this
+     32-bit generator is the same as for the one-bit generator.
+     Moreover, this independence means that all 32-bit patterns are
+     equally likely, and in particular that 0 is an allowed random
+     value.  (We are grateful to Heiko Bauke for clarifying for us these
+     properties of GFSR random number generators.)
+
+     For more information see,
+          Robert M. Ziff, "Four-tap shift-register-sequence
+          random-number generators", `Computers in Physics', 12(4),
+          Jul/Aug 1998, pp 385-392.
+
+
+File: gsl-ref.info,  Node: Unix random number generators,  Next: Other random number generators,  Prev: Random number generator algorithms,  Up: Random Number Generation
+
+17.10 Unix random number generators
+===================================
+
+The standard Unix random number generators `rand', `random' and
+`rand48' are provided as part of GSL. Although these generators are
+widely available individually often they aren't all available on the
+same platform.  This makes it difficult to write portable code using
+them and so we have included the complete set of Unix generators in GSL
+for convenience.  Note that these generators don't produce high-quality
+randomness and aren't suitable for work requiring accurate statistics.
+However, if you won't be measuring statistical quantities and just want
+to introduce some variation into your program then these generators are
+quite acceptable.
+
+ -- Generator: gsl_rng_rand
+     This is the BSD `rand' generator.  Its sequence is
+
+          x_{n+1} = (a x_n + c) mod m
+
+     with a = 1103515245, c = 12345 and m = 2^31.  The seed specifies
+     the initial value, x_1.  The period of this generator is 2^31, and
+     it uses 1 word of storage per generator.
+
+ -- Generator: gsl_rng_random_bsd
+ -- Generator: gsl_rng_random_libc5
+ -- Generator: gsl_rng_random_glibc2
+     These generators implement the `random' family of functions, a set
+     of linear feedback shift register generators originally used in BSD
+     Unix.  There are several versions of `random' in use today: the
+     original BSD version (e.g. on SunOS4), a libc5 version (found on
+     older GNU/Linux systems) and a glibc2 version.  Each version uses a
+     different seeding procedure, and thus produces different sequences.
+
+     The original BSD routines accepted a variable length buffer for the
+     generator state, with longer buffers providing higher-quality
+     randomness.  The `random' function implemented algorithms for
+     buffer lengths of 8, 32, 64, 128 and 256 bytes, and the algorithm
+     with the largest length that would fit into the user-supplied
+     buffer was used.  To support these algorithms additional
+     generators are available with the following names,
+
+          gsl_rng_random8_bsd
+          gsl_rng_random32_bsd
+          gsl_rng_random64_bsd
+          gsl_rng_random128_bsd
+          gsl_rng_random256_bsd
+
+     where the numeric suffix indicates the buffer length.  The
+     original BSD `random' function used a 128-byte default buffer and
+     so `gsl_rng_random_bsd' has been made equivalent to
+     `gsl_rng_random128_bsd'.  Corresponding versions of the `libc5'
+     and `glibc2' generators are also available, with the names
+     `gsl_rng_random8_libc5', `gsl_rng_random8_glibc2', etc.
+
+ -- Generator: gsl_rng_rand48
+     This is the Unix `rand48' generator.  Its sequence is
+
+          x_{n+1} = (a x_n + c) mod m
+
+     defined on 48-bit unsigned integers with a = 25214903917, c = 11
+     and m = 2^48.  The seed specifies the upper 32 bits of the initial
+     value, x_1, with the lower 16 bits set to `0x330E'.  The function
+     `gsl_rng_get' returns the upper 32 bits from each term of the
+     sequence.  This does not have a direct parallel in the original
+     `rand48' functions, but forcing the result to type `long int'
+     reproduces the output of `mrand48'.  The function
+     `gsl_rng_uniform' uses the full 48 bits of internal state to return
+     the double precision number x_n/m, which is equivalent to the
+     function `drand48'.  Note that some versions of the GNU C Library
+     contained a bug in `mrand48' function which caused it to produce
+     different results (only the lower 16-bits of the return value were
+     set).
+
+
+File: gsl-ref.info,  Node: Other random number generators,  Next: Random Number Generator Performance,  Prev: Unix random number generators,  Up: Random Number Generation
+
+17.11 Other random number generators
+====================================
+
+The generators in this section are provided for compatibility with
+existing libraries.  If you are converting an existing program to use
+GSL then you can select these generators to check your new
+implementation against the original one, using the same random number
+generator.  After verifying that your new program reproduces the
+original results you can then switch to a higher-quality generator.
+
+   Note that most of the generators in this section are based on single
+linear congruence relations, which are the least sophisticated type of
+generator.  In particular, linear congruences have poor properties when
+used with a non-prime modulus, as several of these routines do (e.g.
+with a power of two modulus, 2^31 or 2^32).  This leads to periodicity
+in the least significant bits of each number, with only the higher bits
+having any randomness.  Thus if you want to produce a random bitstream
+it is best to avoid using the least significant bits.
+
+ -- Generator: gsl_rng_ranf
+     This is the CRAY random number generator `RANF'.  Its sequence is
+
+          x_{n+1} = (a x_n) mod m
+
+     defined on 48-bit unsigned integers with a = 44485709377909 and m
+     = 2^48.  The seed specifies the lower 32 bits of the initial value,
+     x_1, with the lowest bit set to prevent the seed taking an even
+     value.  The upper 16 bits of x_1 are set to 0. A consequence of
+     this procedure is that the pairs of seeds 2 and 3, 4 and 5, etc
+     produce the same sequences.
+
+     The generator compatible with the CRAY MATHLIB routine RANF. It
+     produces double precision floating point numbers which should be
+     identical to those from the original RANF.
+
+     There is a subtlety in the implementation of the seeding.  The
+     initial state is reversed through one step, by multiplying by the
+     modular inverse of a mod m.  This is done for compatibility with
+     the original CRAY implementation.
+
+     Note that you can only seed the generator with integers up to
+     2^32, while the original CRAY implementation uses non-portable
+     wide integers which can cover all 2^48 states of the generator.
+
+     The function `gsl_rng_get' returns the upper 32 bits from each term
+     of the sequence.  The function `gsl_rng_uniform' uses the full 48
+     bits to return the double precision number x_n/m.
+
+     The period of this generator is 2^46.
+
+ -- Generator: gsl_rng_ranmar
+     This is the RANMAR lagged-fibonacci generator of Marsaglia, Zaman
+     and Tsang.  It is a 24-bit generator, originally designed for
+     single-precision IEEE floating point numbers.  It was included in
+     the CERNLIB high-energy physics library.
+
+ -- Generator: gsl_rng_r250
+     This is the shift-register generator of Kirkpatrick and Stoll.  The
+     sequence is based on the recurrence
+
+          x_n = x_{n-103} ^^ x_{n-250}
+
+     where ^^ denotes "exclusive-or", defined on 32-bit words.  The
+     period of this generator is about 2^250 and it uses 250 words of
+     state per generator.
+
+     For more information see,
+          S. Kirkpatrick and E. Stoll, "A very fast shift-register
+          sequence random number generator", `Journal of Computational
+          Physics', 40, 517-526 (1981)
+
+ -- Generator: gsl_rng_tt800
+     This is an earlier version of the twisted generalized feedback
+     shift-register generator, and has been superseded by the
+     development of MT19937.  However, it is still an acceptable
+     generator in its own right.  It has a period of 2^800 and uses 33
+     words of storage per generator.
+
+     For more information see,
+          Makoto Matsumoto and Yoshiharu Kurita, "Twisted GFSR
+          Generators II", `ACM Transactions on Modelling and Computer
+          Simulation', Vol. 4, No. 3, 1994, pages 254-266.
+
+ -- Generator: gsl_rng_vax
+     This is the VAX generator `MTH$RANDOM'.  Its sequence is,
+
+          x_{n+1} = (a x_n + c) mod m
+
+     with a = 69069, c = 1 and m = 2^32.  The seed specifies the
+     initial value, x_1.  The period of this generator is 2^32 and it
+     uses 1 word of storage per generator.
+
+ -- Generator: gsl_rng_transputer
+     This is the random number generator from the INMOS Transputer
+     Development system.  Its sequence is,
+
+          x_{n+1} = (a x_n) mod m
+
+     with a = 1664525 and m = 2^32.  The seed specifies the initial
+     value, x_1.
+
+ -- Generator: gsl_rng_randu
+     This is the IBM `RANDU' generator.  Its sequence is
+
+          x_{n+1} = (a x_n) mod m
+
+     with a = 65539 and m = 2^31.  The seed specifies the initial value,
+     x_1.  The period of this generator was only 2^29.  It has become a
+     textbook example of a poor generator.
+
+ -- Generator: gsl_rng_minstd
+     This is Park and Miller's "minimal standard" MINSTD generator, a
+     simple linear congruence which takes care to avoid the major
+     pitfalls of such algorithms.  Its sequence is,
+
+          x_{n+1} = (a x_n) mod m
+
+     with a = 16807 and m = 2^31 - 1 = 2147483647.  The seed specifies
+     the initial value, x_1.  The period of this generator is about
+     2^31.
+
+     This generator is used in the IMSL Library (subroutine RNUN) and in
+     MATLAB (the RAND function).  It is also sometimes known by the
+     acronym "GGL" (I'm not sure what that stands for).
+
+     For more information see,
+          Park and Miller, "Random Number Generators: Good ones are
+          hard to find", `Communications of the ACM', October 1988,
+          Volume 31, No 10, pages 1192-1201.
+
+ -- Generator: gsl_rng_uni
+ -- Generator: gsl_rng_uni32
+     This is a reimplementation of the 16-bit SLATEC random number
+     generator RUNIF. A generalization of the generator to 32 bits is
+     provided by `gsl_rng_uni32'.  The original source code is
+     available from NETLIB.
+
+ -- Generator: gsl_rng_slatec
+     This is the SLATEC random number generator RAND. It is ancient.
+     The original source code is available from NETLIB.
+
+ -- Generator: gsl_rng_zuf
+     This is the ZUFALL lagged Fibonacci series generator of Peterson.
+     Its sequence is,
+
+          t = u_{n-273} + u_{n-607}
+          u_n  = t - floor(t)
+
+     The original source code is available from NETLIB.  For more
+     information see,
+          W. Petersen, "Lagged Fibonacci Random Number Generators for
+          the NEC SX-3", `International Journal of High Speed
+          Computing' (1994).
+
+ -- Generator: gsl_rng_knuthran2
+     This is a second-order multiple recursive generator described by
+     Knuth in `Seminumerical Algorithms', 3rd Ed., page 108.  Its
+     sequence is,
+
+          x_n = (a_1 x_{n-1} + a_2 x_{n-2}) mod m
+
+     with a_1 = 271828183, a_2 = 314159269, and m = 2^31 - 1.
+
+ -- Generator: gsl_rng_knuthran2002
+ -- Generator: gsl_rng_knuthran
+     This is a second-order multiple recursive generator described by
+     Knuth in `Seminumerical Algorithms', 3rd Ed., Section 3.6.  Knuth
+     provides its C code.  The updated routine `gsl_rng_knuthran2002'
+     is from the revised 9th printing and corrects some weaknesses in
+     the earlier version, which is implemented as `gsl_rng_knuthran'.
+
+ -- Generator: gsl_rng_borosh13
+ -- Generator: gsl_rng_fishman18
+ -- Generator: gsl_rng_fishman20
+ -- Generator: gsl_rng_lecuyer21
+ -- Generator: gsl_rng_waterman14
+     These multiplicative generators are taken from Knuth's
+     `Seminumerical Algorithms', 3rd Ed., pages 106-108. Their sequence
+     is,
+
+          x_{n+1} = (a x_n) mod m
+
+     where the seed specifies the initial value, x_1.  The parameters a
+     and m are as follows, Borosh-Niederreiter: a = 1812433253, m =
+     2^32, Fishman18: a = 62089911, m = 2^31 - 1, Fishman20: a = 48271,
+     m = 2^31 - 1, L'Ecuyer: a = 40692, m = 2^31 - 249, Waterman: a =
+     1566083941, m = 2^32.
+
+ -- Generator: gsl_rng_fishman2x
+     This is the L'Ecuyer-Fishman random number generator. It is taken
+     from Knuth's `Seminumerical Algorithms', 3rd Ed., page 108. Its
+     sequence is,
+
+          z_{n+1} = (x_n - y_n) mod m
+
+     with m = 2^31 - 1.  x_n and y_n are given by the `fishman20' and
+     `lecuyer21' algorithms.  The seed specifies the initial value, x_1.
+
+
+ -- Generator: gsl_rng_coveyou
+     This is the Coveyou random number generator. It is taken from
+     Knuth's `Seminumerical Algorithms', 3rd Ed., Section 3.2.2. Its
+     sequence is,
+
+          x_{n+1} = (x_n (x_n + 1)) mod m
+
+     with m = 2^32.  The seed specifies the initial value, x_1.
+
+
+File: gsl-ref.info,  Node: Random Number Generator Performance,  Next: Random Number Generator Examples,  Prev: Other random number generators,  Up: Random Number Generation
+
+17.12 Performance
+=================
+
+The following table shows the relative performance of a selection the
+available random number generators.  The fastest simulation quality
+generators are `taus', `gfsr4' and `mt19937'.  The generators which
+offer the best mathematically-proven quality are those based on the
+RANLUX algorithm.
+
+     1754 k ints/sec,    870 k doubles/sec, taus
+     1613 k ints/sec,    855 k doubles/sec, gfsr4
+     1370 k ints/sec,    769 k doubles/sec, mt19937
+      565 k ints/sec,    571 k doubles/sec, ranlxs0
+      400 k ints/sec,    405 k doubles/sec, ranlxs1
+      490 k ints/sec,    389 k doubles/sec, mrg
+      407 k ints/sec,    297 k doubles/sec, ranlux
+      243 k ints/sec,    254 k doubles/sec, ranlxd1
+      251 k ints/sec,    253 k doubles/sec, ranlxs2
+      238 k ints/sec,    215 k doubles/sec, cmrg
+      247 k ints/sec,    198 k doubles/sec, ranlux389
+      141 k ints/sec,    140 k doubles/sec, ranlxd2
+
+     1852 k ints/sec,    935 k doubles/sec, ran3
+      813 k ints/sec,    575 k doubles/sec, ran0
+      787 k ints/sec,    476 k doubles/sec, ran1
+      379 k ints/sec,    292 k doubles/sec, ran2
+
+
+File: gsl-ref.info,  Node: Random Number Generator Examples,  Next: Random Number References and Further Reading,  Prev: Random Number Generator Performance,  Up: Random Number Generation
+
+17.13 Examples
+==============
+
+The following program demonstrates the use of a random number generator
+to produce uniform random numbers in the range [0.0, 1.0),
+
+     #include <stdio.h>
+     #include <gsl/gsl_rng.h>
+
+     int
+     main (void)
+     {
+       const gsl_rng_type * T;
+       gsl_rng * r;
+
+       int i, n = 10;
+
+       gsl_rng_env_setup();
+
+       T = gsl_rng_default;
+       r = gsl_rng_alloc (T);
+
+       for (i = 0; i < n; i++)
+         {
+           double u = gsl_rng_uniform (r);
+           printf ("%.5f\n", u);
+         }
+
+       gsl_rng_free (r);
+
+       return 0;
+     }
+
+Here is the output of the program,
+
+     $ ./a.out
+     0.99974
+     0.16291
+     0.28262
+     0.94720
+     0.23166
+     0.48497
+     0.95748
+     0.74431
+     0.54004
+     0.73995
+
+The numbers depend on the seed used by the generator.  The default seed
+can be changed with the `GSL_RNG_SEED' environment variable to produce
+a different stream of numbers.  The generator itself can be changed
+using the environment variable `GSL_RNG_TYPE'.  Here is the output of
+the program using a seed value of 123 and the multiple-recursive
+generator `mrg',
+
+     $ GSL_RNG_SEED=123 GSL_RNG_TYPE=mrg ./a.out
+     GSL_RNG_TYPE=mrg
+     GSL_RNG_SEED=123
+     0.33050
+     0.86631
+     0.32982
+     0.67620
+     0.53391
+     0.06457
+     0.16847
+     0.70229
+     0.04371
+     0.86374
+
+
+File: gsl-ref.info,  Node: Random Number References and Further Reading,  Next: Random Number Acknowledgements,  Prev: Random Number Generator Examples,  Up: Random Number Generation
+
+17.14 References and Further Reading
+====================================
+
+The subject of random number generation and testing is reviewed
+extensively in Knuth's `Seminumerical Algorithms'.
+
+     Donald E. Knuth, `The Art of Computer Programming: Seminumerical
+     Algorithms' (Vol 2, 3rd Ed, 1997), Addison-Wesley, ISBN 0201896842.
+
+Further information is available in the review paper written by Pierre
+L'Ecuyer,
+
+     P. L'Ecuyer, "Random Number Generation", Chapter 4 of the Handbook
+     on Simulation, Jerry Banks Ed., Wiley, 1998, 93-137.
+
+     `http://www.iro.umontreal.ca/~lecuyer/papers.html' in the file
+     `handsim.ps'.
+
+The source code for the DIEHARD random number generator tests is also
+available online,
+
+     `DIEHARD source code' G. Marsaglia,
+
+     `http://stat.fsu.edu/pub/diehard/'
+
+A comprehensive set of random number generator tests is available from
+NIST,
+
+     NIST Special Publication 800-22, "A Statistical Test Suite for the
+     Validation of Random Number Generators and Pseudo Random Number
+     Generators for Cryptographic Applications".
+
+     `http://csrc.nist.gov/rng/'
+
+
+File: gsl-ref.info,  Node: Random Number Acknowledgements,  Prev: Random Number References and Further Reading,  Up: Random Number Generation
+
+17.15 Acknowledgements
+======================
+
+Thanks to Makoto Matsumoto, Takuji Nishimura and Yoshiharu Kurita for
+making the source code to their generators (MT19937, MM&TN; TT800,
+MM&YK) available under the GNU General Public License.  Thanks to Martin
+Lu"scher for providing notes and source code for the RANLXS and RANLXD
+generators.
+
+
+File: gsl-ref.info,  Node: Quasi-Random Sequences,  Next: Random Number Distributions,  Prev: Random Number Generation,  Up: Top
+
+18 Quasi-Random Sequences
+*************************
+
+This chapter describes functions for generating quasi-random sequences
+in arbitrary dimensions.  A quasi-random sequence progressively covers a
+d-dimensional space with a set of points that are uniformly
+distributed.  Quasi-random sequences are also known as low-discrepancy
+sequences.  The quasi-random sequence generators use an interface that
+is similar to the interface for random number generators, except that
+seeding is not required--each generator produces a single sequence.
+
+   The functions described in this section are declared in the header
+file `gsl_qrng.h'.
+
+* Menu:
+
+* Quasi-random number generator initialization::
+* Sampling from a quasi-random number generator::
+* Auxiliary quasi-random number generator functions::
+* Saving and resorting quasi-random number generator state::
+* Quasi-random number generator algorithms::
+* Quasi-random number generator examples::
+* Quasi-random number references::
+
+
+File: gsl-ref.info,  Node: Quasi-random number generator initialization,  Next: Sampling from a quasi-random number generator,  Up: Quasi-Random Sequences
+
+18.1 Quasi-random number generator initialization
+=================================================
+
+ -- Function: gsl_qrng * gsl_qrng_alloc (const gsl_qrng_type * T,
+          unsigned int D)
+     This function returns a pointer to a newly-created instance of a
+     quasi-random sequence generator of type T and dimension D.  If
+     there is insufficient memory to create the generator then the
+     function returns a null pointer and the error handler is invoked
+     with an error code of `GSL_ENOMEM'.
+
+ -- Function: void gsl_qrng_free (gsl_qrng * Q)
+     This function frees all the memory associated with the generator Q.
+
+ -- Function: void gsl_qrng_init (gsl_qrng * Q)
+     This function reinitializes the generator Q to its starting point.
+     Note that quasi-random sequences do not use a seed and always
+     produce the same set of values.
+
+
+File: gsl-ref.info,  Node: Sampling from a quasi-random number generator,  Next: Auxiliary quasi-random number generator functions,  Prev: Quasi-random number generator initialization,  Up: Quasi-Random Sequences
+
+18.2 Sampling from a quasi-random number generator
+==================================================
+
+ -- Function: int gsl_qrng_get (const gsl_qrng * Q, double X[])
+     This function stores the next point from the sequence generator Q
+     in the array X.  The space available for X must match the
+     dimension of the generator.  The point X will lie in the range 0 <
+     x_i < 1 for each x_i.
+
+
+File: gsl-ref.info,  Node: Auxiliary quasi-random number generator functions,  Next: Saving and resorting quasi-random number generator state,  Prev: Sampling from a quasi-random number generator,  Up: Quasi-Random Sequences
+
+18.3 Auxiliary quasi-random number generator functions
+======================================================
+
+ -- Function: const char * gsl_qrng_name (const gsl_qrng * Q)
+     This function returns a pointer to the name of the generator.
+
+ -- Function: size_t gsl_qrng_size (const gsl_qrng * Q)
+ -- Function: void * gsl_qrng_state (const gsl_qrng * Q)
+     These functions return a pointer to the state of generator R and
+     its size.  You can use this information to access the state
+     directly.  For example, the following code will write the state of
+     a generator to a stream,
+
+          void * state = gsl_qrng_state (q);
+          size_t n = gsl_qrng_size (q);
+          fwrite (state, n, 1, stream);
+
+
+File: gsl-ref.info,  Node: Saving and resorting quasi-random number generator state,  Next: Quasi-random number generator algorithms,  Prev: Auxiliary quasi-random number generator functions,  Up: Quasi-Random Sequences
+
+18.4 Saving and resorting quasi-random number generator state
+=============================================================
+
+ -- Function: int gsl_qrng_memcpy (gsl_qrng * DEST, const gsl_qrng *
+          SRC)
+     This function copies the quasi-random sequence generator SRC into
+     the pre-existing generator DEST, making DEST into an exact copy of
+     SRC.  The two generators must be of the same type.
+
+ -- Function: gsl_qrng * gsl_qrng_clone (const gsl_qrng * Q)
+     This function returns a pointer to a newly created generator which
+     is an exact copy of the generator Q.
+
+
+File: gsl-ref.info,  Node: Quasi-random number generator algorithms,  Next: Quasi-random number generator examples,  Prev: Saving and resorting quasi-random number generator state,  Up: Quasi-Random Sequences
+
+18.5 Quasi-random number generator algorithms
+=============================================
+
+The following quasi-random sequence algorithms are available,
+
+ -- Generator: gsl_qrng_niederreiter_2
+     This generator uses the algorithm described in Bratley, Fox,
+     Niederreiter, `ACM Trans. Model. Comp. Sim.' 2, 195 (1992). It is
+     valid up to 12 dimensions.
+
+ -- Generator: gsl_qrng_sobol
+     This generator uses the Sobol sequence described in Antonov,
+     Saleev, `USSR Comput. Maths. Math. Phys.' 19, 252 (1980). It is
+     valid up to 40 dimensions.
+
+
+File: gsl-ref.info,  Node: Quasi-random number generator examples,  Next: Quasi-random number references,  Prev: Quasi-random number generator algorithms,  Up: Quasi-Random Sequences
+
+18.6 Examples
+=============
+
+The following program prints the first 1024 points of the 2-dimensional
+Sobol sequence.
+
+     #include <stdio.h>
+     #include <gsl/gsl_qrng.h>
+
+     int
+     main (void)
+     {
+       int i;
+       gsl_qrng * q = gsl_qrng_alloc (gsl_qrng_sobol, 2);
+
+       for (i = 0; i < 1024; i++)
+         {
+           double v[2];
+           gsl_qrng_get (q, v);
+           printf ("%.5f %.5f\n", v[0], v[1]);
+         }
+
+       gsl_qrng_free (q);
+       return 0;
+     }
+
+Here is the output from the program,
+
+     $ ./a.out
+     0.50000 0.50000
+     0.75000 0.25000
+     0.25000 0.75000
+     0.37500 0.37500
+     0.87500 0.87500
+     0.62500 0.12500
+     0.12500 0.62500
+     ....
+
+It can be seen that successive points progressively fill-in the spaces
+between previous points.
+
+
+File: gsl-ref.info,  Node: Quasi-random number references,  Prev: Quasi-random number generator examples,  Up: Quasi-Random Sequences
+
+18.7 References
+===============
+
+The implementations of the quasi-random sequence routines are based on
+the algorithms described in the following paper,
+
+     P. Bratley and B.L. Fox and H. Niederreiter, "Algorithm 738:
+     Programs to Generate Niederreiter's Low-discrepancy Sequences",
+     `ACM Transactions on Mathematical Software', Vol. 20, No. 4,
+     December, 1994, p. 494-495.
+
+
+File: gsl-ref.info,  Node: Random Number Distributions,  Next: Statistics,  Prev: Quasi-Random Sequences,  Up: Top
+
+19 Random Number Distributions
+******************************
+
+This chapter describes functions for generating random variates and
+computing their probability distributions.  Samples from the
+distributions described in this chapter can be obtained using any of the
+random number generators in the library as an underlying source of
+randomness.
+
+   In the simplest cases a non-uniform distribution can be obtained
+analytically from the uniform distribution of a random number generator
+by applying an appropriate transformation.  This method uses one call to
+the random number generator.  More complicated distributions are created
+by the "acceptance-rejection" method, which compares the desired
+distribution against a distribution which is similar and known
+analytically.  This usually requires several samples from the generator.
+
+   The library also provides cumulative distribution functions and
+inverse cumulative distribution functions, sometimes referred to as
+quantile functions.  The cumulative distribution functions and their
+inverses are computed separately for the upper and lower tails of the
+distribution, allowing full accuracy to be retained for small results.
+
+   The functions for random variates and probability density functions
+described in this section are declared in `gsl_randist.h'.  The
+corresponding cumulative distribution functions are declared in
+`gsl_cdf.h'.
+
+   Note that the discrete random variate functions always return a
+value of type `unsigned int', and on most platforms this has a maximum
+value of 2^32-1 ~=~ 4.29e9. They should only be called with a safe
+range of parameters (where there is a negligible probability of a
+variate exceeding this limit) to prevent incorrect results due to
+overflow.
+
+* Menu:
+
+* Random Number Distribution Introduction::
+* The Gaussian Distribution::
+* The Gaussian Tail Distribution::
+* The Bivariate Gaussian Distribution::
+* The Exponential Distribution::
+* The Laplace Distribution::
+* The Exponential Power Distribution::
+* The Cauchy Distribution::
+* The Rayleigh Distribution::
+* The Rayleigh Tail Distribution::
+* The Landau Distribution::
+* The Levy alpha-Stable Distributions::
+* The Levy skew alpha-Stable Distribution::
+* The Gamma Distribution::
+* The Flat (Uniform) Distribution::
+* The Lognormal Distribution::
+* The Chi-squared Distribution::
+* The F-distribution::
+* The t-distribution::
+* The Beta Distribution::
+* The Logistic Distribution::
+* The Pareto Distribution::
+* Spherical Vector Distributions::
+* The Weibull Distribution::
+* The Type-1 Gumbel Distribution::
+* The Type-2 Gumbel Distribution::
+* The Dirichlet Distribution::
+* General Discrete Distributions::
+* The Poisson Distribution::
+* The Bernoulli Distribution::
+* The Binomial Distribution::
+* The Multinomial Distribution::
+* The Negative Binomial Distribution::
+* The Pascal Distribution::
+* The Geometric Distribution::
+* The Hypergeometric Distribution::
+* The Logarithmic Distribution::
+* Shuffling and Sampling::
+* Random Number Distribution Examples::
+* Random Number Distribution References and Further Reading::
+
+
+File: gsl-ref.info,  Node: Random Number Distribution Introduction,  Next: The Gaussian Distribution,  Up: Random Number Distributions
+
+19.1 Introduction
+=================
+
+Continuous random number distributions are defined by a probability
+density function, p(x), such that the probability of x occurring in the
+infinitesimal range x to x+dx is p dx.
+
+   The cumulative distribution function for the lower tail P(x) is
+defined by the integral,
+
+     P(x) = \int_{-\infty}^{x} dx' p(x')
+
+and gives the probability of a variate taking a value less than x.
+
+   The cumulative distribution function for the upper tail Q(x) is
+defined by the integral,
+
+     Q(x) = \int_{x}^{+\infty} dx' p(x')
+
+and gives the probability of a variate taking a value greater than x.
+
+   The upper and lower cumulative distribution functions are related by
+P(x) + Q(x) = 1 and satisfy 0 <= P(x) <= 1, 0 <= Q(x) <= 1.
+
+   The inverse cumulative distributions, x=P^{-1}(P) and x=Q^{-1}(Q)
+give the values of x which correspond to a specific value of P or Q.
+They can be used to find confidence limits from probability values.
+
+   For discrete distributions the probability of sampling the integer
+value k is given by p(k), where \sum_k p(k) = 1.  The cumulative
+distribution for the lower tail P(k) of a discrete distribution is
+defined as,
+
+     P(k) = \sum_{i <= k} p(i)
+
+where the sum is over the allowed range of the distribution less than
+or equal to k.
+
+   The cumulative distribution for the upper tail of a discrete
+distribution Q(k) is defined as
+
+     Q(k) = \sum_{i > k} p(i)
+
+giving the sum of probabilities for all values greater than k.  These
+two definitions satisfy the identity P(k)+Q(k)=1.
+
+   If the range of the distribution is 1 to n inclusive then P(n)=1,
+Q(n)=0 while P(1) = p(1), Q(1)=1-p(1).
+
+
+File: gsl-ref.info,  Node: The Gaussian Distribution,  Next: The Gaussian Tail Distribution,  Prev: Random Number Distribution Introduction,  Up: Random Number Distributions
+
+19.2 The Gaussian Distribution
+==============================
+
+ -- Function: double gsl_ran_gaussian (const gsl_rng * R, double SIGMA)
+     This function returns a Gaussian random variate, with mean zero and
+     standard deviation SIGMA.  The probability distribution for
+     Gaussian random variates is,
+
+          p(x) dx = {1 \over \sqrt{2 \pi \sigma^2}} \exp (-x^2 / 2\sigma^2) dx
+
+     for x in the range -\infty to +\infty.  Use the transformation z =
+     \mu + x on the numbers returned by `gsl_ran_gaussian' to obtain a
+     Gaussian distribution with mean \mu.  This function uses the
+     Box-Mueller algorithm which requires two calls to the random
+     number generator R.
+
+ -- Function: double gsl_ran_gaussian_pdf (double X, double SIGMA)
+     This function computes the probability density p(x) at X for a
+     Gaussian distribution with standard deviation SIGMA, using the
+     formula given above.
+
+
+ -- Function: double gsl_ran_gaussian_ziggurat (const gsl_rng * R,
+          double SIGMA)
+ -- Function: double gsl_ran_gaussian_ratio_method (const gsl_rng * R,
+          double SIGMA)
+     This function computes a Gaussian random variate using the
+     alternative Marsaglia-Tsang ziggurat and Kinderman-Monahan-Leva
+     ratio methods.  The Ziggurat algorithm is the fastest available
+     algorithm in most cases.
+
+ -- Function: double gsl_ran_ugaussian (const gsl_rng * R)
+ -- Function: double gsl_ran_ugaussian_pdf (double X)
+ -- Function: double gsl_ran_ugaussian_ratio_method (const gsl_rng * R)
+     These functions compute results for the unit Gaussian
+     distribution.  They are equivalent to the functions above with a
+     standard deviation of one, SIGMA = 1.
+
+ -- Function: double gsl_cdf_gaussian_P (double X, double SIGMA)
+ -- Function: double gsl_cdf_gaussian_Q (double X, double SIGMA)
+ -- Function: double gsl_cdf_gaussian_Pinv (double P, double SIGMA)
+ -- Function: double gsl_cdf_gaussian_Qinv (double Q, double SIGMA)
+     These functions compute the cumulative distribution functions
+     P(x), Q(x) and their inverses for the Gaussian distribution with
+     standard deviation SIGMA.
+
+ -- Function: double gsl_cdf_ugaussian_P (double X)
+ -- Function: double gsl_cdf_ugaussian_Q (double X)
+ -- Function: double gsl_cdf_ugaussian_Pinv (double P)
+ -- Function: double gsl_cdf_ugaussian_Qinv (double Q)
+     These functions compute the cumulative distribution functions
+     P(x), Q(x) and their inverses for the unit Gaussian distribution.
+
+
+File: gsl-ref.info,  Node: The Gaussian Tail Distribution,  Next: The Bivariate Gaussian Distribution,  Prev: The Gaussian Distribution,  Up: Random Number Distributions
+
+19.3 The Gaussian Tail Distribution
+===================================
+
+ -- Function: double gsl_ran_gaussian_tail (const gsl_rng * R, double
+          A, double SIGMA)
+     This function provides random variates from the upper tail of a
+     Gaussian distribution with standard deviation SIGMA.  The values
+     returned are larger than the lower limit A, which must be
+     positive.  The method is based on Marsaglia's famous
+     rectangle-wedge-tail algorithm (Ann.  Math. Stat. 32, 894-899
+     (1961)), with this aspect explained in Knuth, v2, 3rd ed, p139,586
+     (exercise 11).
+
+     The probability distribution for Gaussian tail random variates is,
+
+          p(x) dx = {1 \over N(a;\sigma) \sqrt{2 \pi \sigma^2}} \exp (- x^2/(2 \sigma^2)) dx
+
+     for x > a where N(a;\sigma) is the normalization constant,
+
+          N(a;\sigma) = (1/2) erfc(a / sqrt(2 sigma^2)).
+
+
+ -- Function: double gsl_ran_gaussian_tail_pdf (double X, double A,
+          double SIGMA)
+     This function computes the probability density p(x) at X for a
+     Gaussian tail distribution with standard deviation SIGMA and lower
+     limit A, using the formula given above.
+
+
+ -- Function: double gsl_ran_ugaussian_tail (const gsl_rng * R, double
+          A)
+ -- Function: double gsl_ran_ugaussian_tail_pdf (double X, double A)
+     These functions compute results for the tail of a unit Gaussian
+     distribution.  They are equivalent to the functions above with a
+     standard deviation of one, SIGMA = 1.
+
+
+File: gsl-ref.info,  Node: The Bivariate Gaussian Distribution,  Next: The Exponential Distribution,  Prev: The Gaussian Tail Distribution,  Up: Random Number Distributions
+
+19.4 The Bivariate Gaussian Distribution
+========================================
+
+ -- Function: void gsl_ran_bivariate_gaussian (const gsl_rng * R,
+          double SIGMA_X, double SIGMA_Y, double RHO, double * X,
+          double * Y)
+     This function generates a pair of correlated Gaussian variates,
+     with mean zero, correlation coefficient RHO and standard deviations
+     SIGMA_X and SIGMA_Y in the x and y directions.  The probability
+     distribution for bivariate Gaussian random variates is,
+
+          p(x,y) dx dy = {1 \over 2 \pi \sigma_x \sigma_y \sqrt{1-\rho^2}} \exp (-(x^2/\sigma_x^2 + y^2/\sigma_y^2 - 2 \rho x y/(\sigma_x\sigma_y))/2(1-\rho^2)) dx dy
+
+     for x,y in the range -\infty to +\infty.  The correlation
+     coefficient RHO should lie between 1 and -1.
+
+ -- Function: double gsl_ran_bivariate_gaussian_pdf (double X, double
+          Y, double SIGMA_X, double SIGMA_Y, double RHO)
+     This function computes the probability density p(x,y) at (X,Y) for
+     a bivariate Gaussian distribution with standard deviations
+     SIGMA_X, SIGMA_Y and correlation coefficient RHO, using the
+     formula given above.
+
+
+
+File: gsl-ref.info,  Node: The Exponential Distribution,  Next: The Laplace Distribution,  Prev: The Bivariate Gaussian Distribution,  Up: Random Number Distributions
+
+19.5 The Exponential Distribution
+=================================
+
+ -- Function: double gsl_ran_exponential (const gsl_rng * R, double MU)
+     This function returns a random variate from the exponential
+     distribution with mean MU. The distribution is,
+
+          p(x) dx = {1 \over \mu} \exp(-x/\mu) dx
+
+     for x >= 0.
+
+ -- Function: double gsl_ran_exponential_pdf (double X, double MU)
+     This function computes the probability density p(x) at X for an
+     exponential distribution with mean MU, using the formula given
+     above.
+
+
+ -- Function: double gsl_cdf_exponential_P (double X, double MU)
+ -- Function: double gsl_cdf_exponential_Q (double X, double MU)
+ -- Function: double gsl_cdf_exponential_Pinv (double P, double MU)
+ -- Function: double gsl_cdf_exponential_Qinv (double Q, double MU)
+     These functions compute the cumulative distribution functions
+     P(x), Q(x) and their inverses for the exponential distribution
+     with mean MU.
+
+
+File: gsl-ref.info,  Node: The Laplace Distribution,  Next: The Exponential Power Distribution,  Prev: The Exponential Distribution,  Up: Random Number Distributions
+
+19.6 The Laplace Distribution
+=============================
+
+ -- Function: double gsl_ran_laplace (const gsl_rng * R, double A)
+     This function returns a random variate from the Laplace
+     distribution with width A.  The distribution is,
+
+          p(x) dx = {1 \over 2 a}  \exp(-|x/a|) dx
+
+     for -\infty < x < \infty.
+
+ -- Function: double gsl_ran_laplace_pdf (double X, double A)
+     This function computes the probability density p(x) at X for a
+     Laplace distribution with width A, using the formula given above.
+
+
+ -- Function: double gsl_cdf_laplace_P (double X, double A)
+ -- Function: double gsl_cdf_laplace_Q (double X, double A)
+ -- Function: double gsl_cdf_laplace_Pinv (double P, double A)
+ -- Function: double gsl_cdf_laplace_Qinv (double Q, double A)
+     These functions compute the cumulative distribution functions
+     P(x), Q(x) and their inverses for the Laplace distribution with
+     width A.
+
+
+File: gsl-ref.info,  Node: The Exponential Power Distribution,  Next: The Cauchy Distribution,  Prev: The Laplace Distribution,  Up: Random Number Distributions
+
+19.7 The Exponential Power Distribution
+=======================================
+
+ -- Function: double gsl_ran_exppow (const gsl_rng * R, double A,
+          double B)
+     This function returns a random variate from the exponential power
+     distribution with scale parameter A and exponent B.  The
+     distribution is,
+
+          p(x) dx = {1 \over 2 a \Gamma(1+1/b)} \exp(-|x/a|^b) dx
+
+     for x >= 0.  For b = 1 this reduces to the Laplace distribution.
+     For b = 2 it has the same form as a gaussian distribution, but
+     with a = \sqrt{2} \sigma.
+
+ -- Function: double gsl_ran_exppow_pdf (double X, double A, double B)
+     This function computes the probability density p(x) at X for an
+     exponential power distribution with scale parameter A and exponent
+     B, using the formula given above.
+
+
+ -- Function: double gsl_cdf_exppow_P (double X, double A, double B)
+ -- Function: double gsl_cdf_exppow_Q (double X, double A, double B)
+     These functions compute the cumulative distribution functions
+     P(x), Q(x) for the exponential power distribution with parameters
+     A and B.
+
+
+File: gsl-ref.info,  Node: The Cauchy Distribution,  Next: The Rayleigh Distribution,  Prev: The Exponential Power Distribution,  Up: Random Number Distributions
+
+19.8 The Cauchy Distribution
+============================
+
+ -- Function: double gsl_ran_cauchy (const gsl_rng * R, double A)
+     This function returns a random variate from the Cauchy
+     distribution with scale parameter A.  The probability distribution
+     for Cauchy random variates is,
+
+          p(x) dx = {1 \over a\pi (1 + (x/a)^2) } dx
+
+     for x in the range -\infty to +\infty.  The Cauchy distribution is
+     also known as the Lorentz distribution.
+
+ -- Function: double gsl_ran_cauchy_pdf (double X, double A)
+     This function computes the probability density p(x) at X for a
+     Cauchy distribution with scale parameter A, using the formula
+     given above.
+
+
+ -- Function: double gsl_cdf_cauchy_P (double X, double A)
+ -- Function: double gsl_cdf_cauchy_Q (double X, double A)
+ -- Function: double gsl_cdf_cauchy_Pinv (double P, double A)
+ -- Function: double gsl_cdf_cauchy_Qinv (double Q, double A)
+     These functions compute the cumulative distribution functions
+     P(x), Q(x) and their inverses for the Cauchy distribution with
+     scale parameter A.
+
+
+File: gsl-ref.info,  Node: The Rayleigh Distribution,  Next: The Rayleigh Tail Distribution,  Prev: The Cauchy Distribution,  Up: Random Number Distributions
+
+19.9 The Rayleigh Distribution
+==============================
+
+ -- Function: double gsl_ran_rayleigh (const gsl_rng * R, double SIGMA)
+     This function returns a random variate from the Rayleigh
+     distribution with scale parameter SIGMA.  The distribution is,
+
+          p(x) dx = {x \over \sigma^2} \exp(- x^2/(2 \sigma^2)) dx
+
+     for x > 0.
+
+ -- Function: double gsl_ran_rayleigh_pdf (double X, double SIGMA)
+     This function computes the probability density p(x) at X for a
+     Rayleigh distribution with scale parameter SIGMA, using the
+     formula given above.
+
+
+ -- Function: double gsl_cdf_rayleigh_P (double X, double SIGMA)
+ -- Function: double gsl_cdf_rayleigh_Q (double X, double SIGMA)
+ -- Function: double gsl_cdf_rayleigh_Pinv (double P, double SIGMA)
+ -- Function: double gsl_cdf_rayleigh_Qinv (double Q, double SIGMA)
+     These functions compute the cumulative distribution functions
+     P(x), Q(x) and their inverses for the Rayleigh distribution with
+     scale parameter SIGMA.
+
+
+File: gsl-ref.info,  Node: The Rayleigh Tail Distribution,  Next: The Landau Distribution,  Prev: The Rayleigh Distribution,  Up: Random Number Distributions
+
+19.10 The Rayleigh Tail Distribution
+====================================
+
+ -- Function: double gsl_ran_rayleigh_tail (const gsl_rng * R, double
+          A, double SIGMA)
+     This function returns a random variate from the tail of the
+     Rayleigh distribution with scale parameter SIGMA and a lower limit
+     of A.  The distribution is,
+
+          p(x) dx = {x \over \sigma^2} \exp ((a^2 - x^2) /(2 \sigma^2)) dx
+
+     for x > a.
+
+ -- Function: double gsl_ran_rayleigh_tail_pdf (double X, double A,
+          double SIGMA)
+     This function computes the probability density p(x) at X for a
+     Rayleigh tail distribution with scale parameter SIGMA and lower
+     limit A, using the formula given above.
+
+
+
+File: gsl-ref.info,  Node: The Landau Distribution,  Next: The Levy alpha-Stable Distributions,  Prev: The Rayleigh Tail Distribution,  Up: Random Number Distributions
+
+19.11 The Landau Distribution
+=============================
+
+ -- Function: double gsl_ran_landau (const gsl_rng * R)
+     This function returns a random variate from the Landau
+     distribution.  The probability distribution for Landau random
+     variates is defined analytically by the complex integral,
+
+          p(x) = (1/(2 \pi i)) \int_{c-i\infty}^{c+i\infty} ds exp(s log(s) + x s)
+     For numerical purposes it is more convenient to use the following
+     equivalent form of the integral,
+
+          p(x) = (1/\pi) \int_0^\infty dt \exp(-t \log(t) - x t) \sin(\pi t).
+
+ -- Function: double gsl_ran_landau_pdf (double X)
+     This function computes the probability density p(x) at X for the
+     Landau distribution using an approximation to the formula given
+     above.
+
+
+
+File: gsl-ref.info,  Node: The Levy alpha-Stable Distributions,  Next: The Levy skew alpha-Stable Distribution,  Prev: The Landau Distribution,  Up: Random Number Distributions
+
+19.12 The Levy alpha-Stable Distributions
+=========================================
+
+ -- Function: double gsl_ran_levy (const gsl_rng * R, double C, double
+          ALPHA)
+     This function returns a random variate from the Levy symmetric
+     stable distribution with scale C and exponent ALPHA.  The symmetric
+     stable probability distribution is defined by a fourier transform,
+
+          p(x) = {1 \over 2 \pi} \int_{-\infty}^{+\infty} dt \exp(-it x - |c t|^alpha)
+
+     There is no explicit solution for the form of p(x) and the library
+     does not define a corresponding `pdf' function.  For \alpha = 1
+     the distribution reduces to the Cauchy distribution.  For \alpha =
+     2 it is a Gaussian distribution with \sigma = \sqrt{2} c.  For
+     \alpha < 1 the tails of the distribution become extremely wide.
+
+     The algorithm only works for 0 < alpha <= 2.
+
+
+
+File: gsl-ref.info,  Node: The Levy skew alpha-Stable Distribution,  Next: The Gamma Distribution,  Prev: The Levy alpha-Stable Distributions,  Up: Random Number Distributions
+
+19.13 The Levy skew alpha-Stable Distribution
+=============================================
+
+ -- Function: double gsl_ran_levy_skew (const gsl_rng * R, double C,
+          double ALPHA, double BETA)
+     This function returns a random variate from the Levy skew stable
+     distribution with scale C, exponent ALPHA and skewness parameter
+     BETA.  The skewness parameter must lie in the range [-1,1].  The
+     Levy skew stable probability distribution is defined by a fourier
+     transform,
+
+          p(x) = {1 \over 2 \pi} \int_{-\infty}^{+\infty} dt \exp(-it x - |c t|^alpha (1-i beta sign(t) tan(pi alpha/2)))
+
+     When \alpha = 1 the term \tan(\pi \alpha/2) is replaced by
+     -(2/\pi)\log|t|.  There is no explicit solution for the form of
+     p(x) and the library does not define a corresponding `pdf'
+     function.  For \alpha = 2 the distribution reduces to a Gaussian
+     distribution with \sigma = \sqrt{2} c and the skewness parameter
+     has no effect.  For \alpha < 1 the tails of the distribution
+     become extremely wide.  The symmetric distribution corresponds to
+     \beta = 0.
+
+     The algorithm only works for 0 < alpha <= 2.
+
+   The Levy alpha-stable distributions have the property that if N
+alpha-stable variates are drawn from the distribution p(c, \alpha,
+\beta) then the sum Y = X_1 + X_2 + \dots + X_N will also be
+distributed as an alpha-stable variate, p(N^(1/\alpha) c, \alpha,
+\beta).
+
+
+
+File: gsl-ref.info,  Node: The Gamma Distribution,  Next: The Flat (Uniform) Distribution,  Prev: The Levy skew alpha-Stable Distribution,  Up: Random Number Distributions
+
+19.14 The Gamma Distribution
+============================
+
+ -- Function: double gsl_ran_gamma (const gsl_rng * R, double A, double
+          B)
+     This function returns a random variate from the gamma
+     distribution.  The distribution function is,
+
+          p(x) dx = {1 \over \Gamma(a) b^a} x^{a-1} e^{-x/b} dx
+
+     for x > 0.
+
+     The gamma distribution with an integer parameter A is known as the
+     Erlang distribution.
+
+     The variates are computed using the Marsaglia-Tsang fast gamma
+     method.  This function for this method was previously called
+     `gsl_ran_gamma_mt' and can still be accessed using this name.
+
+ -- Function: double gsl_ran_gamma_knuth (const gsl_rng * R, double A,
+          double B)
+     This function returns a gamma variate using the algorithms from
+     Knuth (vol 2).
+
+ -- Function: double gsl_ran_gamma_pdf (double X, double A, double B)
+     This function computes the probability density p(x) at X for a
+     gamma distribution with parameters A and B, using the formula
+     given above.
+
+
+ -- Function: double gsl_cdf_gamma_P (double X, double A, double B)
+ -- Function: double gsl_cdf_gamma_Q (double X, double A, double B)
+ -- Function: double gsl_cdf_gamma_Pinv (double P, double A, double B)
+ -- Function: double gsl_cdf_gamma_Qinv (double Q, double A, double B)
+     These functions compute the cumulative distribution functions
+     P(x), Q(x) and their inverses for the gamma distribution with
+     parameters A and B.
+
+
+File: gsl-ref.info,  Node: The Flat (Uniform) Distribution,  Next: The Lognormal Distribution,  Prev: The Gamma Distribution,  Up: Random Number Distributions
+
+19.15 The Flat (Uniform) Distribution
+=====================================
+
+ -- Function: double gsl_ran_flat (const gsl_rng * R, double A, double
+          B)
+     This function returns a random variate from the flat (uniform)
+     distribution from A to B. The distribution is,
+
+          p(x) dx = {1 \over (b-a)} dx
+
+     if a <= x < b and 0 otherwise.
+
+ -- Function: double gsl_ran_flat_pdf (double X, double A, double B)
+     This function computes the probability density p(x) at X for a
+     uniform distribution from A to B, using the formula given above.
+
+
+ -- Function: double gsl_cdf_flat_P (double X, double A, double B)
+ -- Function: double gsl_cdf_flat_Q (double X, double A, double B)
+ -- Function: double gsl_cdf_flat_Pinv (double P, double A, double B)
+ -- Function: double gsl_cdf_flat_Qinv (double Q, double A, double B)
+     These functions compute the cumulative distribution functions
+     P(x), Q(x) and their inverses for a uniform distribution from A to
+     B.
+
+
+File: gsl-ref.info,  Node: The Lognormal Distribution,  Next: The Chi-squared Distribution,  Prev: The Flat (Uniform) Distribution,  Up: Random Number Distributions
+
+19.16 The Lognormal Distribution
+================================
+
+ -- Function: double gsl_ran_lognormal (const gsl_rng * R, double ZETA,
+          double SIGMA)
+     This function returns a random variate from the lognormal
+     distribution.  The distribution function is,
+
+          p(x) dx = {1 \over x \sqrt{2 \pi \sigma^2} } \exp(-(\ln(x) - \zeta)^2/2 \sigma^2) dx
+
+     for x > 0.
+
+ -- Function: double gsl_ran_lognormal_pdf (double X, double ZETA,
+          double SIGMA)
+     This function computes the probability density p(x) at X for a
+     lognormal distribution with parameters ZETA and SIGMA, using the
+     formula given above.
+
+
+ -- Function: double gsl_cdf_lognormal_P (double X, double ZETA, double
+          SIGMA)
+ -- Function: double gsl_cdf_lognormal_Q (double X, double ZETA, double
+          SIGMA)
+ -- Function: double gsl_cdf_lognormal_Pinv (double P, double ZETA,
+          double SIGMA)
+ -- Function: double gsl_cdf_lognormal_Qinv (double Q, double ZETA,
+          double SIGMA)
+     These functions compute the cumulative distribution functions
+     P(x), Q(x) and their inverses for the lognormal distribution with
+     parameters ZETA and SIGMA.
+
+
+File: gsl-ref.info,  Node: The Chi-squared Distribution,  Next: The F-distribution,  Prev: The Lognormal Distribution,  Up: Random Number Distributions
+
+19.17 The Chi-squared Distribution
+==================================
+
+The chi-squared distribution arises in statistics.  If Y_i are n
+independent gaussian random variates with unit variance then the
+sum-of-squares,
+
+     X_i = \sum_i Y_i^2
+
+has a chi-squared distribution with n degrees of freedom.
+
+ -- Function: double gsl_ran_chisq (const gsl_rng * R, double NU)
+     This function returns a random variate from the chi-squared
+     distribution with NU degrees of freedom. The distribution function
+     is,
+
+          p(x) dx = {1 \over 2 \Gamma(\nu/2) } (x/2)^{\nu/2 - 1} \exp(-x/2) dx
+
+     for x >= 0.
+
+ -- Function: double gsl_ran_chisq_pdf (double X, double NU)
+     This function computes the probability density p(x) at X for a
+     chi-squared distribution with NU degrees of freedom, using the
+     formula given above.
+
+
+ -- Function: double gsl_cdf_chisq_P (double X, double NU)
+ -- Function: double gsl_cdf_chisq_Q (double X, double NU)
+ -- Function: double gsl_cdf_chisq_Pinv (double P, double NU)
+ -- Function: double gsl_cdf_chisq_Qinv (double Q, double NU)
+     These functions compute the cumulative distribution functions
+     P(x), Q(x) and their inverses for the chi-squared distribution
+     with NU degrees of freedom.
+
+
+File: gsl-ref.info,  Node: The F-distribution,  Next: The t-distribution,  Prev: The Chi-squared Distribution,  Up: Random Number Distributions
+
+19.18 The F-distribution
+========================
+
+The F-distribution arises in statistics.  If Y_1 and Y_2 are
+chi-squared deviates with \nu_1 and \nu_2 degrees of freedom then the
+ratio,
+
+     X = { (Y_1 / \nu_1) \over (Y_2 / \nu_2) }
+
+has an F-distribution F(x;\nu_1,\nu_2).
+
+ -- Function: double gsl_ran_fdist (const gsl_rng * R, double NU1,
+          double NU2)
+     This function returns a random variate from the F-distribution
+     with degrees of freedom NU1 and NU2. The distribution function is,
+
+          p(x) dx =
+             { \Gamma((\nu_1 + \nu_2)/2)
+                  \over \Gamma(\nu_1/2) \Gamma(\nu_2/2) }
+             \nu_1^{\nu_1/2} \nu_2^{\nu_2/2}
+             x^{\nu_1/2 - 1} (\nu_2 + \nu_1 x)^{-\nu_1/2 -\nu_2/2}
+
+     for x >= 0.
+
+ -- Function: double gsl_ran_fdist_pdf (double X, double NU1, double
+          NU2)
+     This function computes the probability density p(x) at X for an
+     F-distribution with NU1 and NU2 degrees of freedom, using the
+     formula given above.
+
+
+ -- Function: double gsl_cdf_fdist_P (double X, double NU1, double NU2)
+ -- Function: double gsl_cdf_fdist_Q (double X, double NU1, double NU2)
+ -- Function: double gsl_cdf_fdist_Pinv (double P, double NU1, double
+          NU2)
+ -- Function: double gsl_cdf_fdist_Qinv (double Q, double NU1, double
+          NU2)
+     These functions compute the cumulative distribution functions
+     P(x), Q(x) and their inverses for the F-distribution with NU1 and
+     NU2 degrees of freedom.
+
+
+File: gsl-ref.info,  Node: The t-distribution,  Next: The Beta Distribution,  Prev: The F-distribution,  Up: Random Number Distributions
+
+19.19 The t-distribution
+========================
+
+The t-distribution arises in statistics.  If Y_1 has a normal
+distribution and Y_2 has a chi-squared distribution with \nu degrees of
+freedom then the ratio,
+
+     X = { Y_1 \over \sqrt{Y_2 / \nu} }
+
+has a t-distribution t(x;\nu) with \nu degrees of freedom.
+
+ -- Function: double gsl_ran_tdist (const gsl_rng * R, double NU)
+     This function returns a random variate from the t-distribution.
+     The distribution function is,
+
+          p(x) dx = {\Gamma((\nu + 1)/2) \over \sqrt{\pi \nu} \Gamma(\nu/2)}
+             (1 + x^2/\nu)^{-(\nu + 1)/2} dx
+
+     for -\infty < x < +\infty.
+
+ -- Function: double gsl_ran_tdist_pdf (double X, double NU)
+     This function computes the probability density p(x) at X for a
+     t-distribution with NU degrees of freedom, using the formula given
+     above.
+
+
+ -- Function: double gsl_cdf_tdist_P (double X, double NU)
+ -- Function: double gsl_cdf_tdist_Q (double X, double NU)
+ -- Function: double gsl_cdf_tdist_Pinv (double P, double NU)
+ -- Function: double gsl_cdf_tdist_Qinv (double Q, double NU)
+     These functions compute the cumulative distribution functions
+     P(x), Q(x) and their inverses for the t-distribution with NU
+     degrees of freedom.
+
+
+File: gsl-ref.info,  Node: The Beta Distribution,  Next: The Logistic Distribution,  Prev: The t-distribution,  Up: Random Number Distributions
+
+19.20 The Beta Distribution
+===========================
+
+ -- Function: double gsl_ran_beta (const gsl_rng * R, double A, double
+          B)
+     This function returns a random variate from the beta distribution.
+     The distribution function is,
+
+          p(x) dx = {\Gamma(a+b) \over \Gamma(a) \Gamma(b)} x^{a-1} (1-x)^{b-1} dx
+
+     for 0 <= x <= 1.
+
+ -- Function: double gsl_ran_beta_pdf (double X, double A, double B)
+     This function computes the probability density p(x) at X for a
+     beta distribution with parameters A and B, using the formula given
+     above.
+
+
+ -- Function: double gsl_cdf_beta_P (double X, double A, double B)
+ -- Function: double gsl_cdf_beta_Q (double X, double A, double B)
+ -- Function: double gsl_cdf_beta_Pinv (double P, double A, double B)
+ -- Function: double gsl_cdf_beta_Qinv (double Q, double A, double B)
+     These functions compute the cumulative distribution functions
+     P(x), Q(x) and their inverses for the beta distribution with
+     parameters A and B.
+
+
+File: gsl-ref.info,  Node: The Logistic Distribution,  Next: The Pareto Distribution,  Prev: The Beta Distribution,  Up: Random Number Distributions
+
+19.21 The Logistic Distribution
+===============================
+
+ -- Function: double gsl_ran_logistic (const gsl_rng * R, double A)
+     This function returns a random variate from the logistic
+     distribution.  The distribution function is,
+
+          p(x) dx = { \exp(-x/a) \over a (1 + \exp(-x/a))^2 } dx
+
+     for -\infty < x < +\infty.
+
+ -- Function: double gsl_ran_logistic_pdf (double X, double A)
+     This function computes the probability density p(x) at X for a
+     logistic distribution with scale parameter A, using the formula
+     given above.
+
+
+ -- Function: double gsl_cdf_logistic_P (double X, double A)
+ -- Function: double gsl_cdf_logistic_Q (double X, double A)
+ -- Function: double gsl_cdf_logistic_Pinv (double P, double A)
+ -- Function: double gsl_cdf_logistic_Qinv (double Q, double A)
+     These functions compute the cumulative distribution functions
+     P(x), Q(x) and their inverses for the logistic distribution with
+     scale parameter A.
+
+
+File: gsl-ref.info,  Node: The Pareto Distribution,  Next: Spherical Vector Distributions,  Prev: The Logistic Distribution,  Up: Random Number Distributions
+
+19.22 The Pareto Distribution
+=============================
+
+ -- Function: double gsl_ran_pareto (const gsl_rng * R, double A,
+          double B)
+     This function returns a random variate from the Pareto
+     distribution of order A.  The distribution function is,
+
+          p(x) dx = (a/b) / (x/b)^{a+1} dx
+
+     for x >= b.
+
+ -- Function: double gsl_ran_pareto_pdf (double X, double A, double B)
+     This function computes the probability density p(x) at X for a
+     Pareto distribution with exponent A and scale B, using the formula
+     given above.
+
+
+ -- Function: double gsl_cdf_pareto_P (double X, double A, double B)
+ -- Function: double gsl_cdf_pareto_Q (double X, double A, double B)
+ -- Function: double gsl_cdf_pareto_Pinv (double P, double A, double B)
+ -- Function: double gsl_cdf_pareto_Qinv (double Q, double A, double B)
+     These functions compute the cumulative distribution functions
+     P(x), Q(x) and their inverses for the Pareto distribution with
+     exponent A and scale B.
+
+
+File: gsl-ref.info,  Node: Spherical Vector Distributions,  Next: The Weibull Distribution,  Prev: The Pareto Distribution,  Up: Random Number Distributions
+
+19.23 Spherical Vector Distributions
+====================================
+
+The spherical distributions generate random vectors, located on a
+spherical surface.  They can be used as random directions, for example
+in the steps of a random walk.
+
+ -- Function: void gsl_ran_dir_2d (const gsl_rng * R, double * X,
+          double * Y)
+ -- Function: void gsl_ran_dir_2d_trig_method (const gsl_rng * R,
+          double * X, double * Y)
+     This function returns a random direction vector v = (X,Y) in two
+     dimensions.  The vector is normalized such that |v|^2 = x^2 + y^2
+     = 1.  The obvious way to do this is to take a uniform random
+     number between 0 and 2\pi and let X and Y be the sine and cosine
+     respectively.  Two trig functions would have been expensive in the
+     old days, but with modern hardware implementations, this is
+     sometimes the fastest way to go.  This is the case for the Pentium
+     (but not the case for the Sun Sparcstation).  One can avoid the
+     trig evaluations by choosing X and Y in the interior of a unit
+     circle (choose them at random from the interior of the enclosing
+     square, and then reject those that are outside the unit circle),
+     and then dividing by \sqrt{x^2 + y^2}.  A much cleverer approach,
+     attributed to von Neumann (See Knuth, v2, 3rd ed, p140, exercise
+     23), requires neither trig nor a square root.  In this approach, U
+     and V are chosen at random from the interior of a unit circle, and
+     then x=(u^2-v^2)/(u^2+v^2) and y=2uv/(u^2+v^2).
+
+ -- Function: void gsl_ran_dir_3d (const gsl_rng * R, double * X,
+          double * Y, double * Z)
+     This function returns a random direction vector v = (X,Y,Z) in
+     three dimensions.  The vector is normalized such that |v|^2 = x^2
+     + y^2 + z^2 = 1.  The method employed is due to Robert E. Knop
+     (CACM 13, 326 (1970)), and explained in Knuth, v2, 3rd ed, p136.
+     It uses the surprising fact that the distribution projected along
+     any axis is actually uniform (this is only true for 3 dimensions).
+
+ -- Function: void gsl_ran_dir_nd (const gsl_rng * R, size_t N, double
+          * X)
+     This function returns a random direction vector v =
+     (x_1,x_2,...,x_n) in N dimensions.  The vector is normalized such
+     that |v|^2 = x_1^2 + x_2^2 + ... + x_n^2 = 1.  The method uses the
+     fact that a multivariate gaussian distribution is spherically
+     symmetric.  Each component is generated to have a gaussian
+     distribution, and then the components are normalized.  The method
+     is described by Knuth, v2, 3rd ed, p135-136, and attributed to G.
+     W. Brown, Modern Mathematics for the Engineer (1956).
+
+
+File: gsl-ref.info,  Node: The Weibull Distribution,  Next: The Type-1 Gumbel Distribution,  Prev: Spherical Vector Distributions,  Up: Random Number Distributions
+
+19.24 The Weibull Distribution
+==============================
+
+ -- Function: double gsl_ran_weibull (const gsl_rng * R, double A,
+          double B)
+     This function returns a random variate from the Weibull
+     distribution.  The distribution function is,
+
+          p(x) dx = {b \over a^b} x^{b-1}  \exp(-(x/a)^b) dx
+
+     for x >= 0.
+
+ -- Function: double gsl_ran_weibull_pdf (double X, double A, double B)
+     This function computes the probability density p(x) at X for a
+     Weibull distribution with scale A and exponent B, using the
+     formula given above.
+
+
+ -- Function: double gsl_cdf_weibull_P (double X, double A, double B)
+ -- Function: double gsl_cdf_weibull_Q (double X, double A, double B)
+ -- Function: double gsl_cdf_weibull_Pinv (double P, double A, double B)
+ -- Function: double gsl_cdf_weibull_Qinv (double Q, double A, double B)
+     These functions compute the cumulative distribution functions
+     P(x), Q(x) and their inverses for the Weibull distribution with
+     scale A and exponent B.
+
+
+File: gsl-ref.info,  Node: The Type-1 Gumbel Distribution,  Next: The Type-2 Gumbel Distribution,  Prev: The Weibull Distribution,  Up: Random Number Distributions
+
+19.25 The Type-1 Gumbel Distribution
+====================================
+
+ -- Function: double gsl_ran_gumbel1 (const gsl_rng * R, double A,
+          double B)
+     This function returns  a random variate from the Type-1 Gumbel
+     distribution.  The Type-1 Gumbel distribution function is,
+
+          p(x) dx = a b \exp(-(b \exp(-ax) + ax)) dx
+
+     for -\infty < x < \infty.
+
+ -- Function: double gsl_ran_gumbel1_pdf (double X, double A, double B)
+     This function computes the probability density p(x) at X for a
+     Type-1 Gumbel distribution with parameters A and B, using the
+     formula given above.
+
+
+ -- Function: double gsl_cdf_gumbel1_P (double X, double A, double B)
+ -- Function: double gsl_cdf_gumbel1_Q (double X, double A, double B)
+ -- Function: double gsl_cdf_gumbel1_Pinv (double P, double A, double B)
+ -- Function: double gsl_cdf_gumbel1_Qinv (double Q, double A, double B)
+     These functions compute the cumulative distribution functions
+     P(x), Q(x) and their inverses for the Type-1 Gumbel distribution
+     with parameters A and B.
+
+
+File: gsl-ref.info,  Node: The Type-2 Gumbel Distribution,  Next: The Dirichlet Distribution,  Prev: The Type-1 Gumbel Distribution,  Up: Random Number Distributions
+
+19.26 The Type-2 Gumbel Distribution
+====================================
+
+ -- Function: double gsl_ran_gumbel2 (const gsl_rng * R, double A,
+          double B)
+     This function returns a random variate from the Type-2 Gumbel
+     distribution.  The Type-2 Gumbel distribution function is,
+
+          p(x) dx = a b x^{-a-1} \exp(-b x^{-a}) dx
+
+     for 0 < x < \infty.
+
+ -- Function: double gsl_ran_gumbel2_pdf (double X, double A, double B)
+     This function computes the probability density p(x) at X for a
+     Type-2 Gumbel distribution with parameters A and B, using the
+     formula given above.
+
+
+ -- Function: double gsl_cdf_gumbel2_P (double X, double A, double B)
+ -- Function: double gsl_cdf_gumbel2_Q (double X, double A, double B)
+ -- Function: double gsl_cdf_gumbel2_Pinv (double P, double A, double B)
+ -- Function: double gsl_cdf_gumbel2_Qinv (double Q, double A, double B)
+     These functions compute the cumulative distribution functions
+     P(x), Q(x) and their inverses for the Type-2 Gumbel distribution
+     with parameters A and B.
+
+
+File: gsl-ref.info,  Node: The Dirichlet Distribution,  Next: General Discrete Distributions,  Prev: The Type-2 Gumbel Distribution,  Up: Random Number Distributions
+
+19.27 The Dirichlet Distribution
+================================
+
+ -- Function: void gsl_ran_dirichlet (const gsl_rng * R, size_t K,
+          const double ALPHA[], double THETA[])
+     This function returns an array of K random variates from a
+     Dirichlet distribution of order K-1. The distribution function is
+
+          p(\theta_1, ..., \theta_K) d\theta_1 ... d\theta_K =
+            (1/Z) \prod_{i=1}^K \theta_i^{\alpha_i - 1} \delta(1 -\sum_{i=1}^K \theta_i) d\theta_1 ... d\theta_K
+
+     for theta_i >= 0 and alpha_i >= 0.  The delta function ensures
+     that \sum \theta_i = 1.  The normalization factor Z is
+
+          Z = {\prod_{i=1}^K \Gamma(\alpha_i)} / {\Gamma( \sum_{i=1}^K \alpha_i)}
+
+     The random variates are generated by sampling K values from gamma
+     distributions with parameters a=alpha_i, b=1, and renormalizing.
+     See A.M. Law, W.D. Kelton, `Simulation Modeling and Analysis'
+     (1991).
+
+ -- Function: double gsl_ran_dirichlet_pdf (size_t K, const double
+          ALPHA[], const double THETA[])
+     This function computes the probability density p(\theta_1, ... ,
+     \theta_K) at THETA[K] for a Dirichlet distribution with parameters
+     ALPHA[K], using the formula given above.
+
+ -- Function: double gsl_ran_dirichlet_lnpdf (size_t K, const double
+          ALPHA[], const double THETA[])
+     This function computes the logarithm of the probability density
+     p(\theta_1, ... , \theta_K) for a Dirichlet distribution with
+     parameters ALPHA[K].
+
+
+File: gsl-ref.info,  Node: General Discrete Distributions,  Next: The Poisson Distribution,  Prev: The Dirichlet Distribution,  Up: Random Number Distributions
+
+19.28 General Discrete Distributions
+====================================
+
+Given K discrete events with different probabilities P[k], produce a
+random value k consistent with its probability.
+
+   The obvious way to do this is to preprocess the probability list by
+generating a cumulative probability array with K+1 elements:
+
+       C[0] = 0
+     C[k+1] = C[k]+P[k].
+
+Note that this construction produces C[K]=1.  Now choose a uniform
+deviate u between 0 and 1, and find the value of k such that C[k] <= u
+< C[k+1].  Although this in principle requires of order \log K steps per
+random number generation, they are fast steps, and if you use something
+like \lfloor uK \rfloor as a starting point, you can often do pretty
+well.
+
+   But faster methods have been devised.  Again, the idea is to
+preprocess the probability list, and save the result in some form of
+lookup table; then the individual calls for a random discrete event can
+go rapidly.  An approach invented by G. Marsaglia (Generating discrete
+random numbers in a computer, Comm ACM 6, 37-38 (1963)) is very clever,
+and readers interested in examples of good algorithm design are
+directed to this short and well-written paper.  Unfortunately, for
+large K, Marsaglia's lookup table can be quite large.
+
+   A much better approach is due to Alastair J. Walker (An efficient
+method for generating discrete random variables with general
+distributions, ACM Trans on Mathematical Software 3, 253-256 (1977);
+see also Knuth, v2, 3rd ed, p120-121,139).  This requires two lookup
+tables, one floating point and one integer, but both only of size K.
+After preprocessing, the random numbers are generated in O(1) time,
+even for large K.  The preprocessing suggested by Walker requires
+O(K^2) effort, but that is not actually necessary, and the
+implementation provided here only takes O(K) effort.  In general, more
+preprocessing leads to faster generation of the individual random
+numbers, but a diminishing return is reached pretty early.  Knuth points
+out that the optimal preprocessing is combinatorially difficult for
+large K.
+
+   This method can be used to speed up some of the discrete random
+number generators below, such as the binomial distribution.  To use it
+for something like the Poisson Distribution, a modification would have
+to be made, since it only takes a finite set of K outcomes.
+
+ -- Function: gsl_ran_discrete_t * gsl_ran_discrete_preproc (size_t K,
+          const double * P)
+     This function returns a pointer to a structure that contains the
+     lookup table for the discrete random number generator.  The array
+     P[] contains the probabilities of the discrete events; these array
+     elements must all be positive, but they needn't add up to one (so
+     you can think of them more generally as "weights")--the
+     preprocessor will normalize appropriately.  This return value is
+     used as an argument for the `gsl_ran_discrete' function below.
+
+ -- Function: size_t gsl_ran_discrete (const gsl_rng * R, const
+          gsl_ran_discrete_t * G)
+     After the preprocessor, above, has been called, you use this
+     function to get the discrete random numbers.
+
+ -- Function: double gsl_ran_discrete_pdf (size_t K, const
+          gsl_ran_discrete_t * G)
+     Returns the probability P[k] of observing the variable K.  Since
+     P[k] is not stored as part of the lookup table, it must be
+     recomputed; this computation takes O(K), so if K is large and you
+     care about the original array P[k] used to create the lookup
+     table, then you should just keep this original array P[k] around.
+
+ -- Function: void gsl_ran_discrete_free (gsl_ran_discrete_t * G)
+     De-allocates the lookup table pointed to by G.
+
+
+File: gsl-ref.info,  Node: The Poisson Distribution,  Next: The Bernoulli Distribution,  Prev: General Discrete Distributions,  Up: Random Number Distributions
+
+19.29 The Poisson Distribution
+==============================
+
+ -- Function: unsigned int gsl_ran_poisson (const gsl_rng * R, double
+          MU)
+     This function returns a random integer from the Poisson
+     distribution with mean MU.  The probability distribution for
+     Poisson variates is,
+
+          p(k) = {\mu^k \over k!} \exp(-\mu)
+
+     for k >= 0.
+
+ -- Function: double gsl_ran_poisson_pdf (unsigned int K, double MU)
+     This function computes the probability p(k) of obtaining  K from a
+     Poisson distribution with mean MU, using the formula given above.
+
+
+ -- Function: double gsl_cdf_poisson_P (unsigned int K, double MU)
+ -- Function: double gsl_cdf_poisson_Q (unsigned int K, double MU)
+     These functions compute the cumulative distribution functions
+     P(k), Q(k) for the Poisson distribution with parameter MU.
+
+
+File: gsl-ref.info,  Node: The Bernoulli Distribution,  Next: The Binomial Distribution,  Prev: The Poisson Distribution,  Up: Random Number Distributions
+
+19.30 The Bernoulli Distribution
+================================
+
+ -- Function: unsigned int gsl_ran_bernoulli (const gsl_rng * R, double
+          P)
+     This function returns either 0 or 1, the result of a Bernoulli
+     trial with probability P.  The probability distribution for a
+     Bernoulli trial is,
+
+          p(0) = 1 - p
+          p(1) = p
+
+
+ -- Function: double gsl_ran_bernoulli_pdf (unsigned int K, double P)
+     This function computes the probability p(k) of obtaining K from a
+     Bernoulli distribution with probability parameter P, using the
+     formula given above.
+
+
+
+File: gsl-ref.info,  Node: The Binomial Distribution,  Next: The Multinomial Distribution,  Prev: The Bernoulli Distribution,  Up: Random Number Distributions
+
+19.31 The Binomial Distribution
+===============================
+
+ -- Function: unsigned int gsl_ran_binomial (const gsl_rng * R, double
+          P, unsigned int N)
+     This function returns a random integer from the binomial
+     distribution, the number of successes in N independent trials with
+     probability P.  The probability distribution for binomial variates
+     is,
+
+          p(k) = {n! \over k! (n-k)! } p^k (1-p)^{n-k}
+
+     for 0 <= k <= n.
+
+ -- Function: double gsl_ran_binomial_pdf (unsigned int K, double P,
+          unsigned int N)
+     This function computes the probability p(k) of obtaining K from a
+     binomial distribution with parameters P and N, using the formula
+     given above.
+
+
+ -- Function: double gsl_cdf_binomial_P (unsigned int K, double P,
+          unsigned int N)
+ -- Function: double gsl_cdf_binomial_Q (unsigned int K, double P,
+          unsigned int N)
+     These functions compute the cumulative distribution functions
+     P(k), Q(k)  for the binomial distribution with parameters P and N.
+
+
+File: gsl-ref.info,  Node: The Multinomial Distribution,  Next: The Negative Binomial Distribution,  Prev: The Binomial Distribution,  Up: Random Number Distributions
+
+19.32 The Multinomial Distribution
+==================================
+
+ -- Function: void gsl_ran_multinomial (const gsl_rng * R, size_t K,
+          unsigned int N, const double P[], unsigned int N[])
+     This function computes a random sample N[] from the multinomial
+     distribution formed by N trials from an underlying distribution
+     P[K]. The distribution function for N[] is,
+
+          P(n_1, n_2, ..., n_K) =
+            (N!/(n_1! n_2! ... n_K!)) p_1^n_1 p_2^n_2 ... p_K^n_K
+
+     where (n_1, n_2, ..., n_K) are nonnegative integers with
+     sum_{k=1}^K n_k = N, and (p_1, p_2, ..., p_K) is a probability
+     distribution with \sum p_i = 1.  If the array P[K] is not
+     normalized then its entries will be treated as weights and
+     normalized appropriately.  The arrays N[] and P[] must both be of
+     length K.
+
+     Random variates are generated using the conditional binomial
+     method (see C.S. David, `The computer generation of multinomial
+     random variates', Comp. Stat. Data Anal. 16 (1993) 205-217 for
+     details).
+
+ -- Function: double gsl_ran_multinomial_pdf (size_t K, const double
+          P[], const unsigned int N[])
+     This function computes the probability P(n_1, n_2, ..., n_K) of
+     sampling N[K] from a multinomial distribution with parameters
+     P[K], using the formula given above.
+
+ -- Function: double gsl_ran_multinomial_lnpdf (size_t K, const double
+          P[], const unsigned int N[])
+     This function returns the logarithm of the probability for the
+     multinomial distribution P(n_1, n_2, ..., n_K) with parameters
+     P[K].
+
+
+File: gsl-ref.info,  Node: The Negative Binomial Distribution,  Next: The Pascal Distribution,  Prev: The Multinomial Distribution,  Up: Random Number Distributions
+
+19.33 The Negative Binomial Distribution
+========================================
+
+ -- Function: unsigned int gsl_ran_negative_binomial (const gsl_rng *
+          R, double P, double N)
+     This function returns a random integer from the negative binomial
+     distribution, the number of failures occurring before N successes
+     in independent trials with probability P of success.  The
+     probability distribution for negative binomial variates is,
+
+          p(k) = {\Gamma(n + k) \over \Gamma(k+1) \Gamma(n) } p^n (1-p)^k
+
+     Note that n is not required to be an integer.
+
+ -- Function: double gsl_ran_negative_binomial_pdf (unsigned int K,
+          double P, double N)
+     This function computes the probability p(k) of obtaining K from a
+     negative binomial distribution with parameters P and N, using the
+     formula given above.
+
+
+ -- Function: double gsl_cdf_negative_binomial_P (unsigned int K,
+          double P, double N)
+ -- Function: double gsl_cdf_negative_binomial_Q (unsigned int K,
+          double P, double N)
+     These functions compute the cumulative distribution functions
+     P(k), Q(k) for the negative binomial distribution with parameters
+     P and N.
+
+
+File: gsl-ref.info,  Node: The Pascal Distribution,  Next: The Geometric Distribution,  Prev: The Negative Binomial Distribution,  Up: Random Number Distributions
+
+19.34 The Pascal Distribution
+=============================
+
+ -- Function: unsigned int gsl_ran_pascal (const gsl_rng * R, double P,
+          unsigned int N)
+     This function returns a random integer from the Pascal
+     distribution.  The Pascal distribution is simply a negative
+     binomial distribution with an integer value of n.
+
+          p(k) = {(n + k - 1)! \over k! (n - 1)! } p^n (1-p)^k
+
+     for k >= 0
+
+ -- Function: double gsl_ran_pascal_pdf (unsigned int K, double P,
+          unsigned int N)
+     This function computes the probability p(k) of obtaining K from a
+     Pascal distribution with parameters P and N, using the formula
+     given above.
+
+
+ -- Function: double gsl_cdf_pascal_P (unsigned int K, double P,
+          unsigned int N)
+ -- Function: double gsl_cdf_pascal_Q (unsigned int K, double P,
+          unsigned int N)
+     These functions compute the cumulative distribution functions
+     P(k), Q(k) for the Pascal distribution with parameters P and N.
+
+
+File: gsl-ref.info,  Node: The Geometric Distribution,  Next: The Hypergeometric Distribution,  Prev: The Pascal Distribution,  Up: Random Number Distributions
+
+19.35 The Geometric Distribution
+================================
+
+ -- Function: unsigned int gsl_ran_geometric (const gsl_rng * R, double
+          P)
+     This function returns a random integer from the geometric
+     distribution, the number of independent trials with probability P
+     until the first success.  The probability distribution for
+     geometric variates is,
+
+          p(k) =  p (1-p)^(k-1)
+
+     for k >= 1.  Note that the distribution begins with k=1 with this
+     definition.  There is another convention in which the exponent k-1
+     is replaced by k.
+
+ -- Function: double gsl_ran_geometric_pdf (unsigned int K, double P)
+     This function computes the probability p(k) of obtaining K from a
+     geometric distribution with probability parameter P, using the
+     formula given above.
+
+
+ -- Function: double gsl_cdf_geometric_P (unsigned int K, double P)
+ -- Function: double gsl_cdf_geometric_Q (unsigned int K, double P)
+     These functions compute the cumulative distribution functions
+     P(k), Q(k) for the geometric distribution with parameter P.
+
+
+File: gsl-ref.info,  Node: The Hypergeometric Distribution,  Next: The Logarithmic Distribution,  Prev: The Geometric Distribution,  Up: Random Number Distributions
+
+19.36 The Hypergeometric Distribution
+=====================================
+
+ -- Function: unsigned int gsl_ran_hypergeometric (const gsl_rng * R,
+          unsigned int N1, unsigned int N2, unsigned int T)
+     This function returns a random integer from the hypergeometric
+     distribution.  The probability distribution for hypergeometric
+     random variates is,
+
+          p(k) =  C(n_1, k) C(n_2, t - k) / C(n_1 + n_2, t)
+
+     where C(a,b) = a!/(b!(a-b)!) and t <= n_1 + n_2.  The domain of k
+     is max(0,t-n_2), ..., min(t,n_1).
+
+     If a population contains n_1 elements of "type 1" and n_2 elements
+     of "type 2" then the hypergeometric distribution gives the
+     probability of obtaining k elements of "type 1" in t samples from
+     the population without replacement.
+
+ -- Function: double gsl_ran_hypergeometric_pdf (unsigned int K,
+          unsigned int N1, unsigned int N2, unsigned int T)
+     This function computes the probability p(k) of obtaining K from a
+     hypergeometric distribution with parameters N1, N2, T, using the
+     formula given above.
+
+
+ -- Function: double gsl_cdf_hypergeometric_P (unsigned int K, unsigned
+          int N1, unsigned int N2, unsigned int T)
+ -- Function: double gsl_cdf_hypergeometric_Q (unsigned int K, unsigned
+          int N1, unsigned int N2, unsigned int T)
+     These functions compute the cumulative distribution functions
+     P(k), Q(k) for the hypergeometric distribution with parameters N1,
+     N2 and T.
+
+
+File: gsl-ref.info,  Node: The Logarithmic Distribution,  Next: Shuffling and Sampling,  Prev: The Hypergeometric Distribution,  Up: Random Number Distributions
+
+19.37 The Logarithmic Distribution
+==================================
+
+ -- Function: unsigned int gsl_ran_logarithmic (const gsl_rng * R,
+          double P)
+     This function returns a random integer from the logarithmic
+     distribution.  The probability distribution for logarithmic random
+     variates is,
+
+          p(k) = {-1 \over \log(1-p)} {(p^k \over k)}
+
+     for k >= 1.
+
+ -- Function: double gsl_ran_logarithmic_pdf (unsigned int K, double P)
+     This function computes the probability p(k) of obtaining K from a
+     logarithmic distribution with probability parameter P, using the
+     formula given above.
+
+
+
+File: gsl-ref.info,  Node: Shuffling and Sampling,  Next: Random Number Distribution Examples,  Prev: The Logarithmic Distribution,  Up: Random Number Distributions
+
+19.38 Shuffling and Sampling
+============================
+
+The following functions allow the shuffling and sampling of a set of
+objects.  The algorithms rely on a random number generator as a source
+of randomness and a poor quality generator can lead to correlations in
+the output.  In particular it is important to avoid generators with a
+short period.  For more information see Knuth, v2, 3rd ed, Section
+3.4.2, "Random Sampling and Shuffling".
+
+ -- Function: void gsl_ran_shuffle (const gsl_rng * R, void * BASE,
+          size_t N, size_t SIZE)
+     This function randomly shuffles the order of N objects, each of
+     size SIZE, stored in the array BASE[0..N-1].  The output of the
+     random number generator R is used to produce the permutation.  The
+     algorithm generates all possible n!  permutations with equal
+     probability, assuming a perfect source of random numbers.
+
+     The following code shows how to shuffle the numbers from 0 to 51,
+
+          int a[52];
+
+          for (i = 0; i < 52; i++)
+            {
+              a[i] = i;
+            }
+
+          gsl_ran_shuffle (r, a, 52, sizeof (int));
+
+
+ -- Function: int gsl_ran_choose (const gsl_rng * R, void * DEST,
+          size_t K, void * SRC, size_t N, size_t SIZE)
+     This function fills the array DEST[k] with K objects taken
+     randomly from the N elements of the array SRC[0..N-1].  The
+     objects are each of size SIZE.  The output of the random number
+     generator R is used to make the selection.  The algorithm ensures
+     all possible samples are equally likely, assuming a perfect source
+     of randomness.
+
+     The objects are sampled _without_ replacement, thus each object can
+     only appear once in DEST[k].  It is required that K be less than
+     or equal to `n'.  The objects in DEST will be in the same relative
+     order as those in SRC.  You will need to call `gsl_ran_shuffle(r,
+     dest, n, size)' if you want to randomize the order.
+
+     The following code shows how to select a random sample of three
+     unique numbers from the set 0 to 99,
+
+          double a[3], b[100];
+
+          for (i = 0; i < 100; i++)
+            {
+              b[i] = (double) i;
+            }
+
+          gsl_ran_choose (r, a, 3, b, 100, sizeof (double));
+
+
+ -- Function: void gsl_ran_sample (const gsl_rng * R, void * DEST,
+          size_t K, void * SRC, size_t N, size_t SIZE)
+     This function is like `gsl_ran_choose' but samples K items from
+     the original array of N items SRC with replacement, so the same
+     object can appear more than once in the output sequence DEST.
+     There is no requirement that K be less than N in this case.
+
+
+File: gsl-ref.info,  Node: Random Number Distribution Examples,  Next: Random Number Distribution References and Further Reading,  Prev: Shuffling and Sampling,  Up: Random Number Distributions
+
+19.39 Examples
+==============
+
+The following program demonstrates the use of a random number generator
+to produce variates from a distribution.  It prints 10 samples from the
+Poisson distribution with a mean of 3.
+
+     #include <stdio.h>
+     #include <gsl/gsl_rng.h>
+     #include <gsl/gsl_randist.h>
+
+     int
+     main (void)
+     {
+       const gsl_rng_type * T;
+       gsl_rng * r;
+
+       int i, n = 10;
+       double mu = 3.0;
+
+       /* create a generator chosen by the
+          environment variable GSL_RNG_TYPE */
+
+       gsl_rng_env_setup();
+
+       T = gsl_rng_default;
+       r = gsl_rng_alloc (T);
+
+       /* print n random variates chosen from
+          the poisson distribution with mean
+          parameter mu */
+
+       for (i = 0; i < n; i++)
+         {
+           unsigned int k = gsl_ran_poisson (r, mu);
+           printf (" %u", k);
+         }
+
+       printf ("\n");
+       gsl_rng_free (r);
+       return 0;
+     }
+
+If the library and header files are installed under `/usr/local' (the
+default location) then the program can be compiled with these options,
+
+     $ gcc -Wall demo.c -lgsl -lgslcblas -lm
+
+Here is the output of the program,
+
+     $ ./a.out
+      2 5 5 2 1 0 3 4 1 1
+
+The variates depend on the seed used by the generator.  The seed for the
+default generator type `gsl_rng_default' can be changed with the
+`GSL_RNG_SEED' environment variable to produce a different stream of
+variates,
+
+     $ GSL_RNG_SEED=123 ./a.out
+     GSL_RNG_SEED=123
+      4 5 6 3 3 1 4 2 5 5
+
+The following program generates a random walk in two dimensions.
+
+     #include <stdio.h>
+     #include <gsl/gsl_rng.h>
+     #include <gsl/gsl_randist.h>
+
+     int
+     main (void)
+     {
+       int i;
+       double x = 0, y = 0, dx, dy;
+
+       const gsl_rng_type * T;
+       gsl_rng * r;
+
+       gsl_rng_env_setup();
+       T = gsl_rng_default;
+       r = gsl_rng_alloc (T);
+
+       printf ("%g %g\n", x, y);
+
+       for (i = 0; i < 10; i++)
+         {
+           gsl_ran_dir_2d (r, &dx, &dy);
+           x += dx; y += dy;
+           printf ("%g %g\n", x, y);
+         }
+
+       gsl_rng_free (r);
+       return 0;
+     }
+
+Here is the output from the program, three 10-step random walks from
+the origin,
+
+   The following program computes the upper and lower cumulative
+distribution functions for the standard normal distribution at x=2.
+
+     #include <stdio.h>
+     #include <gsl/gsl_cdf.h>
+
+     int
+     main (void)
+     {
+       double P, Q;
+       double x = 2.0;
+
+       P = gsl_cdf_ugaussian_P (x);
+       printf ("prob(x < %f) = %f\n", x, P);
+
+       Q = gsl_cdf_ugaussian_Q (x);
+       printf ("prob(x > %f) = %f\n", x, Q);
+
+       x = gsl_cdf_ugaussian_Pinv (P);
+       printf ("Pinv(%f) = %f\n", P, x);
+
+       x = gsl_cdf_ugaussian_Qinv (Q);
+       printf ("Qinv(%f) = %f\n", Q, x);
+
+       return 0;
+     }
+
+Here is the output of the program,
+
+     prob(x < 2.000000) = 0.977250
+     prob(x > 2.000000) = 0.022750
+     Pinv(0.977250) = 2.000000
+     Qinv(0.022750) = 2.000000
+
+
+File: gsl-ref.info,  Node: Random Number Distribution References and Further Reading,  Prev: Random Number Distribution Examples,  Up: Random Number Distributions
+
+19.40 References and Further Reading
+====================================
+
+For an encyclopaedic coverage of the subject readers are advised to
+consult the book `Non-Uniform Random Variate Generation' by Luc
+Devroye.  It covers every imaginable distribution and provides hundreds
+of algorithms.
+
+     Luc Devroye, `Non-Uniform Random Variate Generation',
+     Springer-Verlag, ISBN 0-387-96305-7.
+
+The subject of random variate generation is also reviewed by Knuth, who
+describes algorithms for all the major distributions.
+
+     Donald E. Knuth, `The Art of Computer Programming: Seminumerical
+     Algorithms' (Vol 2, 3rd Ed, 1997), Addison-Wesley, ISBN 0201896842.
+
+The Particle Data Group provides a short review of techniques for
+generating distributions of random numbers in the "Monte Carlo" section
+of its Annual Review of Particle Physics.
+
+     `Review of Particle Properties' R.M. Barnett et al., Physical
+     Review D54, 1 (1996) `http://pdg.lbl.gov/'.
+
+The Review of Particle Physics is available online in postscript and pdf
+format.
+
+An overview of methods used to compute cumulative distribution functions
+can be found in `Statistical Computing' by W.J. Kennedy and J.E.
+Gentle. Another general reference is `Elements of Statistical
+Computing' by R.A. Thisted.
+
+     William E. Kennedy and James E. Gentle, `Statistical Computing'
+     (1980), Marcel Dekker, ISBN 0-8247-6898-1.
+
+     Ronald A. Thisted, `Elements of Statistical Computing' (1988),
+     Chapman & Hall, ISBN 0-412-01371-1.
+
+The cumulative distribution functions for the Gaussian distribution are
+based on the following papers,
+
+     `Rational Chebyshev Approximations Using Linear Equations', W.J.
+     Cody, W. Fraser, J.F. Hart. Numerische Mathematik 12, 242-251
+     (1968).
+
+     `Rational Chebyshev Approximations for the Error Function', W.J.
+     Cody. Mathematics of Computation 23, n107, 631-637 (July 1969).
+
+
+File: gsl-ref.info,  Node: Statistics,  Next: Histograms,  Prev: Random Number Distributions,  Up: Top
+
+20 Statistics
+*************
+
+This chapter describes the statistical functions in the library.  The
+basic statistical functions include routines to compute the mean,
+variance and standard deviation.  More advanced functions allow you to
+calculate absolute deviations, skewness, and kurtosis as well as the
+median and arbitrary percentiles.  The algorithms use recurrence
+relations to compute average quantities in a stable way, without large
+intermediate values that might overflow.
+
+   The functions are available in versions for datasets in the standard
+floating-point and integer types.  The versions for double precision
+floating-point data have the prefix `gsl_stats' and are declared in the
+header file `gsl_statistics_double.h'.  The versions for integer data
+have the prefix `gsl_stats_int' and are declared in the header file
+`gsl_statistics_int.h'.
+
+* Menu:
+
+* Mean and standard deviation and variance::
+* Absolute deviation::
+* Higher moments (skewness and kurtosis)::
+* Autocorrelation::
+* Covariance::
+* Weighted Samples::
+* Maximum and Minimum values::
+* Median and Percentiles::
+* Example statistical programs::
+* Statistics References and Further Reading::
+
+
+File: gsl-ref.info,  Node: Mean and standard deviation and variance,  Next: Absolute deviation,  Up: Statistics
+
+20.1 Mean, Standard Deviation and Variance
+==========================================
+
+ -- Function: double gsl_stats_mean (const double DATA[], size_t
+          STRIDE, size_t N)
+     This function returns the arithmetic mean of DATA, a dataset of
+     length N with stride STRIDE.  The arithmetic mean, or "sample
+     mean", is denoted by \Hat\mu and defined as,
+
+          \Hat\mu = (1/N) \sum x_i
+
+     where x_i are the elements of the dataset DATA.  For samples drawn
+     from a gaussian distribution the variance of \Hat\mu is \sigma^2 /
+     N.
+
+ -- Function: double gsl_stats_variance (const double DATA[], size_t
+          STRIDE, size_t N)
+     This function returns the estimated, or "sample", variance of
+     DATA, a dataset of length N with stride STRIDE.  The estimated
+     variance is denoted by \Hat\sigma^2 and is defined by,
+
+          \Hat\sigma^2 = (1/(N-1)) \sum (x_i - \Hat\mu)^2
+
+     where x_i are the elements of the dataset DATA.  Note that the
+     normalization factor of 1/(N-1) results from the derivation of
+     \Hat\sigma^2 as an unbiased estimator of the population variance
+     \sigma^2.  For samples drawn from a gaussian distribution the
+     variance of \Hat\sigma^2 itself is 2 \sigma^4 / N.
+
+     This function computes the mean via a call to `gsl_stats_mean'.  If
+     you have already computed the mean then you can pass it directly to
+     `gsl_stats_variance_m'.
+
+ -- Function: double gsl_stats_variance_m (const double DATA[], size_t
+          STRIDE, size_t N, double MEAN)
+     This function returns the sample variance of DATA relative to the
+     given value of MEAN.  The function is computed with \Hat\mu
+     replaced by the value of MEAN that you supply,
+
+          \Hat\sigma^2 = (1/(N-1)) \sum (x_i - mean)^2
+
+ -- Function: double gsl_stats_sd (const double DATA[], size_t STRIDE,
+          size_t N)
+ -- Function: double gsl_stats_sd_m (const double DATA[], size_t
+          STRIDE, size_t N, double MEAN)
+     The standard deviation is defined as the square root of the
+     variance.  These functions return the square root of the
+     corresponding variance functions above.
+
+ -- Function: double gsl_stats_variance_with_fixed_mean (const double
+          DATA[], size_t STRIDE, size_t N, double MEAN)
+     This function computes an unbiased estimate of the variance of
+     DATA when the population mean MEAN of the underlying distribution
+     is known _a priori_.  In this case the estimator for the variance
+     uses the factor 1/N and the sample mean \Hat\mu is replaced by the
+     known population mean \mu,
+
+          \Hat\sigma^2 = (1/N) \sum (x_i - \mu)^2
+
+ -- Function: double gsl_stats_sd_with_fixed_mean (const double DATA[],
+          size_t STRIDE, size_t N, double MEAN)
+     This function calculates the standard deviation of DATA for a
+     fixed population mean MEAN.  The result is the square root of the
+     corresponding variance function.
+
+
+File: gsl-ref.info,  Node: Absolute deviation,  Next: Higher moments (skewness and kurtosis),  Prev: Mean and standard deviation and variance,  Up: Statistics
+
+20.2 Absolute deviation
+=======================
+
+ -- Function: double gsl_stats_absdev (const double DATA[], size_t
+          STRIDE, size_t N)
+     This function computes the absolute deviation from the mean of
+     DATA, a dataset of length N with stride STRIDE.  The absolute
+     deviation from the mean is defined as,
+
+          absdev  = (1/N) \sum |x_i - \Hat\mu|
+
+     where x_i are the elements of the dataset DATA.  The absolute
+     deviation from the mean provides a more robust measure of the
+     width of a distribution than the variance.  This function computes
+     the mean of DATA via a call to `gsl_stats_mean'.
+
+ -- Function: double gsl_stats_absdev_m (const double DATA[], size_t
+          STRIDE, size_t N, double MEAN)
+     This function computes the absolute deviation of the dataset DATA
+     relative to the given value of MEAN,
+
+          absdev  = (1/N) \sum |x_i - mean|
+
+     This function is useful if you have already computed the mean of
+     DATA (and want to avoid recomputing it), or wish to calculate the
+     absolute deviation relative to another value (such as zero, or the
+     median).
+
+
+File: gsl-ref.info,  Node: Higher moments (skewness and kurtosis),  Next: Autocorrelation,  Prev: Absolute deviation,  Up: Statistics
+
+20.3 Higher moments (skewness and kurtosis)
+===========================================
+
+ -- Function: double gsl_stats_skew (const double DATA[], size_t
+          STRIDE, size_t N)
+     This function computes the skewness of DATA, a dataset of length N
+     with stride STRIDE.  The skewness is defined as,
+
+          skew = (1/N) \sum ((x_i - \Hat\mu)/\Hat\sigma)^3
+
+     where x_i are the elements of the dataset DATA.  The skewness
+     measures the asymmetry of the tails of a distribution.
+
+     The function computes the mean and estimated standard deviation of
+     DATA via calls to `gsl_stats_mean' and `gsl_stats_sd'.
+
+ -- Function: double gsl_stats_skew_m_sd (const double DATA[], size_t
+          STRIDE, size_t N, double MEAN, double SD)
+     This function computes the skewness of the dataset DATA using the
+     given values of the mean MEAN and standard deviation SD,
+
+          skew = (1/N) \sum ((x_i - mean)/sd)^3
+
+     These functions are useful if you have already computed the mean
+     and standard deviation of DATA and want to avoid recomputing them.
+
+ -- Function: double gsl_stats_kurtosis (const double DATA[], size_t
+          STRIDE, size_t N)
+     This function computes the kurtosis of DATA, a dataset of length N
+     with stride STRIDE.  The kurtosis is defined as,
+
+          kurtosis = ((1/N) \sum ((x_i - \Hat\mu)/\Hat\sigma)^4)  - 3
+
+     The kurtosis measures how sharply peaked a distribution is,
+     relative to its width.  The kurtosis is normalized to zero for a
+     gaussian distribution.
+
+ -- Function: double gsl_stats_kurtosis_m_sd (const double DATA[],
+          size_t STRIDE, size_t N, double MEAN, double SD)
+     This function computes the kurtosis of the dataset DATA using the
+     given values of the mean MEAN and standard deviation SD,
+
+          kurtosis = ((1/N) \sum ((x_i - mean)/sd)^4) - 3
+
+     This function is useful if you have already computed the mean and
+     standard deviation of DATA and want to avoid recomputing them.
+
+
+File: gsl-ref.info,  Node: Autocorrelation,  Next: Covariance,  Prev: Higher moments (skewness and kurtosis),  Up: Statistics
+
+20.4 Autocorrelation
+====================
+
+ -- Function: double gsl_stats_lag1_autocorrelation (const double
+          DATA[], const size_t STRIDE, const size_t N)
+     This function computes the lag-1 autocorrelation of the dataset
+     DATA.
+
+          a_1 = {\sum_{i = 1}^{n} (x_{i} - \Hat\mu) (x_{i-1} - \Hat\mu)
+                 \over
+                 \sum_{i = 1}^{n} (x_{i} - \Hat\mu) (x_{i} - \Hat\mu)}
+
+ -- Function: double gsl_stats_lag1_autocorrelation_m (const double
+          DATA[], const size_t STRIDE, const size_t N, const double
+          MEAN)
+     This function computes the lag-1 autocorrelation of the dataset
+     DATA using the given value of the mean MEAN.
+
+
+
+File: gsl-ref.info,  Node: Covariance,  Next: Weighted Samples,  Prev: Autocorrelation,  Up: Statistics
+
+20.5 Covariance
+===============
+
+ -- Function: double gsl_stats_covariance (const double DATA1[], const
+          size_t STRIDE1, const double DATA2[], const size_t STRIDE2,
+          const size_t N)
+     This function computes the covariance of the datasets DATA1 and
+     DATA2 which must both be of the same length N.
+
+          covar = (1/(n - 1)) \sum_{i = 1}^{n} (x_i - \Hat x) (y_i - \Hat y)
+
+ -- Function: double gsl_stats_covariance_m (const double DATA1[],
+          const size_t STRIDE1, const double DATA2[], const size_t
+          STRIDE2, const size_t N, const double MEAN1, const double
+          MEAN2)
+     This function computes the covariance of the datasets DATA1 and
+     DATA2 using the given values of the means, MEAN1 and MEAN2.  This
+     is useful if you have already computed the means of DATA1 and
+     DATA2 and want to avoid recomputing them.
+
+
+File: gsl-ref.info,  Node: Weighted Samples,  Next: Maximum and Minimum values,  Prev: Covariance,  Up: Statistics
+
+20.6 Weighted Samples
+=====================
+
+The functions described in this section allow the computation of
+statistics for weighted samples.  The functions accept an array of
+samples, x_i, with associated weights, w_i.  Each sample x_i is
+considered as having been drawn from a Gaussian distribution with
+variance \sigma_i^2.  The sample weight w_i is defined as the
+reciprocal of this variance, w_i = 1/\sigma_i^2.  Setting a weight to
+zero corresponds to removing a sample from a dataset.
+
+ -- Function: double gsl_stats_wmean (const double W[], size_t WSTRIDE,
+          const double DATA[], size_t STRIDE, size_t N)
+     This function returns the weighted mean of the dataset DATA with
+     stride STRIDE and length N, using the set of weights W with stride
+     WSTRIDE and length N.  The weighted mean is defined as,
+
+          \Hat\mu = (\sum w_i x_i) / (\sum w_i)
+
+ -- Function: double gsl_stats_wvariance (const double W[], size_t
+          WSTRIDE, const double DATA[], size_t STRIDE, size_t N)
+     This function returns the estimated variance of the dataset DATA
+     with stride STRIDE and length N, using the set of weights W with
+     stride WSTRIDE and length N.  The estimated variance of a weighted
+     dataset is defined as,
+
+          \Hat\sigma^2 = ((\sum w_i)/((\sum w_i)^2 - \sum (w_i^2)))
+                          \sum w_i (x_i - \Hat\mu)^2
+
+     Note that this expression reduces to an unweighted variance with
+     the familiar 1/(N-1) factor when there are N equal non-zero
+     weights.
+
+ -- Function: double gsl_stats_wvariance_m (const double W[], size_t
+          WSTRIDE, const double DATA[], size_t STRIDE, size_t N, double
+          WMEAN)
+     This function returns the estimated variance of the weighted
+     dataset DATA using the given weighted mean WMEAN.
+
+ -- Function: double gsl_stats_wsd (const double W[], size_t WSTRIDE,
+          const double DATA[], size_t STRIDE, size_t N)
+     The standard deviation is defined as the square root of the
+     variance.  This function returns the square root of the
+     corresponding variance function `gsl_stats_wvariance' above.
+
+ -- Function: double gsl_stats_wsd_m (const double W[], size_t WSTRIDE,
+          const double DATA[], size_t STRIDE, size_t N, double WMEAN)
+     This function returns the square root of the corresponding variance
+     function `gsl_stats_wvariance_m' above.
+
+ -- Function: double gsl_stats_wvariance_with_fixed_mean (const double
+          W[], size_t WSTRIDE, const double DATA[], size_t STRIDE,
+          size_t N, const double MEAN)
+     This function computes an unbiased estimate of the variance of
+     weighted dataset DATA when the population mean MEAN of the
+     underlying distribution is known _a priori_.  In this case the
+     estimator for the variance replaces the sample mean \Hat\mu by the
+     known population mean \mu,
+
+          \Hat\sigma^2 = (\sum w_i (x_i - \mu)^2) / (\sum w_i)
+
+ -- Function: double gsl_stats_wsd_with_fixed_mean (const double W[],
+          size_t WSTRIDE, const double DATA[], size_t STRIDE, size_t N,
+          const double MEAN)
+     The standard deviation is defined as the square root of the
+     variance.  This function returns the square root of the
+     corresponding variance function above.
+
+ -- Function: double gsl_stats_wabsdev (const double W[], size_t
+          WSTRIDE, const double DATA[], size_t STRIDE, size_t N)
+     This function computes the weighted absolute deviation from the
+     weighted mean of DATA.  The absolute deviation from the mean is
+     defined as,
+
+          absdev = (\sum w_i |x_i - \Hat\mu|) / (\sum w_i)
+
+ -- Function: double gsl_stats_wabsdev_m (const double W[], size_t
+          WSTRIDE, const double DATA[], size_t STRIDE, size_t N, double
+          WMEAN)
+     This function computes the absolute deviation of the weighted
+     dataset DATA about the given weighted mean WMEAN.
+
+ -- Function: double gsl_stats_wskew (const double W[], size_t WSTRIDE,
+          const double DATA[], size_t STRIDE, size_t N)
+     This function computes the weighted skewness of the dataset DATA.
+
+          skew = (\sum w_i ((x_i - xbar)/\sigma)^3) / (\sum w_i)
+
+ -- Function: double gsl_stats_wskew_m_sd (const double W[], size_t
+          WSTRIDE, const double DATA[], size_t STRIDE, size_t N, double
+          WMEAN, double WSD)
+     This function computes the weighted skewness of the dataset DATA
+     using the given values of the weighted mean and weighted standard
+     deviation, WMEAN and WSD.
+
+ -- Function: double gsl_stats_wkurtosis (const double W[], size_t
+          WSTRIDE, const double DATA[], size_t STRIDE, size_t N)
+     This function computes the weighted kurtosis of the dataset DATA.
+
+          kurtosis = ((\sum w_i ((x_i - xbar)/sigma)^4) / (\sum w_i)) - 3
+
+ -- Function: double gsl_stats_wkurtosis_m_sd (const double W[], size_t
+          WSTRIDE, const double DATA[], size_t STRIDE, size_t N, double
+          WMEAN, double WSD)
+     This function computes the weighted kurtosis of the dataset DATA
+     using the given values of the weighted mean and weighted standard
+     deviation, WMEAN and WSD.
+
diff --git a/gsl-1.9/doc/gsl-ref.info-3 b/gsl-1.9/doc/gsl-ref.info-3
new file mode 100644
index 0000000..d30fefa
--- /dev/null
+++ b/gsl-1.9/doc/gsl-ref.info-3
@@ -0,0 +1,7520 @@
+This is gsl-ref.info, produced by makeinfo version 4.8 from
+gsl-ref.texi.
+
+INFO-DIR-SECTION Scientific software
+START-INFO-DIR-ENTRY
+* gsl-ref: (gsl-ref).                   GNU Scientific Library - Reference
+END-INFO-DIR-ENTRY
+
+   Copyright (C) 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004,
+2005, 2006, 2007 The GSL Team.
+
+   Permission is granted to copy, distribute and/or modify this document
+under the terms of the GNU Free Documentation License, Version 1.2 or
+any later version published by the Free Software Foundation; with the
+Invariant Sections being "GNU General Public License" and "Free Software
+Needs Free Documentation", the Front-Cover text being "A GNU Manual",
+and with the Back-Cover Text being (a) (see below).  A copy of the
+license is included in the section entitled "GNU Free Documentation
+License".
+
+   (a) The Back-Cover Text is: "You have freedom to copy and modify this
+GNU Manual, like GNU software."
+
+
+File: gsl-ref.info,  Node: Maximum and Minimum values,  Next: Median and Percentiles,  Prev: Weighted Samples,  Up: Statistics
+
+20.7 Maximum and Minimum values
+===============================
+
+The following functions find the maximum and minimum values of a
+dataset (or their indices).  If the data contains `NaN's then a `NaN'
+will be returned, since the maximum or minimum value is undefined.  For
+functions which return an index, the location of the first `NaN' in the
+array is returned.
+
+ -- Function: double gsl_stats_max (const double DATA[], size_t STRIDE,
+          size_t N)
+     This function returns the maximum value in DATA, a dataset of
+     length N with stride STRIDE.  The maximum value is defined as the
+     value of the element x_i which satisfies x_i >= x_j for all j.
+
+     If you want instead to find the element with the largest absolute
+     magnitude you will need to apply `fabs' or `abs' to your data
+     before calling this function.
+
+ -- Function: double gsl_stats_min (const double DATA[], size_t STRIDE,
+          size_t N)
+     This function returns the minimum value in DATA, a dataset of
+     length N with stride STRIDE.  The minimum value is defined as the
+     value of the element x_i which satisfies x_i <= x_j for all j.
+
+     If you want instead to find the element with the smallest absolute
+     magnitude you will need to apply `fabs' or `abs' to your data
+     before calling this function.
+
+ -- Function: void gsl_stats_minmax (double * MIN, double * MAX, const
+          double DATA[], size_t STRIDE, size_t N)
+     This function finds both the minimum and maximum values MIN, MAX
+     in DATA in a single pass.
+
+ -- Function: size_t gsl_stats_max_index (const double DATA[], size_t
+          STRIDE, size_t N)
+     This function returns the index of the maximum value in DATA, a
+     dataset of length N with stride STRIDE.  The maximum value is
+     defined as the value of the element x_i which satisfies x_i >= x_j
+     for all j.  When there are several equal maximum elements then the
+     first one is chosen.
+
+ -- Function: size_t gsl_stats_min_index (const double DATA[], size_t
+          STRIDE, size_t N)
+     This function returns the index of the minimum value in DATA, a
+     dataset of length N with stride STRIDE.  The minimum value is
+     defined as the value of the element x_i which satisfies x_i >= x_j
+     for all j.  When there are several equal minimum elements then the
+     first one is chosen.
+
+ -- Function: void gsl_stats_minmax_index (size_t * MIN_INDEX, size_t *
+          MAX_INDEX, const double DATA[], size_t STRIDE, size_t N)
+     This function returns the indexes MIN_INDEX, MAX_INDEX of the
+     minimum and maximum values in DATA in a single pass.
+
+
+File: gsl-ref.info,  Node: Median and Percentiles,  Next: Example statistical programs,  Prev: Maximum and Minimum values,  Up: Statistics
+
+20.8 Median and Percentiles
+===========================
+
+The median and percentile functions described in this section operate on
+sorted data.  For convenience we use "quantiles", measured on a scale
+of 0 to 1, instead of percentiles (which use a scale of 0 to 100).
+
+ -- Function: double gsl_stats_median_from_sorted_data (const double
+          SORTED_DATA[], size_t STRIDE, size_t N)
+     This function returns the median value of SORTED_DATA, a dataset
+     of length N with stride STRIDE.  The elements of the array must be
+     in ascending numerical order.  There are no checks to see whether
+     the data are sorted, so the function `gsl_sort' should always be
+     used first.
+
+     When the dataset has an odd number of elements the median is the
+     value of element (n-1)/2.  When the dataset has an even number of
+     elements the median is the mean of the two nearest middle values,
+     elements (n-1)/2 and n/2.  Since the algorithm for computing the
+     median involves interpolation this function always returns a
+     floating-point number, even for integer data types.
+
+ -- Function: double gsl_stats_quantile_from_sorted_data (const double
+          SORTED_DATA[], size_t STRIDE, size_t N, double F)
+     This function returns a quantile value of SORTED_DATA, a
+     double-precision array of length N with stride STRIDE.  The
+     elements of the array must be in ascending numerical order.  The
+     quantile is determined by the F, a fraction between 0 and 1.  For
+     example, to compute the value of the 75th percentile F should have
+     the value 0.75.
+
+     There are no checks to see whether the data are sorted, so the
+     function `gsl_sort' should always be used first.
+
+     The quantile is found by interpolation, using the formula
+
+          quantile = (1 - \delta) x_i + \delta x_{i+1}
+
+     where i is `floor'((n - 1)f) and \delta is (n-1)f - i.
+
+     Thus the minimum value of the array (`data[0*stride]') is given by
+     F equal to zero, the maximum value (`data[(n-1)*stride]') is given
+     by F equal to one and the median value is given by F equal to 0.5.
+     Since the algorithm for computing quantiles involves
+     interpolation this function always returns a floating-point
+     number, even for integer data types.
+
+
+File: gsl-ref.info,  Node: Example statistical programs,  Next: Statistics References and Further Reading,  Prev: Median and Percentiles,  Up: Statistics
+
+20.9 Examples
+=============
+
+Here is a basic example of how to use the statistical functions:
+
+     #include <stdio.h>
+     #include <gsl/gsl_statistics.h>
+
+     int
+     main(void)
+     {
+       double data[5] = {17.2, 18.1, 16.5, 18.3, 12.6};
+       double mean, variance, largest, smallest;
+
+       mean     = gsl_stats_mean(data, 1, 5);
+       variance = gsl_stats_variance(data, 1, 5);
+       largest  = gsl_stats_max(data, 1, 5);
+       smallest = gsl_stats_min(data, 1, 5);
+
+       printf ("The dataset is %g, %g, %g, %g, %g\n",
+              data[0], data[1], data[2], data[3], data[4]);
+
+       printf ("The sample mean is %g\n", mean);
+       printf ("The estimated variance is %g\n", variance);
+       printf ("The largest value is %g\n", largest);
+       printf ("The smallest value is %g\n", smallest);
+       return 0;
+     }
+
+   The program should produce the following output,
+
+     The dataset is 17.2, 18.1, 16.5, 18.3, 12.6
+     The sample mean is 16.54
+     The estimated variance is 4.2984
+     The largest value is 18.3
+     The smallest value is 12.6
+
+   Here is an example using sorted data,
+
+     #include <stdio.h>
+     #include <gsl/gsl_sort.h>
+     #include <gsl/gsl_statistics.h>
+
+     int
+     main(void)
+     {
+       double data[5] = {17.2, 18.1, 16.5, 18.3, 12.6};
+       double median, upperq, lowerq;
+
+       printf ("Original dataset:  %g, %g, %g, %g, %g\n",
+              data[0], data[1], data[2], data[3], data[4]);
+
+       gsl_sort (data, 1, 5);
+
+       printf ("Sorted dataset: %g, %g, %g, %g, %g\n",
+              data[0], data[1], data[2], data[3], data[4]);
+
+       median
+         = gsl_stats_median_from_sorted_data (data,
+                                              1, 5);
+
+       upperq
+         = gsl_stats_quantile_from_sorted_data (data,
+                                                1, 5,
+                                                0.75);
+       lowerq
+         = gsl_stats_quantile_from_sorted_data (data,
+                                                1, 5,
+                                                0.25);
+
+       printf ("The median is %g\n", median);
+       printf ("The upper quartile is %g\n", upperq);
+       printf ("The lower quartile is %g\n", lowerq);
+       return 0;
+     }
+
+   This program should produce the following output,
+
+     Original dataset: 17.2, 18.1, 16.5, 18.3, 12.6
+     Sorted dataset: 12.6, 16.5, 17.2, 18.1, 18.3
+     The median is 17.2
+     The upper quartile is 18.1
+     The lower quartile is 16.5
+
+
+File: gsl-ref.info,  Node: Statistics References and Further Reading,  Prev: Example statistical programs,  Up: Statistics
+
+20.10 References and Further Reading
+====================================
+
+The standard reference for almost any topic in statistics is the
+multi-volume `Advanced Theory of Statistics' by Kendall and Stuart.
+
+     Maurice Kendall, Alan Stuart, and J. Keith Ord.  `The Advanced
+     Theory of Statistics' (multiple volumes) reprinted as `Kendall's
+     Advanced Theory of Statistics'.  Wiley, ISBN 047023380X.
+
+Many statistical concepts can be more easily understood by a Bayesian
+approach.  The following book by Gelman, Carlin, Stern and Rubin gives a
+comprehensive coverage of the subject.
+
+     Andrew Gelman, John B. Carlin, Hal S. Stern, Donald B. Rubin.
+     `Bayesian Data Analysis'.  Chapman & Hall, ISBN 0412039915.
+
+For physicists the Particle Data Group provides useful reviews of
+Probability and Statistics in the "Mathematical Tools" section of its
+Annual Review of Particle Physics.
+
+     `Review of Particle Properties' R.M. Barnett et al., Physical
+     Review D54, 1 (1996)
+
+The Review of Particle Physics is available online at the website
+`http://pdg.lbl.gov/'.
+
+
+File: gsl-ref.info,  Node: Histograms,  Next: N-tuples,  Prev: Statistics,  Up: Top
+
+21 Histograms
+*************
+
+This chapter describes functions for creating histograms.  Histograms
+provide a convenient way of summarizing the distribution of a set of
+data. A histogram consists of a set of "bins" which count the number of
+events falling into a given range of a continuous variable x.  In GSL
+the bins of a histogram contain floating-point numbers, so they can be
+used to record both integer and non-integer distributions.  The bins
+can use arbitrary sets of ranges (uniformly spaced bins are the
+default).  Both one and two-dimensional histograms are supported.
+
+   Once a histogram has been created it can also be converted into a
+probability distribution function.  The library provides efficient
+routines for selecting random samples from probability distributions.
+This can be useful for generating simulations based on real data.
+
+   The functions are declared in the header files `gsl_histogram.h' and
+`gsl_histogram2d.h'.
+
+* Menu:
+
+* The histogram struct::
+* Histogram allocation::
+* Copying Histograms::
+* Updating and accessing histogram elements::
+* Searching histogram ranges::
+* Histogram Statistics::
+* Histogram Operations::
+* Reading and writing histograms::
+* Resampling from histograms::
+* The histogram probability distribution struct::
+* Example programs for histograms::
+* Two dimensional histograms::
+* The 2D histogram struct::
+* 2D Histogram allocation::
+* Copying 2D Histograms::
+* Updating and accessing 2D histogram elements::
+* Searching 2D histogram ranges::
+* 2D Histogram Statistics::
+* 2D Histogram Operations::
+* Reading and writing 2D histograms::
+* Resampling from 2D histograms::
+* Example programs for 2D histograms::
+
+
+File: gsl-ref.info,  Node: The histogram struct,  Next: Histogram allocation,  Up: Histograms
+
+21.1 The histogram struct
+=========================
+
+A histogram is defined by the following struct,
+
+ -- Data Type: gsl_histogram
+    `size_t n'
+          This is the number of histogram bins
+
+    `double * range'
+          The ranges of the bins are stored in an array of N+1 elements
+          pointed to by RANGE.
+
+    `double * bin'
+          The counts for each bin are stored in an array of N elements
+          pointed to by BIN.  The bins are floating-point numbers, so
+          you can increment them by non-integer values if necessary.
+
+The range for BIN[i] is given by RANGE[i] to RANGE[i+1].  For n bins
+there are n+1 entries in the array RANGE.  Each bin is inclusive at the
+lower end and exclusive at the upper end.  Mathematically this means
+that the bins are defined by the following inequality,
+     bin[i] corresponds to range[i] <= x < range[i+1]
+
+Here is a diagram of the correspondence between ranges and bins on the
+number-line for x,
+
+
+          [ bin[0] )[ bin[1] )[ bin[2] )[ bin[3] )[ bin[4] )
+       ---|---------|---------|---------|---------|---------|---  x
+        r[0]      r[1]      r[2]      r[3]      r[4]      r[5]
+
+In this picture the values of the RANGE array are denoted by r.  On the
+left-hand side of each bin the square bracket `[' denotes an inclusive
+lower bound (r <= x), and the round parentheses `)' on the right-hand
+side denote an exclusive upper bound (x < r).  Thus any samples which
+fall on the upper end of the histogram are excluded.  If you want to
+include this value for the last bin you will need to add an extra bin
+to your histogram.
+
+   The `gsl_histogram' struct and its associated functions are defined
+in the header file `gsl_histogram.h'.
+
+
+File: gsl-ref.info,  Node: Histogram allocation,  Next: Copying Histograms,  Prev: The histogram struct,  Up: Histograms
+
+21.2 Histogram allocation
+=========================
+
+The functions for allocating memory to a histogram follow the style of
+`malloc' and `free'.  In addition they also perform their own error
+checking.  If there is insufficient memory available to allocate a
+histogram then the functions call the error handler (with an error
+number of `GSL_ENOMEM') in addition to returning a null pointer.  Thus
+if you use the library error handler to abort your program then it
+isn't necessary to check every histogram `alloc'.
+
+ -- Function: gsl_histogram * gsl_histogram_alloc (size_t N)
+     This function allocates memory for a histogram with N bins, and
+     returns a pointer to a newly created `gsl_histogram' struct.  If
+     insufficient memory is available a null pointer is returned and the
+     error handler is invoked with an error code of `GSL_ENOMEM'. The
+     bins and ranges are not initialized, and should be prepared using
+     one of the range-setting functions below in order to make the
+     histogram ready for use.
+
+ -- Function: int gsl_histogram_set_ranges (gsl_histogram * H, const
+          double RANGE[], size_t SIZE)
+     This function sets the ranges of the existing histogram H using
+     the array RANGE of size SIZE.  The values of the histogram bins
+     are reset to zero.  The `range' array should contain the desired
+     bin limits.  The ranges can be arbitrary, subject to the
+     restriction that they are monotonically increasing.
+
+     The following example shows how to create a histogram with
+     logarithmic bins with ranges [1,10), [10,100) and [100,1000).
+
+          gsl_histogram * h = gsl_histogram_alloc (3);
+
+          /* bin[0] covers the range 1 <= x < 10 */
+          /* bin[1] covers the range 10 <= x < 100 */
+          /* bin[2] covers the range 100 <= x < 1000 */
+
+          double range[4] = { 1.0, 10.0, 100.0, 1000.0 };
+
+          gsl_histogram_set_ranges (h, range, 4);
+
+     Note that the size of the RANGE array should be defined to be one
+     element bigger than the number of bins.  The additional element is
+     required for the upper value of the final bin.
+
+ -- Function: int gsl_histogram_set_ranges_uniform (gsl_histogram * H,
+          double XMIN, double XMAX)
+     This function sets the ranges of the existing histogram H to cover
+     the range XMIN to XMAX uniformly.  The values of the histogram
+     bins are reset to zero.  The bin ranges are shown in the table
+     below,
+          bin[0] corresponds to xmin <= x < xmin + d
+          bin[1] corresponds to xmin + d <= x < xmin + 2 d
+          ......
+          bin[n-1] corresponds to xmin + (n-1)d <= x < xmax
+
+     where d is the bin spacing, d = (xmax-xmin)/n.
+
+ -- Function: void gsl_histogram_free (gsl_histogram * H)
+     This function frees the histogram H and all of the memory
+     associated with it.
+
+
+File: gsl-ref.info,  Node: Copying Histograms,  Next: Updating and accessing histogram elements,  Prev: Histogram allocation,  Up: Histograms
+
+21.3 Copying Histograms
+=======================
+
+ -- Function: int gsl_histogram_memcpy (gsl_histogram * DEST, const
+          gsl_histogram * SRC)
+     This function copies the histogram SRC into the pre-existing
+     histogram DEST, making DEST into an exact copy of SRC.  The two
+     histograms must be of the same size.
+
+ -- Function: gsl_histogram * gsl_histogram_clone (const gsl_histogram
+          * SRC)
+     This function returns a pointer to a newly created histogram which
+     is an exact copy of the histogram SRC.
+
+
+File: gsl-ref.info,  Node: Updating and accessing histogram elements,  Next: Searching histogram ranges,  Prev: Copying Histograms,  Up: Histograms
+
+21.4 Updating and accessing histogram elements
+==============================================
+
+There are two ways to access histogram bins, either by specifying an x
+coordinate or by using the bin-index directly.  The functions for
+accessing the histogram through x coordinates use a binary search to
+identify the bin which covers the appropriate range.
+
+ -- Function: int gsl_histogram_increment (gsl_histogram * H, double X)
+     This function updates the histogram H by adding one (1.0) to the
+     bin whose range contains the coordinate X.
+
+     If X lies in the valid range of the histogram then the function
+     returns zero to indicate success.  If X is less than the lower
+     limit of the histogram then the function returns `GSL_EDOM', and
+     none of bins are modified.  Similarly, if the value of X is greater
+     than or equal to the upper limit of the histogram then the function
+     returns `GSL_EDOM', and none of the bins are modified.  The error
+     handler is not called, however, since it is often necessary to
+     compute histograms for a small range of a larger dataset, ignoring
+     the values outside the range of interest.
+
+ -- Function: int gsl_histogram_accumulate (gsl_histogram * H, double
+          X, double WEIGHT)
+     This function is similar to `gsl_histogram_increment' but increases
+     the value of the appropriate bin in the histogram H by the
+     floating-point number WEIGHT.
+
+ -- Function: double gsl_histogram_get (const gsl_histogram * H, size_t
+          I)
+     This function returns the contents of the I-th bin of the histogram
+     H.  If I lies outside the valid range of indices for the histogram
+     then the error handler is called with an error code of `GSL_EDOM'
+     and the function returns 0.
+
+ -- Function: int gsl_histogram_get_range (const gsl_histogram * H,
+          size_t I, double * LOWER, double * UPPER)
+     This function finds the upper and lower range limits of the I-th
+     bin of the histogram H.  If the index I is valid then the
+     corresponding range limits are stored in LOWER and UPPER.  The
+     lower limit is inclusive (i.e. events with this coordinate are
+     included in the bin) and the upper limit is exclusive (i.e. events
+     with the coordinate of the upper limit are excluded and fall in the
+     neighboring higher bin, if it exists).  The function returns 0 to
+     indicate success.  If I lies outside the valid range of indices for
+     the histogram then the error handler is called and the function
+     returns an error code of `GSL_EDOM'.
+
+ -- Function: double gsl_histogram_max (const gsl_histogram * H)
+ -- Function: double gsl_histogram_min (const gsl_histogram * H)
+ -- Function: size_t gsl_histogram_bins (const gsl_histogram * H)
+     These functions return the maximum upper and minimum lower range
+     limits and the number of bins of the histogram H.  They provide a
+     way of determining these values without accessing the
+     `gsl_histogram' struct directly.
+
+ -- Function: void gsl_histogram_reset (gsl_histogram * H)
+     This function resets all the bins in the histogram H to zero.
+
+
+File: gsl-ref.info,  Node: Searching histogram ranges,  Next: Histogram Statistics,  Prev: Updating and accessing histogram elements,  Up: Histograms
+
+21.5 Searching histogram ranges
+===============================
+
+The following functions are used by the access and update routines to
+locate the bin which corresponds to a given x coordinate.
+
+ -- Function: int gsl_histogram_find (const gsl_histogram * H, double
+          X, size_t * I)
+     This function finds and sets the index I to the bin number which
+     covers the coordinate X in the histogram H.  The bin is located
+     using a binary search. The search includes an optimization for
+     histograms with uniform range, and will return the correct bin
+     immediately in this case.  If X is found in the range of the
+     histogram then the function sets the index I and returns
+     `GSL_SUCCESS'.  If X lies outside the valid range of the histogram
+     then the function returns `GSL_EDOM' and the error handler is
+     invoked.
+
+
+File: gsl-ref.info,  Node: Histogram Statistics,  Next: Histogram Operations,  Prev: Searching histogram ranges,  Up: Histograms
+
+21.6 Histogram Statistics
+=========================
+
+ -- Function: double gsl_histogram_max_val (const gsl_histogram * H)
+     This function returns the maximum value contained in the histogram
+     bins.
+
+ -- Function: size_t gsl_histogram_max_bin (const gsl_histogram * H)
+     This function returns the index of the bin containing the maximum
+     value. In the case where several bins contain the same maximum
+     value the smallest index is returned.
+
+ -- Function: double gsl_histogram_min_val (const gsl_histogram * H)
+     This function returns the minimum value contained in the histogram
+     bins.
+
+ -- Function: size_t gsl_histogram_min_bin (const gsl_histogram * H)
+     This function returns the index of the bin containing the minimum
+     value. In the case where several bins contain the same maximum
+     value the smallest index is returned.
+
+ -- Function: double gsl_histogram_mean (const gsl_histogram * H)
+     This function returns the mean of the histogrammed variable, where
+     the histogram is regarded as a probability distribution. Negative
+     bin values are ignored for the purposes of this calculation.  The
+     accuracy of the result is limited by the bin width.
+
+ -- Function: double gsl_histogram_sigma (const gsl_histogram * H)
+     This function returns the standard deviation of the histogrammed
+     variable, where the histogram is regarded as a probability
+     distribution. Negative bin values are ignored for the purposes of
+     this calculation. The accuracy of the result is limited by the bin
+     width.
+
+ -- Function: double gsl_histogram_sum (const gsl_histogram * H)
+     This function returns the sum of all bin values. Negative bin
+     values are included in the sum.
+
+
+File: gsl-ref.info,  Node: Histogram Operations,  Next: Reading and writing histograms,  Prev: Histogram Statistics,  Up: Histograms
+
+21.7 Histogram Operations
+=========================
+
+ -- Function: int gsl_histogram_equal_bins_p (const gsl_histogram * H1,
+          const gsl_histogram * H2)
+     This function returns 1 if the all of the individual bin ranges of
+     the two histograms are identical, and 0 otherwise.
+
+ -- Function: int gsl_histogram_add (gsl_histogram * H1, const
+          gsl_histogram * H2)
+     This function adds the contents of the bins in histogram H2 to the
+     corresponding bins of histogram H1,  i.e. h'_1(i) = h_1(i) +
+     h_2(i).  The two histograms must have identical bin ranges.
+
+ -- Function: int gsl_histogram_sub (gsl_histogram * H1, const
+          gsl_histogram * H2)
+     This function subtracts the contents of the bins in histogram H2
+     from the corresponding bins of histogram H1, i.e. h'_1(i) = h_1(i)
+     - h_2(i).  The two histograms must have identical bin ranges.
+
+ -- Function: int gsl_histogram_mul (gsl_histogram * H1, const
+          gsl_histogram * H2)
+     This function multiplies the contents of the bins of histogram H1
+     by the contents of the corresponding bins in histogram H2, i.e.
+     h'_1(i) = h_1(i) * h_2(i).  The two histograms must have identical
+     bin ranges.
+
+ -- Function: int gsl_histogram_div (gsl_histogram * H1, const
+          gsl_histogram * H2)
+     This function divides the contents of the bins of histogram H1 by
+     the contents of the corresponding bins in histogram H2, i.e.
+     h'_1(i) = h_1(i) / h_2(i).  The two histograms must have identical
+     bin ranges.
+
+ -- Function: int gsl_histogram_scale (gsl_histogram * H, double SCALE)
+     This function multiplies the contents of the bins of histogram H
+     by the constant SCALE, i.e. h'_1(i) = h_1(i) * scale.
+
+ -- Function: int gsl_histogram_shift (gsl_histogram * H, double OFFSET)
+     This function shifts the contents of the bins of histogram H by
+     the constant OFFSET, i.e. h'_1(i) = h_1(i) + offset.
+
+
+File: gsl-ref.info,  Node: Reading and writing histograms,  Next: Resampling from histograms,  Prev: Histogram Operations,  Up: Histograms
+
+21.8 Reading and writing histograms
+===================================
+
+The library provides functions for reading and writing histograms to a
+file as binary data or formatted text.
+
+ -- Function: int gsl_histogram_fwrite (FILE * STREAM, const
+          gsl_histogram * H)
+     This function writes the ranges and bins of the histogram H to the
+     stream STREAM in binary format.  The return value is 0 for success
+     and `GSL_EFAILED' if there was a problem writing to the file.
+     Since the data is written in the native binary format it may not
+     be portable between different architectures.
+
+ -- Function: int gsl_histogram_fread (FILE * STREAM, gsl_histogram * H)
+     This function reads into the histogram H from the open stream
+     STREAM in binary format.  The histogram H must be preallocated
+     with the correct size since the function uses the number of bins
+     in H to determine how many bytes to read.  The return value is 0
+     for success and `GSL_EFAILED' if there was a problem reading from
+     the file.  The data is assumed to have been written in the native
+     binary format on the same architecture.
+
+ -- Function: int gsl_histogram_fprintf (FILE * STREAM, const
+          gsl_histogram * H, const char * RANGE_FORMAT, const char *
+          BIN_FORMAT)
+     This function writes the ranges and bins of the histogram H
+     line-by-line to the stream STREAM using the format specifiers
+     RANGE_FORMAT and BIN_FORMAT.  These should be one of the `%g',
+     `%e' or `%f' formats for floating point numbers.  The function
+     returns 0 for success and `GSL_EFAILED' if there was a problem
+     writing to the file.  The histogram output is formatted in three
+     columns, and the columns are separated by spaces, like this,
+
+          range[0] range[1] bin[0]
+          range[1] range[2] bin[1]
+          range[2] range[3] bin[2]
+          ....
+          range[n-1] range[n] bin[n-1]
+
+     The values of the ranges are formatted using RANGE_FORMAT and the
+     value of the bins are formatted using BIN_FORMAT.  Each line
+     contains the lower and upper limit of the range of the bins and the
+     value of the bin itself.  Since the upper limit of one bin is the
+     lower limit of the next there is duplication of these values
+     between lines but this allows the histogram to be manipulated with
+     line-oriented tools.
+
+ -- Function: int gsl_histogram_fscanf (FILE * STREAM, gsl_histogram *
+          H)
+     This function reads formatted data from the stream STREAM into the
+     histogram H.  The data is assumed to be in the three-column format
+     used by `gsl_histogram_fprintf'.  The histogram H must be
+     preallocated with the correct length since the function uses the
+     size of H to determine how many numbers to read.  The function
+     returns 0 for success and `GSL_EFAILED' if there was a problem
+     reading from the file.
+
+
+File: gsl-ref.info,  Node: Resampling from histograms,  Next: The histogram probability distribution struct,  Prev: Reading and writing histograms,  Up: Histograms
+
+21.9 Resampling from histograms
+===============================
+
+A histogram made by counting events can be regarded as a measurement of
+a probability distribution.  Allowing for statistical error, the height
+of each bin represents the probability of an event where the value of x
+falls in the range of that bin.  The probability distribution function
+has the one-dimensional form p(x)dx where,
+
+     p(x) = n_i/ (N w_i)
+
+In this equation n_i is the number of events in the bin which contains
+x, w_i is the width of the bin and N is the total number of events.
+The distribution of events within each bin is assumed to be uniform.
+
+
+File: gsl-ref.info,  Node: The histogram probability distribution struct,  Next: Example programs for histograms,  Prev: Resampling from histograms,  Up: Histograms
+
+21.10 The histogram probability distribution struct
+===================================================
+
+The probability distribution function for a histogram consists of a set
+of "bins" which measure the probability of an event falling into a
+given range of a continuous variable x. A probability distribution
+function is defined by the following struct, which actually stores the
+cumulative probability distribution function.  This is the natural
+quantity for generating samples via the inverse transform method,
+because there is a one-to-one mapping between the cumulative
+probability distribution and the range [0,1].  It can be shown that by
+taking a uniform random number in this range and finding its
+corresponding coordinate in the cumulative probability distribution we
+obtain samples with the desired probability distribution.
+
+ -- Data Type: gsl_histogram_pdf
+    `size_t n'
+          This is the number of bins used to approximate the probability
+          distribution function.
+
+    `double * range'
+          The ranges of the bins are stored in an array of N+1 elements
+          pointed to by RANGE.
+
+    `double * sum'
+          The cumulative probability for the bins is stored in an array
+          of N elements pointed to by SUM.
+
+The following functions allow you to create a `gsl_histogram_pdf'
+struct which represents this probability distribution and generate
+random samples from it.
+
+ -- Function: gsl_histogram_pdf * gsl_histogram_pdf_alloc (size_t N)
+     This function allocates memory for a probability distribution with
+     N bins and returns a pointer to a newly initialized
+     `gsl_histogram_pdf' struct. If insufficient memory is available a
+     null pointer is returned and the error handler is invoked with an
+     error code of `GSL_ENOMEM'.
+
+ -- Function: int gsl_histogram_pdf_init (gsl_histogram_pdf * P, const
+          gsl_histogram * H)
+     This function initializes the probability distribution P with the
+     contents of the histogram H. If any of the bins of H are negative
+     then the error handler is invoked with an error code of `GSL_EDOM'
+     because a probability distribution cannot contain negative values.
+
+ -- Function: void gsl_histogram_pdf_free (gsl_histogram_pdf * P)
+     This function frees the probability distribution function P and
+     all of the memory associated with it.
+
+ -- Function: double gsl_histogram_pdf_sample (const gsl_histogram_pdf
+          * P, double R)
+     This function uses R, a uniform random number between zero and
+     one, to compute a single random sample from the probability
+     distribution P.  The algorithm used to compute the sample s is
+     given by the following formula,
+
+          s = range[i] + delta * (range[i+1] - range[i])
+
+     where i is the index which satisfies sum[i] <=  r < sum[i+1] and
+     delta is (r - sum[i])/(sum[i+1] - sum[i]).
+
+
+File: gsl-ref.info,  Node: Example programs for histograms,  Next: Two dimensional histograms,  Prev: The histogram probability distribution struct,  Up: Histograms
+
+21.11 Example programs for histograms
+=====================================
+
+The following program shows how to make a simple histogram of a column
+of numerical data supplied on `stdin'.  The program takes three
+arguments, specifying the upper and lower bounds of the histogram and
+the number of bins.  It then reads numbers from `stdin', one line at a
+time, and adds them to the histogram.  When there is no more data to
+read it prints out the accumulated histogram using
+`gsl_histogram_fprintf'.
+
+     #include <stdio.h>
+     #include <stdlib.h>
+     #include <gsl/gsl_histogram.h>
+
+     int
+     main (int argc, char **argv)
+     {
+       double a, b;
+       size_t n;
+
+       if (argc != 4)
+         {
+           printf ("Usage: gsl-histogram xmin xmax n\n"
+                   "Computes a histogram of the data "
+                   "on stdin using n bins from xmin "
+                   "to xmax\n");
+           exit (0);
+         }
+
+       a = atof (argv[1]);
+       b = atof (argv[2]);
+       n = atoi (argv[3]);
+
+       {
+         double x;
+         gsl_histogram * h = gsl_histogram_alloc (n);
+         gsl_histogram_set_ranges_uniform (h, a, b);
+
+         while (fscanf (stdin, "%lg", &x) == 1)
+           {
+             gsl_histogram_increment (h, x);
+           }
+         gsl_histogram_fprintf (stdout, h, "%g", "%g");
+         gsl_histogram_free (h);
+       }
+       exit (0);
+     }
+
+Here is an example of the program in use.  We generate 10000 random
+samples from a Cauchy distribution with a width of 30 and histogram
+them over the range -100 to 100, using 200 bins.
+
+     $ gsl-randist 0 10000 cauchy 30
+        | gsl-histogram -100 100 200 > histogram.dat
+
+A plot of the resulting histogram shows the familiar shape of the
+Cauchy distribution and the fluctuations caused by the finite sample
+size.
+
+     $ awk '{print $1, $3 ; print $2, $3}' histogram.dat
+        | graph -T X
+
+
+File: gsl-ref.info,  Node: Two dimensional histograms,  Next: The 2D histogram struct,  Prev: Example programs for histograms,  Up: Histograms
+
+21.12 Two dimensional histograms
+================================
+
+A two dimensional histogram consists of a set of "bins" which count the
+number of events falling in a given area of the (x,y) plane.  The
+simplest way to use a two dimensional histogram is to record
+two-dimensional position information, n(x,y).  Another possibility is
+to form a "joint distribution" by recording related variables.  For
+example a detector might record both the position of an event (x) and
+the amount of energy it deposited E.  These could be histogrammed as
+the joint distribution n(x,E).
+
+
+File: gsl-ref.info,  Node: The 2D histogram struct,  Next: 2D Histogram allocation,  Prev: Two dimensional histograms,  Up: Histograms
+
+21.13 The 2D histogram struct
+=============================
+
+Two dimensional histograms are defined by the following struct,
+
+ -- Data Type: gsl_histogram2d
+    `size_t nx, ny'
+          This is the number of histogram bins in the x and y
+          directions.
+
+    `double * xrange'
+          The ranges of the bins in the x-direction are stored in an
+          array of NX + 1 elements pointed to by XRANGE.
+
+    `double * yrange'
+          The ranges of the bins in the y-direction are stored in an
+          array of NY + 1 elements pointed to by YRANGE.
+
+    `double * bin'
+          The counts for each bin are stored in an array pointed to by
+          BIN.  The bins are floating-point numbers, so you can
+          increment them by non-integer values if necessary.  The array
+          BIN stores the two dimensional array of bins in a single
+          block of memory according to the mapping `bin(i,j)' = `bin[i
+          * ny + j]'.
+
+The range for `bin(i,j)' is given by `xrange[i]' to `xrange[i+1]' in
+the x-direction and `yrange[j]' to `yrange[j+1]' in the y-direction.
+Each bin is inclusive at the lower end and exclusive at the upper end.
+Mathematically this means that the bins are defined by the following
+inequality,
+     bin(i,j) corresponds to xrange[i] <= x < xrange[i+1]
+                         and yrange[j] <= y < yrange[j+1]
+
+Note that any samples which fall on the upper sides of the histogram are
+excluded.  If you want to include these values for the side bins you
+will need to add an extra row or column to your histogram.
+
+   The `gsl_histogram2d' struct and its associated functions are
+defined in the header file `gsl_histogram2d.h'.
+
+
+File: gsl-ref.info,  Node: 2D Histogram allocation,  Next: Copying 2D Histograms,  Prev: The 2D histogram struct,  Up: Histograms
+
+21.14 2D Histogram allocation
+=============================
+
+The functions for allocating memory to a 2D histogram follow the style
+of `malloc' and `free'.  In addition they also perform their own error
+checking.  If there is insufficient memory available to allocate a
+histogram then the functions call the error handler (with an error
+number of `GSL_ENOMEM') in addition to returning a null pointer.  Thus
+if you use the library error handler to abort your program then it
+isn't necessary to check every 2D histogram `alloc'.
+
+ -- Function: gsl_histogram2d * gsl_histogram2d_alloc (size_t NX,
+          size_t NY)
+     This function allocates memory for a two-dimensional histogram with
+     NX bins in the x direction and NY bins in the y direction.  The
+     function returns a pointer to a newly created `gsl_histogram2d'
+     struct. If insufficient memory is available a null pointer is
+     returned and the error handler is invoked with an error code of
+     `GSL_ENOMEM'. The bins and ranges must be initialized with one of
+     the functions below before the histogram is ready for use.
+
+ -- Function: int gsl_histogram2d_set_ranges (gsl_histogram2d * H,
+          const double XRANGE[], size_t XSIZE, const double YRANGE[],
+          size_t YSIZE)
+     This function sets the ranges of the existing histogram H using
+     the arrays XRANGE and YRANGE of size XSIZE and YSIZE respectively.
+     The values of the histogram bins are reset to zero.
+
+ -- Function: int gsl_histogram2d_set_ranges_uniform (gsl_histogram2d *
+          H, double XMIN, double XMAX, double YMIN, double YMAX)
+     This function sets the ranges of the existing histogram H to cover
+     the ranges XMIN to XMAX and YMIN to YMAX uniformly.  The values of
+     the histogram bins are reset to zero.
+
+ -- Function: void gsl_histogram2d_free (gsl_histogram2d * H)
+     This function frees the 2D histogram H and all of the memory
+     associated with it.
+
+
+File: gsl-ref.info,  Node: Copying 2D Histograms,  Next: Updating and accessing 2D histogram elements,  Prev: 2D Histogram allocation,  Up: Histograms
+
+21.15 Copying 2D Histograms
+===========================
+
+ -- Function: int gsl_histogram2d_memcpy (gsl_histogram2d * DEST, const
+          gsl_histogram2d * SRC)
+     This function copies the histogram SRC into the pre-existing
+     histogram DEST, making DEST into an exact copy of SRC.  The two
+     histograms must be of the same size.
+
+ -- Function: gsl_histogram2d * gsl_histogram2d_clone (const
+          gsl_histogram2d * SRC)
+     This function returns a pointer to a newly created histogram which
+     is an exact copy of the histogram SRC.
+
+
+File: gsl-ref.info,  Node: Updating and accessing 2D histogram elements,  Next: Searching 2D histogram ranges,  Prev: Copying 2D Histograms,  Up: Histograms
+
+21.16 Updating and accessing 2D histogram elements
+==================================================
+
+You can access the bins of a two-dimensional histogram either by
+specifying a pair of (x,y) coordinates or by using the bin indices
+(i,j) directly.  The functions for accessing the histogram through
+(x,y) coordinates use binary searches in the x and y directions to
+identify the bin which covers the appropriate range.
+
+ -- Function: int gsl_histogram2d_increment (gsl_histogram2d * H,
+          double X, double Y)
+     This function updates the histogram H by adding one (1.0) to the
+     bin whose x and y ranges contain the coordinates (X,Y).
+
+     If the point (x,y) lies inside the valid ranges of the histogram
+     then the function returns zero to indicate success.  If (x,y) lies
+     outside the limits of the histogram then the function returns
+     `GSL_EDOM', and none of the bins are modified.  The error handler
+     is not called, since it is often necessary to compute histograms
+     for a small range of a larger dataset, ignoring any coordinates
+     outside the range of interest.
+
+ -- Function: int gsl_histogram2d_accumulate (gsl_histogram2d * H,
+          double X, double Y, double WEIGHT)
+     This function is similar to `gsl_histogram2d_increment' but
+     increases the value of the appropriate bin in the histogram H by
+     the floating-point number WEIGHT.
+
+ -- Function: double gsl_histogram2d_get (const gsl_histogram2d * H,
+          size_t I, size_t J)
+     This function returns the contents of the (I,J)-th bin of the
+     histogram H.  If (I,J) lies outside the valid range of indices for
+     the histogram then the error handler is called with an error code
+     of `GSL_EDOM' and the function returns 0.
+
+ -- Function: int gsl_histogram2d_get_xrange (const gsl_histogram2d *
+          H, size_t I, double * XLOWER, double * XUPPER)
+ -- Function: int gsl_histogram2d_get_yrange (const gsl_histogram2d *
+          H, size_t J, double * YLOWER, double * YUPPER)
+     These functions find the upper and lower range limits of the I-th
+     and J-th bins in the x and y directions of the histogram H.  The
+     range limits are stored in XLOWER and XUPPER or YLOWER and YUPPER.
+     The lower limits are inclusive (i.e. events with these
+     coordinates are included in the bin) and the upper limits are
+     exclusive (i.e. events with the value of the upper limit are not
+     included and fall in the neighboring higher bin, if it exists).
+     The functions return 0 to indicate success.  If I or J lies
+     outside the valid range of indices for the histogram then the
+     error handler is called with an error code of `GSL_EDOM'.
+
+ -- Function: double gsl_histogram2d_xmax (const gsl_histogram2d * H)
+ -- Function: double gsl_histogram2d_xmin (const gsl_histogram2d * H)
+ -- Function: size_t gsl_histogram2d_nx (const gsl_histogram2d * H)
+ -- Function: double gsl_histogram2d_ymax (const gsl_histogram2d * H)
+ -- Function: double gsl_histogram2d_ymin (const gsl_histogram2d * H)
+ -- Function: size_t gsl_histogram2d_ny (const gsl_histogram2d * H)
+     These functions return the maximum upper and minimum lower range
+     limits and the number of bins for the x and y directions of the
+     histogram H.  They provide a way of determining these values
+     without accessing the `gsl_histogram2d' struct directly.
+
+ -- Function: void gsl_histogram2d_reset (gsl_histogram2d * H)
+     This function resets all the bins of the histogram H to zero.
+
+
+File: gsl-ref.info,  Node: Searching 2D histogram ranges,  Next: 2D Histogram Statistics,  Prev: Updating and accessing 2D histogram elements,  Up: Histograms
+
+21.17 Searching 2D histogram ranges
+===================================
+
+The following functions are used by the access and update routines to
+locate the bin which corresponds to a given (x,y) coordinate.
+
+ -- Function: int gsl_histogram2d_find (const gsl_histogram2d * H,
+          double X, double Y, size_t * I, size_t * J)
+     This function finds and sets the indices I and J to the to the bin
+     which covers the coordinates (X,Y). The bin is located using a
+     binary search.  The search includes an optimization for histograms
+     with uniform ranges, and will return the correct bin immediately
+     in this case. If (x,y) is found then the function sets the indices
+     (I,J) and returns `GSL_SUCCESS'.  If (x,y) lies outside the valid
+     range of the histogram then the function returns `GSL_EDOM' and
+     the error handler is invoked.
+
+
+File: gsl-ref.info,  Node: 2D Histogram Statistics,  Next: 2D Histogram Operations,  Prev: Searching 2D histogram ranges,  Up: Histograms
+
+21.18 2D Histogram Statistics
+=============================
+
+ -- Function: double gsl_histogram2d_max_val (const gsl_histogram2d * H)
+     This function returns the maximum value contained in the histogram
+     bins.
+
+ -- Function: void gsl_histogram2d_max_bin (const gsl_histogram2d * H,
+          size_t * I, size_t * J)
+     This function finds the indices of the bin containing the maximum
+     value in the histogram H and stores the result in (I,J). In the
+     case where several bins contain the same maximum value the first
+     bin found is returned.
+
+ -- Function: double gsl_histogram2d_min_val (const gsl_histogram2d * H)
+     This function returns the minimum value contained in the histogram
+     bins.
+
+ -- Function: void gsl_histogram2d_min_bin (const gsl_histogram2d * H,
+          size_t * I, size_t * J)
+     This function finds the indices of the bin containing the minimum
+     value in the histogram H and stores the result in (I,J). In the
+     case where several bins contain the same maximum value the first
+     bin found is returned.
+
+ -- Function: double gsl_histogram2d_xmean (const gsl_histogram2d * H)
+     This function returns the mean of the histogrammed x variable,
+     where the histogram is regarded as a probability distribution.
+     Negative bin values are ignored for the purposes of this
+     calculation.
+
+ -- Function: double gsl_histogram2d_ymean (const gsl_histogram2d * H)
+     This function returns the mean of the histogrammed y variable,
+     where the histogram is regarded as a probability distribution.
+     Negative bin values are ignored for the purposes of this
+     calculation.
+
+ -- Function: double gsl_histogram2d_xsigma (const gsl_histogram2d * H)
+     This function returns the standard deviation of the histogrammed x
+     variable, where the histogram is regarded as a probability
+     distribution. Negative bin values are ignored for the purposes of
+     this calculation.
+
+ -- Function: double gsl_histogram2d_ysigma (const gsl_histogram2d * H)
+     This function returns the standard deviation of the histogrammed y
+     variable, where the histogram is regarded as a probability
+     distribution. Negative bin values are ignored for the purposes of
+     this calculation.
+
+ -- Function: double gsl_histogram2d_cov (const gsl_histogram2d * H)
+     This function returns the covariance of the histogrammed x and y
+     variables, where the histogram is regarded as a probability
+     distribution. Negative bin values are ignored for the purposes of
+     this calculation.
+
+ -- Function: double gsl_histogram2d_sum (const gsl_histogram2d * H)
+     This function returns the sum of all bin values. Negative bin
+     values are included in the sum.
+
+
+File: gsl-ref.info,  Node: 2D Histogram Operations,  Next: Reading and writing 2D histograms,  Prev: 2D Histogram Statistics,  Up: Histograms
+
+21.19 2D Histogram Operations
+=============================
+
+ -- Function: int gsl_histogram2d_equal_bins_p (const gsl_histogram2d *
+          H1, const gsl_histogram2d * H2)
+     This function returns 1 if all the individual bin ranges of the two
+     histograms are identical, and 0 otherwise.
+
+ -- Function: int gsl_histogram2d_add (gsl_histogram2d * H1, const
+          gsl_histogram2d * H2)
+     This function adds the contents of the bins in histogram H2 to the
+     corresponding bins of histogram H1, i.e. h'_1(i,j) = h_1(i,j) +
+     h_2(i,j).  The two histograms must have identical bin ranges.
+
+ -- Function: int gsl_histogram2d_sub (gsl_histogram2d * H1, const
+          gsl_histogram2d * H2)
+     This function subtracts the contents of the bins in histogram H2
+     from the corresponding bins of histogram H1, i.e. h'_1(i,j) =
+     h_1(i,j) - h_2(i,j).  The two histograms must have identical bin
+     ranges.
+
+ -- Function: int gsl_histogram2d_mul (gsl_histogram2d * H1, const
+          gsl_histogram2d * H2)
+     This function multiplies the contents of the bins of histogram H1
+     by the contents of the corresponding bins in histogram H2, i.e.
+     h'_1(i,j) = h_1(i,j) * h_2(i,j).  The two histograms must have
+     identical bin ranges.
+
+ -- Function: int gsl_histogram2d_div (gsl_histogram2d * H1, const
+          gsl_histogram2d * H2)
+     This function divides the contents of the bins of histogram H1 by
+     the contents of the corresponding bins in histogram H2, i.e.
+     h'_1(i,j) = h_1(i,j) / h_2(i,j).  The two histograms must have
+     identical bin ranges.
+
+ -- Function: int gsl_histogram2d_scale (gsl_histogram2d * H, double
+          SCALE)
+     This function multiplies the contents of the bins of histogram H
+     by the constant SCALE, i.e. h'_1(i,j) = h_1(i,j) scale.
+
+ -- Function: int gsl_histogram2d_shift (gsl_histogram2d * H, double
+          OFFSET)
+     This function shifts the contents of the bins of histogram H by
+     the constant OFFSET, i.e. h'_1(i,j) = h_1(i,j) + offset.
+
+
+File: gsl-ref.info,  Node: Reading and writing 2D histograms,  Next: Resampling from 2D histograms,  Prev: 2D Histogram Operations,  Up: Histograms
+
+21.20 Reading and writing 2D histograms
+=======================================
+
+The library provides functions for reading and writing two dimensional
+histograms to a file as binary data or formatted text.
+
+ -- Function: int gsl_histogram2d_fwrite (FILE * STREAM, const
+          gsl_histogram2d * H)
+     This function writes the ranges and bins of the histogram H to the
+     stream STREAM in binary format.  The return value is 0 for success
+     and `GSL_EFAILED' if there was a problem writing to the file.
+     Since the data is written in the native binary format it may not
+     be portable between different architectures.
+
+ -- Function: int gsl_histogram2d_fread (FILE * STREAM, gsl_histogram2d
+          * H)
+     This function reads into the histogram H from the stream STREAM in
+     binary format.  The histogram H must be preallocated with the
+     correct size since the function uses the number of x and y bins in
+     H to determine how many bytes to read.  The return value is 0 for
+     success and `GSL_EFAILED' if there was a problem reading from the
+     file.  The data is assumed to have been written in the native
+     binary format on the same architecture.
+
+ -- Function: int gsl_histogram2d_fprintf (FILE * STREAM, const
+          gsl_histogram2d * H, const char * RANGE_FORMAT, const char *
+          BIN_FORMAT)
+     This function writes the ranges and bins of the histogram H
+     line-by-line to the stream STREAM using the format specifiers
+     RANGE_FORMAT and BIN_FORMAT.  These should be one of the `%g',
+     `%e' or `%f' formats for floating point numbers.  The function
+     returns 0 for success and `GSL_EFAILED' if there was a problem
+     writing to the file.  The histogram output is formatted in five
+     columns, and the columns are separated by spaces, like this,
+
+          xrange[0] xrange[1] yrange[0] yrange[1] bin(0,0)
+          xrange[0] xrange[1] yrange[1] yrange[2] bin(0,1)
+          xrange[0] xrange[1] yrange[2] yrange[3] bin(0,2)
+          ....
+          xrange[0] xrange[1] yrange[ny-1] yrange[ny] bin(0,ny-1)
+
+          xrange[1] xrange[2] yrange[0] yrange[1] bin(1,0)
+          xrange[1] xrange[2] yrange[1] yrange[2] bin(1,1)
+          xrange[1] xrange[2] yrange[1] yrange[2] bin(1,2)
+          ....
+          xrange[1] xrange[2] yrange[ny-1] yrange[ny] bin(1,ny-1)
+
+          ....
+
+          xrange[nx-1] xrange[nx] yrange[0] yrange[1] bin(nx-1,0)
+          xrange[nx-1] xrange[nx] yrange[1] yrange[2] bin(nx-1,1)
+          xrange[nx-1] xrange[nx] yrange[1] yrange[2] bin(nx-1,2)
+          ....
+          xrange[nx-1] xrange[nx] yrange[ny-1] yrange[ny] bin(nx-1,ny-1)
+
+     Each line contains the lower and upper limits of the bin and the
+     contents of the bin.  Since the upper limits of the each bin are
+     the lower limits of the neighboring bins there is duplication of
+     these values but this allows the histogram to be manipulated with
+     line-oriented tools.
+
+ -- Function: int gsl_histogram2d_fscanf (FILE * STREAM,
+          gsl_histogram2d * H)
+     This function reads formatted data from the stream STREAM into the
+     histogram H.  The data is assumed to be in the five-column format
+     used by `gsl_histogram_fprintf'.  The histogram H must be
+     preallocated with the correct lengths since the function uses the
+     sizes of H to determine how many numbers to read.  The function
+     returns 0 for success and `GSL_EFAILED' if there was a problem
+     reading from the file.
+
+
+File: gsl-ref.info,  Node: Resampling from 2D histograms,  Next: Example programs for 2D histograms,  Prev: Reading and writing 2D histograms,  Up: Histograms
+
+21.21 Resampling from 2D histograms
+===================================
+
+As in the one-dimensional case, a two-dimensional histogram made by
+counting events can be regarded as a measurement of a probability
+distribution.  Allowing for statistical error, the height of each bin
+represents the probability of an event where (x,y) falls in the range
+of that bin.  For a two-dimensional histogram the probability
+distribution takes the form p(x,y) dx dy where,
+
+     p(x,y) = n_{ij}/ (N A_{ij})
+
+In this equation n_{ij} is the number of events in the bin which
+contains (x,y), A_{ij} is the area of the bin and N is the total number
+of events.  The distribution of events within each bin is assumed to be
+uniform.
+
+ -- Data Type: gsl_histogram2d_pdf
+    `size_t nx, ny'
+          This is the number of histogram bins used to approximate the
+          probability distribution function in the x and y directions.
+
+    `double * xrange'
+          The ranges of the bins in the x-direction are stored in an
+          array of NX + 1 elements pointed to by XRANGE.
+
+    `double * yrange'
+          The ranges of the bins in the y-direction are stored in an
+          array of NY + 1 pointed to by YRANGE.
+
+    `double * sum'
+          The cumulative probability for the bins is stored in an array
+          of NX*NY elements pointed to by SUM.
+
+The following functions allow you to create a `gsl_histogram2d_pdf'
+struct which represents a two dimensional probability distribution and
+generate random samples from it.
+
+ -- Function: gsl_histogram2d_pdf * gsl_histogram2d_pdf_alloc (size_t
+          NX, size_t NY)
+     This function allocates memory for a two-dimensional probability
+     distribution of size NX-by-NY and returns a pointer to a newly
+     initialized `gsl_histogram2d_pdf' struct. If insufficient memory
+     is available a null pointer is returned and the error handler is
+     invoked with an error code of `GSL_ENOMEM'.
+
+ -- Function: int gsl_histogram2d_pdf_init (gsl_histogram2d_pdf * P,
+          const gsl_histogram2d * H)
+     This function initializes the two-dimensional probability
+     distribution calculated P from the histogram H.  If any of the
+     bins of H are negative then the error handler is invoked with an
+     error code of `GSL_EDOM' because a probability distribution cannot
+     contain negative values.
+
+ -- Function: void gsl_histogram2d_pdf_free (gsl_histogram2d_pdf * P)
+     This function frees the two-dimensional probability distribution
+     function P and all of the memory associated with it.
+
+ -- Function: int gsl_histogram2d_pdf_sample (const gsl_histogram2d_pdf
+          * P, double R1, double R2, double * X, double * Y)
+     This function uses two uniform random numbers between zero and one,
+     R1 and R2, to compute a single random sample from the
+     two-dimensional probability distribution P.
+
+
+File: gsl-ref.info,  Node: Example programs for 2D histograms,  Prev: Resampling from 2D histograms,  Up: Histograms
+
+21.22 Example programs for 2D histograms
+========================================
+
+This program demonstrates two features of two-dimensional histograms.
+First a 10-by-10 two-dimensional histogram is created with x and y
+running from 0 to 1.  Then a few sample points are added to the
+histogram, at (0.3,0.3) with a height of 1, at (0.8,0.1) with a height
+of 5 and at (0.7,0.9) with a height of 0.5.  This histogram with three
+events is used to generate a random sample of 1000 simulated events,
+which are printed out.
+
+     #include <stdio.h>
+     #include <gsl/gsl_rng.h>
+     #include <gsl/gsl_histogram2d.h>
+
+     int
+     main (void)
+     {
+       const gsl_rng_type * T;
+       gsl_rng * r;
+
+       gsl_histogram2d * h = gsl_histogram2d_alloc (10, 10);
+
+       gsl_histogram2d_set_ranges_uniform (h,
+                                           0.0, 1.0,
+                                           0.0, 1.0);
+
+       gsl_histogram2d_accumulate (h, 0.3, 0.3, 1);
+       gsl_histogram2d_accumulate (h, 0.8, 0.1, 5);
+       gsl_histogram2d_accumulate (h, 0.7, 0.9, 0.5);
+
+       gsl_rng_env_setup ();
+
+       T = gsl_rng_default;
+       r = gsl_rng_alloc (T);
+
+       {
+         int i;
+         gsl_histogram2d_pdf * p
+           = gsl_histogram2d_pdf_alloc (h->nx, h->ny);
+
+         gsl_histogram2d_pdf_init (p, h);
+
+         for (i = 0; i < 1000; i++) {
+           double x, y;
+           double u = gsl_rng_uniform (r);
+           double v = gsl_rng_uniform (r);
+
+           gsl_histogram2d_pdf_sample (p, u, v, &x, &y);
+
+           printf ("%g %g\n", x, y);
+         }
+
+         gsl_histogram2d_pdf_free (p);
+       }
+
+       gsl_histogram2d_free (h);
+       gsl_rng_free (r);
+
+       return 0;
+     }
+
+
+
+File: gsl-ref.info,  Node: N-tuples,  Next: Monte Carlo Integration,  Prev: Histograms,  Up: Top
+
+22 N-tuples
+***********
+
+This chapter describes functions for creating and manipulating
+"ntuples", sets of values associated with events.  The ntuples are
+stored in files. Their values can be extracted in any combination and
+"booked" in a histogram using a selection function.
+
+   The values to be stored are held in a user-defined data structure,
+and an ntuple is created associating this data structure with a file.
+The values are then written to the file (normally inside a loop) using
+the ntuple functions described below.
+
+   A histogram can be created from ntuple data by providing a selection
+function and a value function.  The selection function specifies whether
+an event should be included in the subset to be analyzed or not. The
+value function computes the entry to be added to the histogram for each
+event.
+
+   All the ntuple functions are defined in the header file
+`gsl_ntuple.h'
+
+* Menu:
+
+* The ntuple struct::
+* Creating ntuples::
+* Opening an existing ntuple file::
+* Writing ntuples::
+* Reading ntuples ::
+* Closing an ntuple file::
+* Histogramming ntuple values::
+* Example ntuple programs::
+* Ntuple References and Further Reading::
+
+
+File: gsl-ref.info,  Node: The ntuple struct,  Next: Creating ntuples,  Up: N-tuples
+
+22.1 The ntuple struct
+======================
+
+Ntuples are manipulated using the `gsl_ntuple' struct. This struct
+contains information on the file where the ntuple data is stored, a
+pointer to the current ntuple data row and the size of the user-defined
+ntuple data struct.
+
+     typedef struct {
+         FILE * file;
+         void * ntuple_data;
+         size_t size;
+     } gsl_ntuple;
+
+
+File: gsl-ref.info,  Node: Creating ntuples,  Next: Opening an existing ntuple file,  Prev: The ntuple struct,  Up: N-tuples
+
+22.2 Creating ntuples
+=====================
+
+ -- Function: gsl_ntuple * gsl_ntuple_create (char * FILENAME, void *
+          NTUPLE_DATA, size_t SIZE)
+     This function creates a new write-only ntuple file FILENAME for
+     ntuples of size SIZE and returns a pointer to the newly created
+     ntuple struct.  Any existing file with the same name is truncated
+     to zero length and overwritten.  A pointer to memory for the
+     current ntuple row NTUPLE_DATA must be supplied--this is used to
+     copy ntuples in and out of the file.
+
+
+File: gsl-ref.info,  Node: Opening an existing ntuple file,  Next: Writing ntuples,  Prev: Creating ntuples,  Up: N-tuples
+
+22.3 Opening an existing ntuple file
+====================================
+
+ -- Function: gsl_ntuple * gsl_ntuple_open (char * FILENAME, void *
+          NTUPLE_DATA, size_t SIZE)
+     This function opens an existing ntuple file FILENAME for reading
+     and returns a pointer to a corresponding ntuple struct. The
+     ntuples in the file must have size SIZE.  A pointer to memory for
+     the current ntuple row NTUPLE_DATA must be supplied--this is used
+     to copy ntuples in and out of the file.
+
+
+File: gsl-ref.info,  Node: Writing ntuples,  Next: Reading ntuples,  Prev: Opening an existing ntuple file,  Up: N-tuples
+
+22.4 Writing ntuples
+====================
+
+ -- Function: int gsl_ntuple_write (gsl_ntuple * NTUPLE)
+     This function writes the current ntuple NTUPLE->NTUPLE_DATA of
+     size NTUPLE->SIZE to the corresponding file.
+
+ -- Function: int gsl_ntuple_bookdata (gsl_ntuple * NTUPLE)
+     This function is a synonym for `gsl_ntuple_write'.
+
+
+File: gsl-ref.info,  Node: Reading ntuples,  Next: Closing an ntuple file,  Prev: Writing ntuples,  Up: N-tuples
+
+22.5 Reading ntuples
+====================
+
+ -- Function: int gsl_ntuple_read (gsl_ntuple * NTUPLE)
+     This function reads the current row of the ntuple file for NTUPLE
+     and stores the values in NTUPLE->DATA.
+
+
+File: gsl-ref.info,  Node: Closing an ntuple file,  Next: Histogramming ntuple values,  Prev: Reading ntuples,  Up: N-tuples
+
+22.6 Closing an ntuple file
+===========================
+
+ -- Function: int gsl_ntuple_close (gsl_ntuple * NTUPLE)
+     This function closes the ntuple file NTUPLE and frees its
+     associated allocated memory.
+
+
+File: gsl-ref.info,  Node: Histogramming ntuple values,  Next: Example ntuple programs,  Prev: Closing an ntuple file,  Up: N-tuples
+
+22.7 Histogramming ntuple values
+================================
+
+Once an ntuple has been created its contents can be histogrammed in
+various ways using the function `gsl_ntuple_project'.  Two user-defined
+functions must be provided, a function to select events and a function
+to compute scalar values. The selection function and the value function
+both accept the ntuple row as a first argument and other parameters as
+a second argument.
+
+   The "selection function" determines which ntuple rows are selected
+for histogramming.  It is defined by the following struct,
+     typedef struct {
+       int (* function) (void * ntuple_data, void * params);
+       void * params;
+     } gsl_ntuple_select_fn;
+
+The struct component FUNCTION should return a non-zero value for each
+ntuple row that is to be included in the histogram.
+
+   The "value function" computes scalar values for those ntuple rows
+selected by the selection function,
+     typedef struct {
+       double (* function) (void * ntuple_data, void * params);
+       void * params;
+     } gsl_ntuple_value_fn;
+
+In this case the struct component FUNCTION should return the value to
+be added to the histogram for the ntuple row.
+
+ -- Function: int gsl_ntuple_project (gsl_histogram * H, gsl_ntuple *
+          NTUPLE, gsl_ntuple_value_fn * VALUE_FUNC,
+          gsl_ntuple_select_fn * SELECT_FUNC)
+     This function updates the histogram H from the ntuple NTUPLE using
+     the functions VALUE_FUNC and SELECT_FUNC. For each ntuple row
+     where the selection function SELECT_FUNC is non-zero the
+     corresponding value of that row is computed using the function
+     VALUE_FUNC and added to the histogram.  Those ntuple rows where
+     SELECT_FUNC returns zero are ignored.  New entries are added to
+     the histogram, so subsequent calls can be used to accumulate
+     further data in the same histogram.
+
+
+File: gsl-ref.info,  Node: Example ntuple programs,  Next: Ntuple References and Further Reading,  Prev: Histogramming ntuple values,  Up: N-tuples
+
+22.8 Examples
+=============
+
+The following example programs demonstrate the use of ntuples in
+managing a large dataset.  The first program creates a set of 10,000
+simulated "events", each with 3 associated values (x,y,z).  These are
+generated from a gaussian distribution with unit variance, for
+demonstration purposes, and written to the ntuple file `test.dat'.
+
+     #include <gsl/gsl_ntuple.h>
+     #include <gsl/gsl_rng.h>
+     #include <gsl/gsl_randist.h>
+
+     struct data
+     {
+       double x;
+       double y;
+       double z;
+     };
+
+     int
+     main (void)
+     {
+       const gsl_rng_type * T;
+       gsl_rng * r;
+
+       struct data ntuple_row;
+       int i;
+
+       gsl_ntuple *ntuple
+         = gsl_ntuple_create ("test.dat", &ntuple_row,
+                              sizeof (ntuple_row));
+
+       gsl_rng_env_setup ();
+
+       T = gsl_rng_default;
+       r = gsl_rng_alloc (T);
+
+       for (i = 0; i < 10000; i++)
+         {
+           ntuple_row.x = gsl_ran_ugaussian (r);
+           ntuple_row.y = gsl_ran_ugaussian (r);
+           ntuple_row.z = gsl_ran_ugaussian (r);
+
+           gsl_ntuple_write (ntuple);
+         }
+
+       gsl_ntuple_close (ntuple);
+       gsl_rng_free (r);
+
+       return 0;
+     }
+
+The next program analyses the ntuple data in the file `test.dat'.  The
+analysis procedure is to compute the squared-magnitude of each event,
+E^2=x^2+y^2+z^2, and select only those which exceed a lower limit of
+1.5.  The selected events are then histogrammed using their E^2 values.
+
+     #include <math.h>
+     #include <gsl/gsl_ntuple.h>
+     #include <gsl/gsl_histogram.h>
+
+     struct data
+     {
+       double x;
+       double y;
+       double z;
+     };
+
+     int sel_func (void *ntuple_data, void *params);
+     double val_func (void *ntuple_data, void *params);
+
+     int
+     main (void)
+     {
+       struct data ntuple_row;
+
+       gsl_ntuple *ntuple
+         = gsl_ntuple_open ("test.dat", &ntuple_row,
+                            sizeof (ntuple_row));
+       double lower = 1.5;
+
+       gsl_ntuple_select_fn S;
+       gsl_ntuple_value_fn V;
+
+       gsl_histogram *h = gsl_histogram_alloc (100);
+       gsl_histogram_set_ranges_uniform(h, 0.0, 10.0);
+
+       S.function = &sel_func;
+       S.params = &lower;
+
+       V.function = &val_func;
+       V.params = 0;
+
+       gsl_ntuple_project (h, ntuple, &V, &S);
+       gsl_histogram_fprintf (stdout, h, "%f", "%f");
+       gsl_histogram_free (h);
+       gsl_ntuple_close (ntuple);
+
+       return 0;
+     }
+
+     int
+     sel_func (void *ntuple_data, void *params)
+     {
+       struct data * data = (struct data *) ntuple_data;
+       double x, y, z, E2, scale;
+       scale = *(double *) params;
+
+       x = data->x;
+       y = data->y;
+       z = data->z;
+
+       E2 = x * x + y * y + z * z;
+
+       return E2 > scale;
+     }
+
+     double
+     val_func (void *ntuple_data, void *params)
+     {
+       struct data * data = (struct data *) ntuple_data;
+       double x, y, z;
+
+       x = data->x;
+       y = data->y;
+       z = data->z;
+
+       return x * x + y * y + z * z;
+     }
+
+   The following plot shows the distribution of the selected events.
+Note the cut-off at the lower bound.
+
+
+File: gsl-ref.info,  Node: Ntuple References and Further Reading,  Prev: Example ntuple programs,  Up: N-tuples
+
+22.9 References and Further Reading
+===================================
+
+Further information on the use of ntuples can be found in the
+documentation for the CERN packages PAW and HBOOK (available online).
+
+
+File: gsl-ref.info,  Node: Monte Carlo Integration,  Next: Simulated Annealing,  Prev: N-tuples,  Up: Top
+
+23 Monte Carlo Integration
+**************************
+
+This chapter describes routines for multidimensional Monte Carlo
+integration.  These include the traditional Monte Carlo method and
+adaptive algorithms such as VEGAS and MISER which use importance
+sampling and stratified sampling techniques. Each algorithm computes an
+estimate of a multidimensional definite integral of the form,
+
+     I = \int_xl^xu dx \int_yl^yu  dy ...  f(x, y, ...)
+
+over a hypercubic region ((x_l,x_u), (y_l,y_u), ...) using a fixed
+number of function calls.  The routines also provide a statistical
+estimate of the error on the result.  This error estimate should be
+taken as a guide rather than as a strict error bound--random sampling
+of the region may not uncover all the important features of the
+function, resulting in an underestimate of the error.
+
+   The functions are defined in separate header files for each routine,
+`gsl_monte_plain.h', `gsl_monte_miser.h' and `gsl_monte_vegas.h'.
+
+* Menu:
+
+* Monte Carlo Interface::
+* PLAIN Monte Carlo::
+* MISER::
+* VEGAS::
+* Monte Carlo Examples::
+* Monte Carlo Integration References and Further Reading::
+
+
+File: gsl-ref.info,  Node: Monte Carlo Interface,  Next: PLAIN Monte Carlo,  Up: Monte Carlo Integration
+
+23.1 Interface
+==============
+
+All of the Monte Carlo integration routines use the same general form of
+interface.  There is an allocator to allocate memory for control
+variables and workspace, a routine to initialize those control
+variables, the integrator itself, and a function to free the space when
+done.
+
+   Each integration function requires a random number generator to be
+supplied, and returns an estimate of the integral and its standard
+deviation.  The accuracy of the result is determined by the number of
+function calls specified by the user.  If a known level of accuracy is
+required this can be achieved by calling the integrator several times
+and averaging the individual results until the desired accuracy is
+obtained.
+
+   Random sample points used within the Monte Carlo routines are always
+chosen strictly within the integration region, so that endpoint
+singularities are automatically avoided.
+
+   The function to be integrated has its own datatype, defined in the
+header file `gsl_monte.h'.
+
+ -- Data Type: gsl_monte_function
+     This data type defines a general function with parameters for Monte
+     Carlo integration.
+
+    `double (* f) (double * X, size_t DIM, void * PARAMS)'
+          this function should return the value f(x,params) for the
+          argument X and parameters PARAMS, where X is an array of size
+          DIM giving the coordinates of the point where the function is
+          to be evaluated.
+
+    `size_t dim'
+          the number of dimensions for X.
+
+    `void * params'
+          a pointer to the parameters of the function.
+
+Here is an example for a quadratic function in two dimensions,
+
+     f(x,y) = a x^2 + b x y + c y^2
+
+with a = 3, b = 2, c = 1.  The following code defines a
+`gsl_monte_function' `F' which you could pass to an integrator:
+
+     struct my_f_params { double a; double b; double c; };
+
+     double
+     my_f (double x[], size_t dim, void * p) {
+        struct my_f_params * fp = (struct my_f_params *)p;
+
+        if (dim != 2)
+           {
+             fprintf (stderr, "error: dim != 2");
+             abort ();
+           }
+
+        return  fp->a * x[0] * x[0]
+                  + fp->b * x[0] * x[1]
+                    + fp->c * x[1] * x[1];
+     }
+
+     gsl_monte_function F;
+     struct my_f_params params = { 3.0, 2.0, 1.0 };
+
+     F.f = &my_f;
+     F.dim = 2;
+     F.params = &params;
+
+The function f(x) can be evaluated using the following macro,
+
+     #define GSL_MONTE_FN_EVAL(F,x)
+         (*((F)->f))(x,(F)->dim,(F)->params)
+
+
+File: gsl-ref.info,  Node: PLAIN Monte Carlo,  Next: MISER,  Prev: Monte Carlo Interface,  Up: Monte Carlo Integration
+
+23.2 PLAIN Monte Carlo
+======================
+
+The plain Monte Carlo algorithm samples points randomly from the
+integration region to estimate the integral and its error.  Using this
+algorithm the estimate of the integral E(f; N) for N randomly
+distributed points x_i is given by,
+
+     E(f; N) = =  V <f> = (V / N) \sum_i^N f(x_i)
+
+where V is the volume of the integration region.  The error on this
+estimate \sigma(E;N) is calculated from the estimated variance of the
+mean,
+
+     \sigma^2 (E; N) = (V / N) \sum_i^N (f(x_i) -  <f>)^2.
+
+For large N this variance decreases asymptotically as \Var(f)/N, where
+\Var(f) is the true variance of the function over the integration
+region.  The error estimate itself should decrease as
+\sigma(f)/\sqrt{N}.  The familiar law of errors decreasing as
+1/\sqrt{N} applies--to reduce the error by a factor of 10 requires a
+100-fold increase in the number of sample points.
+
+   The functions described in this section are declared in the header
+file `gsl_monte_plain.h'.
+
+ -- Function: gsl_monte_plain_state * gsl_monte_plain_alloc (size_t DIM)
+     This function allocates and initializes a workspace for Monte Carlo
+     integration in DIM dimensions.
+
+ -- Function: int gsl_monte_plain_init (gsl_monte_plain_state* S)
+     This function initializes a previously allocated integration state.
+     This allows an existing workspace to be reused for different
+     integrations.
+
+ -- Function: int gsl_monte_plain_integrate (gsl_monte_function * F,
+          double * XL, double * XU, size_t DIM, size_t CALLS, gsl_rng *
+          R, gsl_monte_plain_state * S, double * RESULT, double *
+          ABSERR)
+     This routines uses the plain Monte Carlo algorithm to integrate the
+     function F over the DIM-dimensional hypercubic region defined by
+     the lower and upper limits in the arrays XL and XU, each of size
+     DIM.  The integration uses a fixed number of function calls CALLS,
+     and obtains random sampling points using the random number
+     generator R. A previously allocated workspace S must be supplied.
+     The result of the integration is returned in RESULT, with an
+     estimated absolute error ABSERR.
+
+ -- Function: void gsl_monte_plain_free (gsl_monte_plain_state * S)
+     This function frees the memory associated with the integrator state
+     S.
+
+
+File: gsl-ref.info,  Node: MISER,  Next: VEGAS,  Prev: PLAIN Monte Carlo,  Up: Monte Carlo Integration
+
+23.3 MISER
+==========
+
+The MISER algorithm of Press and Farrar is based on recursive
+stratified sampling.  This technique aims to reduce the overall
+integration error by concentrating integration points in the regions of
+highest variance.
+
+   The idea of stratified sampling begins with the observation that for
+two disjoint regions a and b with Monte Carlo estimates of the integral
+E_a(f) and E_b(f) and variances \sigma_a^2(f) and \sigma_b^2(f), the
+variance \Var(f) of the combined estimate E(f) = (1/2) (E_a(f) + E_b(f))
+is given by,
+
+     \Var(f) = (\sigma_a^2(f) / 4 N_a) + (\sigma_b^2(f) / 4 N_b).
+
+It can be shown that this variance is minimized by distributing the
+points such that,
+
+     N_a / (N_a + N_b) = \sigma_a / (\sigma_a + \sigma_b).
+
+Hence the smallest error estimate is obtained by allocating sample
+points in proportion to the standard deviation of the function in each
+sub-region.
+
+   The MISER algorithm proceeds by bisecting the integration region
+along one coordinate axis to give two sub-regions at each step.  The
+direction is chosen by examining all d possible bisections and
+selecting the one which will minimize the combined variance of the two
+sub-regions.  The variance in the sub-regions is estimated by sampling
+with a fraction of the total number of points available to the current
+step.  The same procedure is then repeated recursively for each of the
+two half-spaces from the best bisection. The remaining sample points are
+allocated to the sub-regions using the formula for N_a and N_b.  This
+recursive allocation of integration points continues down to a
+user-specified depth where each sub-region is integrated using a plain
+Monte Carlo estimate.  These individual values and their error
+estimates are then combined upwards to give an overall result and an
+estimate of its error.
+
+   The functions described in this section are declared in the header
+file `gsl_monte_miser.h'.
+
+ -- Function: gsl_monte_miser_state * gsl_monte_miser_alloc (size_t DIM)
+     This function allocates and initializes a workspace for Monte Carlo
+     integration in DIM dimensions.  The workspace is used to maintain
+     the state of the integration.
+
+ -- Function: int gsl_monte_miser_init (gsl_monte_miser_state* S)
+     This function initializes a previously allocated integration state.
+     This allows an existing workspace to be reused for different
+     integrations.
+
+ -- Function: int gsl_monte_miser_integrate (gsl_monte_function * F,
+          double * XL, double * XU, size_t DIM, size_t CALLS, gsl_rng *
+          R, gsl_monte_miser_state * S, double * RESULT, double *
+          ABSERR)
+     This routines uses the MISER Monte Carlo algorithm to integrate the
+     function F over the DIM-dimensional hypercubic region defined by
+     the lower and upper limits in the arrays XL and XU, each of size
+     DIM.  The integration uses a fixed number of function calls CALLS,
+     and obtains random sampling points using the random number
+     generator R. A previously allocated workspace S must be supplied.
+     The result of the integration is returned in RESULT, with an
+     estimated absolute error ABSERR.
+
+ -- Function: void gsl_monte_miser_free (gsl_monte_miser_state * S)
+     This function frees the memory associated with the integrator state
+     S.
+
+   The MISER algorithm has several configurable parameters. The
+following variables can be accessed through the `gsl_monte_miser_state'
+struct,
+
+ -- Variable: double estimate_frac
+     This parameter specifies the fraction of the currently available
+     number of function calls which are allocated to estimating the
+     variance at each recursive step. The default value is 0.1.
+
+ -- Variable: size_t min_calls
+     This parameter specifies the minimum number of function calls
+     required for each estimate of the variance. If the number of
+     function calls allocated to the estimate using ESTIMATE_FRAC falls
+     below MIN_CALLS then MIN_CALLS are used instead.  This ensures
+     that each estimate maintains a reasonable level of accuracy.  The
+     default value of MIN_CALLS is `16 * dim'.
+
+ -- Variable: size_t min_calls_per_bisection
+     This parameter specifies the minimum number of function calls
+     required to proceed with a bisection step.  When a recursive step
+     has fewer calls available than MIN_CALLS_PER_BISECTION it performs
+     a plain Monte Carlo estimate of the current sub-region and
+     terminates its branch of the recursion.  The default value of this
+     parameter is `32 * min_calls'.
+
+ -- Variable: double alpha
+     This parameter controls how the estimated variances for the two
+     sub-regions of a bisection are combined when allocating points.
+     With recursive sampling the overall variance should scale better
+     than 1/N, since the values from the sub-regions will be obtained
+     using a procedure which explicitly minimizes their variance.  To
+     accommodate this behavior the MISER algorithm allows the total
+     variance to depend on a scaling parameter \alpha,
+
+          \Var(f) = {\sigma_a \over N_a^\alpha} + {\sigma_b \over N_b^\alpha}.
+
+     The authors of the original paper describing MISER recommend the
+     value \alpha = 2 as a good choice, obtained from numerical
+     experiments, and this is used as the default value in this
+     implementation.
+
+ -- Variable: double dither
+     This parameter introduces a random fractional variation of size
+     DITHER into each bisection, which can be used to break the
+     symmetry of integrands which are concentrated near the exact
+     center of the hypercubic integration region.  The default value of
+     dither is zero, so no variation is introduced. If needed, a
+     typical value of DITHER is 0.1.
+
+
+File: gsl-ref.info,  Node: VEGAS,  Next: Monte Carlo Examples,  Prev: MISER,  Up: Monte Carlo Integration
+
+23.4 VEGAS
+==========
+
+The VEGAS algorithm of Lepage is based on importance sampling.  It
+samples points from the probability distribution described by the
+function |f|, so that the points are concentrated in the regions that
+make the largest contribution to the integral.
+
+   In general, if the Monte Carlo integral of f is sampled with points
+distributed according to a probability distribution described by the
+function g, we obtain an estimate E_g(f; N),
+
+     E_g(f; N) = E(f/g; N)
+
+with a corresponding variance,
+
+     \Var_g(f; N) = \Var(f/g; N).
+
+If the probability distribution is chosen as g = |f|/I(|f|) then it can
+be shown that the variance V_g(f; N) vanishes, and the error in the
+estimate will be zero.  In practice it is not possible to sample from
+the exact distribution g for an arbitrary function, so importance
+sampling algorithms aim to produce efficient approximations to the
+desired distribution.
+
+   The VEGAS algorithm approximates the exact distribution by making a
+number of passes over the integration region while histogramming the
+function f. Each histogram is used to define a sampling distribution
+for the next pass.  Asymptotically this procedure converges to the
+desired distribution. In order to avoid the number of histogram bins
+growing like K^d the probability distribution is approximated by a
+separable function: g(x_1, x_2, ...) = g_1(x_1) g_2(x_2) ...  so that
+the number of bins required is only Kd.  This is equivalent to locating
+the peaks of the function from the projections of the integrand onto
+the coordinate axes.  The efficiency of VEGAS depends on the validity
+of this assumption.  It is most efficient when the peaks of the
+integrand are well-localized.  If an integrand can be rewritten in a
+form which is approximately separable this will increase the efficiency
+of integration with VEGAS.
+
+   VEGAS incorporates a number of additional features, and combines both
+stratified sampling and importance sampling.  The integration region is
+divided into a number of "boxes", with each box getting a fixed number
+of points (the goal is 2).  Each box can then have a fractional number
+of bins, but if the ratio of bins-per-box is less than two, Vegas
+switches to a kind variance reduction (rather than importance sampling).
+
+ -- Function: gsl_monte_vegas_state * gsl_monte_vegas_alloc (size_t DIM)
+     This function allocates and initializes a workspace for Monte Carlo
+     integration in DIM dimensions.  The workspace is used to maintain
+     the state of the integration.
+
+ -- Function: int gsl_monte_vegas_init (gsl_monte_vegas_state* S)
+     This function initializes a previously allocated integration state.
+     This allows an existing workspace to be reused for different
+     integrations.
+
+ -- Function: int gsl_monte_vegas_integrate (gsl_monte_function * F,
+          double * XL, double * XU, size_t DIM, size_t CALLS, gsl_rng *
+          R, gsl_monte_vegas_state * S, double * RESULT, double *
+          ABSERR)
+     This routines uses the VEGAS Monte Carlo algorithm to integrate the
+     function F over the DIM-dimensional hypercubic region defined by
+     the lower and upper limits in the arrays XL and XU, each of size
+     DIM.  The integration uses a fixed number of function calls CALLS,
+     and obtains random sampling points using the random number
+     generator R. A previously allocated workspace S must be supplied.
+     The result of the integration is returned in RESULT, with an
+     estimated absolute error ABSERR.  The result and its error
+     estimate are based on a weighted average of independent samples.
+     The chi-squared per degree of freedom for the weighted average is
+     returned via the state struct component, S->CHISQ, and must be
+     consistent with 1 for the weighted average to be reliable.
+
+ -- Function: void gsl_monte_vegas_free (gsl_monte_vegas_state * S)
+     This function frees the memory associated with the integrator state
+     S.
+
+   The VEGAS algorithm computes a number of independent estimates of the
+integral internally, according to the `iterations' parameter described
+below, and returns their weighted average.  Random sampling of the
+integrand can occasionally produce an estimate where the error is zero,
+particularly if the function is constant in some regions. An estimate
+with zero error causes the weighted average to break down and must be
+handled separately. In the original Fortran implementations of VEGAS
+the error estimate is made non-zero by substituting a small value
+(typically `1e-30').  The implementation in GSL differs from this and
+avoids the use of an arbitrary constant--it either assigns the value a
+weight which is the average weight of the preceding estimates or
+discards it according to the following procedure,
+
+current estimate has zero error, weighted average has finite error
+     The current estimate is assigned a weight which is the average
+     weight of the preceding estimates.
+
+current estimate has finite error, previous estimates had zero error
+     The previous estimates are discarded and the weighted averaging
+     procedure begins with the current estimate.
+
+current estimate has zero error, previous estimates had zero error
+     The estimates are averaged using the arithmetic mean, but no error
+     is computed.
+
+   The VEGAS algorithm is highly configurable. The following variables
+can be accessed through the `gsl_monte_vegas_state' struct,
+
+ -- Variable: double result
+ -- Variable: double sigma
+     These parameters contain the raw value of the integral RESULT and
+     its error SIGMA from the last iteration of the algorithm.
+
+ -- Variable: double chisq
+     This parameter gives the chi-squared per degree of freedom for the
+     weighted estimate of the integral.  The value of CHISQ should be
+     close to 1.  A value of CHISQ which differs significantly from 1
+     indicates that the values from different iterations are
+     inconsistent.  In this case the weighted error will be
+     under-estimated, and further iterations of the algorithm are
+     needed to obtain reliable results.
+
+ -- Variable: double alpha
+     The parameter `alpha' controls the stiffness of the rebinning
+     algorithm.  It is typically set between one and two. A value of
+     zero prevents rebinning of the grid.  The default value is 1.5.
+
+ -- Variable: size_t iterations
+     The number of iterations to perform for each call to the routine.
+     The default value is 5 iterations.
+
+ -- Variable: int stage
+     Setting this determines the "stage" of the calculation.  Normally,
+     `stage = 0' which begins with a new uniform grid and empty weighted
+     average.  Calling vegas with `stage = 1' retains the grid from the
+     previous run but discards the weighted average, so that one can
+     "tune" the grid using a relatively small number of points and then
+     do a large run with `stage = 1' on the optimized grid.  Setting
+     `stage = 2' keeps the grid and the weighted average from the
+     previous run, but may increase (or decrease) the number of
+     histogram bins in the grid depending on the number of calls
+     available.  Choosing `stage = 3' enters at the main loop, so that
+     nothing is changed, and is equivalent to performing additional
+     iterations in a previous call.
+
+ -- Variable: int mode
+     The possible choices are `GSL_VEGAS_MODE_IMPORTANCE',
+     `GSL_VEGAS_MODE_STRATIFIED', `GSL_VEGAS_MODE_IMPORTANCE_ONLY'.
+     This determines whether VEGAS will use importance sampling or
+     stratified sampling, or whether it can pick on its own.  In low
+     dimensions VEGAS uses strict stratified sampling (more precisely,
+     stratified sampling is chosen if there are fewer than 2 bins per
+     box).
+
+ -- Variable: int verbose
+ -- Variable: FILE * ostream
+     These parameters set the level of information printed by VEGAS. All
+     information is written to the stream OSTREAM.  The default setting
+     of VERBOSE is `-1', which turns off all output.  A VERBOSE value
+     of `0' prints summary information about the weighted average and
+     final result, while a value of `1' also displays the grid
+     coordinates.  A value of `2' prints information from the rebinning
+     procedure for each iteration.
+
+
+File: gsl-ref.info,  Node: Monte Carlo Examples,  Next: Monte Carlo Integration References and Further Reading,  Prev: VEGAS,  Up: Monte Carlo Integration
+
+23.5 Examples
+=============
+
+The example program below uses the Monte Carlo routines to estimate the
+value of the following 3-dimensional integral from the theory of random
+walks,
+
+     I = \int_{-pi}^{+pi} {dk_x/(2 pi)}
+         \int_{-pi}^{+pi} {dk_y/(2 pi)}
+         \int_{-pi}^{+pi} {dk_z/(2 pi)}
+          1 / (1 - cos(k_x)cos(k_y)cos(k_z)).
+
+The analytic value of this integral can be shown to be I =
+\Gamma(1/4)^4/(4 \pi^3) = 1.393203929685676859....  The integral gives
+the mean time spent at the origin by a random walk on a body-centered
+cubic lattice in three dimensions.
+
+   For simplicity we will compute the integral over the region (0,0,0)
+to (\pi,\pi,\pi) and multiply by 8 to obtain the full result.  The
+integral is slowly varying in the middle of the region but has
+integrable singularities at the corners (0,0,0), (0,\pi,\pi),
+(\pi,0,\pi) and (\pi,\pi,0).  The Monte Carlo routines only select
+points which are strictly within the integration region and so no
+special measures are needed to avoid these singularities.
+
+     #include <stdlib.h>
+     #include <gsl/gsl_math.h>
+     #include <gsl/gsl_monte.h>
+     #include <gsl/gsl_monte_plain.h>
+     #include <gsl/gsl_monte_miser.h>
+     #include <gsl/gsl_monte_vegas.h>
+
+     /* Computation of the integral,
+
+           I = int (dx dy dz)/(2pi)^3  1/(1-cos(x)cos(y)cos(z))
+
+        over (-pi,-pi,-pi) to (+pi, +pi, +pi).  The exact answer
+        is Gamma(1/4)^4/(4 pi^3).  This example is taken from
+        C.Itzykson, J.M.Drouffe, "Statistical Field Theory -
+        Volume 1", Section 1.1, p21, which cites the original
+        paper M.L.Glasser, I.J.Zucker, Proc.Natl.Acad.Sci.USA 74
+        1800 (1977) */
+
+     /* For simplicity we compute the integral over the region
+        (0,0,0) -> (pi,pi,pi) and multiply by 8 */
+
+     double exact = 1.3932039296856768591842462603255;
+
+     double
+     g (double *k, size_t dim, void *params)
+     {
+       double A = 1.0 / (M_PI * M_PI * M_PI);
+       return A / (1.0 - cos (k[0]) * cos (k[1]) * cos (k[2]));
+     }
+
+     void
+     display_results (char *title, double result, double error)
+     {
+       printf ("%s ==================\n", title);
+       printf ("result = % .6f\n", result);
+       printf ("sigma  = % .6f\n", error);
+       printf ("exact  = % .6f\n", exact);
+       printf ("error  = % .6f = %.1g sigma\n", result - exact,
+               fabs (result - exact) / error);
+     }
+
+     int
+     main (void)
+     {
+       double res, err;
+
+       double xl[3] = { 0, 0, 0 };
+       double xu[3] = { M_PI, M_PI, M_PI };
+
+       const gsl_rng_type *T;
+       gsl_rng *r;
+
+       gsl_monte_function G = { &g, 3, 0 };
+
+       size_t calls = 500000;
+
+       gsl_rng_env_setup ();
+
+       T = gsl_rng_default;
+       r = gsl_rng_alloc (T);
+
+       {
+         gsl_monte_plain_state *s = gsl_monte_plain_alloc (3);
+         gsl_monte_plain_integrate (&G, xl, xu, 3, calls, r, s,
+                                    &res, &err);
+         gsl_monte_plain_free (s);
+
+         display_results ("plain", res, err);
+       }
+
+       {
+         gsl_monte_miser_state *s = gsl_monte_miser_alloc (3);
+         gsl_monte_miser_integrate (&G, xl, xu, 3, calls, r, s,
+                                    &res, &err);
+         gsl_monte_miser_free (s);
+
+         display_results ("miser", res, err);
+       }
+
+       {
+         gsl_monte_vegas_state *s = gsl_monte_vegas_alloc (3);
+
+         gsl_monte_vegas_integrate (&G, xl, xu, 3, 10000, r, s,
+                                    &res, &err);
+         display_results ("vegas warm-up", res, err);
+
+         printf ("converging...\n");
+
+         do
+           {
+             gsl_monte_vegas_integrate (&G, xl, xu, 3, calls/5, r, s,
+                                        &res, &err);
+             printf ("result = % .6f sigma = % .6f "
+                     "chisq/dof = %.1f\n", res, err, s->chisq);
+           }
+         while (fabs (s->chisq - 1.0) > 0.5);
+
+         display_results ("vegas final", res, err);
+
+         gsl_monte_vegas_free (s);
+       }
+
+       gsl_rng_free (r);
+
+       return 0;
+     }
+
+With 500,000 function calls the plain Monte Carlo algorithm achieves a
+fractional error of 0.6%.  The estimated error `sigma' is consistent
+with the actual error, and the computed result differs from the true
+result by about one standard deviation,
+
+     plain ==================
+     result =  1.385867
+     sigma  =  0.007938
+     exact  =  1.393204
+     error  = -0.007337 = 0.9 sigma
+
+The MISER algorithm reduces the error by a factor of two, and also
+correctly estimates the error,
+
+     miser ==================
+     result =  1.390656
+     sigma  =  0.003743
+     exact  =  1.393204
+     error  = -0.002548 = 0.7 sigma
+
+In the case of the VEGAS algorithm the program uses an initial warm-up
+run of 10,000 function calls to prepare, or "warm up", the grid.  This
+is followed by a main run with five iterations of 100,000 function
+calls. The chi-squared per degree of freedom for the five iterations are
+checked for consistency with 1, and the run is repeated if the results
+have not converged. In this case the estimates are consistent on the
+first pass.
+
+     vegas warm-up ==================
+     result =  1.386925
+     sigma  =  0.002651
+     exact  =  1.393204
+     error  = -0.006278 = 2 sigma
+     converging...
+     result =  1.392957 sigma =  0.000452 chisq/dof = 1.1
+     vegas final ==================
+     result =  1.392957
+     sigma  =  0.000452
+     exact  =  1.393204
+     error  = -0.000247 = 0.5 sigma
+
+If the value of `chisq' had differed significantly from 1 it would
+indicate inconsistent results, with a correspondingly underestimated
+error.  The final estimate from VEGAS (using a similar number of
+function calls) is significantly more accurate than the other two
+algorithms.
+
+
+File: gsl-ref.info,  Node: Monte Carlo Integration References and Further Reading,  Prev: Monte Carlo Examples,  Up: Monte Carlo Integration
+
+23.6 References and Further Reading
+===================================
+
+The MISER algorithm is described in the following article by Press and
+Farrar,
+
+     W.H. Press, G.R. Farrar, `Recursive Stratified Sampling for
+     Multidimensional Monte Carlo Integration', Computers in Physics,
+     v4 (1990), pp190-195.
+
+The VEGAS algorithm is described in the following papers,
+
+     G.P. Lepage, `A New Algorithm for Adaptive Multidimensional
+     Integration', Journal of Computational Physics 27, 192-203, (1978)
+
+     G.P. Lepage, `VEGAS: An Adaptive Multi-dimensional Integration
+     Program', Cornell preprint CLNS 80-447, March 1980
+
+
+File: gsl-ref.info,  Node: Simulated Annealing,  Next: Ordinary Differential Equations,  Prev: Monte Carlo Integration,  Up: Top
+
+24 Simulated Annealing
+**********************
+
+Stochastic search techniques are used when the structure of a space is
+not well understood or is not smooth, so that techniques like Newton's
+method (which requires calculating Jacobian derivative matrices) cannot
+be used. In particular, these techniques are frequently used to solve
+combinatorial optimization problems, such as the traveling salesman
+problem.
+
+   The goal is to find a point in the space at which a real valued
+"energy function" (or "cost function") is minimized.  Simulated
+annealing is a minimization technique which has given good results in
+avoiding local minima; it is based on the idea of taking a random walk
+through the space at successively lower temperatures, where the
+probability of taking a step is given by a Boltzmann distribution.
+
+   The functions described in this chapter are declared in the header
+file `gsl_siman.h'.
+
+* Menu:
+
+* Simulated Annealing algorithm::
+* Simulated Annealing functions::
+* Examples with Simulated Annealing::
+* Simulated Annealing References and Further Reading::
+
+
+File: gsl-ref.info,  Node: Simulated Annealing algorithm,  Next: Simulated Annealing functions,  Up: Simulated Annealing
+
+24.1 Simulated Annealing algorithm
+==================================
+
+The simulated annealing algorithm takes random walks through the problem
+space, looking for points with low energies; in these random walks, the
+probability of taking a step is determined by the Boltzmann
+distribution,
+
+     p = e^{-(E_{i+1} - E_i)/(kT)}
+
+if E_{i+1} > E_i, and p = 1 when E_{i+1} <= E_i.
+
+   In other words, a step will occur if the new energy is lower.  If
+the new energy is higher, the transition can still occur, and its
+likelihood is proportional to the temperature T and inversely
+proportional to the energy difference E_{i+1} - E_i.
+
+   The temperature T is initially set to a high value, and a random
+walk is carried out at that temperature.  Then the temperature is
+lowered very slightly according to a "cooling schedule", for example: T
+-> T/mu_T where \mu_T is slightly greater than 1.  
+
+   The slight probability of taking a step that gives higher energy is
+what allows simulated annealing to frequently get out of local minima.
+
+
+File: gsl-ref.info,  Node: Simulated Annealing functions,  Next: Examples with Simulated Annealing,  Prev: Simulated Annealing algorithm,  Up: Simulated Annealing
+
+24.2 Simulated Annealing functions
+==================================
+
+ -- Function: void gsl_siman_solve (const gsl_rng * R, void * X0_P,
+          gsl_siman_Efunc_t EF, gsl_siman_step_t TAKE_STEP,
+          gsl_siman_metric_t DISTANCE, gsl_siman_print_t
+          PRINT_POSITION, gsl_siman_copy_t COPYFUNC,
+          gsl_siman_copy_construct_t COPY_CONSTRUCTOR,
+          gsl_siman_destroy_t DESTRUCTOR, size_t ELEMENT_SIZE,
+          gsl_siman_params_t PARAMS)
+     This function performs a simulated annealing search through a given
+     space.  The space is specified by providing the functions EF and
+     DISTANCE.  The simulated annealing steps are generated using the
+     random number generator R and the function TAKE_STEP.
+
+     The starting configuration of the system should be given by X0_P.
+     The routine offers two modes for updating configurations, a
+     fixed-size mode and a variable-size mode.  In the fixed-size mode
+     the configuration is stored as a single block of memory of size
+     ELEMENT_SIZE.  Copies of this configuration are created, copied
+     and destroyed internally using the standard library functions
+     `malloc', `memcpy' and `free'.  The function pointers COPYFUNC,
+     COPY_CONSTRUCTOR and DESTRUCTOR should be null pointers in
+     fixed-size mode.  In the variable-size mode the functions
+     COPYFUNC, COPY_CONSTRUCTOR and DESTRUCTOR are used to create, copy
+     and destroy configurations internally.  The variable ELEMENT_SIZE
+     should be zero in the variable-size mode.
+
+     The PARAMS structure (described below) controls the run by
+     providing the temperature schedule and other tunable parameters to
+     the algorithm.
+
+     On exit the best result achieved during the search is placed in
+     `*X0_P'.  If the annealing process has been successful this should
+     be a good approximation to the optimal point in the space.
+
+     If the function pointer PRINT_POSITION is not null, a debugging
+     log will be printed to `stdout' with the following columns:
+
+          number_of_iterations temperature x x-(*x0_p) Ef(x)
+
+     and the output of the function PRINT_POSITION itself.  If
+     PRINT_POSITION is null then no information is printed.
+
+The simulated annealing routines require several user-specified
+functions to define the configuration space and energy function.  The
+prototypes for these functions are given below.
+
+ -- Data Type: gsl_siman_Efunc_t
+     This function type should return the energy of a configuration XP.
+
+          double (*gsl_siman_Efunc_t) (void *xp)
+
+ -- Data Type: gsl_siman_step_t
+     This function type should modify the configuration XP using a
+     random step taken from the generator R, up to a maximum distance of
+     STEP_SIZE.
+
+          void (*gsl_siman_step_t) (const gsl_rng *r, void *xp,
+                                    double step_size)
+
+ -- Data Type: gsl_siman_metric_t
+     This function type should return the distance between two
+     configurations XP and YP.
+
+          double (*gsl_siman_metric_t) (void *xp, void *yp)
+
+ -- Data Type: gsl_siman_print_t
+     This function type should print the contents of the configuration
+     XP.
+
+          void (*gsl_siman_print_t) (void *xp)
+
+ -- Data Type: gsl_siman_copy_t
+     This function type should copy the configuration SOURCE into DEST.
+
+          void (*gsl_siman_copy_t) (void *source, void *dest)
+
+ -- Data Type: gsl_siman_copy_construct_t
+     This function type should create a new copy of the configuration
+     XP.
+
+          void * (*gsl_siman_copy_construct_t) (void *xp)
+
+ -- Data Type: gsl_siman_destroy_t
+     This function type should destroy the configuration XP, freeing its
+     memory.
+
+          void (*gsl_siman_destroy_t) (void *xp)
+
+ -- Data Type: gsl_siman_params_t
+     These are the parameters that control a run of `gsl_siman_solve'.
+     This structure contains all the information needed to control the
+     search, beyond the energy function, the step function and the
+     initial guess.
+
+    `int n_tries'
+          The number of points to try for each step.
+
+    `int iters_fixed_T'
+          The number of iterations at each temperature.
+
+    `double step_size'
+          The maximum step size in the random walk.
+
+    `double k, t_initial, mu_t, t_min'
+          The parameters of the Boltzmann distribution and cooling
+          schedule.
+
+
+File: gsl-ref.info,  Node: Examples with Simulated Annealing,  Next: Simulated Annealing References and Further Reading,  Prev: Simulated Annealing functions,  Up: Simulated Annealing
+
+24.3 Examples
+=============
+
+The simulated annealing package is clumsy, and it has to be because it
+is written in C, for C callers, and tries to be polymorphic at the same
+time.  But here we provide some examples which can be pasted into your
+application with little change and should make things easier.
+
+* Menu:
+
+* Trivial example::
+* Traveling Salesman Problem::
+
+
+File: gsl-ref.info,  Node: Trivial example,  Next: Traveling Salesman Problem,  Up: Examples with Simulated Annealing
+
+24.3.1 Trivial example
+----------------------
+
+The first example, in one dimensional cartesian space, sets up an energy
+function which is a damped sine wave; this has many local minima, but
+only one global minimum, somewhere between 1.0 and 1.5.  The initial
+guess given is 15.5, which is several local minima away from the global
+minimum.
+
+     #include <math.h>
+     #include <stdlib.h>
+     #include <string.h>
+     #include <gsl/gsl_siman.h>
+
+     /* set up parameters for this simulated annealing run */
+
+     /* how many points do we try before stepping */
+     #define N_TRIES 200
+
+     /* how many iterations for each T? */
+     #define ITERS_FIXED_T 1000
+
+     /* max step size in random walk */
+     #define STEP_SIZE 1.0
+
+     /* Boltzmann constant */
+     #define K 1.0
+
+     /* initial temperature */
+     #define T_INITIAL 0.008
+
+     /* damping factor for temperature */
+     #define MU_T 1.003
+     #define T_MIN 2.0e-6
+
+     gsl_siman_params_t params
+       = {N_TRIES, ITERS_FIXED_T, STEP_SIZE,
+          K, T_INITIAL, MU_T, T_MIN};
+
+     /* now some functions to test in one dimension */
+     double E1(void *xp)
+     {
+       double x = * ((double *) xp);
+
+       return exp(-pow((x-1.0),2.0))*sin(8*x);
+     }
+
+     double M1(void *xp, void *yp)
+     {
+       double x = *((double *) xp);
+       double y = *((double *) yp);
+
+       return fabs(x - y);
+     }
+
+     void S1(const gsl_rng * r, void *xp, double step_size)
+     {
+       double old_x = *((double *) xp);
+       double new_x;
+
+       double u = gsl_rng_uniform(r);
+       new_x = u * 2 * step_size - step_size + old_x;
+
+       memcpy(xp, &new_x, sizeof(new_x));
+     }
+
+     void P1(void *xp)
+     {
+       printf ("%12g", *((double *) xp));
+     }
+
+     int
+     main(int argc, char *argv[])
+     {
+       const gsl_rng_type * T;
+       gsl_rng * r;
+
+       double x_initial = 15.5;
+
+       gsl_rng_env_setup();
+
+       T = gsl_rng_default;
+       r = gsl_rng_alloc(T);
+
+       gsl_siman_solve(r, &x_initial, E1, S1, M1, P1,
+                       NULL, NULL, NULL,
+                       sizeof(double), params);
+
+       gsl_rng_free (r);
+       return 0;
+     }
+
+   Here are a couple of plots that are generated by running
+`siman_test' in the following way:
+
+     $ ./siman_test | awk '!/^#/ {print $1, $4}'
+      | graph -y 1.34 1.4 -W0 -X generation -Y position
+      | plot -Tps > siman-test.eps
+     $ ./siman_test | awk '!/^#/ {print $1, $4}'
+      | graph -y -0.88 -0.83 -W0 -X generation -Y energy
+      | plot -Tps > siman-energy.eps
+
+
+File: gsl-ref.info,  Node: Traveling Salesman Problem,  Prev: Trivial example,  Up: Examples with Simulated Annealing
+
+24.3.2 Traveling Salesman Problem
+---------------------------------
+
+The TSP ("Traveling Salesman Problem") is the classic combinatorial
+optimization problem.  I have provided a very simple version of it,
+based on the coordinates of twelve cities in the southwestern United
+States.  This should maybe be called the "Flying Salesman Problem",
+since I am using the great-circle distance between cities, rather than
+the driving distance.  Also: I assume the earth is a sphere, so I don't
+use geoid distances.
+
+   The `gsl_siman_solve' routine finds a route which is 3490.62
+Kilometers long; this is confirmed by an exhaustive search of all
+possible routes with the same initial city.
+
+   The full code can be found in `siman/siman_tsp.c', but I include
+here some plots generated in the following way:
+
+     $ ./siman_tsp > tsp.output
+     $ grep -v "^#" tsp.output
+      | awk '{print $1, $NF}'
+      | graph -y 3300 6500 -W0 -X generation -Y distance
+         -L "TSP - 12 southwest cities"
+      | plot -Tps > 12-cities.eps
+     $ grep initial_city_coord tsp.output
+       | awk '{print $2, $3}'
+       | graph -X "longitude (- means west)" -Y "latitude"
+          -L "TSP - initial-order" -f 0.03 -S 1 0.1
+       | plot -Tps > initial-route.eps
+     $ grep final_city_coord tsp.output
+       | awk '{print $2, $3}'
+       | graph -X "longitude (- means west)" -Y "latitude"
+          -L "TSP - final-order" -f 0.03 -S 1 0.1
+       | plot -Tps > final-route.eps
+
+This is the output showing the initial order of the cities; longitude is
+negative, since it is west and I want the plot to look like a map.
+
+     # initial coordinates of cities (longitude and latitude)
+     ###initial_city_coord: -105.95 35.68 Santa Fe
+     ###initial_city_coord: -112.07 33.54 Phoenix
+     ###initial_city_coord: -106.62 35.12 Albuquerque
+     ###initial_city_coord: -103.2 34.41 Clovis
+     ###initial_city_coord: -107.87 37.29 Durango
+     ###initial_city_coord: -96.77 32.79 Dallas
+     ###initial_city_coord: -105.92 35.77 Tesuque
+     ###initial_city_coord: -107.84 35.15 Grants
+     ###initial_city_coord: -106.28 35.89 Los Alamos
+     ###initial_city_coord: -106.76 32.34 Las Cruces
+     ###initial_city_coord: -108.58 37.35 Cortez
+     ###initial_city_coord: -108.74 35.52 Gallup
+     ###initial_city_coord: -105.95 35.68 Santa Fe
+
+   The optimal route turns out to be:
+
+     # final coordinates of cities (longitude and latitude)
+     ###final_city_coord: -105.95 35.68 Santa Fe
+     ###final_city_coord: -106.28 35.89 Los Alamos
+     ###final_city_coord: -106.62 35.12 Albuquerque
+     ###final_city_coord: -107.84 35.15 Grants
+     ###final_city_coord: -107.87 37.29 Durango
+     ###final_city_coord: -108.58 37.35 Cortez
+     ###final_city_coord: -108.74 35.52 Gallup
+     ###final_city_coord: -112.07 33.54 Phoenix
+     ###final_city_coord: -106.76 32.34 Las Cruces
+     ###final_city_coord: -96.77 32.79 Dallas
+     ###final_city_coord: -103.2 34.41 Clovis
+     ###final_city_coord: -105.92 35.77 Tesuque
+     ###final_city_coord: -105.95 35.68 Santa Fe
+
+Here's a plot of the cost function (energy) versus generation (point in
+the calculation at which a new temperature is set) for this problem:
+
+
+File: gsl-ref.info,  Node: Simulated Annealing References and Further Reading,  Prev: Examples with Simulated Annealing,  Up: Simulated Annealing
+
+24.4 References and Further Reading
+===================================
+
+Further information is available in the following book,
+
+     `Modern Heuristic Techniques for Combinatorial Problems', Colin R.
+     Reeves (ed.), McGraw-Hill, 1995 (ISBN 0-07-709239-2).
+
+
+File: gsl-ref.info,  Node: Ordinary Differential Equations,  Next: Interpolation,  Prev: Simulated Annealing,  Up: Top
+
+25 Ordinary Differential Equations
+**********************************
+
+This chapter describes functions for solving ordinary differential
+equation (ODE) initial value problems.  The library provides a variety
+of low-level methods, such as Runge-Kutta and Bulirsch-Stoer routines,
+and higher-level components for adaptive step-size control.  The
+components can be combined by the user to achieve the desired solution,
+with full access to any intermediate steps.
+
+   These functions are declared in the header file `gsl_odeiv.h'.
+
+* Menu:
+
+* Defining the ODE System::
+* Stepping Functions::
+* Adaptive Step-size Control::
+* Evolution::
+* ODE Example programs::
+* ODE References and Further Reading::
+
+
+File: gsl-ref.info,  Node: Defining the ODE System,  Next: Stepping Functions,  Up: Ordinary Differential Equations
+
+25.1 Defining the ODE System
+============================
+
+The routines solve the general n-dimensional first-order system,
+
+     dy_i(t)/dt = f_i(t, y_1(t), ..., y_n(t))
+
+for i = 1, \dots, n.  The stepping functions rely on the vector of
+derivatives f_i and the Jacobian matrix, J_{ij} = df_i(t,y(t)) / dy_j.
+A system of equations is defined using the `gsl_odeiv_system' datatype.
+
+ -- Data Type: gsl_odeiv_system
+     This data type defines a general ODE system with arbitrary
+     parameters.
+
+    `int (* function) (double t, const double y[], double dydt[], void * params)'
+          This function should store the vector elements
+          f_i(t,y,params) in the array DYDT, for arguments (T,Y) and
+          parameters PARAMS.  The function should return `GSL_SUCCESS'
+          if the calculation was completed successfully. Any other
+          return value indicates an error.
+
+    `int (* jacobian) (double t, const double y[], double * dfdy, double dfdt[], void * params);'
+          This function should store the vector of derivative elements
+          df_i(t,y,params)/dt in the array DFDT and the Jacobian matrix
+          J_{ij} in the array DFDY, regarded as a row-ordered matrix
+          `J(i,j) = dfdy[i * dimension + j]' where `dimension' is the
+          dimension of the system.  The function should return
+          `GSL_SUCCESS' if the calculation was completed successfully.
+          Any other return value indicates an error.
+
+          Some of the simpler solver algorithms do not make use of the
+          Jacobian matrix, so it is not always strictly necessary to
+          provide it (the `jacobian' element of the struct can be
+          replaced by a null pointer for those algorithms).  However,
+          it is useful to provide the Jacobian to allow the solver
+          algorithms to be interchanged--the best algorithms make use
+          of the Jacobian.
+
+    `size_t dimension;'
+          This is the dimension of the system of equations.
+
+    `void * params'
+          This is a pointer to the arbitrary parameters of the system.
+
+
+File: gsl-ref.info,  Node: Stepping Functions,  Next: Adaptive Step-size Control,  Prev: Defining the ODE System,  Up: Ordinary Differential Equations
+
+25.2 Stepping Functions
+=======================
+
+The lowest level components are the "stepping functions" which advance
+a solution from time t to t+h for a fixed step-size h and estimate the
+resulting local error.
+
+ -- Function: gsl_odeiv_step * gsl_odeiv_step_alloc (const
+          gsl_odeiv_step_type * T, size_t DIM)
+     This function returns a pointer to a newly allocated instance of a
+     stepping function of type T for a system of DIM dimensions.
+
+ -- Function: int gsl_odeiv_step_reset (gsl_odeiv_step * S)
+     This function resets the stepping function S.  It should be used
+     whenever the next use of S will not be a continuation of a
+     previous step.
+
+ -- Function: void gsl_odeiv_step_free (gsl_odeiv_step * S)
+     This function frees all the memory associated with the stepping
+     function S.
+
+ -- Function: const char * gsl_odeiv_step_name (const gsl_odeiv_step *
+          S)
+     This function returns a pointer to the name of the stepping
+     function.  For example,
+
+          printf ("step method is '%s'\n",
+                   gsl_odeiv_step_name (s));
+
+     would print something like `step method is 'rk4''.
+
+ -- Function: unsigned int gsl_odeiv_step_order (const gsl_odeiv_step *
+          S)
+     This function returns the order of the stepping function on the
+     previous step.  This order can vary if the stepping function
+     itself is adaptive.
+
+ -- Function: int gsl_odeiv_step_apply (gsl_odeiv_step * S, double T,
+          double H, double Y[], double YERR[], const double DYDT_IN[],
+          double DYDT_OUT[], const gsl_odeiv_system * DYDT)
+     This function applies the stepping function S to the system of
+     equations defined by DYDT, using the step size H to advance the
+     system from time T and state Y to time T+H.  The new state of the
+     system is stored in Y on output, with an estimate of the absolute
+     error in each component stored in YERR.  If the argument DYDT_IN
+     is not null it should point an array containing the derivatives
+     for the system at time T on input. This is optional as the
+     derivatives will be computed internally if they are not provided,
+     but allows the reuse of existing derivative information.  On
+     output the new derivatives of the system at time T+H will be
+     stored in DYDT_OUT if it is not null.
+
+     If the user-supplied functions defined in the system DYDT return a
+     status other than `GSL_SUCCESS' the step will be aborted.  In this
+     case, the elements of Y will be restored to their pre-step values
+     and the error code from the user-supplied function will be
+     returned.  The step-size H will be set to the step-size which
+     caused the error.  If the function is called again with a smaller
+     step-size, e.g. H/10, it should be possible to get closer to any
+     singularity.  To distinguish between error codes from the
+     user-supplied functions and those from `gsl_odeiv_step_apply'
+     itself, any user-defined return values should be distinct from the
+     standard GSL error codes.
+
+   The following algorithms are available,
+
+ -- Step Type: gsl_odeiv_step_rk2
+     Embedded Runge-Kutta (2, 3) method.
+
+ -- Step Type: gsl_odeiv_step_rk4
+     4th order (classical) Runge-Kutta.
+
+ -- Step Type: gsl_odeiv_step_rkf45
+     Embedded Runge-Kutta-Fehlberg (4, 5) method.  This method is a good
+     general-purpose integrator.
+
+ -- Step Type: gsl_odeiv_step_rkck
+     Embedded Runge-Kutta Cash-Karp (4, 5) method.
+
+ -- Step Type: gsl_odeiv_step_rk8pd
+     Embedded Runge-Kutta Prince-Dormand (8,9) method.
+
+ -- Step Type: gsl_odeiv_step_rk2imp
+     Implicit 2nd order Runge-Kutta at Gaussian points.
+
+ -- Step Type: gsl_odeiv_step_rk4imp
+     Implicit 4th order Runge-Kutta at Gaussian points.
+
+ -- Step Type: gsl_odeiv_step_bsimp
+     Implicit Bulirsch-Stoer method of Bader and Deuflhard.  This
+     algorithm requires the Jacobian.
+
+ -- Step Type: gsl_odeiv_step_gear1
+     M=1 implicit Gear method.
+
+ -- Step Type: gsl_odeiv_step_gear2
+     M=2 implicit Gear method.
+
+
+File: gsl-ref.info,  Node: Adaptive Step-size Control,  Next: Evolution,  Prev: Stepping Functions,  Up: Ordinary Differential Equations
+
+25.3 Adaptive Step-size Control
+===============================
+
+The control function examines the proposed change to the solution
+produced by a stepping function and attempts to determine the optimal
+step-size for a user-specified level of error.
+
+ -- Function: gsl_odeiv_control * gsl_odeiv_control_standard_new
+          (double EPS_ABS, double EPS_REL, double A_Y, double A_DYDT)
+     The standard control object is a four parameter heuristic based on
+     absolute and relative errors EPS_ABS and EPS_REL, and scaling
+     factors A_Y and A_DYDT for the system state y(t) and derivatives
+     y'(t) respectively.
+
+     The step-size adjustment procedure for this method begins by
+     computing the desired error level D_i for each component,
+
+          D_i = eps_abs + eps_rel * (a_y |y_i| + a_dydt h |y'_i|)
+
+     and comparing it with the observed error E_i = |yerr_i|.  If the
+     observed error E exceeds the desired error level D by more than
+     10% for any component then the method reduces the step-size by an
+     appropriate factor,
+
+          h_new = h_old * S * (E/D)^(-1/q)
+
+     where q is the consistency order of the method (e.g. q=4 for 4(5)
+     embedded RK), and S is a safety factor of 0.9. The ratio E/D is
+     taken to be the maximum of the ratios E_i/D_i.
+
+     If the observed error E is less than 50% of the desired error
+     level D for the maximum ratio E_i/D_i then the algorithm takes the
+     opportunity to increase the step-size to bring the error in line
+     with the desired level,
+
+          h_new = h_old * S * (E/D)^(-1/(q+1))
+
+     This encompasses all the standard error scaling methods. To avoid
+     uncontrolled changes in the stepsize, the overall scaling factor is
+     limited to the range 1/5 to 5.
+
+ -- Function: gsl_odeiv_control * gsl_odeiv_control_y_new (double
+          EPS_ABS, double EPS_REL)
+     This function creates a new control object which will keep the
+     local error on each step within an absolute error of EPS_ABS and
+     relative error of EPS_REL with respect to the solution y_i(t).
+     This is equivalent to the standard control object with A_Y=1 and
+     A_DYDT=0.
+
+ -- Function: gsl_odeiv_control * gsl_odeiv_control_yp_new (double
+          EPS_ABS, double EPS_REL)
+     This function creates a new control object which will keep the
+     local error on each step within an absolute error of EPS_ABS and
+     relative error of EPS_REL with respect to the derivatives of the
+     solution y'_i(t).  This is equivalent to the standard control
+     object with A_Y=0 and A_DYDT=1.
+
+ -- Function: gsl_odeiv_control * gsl_odeiv_control_scaled_new (double
+          EPS_ABS, double EPS_REL, double A_Y, double A_DYDT, const
+          double SCALE_ABS[], size_t DIM)
+     This function creates a new control object which uses the same
+     algorithm as `gsl_odeiv_control_standard_new' but with an absolute
+     error which is scaled for each component by the array SCALE_ABS.
+     The formula for D_i for this control object is,
+
+          D_i = eps_abs * s_i + eps_rel * (a_y |y_i| + a_dydt h |y'_i|)
+
+     where s_i is the i-th component of the array SCALE_ABS.  The same
+     error control heuristic is used by the Matlab ODE suite.
+
+ -- Function: gsl_odeiv_control * gsl_odeiv_control_alloc (const
+          gsl_odeiv_control_type * T)
+     This function returns a pointer to a newly allocated instance of a
+     control function of type T.  This function is only needed for
+     defining new types of control functions.  For most purposes the
+     standard control functions described above should be sufficient.
+
+ -- Function: int gsl_odeiv_control_init (gsl_odeiv_control * C, double
+          EPS_ABS, double EPS_REL, double A_Y, double A_DYDT)
+     This function initializes the control function C with the
+     parameters EPS_ABS (absolute error), EPS_REL (relative error), A_Y
+     (scaling factor for y) and A_DYDT (scaling factor for derivatives).
+
+ -- Function: void gsl_odeiv_control_free (gsl_odeiv_control * C)
+     This function frees all the memory associated with the control
+     function C.
+
+ -- Function: int gsl_odeiv_control_hadjust (gsl_odeiv_control * C,
+          gsl_odeiv_step * S, const double Y[], const double YERR[],
+          const double DYDT[], double * H)
+     This function adjusts the step-size H using the control function
+     C, and the current values of Y, YERR and DYDT.  The stepping
+     function STEP is also needed to determine the order of the method.
+     If the error in the y-values YERR is found to be too large then
+     the step-size H is reduced and the function returns
+     `GSL_ODEIV_HADJ_DEC'.  If the error is sufficiently small then H
+     may be increased and `GSL_ODEIV_HADJ_INC' is returned.  The
+     function returns `GSL_ODEIV_HADJ_NIL' if the step-size is
+     unchanged.  The goal of the function is to estimate the largest
+     step-size which satisfies the user-specified accuracy requirements
+     for the current point.
+
+ -- Function: const char * gsl_odeiv_control_name (const
+          gsl_odeiv_control * C)
+     This function returns a pointer to the name of the control
+     function.  For example,
+
+          printf ("control method is '%s'\n",
+                  gsl_odeiv_control_name (c));
+
+     would print something like `control method is 'standard''
+
+
+File: gsl-ref.info,  Node: Evolution,  Next: ODE Example programs,  Prev: Adaptive Step-size Control,  Up: Ordinary Differential Equations
+
+25.4 Evolution
+==============
+
+The highest level of the system is the evolution function which combines
+the results of a stepping function and control function to reliably
+advance the solution forward over an interval (t_0, t_1).  If the
+control function signals that the step-size should be decreased the
+evolution function backs out of the current step and tries the proposed
+smaller step-size.  This process is continued until an acceptable
+step-size is found.
+
+ -- Function: gsl_odeiv_evolve * gsl_odeiv_evolve_alloc (size_t DIM)
+     This function returns a pointer to a newly allocated instance of an
+     evolution function for a system of DIM dimensions.
+
+ -- Function: int gsl_odeiv_evolve_apply (gsl_odeiv_evolve * E,
+          gsl_odeiv_control * CON, gsl_odeiv_step * STEP, const
+          gsl_odeiv_system * DYDT, double * T, double T1, double * H,
+          double Y[])
+     This function advances the system (E, DYDT) from time T and
+     position Y using the stepping function STEP.  The new time and
+     position are stored in T and Y on output.  The initial step-size
+     is taken as H, but this will be modified using the control
+     function C to achieve the appropriate error bound if necessary.
+     The routine may make several calls to STEP in order to determine
+     the optimum step-size. An estimate of the local error for the step
+     can be obtained from the components of the array `E->yerr[]'.  If
+     the step-size has been changed the value of H will be modified on
+     output.  The maximum time T1 is guaranteed not to be exceeded by
+     the time-step.  On the final time-step the value of T will be set
+     to T1 exactly.
+
+     If the user-supplied functions defined in the system DYDT return a
+     status other than `GSL_SUCCESS' the step will be aborted.  In this
+     case,  T and Y will be restored to their pre-step values and the
+     error code from the user-supplied function will be returned.  To
+     distinguish between error codes from the user-supplied functions
+     and those from `gsl_odeiv_evolve_apply' itself, any user-defined
+     return values should be distinct from the standard GSL error codes.
+
+ -- Function: int gsl_odeiv_evolve_reset (gsl_odeiv_evolve * E)
+     This function resets the evolution function E.  It should be used
+     whenever the next use of E will not be a continuation of a
+     previous step.
+
+ -- Function: void gsl_odeiv_evolve_free (gsl_odeiv_evolve * E)
+     This function frees all the memory associated with the evolution
+     function E.
+
+
+File: gsl-ref.info,  Node: ODE Example programs,  Next: ODE References and Further Reading,  Prev: Evolution,  Up: Ordinary Differential Equations
+
+25.5 Examples
+=============
+
+The following program solves the second-order nonlinear Van der Pol
+oscillator equation,
+
+     x''(t) + \mu x'(t) (x(t)^2 - 1) + x(t) = 0
+
+This can be converted into a first order system suitable for use with
+the routines described in this chapter by introducing a separate
+variable for the velocity, y = x'(t),
+
+     x' = y
+     y' = -x + \mu y (1-x^2)
+
+The program begins by defining functions for these derivatives and
+their Jacobian,
+
+     #include <stdio.h>
+     #include <gsl/gsl_errno.h>
+     #include <gsl/gsl_matrix.h>
+     #include <gsl/gsl_odeiv.h>
+
+     int
+     func (double t, const double y[], double f[],
+           void *params)
+     {
+       double mu = *(double *)params;
+       f[0] = y[1];
+       f[1] = -y[0] - mu*y[1]*(y[0]*y[0] - 1);
+       return GSL_SUCCESS;
+     }
+
+     int
+     jac (double t, const double y[], double *dfdy,
+          double dfdt[], void *params)
+     {
+       double mu = *(double *)params;
+       gsl_matrix_view dfdy_mat
+         = gsl_matrix_view_array (dfdy, 2, 2);
+       gsl_matrix * m = &dfdy_mat.matrix;
+       gsl_matrix_set (m, 0, 0, 0.0);
+       gsl_matrix_set (m, 0, 1, 1.0);
+       gsl_matrix_set (m, 1, 0, -2.0*mu*y[0]*y[1] - 1.0);
+       gsl_matrix_set (m, 1, 1, -mu*(y[0]*y[0] - 1.0));
+       dfdt[0] = 0.0;
+       dfdt[1] = 0.0;
+       return GSL_SUCCESS;
+     }
+
+     int
+     main (void)
+     {
+       const gsl_odeiv_step_type * T
+         = gsl_odeiv_step_rk8pd;
+
+       gsl_odeiv_step * s
+         = gsl_odeiv_step_alloc (T, 2);
+       gsl_odeiv_control * c
+         = gsl_odeiv_control_y_new (1e-6, 0.0);
+       gsl_odeiv_evolve * e
+         = gsl_odeiv_evolve_alloc (2);
+
+       double mu = 10;
+       gsl_odeiv_system sys = {func, jac, 2, &mu};
+
+       double t = 0.0, t1 = 100.0;
+       double h = 1e-6;
+       double y[2] = { 1.0, 0.0 };
+
+       while (t < t1)
+         {
+           int status = gsl_odeiv_evolve_apply (e, c, s,
+                                                &sys,
+                                                &t, t1,
+                                                &h, y);
+
+           if (status != GSL_SUCCESS)
+               break;
+
+           printf ("%.5e %.5e %.5e\n", t, y[0], y[1]);
+         }
+
+       gsl_odeiv_evolve_free (e);
+       gsl_odeiv_control_free (c);
+       gsl_odeiv_step_free (s);
+       return 0;
+     }
+
+For functions with multiple parameters, the appropriate information can
+be passed in through the PARAMS argument using a pointer to a struct.
+
+   The main loop of the program evolves the solution from (y, y') = (1,
+0) at t=0 to t=100.  The step-size h is automatically adjusted by the
+controller to maintain an absolute accuracy of 10^{-6} in the function
+values Y.
+
+To obtain the values at regular intervals, rather than the variable
+spacings chosen by the control function, the main loop can be modified
+to advance the solution from one point to the next.  For example, the
+following main loop prints the solution at the fixed points t = 0, 1,
+2, \dots, 100,
+
+       for (i = 1; i <= 100; i++)
+         {
+           double ti = i * t1 / 100.0;
+
+           while (t < ti)
+             {
+               gsl_odeiv_evolve_apply (e, c, s,
+                                       &sys,
+                                       &t, ti, &h,
+                                       y);
+             }
+
+           printf ("%.5e %.5e %.5e\n", t, y[0], y[1]);
+         }
+
+It is also possible to work with a non-adaptive integrator, using only
+the stepping function itself.  The following program uses the `rk4'
+fourth-order Runge-Kutta stepping function with a fixed stepsize of
+0.01,
+
+     int
+     main (void)
+     {
+       const gsl_odeiv_step_type * T
+         = gsl_odeiv_step_rk4;
+
+       gsl_odeiv_step * s
+         = gsl_odeiv_step_alloc (T, 2);
+
+       double mu = 10;
+       gsl_odeiv_system sys = {func, jac, 2, &mu};
+
+       double t = 0.0, t1 = 100.0;
+       double h = 1e-2;
+       double y[2] = { 1.0, 0.0 }, y_err[2];
+       double dydt_in[2], dydt_out[2];
+
+       /* initialise dydt_in from system parameters */
+       GSL_ODEIV_FN_EVAL(&sys, t, y, dydt_in);
+
+       while (t < t1)
+         {
+           int status = gsl_odeiv_step_apply (s, t, h,
+                                              y, y_err,
+                                              dydt_in,
+                                              dydt_out,
+                                              &sys);
+
+           if (status != GSL_SUCCESS)
+               break;
+
+           dydt_in[0] = dydt_out[0];
+           dydt_in[1] = dydt_out[1];
+
+           t += h;
+
+           printf ("%.5e %.5e %.5e\n", t, y[0], y[1]);
+         }
+
+       gsl_odeiv_step_free (s);
+       return 0;
+     }
+
+The derivatives must be initialized for the starting point t=0 before
+the first step is taken.  Subsequent steps use the output derivatives
+DYDT_OUT as inputs to the next step by copying their values into
+DYDT_IN.
+
+
+File: gsl-ref.info,  Node: ODE References and Further Reading,  Prev: ODE Example programs,  Up: Ordinary Differential Equations
+
+25.6 References and Further Reading
+===================================
+
+Many of the basic Runge-Kutta formulas can be found in the Handbook of
+Mathematical Functions,
+
+     Abramowitz & Stegun (eds.), `Handbook of Mathematical Functions',
+     Section 25.5.
+
+The implicit Bulirsch-Stoer algorithm `bsimp' is described in the
+following paper,
+
+     G. Bader and P. Deuflhard, "A Semi-Implicit Mid-Point Rule for
+     Stiff Systems of Ordinary Differential Equations.", Numer. Math.
+     41, 373-398, 1983.
+
+
+File: gsl-ref.info,  Node: Interpolation,  Next: Numerical Differentiation,  Prev: Ordinary Differential Equations,  Up: Top
+
+26 Interpolation
+****************
+
+This chapter describes functions for performing interpolation.  The
+library provides a variety of interpolation methods, including Cubic
+splines and Akima splines.  The interpolation types are interchangeable,
+allowing different methods to be used without recompiling.
+Interpolations can be defined for both normal and periodic boundary
+conditions.  Additional functions are available for computing
+derivatives and integrals of interpolating functions.
+
+   The functions described in this section are declared in the header
+files `gsl_interp.h' and `gsl_spline.h'.
+
+* Menu:
+
+* Introduction to Interpolation::
+* Interpolation Functions::
+* Interpolation Types::
+* Index Look-up and Acceleration::
+* Evaluation of Interpolating Functions::
+* Higher-level Interface::
+* Interpolation Example programs::
+* Interpolation References and Further Reading::
+
+
+File: gsl-ref.info,  Node: Introduction to Interpolation,  Next: Interpolation Functions,  Up: Interpolation
+
+26.1 Introduction
+=================
+
+Given a set of data points (x_1, y_1) \dots (x_n, y_n) the routines
+described in this section compute a continuous interpolating function
+y(x) such that y(x_i) = y_i.  The interpolation is piecewise smooth,
+and its behavior at the end-points is determined by the type of
+interpolation used.
+
+
+File: gsl-ref.info,  Node: Interpolation Functions,  Next: Interpolation Types,  Prev: Introduction to Interpolation,  Up: Interpolation
+
+26.2 Interpolation Functions
+============================
+
+The interpolation function for a given dataset is stored in a
+`gsl_interp' object.  These are created by the following functions.
+
+ -- Function: gsl_interp * gsl_interp_alloc (const gsl_interp_type * T,
+          size_t SIZE)
+     This function returns a pointer to a newly allocated interpolation
+     object of type T for SIZE data-points.
+
+ -- Function: int gsl_interp_init (gsl_interp * INTERP, const double
+          XA[], const double YA[], size_t SIZE)
+     This function initializes the interpolation object INTERP for the
+     data (XA,YA) where XA and YA are arrays of size SIZE.  The
+     interpolation object (`gsl_interp') does not save the data arrays
+     XA and YA and only stores the static state computed from the data.
+     The XA data array is always assumed to be strictly ordered; the
+     behavior for other arrangements is not defined.
+
+ -- Function: void gsl_interp_free (gsl_interp * INTERP)
+     This function frees the interpolation object INTERP.
+
+
+File: gsl-ref.info,  Node: Interpolation Types,  Next: Index Look-up and Acceleration,  Prev: Interpolation Functions,  Up: Interpolation
+
+26.3 Interpolation Types
+========================
+
+The interpolation library provides five interpolation types:
+
+ -- Interpolation Type: gsl_interp_linear
+     Linear interpolation.  This interpolation method does not require
+     any additional memory.
+
+ -- Interpolation Type: gsl_interp_polynomial
+     Polynomial interpolation.  This method should only be used for
+     interpolating small numbers of points because polynomial
+     interpolation introduces large oscillations, even for well-behaved
+     datasets.  The number of terms in the interpolating polynomial is
+     equal to the number of points.
+
+ -- Interpolation Type: gsl_interp_cspline
+     Cubic spline with natural boundary conditions.  The resulting
+     curve is piecewise cubic on each interval, with matching first and
+     second derivatives at the supplied data-points.  The second
+     derivative is chosen to be zero at the first point and last point.
+
+ -- Interpolation Type: gsl_interp_cspline_periodic
+     Cubic spline with periodic boundary conditions.  The resulting
+     curve is piecewise cubic on each interval, with matching first and
+     second derivatives at the supplied data-points.  The derivatives
+     at the first and last points are also matched.  Note that the last
+     point in the data must have the same y-value as the first point,
+     otherwise the resulting periodic interpolation will have a
+     discontinuity at the boundary.
+
+
+ -- Interpolation Type: gsl_interp_akima
+     Non-rounded Akima spline with natural boundary conditions.  This
+     method uses the non-rounded corner algorithm of Wodicka.
+
+ -- Interpolation Type: gsl_interp_akima_periodic
+     Non-rounded Akima spline with periodic boundary conditions.  This
+     method uses the non-rounded corner algorithm of Wodicka.
+
+   The following related functions are available:
+
+ -- Function: const char * gsl_interp_name (const gsl_interp * INTERP)
+     This function returns the name of the interpolation type used by
+     INTERP.  For example,
+
+          printf ("interp uses '%s' interpolation.\n",
+                  gsl_interp_name (interp));
+
+     would print something like,
+
+          interp uses 'cspline' interpolation.
+
+ -- Function: unsigned int gsl_interp_min_size (const gsl_interp *
+          INTERP)
+     This function returns the minimum number of points required by the
+     interpolation type of INTERP.  For example, Akima spline
+     interpolation requires a minimum of 5 points.
+
+
+File: gsl-ref.info,  Node: Index Look-up and Acceleration,  Next: Evaluation of Interpolating Functions,  Prev: Interpolation Types,  Up: Interpolation
+
+26.4 Index Look-up and Acceleration
+===================================
+
+The state of searches can be stored in a `gsl_interp_accel' object,
+which is a kind of iterator for interpolation lookups.  It caches the
+previous value of an index lookup.  When the subsequent interpolation
+point falls in the same interval its index value can be returned
+immediately.
+
+ -- Function: size_t gsl_interp_bsearch (const double X_ARRAY[], double
+          X, size_t INDEX_LO, size_t INDEX_HI)
+     This function returns the index i of the array X_ARRAY such that
+     `x_array[i] <= x < x_array[i+1]'.  The index is searched for in
+     the range [INDEX_LO,INDEX_HI].
+
+ -- Function: gsl_interp_accel * gsl_interp_accel_alloc (void)
+     This function returns a pointer to an accelerator object, which is
+     a kind of iterator for interpolation lookups.  It tracks the state
+     of lookups, thus allowing for application of various acceleration
+     strategies.
+
+ -- Function: size_t gsl_interp_accel_find (gsl_interp_accel * A, const
+          double X_ARRAY[], size_t SIZE, double X)
+     This function performs a lookup action on the data array X_ARRAY
+     of size SIZE, using the given accelerator A.  This is how lookups
+     are performed during evaluation of an interpolation.  The function
+     returns an index i such that `x_array[i] <= x < x_array[i+1]'.
+
+ -- Function: void gsl_interp_accel_free (gsl_interp_accel* ACC)
+     This function frees the accelerator object ACC.
+
+
+File: gsl-ref.info,  Node: Evaluation of Interpolating Functions,  Next: Higher-level Interface,  Prev: Index Look-up and Acceleration,  Up: Interpolation
+
+26.5 Evaluation of Interpolating Functions
+==========================================
+
+ -- Function: double gsl_interp_eval (const gsl_interp * INTERP, const
+          double XA[], const double YA[], double X, gsl_interp_accel *
+          ACC)
+ -- Function: int gsl_interp_eval_e (const gsl_interp * INTERP, const
+          double XA[], const double YA[], double X, gsl_interp_accel *
+          ACC, double * Y)
+     These functions return the interpolated value of Y for a given
+     point X, using the interpolation object INTERP, data arrays XA and
+     YA and the accelerator ACC.
+
+ -- Function: double gsl_interp_eval_deriv (const gsl_interp * INTERP,
+          const double XA[], const double YA[], double X,
+          gsl_interp_accel * ACC)
+ -- Function: int gsl_interp_eval_deriv_e (const gsl_interp * INTERP,
+          const double XA[], const double YA[], double X,
+          gsl_interp_accel * ACC, double * D)
+     These functions return the derivative D of an interpolated
+     function for a given point X, using the interpolation object
+     INTERP, data arrays XA and YA and the accelerator ACC.
+
+ -- Function: double gsl_interp_eval_deriv2 (const gsl_interp * INTERP,
+          const double XA[], const double YA[], double X,
+          gsl_interp_accel * ACC)
+ -- Function: int gsl_interp_eval_deriv2_e (const gsl_interp * INTERP,
+          const double XA[], const double YA[], double X,
+          gsl_interp_accel * ACC, double * D2)
+     These functions return the second derivative D2 of an interpolated
+     function for a given point X, using the interpolation object
+     INTERP, data arrays XA and YA and the accelerator ACC.
+
+ -- Function: double gsl_interp_eval_integ (const gsl_interp * INTERP,
+          const double XA[], const double YA[], double A, double B,
+          gsl_interp_accel * ACC)
+ -- Function: int gsl_interp_eval_integ_e (const gsl_interp * INTERP,
+          const double XA[], const double YA[], double A, double B,
+          gsl_interp_accel * ACC, double * RESULT)
+     These functions return the numerical integral RESULT of an
+     interpolated function over the range [A, B], using the
+     interpolation object INTERP, data arrays XA and YA and the
+     accelerator ACC.
+
+
+File: gsl-ref.info,  Node: Higher-level Interface,  Next: Interpolation Example programs,  Prev: Evaluation of Interpolating Functions,  Up: Interpolation
+
+26.6 Higher-level Interface
+===========================
+
+The functions described in the previous sections required the user to
+supply pointers to the x and y arrays on each call.  The following
+functions are equivalent to the corresponding `gsl_interp' functions
+but maintain a copy of this data in the `gsl_spline' object.  This
+removes the need to pass both XA and YA as arguments on each
+evaluation. These functions are defined in the header file
+`gsl_spline.h'.
+
+ -- Function: gsl_spline * gsl_spline_alloc (const gsl_interp_type * T,
+          size_t SIZE)
+
+ -- Function: int gsl_spline_init (gsl_spline * SPLINE, const double
+          XA[], const double YA[], size_t SIZE)
+
+ -- Function: void gsl_spline_free (gsl_spline * SPLINE)
+
+ -- Function: const char * gsl_spline_name (const gsl_spline * SPLINE)
+
+ -- Function: unsigned int gsl_spline_min_size (const gsl_spline *
+          SPLINE)
+
+ -- Function: double gsl_spline_eval (const gsl_spline * SPLINE, double
+          X, gsl_interp_accel * ACC)
+ -- Function: int gsl_spline_eval_e (const gsl_spline * SPLINE, double
+          X, gsl_interp_accel * ACC, double * Y)
+
+ -- Function: double gsl_spline_eval_deriv (const gsl_spline * SPLINE,
+          double X, gsl_interp_accel * ACC)
+ -- Function: int gsl_spline_eval_deriv_e (const gsl_spline * SPLINE,
+          double X, gsl_interp_accel * ACC, double * D)
+
+ -- Function: double gsl_spline_eval_deriv2 (const gsl_spline * SPLINE,
+          double X, gsl_interp_accel * ACC)
+ -- Function: int gsl_spline_eval_deriv2_e (const gsl_spline * SPLINE,
+          double X, gsl_interp_accel * ACC, double * D2)
+
+ -- Function: double gsl_spline_eval_integ (const gsl_spline * SPLINE,
+          double A, double B, gsl_interp_accel * ACC)
+ -- Function: int gsl_spline_eval_integ_e (const gsl_spline * SPLINE,
+          double A, double B, gsl_interp_accel * ACC, double * RESULT)
+
+
+File: gsl-ref.info,  Node: Interpolation Example programs,  Next: Interpolation References and Further Reading,  Prev: Higher-level Interface,  Up: Interpolation
+
+26.7 Examples
+=============
+
+The following program demonstrates the use of the interpolation and
+spline functions.  It computes a cubic spline interpolation of the
+10-point dataset (x_i, y_i) where x_i = i + \sin(i)/2 and y_i = i +
+\cos(i^2) for i = 0 \dots 9.
+
+     #include <stdlib.h>
+     #include <stdio.h>
+     #include <math.h>
+     #include <gsl/gsl_errno.h>
+     #include <gsl/gsl_spline.h>
+
+     int
+     main (void)
+     {
+       int i;
+       double xi, yi, x[10], y[10];
+
+       printf ("#m=0,S=2\n");
+
+       for (i = 0; i < 10; i++)
+         {
+           x[i] = i + 0.5 * sin (i);
+           y[i] = i + cos (i * i);
+           printf ("%g %g\n", x[i], y[i]);
+         }
+
+       printf ("#m=1,S=0\n");
+
+       {
+         gsl_interp_accel *acc
+           = gsl_interp_accel_alloc ();
+         gsl_spline *spline
+           = gsl_spline_alloc (gsl_interp_cspline, 10);
+
+         gsl_spline_init (spline, x, y, 10);
+
+         for (xi = x[0]; xi < x[9]; xi += 0.01)
+           {
+             yi = gsl_spline_eval (spline, xi, acc);
+             printf ("%g %g\n", xi, yi);
+           }
+         gsl_spline_free (spline);
+         gsl_interp_accel_free (acc);
+       }
+       return 0;
+     }
+
+The output is designed to be used with the GNU plotutils `graph'
+program,
+
+     $ ./a.out > interp.dat
+     $ graph -T ps < interp.dat > interp.ps
+
+The result shows a smooth interpolation of the original points.  The
+interpolation method can changed simply by varying the first argument of
+`gsl_spline_alloc'.
+
+   The next program demonstrates a periodic cubic spline with 4 data
+points.  Note that the first and last points must be supplied with the
+same y-value for a periodic spline.
+
+     #include <stdlib.h>
+     #include <stdio.h>
+     #include <math.h>
+     #include <gsl/gsl_errno.h>
+     #include <gsl/gsl_spline.h>
+
+     int
+     main (void)
+     {
+       int N = 4;
+       double x[4] = {0.00, 0.10,  0.27,  0.30};
+       double y[4] = {0.15, 0.70, -0.10,  0.15}; /* Note: first = last
+                                                    for periodic data */
+
+       gsl_interp_accel *acc = gsl_interp_accel_alloc ();
+       const gsl_interp_type *t = gsl_interp_cspline_periodic;
+       gsl_spline *spline = gsl_spline_alloc (t, N);
+
+       int i; double xi, yi;
+
+       printf ("#m=0,S=5\n");
+       for (i = 0; i < N; i++)
+         {
+           printf ("%g %g\n", x[i], y[i]);
+         }
+
+       printf ("#m=1,S=0\n");
+       gsl_spline_init (spline, x, y, N);
+
+       for (i = 0; i <= 100; i++)
+         {
+           xi = (1 - i / 100.0) * x[0] + (i / 100.0) * x[N-1];
+           yi = gsl_spline_eval (spline, xi, acc);
+           printf ("%g %g\n", xi, yi);
+         }
+
+       gsl_spline_free (spline);
+       gsl_interp_accel_free (acc);
+       return 0;
+     }
+
+The output can be plotted with GNU `graph'.
+
+     $ ./a.out > interp.dat
+     $ graph -T ps < interp.dat > interp.ps
+
+The result shows a periodic interpolation of the original points. The
+slope of the fitted curve is the same at the beginning and end of the
+data, and the second derivative is also.
+
+
+File: gsl-ref.info,  Node: Interpolation References and Further Reading,  Prev: Interpolation Example programs,  Up: Interpolation
+
+26.8 References and Further Reading
+===================================
+
+Descriptions of the interpolation algorithms and further references can
+be found in the following books:
+
+     C.W. Ueberhuber, `Numerical Computation (Volume 1), Chapter 9
+     "Interpolation"', Springer (1997), ISBN 3-540-62058-3.
+
+     D.M. Young, R.T. Gregory `A Survey of Numerical Mathematics
+     (Volume 1), Chapter 6.8', Dover (1988), ISBN 0-486-65691-8.
+
+
+
+File: gsl-ref.info,  Node: Numerical Differentiation,  Next: Chebyshev Approximations,  Prev: Interpolation,  Up: Top
+
+27 Numerical Differentiation
+****************************
+
+The functions described in this chapter compute numerical derivatives by
+finite differencing.  An adaptive algorithm is used to find the best
+choice of finite difference and to estimate the error in the derivative.
+These functions are declared in the header file `gsl_deriv.h'.
+
+* Menu:
+
+* Numerical Differentiation functions::
+* Numerical Differentiation Examples::
+* Numerical Differentiation References::
+
+
+File: gsl-ref.info,  Node: Numerical Differentiation functions,  Next: Numerical Differentiation Examples,  Up: Numerical Differentiation
+
+27.1 Functions
+==============
+
+ -- Function: int gsl_deriv_central (const gsl_function * F, double X,
+          double H, double * RESULT, double * ABSERR)
+     This function computes the numerical derivative of the function F
+     at the point X using an adaptive central difference algorithm with
+     a step-size of H.   The derivative is returned in RESULT and an
+     estimate of its absolute error is returned in ABSERR.
+
+     The initial value of H is used to estimate an optimal step-size,
+     based on the scaling of the truncation error and round-off error
+     in the derivative calculation.  The derivative is computed using a
+     5-point rule for equally spaced abscissae at x-h, x-h/2, x, x+h/2,
+     x+h, with an error estimate taken from the difference between the
+     5-point rule and the corresponding 3-point rule x-h, x, x+h.  Note
+     that the value of the function at x does not contribute to the
+     derivative calculation, so only 4-points are actually used.
+
+ -- Function: int gsl_deriv_forward (const gsl_function * F, double X,
+          double H, double * RESULT, double * ABSERR)
+     This function computes the numerical derivative of the function F
+     at the point X using an adaptive forward difference algorithm with
+     a step-size of H. The function is evaluated only at points greater
+     than X, and never at X itself.  The derivative is returned in
+     RESULT and an estimate of its absolute error is returned in
+     ABSERR.  This function should be used if f(x) has a discontinuity
+     at X, or is undefined for values less than X.
+
+     The initial value of H is used to estimate an optimal step-size,
+     based on the scaling of the truncation error and round-off error
+     in the derivative calculation.  The derivative at x is computed
+     using an "open" 4-point rule for equally spaced abscissae at x+h/4,
+     x+h/2, x+3h/4, x+h, with an error estimate taken from the
+     difference between the 4-point rule and the corresponding 2-point
+     rule x+h/2, x+h.
+
+ -- Function: int gsl_deriv_backward (const gsl_function * F, double X,
+          double H, double * RESULT, double * ABSERR)
+     This function computes the numerical derivative of the function F
+     at the point X using an adaptive backward difference algorithm
+     with a step-size of H. The function is evaluated only at points
+     less than X, and never at X itself.  The derivative is returned in
+     RESULT and an estimate of its absolute error is returned in
+     ABSERR.  This function should be used if f(x) has a discontinuity
+     at X, or is undefined for values greater than X.
+
+     This function is equivalent to calling `gsl_deriv_forward' with a
+     negative step-size.
+
+
+File: gsl-ref.info,  Node: Numerical Differentiation Examples,  Next: Numerical Differentiation References,  Prev: Numerical Differentiation functions,  Up: Numerical Differentiation
+
+27.2 Examples
+=============
+
+The following code estimates the derivative of the function f(x) =
+x^{3/2} at x=2 and at x=0.  The function f(x) is undefined for x<0 so
+the derivative at x=0 is computed using `gsl_deriv_forward'.
+
+     #include <stdio.h>
+     #include <gsl/gsl_math.h>
+     #include <gsl/gsl_deriv.h>
+
+     double f (double x, void * params)
+     {
+       return pow (x, 1.5);
+     }
+
+     int
+     main (void)
+     {
+       gsl_function F;
+       double result, abserr;
+
+       F.function = &f;
+       F.params = 0;
+
+       printf ("f(x) = x^(3/2)\n");
+
+       gsl_deriv_central (&F, 2.0, 1e-8, &result, &abserr);
+       printf ("x = 2.0\n");
+       printf ("f'(x) = %.10f +/- %.10f\n", result, abserr);
+       printf ("exact = %.10f\n\n", 1.5 * sqrt(2.0));
+
+       gsl_deriv_forward (&F, 0.0, 1e-8, &result, &abserr);
+       printf ("x = 0.0\n");
+       printf ("f'(x) = %.10f +/- %.10f\n", result, abserr);
+       printf ("exact = %.10f\n", 0.0);
+
+       return 0;
+     }
+
+Here is the output of the program,
+
+     $ ./a.out
+     f(x) = x^(3/2)
+     x = 2.0
+     f'(x) = 2.1213203120 +/- 0.0000004064
+     exact = 2.1213203436
+
+     x = 0.0
+     f'(x) = 0.0000000160 +/- 0.0000000339
+     exact = 0.0000000000
+
+
+File: gsl-ref.info,  Node: Numerical Differentiation References,  Prev: Numerical Differentiation Examples,  Up: Numerical Differentiation
+
+27.3 References and Further Reading
+===================================
+
+The algorithms used by these functions are described in the following
+sources:
+
+     Abramowitz and Stegun, `Handbook of Mathematical Functions',
+     Section 25.3.4, and Table 25.5 (Coefficients for Differentiation).
+
+     S.D. Conte and Carl de Boor, `Elementary Numerical Analysis: An
+     Algorithmic Approach', McGraw-Hill, 1972.
+
+
+File: gsl-ref.info,  Node: Chebyshev Approximations,  Next: Series Acceleration,  Prev: Numerical Differentiation,  Up: Top
+
+28 Chebyshev Approximations
+***************************
+
+This chapter describes routines for computing Chebyshev approximations
+to univariate functions.  A Chebyshev approximation is a truncation of
+the series f(x) = \sum c_n T_n(x), where the Chebyshev polynomials
+T_n(x) = \cos(n \arccos x) provide an orthogonal basis of polynomials
+on the interval [-1,1] with the weight function 1 / \sqrt{1-x^2}.  The
+first few Chebyshev polynomials are, T_0(x) = 1, T_1(x) = x, T_2(x) = 2
+x^2 - 1.  For further information see Abramowitz & Stegun, Chapter 22.
+
+   The functions described in this chapter are declared in the header
+file `gsl_chebyshev.h'.
+
+* Menu:
+
+* Chebyshev Definitions::
+* Creation and Calculation of Chebyshev Series::
+* Chebyshev Series Evaluation::
+* Derivatives and Integrals::
+* Chebyshev Approximation examples::
+* Chebyshev Approximation References and Further Reading::
+
+
+File: gsl-ref.info,  Node: Chebyshev Definitions,  Next: Creation and Calculation of Chebyshev Series,  Up: Chebyshev Approximations
+
+28.1 Definitions
+================
+
+A Chebyshev series  is stored using the following structure,
+
+     typedef struct
+     {
+       double * c;   /* coefficients  c[0] .. c[order] */
+       int order;    /* order of expansion             */
+       double a;     /* lower interval point           */
+       double b;     /* upper interval point           */
+       ...
+     } gsl_cheb_series
+
+The approximation is made over the range [a,b] using ORDER+1 terms,
+including the coefficient c[0].  The series is computed using the
+following convention,
+
+     f(x) = (c_0 / 2) + \sum_{n=1} c_n T_n(x)
+
+which is needed when accessing the coefficients directly.
+
+
+File: gsl-ref.info,  Node: Creation and Calculation of Chebyshev Series,  Next: Chebyshev Series Evaluation,  Prev: Chebyshev Definitions,  Up: Chebyshev Approximations
+
+28.2 Creation and Calculation of Chebyshev Series
+=================================================
+
+ -- Function: gsl_cheb_series * gsl_cheb_alloc (const size_t N)
+     This function allocates space for a Chebyshev series of order N
+     and returns a pointer to a new `gsl_cheb_series' struct.
+
+ -- Function: void gsl_cheb_free (gsl_cheb_series * CS)
+     This function frees a previously allocated Chebyshev series CS.
+
+ -- Function: int gsl_cheb_init (gsl_cheb_series * CS, const
+          gsl_function * F, const double A, const double B)
+     This function computes the Chebyshev approximation CS for the
+     function F over the range (a,b) to the previously specified order.
+     The computation of the Chebyshev approximation is an O(n^2)
+     process, and requires n function evaluations.
+
+
+File: gsl-ref.info,  Node: Chebyshev Series Evaluation,  Next: Derivatives and Integrals,  Prev: Creation and Calculation of Chebyshev Series,  Up: Chebyshev Approximations
+
+28.3 Chebyshev Series Evaluation
+================================
+
+ -- Function: double gsl_cheb_eval (const gsl_cheb_series * CS, double
+          X)
+     This function evaluates the Chebyshev series CS at a given point X.
+
+ -- Function: int gsl_cheb_eval_err (const gsl_cheb_series * CS, const
+          double X, double * RESULT, double * ABSERR)
+     This function computes the Chebyshev series CS at a given point X,
+     estimating both the series RESULT and its absolute error ABSERR.
+     The error estimate is made from the first neglected term in the
+     series.
+
+ -- Function: double gsl_cheb_eval_n (const gsl_cheb_series * CS,
+          size_t ORDER, double X)
+     This function evaluates the Chebyshev series CS at a given point
+     N, to (at most) the given order ORDER.
+
+ -- Function: int gsl_cheb_eval_n_err (const gsl_cheb_series * CS,
+          const size_t ORDER, const double X, double * RESULT, double *
+          ABSERR)
+     This function evaluates a Chebyshev series CS at a given point X,
+     estimating both the series RESULT and its absolute error ABSERR,
+     to (at most) the given order ORDER.  The error estimate is made
+     from the first neglected term in the series.
+
+
+File: gsl-ref.info,  Node: Derivatives and Integrals,  Next: Chebyshev Approximation examples,  Prev: Chebyshev Series Evaluation,  Up: Chebyshev Approximations
+
+28.4 Derivatives and Integrals
+==============================
+
+The following functions allow a Chebyshev series to be differentiated or
+integrated, producing a new Chebyshev series.  Note that the error
+estimate produced by evaluating the derivative series will be
+underestimated due to the contribution of higher order terms being
+neglected.
+
+ -- Function: int gsl_cheb_calc_deriv (gsl_cheb_series * DERIV, const
+          gsl_cheb_series * CS)
+     This function computes the derivative of the series CS, storing
+     the derivative coefficients in the previously allocated DERIV.
+     The two series CS and DERIV must have been allocated with the same
+     order.
+
+ -- Function: int gsl_cheb_calc_integ (gsl_cheb_series * INTEG, const
+          gsl_cheb_series * CS)
+     This function computes the integral of the series CS, storing the
+     integral coefficients in the previously allocated INTEG.  The two
+     series CS and INTEG must have been allocated with the same order.
+     The lower limit of the integration is taken to be the left hand
+     end of the range A.
+
+
+File: gsl-ref.info,  Node: Chebyshev Approximation examples,  Next: Chebyshev Approximation References and Further Reading,  Prev: Derivatives and Integrals,  Up: Chebyshev Approximations
+
+28.5 Examples
+=============
+
+The following example program computes Chebyshev approximations to a
+step function.  This is an extremely difficult approximation to make,
+due to the discontinuity, and was chosen as an example where
+approximation error is visible.  For smooth functions the Chebyshev
+approximation converges extremely rapidly and errors would not be
+visible.
+
+     #include <stdio.h>
+     #include <gsl/gsl_math.h>
+     #include <gsl/gsl_chebyshev.h>
+
+     double
+     f (double x, void *p)
+     {
+       if (x < 0.5)
+         return 0.25;
+       else
+         return 0.75;
+     }
+
+     int
+     main (void)
+     {
+       int i, n = 10000;
+
+       gsl_cheb_series *cs = gsl_cheb_alloc (40);
+
+       gsl_function F;
+
+       F.function = f;
+       F.params = 0;
+
+       gsl_cheb_init (cs, &F, 0.0, 1.0);
+
+       for (i = 0; i < n; i++)
+         {
+           double x = i / (double)n;
+           double r10 = gsl_cheb_eval_n (cs, 10, x);
+           double r40 = gsl_cheb_eval (cs, x);
+           printf ("%g %g %g %g\n",
+                   x, GSL_FN_EVAL (&F, x), r10, r40);
+         }
+
+       gsl_cheb_free (cs);
+
+       return 0;
+     }
+
+The output from the program gives the original function, 10-th order
+approximation and 40-th order approximation, all sampled at intervals of
+0.001 in x.
+
+
+File: gsl-ref.info,  Node: Chebyshev Approximation References and Further Reading,  Prev: Chebyshev Approximation examples,  Up: Chebyshev Approximations
+
+28.6 References and Further Reading
+===================================
+
+The following paper describes the use of Chebyshev series,
+
+     R. Broucke, "Ten Subroutines for the Manipulation of Chebyshev
+     Series [C1] (Algorithm 446)". `Communications of the ACM' 16(4),
+     254-256 (1973)
+
+
+File: gsl-ref.info,  Node: Series Acceleration,  Next: Wavelet Transforms,  Prev: Chebyshev Approximations,  Up: Top
+
+29 Series Acceleration
+**********************
+
+The functions described in this chapter accelerate the convergence of a
+series using the Levin u-transform.  This method takes a small number of
+terms from the start of a series and uses a systematic approximation to
+compute an extrapolated value and an estimate of its error.  The
+u-transform works for both convergent and divergent series, including
+asymptotic series.
+
+   These functions are declared in the header file `gsl_sum.h'.
+
+* Menu:
+
+* Acceleration functions::
+* Acceleration functions without error estimation::
+* Example of accelerating a series::
+* Series Acceleration References::
+
+
+File: gsl-ref.info,  Node: Acceleration functions,  Next: Acceleration functions without error estimation,  Up: Series Acceleration
+
+29.1 Acceleration functions
+===========================
+
+The following functions compute the full Levin u-transform of a series
+with its error estimate.  The error estimate is computed by propagating
+rounding errors from each term through to the final extrapolation.
+
+   These functions are intended for summing analytic series where each
+term is known to high accuracy, and the rounding errors are assumed to
+originate from finite precision. They are taken to be relative errors of
+order `GSL_DBL_EPSILON' for each term.
+
+   The calculation of the error in the extrapolated value is an O(N^2)
+process, which is expensive in time and memory.  A faster but less
+reliable method which estimates the error from the convergence of the
+extrapolated value is described in the next section.  For the method
+described here a full table of intermediate values and derivatives
+through to O(N) must be computed and stored, but this does give a
+reliable error estimate.
+
+ -- Function: gsl_sum_levin_u_workspace * gsl_sum_levin_u_alloc (size_t
+          N)
+     This function allocates a workspace for a Levin u-transform of N
+     terms.  The size of the workspace is O(2n^2 + 3n).
+
+ -- Function: void gsl_sum_levin_u_free (gsl_sum_levin_u_workspace * W)
+     This function frees the memory associated with the workspace W.
+
+ -- Function: int gsl_sum_levin_u_accel (const double * ARRAY, size_t
+          ARRAY_SIZE, gsl_sum_levin_u_workspace * W, double *
+          SUM_ACCEL, double * ABSERR)
+     This function takes the terms of a series in ARRAY of size
+     ARRAY_SIZE and computes the extrapolated limit of the series using
+     a Levin u-transform.  Additional working space must be provided in
+     W.  The extrapolated sum is stored in SUM_ACCEL, with an estimate
+     of the absolute error stored in ABSERR.  The actual term-by-term
+     sum is returned in `w->sum_plain'. The algorithm calculates the
+     truncation error (the difference between two successive
+     extrapolations) and round-off error (propagated from the individual
+     terms) to choose an optimal number of terms for the extrapolation.
+     All the terms of the series passed in through ARRAY should be
+     non-zero.
+
+
+File: gsl-ref.info,  Node: Acceleration functions without error estimation,  Next: Example of accelerating a series,  Prev: Acceleration functions,  Up: Series Acceleration
+
+29.2 Acceleration functions without error estimation
+====================================================
+
+The functions described in this section compute the Levin u-transform of
+series and attempt to estimate the error from the "truncation error" in
+the extrapolation, the difference between the final two approximations.
+Using this method avoids the need to compute an intermediate table of
+derivatives because the error is estimated from the behavior of the
+extrapolated value itself. Consequently this algorithm is an O(N)
+process and only requires O(N) terms of storage.  If the series
+converges sufficiently fast then this procedure can be acceptable.  It
+is appropriate to use this method when there is a need to compute many
+extrapolations of series with similar convergence properties at
+high-speed.  For example, when numerically integrating a function
+defined by a parameterized series where the parameter varies only
+slightly. A reliable error estimate should be computed first using the
+full algorithm described above in order to verify the consistency of the
+results.
+
+ -- Function: gsl_sum_levin_utrunc_workspace *
+gsl_sum_levin_utrunc_alloc (size_t N)
+     This function allocates a workspace for a Levin u-transform of N
+     terms, without error estimation.  The size of the workspace is
+     O(3n).
+
+ -- Function: void gsl_sum_levin_utrunc_free
+          (gsl_sum_levin_utrunc_workspace * W)
+     This function frees the memory associated with the workspace W.
+
+ -- Function: int gsl_sum_levin_utrunc_accel (const double * ARRAY,
+          size_t ARRAY_SIZE, gsl_sum_levin_utrunc_workspace * W, double
+          * SUM_ACCEL, double * ABSERR_TRUNC)
+     This function takes the terms of a series in ARRAY of size
+     ARRAY_SIZE and computes the extrapolated limit of the series using
+     a Levin u-transform.  Additional working space must be provided in
+     W.  The extrapolated sum is stored in SUM_ACCEL.  The actual
+     term-by-term sum is returned in `w->sum_plain'. The algorithm
+     terminates when the difference between two successive
+     extrapolations reaches a minimum or is sufficiently small. The
+     difference between these two values is used as estimate of the
+     error and is stored in ABSERR_TRUNC.  To improve the reliability
+     of the algorithm the extrapolated values are replaced by moving
+     averages when calculating the truncation error, smoothing out any
+     fluctuations.
+
+
+File: gsl-ref.info,  Node: Example of accelerating a series,  Next: Series Acceleration References,  Prev: Acceleration functions without error estimation,  Up: Series Acceleration
+
+29.3 Examples
+=============
+
+The following code calculates an estimate of \zeta(2) = \pi^2 / 6 using
+the series,
+
+     \zeta(2) = 1 + 1/2^2 + 1/3^2 + 1/4^2 + ...
+
+After N terms the error in the sum is O(1/N), making direct summation
+of the series converge slowly.
+
+     #include <stdio.h>
+     #include <gsl/gsl_math.h>
+     #include <gsl/gsl_sum.h>
+
+     #define N 20
+
+     int
+     main (void)
+     {
+       double t[N];
+       double sum_accel, err;
+       double sum = 0;
+       int n;
+
+       gsl_sum_levin_u_workspace * w
+         = gsl_sum_levin_u_alloc (N);
+
+       const double zeta_2 = M_PI * M_PI / 6.0;
+
+       /* terms for zeta(2) = \sum_{n=1}^{\infty} 1/n^2 */
+
+       for (n = 0; n < N; n++)
+         {
+           double np1 = n + 1.0;
+           t[n] = 1.0 / (np1 * np1);
+           sum += t[n];
+         }
+
+       gsl_sum_levin_u_accel (t, N, w, &sum_accel, &err);
+
+       printf ("term-by-term sum = % .16f using %d terms\n",
+               sum, N);
+
+       printf ("term-by-term sum = % .16f using %d terms\n",
+               w->sum_plain, w->terms_used);
+
+       printf ("exact value      = % .16f\n", zeta_2);
+       printf ("accelerated sum  = % .16f using %d terms\n",
+               sum_accel, w->terms_used);
+
+       printf ("estimated error  = % .16f\n", err);
+       printf ("actual error     = % .16f\n",
+               sum_accel - zeta_2);
+
+       gsl_sum_levin_u_free (w);
+       return 0;
+     }
+
+The output below shows that the Levin u-transform is able to obtain an
+estimate of the sum to 1 part in 10^10 using the first eleven terms of
+the series.  The error estimate returned by the function is also
+accurate, giving the correct number of significant digits.
+
+     $ ./a.out
+     term-by-term sum =  1.5961632439130233 using 20 terms
+     term-by-term sum =  1.5759958390005426 using 13 terms
+     exact value      =  1.6449340668482264
+     accelerated sum  =  1.6449340668166479 using 13 terms
+     estimated error  =  0.0000000000508580
+     actual error     = -0.0000000000315785
+
+Note that a direct summation of this series would require 10^10 terms
+to achieve the same precision as the accelerated sum does in 13 terms.
+
+
+File: gsl-ref.info,  Node: Series Acceleration References,  Prev: Example of accelerating a series,  Up: Series Acceleration
+
+29.4 References and Further Reading
+===================================
+
+The algorithms used by these functions are described in the following
+papers,
+
+     T. Fessler, W.F. Ford, D.A. Smith, HURRY: An acceleration
+     algorithm for scalar sequences and series `ACM Transactions on
+     Mathematical Software', 9(3):346-354, 1983.  and Algorithm 602
+     9(3):355-357, 1983.
+
+The theory of the u-transform was presented by Levin,
+
+     D. Levin, Development of Non-Linear Transformations for Improving
+     Convergence of Sequences, `Intern. J. Computer Math.' B3:371-388,
+     1973.
+
+A review paper on the Levin Transform is available online,
+     Herbert H. H. Homeier, Scalar Levin-Type Sequence Transformations,
+     `http://arxiv.org/abs/math/0005209'.
+
+
+File: gsl-ref.info,  Node: Wavelet Transforms,  Next: Discrete Hankel Transforms,  Prev: Series Acceleration,  Up: Top
+
+30 Wavelet Transforms
+*********************
+
+This chapter describes functions for performing Discrete Wavelet
+Transforms (DWTs).  The library includes wavelets for real data in both
+one and two dimensions.  The wavelet functions are declared in the
+header files `gsl_wavelet.h' and `gsl_wavelet2d.h'.
+
+* Menu:
+
+* DWT Definitions::
+* DWT Initialization::
+* DWT Transform Functions::
+* DWT Examples::
+* DWT References::
+
+
+File: gsl-ref.info,  Node: DWT Definitions,  Next: DWT Initialization,  Up: Wavelet Transforms
+
+30.1 Definitions
+================
+
+The continuous wavelet transform and its inverse are defined by the
+relations,
+
+     w(s,\tau) = \int f(t) * \psi^*_{s,\tau}(t) dt
+
+and,
+
+     f(t) = \int \int_{-\infty}^\infty w(s, \tau) * \psi_{s,\tau}(t) d\tau ds
+
+where the basis functions \psi_{s,\tau} are obtained by scaling and
+translation from a single function, referred to as the "mother wavelet".
+
+   The discrete version of the wavelet transform acts on equally-spaced
+samples, with fixed scaling and translation steps (s, \tau).  The
+frequency and time axes are sampled "dyadically" on scales of 2^j
+through a level parameter j.  The resulting family of functions
+{\psi_{j,n}} constitutes an orthonormal basis for square-integrable
+signals.
+
+   The discrete wavelet transform is an O(N) algorithm, and is also
+referred to as the "fast wavelet transform".
+
+
+File: gsl-ref.info,  Node: DWT Initialization,  Next: DWT Transform Functions,  Prev: DWT Definitions,  Up: Wavelet Transforms
+
+30.2 Initialization
+===================
+
+The `gsl_wavelet' structure contains the filter coefficients defining
+the wavelet and any associated offset parameters.
+
+ -- Function: gsl_wavelet * gsl_wavelet_alloc (const gsl_wavelet_type *
+          T, size_t K)
+     This function allocates and initializes a wavelet object of type
+     T.  The parameter K selects the specific member of the wavelet
+     family.  A null pointer is returned if insufficient memory is
+     available or if a unsupported member is selected.
+
+   The following wavelet types are implemented:
+
+ -- Wavelet: gsl_wavelet_daubechies
+ -- Wavelet: gsl_wavelet_daubechies_centered
+     The is the Daubechies wavelet family of maximum phase with k/2
+     vanishing moments.  The implemented wavelets are k=4, 6, ..., 20,
+     with K even.
+
+ -- Wavelet: gsl_wavelet_haar
+ -- Wavelet: gsl_wavelet_haar_centered
+     This is the Haar wavelet.  The only valid choice of k for the Haar
+     wavelet is k=2.
+
+ -- Wavelet: gsl_wavelet_bspline
+ -- Wavelet: gsl_wavelet_bspline_centered
+     This is the biorthogonal B-spline wavelet family of order (i,j).
+     The implemented values of k = 100*i + j are 103, 105, 202, 204,
+     206, 208, 301, 303, 305 307, 309.
+
+The centered forms of the wavelets align the coefficients of the various
+sub-bands on edges.  Thus the resulting visualization of the
+coefficients of the wavelet transform in the phase plane is easier to
+understand.
+
+ -- Function: const char * gsl_wavelet_name (const gsl_wavelet * W)
+     This function returns a pointer to the name of the wavelet family
+     for W.
+
+ -- Function: void gsl_wavelet_free (gsl_wavelet * W)
+     This function frees the wavelet object W.
+
+   The `gsl_wavelet_workspace' structure contains scratch space of the
+same size as the input data and is used to hold intermediate results
+during the transform.
+
+ -- Function: gsl_wavelet_workspace * gsl_wavelet_workspace_alloc
+          (size_t N)
+     This function allocates a workspace for the discrete wavelet
+     transform.  To perform a one-dimensional transform on N elements,
+     a workspace of size N must be provided.  For two-dimensional
+     transforms of N-by-N matrices it is sufficient to allocate a
+     workspace of size N, since the transform operates on individual
+     rows and columns.
+
+ -- Function: void gsl_wavelet_workspace_free (gsl_wavelet_workspace *
+          WORK)
+     This function frees the allocated workspace WORK.
+
+
+File: gsl-ref.info,  Node: DWT Transform Functions,  Next: DWT Examples,  Prev: DWT Initialization,  Up: Wavelet Transforms
+
+30.3 Transform Functions
+========================
+
+This sections describes the actual functions performing the discrete
+wavelet transform.  Note that the transforms use periodic boundary
+conditions.  If the signal is not periodic in the sample length then
+spurious coefficients will appear at the beginning and end of each level
+of the transform.
+
+* Menu:
+
+* DWT in one dimension::
+* DWT in two dimension::
+
+
+File: gsl-ref.info,  Node: DWT in one dimension,  Next: DWT in two dimension,  Up: DWT Transform Functions
+
+30.3.1 Wavelet transforms in one dimension
+------------------------------------------
+
+ -- Function: int gsl_wavelet_transform (const gsl_wavelet * W, double
+          * DATA, size_t STRIDE, size_t N, gsl_wavelet_direction DIR,
+          gsl_wavelet_workspace * WORK)
+ -- Function: int gsl_wavelet_transform_forward (const gsl_wavelet * W,
+          double * DATA, size_t STRIDE, size_t N, gsl_wavelet_workspace
+          * WORK)
+ -- Function: int gsl_wavelet_transform_inverse (const gsl_wavelet * W,
+          double * DATA, size_t STRIDE, size_t N, gsl_wavelet_workspace
+          * WORK)
+     These functions compute in-place forward and inverse discrete
+     wavelet transforms of length N with stride STRIDE on the array
+     DATA. The length of the transform N is restricted to powers of
+     two.  For the `transform' version of the function the argument DIR
+     can be either `forward' (+1) or `backward' (-1).  A workspace WORK
+     of length N must be provided.
+
+     For the forward transform, the elements of the original array are
+     replaced by the discrete wavelet transform f_i -> w_{j,k} in a
+     packed triangular storage layout, where J is the index of the level
+     j = 0 ... J-1 and K is the index of the coefficient within each
+     level, k = 0 ... (2^j)-1.  The total number of levels is J =
+     \log_2(n).  The output data has the following form,
+
+          (s_{-1,0}, d_{0,0}, d_{1,0}, d_{1,1}, d_{2,0}, ...,
+            d_{j,k}, ..., d_{J-1,2^{J-1}-1})
+
+     where the first element is the smoothing coefficient s_{-1,0},
+     followed by the detail coefficients d_{j,k} for each level j.  The
+     backward transform inverts these coefficients to obtain the
+     original data.
+
+     These functions return a status of `GSL_SUCCESS' upon successful
+     completion.  `GSL_EINVAL' is returned if N is not an integer power
+     of 2 or if insufficient workspace is provided.
+
+
+File: gsl-ref.info,  Node: DWT in two dimension,  Prev: DWT in one dimension,  Up: DWT Transform Functions
+
+30.3.2 Wavelet transforms in two dimension
+------------------------------------------
+
+The library provides functions to perform two-dimensional discrete
+wavelet transforms on square matrices.  The matrix dimensions must be an
+integer power of two.  There are two possible orderings of the rows and
+columns in the two-dimensional wavelet transform, referred to as the
+"standard" and "non-standard" forms.
+
+   The "standard" transform performs a complete discrete wavelet
+transform on the rows of the matrix, followed by a separate complete
+discrete wavelet transform on the columns of the resulting
+row-transformed matrix.  This procedure uses the same ordering as a
+two-dimensional fourier transform.
+
+   The "non-standard" transform is performed in interleaved passes on
+the rows and columns of the matrix for each level of the transform.  The
+first level of the transform is applied to the matrix rows, and then to
+the matrix columns.  This procedure is then repeated across the rows and
+columns of the data for the subsequent levels of the transform, until
+the full discrete wavelet transform is complete.  The non-standard form
+of the discrete wavelet transform is typically used in image analysis.
+
+   The functions described in this section are declared in the header
+file `gsl_wavelet2d.h'.
+
+ -- Function: int gsl_wavelet2d_transform (const gsl_wavelet * W,
+          double * DATA, size_t TDA, size_t SIZE1, size_t SIZE2,
+          gsl_wavelet_direction DIR, gsl_wavelet_workspace * WORK)
+ -- Function: int gsl_wavelet2d_transform_forward (const gsl_wavelet *
+          W, double * DATA, size_t TDA, size_t SIZE1, size_t SIZE2,
+          gsl_wavelet_workspace * WORK)
+ -- Function: int gsl_wavelet2d_transform_inverse (const gsl_wavelet *
+          W, double * DATA, size_t TDA, size_t SIZE1, size_t SIZE2,
+          gsl_wavelet_workspace * WORK)
+     These functions compute two-dimensional in-place forward and
+     inverse discrete wavelet transforms in standard and non-standard
+     forms on the array DATA stored in row-major form with dimensions
+     SIZE1 and SIZE2 and physical row length TDA.  The dimensions must
+     be equal (square matrix) and are restricted to powers of two.  For
+     the `transform' version of the function the argument DIR can be
+     either `forward' (+1) or `backward' (-1).  A workspace WORK of the
+     appropriate size must be provided.  On exit, the appropriate
+     elements of the array DATA are replaced by their two-dimensional
+     wavelet transform.
+
+     The functions return a status of `GSL_SUCCESS' upon successful
+     completion.  `GSL_EINVAL' is returned if SIZE1 and SIZE2 are not
+     equal and integer powers of 2, or if insufficient workspace is
+     provided.
+
+ -- Function: int gsl_wavelet2d_transform_matrix (const gsl_wavelet *
+          W, gsl_matrix * M, gsl_wavelet_direction DIR,
+          gsl_wavelet_workspace * WORK)
+ -- Function: int gsl_wavelet2d_transform_matrix_forward (const
+          gsl_wavelet * W, gsl_matrix * M, gsl_wavelet_workspace * WORK)
+ -- Function: int gsl_wavelet2d_transform_matrix_inverse (const
+          gsl_wavelet * W, gsl_matrix * M, gsl_wavelet_workspace * WORK)
+     These functions compute the two-dimensional in-place wavelet
+     transform on a matrix A.
+
+ -- Function: int gsl_wavelet2d_nstransform (const gsl_wavelet * W,
+          double * DATA, size_t TDA, size_t SIZE1, size_t SIZE2,
+          gsl_wavelet_direction DIR, gsl_wavelet_workspace * WORK)
+ -- Function: int gsl_wavelet2d_nstransform_forward (const gsl_wavelet
+          * W, double * DATA, size_t TDA, size_t SIZE1, size_t SIZE2,
+          gsl_wavelet_workspace * WORK)
+ -- Function: int gsl_wavelet2d_nstransform_inverse (const gsl_wavelet
+          * W, double * DATA, size_t TDA, size_t SIZE1, size_t SIZE2,
+          gsl_wavelet_workspace * WORK)
+     These functions compute the two-dimensional wavelet transform in
+     non-standard form.
+
+ -- Function: int gsl_wavelet2d_nstransform_matrix (const gsl_wavelet *
+          W, gsl_matrix * M, gsl_wavelet_direction DIR,
+          gsl_wavelet_workspace * WORK)
+ -- Function: int gsl_wavelet2d_nstransform_matrix_forward (const
+          gsl_wavelet * W, gsl_matrix * M, gsl_wavelet_workspace * WORK)
+ -- Function: int gsl_wavelet2d_nstransform_matrix_inverse (const
+          gsl_wavelet * W, gsl_matrix * M, gsl_wavelet_workspace * WORK)
+     These functions compute the non-standard form of the
+     two-dimensional in-place wavelet transform on a matrix A.
+
+
+File: gsl-ref.info,  Node: DWT Examples,  Next: DWT References,  Prev: DWT Transform Functions,  Up: Wavelet Transforms
+
+30.4 Examples
+=============
+
+The following program demonstrates the use of the one-dimensional
+wavelet transform functions.  It computes an approximation to an input
+signal (of length 256) using the 20 largest components of the wavelet
+transform, while setting the others to zero.
+
+     #include <stdio.h>
+     #include <math.h>
+     #include <gsl/gsl_sort.h>
+     #include <gsl/gsl_wavelet.h>
+
+     int
+     main (int argc, char **argv)
+     {
+       int i, n = 256, nc = 20;
+       double *data = malloc (n * sizeof (double));
+       double *abscoeff = malloc (n * sizeof (double));
+       size_t *p = malloc (n * sizeof (size_t));
+
+       FILE * f;
+       gsl_wavelet *w;
+       gsl_wavelet_workspace *work;
+
+       w = gsl_wavelet_alloc (gsl_wavelet_daubechies, 4);
+       work = gsl_wavelet_workspace_alloc (n);
+
+       f = fopen (argv[1], "r");
+       for (i = 0; i < n; i++)
+         {
+           fscanf (f, "%lg", &data[i]);
+         }
+       fclose (f);
+
+       gsl_wavelet_transform_forward (w, data, 1, n, work);
+
+       for (i = 0; i < n; i++)
+         {
+           abscoeff[i] = fabs (data[i]);
+         }
+
+       gsl_sort_index (p, abscoeff, 1, n);
+
+       for (i = 0; (i + nc) < n; i++)
+         data[p[i]] = 0;
+
+       gsl_wavelet_transform_inverse (w, data, 1, n, work);
+
+       for (i = 0; i < n; i++)
+         {
+           printf ("%g\n", data[i]);
+         }
+
+       gsl_wavelet_free (w);
+       gsl_wavelet_workspace_free (work);
+
+       free (data);
+       free (abscoeff);
+       free (p);
+       return 0;
+     }
+
+The output can be used with the GNU plotutils `graph' program,
+
+     $ ./a.out ecg.dat > dwt.dat
+     $ graph -T ps -x 0 256 32 -h 0.3 -a dwt.dat > dwt.ps
+
+
+File: gsl-ref.info,  Node: DWT References,  Prev: DWT Examples,  Up: Wavelet Transforms
+
+30.5 References and Further Reading
+===================================
+
+The mathematical background to wavelet transforms is covered in the
+original lectures by Daubechies,
+
+     Ingrid Daubechies.  Ten Lectures on Wavelets.  `CBMS-NSF Regional
+     Conference Series in Applied Mathematics' (1992), SIAM, ISBN
+     0898712742.
+
+An easy to read introduction to the subject with an emphasis on the
+application of the wavelet transform in various branches of science is,
+
+     Paul S. Addison. `The Illustrated Wavelet Transform Handbook'.
+     Institute of Physics Publishing (2002), ISBN 0750306920.
+
+For extensive coverage of signal analysis by wavelets, wavelet packets
+and local cosine bases see,
+
+     S. G. Mallat.  `A wavelet tour of signal processing' (Second
+     edition). Academic Press (1999), ISBN 012466606X.
+
+The concept of multiresolution analysis underlying the wavelet transform
+is described in,
+
+     S. G. Mallat.  Multiresolution Approximations and Wavelet
+     Orthonormal Bases of L^2(R).  `Transactions of the American
+     Mathematical Society', 315(1), 1989, 69-87.
+
+     S. G. Mallat.  A Theory for Multiresolution Signal
+     Decomposition--The Wavelet Representation.  `IEEE Transactions on
+     Pattern Analysis and Machine Intelligence', 11, 1989, 674-693.
+
+The coefficients for the individual wavelet families implemented by the
+library can be found in the following papers,
+
+     I. Daubechies.  Orthonormal Bases of Compactly Supported Wavelets.
+     `Communications on Pure and Applied Mathematics', 41 (1988)
+     909-996.
+
+     A. Cohen, I. Daubechies, and J.-C. Feauveau.  Biorthogonal Bases
+     of Compactly Supported Wavelets.  `Communications on Pure and
+     Applied Mathematics', 45 (1992) 485-560.
+
+The PhysioNet archive of physiological datasets can be found online at
+`http://www.physionet.org/' and is described in the following paper,
+
+     Goldberger et al.  PhysioBank, PhysioToolkit, and PhysioNet:
+     Components of a New Research Resource for Complex Physiologic
+     Signals.  `Circulation' 101(23):e215-e220 2000.
+
+
+File: gsl-ref.info,  Node: Discrete Hankel Transforms,  Next: One dimensional Root-Finding,  Prev: Wavelet Transforms,  Up: Top
+
+31 Discrete Hankel Transforms
+*****************************
+
+This chapter describes functions for performing Discrete Hankel
+Transforms (DHTs).  The functions are declared in the header file
+`gsl_dht.h'.
+
+* Menu:
+
+* Discrete Hankel Transform Definition::
+* Discrete Hankel Transform Functions::
+* Discrete Hankel Transform References::
+
+
+File: gsl-ref.info,  Node: Discrete Hankel Transform Definition,  Next: Discrete Hankel Transform Functions,  Up: Discrete Hankel Transforms
+
+31.1 Definitions
+================
+
+The discrete Hankel transform acts on a vector of sampled data, where
+the samples are assumed to have been taken at points related to the
+zeroes of a Bessel function of fixed order; compare this to the case of
+the discrete Fourier transform, where samples are taken at points
+related to the zeroes of the sine or cosine function.
+
+   Specifically, let f(t) be a function on the unit interval.  Then the
+finite \nu-Hankel transform of f(t) is defined to be the set of numbers
+g_m given by,
+     g_m = \int_0^1 t dt J_\nu(j_(\nu,m)t) f(t),
+
+so that,
+     f(t) = \sum_{m=1}^\infty (2 J_\nu(j_(\nu,m)x) / J_(\nu+1)(j_(\nu,m))^2) g_m.
+
+Suppose that f is band-limited in the sense that g_m=0 for m > M. Then
+we have the following fundamental sampling theorem.
+     g_m = (2 / j_(\nu,M)^2)
+           \sum_{k=1}^{M-1} f(j_(\nu,k)/j_(\nu,M))
+               (J_\nu(j_(\nu,m) j_(\nu,k) / j_(\nu,M)) / J_(\nu+1)(j_(\nu,k))^2).
+
+It is this discrete expression which defines the discrete Hankel
+transform. The kernel in the summation above defines the matrix of the
+\nu-Hankel transform of size M-1.  The coefficients of this matrix,
+being dependent on \nu and M, must be precomputed and stored; the
+`gsl_dht' object encapsulates this data.  The allocation function
+`gsl_dht_alloc' returns a `gsl_dht' object which must be properly
+initialized with `gsl_dht_init' before it can be used to perform
+transforms on data sample vectors, for fixed \nu and M, using the
+`gsl_dht_apply' function. The implementation allows a scaling of the
+fundamental interval, for convenience, so that one can assume the
+function is defined on the interval [0,X], rather than the unit
+interval.
+
+   Notice that by assumption f(t) vanishes at the endpoints of the
+interval, consistent with the inversion formula and the sampling
+formula given above. Therefore, this transform corresponds to an
+orthogonal expansion in eigenfunctions of the Dirichlet problem for the
+Bessel differential equation.
+
+
+File: gsl-ref.info,  Node: Discrete Hankel Transform Functions,  Next: Discrete Hankel Transform References,  Prev: Discrete Hankel Transform Definition,  Up: Discrete Hankel Transforms
+
+31.2 Functions
+==============
+
+ -- Function: gsl_dht * gsl_dht_alloc (size_t SIZE)
+     This function allocates a Discrete Hankel transform object of size
+     SIZE.
+
+ -- Function: int gsl_dht_init (gsl_dht * T, double NU, double XMAX)
+     This function initializes the transform T for the given values of
+     NU and X.
+
+ -- Function: gsl_dht * gsl_dht_new (size_t SIZE, double NU, double
+          XMAX)
+     This function allocates a Discrete Hankel transform object of size
+     SIZE and initializes it for the given values of NU and X.
+
+ -- Function: void gsl_dht_free (gsl_dht * T)
+     This function frees the transform T.
+
+ -- Function: int gsl_dht_apply (const gsl_dht * T, double * F_IN,
+          double * F_OUT)
+     This function applies the transform T to the array F_IN whose size
+     is equal to the size of the transform.  The result is stored in
+     the array F_OUT which must be of the same length.
+
+ -- Function: double gsl_dht_x_sample (const gsl_dht * T, int N)
+     This function returns the value of the N-th sample point in the
+     unit interval, (j_{\nu,n+1}/j_{\nu,M}) X. These are the points
+     where the function f(t) is assumed to be sampled.
+
+ -- Function: double gsl_dht_k_sample (const gsl_dht * T, int N)
+     This function returns the value of the N-th sample point in
+     "k-space", j_{\nu,n+1}/X.
+
+
+File: gsl-ref.info,  Node: Discrete Hankel Transform References,  Prev: Discrete Hankel Transform Functions,  Up: Discrete Hankel Transforms
+
+31.3 References and Further Reading
+===================================
+
+The algorithms used by these functions are described in the following
+papers,
+
+     H. Fisk Johnson, Comp. Phys. Comm. 43, 181 (1987).
+
+     D. Lemoine, J. Chem. Phys. 101, 3936 (1994).
+
+
+File: gsl-ref.info,  Node: One dimensional Root-Finding,  Next: One dimensional Minimization,  Prev: Discrete Hankel Transforms,  Up: Top
+
+32 One dimensional Root-Finding
+*******************************
+
+This chapter describes routines for finding roots of arbitrary
+one-dimensional functions.  The library provides low level components
+for a variety of iterative solvers and convergence tests.  These can be
+combined by the user to achieve the desired solution, with full access
+to the intermediate steps of the iteration.  Each class of methods uses
+the same framework, so that you can switch between solvers at runtime
+without needing to recompile your program.  Each instance of a solver
+keeps track of its own state, allowing the solvers to be used in
+multi-threaded programs.
+
+   The header file `gsl_roots.h' contains prototypes for the root
+finding functions and related declarations.
+
+* Menu:
+
+* Root Finding Overview::
+* Root Finding Caveats::
+* Initializing the Solver::
+* Providing the function to solve::
+* Search Bounds and Guesses::
+* Root Finding Iteration::
+* Search Stopping Parameters::
+* Root Bracketing Algorithms::
+* Root Finding Algorithms using Derivatives::
+* Root Finding Examples::
+* Root Finding References and Further Reading::
+
+
+File: gsl-ref.info,  Node: Root Finding Overview,  Next: Root Finding Caveats,  Up: One dimensional Root-Finding
+
+32.1 Overview
+=============
+
+One-dimensional root finding algorithms can be divided into two classes,
+"root bracketing" and "root polishing".  Algorithms which proceed by
+bracketing a root are guaranteed to converge.  Bracketing algorithms
+begin with a bounded region known to contain a root.  The size of this
+bounded region is reduced, iteratively, until it encloses the root to a
+desired tolerance.  This provides a rigorous error estimate for the
+location of the root.
+
+   The technique of "root polishing" attempts to improve an initial
+guess to the root.  These algorithms converge only if started "close
+enough" to a root, and sacrifice a rigorous error bound for speed.  By
+approximating the behavior of a function in the vicinity of a root they
+attempt to find a higher order improvement of an initial guess.  When
+the behavior of the function is compatible with the algorithm and a good
+initial guess is available a polishing algorithm can provide rapid
+convergence.
+
+   In GSL both types of algorithm are available in similar frameworks.
+The user provides a high-level driver for the algorithms, and the
+library provides the individual functions necessary for each of the
+steps.  There are three main phases of the iteration.  The steps are,
+
+   * initialize solver state, S, for algorithm T
+
+   * update S using the iteration T
+
+   * test S for convergence, and repeat iteration if necessary
+
+The state for bracketing solvers is held in a `gsl_root_fsolver'
+struct.  The updating procedure uses only function evaluations (not
+derivatives).  The state for root polishing solvers is held in a
+`gsl_root_fdfsolver' struct.  The updates require both the function and
+its derivative (hence the name `fdf') to be supplied by the user.
+
+
+File: gsl-ref.info,  Node: Root Finding Caveats,  Next: Initializing the Solver,  Prev: Root Finding Overview,  Up: One dimensional Root-Finding
+
+32.2 Caveats
+============
+
+Note that root finding functions can only search for one root at a time.
+When there are several roots in the search area, the first root to be
+found will be returned; however it is difficult to predict which of the
+roots this will be. _In most cases, no error will be reported if you
+try to find a root in an area where there is more than one._
+
+   Care must be taken when a function may have a multiple root (such as
+f(x) = (x-x_0)^2 or f(x) = (x-x_0)^3).  It is not possible to use
+root-bracketing algorithms on even-multiplicity roots.  For these
+algorithms the initial interval must contain a zero-crossing, where the
+function is negative at one end of the interval and positive at the
+other end.  Roots with even-multiplicity do not cross zero, but only
+touch it instantaneously.  Algorithms based on root bracketing will
+still work for odd-multiplicity roots (e.g. cubic, quintic, ...).  Root
+polishing algorithms generally work with higher multiplicity roots, but
+at a reduced rate of convergence.  In these cases the "Steffenson
+algorithm" can be used to accelerate the convergence of multiple roots.
+
+   While it is not absolutely required that f have a root within the
+search region, numerical root finding functions should not be used
+haphazardly to check for the _existence_ of roots.  There are better
+ways to do this.  Because it is easy to create situations where
+numerical root finders can fail, it is a bad idea to throw a root
+finder at a function you do not know much about.  In general it is best
+to examine the function visually by plotting before searching for a
+root.
+
+
+File: gsl-ref.info,  Node: Initializing the Solver,  Next: Providing the function to solve,  Prev: Root Finding Caveats,  Up: One dimensional Root-Finding
+
+32.3 Initializing the Solver
+============================
+
+ -- Function: gsl_root_fsolver * gsl_root_fsolver_alloc (const
+          gsl_root_fsolver_type * T)
+     This function returns a pointer to a newly allocated instance of a
+     solver of type T.  For example, the following code creates an
+     instance of a bisection solver,
+
+          const gsl_root_fsolver_type * T
+            = gsl_root_fsolver_bisection;
+          gsl_root_fsolver * s
+            = gsl_root_fsolver_alloc (T);
+
+     If there is insufficient memory to create the solver then the
+     function returns a null pointer and the error handler is invoked
+     with an error code of `GSL_ENOMEM'.
+
+ -- Function: gsl_root_fdfsolver * gsl_root_fdfsolver_alloc (const
+          gsl_root_fdfsolver_type * T)
+     This function returns a pointer to a newly allocated instance of a
+     derivative-based solver of type T.  For example, the following
+     code creates an instance of a Newton-Raphson solver,
+
+          const gsl_root_fdfsolver_type * T
+            = gsl_root_fdfsolver_newton;
+          gsl_root_fdfsolver * s
+            = gsl_root_fdfsolver_alloc (T);
+
+     If there is insufficient memory to create the solver then the
+     function returns a null pointer and the error handler is invoked
+     with an error code of `GSL_ENOMEM'.
+
+ -- Function: int gsl_root_fsolver_set (gsl_root_fsolver * S,
+          gsl_function * F, double X_LOWER, double X_UPPER)
+     This function initializes, or reinitializes, an existing solver S
+     to use the function F and the initial search interval [X_LOWER,
+     X_UPPER].
+
+ -- Function: int gsl_root_fdfsolver_set (gsl_root_fdfsolver * S,
+          gsl_function_fdf * FDF, double ROOT)
+     This function initializes, or reinitializes, an existing solver S
+     to use the function and derivative FDF and the initial guess ROOT.
+
+ -- Function: void gsl_root_fsolver_free (gsl_root_fsolver * S)
+ -- Function: void gsl_root_fdfsolver_free (gsl_root_fdfsolver * S)
+     These functions free all the memory associated with the solver S.
+
+ -- Function: const char * gsl_root_fsolver_name (const
+          gsl_root_fsolver * S)
+ -- Function: const char * gsl_root_fdfsolver_name (const
+          gsl_root_fdfsolver * S)
+     These functions return a pointer to the name of the solver.  For
+     example,
+
+          printf ("s is a '%s' solver\n",
+                  gsl_root_fsolver_name (s));
+
+     would print something like `s is a 'bisection' solver'.
+
+
+File: gsl-ref.info,  Node: Providing the function to solve,  Next: Search Bounds and Guesses,  Prev: Initializing the Solver,  Up: One dimensional Root-Finding
+
+32.4 Providing the function to solve
+====================================
+
+You must provide a continuous function of one variable for the root
+finders to operate on, and, sometimes, its first derivative.  In order
+to allow for general parameters the functions are defined by the
+following data types:
+
+ -- Data Type: gsl_function
+     This data type defines a general function with parameters.
+
+    `double (* function) (double X, void * PARAMS)'
+          this function should return the value f(x,params) for
+          argument X and parameters PARAMS
+
+    `void * params'
+          a pointer to the parameters of the function
+
+   Here is an example for the general quadratic function,
+
+     f(x) = a x^2 + b x + c
+
+with a = 3, b = 2, c = 1.  The following code defines a `gsl_function'
+`F' which you could pass to a root finder:
+
+     struct my_f_params { double a; double b; double c; };
+
+     double
+     my_f (double x, void * p) {
+        struct my_f_params * params
+          = (struct my_f_params *)p;
+        double a = (params->a);
+        double b = (params->b);
+        double c = (params->c);
+
+        return  (a * x + b) * x + c;
+     }
+
+     gsl_function F;
+     struct my_f_params params = { 3.0, 2.0, 1.0 };
+
+     F.function = &my_f;
+     F.params = &params;
+
+The function f(x) can be evaluated using the following macro,
+
+     #define GSL_FN_EVAL(F,x)
+         (*((F)->function))(x,(F)->params)
+
+ -- Data Type: gsl_function_fdf
+     This data type defines a general function with parameters and its
+     first derivative.
+
+    `double (* f) (double X, void * PARAMS)'
+          this function should return the value of f(x,params) for
+          argument X and parameters PARAMS
+
+    `double (* df) (double X, void * PARAMS)'
+          this function should return the value of the derivative of F
+          with respect to X, f'(x,params), for argument X and
+          parameters PARAMS
+
+    `void (* fdf) (double X, void * PARAMS, double * F, double * Df)'
+          this function should set the values of the function F to
+          f(x,params) and its derivative DF to f'(x,params) for
+          argument X and parameters PARAMS.  This function provides an
+          optimization of the separate functions for f(x) and f'(x)--it
+          is always faster to compute the function and its derivative
+          at the same time.
+
+    `void * params'
+          a pointer to the parameters of the function
+
+   Here is an example where f(x) = 2\exp(2x):
+
+     double
+     my_f (double x, void * params)
+     {
+        return exp (2 * x);
+     }
+
+     double
+     my_df (double x, void * params)
+     {
+        return 2 * exp (2 * x);
+     }
+
+     void
+     my_fdf (double x, void * params,
+             double * f, double * df)
+     {
+        double t = exp (2 * x);
+
+        *f = t;
+        *df = 2 * t;   /* uses existing value */
+     }
+
+     gsl_function_fdf FDF;
+
+     FDF.f = &my_f;
+     FDF.df = &my_df;
+     FDF.fdf = &my_fdf;
+     FDF.params = 0;
+
+The function f(x) can be evaluated using the following macro,
+
+     #define GSL_FN_FDF_EVAL_F(FDF,x)
+          (*((FDF)->f))(x,(FDF)->params)
+
+The derivative f'(x) can be evaluated using the following macro,
+
+     #define GSL_FN_FDF_EVAL_DF(FDF,x)
+          (*((FDF)->df))(x,(FDF)->params)
+
+and both the function y = f(x) and its derivative dy = f'(x) can be
+evaluated at the same time using the following macro,
+
+     #define GSL_FN_FDF_EVAL_F_DF(FDF,x,y,dy)
+          (*((FDF)->fdf))(x,(FDF)->params,(y),(dy))
+
+The macro stores f(x) in its Y argument and f'(x) in its DY
+argument--both of these should be pointers to `double'.
+
+
+File: gsl-ref.info,  Node: Search Bounds and Guesses,  Next: Root Finding Iteration,  Prev: Providing the function to solve,  Up: One dimensional Root-Finding
+
+32.5 Search Bounds and Guesses
+==============================
+
+You provide either search bounds or an initial guess; this section
+explains how search bounds and guesses work and how function arguments
+control them.
+
+   A guess is simply an x value which is iterated until it is within
+the desired precision of a root.  It takes the form of a `double'.
+
+   Search bounds are the endpoints of a interval which is iterated until
+the length of the interval is smaller than the requested precision.  The
+interval is defined by two values, the lower limit and the upper limit.
+Whether the endpoints are intended to be included in the interval or not
+depends on the context in which the interval is used.
+
+
+File: gsl-ref.info,  Node: Root Finding Iteration,  Next: Search Stopping Parameters,  Prev: Search Bounds and Guesses,  Up: One dimensional Root-Finding
+
+32.6 Iteration
+==============
+
+The following functions drive the iteration of each algorithm.  Each
+function performs one iteration to update the state of any solver of the
+corresponding type.  The same functions work for all solvers so that
+different methods can be substituted at runtime without modifications to
+the code.
+
+ -- Function: int gsl_root_fsolver_iterate (gsl_root_fsolver * S)
+ -- Function: int gsl_root_fdfsolver_iterate (gsl_root_fdfsolver * S)
+     These functions perform a single iteration of the solver S.  If the
+     iteration encounters an unexpected problem then an error code will
+     be returned,
+
+    `GSL_EBADFUNC'
+          the iteration encountered a singular point where the function
+          or its derivative evaluated to `Inf' or `NaN'.
+
+    `GSL_EZERODIV'
+          the derivative of the function vanished at the iteration
+          point, preventing the algorithm from continuing without a
+          division by zero.
+
+   The solver maintains a current best estimate of the root at all
+times.  The bracketing solvers also keep track of the current best
+interval bounding the root.  This information can be accessed with the
+following auxiliary functions,
+
+ -- Function: double gsl_root_fsolver_root (const gsl_root_fsolver * S)
+ -- Function: double gsl_root_fdfsolver_root (const gsl_root_fdfsolver
+          * S)
+     These functions return the current estimate of the root for the
+     solver S.
+
+ -- Function: double gsl_root_fsolver_x_lower (const gsl_root_fsolver *
+          S)
+ -- Function: double gsl_root_fsolver_x_upper (const gsl_root_fsolver *
+          S)
+     These functions return the current bracketing interval for the
+     solver S.
+
+
+File: gsl-ref.info,  Node: Search Stopping Parameters,  Next: Root Bracketing Algorithms,  Prev: Root Finding Iteration,  Up: One dimensional Root-Finding
+
+32.7 Search Stopping Parameters
+===============================
+
+A root finding procedure should stop when one of the following
+conditions is true:
+
+   * A root has been found to within the user-specified precision.
+
+   * A user-specified maximum number of iterations has been reached.
+
+   * An error has occurred.
+
+The handling of these conditions is under user control.  The functions
+below allow the user to test the precision of the current result in
+several standard ways.
+
+ -- Function: int gsl_root_test_interval (double X_LOWER, double
+          X_UPPER, double EPSABS, double EPSREL)
+     This function tests for the convergence of the interval [X_LOWER,
+     X_UPPER] with absolute error EPSABS and relative error EPSREL.
+     The test returns `GSL_SUCCESS' if the following condition is
+     achieved,
+
+          |a - b| < epsabs + epsrel min(|a|,|b|)
+
+     when the interval x = [a,b] does not include the origin.  If the
+     interval includes the origin then \min(|a|,|b|) is replaced by
+     zero (which is the minimum value of |x| over the interval).  This
+     ensures that the relative error is accurately estimated for roots
+     close to the origin.
+
+     This condition on the interval also implies that any estimate of
+     the root r in the interval satisfies the same condition with
+     respect to the true root r^*,
+
+          |r - r^*| < epsabs + epsrel r^*
+
+     assuming that the true root r^* is contained within the interval.
+
+ -- Function: int gsl_root_test_delta (double X1, double X0, double
+          EPSABS, double EPSREL)
+     This function tests for the convergence of the sequence ..., X0,
+     X1 with absolute error EPSABS and relative error EPSREL.  The test
+     returns `GSL_SUCCESS' if the following condition is achieved,
+
+          |x_1 - x_0| < epsabs + epsrel |x_1|
+
+     and returns `GSL_CONTINUE' otherwise.
+
+ -- Function: int gsl_root_test_residual (double F, double EPSABS)
+     This function tests the residual value F against the absolute
+     error bound EPSABS.  The test returns `GSL_SUCCESS' if the
+     following condition is achieved,
+
+          |f| < epsabs
+
+     and returns `GSL_CONTINUE' otherwise.  This criterion is suitable
+     for situations where the precise location of the root, x, is
+     unimportant provided a value can be found where the residual,
+     |f(x)|, is small enough.
+
+
+File: gsl-ref.info,  Node: Root Bracketing Algorithms,  Next: Root Finding Algorithms using Derivatives,  Prev: Search Stopping Parameters,  Up: One dimensional Root-Finding
+
+32.8 Root Bracketing Algorithms
+===============================
+
+The root bracketing algorithms described in this section require an
+initial interval which is guaranteed to contain a root--if a and b are
+the endpoints of the interval then f(a) must differ in sign from f(b).
+This ensures that the function crosses zero at least once in the
+interval.  If a valid initial interval is used then these algorithm
+cannot fail, provided the function is well-behaved.
+
+   Note that a bracketing algorithm cannot find roots of even degree,
+since these do not cross the x-axis.
+
+ -- Solver: gsl_root_fsolver_bisection
+     The "bisection algorithm" is the simplest method of bracketing the
+     roots of a function.   It is the slowest algorithm provided by the
+     library, with linear convergence.
+
+     On each iteration, the interval is bisected and the value of the
+     function at the midpoint is calculated.  The sign of this value is
+     used to determine which half of the interval does not contain a
+     root.  That half is discarded to give a new, smaller interval
+     containing the root.  This procedure can be continued indefinitely
+     until the interval is sufficiently small.
+
+     At any time the current estimate of the root is taken as the
+     midpoint of the interval.
+
+
+ -- Solver: gsl_root_fsolver_falsepos
+     The "false position algorithm" is a method of finding roots based
+     on linear interpolation.  Its convergence is linear, but it is
+     usually faster than bisection.
+
+     On each iteration a line is drawn between the endpoints (a,f(a))
+     and (b,f(b)) and the point where this line crosses the x-axis
+     taken as a "midpoint".  The value of the function at this point is
+     calculated and its sign is used to determine which side of the
+     interval does not contain a root.  That side is discarded to give a
+     new, smaller interval containing the root.  This procedure can be
+     continued indefinitely until the interval is sufficiently small.
+
+     The best estimate of the root is taken from the linear
+     interpolation of the interval on the current iteration.
+
+
+ -- Solver: gsl_root_fsolver_brent
+     The "Brent-Dekker method" (referred to here as "Brent's method")
+     combines an interpolation strategy with the bisection algorithm.
+     This produces a fast algorithm which is still robust.
+
+     On each iteration Brent's method approximates the function using an
+     interpolating curve.  On the first iteration this is a linear
+     interpolation of the two endpoints.  For subsequent iterations the
+     algorithm uses an inverse quadratic fit to the last three points,
+     for higher accuracy.  The intercept of the interpolating curve
+     with the x-axis is taken as a guess for the root.  If it lies
+     within the bounds of the current interval then the interpolating
+     point is accepted, and used to generate a smaller interval.  If
+     the interpolating point is not accepted then the algorithm falls
+     back to an ordinary bisection step.
+
+     The best estimate of the root is taken from the most recent
+     interpolation or bisection.
+
+
+File: gsl-ref.info,  Node: Root Finding Algorithms using Derivatives,  Next: Root Finding Examples,  Prev: Root Bracketing Algorithms,  Up: One dimensional Root-Finding
+
+32.9 Root Finding Algorithms using Derivatives
+==============================================
+
+The root polishing algorithms described in this section require an
+initial guess for the location of the root.  There is no absolute
+guarantee of convergence--the function must be suitable for this
+technique and the initial guess must be sufficiently close to the root
+for it to work.  When these conditions are satisfied then convergence is
+quadratic.
+
+   These algorithms make use of both the function and its derivative.
+
+ -- Derivative Solver: gsl_root_fdfsolver_newton
+     Newton's Method is the standard root-polishing algorithm.  The
+     algorithm begins with an initial guess for the location of the
+     root.  On each iteration, a line tangent to the function f is
+     drawn at that position.  The point where this line crosses the
+     x-axis becomes the new guess.  The iteration is defined by the
+     following sequence,
+
+          x_{i+1} = x_i - f(x_i)/f'(x_i)
+
+     Newton's method converges quadratically for single roots, and
+     linearly for multiple roots.
+
+
+ -- Derivative Solver: gsl_root_fdfsolver_secant
+     The "secant method" is a simplified version of Newton's method
+     which does not require the computation of the derivative on every
+     step.
+
+     On its first iteration the algorithm begins with Newton's method,
+     using the derivative to compute a first step,
+
+          x_1 = x_0 - f(x_0)/f'(x_0)
+
+     Subsequent iterations avoid the evaluation of the derivative by
+     replacing it with a numerical estimate, the slope of the line
+     through the previous two points,
+
+          x_{i+1} = x_i f(x_i) / f'_{est} where
+           f'_{est} = (f(x_i) - f(x_{i-1})/(x_i - x_{i-1})
+
+     When the derivative does not change significantly in the vicinity
+     of the root the secant method gives a useful saving.
+     Asymptotically the secant method is faster than Newton's method
+     whenever the cost of evaluating the derivative is more than 0.44
+     times the cost of evaluating the function itself.  As with all
+     methods of computing a numerical derivative the estimate can
+     suffer from cancellation errors if the separation of the points
+     becomes too small.
+
+     On single roots, the method has a convergence of order (1 + \sqrt
+     5)/2 (approximately 1.62).  It converges linearly for multiple
+     roots.
+
+
+ -- Derivative Solver: gsl_root_fdfsolver_steffenson
+     The "Steffenson Method" provides the fastest convergence of all the
+     routines.  It combines the basic Newton algorithm with an Aitken
+     "delta-squared" acceleration.  If the Newton iterates are x_i then
+     the acceleration procedure generates a new sequence R_i,
+
+          R_i = x_i - (x_{i+1} - x_i)^2 / (x_{i+2} - 2 x_{i+1} + x_{i})
+
+     which converges faster than the original sequence under reasonable
+     conditions.  The new sequence requires three terms before it can
+     produce its first value so the method returns accelerated values
+     on the second and subsequent iterations.  On the first iteration
+     it returns the ordinary Newton estimate.  The Newton iterate is
+     also returned if the denominator of the acceleration term ever
+     becomes zero.
+
+     As with all acceleration procedures this method can become
+     unstable if the function is not well-behaved.
+
+
+File: gsl-ref.info,  Node: Root Finding Examples,  Next: Root Finding References and Further Reading,  Prev: Root Finding Algorithms using Derivatives,  Up: One dimensional Root-Finding
+
+32.10 Examples
+==============
+
+For any root finding algorithm we need to prepare the function to be
+solved.  For this example we will use the general quadratic equation
+described earlier.  We first need a header file (`demo_fn.h') to define
+the function parameters,
+
+     struct quadratic_params
+       {
+         double a, b, c;
+       };
+
+     double quadratic (double x, void *params);
+     double quadratic_deriv (double x, void *params);
+     void quadratic_fdf (double x, void *params,
+                         double *y, double *dy);
+
+We place the function definitions in a separate file (`demo_fn.c'),
+
+     double
+     quadratic (double x, void *params)
+     {
+       struct quadratic_params *p
+         = (struct quadratic_params *) params;
+
+       double a = p->a;
+       double b = p->b;
+       double c = p->c;
+
+       return (a * x + b) * x + c;
+     }
+
+     double
+     quadratic_deriv (double x, void *params)
+     {
+       struct quadratic_params *p
+         = (struct quadratic_params *) params;
+
+       double a = p->a;
+       double b = p->b;
+       double c = p->c;
+
+       return 2.0 * a * x + b;
+     }
+
+     void
+     quadratic_fdf (double x, void *params,
+                    double *y, double *dy)
+     {
+       struct quadratic_params *p
+         = (struct quadratic_params *) params;
+
+       double a = p->a;
+       double b = p->b;
+       double c = p->c;
+
+       *y = (a * x + b) * x + c;
+       *dy = 2.0 * a * x + b;
+     }
+
+The first program uses the function solver `gsl_root_fsolver_brent' for
+Brent's method and the general quadratic defined above to solve the
+following equation,
+
+     x^2 - 5 = 0
+
+with solution x = \sqrt 5 = 2.236068...
+
+     #include <stdio.h>
+     #include <gsl/gsl_errno.h>
+     #include <gsl/gsl_math.h>
+     #include <gsl/gsl_roots.h>
+
+     #include "demo_fn.h"
+     #include "demo_fn.c"
+
+     int
+     main (void)
+     {
+       int status;
+       int iter = 0, max_iter = 100;
+       const gsl_root_fsolver_type *T;
+       gsl_root_fsolver *s;
+       double r = 0, r_expected = sqrt (5.0);
+       double x_lo = 0.0, x_hi = 5.0;
+       gsl_function F;
+       struct quadratic_params params = {1.0, 0.0, -5.0};
+
+       F.function = &quadratic;
+       F.params = &params;
+
+       T = gsl_root_fsolver_brent;
+       s = gsl_root_fsolver_alloc (T);
+       gsl_root_fsolver_set (s, &F, x_lo, x_hi);
+
+       printf ("using %s method\n",
+               gsl_root_fsolver_name (s));
+
+       printf ("%5s [%9s, %9s] %9s %10s %9s\n",
+               "iter", "lower", "upper", "root",
+               "err", "err(est)");
+
+       do
+         {
+           iter++;
+           status = gsl_root_fsolver_iterate (s);
+           r = gsl_root_fsolver_root (s);
+           x_lo = gsl_root_fsolver_x_lower (s);
+           x_hi = gsl_root_fsolver_x_upper (s);
+           status = gsl_root_test_interval (x_lo, x_hi,
+                                            0, 0.001);
+
+           if (status == GSL_SUCCESS)
+             printf ("Converged:\n");
+
+           printf ("%5d [%.7f, %.7f] %.7f %+.7f %.7f\n",
+                   iter, x_lo, x_hi,
+                   r, r - r_expected,
+                   x_hi - x_lo);
+         }
+       while (status == GSL_CONTINUE && iter < max_iter);
+
+       gsl_root_fsolver_free (s);
+
+       return status;
+     }
+
+Here are the results of the iterations,
+
+     $ ./a.out
+     using brent method
+      iter [    lower,     upper]      root        err  err(est)
+         1 [1.0000000, 5.0000000] 1.0000000 -1.2360680 4.0000000
+         2 [1.0000000, 3.0000000] 3.0000000 +0.7639320 2.0000000
+         3 [2.0000000, 3.0000000] 2.0000000 -0.2360680 1.0000000
+         4 [2.2000000, 3.0000000] 2.2000000 -0.0360680 0.8000000
+         5 [2.2000000, 2.2366300] 2.2366300 +0.0005621 0.0366300
+     Converged:
+         6 [2.2360634, 2.2366300] 2.2360634 -0.0000046 0.0005666
+
+If the program is modified to use the bisection solver instead of
+Brent's method, by changing `gsl_root_fsolver_brent' to
+`gsl_root_fsolver_bisection' the slower convergence of the Bisection
+method can be observed,
+
+     $ ./a.out
+     using bisection method
+      iter [    lower,     upper]      root        err  err(est)
+         1 [0.0000000, 2.5000000] 1.2500000 -0.9860680 2.5000000
+         2 [1.2500000, 2.5000000] 1.8750000 -0.3610680 1.2500000
+         3 [1.8750000, 2.5000000] 2.1875000 -0.0485680 0.6250000
+         4 [2.1875000, 2.5000000] 2.3437500 +0.1076820 0.3125000
+         5 [2.1875000, 2.3437500] 2.2656250 +0.0295570 0.1562500
+         6 [2.1875000, 2.2656250] 2.2265625 -0.0095055 0.0781250
+         7 [2.2265625, 2.2656250] 2.2460938 +0.0100258 0.0390625
+         8 [2.2265625, 2.2460938] 2.2363281 +0.0002601 0.0195312
+         9 [2.2265625, 2.2363281] 2.2314453 -0.0046227 0.0097656
+        10 [2.2314453, 2.2363281] 2.2338867 -0.0021813 0.0048828
+        11 [2.2338867, 2.2363281] 2.2351074 -0.0009606 0.0024414
+     Converged:
+        12 [2.2351074, 2.2363281] 2.2357178 -0.0003502 0.0012207
+
+   The next program solves the same function using a derivative solver
+instead.
+
+     #include <stdio.h>
+     #include <gsl/gsl_errno.h>
+     #include <gsl/gsl_math.h>
+     #include <gsl/gsl_roots.h>
+
+     #include "demo_fn.h"
+     #include "demo_fn.c"
+
+     int
+     main (void)
+     {
+       int status;
+       int iter = 0, max_iter = 100;
+       const gsl_root_fdfsolver_type *T;
+       gsl_root_fdfsolver *s;
+       double x0, x = 5.0, r_expected = sqrt (5.0);
+       gsl_function_fdf FDF;
+       struct quadratic_params params = {1.0, 0.0, -5.0};
+
+       FDF.f = &quadratic;
+       FDF.df = &quadratic_deriv;
+       FDF.fdf = &quadratic_fdf;
+       FDF.params = &params;
+
+       T = gsl_root_fdfsolver_newton;
+       s = gsl_root_fdfsolver_alloc (T);
+       gsl_root_fdfsolver_set (s, &FDF, x);
+
+       printf ("using %s method\n",
+               gsl_root_fdfsolver_name (s));
+
+       printf ("%-5s %10s %10s %10s\n",
+               "iter", "root", "err", "err(est)");
+       do
+         {
+           iter++;
+           status = gsl_root_fdfsolver_iterate (s);
+           x0 = x;
+           x = gsl_root_fdfsolver_root (s);
+           status = gsl_root_test_delta (x, x0, 0, 1e-3);
+
+           if (status == GSL_SUCCESS)
+             printf ("Converged:\n");
+
+           printf ("%5d %10.7f %+10.7f %10.7f\n",
+                   iter, x, x - r_expected, x - x0);
+         }
+       while (status == GSL_CONTINUE && iter < max_iter);
+
+       gsl_root_fdfsolver_free (s);
+       return status;
+     }
+
+Here are the results for Newton's method,
+
+     $ ./a.out
+     using newton method
+     iter        root        err   err(est)
+         1  3.0000000 +0.7639320 -2.0000000
+         2  2.3333333 +0.0972654 -0.6666667
+         3  2.2380952 +0.0020273 -0.0952381
+     Converged:
+         4  2.2360689 +0.0000009 -0.0020263
+
+Note that the error can be estimated more accurately by taking the
+difference between the current iterate and next iterate rather than the
+previous iterate.  The other derivative solvers can be investigated by
+changing `gsl_root_fdfsolver_newton' to `gsl_root_fdfsolver_secant' or
+`gsl_root_fdfsolver_steffenson'.
+
+
+File: gsl-ref.info,  Node: Root Finding References and Further Reading,  Prev: Root Finding Examples,  Up: One dimensional Root-Finding
+
+32.11 References and Further Reading
+====================================
+
+For information on the Brent-Dekker algorithm see the following two
+papers,
+
+     R. P. Brent, "An algorithm with guaranteed convergence for finding
+     a zero of a function", `Computer Journal', 14 (1971) 422-425
+
+     J. C. P. Bus and T. J. Dekker, "Two Efficient Algorithms with
+     Guaranteed Convergence for Finding a Zero of a Function", `ACM
+     Transactions of Mathematical Software', Vol. 1 No. 4 (1975) 330-345
+
+
+File: gsl-ref.info,  Node: One dimensional Minimization,  Next: Multidimensional Root-Finding,  Prev: One dimensional Root-Finding,  Up: Top
+
+33 One dimensional Minimization
+*******************************
+
+This chapter describes routines for finding minima of arbitrary
+one-dimensional functions.  The library provides low level components
+for a variety of iterative minimizers and convergence tests.  These can
+be combined by the user to achieve the desired solution, with full
+access to the intermediate steps of the algorithms.  Each class of
+methods uses the same framework, so that you can switch between
+minimizers at runtime without needing to recompile your program.  Each
+instance of a minimizer keeps track of its own state, allowing the
+minimizers to be used in multi-threaded programs.
+
+   The header file `gsl_min.h' contains prototypes for the minimization
+functions and related declarations.  To use the minimization algorithms
+to find the maximum of a function simply invert its sign.
+
+* Menu:
+
+* Minimization Overview::
+* Minimization Caveats::
+* Initializing the Minimizer::
+* Providing the function to minimize::
+* Minimization Iteration::
+* Minimization Stopping Parameters::
+* Minimization Algorithms::
+* Minimization Examples::
+* Minimization References and Further Reading::
+
+
+File: gsl-ref.info,  Node: Minimization Overview,  Next: Minimization Caveats,  Up: One dimensional Minimization
+
+33.1 Overview
+=============
+
+The minimization algorithms begin with a bounded region known to contain
+a minimum.  The region is described by a lower bound a and an upper
+bound b, with an estimate of the location of the minimum x.
+
+The value of the function at x must be less than the value of the
+function at the ends of the interval,
+
+     f(a) > f(x) < f(b)
+
+This condition guarantees that a minimum is contained somewhere within
+the interval.  On each iteration a new point x' is selected using one
+of the available algorithms.  If the new point is a better estimate of
+the minimum, i.e. where f(x') < f(x), then the current estimate of the
+minimum x is updated.  The new point also allows the size of the
+bounded interval to be reduced, by choosing the most compact set of
+points which satisfies the constraint f(a) > f(x) < f(b).  The interval
+is reduced until it encloses the true minimum to a desired tolerance.
+This provides a best estimate of the location of the minimum and a
+rigorous error estimate.
+
+   Several bracketing algorithms are available within a single
+framework.  The user provides a high-level driver for the algorithm,
+and the library provides the individual functions necessary for each of
+the steps.  There are three main phases of the iteration.  The steps
+are,
+
+   * initialize minimizer state, S, for algorithm T
+
+   * update S using the iteration T
+
+   * test S for convergence, and repeat iteration if necessary
+
+The state for the minimizers is held in a `gsl_min_fminimizer' struct.
+The updating procedure uses only function evaluations (not derivatives).
+
+
+File: gsl-ref.info,  Node: Minimization Caveats,  Next: Initializing the Minimizer,  Prev: Minimization Overview,  Up: One dimensional Minimization
+
+33.2 Caveats
+============
+
+Note that minimization functions can only search for one minimum at a
+time.  When there are several minima in the search area, the first
+minimum to be found will be returned; however it is difficult to predict
+which of the minima this will be. _In most cases, no error will be
+reported if you try to find a minimum in an area where there is more
+than one._
+
+   With all minimization algorithms it can be difficult to determine the
+location of the minimum to full numerical precision.  The behavior of
+the function in the region of the minimum x^* can be approximated by a
+Taylor expansion,
+
+     y = f(x^*) + (1/2) f''(x^*) (x - x^*)^2
+
+and the second term of this expansion can be lost when added to the
+first term at finite precision.  This magnifies the error in locating
+x^*, making it proportional to \sqrt \epsilon (where \epsilon is the
+relative accuracy of the floating point numbers).  For functions with
+higher order minima, such as x^4, the magnification of the error is
+correspondingly worse.  The best that can be achieved is to converge to
+the limit of numerical accuracy in the function values, rather than the
+location of the minimum itself.
+
+
+File: gsl-ref.info,  Node: Initializing the Minimizer,  Next: Providing the function to minimize,  Prev: Minimization Caveats,  Up: One dimensional Minimization
+
+33.3 Initializing the Minimizer
+===============================
+
+ -- Function: gsl_min_fminimizer * gsl_min_fminimizer_alloc (const
+          gsl_min_fminimizer_type * T)
+     This function returns a pointer to a newly allocated instance of a
+     minimizer of type T.  For example, the following code creates an
+     instance of a golden section minimizer,
+
+          const gsl_min_fminimizer_type * T
+            = gsl_min_fminimizer_goldensection;
+          gsl_min_fminimizer * s
+            = gsl_min_fminimizer_alloc (T);
+
+     If there is insufficient memory to create the minimizer then the
+     function returns a null pointer and the error handler is invoked
+     with an error code of `GSL_ENOMEM'.
+
+ -- Function: int gsl_min_fminimizer_set (gsl_min_fminimizer * S,
+          gsl_function * F, double X_MINIMUM, double X_LOWER, double
+          X_UPPER)
+     This function sets, or resets, an existing minimizer S to use the
+     function F and the initial search interval [X_LOWER, X_UPPER],
+     with a guess for the location of the minimum X_MINIMUM.
+
+     If the interval given does not contain a minimum, then the function
+     returns an error code of `GSL_EINVAL'.
+
+ -- Function: int gsl_min_fminimizer_set_with_values
+          (gsl_min_fminimizer * S, gsl_function * F, double X_MINIMUM,
+          double F_MINIMUM, double X_LOWER, double F_LOWER, double
+          X_UPPER, double F_UPPER)
+     This function is equivalent to `gsl_min_fminimizer_set' but uses
+     the values F_MINIMUM, F_LOWER and F_UPPER instead of computing
+     `f(x_minimum)', `f(x_lower)' and `f(x_upper)'.
+
+ -- Function: void gsl_min_fminimizer_free (gsl_min_fminimizer * S)
+     This function frees all the memory associated with the minimizer S.
+
+ -- Function: const char * gsl_min_fminimizer_name (const
+          gsl_min_fminimizer * S)
+     This function returns a pointer to the name of the minimizer.  For
+     example,
+
+          printf ("s is a '%s' minimizer\n",
+                  gsl_min_fminimizer_name (s));
+
+     would print something like `s is a 'brent' minimizer'.
+
+
+File: gsl-ref.info,  Node: Providing the function to minimize,  Next: Minimization Iteration,  Prev: Initializing the Minimizer,  Up: One dimensional Minimization
+
+33.4 Providing the function to minimize
+=======================================
+
+You must provide a continuous function of one variable for the
+minimizers to operate on.  In order to allow for general parameters the
+functions are defined by a `gsl_function' data type (*note Providing
+the function to solve::).
+
+
+File: gsl-ref.info,  Node: Minimization Iteration,  Next: Minimization Stopping Parameters,  Prev: Providing the function to minimize,  Up: One dimensional Minimization
+
+33.5 Iteration
+==============
+
+The following functions drive the iteration of each algorithm.  Each
+function performs one iteration to update the state of any minimizer of
+the corresponding type.  The same functions work for all minimizers so
+that different methods can be substituted at runtime without
+modifications to the code.
+
+ -- Function: int gsl_min_fminimizer_iterate (gsl_min_fminimizer * S)
+     This function performs a single iteration of the minimizer S.  If
+     the iteration encounters an unexpected problem then an error code
+     will be returned,
+
+    `GSL_EBADFUNC'
+          the iteration encountered a singular point where the function
+          evaluated to `Inf' or `NaN'.
+
+    `GSL_FAILURE'
+          the algorithm could not improve the current best
+          approximation or bounding interval.
+
+   The minimizer maintains a current best estimate of the position of
+the minimum at all times, and the current interval bounding the minimum.
+This information can be accessed with the following auxiliary functions,
+
+ -- Function: double gsl_min_fminimizer_x_minimum (const
+          gsl_min_fminimizer * S)
+     This function returns the current estimate of the position of the
+     minimum for the minimizer S.
+
+ -- Function: double gsl_min_fminimizer_x_upper (const
+          gsl_min_fminimizer * S)
+ -- Function: double gsl_min_fminimizer_x_lower (const
+          gsl_min_fminimizer * S)
+     These functions return the current upper and lower bound of the
+     interval for the minimizer S.
+
+ -- Function: double gsl_min_fminimizer_f_minimum (const
+          gsl_min_fminimizer * S)
+ -- Function: double gsl_min_fminimizer_f_upper (const
+          gsl_min_fminimizer * S)
+ -- Function: double gsl_min_fminimizer_f_lower (const
+          gsl_min_fminimizer * S)
+     These functions return the value of the function at the current
+     estimate of the minimum and at the upper and lower bounds of the
+     interval for the minimizer S.
+
+
+File: gsl-ref.info,  Node: Minimization Stopping Parameters,  Next: Minimization Algorithms,  Prev: Minimization Iteration,  Up: One dimensional Minimization
+
+33.6 Stopping Parameters
+========================
+
+A minimization procedure should stop when one of the following
+conditions is true:
+
+   * A minimum has been found to within the user-specified precision.
+
+   * A user-specified maximum number of iterations has been reached.
+
+   * An error has occurred.
+
+The handling of these conditions is under user control.  The function
+below allows the user to test the precision of the current result.
+
+ -- Function: int gsl_min_test_interval (double X_LOWER, double
+          X_UPPER, double EPSABS, double EPSREL)
+     This function tests for the convergence of the interval [X_LOWER,
+     X_UPPER] with absolute error EPSABS and relative error EPSREL.
+     The test returns `GSL_SUCCESS' if the following condition is
+     achieved,
+
+          |a - b| < epsabs + epsrel min(|a|,|b|)
+
+     when the interval x = [a,b] does not include the origin.  If the
+     interval includes the origin then \min(|a|,|b|) is replaced by
+     zero (which is the minimum value of |x| over the interval).  This
+     ensures that the relative error is accurately estimated for minima
+     close to the origin.
+
+     This condition on the interval also implies that any estimate of
+     the minimum x_m in the interval satisfies the same condition with
+     respect to the true minimum x_m^*,
+
+          |x_m - x_m^*| < epsabs + epsrel x_m^*
+
+     assuming that the true minimum x_m^* is contained within the
+     interval.
+
+
+File: gsl-ref.info,  Node: Minimization Algorithms,  Next: Minimization Examples,  Prev: Minimization Stopping Parameters,  Up: One dimensional Minimization
+
+33.7 Minimization Algorithms
+============================
+
+The minimization algorithms described in this section require an initial
+interval which is guaranteed to contain a minimum--if a and b are the
+endpoints of the interval and x is an estimate of the minimum then f(a)
+> f(x) < f(b).  This ensures that the function has at least one minimum
+somewhere in the interval.  If a valid initial interval is used then
+these algorithm cannot fail, provided the function is well-behaved.
+
+ -- Minimizer: gsl_min_fminimizer_goldensection
+     The "golden section algorithm" is the simplest method of bracketing
+     the minimum of a function.  It is the slowest algorithm provided
+     by the library, with linear convergence.
+
+     On each iteration, the algorithm first compares the subintervals
+     from the endpoints to the current minimum.  The larger subinterval
+     is divided in a golden section (using the famous ratio (3-\sqrt
+     5)/2 = 0.3189660...) and the value of the function at this new
+     point is calculated.  The new value is used with the constraint
+     f(a') > f(x') < f(b') to a select new interval containing the
+     minimum, by discarding the least useful point.  This procedure can
+     be continued indefinitely until the interval is sufficiently
+     small.  Choosing the golden section as the bisection ratio can be
+     shown to provide the fastest convergence for this type of
+     algorithm.
+
+
+ -- Minimizer: gsl_min_fminimizer_brent
+     The "Brent minimization algorithm" combines a parabolic
+     interpolation with the golden section algorithm.  This produces a
+     fast algorithm which is still robust.
+
+     The outline of the algorithm can be summarized as follows: on each
+     iteration Brent's method approximates the function using an
+     interpolating parabola through three existing points.  The minimum
+     of the parabola is taken as a guess for the minimum.  If it lies
+     within the bounds of the current interval then the interpolating
+     point is accepted, and used to generate a smaller interval.  If
+     the interpolating point is not accepted then the algorithm falls
+     back to an ordinary golden section step.  The full details of
+     Brent's method include some additional checks to improve
+     convergence.
+
+
+File: gsl-ref.info,  Node: Minimization Examples,  Next: Minimization References and Further Reading,  Prev: Minimization Algorithms,  Up: One dimensional Minimization
+
+33.8 Examples
+=============
+
+The following program uses the Brent algorithm to find the minimum of
+the function f(x) = \cos(x) + 1, which occurs at x = \pi.  The starting
+interval is (0,6), with an initial guess for the minimum of 2.
+
+     #include <stdio.h>
+     #include <gsl/gsl_errno.h>
+     #include <gsl/gsl_math.h>
+     #include <gsl/gsl_min.h>
+
+     double fn1 (double x, void * params)
+     {
+       return cos(x) + 1.0;
+     }
+
+     int
+     main (void)
+     {
+       int status;
+       int iter = 0, max_iter = 100;
+       const gsl_min_fminimizer_type *T;
+       gsl_min_fminimizer *s;
+       double m = 2.0, m_expected = M_PI;
+       double a = 0.0, b = 6.0;
+       gsl_function F;
+
+       F.function = &fn1;
+       F.params = 0;
+
+       T = gsl_min_fminimizer_brent;
+       s = gsl_min_fminimizer_alloc (T);
+       gsl_min_fminimizer_set (s, &F, m, a, b);
+
+       printf ("using %s method\n",
+               gsl_min_fminimizer_name (s));
+
+       printf ("%5s [%9s, %9s] %9s %10s %9s\n",
+               "iter", "lower", "upper", "min",
+               "err", "err(est)");
+
+       printf ("%5d [%.7f, %.7f] %.7f %+.7f %.7f\n",
+               iter, a, b,
+               m, m - m_expected, b - a);
+
+       do
+         {
+           iter++;
+           status = gsl_min_fminimizer_iterate (s);
+
+           m = gsl_min_fminimizer_x_minimum (s);
+           a = gsl_min_fminimizer_x_lower (s);
+           b = gsl_min_fminimizer_x_upper (s);
+
+           status
+             = gsl_min_test_interval (a, b, 0.001, 0.0);
+
+           if (status == GSL_SUCCESS)
+             printf ("Converged:\n");
+
+           printf ("%5d [%.7f, %.7f] "
+                   "%.7f %.7f %+.7f %.7f\n",
+                   iter, a, b,
+                   m, m_expected, m - m_expected, b - a);
+         }
+       while (status == GSL_CONTINUE && iter < max_iter);
+
+       gsl_min_fminimizer_free (s);
+
+       return status;
+     }
+
+Here are the results of the minimization procedure.
+
+     $ ./a.out
+         0 [0.0000000, 6.0000000] 2.0000000 -1.1415927 6.0000000
+         1 [2.0000000, 6.0000000] 3.2758640 +0.1342713 4.0000000
+         2 [2.0000000, 3.2831929] 3.2758640 +0.1342713 1.2831929
+         3 [2.8689068, 3.2831929] 3.2758640 +0.1342713 0.4142862
+         4 [2.8689068, 3.2831929] 3.2758640 +0.1342713 0.4142862
+         5 [2.8689068, 3.2758640] 3.1460585 +0.0044658 0.4069572
+         6 [3.1346075, 3.2758640] 3.1460585 +0.0044658 0.1412565
+         7 [3.1346075, 3.1874620] 3.1460585 +0.0044658 0.0528545
+         8 [3.1346075, 3.1460585] 3.1460585 +0.0044658 0.0114510
+         9 [3.1346075, 3.1460585] 3.1424060 +0.0008133 0.0114510
+        10 [3.1346075, 3.1424060] 3.1415885 -0.0000041 0.0077985
+     Converged:
+        11 [3.1415885, 3.1424060] 3.1415927 -0.0000000 0.0008175
+
+
+File: gsl-ref.info,  Node: Minimization References and Further Reading,  Prev: Minimization Examples,  Up: One dimensional Minimization
+
+33.9 References and Further Reading
+===================================
+
+Further information on Brent's algorithm is available in the following
+book,
+
+     Richard Brent, `Algorithms for minimization without derivatives',
+     Prentice-Hall (1973), republished by Dover in paperback (2002),
+     ISBN 0-486-41998-3.
+
+
+File: gsl-ref.info,  Node: Multidimensional Root-Finding,  Next: Multidimensional Minimization,  Prev: One dimensional Minimization,  Up: Top
+
+34 Multidimensional Root-Finding
+********************************
+
+This chapter describes functions for multidimensional root-finding
+(solving nonlinear systems with n equations in n unknowns).  The
+library provides low level components for a variety of iterative
+solvers and convergence tests.  These can be combined by the user to
+achieve the desired solution, with full access to the intermediate
+steps of the iteration.  Each class of methods uses the same framework,
+so that you can switch between solvers at runtime without needing to
+recompile your program.  Each instance of a solver keeps track of its
+own state, allowing the solvers to be used in multi-threaded programs.
+The solvers are based on the original Fortran library MINPACK.
+
+   The header file `gsl_multiroots.h' contains prototypes for the
+multidimensional root finding functions and related declarations.
+
+* Menu:
+
+* Overview of Multidimensional Root Finding::
+* Initializing the Multidimensional Solver::
+* Providing the multidimensional system of equations to solve::
+* Iteration of the multidimensional solver::
+* Search Stopping Parameters for the multidimensional solver::
+* Algorithms using Derivatives::
+* Algorithms without Derivatives::
+* Example programs for Multidimensional Root finding::
+* References and Further Reading for Multidimensional Root Finding::
+
+
+File: gsl-ref.info,  Node: Overview of Multidimensional Root Finding,  Next: Initializing the Multidimensional Solver,  Up: Multidimensional Root-Finding
+
+34.1 Overview
+=============
+
+The problem of multidimensional root finding requires the simultaneous
+solution of n equations, f_i, in n variables, x_i,
+
+     f_i (x_1, ..., x_n) = 0    for i = 1 ... n.
+
+In general there are no bracketing methods available for n dimensional
+systems, and no way of knowing whether any solutions exist.  All
+algorithms proceed from an initial guess using a variant of the Newton
+iteration,
+
+     x -> x' = x - J^{-1} f(x)
+
+where x, f are vector quantities and J is the Jacobian matrix J_{ij} =
+d f_i / d x_j.  Additional strategies can be used to enlarge the region
+of convergence.  These include requiring a decrease in the norm |f| on
+each step proposed by Newton's method, or taking steepest-descent steps
+in the direction of the negative gradient of |f|.
+
+   Several root-finding algorithms are available within a single
+framework.  The user provides a high-level driver for the algorithms,
+and the library provides the individual functions necessary for each of
+the steps.  There are three main phases of the iteration.  The steps
+are,
+
+   * initialize solver state, S, for algorithm T
+
+   * update S using the iteration T
+
+   * test S for convergence, and repeat iteration if necessary
+
+The evaluation of the Jacobian matrix can be problematic, either because
+programming the derivatives is intractable or because computation of the
+n^2 terms of the matrix becomes too expensive.  For these reasons the
+algorithms provided by the library are divided into two classes
+according to whether the derivatives are available or not.
+
+   The state for solvers with an analytic Jacobian matrix is held in a
+`gsl_multiroot_fdfsolver' struct.  The updating procedure requires both
+the function and its derivatives to be supplied by the user.
+
+   The state for solvers which do not use an analytic Jacobian matrix is
+held in a `gsl_multiroot_fsolver' struct.  The updating procedure uses
+only function evaluations (not derivatives).  The algorithms estimate
+the matrix J or J^{-1} by approximate methods.
+
+
+File: gsl-ref.info,  Node: Initializing the Multidimensional Solver,  Next: Providing the multidimensional system of equations to solve,  Prev: Overview of Multidimensional Root Finding,  Up: Multidimensional Root-Finding
+
+34.2 Initializing the Solver
+============================
+
+The following functions initialize a multidimensional solver, either
+with or without derivatives.  The solver itself depends only on the
+dimension of the problem and the algorithm and can be reused for
+different problems.
+
+ -- Function: gsl_multiroot_fsolver * gsl_multiroot_fsolver_alloc
+          (const gsl_multiroot_fsolver_type * T, size_t N)
+     This function returns a pointer to a newly allocated instance of a
+     solver of type T for a system of N dimensions.  For example, the
+     following code creates an instance of a hybrid solver, to solve a
+     3-dimensional system of equations.
+
+          const gsl_multiroot_fsolver_type * T
+              = gsl_multiroot_fsolver_hybrid;
+          gsl_multiroot_fsolver * s
+              = gsl_multiroot_fsolver_alloc (T, 3);
+
+     If there is insufficient memory to create the solver then the
+     function returns a null pointer and the error handler is invoked
+     with an error code of `GSL_ENOMEM'.
+
+ -- Function: gsl_multiroot_fdfsolver * gsl_multiroot_fdfsolver_alloc
+          (const gsl_multiroot_fdfsolver_type * T, size_t N)
+     This function returns a pointer to a newly allocated instance of a
+     derivative solver of type T for a system of N dimensions.  For
+     example, the following code creates an instance of a
+     Newton-Raphson solver, for a 2-dimensional system of equations.
+
+          const gsl_multiroot_fdfsolver_type * T
+              = gsl_multiroot_fdfsolver_newton;
+          gsl_multiroot_fdfsolver * s =
+              gsl_multiroot_fdfsolver_alloc (T, 2);
+
+     If there is insufficient memory to create the solver then the
+     function returns a null pointer and the error handler is invoked
+     with an error code of `GSL_ENOMEM'.
+
+ -- Function: int gsl_multiroot_fsolver_set (gsl_multiroot_fsolver * S,
+          gsl_multiroot_function * F, gsl_vector * X)
+     This function sets, or resets, an existing solver S to use the
+     function F and the initial guess X.
+
+ -- Function: int gsl_multiroot_fdfsolver_set (gsl_multiroot_fdfsolver
+          * S, gsl_multiroot_function_fdf * FDF, gsl_vector * X)
+     This function sets, or resets, an existing solver S to use the
+     function and derivative FDF and the initial guess X.
+
+ -- Function: void gsl_multiroot_fsolver_free (gsl_multiroot_fsolver *
+          S)
+ -- Function: void gsl_multiroot_fdfsolver_free
+          (gsl_multiroot_fdfsolver * S)
+     These functions free all the memory associated with the solver S.
+
+ -- Function: const char * gsl_multiroot_fsolver_name (const
+          gsl_multiroot_fsolver * S)
+ -- Function: const char * gsl_multiroot_fdfsolver_name (const
+          gsl_multiroot_fdfsolver * S)
+     These functions return a pointer to the name of the solver.  For
+     example,
+
+          printf ("s is a '%s' solver\n",
+                  gsl_multiroot_fdfsolver_name (s));
+
+     would print something like `s is a 'newton' solver'.
+
+
+File: gsl-ref.info,  Node: Providing the multidimensional system of equations to solve,  Next: Iteration of the multidimensional solver,  Prev: Initializing the Multidimensional Solver,  Up: Multidimensional Root-Finding
+
+34.3 Providing the function to solve
+====================================
+
+You must provide n functions of n variables for the root finders to
+operate on.  In order to allow for general parameters the functions are
+defined by the following data types:
+
+ -- Data Type: gsl_multiroot_function
+     This data type defines a general system of functions with
+     parameters.
+
+    `int (* f) (const gsl_vector * X, void * PARAMS, gsl_vector * F)'
+          this function should store the vector result f(x,params) in F
+          for argument X and parameters PARAMS, returning an
+          appropriate error code if the function cannot be computed.
+
+    `size_t n'
+          the dimension of the system, i.e. the number of components of
+          the vectors X and F.
+
+    `void * params'
+          a pointer to the parameters of the function.
+
+Here is an example using Powell's test function,
+
+     f_1(x) = A x_0 x_1 - 1,
+     f_2(x) = exp(-x_0) + exp(-x_1) - (1 + 1/A)
+
+with A = 10^4.  The following code defines a `gsl_multiroot_function'
+system `F' which you could pass to a solver:
+
+     struct powell_params { double A; };
+
+     int
+     powell (gsl_vector * x, void * p, gsl_vector * f) {
+        struct powell_params * params
+          = *(struct powell_params *)p;
+        const double A = (params->A);
+        const double x0 = gsl_vector_get(x,0);
+        const double x1 = gsl_vector_get(x,1);
+
+        gsl_vector_set (f, 0, A * x0 * x1 - 1);
+        gsl_vector_set (f, 1, (exp(-x0) + exp(-x1)
+                               - (1.0 + 1.0/A)));
+        return GSL_SUCCESS
+     }
+
+     gsl_multiroot_function F;
+     struct powell_params params = { 10000.0 };
+
+     F.f = &powell;
+     F.n = 2;
+     F.params = &params;
+
+ -- Data Type: gsl_multiroot_function_fdf
+     This data type defines a general system of functions with
+     parameters and the corresponding Jacobian matrix of derivatives,
+
+    `int (* f) (const gsl_vector * X, void * PARAMS, gsl_vector * F)'
+          this function should store the vector result f(x,params) in F
+          for argument X and parameters PARAMS, returning an
+          appropriate error code if the function cannot be computed.
+
+    `int (* df) (const gsl_vector * X, void * PARAMS, gsl_matrix * J)'
+          this function should store the N-by-N matrix result J_ij = d
+          f_i(x,params) / d x_j in J for argument X and parameters
+          PARAMS, returning an appropriate error code if the function
+          cannot be computed.
+
+    `int (* fdf) (const gsl_vector * X, void * PARAMS, gsl_vector * F, gsl_matrix * J)'
+          This function should set the values of the F and J as above,
+          for arguments X and parameters PARAMS.  This function
+          provides an optimization of the separate functions for f(x)
+          and J(x)--it is always faster to compute the function and its
+          derivative at the same time.
+
+    `size_t n'
+          the dimension of the system, i.e. the number of components of
+          the vectors X and F.
+
+    `void * params'
+          a pointer to the parameters of the function.
+
+The example of Powell's test function defined above can be extended to
+include analytic derivatives using the following code,
+
+     int
+     powell_df (gsl_vector * x, void * p, gsl_matrix * J)
+     {
+        struct powell_params * params
+          = *(struct powell_params *)p;
+        const double A = (params->A);
+        const double x0 = gsl_vector_get(x,0);
+        const double x1 = gsl_vector_get(x,1);
+        gsl_matrix_set (J, 0, 0, A * x1);
+        gsl_matrix_set (J, 0, 1, A * x0);
+        gsl_matrix_set (J, 1, 0, -exp(-x0));
+        gsl_matrix_set (J, 1, 1, -exp(-x1));
+        return GSL_SUCCESS
+     }
+
+     int
+     powell_fdf (gsl_vector * x, void * p,
+                 gsl_matrix * f, gsl_matrix * J) {
+        struct powell_params * params
+          = *(struct powell_params *)p;
+        const double A = (params->A);
+        const double x0 = gsl_vector_get(x,0);
+        const double x1 = gsl_vector_get(x,1);
+
+        const double u0 = exp(-x0);
+        const double u1 = exp(-x1);
+
+        gsl_vector_set (f, 0, A * x0 * x1 - 1);
+        gsl_vector_set (f, 1, u0 + u1 - (1 + 1/A));
+
+        gsl_matrix_set (J, 0, 0, A * x1);
+        gsl_matrix_set (J, 0, 1, A * x0);
+        gsl_matrix_set (J, 1, 0, -u0);
+        gsl_matrix_set (J, 1, 1, -u1);
+        return GSL_SUCCESS
+     }
+
+     gsl_multiroot_function_fdf FDF;
+
+     FDF.f = &powell_f;
+     FDF.df = &powell_df;
+     FDF.fdf = &powell_fdf;
+     FDF.n = 2;
+     FDF.params = 0;
+
+Note that the function `powell_fdf' is able to reuse existing terms
+from the function when calculating the Jacobian, thus saving time.
+
+
+File: gsl-ref.info,  Node: Iteration of the multidimensional solver,  Next: Search Stopping Parameters for the multidimensional solver,  Prev: Providing the multidimensional system of equations to solve,  Up: Multidimensional Root-Finding
+
+34.4 Iteration
+==============
+
+The following functions drive the iteration of each algorithm.  Each
+function performs one iteration to update the state of any solver of the
+corresponding type.  The same functions work for all solvers so that
+different methods can be substituted at runtime without modifications to
+the code.
+
+ -- Function: int gsl_multiroot_fsolver_iterate (gsl_multiroot_fsolver
+          * S)
+ -- Function: int gsl_multiroot_fdfsolver_iterate
+          (gsl_multiroot_fdfsolver * S)
+     These functions perform a single iteration of the solver S.  If the
+     iteration encounters an unexpected problem then an error code will
+     be returned,
+
+    `GSL_EBADFUNC'
+          the iteration encountered a singular point where the function
+          or its derivative evaluated to `Inf' or `NaN'.
+
+    `GSL_ENOPROG'
+          the iteration is not making any progress, preventing the
+          algorithm from continuing.
+
+   The solver maintains a current best estimate of the root at all
+times.  This information can be accessed with the following auxiliary
+functions,
+
+ -- Function: gsl_vector * gsl_multiroot_fsolver_root (const
+          gsl_multiroot_fsolver * S)
+ -- Function: gsl_vector * gsl_multiroot_fdfsolver_root (const
+          gsl_multiroot_fdfsolver * S)
+     These functions return the current estimate of the root for the
+     solver S.
+
+ -- Function: gsl_vector * gsl_multiroot_fsolver_f (const
+          gsl_multiroot_fsolver * S)
+ -- Function: gsl_vector * gsl_multiroot_fdfsolver_f (const
+          gsl_multiroot_fdfsolver * S)
+     These functions return the function value f(x) at the current
+     estimate of the root for the solver S.
+
+ -- Function: gsl_vector * gsl_multiroot_fsolver_dx (const
+          gsl_multiroot_fsolver * S)
+ -- Function: gsl_vector * gsl_multiroot_fdfsolver_dx (const
+          gsl_multiroot_fdfsolver * S)
+     These functions return the last step dx taken by the solver S.
+
+
+File: gsl-ref.info,  Node: Search Stopping Parameters for the multidimensional solver,  Next: Algorithms using Derivatives,  Prev: Iteration of the multidimensional solver,  Up: Multidimensional Root-Finding
+
+34.5 Search Stopping Parameters
+===============================
+
+A root finding procedure should stop when one of the following
+conditions is true:
+
+   * A multidimensional root has been found to within the
+     user-specified precision.
+
+   * A user-specified maximum number of iterations has been reached.
+
+   * An error has occurred.
+
+The handling of these conditions is under user control.  The functions
+below allow the user to test the precision of the current result in
+several standard ways.
+
+ -- Function: int gsl_multiroot_test_delta (const gsl_vector * DX,
+          const gsl_vector * X, double EPSABS, double EPSREL)
+     This function tests for the convergence of the sequence by
+     comparing the last step DX with the absolute error EPSABS and
+     relative error EPSREL to the current position X.  The test returns
+     `GSL_SUCCESS' if the following condition is achieved,
+
+          |dx_i| < epsabs + epsrel |x_i|
+
+     for each component of X and returns `GSL_CONTINUE' otherwise.
+
+ -- Function: int gsl_multiroot_test_residual (const gsl_vector * F,
+          double EPSABS)
+     This function tests the residual value F against the absolute
+     error bound EPSABS.  The test returns `GSL_SUCCESS' if the
+     following condition is achieved,
+
+          \sum_i |f_i| < epsabs
+
+     and returns `GSL_CONTINUE' otherwise.  This criterion is suitable
+     for situations where the precise location of the root, x, is
+     unimportant provided a value can be found where the residual is
+     small enough.
+
+
+File: gsl-ref.info,  Node: Algorithms using Derivatives,  Next: Algorithms without Derivatives,  Prev: Search Stopping Parameters for the multidimensional solver,  Up: Multidimensional Root-Finding
+
+34.6 Algorithms using Derivatives
+=================================
+
+The root finding algorithms described in this section make use of both
+the function and its derivative.  They require an initial guess for the
+location of the root, but there is no absolute guarantee of
+convergence--the function must be suitable for this technique and the
+initial guess must be sufficiently close to the root for it to work.
+When the conditions are satisfied then convergence is quadratic.
+
+ -- Derivative Solver: gsl_multiroot_fdfsolver_hybridsj
+     This is a modified version of Powell's Hybrid method as
+     implemented in the HYBRJ algorithm in MINPACK.  Minpack was
+     written by Jorge J. More', Burton S. Garbow and Kenneth E.
+     Hillstrom.  The Hybrid algorithm retains the fast convergence of
+     Newton's method but will also reduce the residual when Newton's
+     method is unreliable.
+
+     The algorithm uses a generalized trust region to keep each step
+     under control.  In order to be accepted a proposed new position x'
+     must satisfy the condition |D (x' - x)| < \delta, where D is a
+     diagonal scaling matrix and \delta is the size of the trust
+     region.  The components of D are computed internally, using the
+     column norms of the Jacobian to estimate the sensitivity of the
+     residual to each component of x.  This improves the behavior of the
+     algorithm for badly scaled functions.
+
+     On each iteration the algorithm first determines the standard
+     Newton step by solving the system J dx = - f.  If this step falls
+     inside the trust region it is used as a trial step in the next
+     stage.  If not, the algorithm uses the linear combination of the
+     Newton and gradient directions which is predicted to minimize the
+     norm of the function while staying inside the trust region,
+
+          dx = - \alpha J^{-1} f(x) - \beta \nabla |f(x)|^2.
+
+     This combination of Newton and gradient directions is referred to
+     as a "dogleg step".
+
+     The proposed step is now tested by evaluating the function at the
+     resulting point, x'.  If the step reduces the norm of the function
+     sufficiently then it is accepted and size of the trust region is
+     increased.  If the proposed step fails to improve the solution
+     then the size of the trust region is decreased and another trial
+     step is computed.
+
+     The speed of the algorithm is increased by computing the changes
+     to the Jacobian approximately, using a rank-1 update.  If two
+     successive attempts fail to reduce the residual then the full
+     Jacobian is recomputed.  The algorithm also monitors the progress
+     of the solution and returns an error if several steps fail to make
+     any improvement,
+
+    `GSL_ENOPROG'
+          the iteration is not making any progress, preventing the
+          algorithm from continuing.
+
+    `GSL_ENOPROGJ'
+          re-evaluations of the Jacobian indicate that the iteration is
+          not making any progress, preventing the algorithm from
+          continuing.
+
+
+ -- Derivative Solver: gsl_multiroot_fdfsolver_hybridj
+     This algorithm is an unscaled version of `hybridsj'.  The steps are
+     controlled by a spherical trust region |x' - x| < \delta, instead
+     of a generalized region.  This can be useful if the generalized
+     region estimated by `hybridsj' is inappropriate.
+
+ -- Derivative Solver: gsl_multiroot_fdfsolver_newton
+     Newton's Method is the standard root-polishing algorithm.  The
+     algorithm begins with an initial guess for the location of the
+     solution.  On each iteration a linear approximation to the
+     function F is used to estimate the step which will zero all the
+     components of the residual.  The iteration is defined by the
+     following sequence,
+
+          x -> x' = x - J^{-1} f(x)
+
+     where the Jacobian matrix J is computed from the derivative
+     functions provided by F.  The step dx is obtained by solving the
+     linear system,
+
+          J dx = - f(x)
+
+     using LU decomposition.
+
+ -- Derivative Solver: gsl_multiroot_fdfsolver_gnewton
+     This is a modified version of Newton's method which attempts to
+     improve global convergence by requiring every step to reduce the
+     Euclidean norm of the residual, |f(x)|.  If the Newton step leads
+     to an increase in the norm then a reduced step of relative size,
+
+          t = (\sqrt(1 + 6 r) - 1) / (3 r)
+
+     is proposed, with r being the ratio of norms |f(x')|^2/|f(x)|^2.
+     This procedure is repeated until a suitable step size is found.
+
+
+File: gsl-ref.info,  Node: Algorithms without Derivatives,  Next: Example programs for Multidimensional Root finding,  Prev: Algorithms using Derivatives,  Up: Multidimensional Root-Finding
+
+34.7 Algorithms without Derivatives
+===================================
+
+The algorithms described in this section do not require any derivative
+information to be supplied by the user.  Any derivatives needed are
+approximated by finite differences.  Note that if the
+finite-differencing step size chosen by these routines is inappropriate,
+an explicit user-supplied numerical derivative can always be used with
+the algorithms described in the previous section.
+
+ -- Solver: gsl_multiroot_fsolver_hybrids
+     This is a version of the Hybrid algorithm which replaces calls to
+     the Jacobian function by its finite difference approximation.  The
+     finite difference approximation is computed using
+     `gsl_multiroots_fdjac' with a relative step size of
+     `GSL_SQRT_DBL_EPSILON'.  Note that this step size will not be
+     suitable for all problems.
+
+ -- Solver: gsl_multiroot_fsolver_hybrid
+     This is a finite difference version of the Hybrid algorithm without
+     internal scaling.
+
+ -- Solver: gsl_multiroot_fsolver_dnewton
+     The "discrete Newton algorithm" is the simplest method of solving a
+     multidimensional system.  It uses the Newton iteration
+
+          x -> x - J^{-1} f(x)
+
+     where the Jacobian matrix J is approximated by taking finite
+     differences of the function F.  The approximation scheme used by
+     this implementation is,
+
+          J_{ij} = (f_i(x + \delta_j) - f_i(x)) /  \delta_j
+
+     where \delta_j is a step of size \sqrt\epsilon |x_j| with \epsilon
+     being the machine precision (\epsilon \approx 2.22 \times 10^-16).
+     The order of convergence of Newton's algorithm is quadratic, but
+     the finite differences require n^2 function evaluations on each
+     iteration.  The algorithm may become unstable if the finite
+     differences are not a good approximation to the true derivatives.
+
+ -- Solver: gsl_multiroot_fsolver_broyden
+     The "Broyden algorithm" is a version of the discrete Newton
+     algorithm which attempts to avoids the expensive update of the
+     Jacobian matrix on each iteration.  The changes to the Jacobian
+     are also approximated, using a rank-1 update,
+
+          J^{-1} \to J^{-1} - (J^{-1} df - dx) dx^T J^{-1} / dx^T J^{-1} df
+
+     where the vectors dx and df are the changes in x and f.  On the
+     first iteration the inverse Jacobian is estimated using finite
+     differences, as in the discrete Newton algorithm.
+
+     This approximation gives a fast update but is unreliable if the
+     changes are not small, and the estimate of the inverse Jacobian
+     becomes worse as time passes.  The algorithm has a tendency to
+     become unstable unless it starts close to the root.  The Jacobian
+     is refreshed if this instability is detected (consult the source
+     for details).
+
+     This algorithm is included only for demonstration purposes, and is
+     not recommended for serious use.
+
+
+File: gsl-ref.info,  Node: Example programs for Multidimensional Root finding,  Next: References and Further Reading for Multidimensional Root Finding,  Prev: Algorithms without Derivatives,  Up: Multidimensional Root-Finding
+
+34.8 Examples
+=============
+
+The multidimensional solvers are used in a similar way to the
+one-dimensional root finding algorithms.  This first example
+demonstrates the `hybrids' scaled-hybrid algorithm, which does not
+require derivatives. The program solves the Rosenbrock system of
+equations,
+
+     f_1 (x, y) = a (1 - x)
+     f_2 (x, y) = b (y - x^2)
+
+with a = 1, b = 10. The solution of this system lies at (x,y) = (1,1)
+in a narrow valley.
+
+   The first stage of the program is to define the system of equations,
+
+     #include <stdlib.h>
+     #include <stdio.h>
+     #include <gsl/gsl_vector.h>
+     #include <gsl/gsl_multiroots.h>
+
+     struct rparams
+       {
+         double a;
+         double b;
+       };
+
+     int
+     rosenbrock_f (const gsl_vector * x, void *params,
+                   gsl_vector * f)
+     {
+       double a = ((struct rparams *) params)->a;
+       double b = ((struct rparams *) params)->b;
+
+       const double x0 = gsl_vector_get (x, 0);
+       const double x1 = gsl_vector_get (x, 1);
+
+       const double y0 = a * (1 - x0);
+       const double y1 = b * (x1 - x0 * x0);
+
+       gsl_vector_set (f, 0, y0);
+       gsl_vector_set (f, 1, y1);
+
+       return GSL_SUCCESS;
+     }
+
+The main program begins by creating the function object `f', with the
+arguments `(x,y)' and parameters `(a,b)'. The solver `s' is initialized
+to use this function, with the `hybrids' method.
+
+     int
+     main (void)
+     {
+       const gsl_multiroot_fsolver_type *T;
+       gsl_multiroot_fsolver *s;
+
+       int status;
+       size_t i, iter = 0;
+
+       const size_t n = 2;
+       struct rparams p = {1.0, 10.0};
+       gsl_multiroot_function f = {&rosenbrock_f, n, &p};
+
+       double x_init[2] = {-10.0, -5.0};
+       gsl_vector *x = gsl_vector_alloc (n);
+
+       gsl_vector_set (x, 0, x_init[0]);
+       gsl_vector_set (x, 1, x_init[1]);
+
+       T = gsl_multiroot_fsolver_hybrids;
+       s = gsl_multiroot_fsolver_alloc (T, 2);
+       gsl_multiroot_fsolver_set (s, &f, x);
+
+       print_state (iter, s);
+
+       do
+         {
+           iter++;
+           status = gsl_multiroot_fsolver_iterate (s);
+
+           print_state (iter, s);
+
+           if (status)   /* check if solver is stuck */
+             break;
+
+           status =
+             gsl_multiroot_test_residual (s->f, 1e-7);
+         }
+       while (status == GSL_CONTINUE && iter < 1000);
+
+       printf ("status = %s\n", gsl_strerror (status));
+
+       gsl_multiroot_fsolver_free (s);
+       gsl_vector_free (x);
+       return 0;
+     }
+
+Note that it is important to check the return status of each solver
+step, in case the algorithm becomes stuck.  If an error condition is
+detected, indicating that the algorithm cannot proceed, then the error
+can be reported to the user, a new starting point chosen or a different
+algorithm used.
+
+   The intermediate state of the solution is displayed by the following
+function.  The solver state contains the vector `s->x' which is the
+current position, and the vector `s->f' with corresponding function
+values.
+
+     int
+     print_state (size_t iter, gsl_multiroot_fsolver * s)
+     {
+       printf ("iter = %3u x = % .3f % .3f "
+               "f(x) = % .3e % .3e\n",
+               iter,
+               gsl_vector_get (s->x, 0),
+               gsl_vector_get (s->x, 1),
+               gsl_vector_get (s->f, 0),
+               gsl_vector_get (s->f, 1));
+     }
+
+Here are the results of running the program. The algorithm is started at
+(-10,-5) far from the solution.  Since the solution is hidden in a
+narrow valley the earliest steps follow the gradient of the function
+downhill, in an attempt to reduce the large value of the residual. Once
+the root has been approximately located, on iteration 8, the Newton
+behavior takes over and convergence is very rapid.
+
+     iter =  0 x = -10.000  -5.000  f(x) = 1.100e+01 -1.050e+03
+     iter =  1 x = -10.000  -5.000  f(x) = 1.100e+01 -1.050e+03
+     iter =  2 x =  -3.976  24.827  f(x) = 4.976e+00  9.020e+01
+     iter =  3 x =  -3.976  24.827  f(x) = 4.976e+00  9.020e+01
+     iter =  4 x =  -3.976  24.827  f(x) = 4.976e+00  9.020e+01
+     iter =  5 x =  -1.274  -5.680  f(x) = 2.274e+00 -7.302e+01
+     iter =  6 x =  -1.274  -5.680  f(x) = 2.274e+00 -7.302e+01
+     iter =  7 x =   0.249   0.298  f(x) = 7.511e-01  2.359e+00
+     iter =  8 x =   0.249   0.298  f(x) = 7.511e-01  2.359e+00
+     iter =  9 x =   1.000   0.878  f(x) = 1.268e-10 -1.218e+00
+     iter = 10 x =   1.000   0.989  f(x) = 1.124e-11 -1.080e-01
+     iter = 11 x =   1.000   1.000  f(x) = 0.000e+00  0.000e+00
+     status = success
+
+Note that the algorithm does not update the location on every
+iteration. Some iterations are used to adjust the trust-region
+parameter, after trying a step which was found to be divergent, or to
+recompute the Jacobian, when poor convergence behavior is detected.
+
+   The next example program adds derivative information, in order to
+accelerate the solution. There are two derivative functions
+`rosenbrock_df' and `rosenbrock_fdf'. The latter computes both the
+function and its derivative simultaneously. This allows the
+optimization of any common terms.  For simplicity we substitute calls to
+the separate `f' and `df' functions at this point in the code below.
+
+     int
+     rosenbrock_df (const gsl_vector * x, void *params,
+                    gsl_matrix * J)
+     {
+       const double a = ((struct rparams *) params)->a;
+       const double b = ((struct rparams *) params)->b;
+
+       const double x0 = gsl_vector_get (x, 0);
+
+       const double df00 = -a;
+       const double df01 = 0;
+       const double df10 = -2 * b  * x0;
+       const double df11 = b;
+
+       gsl_matrix_set (J, 0, 0, df00);
+       gsl_matrix_set (J, 0, 1, df01);
+       gsl_matrix_set (J, 1, 0, df10);
+       gsl_matrix_set (J, 1, 1, df11);
+
+       return GSL_SUCCESS;
+     }
+
+     int
+     rosenbrock_fdf (const gsl_vector * x, void *params,
+                     gsl_vector * f, gsl_matrix * J)
+     {
+       rosenbrock_f (x, params, f);
+       rosenbrock_df (x, params, J);
+
+       return GSL_SUCCESS;
+     }
+
+The main program now makes calls to the corresponding `fdfsolver'
+versions of the functions,
+
+     int
+     main (void)
+     {
+       const gsl_multiroot_fdfsolver_type *T;
+       gsl_multiroot_fdfsolver *s;
+
+       int status;
+       size_t i, iter = 0;
+
+       const size_t n = 2;
+       struct rparams p = {1.0, 10.0};
+       gsl_multiroot_function_fdf f = {&rosenbrock_f,
+                                       &rosenbrock_df,
+                                       &rosenbrock_fdf,
+                                       n, &p};
+
+       double x_init[2] = {-10.0, -5.0};
+       gsl_vector *x = gsl_vector_alloc (n);
+
+       gsl_vector_set (x, 0, x_init[0]);
+       gsl_vector_set (x, 1, x_init[1]);
+
+       T = gsl_multiroot_fdfsolver_gnewton;
+       s = gsl_multiroot_fdfsolver_alloc (T, n);
+       gsl_multiroot_fdfsolver_set (s, &f, x);
+
+       print_state (iter, s);
+
+       do
+         {
+           iter++;
+
+           status = gsl_multiroot_fdfsolver_iterate (s);
+
+           print_state (iter, s);
+
+           if (status)
+             break;
+
+           status = gsl_multiroot_test_residual (s->f, 1e-7);
+         }
+       while (status == GSL_CONTINUE && iter < 1000);
+
+       printf ("status = %s\n", gsl_strerror (status));
+
+       gsl_multiroot_fdfsolver_free (s);
+       gsl_vector_free (x);
+       return 0;
+     }
+
+The addition of derivative information to the `hybrids' solver does not
+make any significant difference to its behavior, since it able to
+approximate the Jacobian numerically with sufficient accuracy.  To
+illustrate the behavior of a different derivative solver we switch to
+`gnewton'. This is a traditional Newton solver with the constraint that
+it scales back its step if the full step would lead "uphill". Here is
+the output for the `gnewton' algorithm,
+
+     iter = 0 x = -10.000  -5.000 f(x) =  1.100e+01 -1.050e+03
+     iter = 1 x =  -4.231 -65.317 f(x) =  5.231e+00 -8.321e+02
+     iter = 2 x =   1.000 -26.358 f(x) = -8.882e-16 -2.736e+02
+     iter = 3 x =   1.000   1.000 f(x) = -2.220e-16 -4.441e-15
+     status = success
+
+The convergence is much more rapid, but takes a wide excursion out to
+the point (-4.23,-65.3). This could cause the algorithm to go astray in
+a realistic application.  The hybrid algorithm follows the downhill
+path to the solution more reliably.
+
+
+File: gsl-ref.info,  Node: References and Further Reading for Multidimensional Root Finding,  Prev: Example programs for Multidimensional Root finding,  Up: Multidimensional Root-Finding
+
+34.9 References and Further Reading
+===================================
+
+The original version of the Hybrid method is described in the following
+articles by Powell,
+
+     M.J.D. Powell, "A Hybrid Method for Nonlinear Equations" (Chap 6, p
+     87-114) and "A Fortran Subroutine for Solving systems of Nonlinear
+     Algebraic Equations" (Chap 7, p 115-161), in `Numerical Methods for
+     Nonlinear Algebraic Equations', P. Rabinowitz, editor.  Gordon and
+     Breach, 1970.
+
+The following papers are also relevant to the algorithms described in
+this section,
+
+     J.J. More', M.Y. Cosnard, "Numerical Solution of Nonlinear
+     Equations", `ACM Transactions on Mathematical Software', Vol 5, No
+     1, (1979), p 64-85
+
+     C.G. Broyden, "A Class of Methods for Solving Nonlinear
+     Simultaneous Equations", `Mathematics of Computation', Vol 19
+     (1965), p 577-593
+
+     J.J. More', B.S. Garbow, K.E. Hillstrom, "Testing Unconstrained
+     Optimization Software", ACM Transactions on Mathematical Software,
+     Vol 7, No 1 (1981), p 17-41
+
+
+File: gsl-ref.info,  Node: Multidimensional Minimization,  Next: Least-Squares Fitting,  Prev: Multidimensional Root-Finding,  Up: Top
+
+35 Multidimensional Minimization
+********************************
+
+This chapter describes routines for finding minima of arbitrary
+multidimensional functions.  The library provides low level components
+for a variety of iterative minimizers and convergence tests.  These can
+be combined by the user to achieve the desired solution, while providing
+full access to the intermediate steps of the algorithms.  Each class of
+methods uses the same framework, so that you can switch between
+minimizers at runtime without needing to recompile your program.  Each
+instance of a minimizer keeps track of its own state, allowing the
+minimizers to be used in multi-threaded programs. The minimization
+algorithms can be used to maximize a function by inverting its sign.
+
+   The header file `gsl_multimin.h' contains prototypes for the
+minimization functions and related declarations.
+
+* Menu:
+
+* Multimin Overview::
+* Multimin Caveats::
+* Initializing the Multidimensional Minimizer::
+* Providing a function to minimize::
+* Multimin Iteration::
+* Multimin Stopping Criteria::
+* Multimin Algorithms::
+* Multimin Examples::
+* Multimin References and Further Reading::
+
+
+File: gsl-ref.info,  Node: Multimin Overview,  Next: Multimin Caveats,  Up: Multidimensional Minimization
+
+35.1 Overview
+=============
+
+The problem of multidimensional minimization requires finding a point x
+such that the scalar function,
+
+     f(x_1, ..., x_n)
+
+takes a value which is lower than at any neighboring point. For smooth
+functions the gradient g = \nabla f vanishes at the minimum. In general
+there are no bracketing methods available for the minimization of
+n-dimensional functions.  The algorithms proceed from an initial guess
+using a search algorithm which attempts to move in a downhill direction.
+
+   Algorithms making use of the gradient of the function perform a
+one-dimensional line minimisation along this direction until the lowest
+point is found to a suitable tolerance.  The search direction is then
+updated with local information from the function and its derivatives,
+and the whole process repeated until the true n-dimensional minimum is
+found.
+
+   The Nelder-Mead Simplex algorithm applies a different strategy.  It
+maintains n+1 trial parameter vectors as the vertices of a
+n-dimensional simplex.  In each iteration step it tries to improve the
+worst vertex by a simple geometrical transformation until the size of
+the simplex falls below a given tolerance.
+
+   Both types of algorithms use a standard framework. The user provides
+a high-level driver for the algorithms, and the library provides the
+individual functions necessary for each of the steps.  There are three
+main phases of the iteration.  The steps are,
+
+   * initialize minimizer state, S, for algorithm T
+
+   * update S using the iteration T
+
+   * test S for convergence, and repeat iteration if necessary
+
+Each iteration step consists either of an improvement to the
+line-minimisation in the current direction or an update to the search
+direction itself.  The state for the minimizers is held in a
+`gsl_multimin_fdfminimizer' struct or a `gsl_multimin_fminimizer'
+struct.
+
+
+File: gsl-ref.info,  Node: Multimin Caveats,  Next: Initializing the Multidimensional Minimizer,  Prev: Multimin Overview,  Up: Multidimensional Minimization
+
+35.2 Caveats
+============
+
+Note that the minimization algorithms can only search for one local
+minimum at a time.  When there are several local minima in the search
+area, the first minimum to be found will be returned; however it is
+difficult to predict which of the minima this will be.  In most cases,
+no error will be reported if you try to find a local minimum in an area
+where there is more than one.
+
+   It is also important to note that the minimization algorithms find
+local minima; there is no way to determine whether a minimum is a global
+minimum of the function in question.
+
+
+File: gsl-ref.info,  Node: Initializing the Multidimensional Minimizer,  Next: Providing a function to minimize,  Prev: Multimin Caveats,  Up: Multidimensional Minimization
+
+35.3 Initializing the Multidimensional Minimizer
+================================================
+
+The following function initializes a multidimensional minimizer.  The
+minimizer itself depends only on the dimension of the problem and the
+algorithm and can be reused for different problems.
+
+ -- Function: gsl_multimin_fdfminimizer *
+gsl_multimin_fdfminimizer_alloc (const gsl_multimin_fdfminimizer_type *
+          T, size_t N)
+ -- Function: gsl_multimin_fminimizer * gsl_multimin_fminimizer_alloc
+          (const gsl_multimin_fminimizer_type * T, size_t N)
+     This function returns a pointer to a newly allocated instance of a
+     minimizer of type T for an N-dimension function.  If there is
+     insufficient memory to create the minimizer then the function
+     returns a null pointer and the error handler is invoked with an
+     error code of `GSL_ENOMEM'.
+
+ -- Function: int gsl_multimin_fdfminimizer_set
+          (gsl_multimin_fdfminimizer * S, gsl_multimin_function_fdf *
+          FDF, const gsl_vector * X, double STEP_SIZE, double TOL)
+     This function initializes the minimizer S to minimize the function
+     FDF starting from the initial point X.  The size of the first
+     trial step is given by STEP_SIZE.  The accuracy of the line
+     minimization is specified by TOL.  The precise meaning of this
+     parameter depends on the method used.  Typically the line
+     minimization is considered successful if the gradient of the
+     function g is orthogonal to the current search direction p to a
+     relative accuracy of TOL, where dot(p,g) < tol |p| |g|.  A TOL
+     value of 0.1 is suitable for most purposes, since line
+     minimization only needs to be carried out approximately.    Note
+     that setting TOL to zero will force the use of "exact"
+     line-searches, which are extremely expensive.
+
+ -- Function: int gsl_multimin_fminimizer_set (gsl_multimin_fminimizer
+          * S, gsl_multimin_function * F, const gsl_vector * X, const
+          gsl_vector * STEP_SIZE)
+     This function initializes the minimizer S to minimize the function
+     F, starting from the initial point X. The size of the initial
+     trial steps is given in vector STEP_SIZE. The precise meaning of
+     this parameter depends on the method used.
+
+ -- Function: void gsl_multimin_fdfminimizer_free
+          (gsl_multimin_fdfminimizer * S)
+ -- Function: void gsl_multimin_fminimizer_free
+          (gsl_multimin_fminimizer * S)
+     This function frees all the memory associated with the minimizer S.
+
+ -- Function: const char * gsl_multimin_fdfminimizer_name (const
+          gsl_multimin_fdfminimizer * S)
+ -- Function: const char * gsl_multimin_fminimizer_name (const
+          gsl_multimin_fminimizer * S)
+     This function returns a pointer to the name of the minimizer.  For
+     example,
+
+          printf ("s is a '%s' minimizer\n",
+                  gsl_multimin_fdfminimizer_name (s));
+
+     would print something like `s is a 'conjugate_pr' minimizer'.
+
+
+File: gsl-ref.info,  Node: Providing a function to minimize,  Next: Multimin Iteration,  Prev: Initializing the Multidimensional Minimizer,  Up: Multidimensional Minimization
+
+35.4 Providing a function to minimize
+=====================================
+
+You must provide a parametric function of n variables for the
+minimizers to operate on.  You may also need to provide a routine which
+calculates the gradient of the function and a third routine which
+calculates both the function value and the gradient together.  In order
+to allow for general parameters the functions are defined by the
+following data types:
+
+ -- Data Type: gsl_multimin_function_fdf
+     This data type defines a general function of n variables with
+     parameters and the corresponding gradient vector of derivatives,
+
+    `double (* f) (const gsl_vector * X, void * PARAMS)'
+          this function should return the result f(x,params) for
+          argument X and parameters PARAMS.
+
+    `void (* df) (const gsl_vector * X, void * PARAMS, gsl_vector * G)'
+          this function should store the N-dimensional gradient g_i = d
+          f(x,params) / d x_i in the vector G for argument X and
+          parameters PARAMS, returning an appropriate error code if the
+          function cannot be computed.
+
+    `void (* fdf) (const gsl_vector * X, void * PARAMS, double * f, gsl_vector * G)'
+          This function should set the values of the F and G as above,
+          for arguments X and parameters PARAMS.  This function
+          provides an optimization of the separate functions for f(x)
+          and g(x)--it is always faster to compute the function and its
+          derivative at the same time.
+
+    `size_t n'
+          the dimension of the system, i.e. the number of components of
+          the vectors X.
+
+    `void * params'
+          a pointer to the parameters of the function.
+
+ -- Data Type: gsl_multimin_function
+     This data type defines a general function of n variables with
+     parameters,
+
+    `double (* f) (const gsl_vector * X, void * PARAMS)'
+          this function should return the result f(x,params) for
+          argument X and parameters PARAMS.
+
+    `size_t n'
+          the dimension of the system, i.e. the number of components of
+          the vectors X.
+
+    `void * params'
+          a pointer to the parameters of the function.
+
+The following example function defines a simple paraboloid with two
+parameters,
+
+     /* Paraboloid centered on (dp[0],dp[1]) */
+
+     double
+     my_f (const gsl_vector *v, void *params)
+     {
+       double x, y;
+       double *dp = (double *)params;
+
+       x = gsl_vector_get(v, 0);
+       y = gsl_vector_get(v, 1);
+
+       return 10.0 * (x - dp[0]) * (x - dp[0]) +
+                20.0 * (y - dp[1]) * (y - dp[1]) + 30.0;
+     }
+
+     /* The gradient of f, df = (df/dx, df/dy). */
+     void
+     my_df (const gsl_vector *v, void *params,
+            gsl_vector *df)
+     {
+       double x, y;
+       double *dp = (double *)params;
+
+       x = gsl_vector_get(v, 0);
+       y = gsl_vector_get(v, 1);
+
+       gsl_vector_set(df, 0, 20.0 * (x - dp[0]));
+       gsl_vector_set(df, 1, 40.0 * (y - dp[1]));
+     }
+
+     /* Compute both f and df together. */
+     void
+     my_fdf (const gsl_vector *x, void *params,
+             double *f, gsl_vector *df)
+     {
+       *f = my_f(x, params);
+       my_df(x, params, df);
+     }
+
+The function can be initialized using the following code,
+
+     gsl_multimin_function_fdf my_func;
+
+     double p[2] = { 1.0, 2.0 }; /* center at (1,2) */
+
+     my_func.f = &my_f;
+     my_func.df = &my_df;
+     my_func.fdf = &my_fdf;
+     my_func.n = 2;
+     my_func.params = (void *)p;
+
+
+File: gsl-ref.info,  Node: Multimin Iteration,  Next: Multimin Stopping Criteria,  Prev: Providing a function to minimize,  Up: Multidimensional Minimization
+
+35.5 Iteration
+==============
+
+The following function drives the iteration of each algorithm.  The
+function performs one iteration to update the state of the minimizer.
+The same function works for all minimizers so that different methods can
+be substituted at runtime without modifications to the code.
+
+ -- Function: int gsl_multimin_fdfminimizer_iterate
+          (gsl_multimin_fdfminimizer * S)
+ -- Function: int gsl_multimin_fminimizer_iterate
+          (gsl_multimin_fminimizer * S)
+     These functions perform a single iteration of the minimizer S.  If
+     the iteration encounters an unexpected problem then an error code
+     will be returned.
+
+The minimizer maintains a current best estimate of the minimum at all
+times.  This information can be accessed with the following auxiliary
+functions,
+
+ -- Function: gsl_vector * gsl_multimin_fdfminimizer_x (const
+          gsl_multimin_fdfminimizer * S)
+ -- Function: gsl_vector * gsl_multimin_fminimizer_x (const
+          gsl_multimin_fminimizer * S)
+ -- Function: double gsl_multimin_fdfminimizer_minimum (const
+          gsl_multimin_fdfminimizer * S)
+ -- Function: double gsl_multimin_fminimizer_minimum (const
+          gsl_multimin_fminimizer * S)
+ -- Function: gsl_vector * gsl_multimin_fdfminimizer_gradient (const
+          gsl_multimin_fdfminimizer * S)
+ -- Function: double gsl_multimin_fminimizer_size (const
+          gsl_multimin_fminimizer * S)
+     These functions return the current best estimate of the location
+     of the minimum, the value of the function at that point, its
+     gradient, and minimizer specific characteristic size for the
+     minimizer S.
+
+ -- Function: int gsl_multimin_fdfminimizer_restart
+          (gsl_multimin_fdfminimizer * S)
+     This function resets the minimizer S to use the current point as a
+     new starting point.
+
+
+File: gsl-ref.info,  Node: Multimin Stopping Criteria,  Next: Multimin Algorithms,  Prev: Multimin Iteration,  Up: Multidimensional Minimization
+
+35.6 Stopping Criteria
+======================
+
+A minimization procedure should stop when one of the following
+conditions is true:
+
+   * A minimum has been found to within the user-specified precision.
+
+   * A user-specified maximum number of iterations has been reached.
+
+   * An error has occurred.
+
+The handling of these conditions is under user control.  The functions
+below allow the user to test the precision of the current result.
+
+ -- Function: int gsl_multimin_test_gradient (const gsl_vector * G,
+          double EPSABS)
+     This function tests the norm of the gradient G against the
+     absolute tolerance EPSABS. The gradient of a multidimensional
+     function goes to zero at a minimum. The test returns `GSL_SUCCESS'
+     if the following condition is achieved,
+
+          |g| < epsabs
+
+     and returns `GSL_CONTINUE' otherwise.  A suitable choice of EPSABS
+     can be made from the desired accuracy in the function for small
+     variations in x.  The relationship between these quantities is
+     given by \delta f = g \delta x.
+
+ -- Function: int gsl_multimin_test_size (const double SIZE, double
+          EPSABS)
+     This function tests the minimizer specific characteristic size (if
+     applicable to the used minimizer) against absolute tolerance
+     EPSABS.  The test returns `GSL_SUCCESS' if the size is smaller
+     than tolerance, otherwise `GSL_CONTINUE' is returned.
+
+
+File: gsl-ref.info,  Node: Multimin Algorithms,  Next: Multimin Examples,  Prev: Multimin Stopping Criteria,  Up: Multidimensional Minimization
+
+35.7 Algorithms
+===============
+
+There are several minimization methods available. The best choice of
+algorithm depends on the problem.  All of the algorithms use the value
+of the function and its gradient at each evaluation point, except for
+the Simplex algorithm which uses function values only.
+
+ -- Minimizer: gsl_multimin_fdfminimizer_conjugate_fr
+     This is the Fletcher-Reeves conjugate gradient algorithm. The
+     conjugate gradient algorithm proceeds as a succession of line
+     minimizations. The sequence of search directions is used to build
+     up an approximation to the curvature of the function in the
+     neighborhood of the minimum.
+
+     An initial search direction P is chosen using the gradient, and
+     line minimization is carried out in that direction.  The accuracy
+     of the line minimization is specified by the parameter TOL.  The
+     minimum along this line occurs when the function gradient G and
+     the search direction P are orthogonal.  The line minimization
+     terminates when dot(p,g) < tol |p| |g|.  The search direction is
+     updated  using the Fletcher-Reeves formula p' = g' - \beta g where
+     \beta=-|g'|^2/|g|^2, and the line minimization is then repeated
+     for the new search direction.
+
+ -- Minimizer: gsl_multimin_fdfminimizer_conjugate_pr
+     This is the Polak-Ribiere conjugate gradient algorithm.  It is
+     similar to the Fletcher-Reeves method, differing only in the
+     choice of the coefficient \beta. Both methods work well when the
+     evaluation point is close enough to the minimum of the objective
+     function that it is well approximated by a quadratic hypersurface.
+
+ -- Minimizer: gsl_multimin_fdfminimizer_vector_bfgs2
+ -- Minimizer: gsl_multimin_fdfminimizer_vector_bfgs
+     These methods use the vector Broyden-Fletcher-Goldfarb-Shanno
+     (BFGS) algorithm.  This is a quasi-Newton method which builds up
+     an approximation to the second derivatives of the function f using
+     the difference between successive gradient vectors.  By combining
+     the first and second derivatives the algorithm is able to take
+     Newton-type steps towards the function minimum, assuming quadratic
+     behavior in that region.
+
+     The `bfgs2' version of this minimizer is the most efficient
+     version available, and is a faithful implementation of the line
+     minimization scheme described in Fletcher's `Practical Methods of
+     Optimization', Algorithms 2.6.2 and 2.6.4.  It supercedes the
+     original `bfgs' routine and requires substantially fewer function
+     and gradient evaluations.  The user-supplied tolerance TOL
+     corresponds to the parameter \sigma used by Fletcher.  A value of
+     0.1 is recommended for typical use (larger values correspond to
+     less accurate line searches).
+
+
+ -- Minimizer: gsl_multimin_fdfminimizer_steepest_descent
+     The steepest descent algorithm follows the downhill gradient of the
+     function at each step. When a downhill step is successful the
+     step-size is increased by a factor of two.  If the downhill step
+     leads to a higher function value then the algorithm backtracks and
+     the step size is decreased using the parameter TOL.  A suitable
+     value of TOL for most applications is 0.1.  The steepest descent
+     method is inefficient and is included only for demonstration
+     purposes.
+
+ -- Minimizer: gsl_multimin_fminimizer_nmsimplex
+     This is the Simplex algorithm of Nelder and Mead. It constructs n
+     vectors p_i from the starting vector X and the vector STEP_SIZE as
+     follows:
+
+          p_0 = (x_0, x_1, ... , x_n)
+          p_1 = (x_0 + step_size_0, x_1, ... , x_n)
+          p_2 = (x_0, x_1 + step_size_1, ... , x_n)
+          ... = ...
+          p_n = (x_0, x_1, ... , x_n+step_size_n)
+
+     These vectors form the n+1 vertices of a simplex in n dimensions.
+     On each iteration the algorithm tries to improve the parameter
+     vector p_i corresponding to the highest function value by simple
+     geometrical transformations.  These are reflection, reflection
+     followed by expansion, contraction and multiple contraction. Using
+     these transformations the simplex moves through the parameter
+     space towards the minimum, where it contracts itself.
+
+     After each iteration, the best vertex is returned.  Note, that due
+     to the nature of the algorithm not every step improves the current
+     best parameter vector.  Usually several iterations are required.
+
+     The routine calculates the minimizer specific characteristic size
+     as the average distance from the geometrical center of the simplex
+     to all its vertices.  This size can be used as a stopping
+     criteria, as the simplex contracts itself near the minimum. The
+     size is returned by the function `gsl_multimin_fminimizer_size'.
+
+
+File: gsl-ref.info,  Node: Multimin Examples,  Next: Multimin References and Further Reading,  Prev: Multimin Algorithms,  Up: Multidimensional Minimization
+
+35.8 Examples
+=============
+
+This example program finds the minimum of the paraboloid function
+defined earlier.  The location of the minimum is offset from the origin
+in x and y, and the function value at the minimum is non-zero. The main
+program is given below, it requires the example function given earlier
+in this chapter.
+
+     int
+     main (void)
+     {
+       size_t iter = 0;
+       int status;
+
+       const gsl_multimin_fdfminimizer_type *T;
+       gsl_multimin_fdfminimizer *s;
+
+       /* Position of the minimum (1,2). */
+       double par[2] = { 1.0, 2.0 };
+
+       gsl_vector *x;
+       gsl_multimin_function_fdf my_func;
+
+       my_func.f = &my_f;
+       my_func.df = &my_df;
+       my_func.fdf = &my_fdf;
+       my_func.n = 2;
+       my_func.params = &par;
+
+       /* Starting point, x = (5,7) */
+       x = gsl_vector_alloc (2);
+       gsl_vector_set (x, 0, 5.0);
+       gsl_vector_set (x, 1, 7.0);
+
+       T = gsl_multimin_fdfminimizer_conjugate_fr;
+       s = gsl_multimin_fdfminimizer_alloc (T, 2);
+
+       gsl_multimin_fdfminimizer_set (s, &my_func, x, 0.01, 1e-4);
+
+       do
+         {
+           iter++;
+           status = gsl_multimin_fdfminimizer_iterate (s);
+
+           if (status)
+             break;
+
+           status = gsl_multimin_test_gradient (s->gradient, 1e-3);
+
+           if (status == GSL_SUCCESS)
+             printf ("Minimum found at:\n");
+
+           printf ("%5d %.5f %.5f %10.5f\n", iter,
+                   gsl_vector_get (s->x, 0),
+                   gsl_vector_get (s->x, 1),
+                   s->f);
+
+         }
+       while (status == GSL_CONTINUE && iter < 100);
+
+       gsl_multimin_fdfminimizer_free (s);
+       gsl_vector_free (x);
+
+       return 0;
+     }
+
+The initial step-size is chosen as 0.01, a conservative estimate in this
+case, and the line minimization parameter is set at 0.0001.  The program
+terminates when the norm of the gradient has been reduced below 0.001.
+The output of the program is shown below,
+
+              x       y         f
+         1 4.99629 6.99072  687.84780
+         2 4.98886 6.97215  683.55456
+         3 4.97400 6.93501  675.01278
+         4 4.94429 6.86073  658.10798
+         5 4.88487 6.71217  625.01340
+         6 4.76602 6.41506  561.68440
+         7 4.52833 5.82083  446.46694
+         8 4.05295 4.63238  261.79422
+         9 3.10219 2.25548   75.49762
+        10 2.85185 1.62963   67.03704
+        11 2.19088 1.76182   45.31640
+        12 0.86892 2.02622   30.18555
+     Minimum found at:
+        13 1.00000 2.00000   30.00000
+
+Note that the algorithm gradually increases the step size as it
+successfully moves downhill, as can be seen by plotting the successive
+points.
+
+The conjugate gradient algorithm finds the minimum on its second
+direction because the function is purely quadratic. Additional
+iterations would be needed for a more complicated function.
+
+   Here is another example using the Nelder-Mead Simplex algorithm to
+minimize the same example object function, as above.
+
+     int
+     main(void)
+     {
+       size_t np = 2;
+       double par[2] = {1.0, 2.0};
+
+       const gsl_multimin_fminimizer_type *T =
+         gsl_multimin_fminimizer_nmsimplex;
+       gsl_multimin_fminimizer *s = NULL;
+       gsl_vector *ss, *x;
+       gsl_multimin_function minex_func;
+
+       size_t iter = 0, i;
+       int status;
+       double size;
+
+       /* Initial vertex size vector */
+       ss = gsl_vector_alloc (np);
+
+       /* Set all step sizes to 1 */
+       gsl_vector_set_all (ss, 1.0);
+
+       /* Starting point */
+       x = gsl_vector_alloc (np);
+
+       gsl_vector_set (x, 0, 5.0);
+       gsl_vector_set (x, 1, 7.0);
+
+       /* Initialize method and iterate */
+       minex_func.f = &my_f;
+       minex_func.n = np;
+       minex_func.params = (void *)&par;
+
+       s = gsl_multimin_fminimizer_alloc (T, np);
+       gsl_multimin_fminimizer_set (s, &minex_func, x, ss);
+
+       do
+         {
+           iter++;
+           status = gsl_multimin_fminimizer_iterate(s);
+
+           if (status)
+             break;
+
+           size = gsl_multimin_fminimizer_size (s);
+           status = gsl_multimin_test_size (size, 1e-2);
+
+           if (status == GSL_SUCCESS)
+             {
+               printf ("converged to minimum at\n");
+             }
+
+           printf ("%5d ", iter);
+           for (i = 0; i < np; i++)
+             {
+               printf ("%10.3e ", gsl_vector_get (s->x, i));
+             }
+           printf ("f() = %7.3f size = %.3f\n", s->fval, size);
+         }
+       while (status == GSL_CONTINUE && iter < 100);
+
+       gsl_vector_free(x);
+       gsl_vector_free(ss);
+       gsl_multimin_fminimizer_free (s);
+
+       return status;
+     }
+
+The minimum search stops when the Simplex size drops to 0.01. The
+output is shown below.
+
+         1  6.500e+00  5.000e+00 f() = 512.500 size = 1.082
+         2  5.250e+00  4.000e+00 f() = 290.625 size = 1.372
+         3  5.250e+00  4.000e+00 f() = 290.625 size = 1.372
+         4  5.500e+00  1.000e+00 f() = 252.500 size = 1.372
+         5  2.625e+00  3.500e+00 f() = 101.406 size = 1.823
+         6  3.469e+00  1.375e+00 f() = 98.760  size = 1.526
+         7  1.820e+00  3.156e+00 f() = 63.467  size = 1.105
+         8  1.820e+00  3.156e+00 f() = 63.467  size = 1.105
+         9  1.016e+00  2.812e+00 f() = 43.206  size = 1.105
+        10  2.041e+00  2.008e+00 f() = 40.838  size = 0.645
+        11  1.236e+00  1.664e+00 f() = 32.816  size = 0.645
+        12  1.236e+00  1.664e+00 f() = 32.816  size = 0.447
+        13  5.225e-01  1.980e+00 f() = 32.288  size = 0.447
+        14  1.103e+00  2.073e+00 f() = 30.214  size = 0.345
+        15  1.103e+00  2.073e+00 f() = 30.214  size = 0.264
+        16  1.103e+00  2.073e+00 f() = 30.214  size = 0.160
+        17  9.864e-01  1.934e+00 f() = 30.090  size = 0.132
+        18  9.190e-01  1.987e+00 f() = 30.069  size = 0.092
+        19  1.028e+00  2.017e+00 f() = 30.013  size = 0.056
+        20  1.028e+00  2.017e+00 f() = 30.013  size = 0.046
+        21  1.028e+00  2.017e+00 f() = 30.013  size = 0.033
+        22  9.874e-01  1.985e+00 f() = 30.006  size = 0.028
+        23  9.846e-01  1.995e+00 f() = 30.003  size = 0.023
+        24  1.007e+00  2.003e+00 f() = 30.001  size = 0.012
+     converged to minimum at
+        25  1.007e+00  2.003e+00 f() = 30.001  size = 0.010
+
+The simplex size first increases, while the simplex moves towards the
+minimum. After a while the size begins to decrease as the simplex
+contracts around the minimum.
+
+
+File: gsl-ref.info,  Node: Multimin References and Further Reading,  Prev: Multimin Examples,  Up: Multidimensional Minimization
+
+35.9 References and Further Reading
+===================================
+
+The conjugate gradient and BFGS methods are described in detail in the
+following book,
+
+     R. Fletcher, `Practical Methods of Optimization (Second Edition)'
+     Wiley (1987), ISBN 0471915475.
+
+   A brief description of multidimensional minimization algorithms and
+more recent references can be found in,
+
+     C.W. Ueberhuber, `Numerical Computation (Volume 2)', Chapter 14,
+     Section 4.4 "Minimization Methods", p. 325-335, Springer (1997),
+     ISBN 3-540-62057-5.
+
+The simplex algorithm is described in the following paper,
+
+     J.A. Nelder and R. Mead, `A simplex method for function
+     minimization', Computer Journal vol. 7 (1965), 308-315.
+
+
+
+File: gsl-ref.info,  Node: Least-Squares Fitting,  Next: Nonlinear Least-Squares Fitting,  Prev: Multidimensional Minimization,  Up: Top
+
+36 Least-Squares Fitting
+************************
+
+This chapter describes routines for performing least squares fits to
+experimental data using linear combinations of functions.  The data may
+be weighted or unweighted, i.e. with known or unknown errors.  For
+weighted data the functions compute the best fit parameters and their
+associated covariance matrix.  For unweighted data the covariance
+matrix is estimated from the scatter of the points, giving a
+variance-covariance matrix.
+
+   The functions are divided into separate versions for simple one- or
+two-parameter regression and multiple-parameter fits.  The functions
+are declared in the header file `gsl_fit.h'.
+
+* Menu:
+
+* Fitting Overview::
+* Linear regression::
+* Linear fitting without a constant term::
+* Multi-parameter fitting::
+* Fitting Examples::
+* Fitting References and Further Reading::
+
+
+File: gsl-ref.info,  Node: Fitting Overview,  Next: Linear regression,  Up: Least-Squares Fitting
+
+36.1 Overview
+=============
+
+Least-squares fits are found by minimizing \chi^2 (chi-squared), the
+weighted sum of squared residuals over n experimental datapoints (x_i,
+y_i) for the model Y(c,x),
+
+     \chi^2 = \sum_i w_i (y_i - Y(c, x_i))^2
+
+The p parameters of the model are c = {c_0, c_1, ...}.  The weight
+factors w_i are given by w_i = 1/\sigma_i^2, where \sigma_i is the
+experimental error on the data-point y_i.  The errors are assumed to be
+gaussian and uncorrelated.  For unweighted data the chi-squared sum is
+computed without any weight factors.
+
+   The fitting routines return the best-fit parameters c and their p
+\times p covariance matrix.  The covariance matrix measures the
+statistical errors on the best-fit parameters resulting from the errors
+on the data, \sigma_i, and is defined as C_{ab} = <\delta c_a \delta
+c_b> where < > denotes an average over the gaussian error distributions
+of the underlying datapoints.
+
+   The covariance matrix is calculated by error propagation from the
+data errors \sigma_i.  The change in a fitted parameter \delta c_a
+caused by a small change in the data \delta y_i is given by
+
+     \delta c_a = \sum_i (dc_a/dy_i) \delta y_i
+
+allowing the covariance matrix to be written in terms of the errors on
+the data,
+
+     C_{ab} = \sum_{i,j} (dc_a/dy_i) (dc_b/dy_j) <\delta y_i \delta y_j>
+
+For uncorrelated data the fluctuations of the underlying datapoints
+satisfy <\delta y_i \delta y_j> = \sigma_i^2 \delta_{ij}, giving a
+corresponding parameter covariance matrix of
+
+     C_{ab} = \sum_i (1/w_i) (dc_a/dy_i) (dc_b/dy_i)
+
+When computing the covariance matrix for unweighted data, i.e. data
+with unknown errors, the weight factors w_i in this sum are replaced by
+the single estimate w = 1/\sigma^2, where \sigma^2 is the computed
+variance of the residuals about the best-fit model, \sigma^2 = \sum
+(y_i - Y(c,x_i))^2 / (n-p).  This is referred to as the
+"variance-covariance matrix".  
+
+   The standard deviations of the best-fit parameters are given by the
+square root of the corresponding diagonal elements of the covariance
+matrix, \sigma_{c_a} = \sqrt{C_{aa}}.
+
+
+File: gsl-ref.info,  Node: Linear regression,  Next: Linear fitting without a constant term,  Prev: Fitting Overview,  Up: Least-Squares Fitting
+
+36.2 Linear regression
+======================
+
+The functions described in this section can be used to perform
+least-squares fits to a straight line model, Y(c,x) = c_0 + c_1 x.
+
+ -- Function: int gsl_fit_linear (const double * X, const size_t
+          XSTRIDE, const double * Y, const size_t YSTRIDE, size_t N,
+          double * C0, double * C1, double * COV00, double * COV01,
+          double * COV11, double * SUMSQ)
+     This function computes the best-fit linear regression coefficients
+     (C0,C1) of the model Y = c_0 + c_1 X for the dataset (X, Y), two
+     vectors of length N with strides XSTRIDE and YSTRIDE.  The errors
+     on Y are assumed unknown so the variance-covariance matrix for the
+     parameters (C0, C1) is estimated from the scatter of the points
+     around the best-fit line and returned via the parameters (COV00,
+     COV01, COV11).  The sum of squares of the residuals from the
+     best-fit line is returned in SUMSQ.
+
+ -- Function: int gsl_fit_wlinear (const double * X, const size_t
+          XSTRIDE, const double * W, const size_t WSTRIDE, const double
+          * Y, const size_t YSTRIDE, size_t N, double * C0, double *
+          C1, double * COV00, double * COV01, double * COV11, double *
+          CHISQ)
+     This function computes the best-fit linear regression coefficients
+     (C0,C1) of the model Y = c_0 + c_1 X for the weighted dataset (X,
+     Y), two vectors of length N with strides XSTRIDE and YSTRIDE.  The
+     vector W, of length N and stride WSTRIDE, specifies the weight of
+     each datapoint. The weight is the reciprocal of the variance for
+     each datapoint in Y.
+
+     The covariance matrix for the parameters (C0, C1) is computed
+     using the weights and returned via the parameters (COV00, COV01,
+     COV11).  The weighted sum of squares of the residuals from the
+     best-fit line, \chi^2, is returned in CHISQ.
+
+ -- Function: int gsl_fit_linear_est (double X, double C0, double C1,
+          double C00, double C01, double C11, double * Y, double *
+          Y_ERR)
+     This function uses the best-fit linear regression coefficients
+     C0,C1 and their covariance COV00,COV01,COV11 to compute the fitted
+     function Y and its standard deviation Y_ERR for the model Y = c_0
+     + c_1 X at the point X.
+
diff --git a/gsl-1.9/doc/gsl-ref.info-4 b/gsl-1.9/doc/gsl-ref.info-4
new file mode 100644
index 0000000..cedfd25
--- /dev/null
+++ b/gsl-1.9/doc/gsl-ref.info-4
@@ -0,0 +1,4426 @@
+This is gsl-ref.info, produced by makeinfo version 4.8 from
+gsl-ref.texi.
+
+INFO-DIR-SECTION Scientific software
+START-INFO-DIR-ENTRY
+* gsl-ref: (gsl-ref).                   GNU Scientific Library - Reference
+END-INFO-DIR-ENTRY
+
+   Copyright (C) 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004,
+2005, 2006, 2007 The GSL Team.
+
+   Permission is granted to copy, distribute and/or modify this document
+under the terms of the GNU Free Documentation License, Version 1.2 or
+any later version published by the Free Software Foundation; with the
+Invariant Sections being "GNU General Public License" and "Free Software
+Needs Free Documentation", the Front-Cover text being "A GNU Manual",
+and with the Back-Cover Text being (a) (see below).  A copy of the
+license is included in the section entitled "GNU Free Documentation
+License".
+
+   (a) The Back-Cover Text is: "You have freedom to copy and modify this
+GNU Manual, like GNU software."
+
+
+File: gsl-ref.info,  Node: Linear fitting without a constant term,  Next: Multi-parameter fitting,  Prev: Linear regression,  Up: Least-Squares Fitting
+
+36.3 Linear fitting without a constant term
+===========================================
+
+The functions described in this section can be used to perform
+least-squares fits to a straight line model without a constant term, Y
+= c_1 X.
+
+ -- Function: int gsl_fit_mul (const double * X, const size_t XSTRIDE,
+          const double * Y, const size_t YSTRIDE, size_t N, double *
+          C1, double * COV11, double * SUMSQ)
+     This function computes the best-fit linear regression coefficient
+     C1 of the model Y = c_1 X for the datasets (X, Y), two vectors of
+     length N with strides XSTRIDE and YSTRIDE.  The errors on Y are
+     assumed unknown so the variance of the parameter C1 is estimated
+     from the scatter of the points around the best-fit line and
+     returned via the parameter COV11.  The sum of squares of the
+     residuals from the best-fit line is returned in SUMSQ.
+
+ -- Function: int gsl_fit_wmul (const double * X, const size_t XSTRIDE,
+          const double * W, const size_t WSTRIDE, const double * Y,
+          const size_t YSTRIDE, size_t N, double * C1, double * COV11,
+          double * SUMSQ)
+     This function computes the best-fit linear regression coefficient
+     C1 of the model Y = c_1 X for the weighted datasets (X, Y), two
+     vectors of length N with strides XSTRIDE and YSTRIDE.  The vector
+     W, of length N and stride WSTRIDE, specifies the weight of each
+     datapoint. The weight is the reciprocal of the variance for each
+     datapoint in Y.
+
+     The variance of the parameter C1 is computed using the weights and
+     returned via the parameter COV11.  The weighted sum of squares of
+     the residuals from the best-fit line, \chi^2, is returned in CHISQ.
+
+ -- Function: int gsl_fit_mul_est (double X, double C1, double C11,
+          double * Y, double * Y_ERR)
+     This function uses the best-fit linear regression coefficient C1
+     and its covariance COV11 to compute the fitted function Y and its
+     standard deviation Y_ERR for the model Y = c_1 X at the point X.
+
+
+File: gsl-ref.info,  Node: Multi-parameter fitting,  Next: Fitting Examples,  Prev: Linear fitting without a constant term,  Up: Least-Squares Fitting
+
+36.4 Multi-parameter fitting
+============================
+
+The functions described in this section perform least-squares fits to a
+general linear model, y = X c where y is a vector of n observations, X
+is an n by p matrix of predictor variables, and the elements of the
+vector c are the p unknown best-fit parameters which are to be
+estimated.  The chi-squared value is given by \chi^2 = \sum_i w_i (y_i
+- \sum_j X_{ij} c_j)^2.
+
+   This formulation can be used for fits to any number of functions
+and/or variables by preparing the n-by-p matrix X appropriately.  For
+example, to fit to a p-th order polynomial in X, use the following
+matrix,
+
+     X_{ij} = x_i^j
+
+where the index i runs over the observations and the index j runs from
+0 to p-1.
+
+   To fit to a set of p sinusoidal functions with fixed frequencies
+\omega_1, \omega_2, ..., \omega_p, use,
+
+     X_{ij} = sin(\omega_j x_i)
+
+To fit to p independent variables x_1, x_2, ..., x_p, use,
+
+     X_{ij} = x_j(i)
+
+where x_j(i) is the i-th value of the predictor variable x_j.
+
+   The functions described in this section are declared in the header
+file `gsl_multifit.h'.
+
+   The solution of the general linear least-squares system requires an
+additional working space for intermediate results, such as the singular
+value decomposition of the matrix X.
+
+ -- Function: gsl_multifit_linear_workspace * gsl_multifit_linear_alloc
+          (size_t N, size_t P)
+     This function allocates a workspace for fitting a model to N
+     observations using P parameters.
+
+ -- Function: void gsl_multifit_linear_free
+          (gsl_multifit_linear_workspace * WORK)
+     This function frees the memory associated with the workspace W.
+
+ -- Function: int gsl_multifit_linear (const gsl_matrix * X, const
+          gsl_vector * Y, gsl_vector * C, gsl_matrix * COV, double *
+          CHISQ, gsl_multifit_linear_workspace * WORK)
+ -- Function: int gsl_multifit_linear_svd (const gsl_matrix * X, const
+          gsl_vector * Y, double TOL, size_t * RANK, gsl_vector * C,
+          gsl_matrix * COV, double * CHISQ,
+          gsl_multifit_linear_workspace * WORK)
+     These functions compute the best-fit parameters C of the model y =
+     X c for the observations Y and the matrix of predictor variables
+     X.  The variance-covariance matrix of the model parameters COV is
+     estimated from the scatter of the observations about the best-fit.
+     The sum of squares of the residuals from the best-fit, \chi^2, is
+     returned in CHISQ.
+
+     The best-fit is found by singular value decomposition of the matrix
+     X using the preallocated workspace provided in WORK. The modified
+     Golub-Reinsch SVD algorithm is used, with column scaling to
+     improve the accuracy of the singular values. Any components which
+     have zero singular value (to machine precision) are discarded from
+     the fit.  In the second form of the function the components are
+     discarded if the ratio of singular values s_i/s_0 falls below the
+     user-specified tolerance TOL, and the effective rank is returned
+     in RANK.
+
+ -- Function: int gsl_multifit_wlinear (const gsl_matrix * X, const
+          gsl_vector * W, const gsl_vector * Y, gsl_vector * C,
+          gsl_matrix * COV, double * CHISQ,
+          gsl_multifit_linear_workspace * WORK)
+ -- Function: int gsl_multifit_wlinear_svd (const gsl_matrix * X, const
+          gsl_vector * W, const gsl_vector * Y, double TOL, size_t *
+          RANK, gsl_vector * C, gsl_matrix * COV, double * CHISQ,
+          gsl_multifit_linear_workspace * WORK)
+     This function computes the best-fit parameters C of the weighted
+     model y = X c for the observations Y with weights W and the matrix
+     of predictor variables X.  The covariance matrix of the model
+     parameters COV is computed with the given weights.  The weighted
+     sum of squares of the residuals from the best-fit, \chi^2, is
+     returned in CHISQ.
+
+     The best-fit is found by singular value decomposition of the matrix
+     X using the preallocated workspace provided in WORK. Any
+     components which have zero singular value (to machine precision)
+     are discarded from the fit.  In the second form of the function the
+     components are discarded if the ratio of singular values s_i/s_0
+     falls below the user-specified tolerance TOL, and the effective
+     rank is returned in RANK.
+
+ -- Function: int gsl_multifit_linear_est (const gsl_vector * X, const
+          gsl_vector * C, const gsl_matrix * COV, double * Y, double *
+          Y_ERR)
+     This function uses the best-fit multilinear regression coefficients
+     C and their covariance matrix COV to compute the fitted function
+     value Y and its standard deviation Y_ERR for the model y = x.c at
+     the point X.
+
+
+File: gsl-ref.info,  Node: Fitting Examples,  Next: Fitting References and Further Reading,  Prev: Multi-parameter fitting,  Up: Least-Squares Fitting
+
+36.5 Examples
+=============
+
+The following program computes a least squares straight-line fit to a
+simple dataset, and outputs the best-fit line and its associated one
+standard-deviation error bars.
+
+     #include <stdio.h>
+     #include <gsl/gsl_fit.h>
+
+     int
+     main (void)
+     {
+       int i, n = 4;
+       double x[4] = { 1970, 1980, 1990, 2000 };
+       double y[4] = {   12,   11,   14,   13 };
+       double w[4] = {  0.1,  0.2,  0.3,  0.4 };
+
+       double c0, c1, cov00, cov01, cov11, chisq;
+
+       gsl_fit_wlinear (x, 1, w, 1, y, 1, n,
+                        &c0, &c1, &cov00, &cov01, &cov11,
+                        &chisq);
+
+       printf ("# best fit: Y = %g + %g X\n", c0, c1);
+       printf ("# covariance matrix:\n");
+       printf ("# [ %g, %g\n#   %g, %g]\n",
+               cov00, cov01, cov01, cov11);
+       printf ("# chisq = %g\n", chisq);
+
+       for (i = 0; i < n; i++)
+         printf ("data: %g %g %g\n",
+                        x[i], y[i], 1/sqrt(w[i]));
+
+       printf ("\n");
+
+       for (i = -30; i < 130; i++)
+         {
+           double xf = x[0] + (i/100.0) * (x[n-1] - x[0]);
+           double yf, yf_err;
+
+           gsl_fit_linear_est (xf,
+                               c0, c1,
+                               cov00, cov01, cov11,
+                               &yf, &yf_err);
+
+           printf ("fit: %g %g\n", xf, yf);
+           printf ("hi : %g %g\n", xf, yf + yf_err);
+           printf ("lo : %g %g\n", xf, yf - yf_err);
+         }
+       return 0;
+     }
+
+The following commands extract the data from the output of the program
+and display it using the GNU plotutils `graph' utility,
+
+     $ ./demo > tmp
+     $ more tmp
+     # best fit: Y = -106.6 + 0.06 X
+     # covariance matrix:
+     # [ 39602, -19.9
+     #   -19.9, 0.01]
+     # chisq = 0.8
+
+     $ for n in data fit hi lo ;
+        do
+          grep "^$n" tmp | cut -d: -f2 > $n ;
+        done
+     $ graph -T X -X x -Y y -y 0 20 -m 0 -S 2 -Ie data
+          -S 0 -I a -m 1 fit -m 2 hi -m 2 lo
+
+   The next program performs a quadratic fit y = c_0 + c_1 x + c_2 x^2
+to a weighted dataset using the generalised linear fitting function
+`gsl_multifit_wlinear'.  The model matrix X for a quadratic fit is
+given by,
+
+     X = [ 1   , x_0  , x_0^2 ;
+           1   , x_1  , x_1^2 ;
+           1   , x_2  , x_2^2 ;
+           ... , ...  , ...   ]
+
+where the column of ones corresponds to the constant term c_0.  The two
+remaining columns corresponds to the terms c_1 x and c_2 x^2.
+
+   The program reads N lines of data in the format (X, Y, ERR) where
+ERR is the error (standard deviation) in the value Y.
+
+     #include <stdio.h>
+     #include <gsl/gsl_multifit.h>
+
+     int
+     main (int argc, char **argv)
+     {
+       int i, n;
+       double xi, yi, ei, chisq;
+       gsl_matrix *X, *cov;
+       gsl_vector *y, *w, *c;
+
+       if (argc != 2)
+         {
+           fprintf (stderr,"usage: fit n < data\n");
+           exit (-1);
+         }
+
+       n = atoi (argv[1]);
+
+       X = gsl_matrix_alloc (n, 3);
+       y = gsl_vector_alloc (n);
+       w = gsl_vector_alloc (n);
+
+       c = gsl_vector_alloc (3);
+       cov = gsl_matrix_alloc (3, 3);
+
+       for (i = 0; i < n; i++)
+         {
+           int count = fscanf (stdin, "%lg %lg %lg",
+                               &xi, &yi, &ei);
+
+           if (count != 3)
+             {
+               fprintf (stderr, "error reading file\n");
+               exit (-1);
+             }
+
+           printf ("%g %g +/- %g\n", xi, yi, ei);
+
+           gsl_matrix_set (X, i, 0, 1.0);
+           gsl_matrix_set (X, i, 1, xi);
+           gsl_matrix_set (X, i, 2, xi*xi);
+
+           gsl_vector_set (y, i, yi);
+           gsl_vector_set (w, i, 1.0/(ei*ei));
+         }
+
+       {
+         gsl_multifit_linear_workspace * work
+           = gsl_multifit_linear_alloc (n, 3);
+         gsl_multifit_wlinear (X, w, y, c, cov,
+                               &chisq, work);
+         gsl_multifit_linear_free (work);
+       }
+
+     #define C(i) (gsl_vector_get(c,(i)))
+     #define COV(i,j) (gsl_matrix_get(cov,(i),(j)))
+
+       {
+         printf ("# best fit: Y = %g + %g X + %g X^2\n",
+                 C(0), C(1), C(2));
+
+         printf ("# covariance matrix:\n");
+         printf ("[ %+.5e, %+.5e, %+.5e  \n",
+                    COV(0,0), COV(0,1), COV(0,2));
+         printf ("  %+.5e, %+.5e, %+.5e  \n",
+                    COV(1,0), COV(1,1), COV(1,2));
+         printf ("  %+.5e, %+.5e, %+.5e ]\n",
+                    COV(2,0), COV(2,1), COV(2,2));
+         printf ("# chisq = %g\n", chisq);
+       }
+
+       gsl_matrix_free (X);
+       gsl_vector_free (y);
+       gsl_vector_free (w);
+       gsl_vector_free (c);
+       gsl_matrix_free (cov);
+
+       return 0;
+     }
+
+A suitable set of data for fitting can be generated using the following
+program.  It outputs a set of points with gaussian errors from the curve
+y = e^x in the region 0 < x < 2.
+
+     #include <stdio.h>
+     #include <math.h>
+     #include <gsl/gsl_randist.h>
+
+     int
+     main (void)
+     {
+       double x;
+       const gsl_rng_type * T;
+       gsl_rng * r;
+
+       gsl_rng_env_setup ();
+
+       T = gsl_rng_default;
+       r = gsl_rng_alloc (T);
+
+       for (x = 0.1; x < 2; x+= 0.1)
+         {
+           double y0 = exp (x);
+           double sigma = 0.1 * y0;
+           double dy = gsl_ran_gaussian (r, sigma);
+
+           printf ("%g %g %g\n", x, y0 + dy, sigma);
+         }
+
+       gsl_rng_free(r);
+
+       return 0;
+     }
+
+The data can be prepared by running the resulting executable program,
+
+     $ ./generate > exp.dat
+     $ more exp.dat
+     0.1 0.97935 0.110517
+     0.2 1.3359 0.12214
+     0.3 1.52573 0.134986
+     0.4 1.60318 0.149182
+     0.5 1.81731 0.164872
+     0.6 1.92475 0.182212
+     ....
+
+To fit the data use the previous program, with the number of data points
+given as the first argument.  In this case there are 19 data points.
+
+     $ ./fit 19 < exp.dat
+     0.1 0.97935 +/- 0.110517
+     0.2 1.3359 +/- 0.12214
+     ...
+     # best fit: Y = 1.02318 + 0.956201 X + 0.876796 X^2
+     # covariance matrix:
+     [ +1.25612e-02, -3.64387e-02, +1.94389e-02
+       -3.64387e-02, +1.42339e-01, -8.48761e-02
+       +1.94389e-02, -8.48761e-02, +5.60243e-02 ]
+     # chisq = 23.0987
+
+The parameters of the quadratic fit match the coefficients of the
+expansion of e^x, taking into account the errors on the parameters and
+the O(x^3) difference between the exponential and quadratic functions
+for the larger values of x.  The errors on the parameters are given by
+the square-root of the corresponding diagonal elements of the
+covariance matrix.  The chi-squared per degree of freedom is 1.4,
+indicating a reasonable fit to the data.
+
+
+File: gsl-ref.info,  Node: Fitting References and Further Reading,  Prev: Fitting Examples,  Up: Least-Squares Fitting
+
+36.6 References and Further Reading
+===================================
+
+A summary of formulas and techniques for least squares fitting can be
+found in the "Statistics" chapter of the Annual Review of Particle
+Physics prepared by the Particle Data Group,
+
+     `Review of Particle Properties', R.M. Barnett et al., Physical
+     Review D54, 1 (1996) `http://pdg.lbl.gov/'
+
+The Review of Particle Physics is available online at the website given
+above.
+
+   The tests used to prepare these routines are based on the NIST
+Statistical Reference Datasets. The datasets and their documentation are
+available from NIST at the following website,
+
+           `http://www.nist.gov/itl/div898/strd/index.html'.
+
+
+File: gsl-ref.info,  Node: Nonlinear Least-Squares Fitting,  Next: Basis Splines,  Prev: Least-Squares Fitting,  Up: Top
+
+37 Nonlinear Least-Squares Fitting
+**********************************
+
+This chapter describes functions for multidimensional nonlinear
+least-squares fitting.  The library provides low level components for a
+variety of iterative solvers and convergence tests.  These can be
+combined by the user to achieve the desired solution, with full access
+to the intermediate steps of the iteration.  Each class of methods uses
+the same framework, so that you can switch between solvers at runtime
+without needing to recompile your program.  Each instance of a solver
+keeps track of its own state, allowing the solvers to be used in
+multi-threaded programs.
+
+   The header file `gsl_multifit_nlin.h' contains prototypes for the
+multidimensional nonlinear fitting functions and related declarations.
+
+* Menu:
+
+* Overview of Nonlinear Least-Squares Fitting::
+* Initializing the Nonlinear Least-Squares Solver::
+* Providing the Function to be Minimized::
+* Iteration of the Minimization Algorithm::
+* Search Stopping Parameters for Minimization Algorithms::
+* Minimization Algorithms using Derivatives::
+* Minimization Algorithms without Derivatives::
+* Computing the covariance matrix of best fit parameters::
+* Example programs for Nonlinear Least-Squares Fitting::
+* References and Further Reading for Nonlinear Least-Squares Fitting::
+
+
+File: gsl-ref.info,  Node: Overview of Nonlinear Least-Squares Fitting,  Next: Initializing the Nonlinear Least-Squares Solver,  Up: Nonlinear Least-Squares Fitting
+
+37.1 Overview
+=============
+
+The problem of multidimensional nonlinear least-squares fitting requires
+the minimization of the squared residuals of n functions, f_i, in p
+parameters, x_i,
+
+     \Phi(x) = (1/2) || F(x) ||^2
+             = (1/2) \sum_{i=1}^{n} f_i(x_1, ..., x_p)^2
+
+All algorithms proceed from an initial guess using the linearization,
+
+     \psi(p) = || F(x+p) || ~=~ || F(x) + J p ||
+
+where x is the initial point, p is the proposed step and J is the
+Jacobian matrix J_{ij} = d f_i / d x_j.  Additional strategies are used
+to enlarge the region of convergence.  These include requiring a
+decrease in the norm ||F|| on each step or using a trust region to
+avoid steps which fall outside the linear regime.
+
+   To perform a weighted least-squares fit of a nonlinear model Y(x,t)
+to data (t_i, y_i) with independent gaussian errors \sigma_i, use
+function components of the following form,
+
+     f_i = (Y(x, t_i) - y_i) / \sigma_i
+
+Note that the model parameters are denoted by x in this chapter since
+the non-linear least-squares algorithms are described geometrically
+(i.e. finding the minimum of a surface).  The independent variable of
+any data to be fitted is denoted by t.
+
+   With the definition above the Jacobian is J_{ij} =(1 / \sigma_i)  d
+Y_i / d x_j, where Y_i = Y(x,t_i).
+
+
+File: gsl-ref.info,  Node: Initializing the Nonlinear Least-Squares Solver,  Next: Providing the Function to be Minimized,  Prev: Overview of Nonlinear Least-Squares Fitting,  Up: Nonlinear Least-Squares Fitting
+
+37.2 Initializing the Solver
+============================
+
+ -- Function: gsl_multifit_fsolver * gsl_multifit_fsolver_alloc (const
+          gsl_multifit_fsolver_type * T, size_t N, size_t P)
+     This function returns a pointer to a newly allocated instance of a
+     solver of type T for N observations and P parameters.  The number
+     of observations N must be greater than or equal to parameters P.
+
+     If there is insufficient memory to create the solver then the
+     function returns a null pointer and the error handler is invoked
+     with an error code of `GSL_ENOMEM'.
+
+ -- Function: gsl_multifit_fdfsolver * gsl_multifit_fdfsolver_alloc
+          (const gsl_multifit_fdfsolver_type * T, size_t N, size_t P)
+     This function returns a pointer to a newly allocated instance of a
+     derivative solver of type T for N observations and P parameters.
+     For example, the following code creates an instance of a
+     Levenberg-Marquardt solver for 100 data points and 3 parameters,
+
+          const gsl_multifit_fdfsolver_type * T
+              = gsl_multifit_fdfsolver_lmder;
+          gsl_multifit_fdfsolver * s
+              = gsl_multifit_fdfsolver_alloc (T, 100, 3);
+
+     The number of observations N must be greater than or equal to
+     parameters P.
+
+     If there is insufficient memory to create the solver then the
+     function returns a null pointer and the error handler is invoked
+     with an error code of `GSL_ENOMEM'.
+
+ -- Function: int gsl_multifit_fsolver_set (gsl_multifit_fsolver * S,
+          gsl_multifit_function * F, gsl_vector * X)
+     This function initializes, or reinitializes, an existing solver S
+     to use the function F and the initial guess X.
+
+ -- Function: int gsl_multifit_fdfsolver_set (gsl_multifit_fdfsolver *
+          S, gsl_multifit_function_fdf * FDF, gsl_vector * X)
+     This function initializes, or reinitializes, an existing solver S
+     to use the function and derivative FDF and the initial guess X.
+
+ -- Function: void gsl_multifit_fsolver_free (gsl_multifit_fsolver * S)
+ -- Function: void gsl_multifit_fdfsolver_free (gsl_multifit_fdfsolver
+          * S)
+     These functions free all the memory associated with the solver S.
+
+ -- Function: const char * gsl_multifit_fsolver_name (const
+          gsl_multifit_fsolver * S)
+ -- Function: const char * gsl_multifit_fdfsolver_name (const
+          gsl_multifit_fdfsolver * S)
+     These functions return a pointer to the name of the solver.  For
+     example,
+
+          printf ("s is a '%s' solver\n",
+                  gsl_multifit_fdfsolver_name (s));
+
+     would print something like `s is a 'lmder' solver'.
+
+
+File: gsl-ref.info,  Node: Providing the Function to be Minimized,  Next: Iteration of the Minimization Algorithm,  Prev: Initializing the Nonlinear Least-Squares Solver,  Up: Nonlinear Least-Squares Fitting
+
+37.3 Providing the Function to be Minimized
+===========================================
+
+You must provide n functions of p variables for the minimization
+algorithms to operate on.  In order to allow for arbitrary parameters
+the functions are defined by the following data types:
+
+ -- Data Type: gsl_multifit_function
+     This data type defines a general system of functions with
+     arbitrary parameters.
+
+    `int (* f) (const gsl_vector * X, void * PARAMS, gsl_vector * F)'
+          this function should store the vector result f(x,params) in F
+          for argument X and arbitrary parameters PARAMS, returning an
+          appropriate error code if the function cannot be computed.
+
+    `size_t n'
+          the number of functions, i.e. the number of components of the
+          vector F.
+
+    `size_t p'
+          the number of independent variables, i.e. the number of
+          components of the vector X.
+
+    `void * params'
+          a pointer to the arbitrary parameters of the function.
+
+ -- Data Type: gsl_multifit_function_fdf
+     This data type defines a general system of functions with
+     arbitrary parameters and the corresponding Jacobian matrix of
+     derivatives,
+
+    `int (* f) (const gsl_vector * X, void * PARAMS, gsl_vector * F)'
+          this function should store the vector result f(x,params) in F
+          for argument X and arbitrary parameters PARAMS, returning an
+          appropriate error code if the function cannot be computed.
+
+    `int (* df) (const gsl_vector * X, void * PARAMS, gsl_matrix * J)'
+          this function should store the N-by-P matrix result J_ij = d
+          f_i(x,params) / d x_j in J for argument X and arbitrary
+          parameters PARAMS, returning an appropriate error code if the
+          function cannot be computed.
+
+    `int (* fdf) (const gsl_vector * X, void * PARAMS, gsl_vector * F, gsl_matrix * J)'
+          This function should set the values of the F and J as above,
+          for arguments X and arbitrary parameters PARAMS.  This
+          function provides an optimization of the separate functions
+          for f(x) and J(x)--it is always faster to compute the
+          function and its derivative at the same time.
+
+    `size_t n'
+          the number of functions, i.e. the number of components of the
+          vector F.
+
+    `size_t p'
+          the number of independent variables, i.e. the number of
+          components of the vector X.
+
+    `void * params'
+          a pointer to the arbitrary parameters of the function.
+
+   Note that when fitting a non-linear model against experimental data,
+the data is passed to the functions above using the PARAMS argument and
+the trial best-fit parameters through the X argument.
+
+
+File: gsl-ref.info,  Node: Iteration of the Minimization Algorithm,  Next: Search Stopping Parameters for Minimization Algorithms,  Prev: Providing the Function to be Minimized,  Up: Nonlinear Least-Squares Fitting
+
+37.4 Iteration
+==============
+
+The following functions drive the iteration of each algorithm.  Each
+function performs one iteration to update the state of any solver of the
+corresponding type.  The same functions work for all solvers so that
+different methods can be substituted at runtime without modifications to
+the code.
+
+ -- Function: int gsl_multifit_fsolver_iterate (gsl_multifit_fsolver *
+          S)
+ -- Function: int gsl_multifit_fdfsolver_iterate
+          (gsl_multifit_fdfsolver * S)
+     These functions perform a single iteration of the solver S.  If
+     the iteration encounters an unexpected problem then an error code
+     will be returned.  The solver maintains a current estimate of the
+     best-fit parameters at all times.
+
+   The solver struct S contains the following entries, which can be
+used to track the progress of the solution:
+
+`gsl_vector * x'
+     The current position.
+
+`gsl_vector * f'
+     The function value at the current position.
+
+`gsl_vector * dx'
+     The difference between the current position and the previous
+     position, i.e. the last step, taken as a vector.
+
+`gsl_matrix * J'
+     The Jacobian matrix at the current position (for the
+     `gsl_multifit_fdfsolver' struct only)
+
+   The best-fit information also can be accessed with the following
+auxiliary functions,
+
+ -- Function: gsl_vector * gsl_multifit_fsolver_position (const
+          gsl_multifit_fsolver * S)
+ -- Function: gsl_vector * gsl_multifit_fdfsolver_position (const
+          gsl_multifit_fdfsolver * S)
+     These functions return the current position (i.e. best-fit
+     parameters) `s->x' of the solver S.
+
+
+File: gsl-ref.info,  Node: Search Stopping Parameters for Minimization Algorithms,  Next: Minimization Algorithms using Derivatives,  Prev: Iteration of the Minimization Algorithm,  Up: Nonlinear Least-Squares Fitting
+
+37.5 Search Stopping Parameters
+===============================
+
+A minimization procedure should stop when one of the following
+conditions is true:
+
+   * A minimum has been found to within the user-specified precision.
+
+   * A user-specified maximum number of iterations has been reached.
+
+   * An error has occurred.
+
+The handling of these conditions is under user control.  The functions
+below allow the user to test the current estimate of the best-fit
+parameters in several standard ways.
+
+ -- Function: int gsl_multifit_test_delta (const gsl_vector * DX, const
+          gsl_vector * X, double EPSABS, double EPSREL)
+     This function tests for the convergence of the sequence by
+     comparing the last step DX with the absolute error EPSABS and
+     relative error EPSREL to the current position X.  The test returns
+     `GSL_SUCCESS' if the following condition is achieved,
+
+          |dx_i| < epsabs + epsrel |x_i|
+
+     for each component of X and returns `GSL_CONTINUE' otherwise.
+
+ -- Function: int gsl_multifit_test_gradient (const gsl_vector * G,
+          double EPSABS)
+     This function tests the residual gradient G against the absolute
+     error bound EPSABS.  Mathematically, the gradient should be
+     exactly zero at the minimum. The test returns `GSL_SUCCESS' if the
+     following condition is achieved,
+
+          \sum_i |g_i| < epsabs
+
+     and returns `GSL_CONTINUE' otherwise.  This criterion is suitable
+     for situations where the precise location of the minimum, x, is
+     unimportant provided a value can be found where the gradient is
+     small enough.
+
+ -- Function: int gsl_multifit_gradient (const gsl_matrix * J, const
+          gsl_vector * F, gsl_vector * G)
+     This function computes the gradient G of \Phi(x) = (1/2)
+     ||F(x)||^2 from the Jacobian matrix J and the function values F,
+     using the formula g = J^T f.
+
+
+File: gsl-ref.info,  Node: Minimization Algorithms using Derivatives,  Next: Minimization Algorithms without Derivatives,  Prev: Search Stopping Parameters for Minimization Algorithms,  Up: Nonlinear Least-Squares Fitting
+
+37.6 Minimization Algorithms using Derivatives
+==============================================
+
+The minimization algorithms described in this section make use of both
+the function and its derivative.  They require an initial guess for the
+location of the minimum. There is no absolute guarantee of
+convergence--the function must be suitable for this technique and the
+initial guess must be sufficiently close to the minimum for it to work.
+
+ -- Derivative Solver: gsl_multifit_fdfsolver_lmsder
+     This is a robust and efficient version of the Levenberg-Marquardt
+     algorithm as implemented in the scaled LMDER routine in MINPACK.
+     Minpack was written by Jorge J. More', Burton S. Garbow and
+     Kenneth E. Hillstrom.
+
+     The algorithm uses a generalized trust region to keep each step
+     under control.  In order to be accepted a proposed new position x'
+     must satisfy the condition |D (x' - x)| < \delta, where D is a
+     diagonal scaling matrix and \delta is the size of the trust
+     region.  The components of D are computed internally, using the
+     column norms of the Jacobian to estimate the sensitivity of the
+     residual to each component of x.  This improves the behavior of the
+     algorithm for badly scaled functions.
+
+     On each iteration the algorithm attempts to minimize the linear
+     system |F + J p| subject to the constraint |D p| < \Delta.  The
+     solution to this constrained linear system is found using the
+     Levenberg-Marquardt method.
+
+     The proposed step is now tested by evaluating the function at the
+     resulting point, x'.  If the step reduces the norm of the function
+     sufficiently, and follows the predicted behavior of the function
+     within the trust region, then it is accepted and the size of the
+     trust region is increased.  If the proposed step fails to improve
+     the solution, or differs significantly from the expected behavior
+     within the trust region, then the size of the trust region is
+     decreased and another trial step is computed.
+
+     The algorithm also monitors the progress of the solution and
+     returns an error if the changes in the solution are smaller than
+     the machine precision.  The possible error codes are,
+
+    `GSL_ETOLF'
+          the decrease in the function falls below machine precision
+
+    `GSL_ETOLX'
+          the change in the position vector falls below machine
+          precision
+
+    `GSL_ETOLG'
+          the norm of the gradient, relative to the norm of the
+          function, falls below machine precision
+
+     These error codes indicate that further iterations will be
+     unlikely to change the solution from its current value.
+
+
+ -- Derivative Solver: gsl_multifit_fdfsolver_lmder
+     This is an unscaled version of the LMDER algorithm.  The elements
+     of the diagonal scaling matrix D are set to 1.  This algorithm may
+     be useful in circumstances where the scaled version of LMDER
+     converges too slowly, or the function is already scaled
+     appropriately.
+
+
+File: gsl-ref.info,  Node: Minimization Algorithms without Derivatives,  Next: Computing the covariance matrix of best fit parameters,  Prev: Minimization Algorithms using Derivatives,  Up: Nonlinear Least-Squares Fitting
+
+37.7 Minimization Algorithms without Derivatives
+================================================
+
+There are no algorithms implemented in this section at the moment.
+
+
+File: gsl-ref.info,  Node: Computing the covariance matrix of best fit parameters,  Next: Example programs for Nonlinear Least-Squares Fitting,  Prev: Minimization Algorithms without Derivatives,  Up: Nonlinear Least-Squares Fitting
+
+37.8 Computing the covariance matrix of best fit parameters
+===========================================================
+
+ -- Function: int gsl_multifit_covar (const gsl_matrix * J, double
+          EPSREL, gsl_matrix * COVAR)
+     This function uses the Jacobian matrix J to compute the covariance
+     matrix of the best-fit parameters, COVAR.  The parameter EPSREL is
+     used to remove linear-dependent columns when J is rank deficient.
+
+     The covariance matrix is given by,
+
+          covar = (J^T J)^{-1}
+
+     and is computed by QR decomposition of J with column-pivoting.  Any
+     columns of R which satisfy
+
+          |R_{kk}| <= epsrel |R_{11}|
+
+     are considered linearly-dependent and are excluded from the
+     covariance matrix (the corresponding rows and columns of the
+     covariance matrix are set to zero).
+
+     If the minimisation uses the weighted least-squares function f_i =
+     (Y(x, t_i) - y_i) / \sigma_i then the covariance matrix above
+     gives the statistical error on the best-fit parameters resulting
+     from the gaussian errors \sigma_i on the underlying data y_i.
+     This can be verified from the relation \delta f = J \delta c and
+     the fact that the fluctuations in f from the data y_i are
+     normalised by \sigma_i and so satisfy <\delta f \delta f^T> = I.
+
+     For an unweighted least-squares function f_i = (Y(x, t_i) - y_i)
+     the covariance matrix above should be multiplied by the variance
+     of the residuals about the best-fit \sigma^2 = \sum (y_i -
+     Y(x,t_i))^2 / (n-p) to give the variance-covariance matrix
+     \sigma^2 C.  This estimates the statistical error on the best-fit
+     parameters from the scatter of the underlying data.
+
+     For more information about covariance matrices see *Note Fitting
+     Overview::.
+
+
+File: gsl-ref.info,  Node: Example programs for Nonlinear Least-Squares Fitting,  Next: References and Further Reading for Nonlinear Least-Squares Fitting,  Prev: Computing the covariance matrix of best fit parameters,  Up: Nonlinear Least-Squares Fitting
+
+37.9 Examples
+=============
+
+The following example program fits a weighted exponential model with
+background to experimental data, Y = A \exp(-\lambda t) + b. The first
+part of the program sets up the functions `expb_f' and `expb_df' to
+calculate the model and its Jacobian.  The appropriate fitting function
+is given by,
+
+     f_i = ((A \exp(-\lambda t_i) + b) - y_i)/\sigma_i
+
+where we have chosen t_i = i.  The Jacobian matrix J is the derivative
+of these functions with respect to the three parameters (A, \lambda,
+b).  It is given by,
+
+     J_{ij} = d f_i / d x_j
+
+where x_0 = A, x_1 = \lambda and x_2 = b.
+
+     /* expfit.c -- model functions for exponential + background */
+
+     struct data {
+       size_t n;
+       double * y;
+       double * sigma;
+     };
+
+     int
+     expb_f (const gsl_vector * x, void *data,
+             gsl_vector * f)
+     {
+       size_t n = ((struct data *)data)->n;
+       double *y = ((struct data *)data)->y;
+       double *sigma = ((struct data *) data)->sigma;
+
+       double A = gsl_vector_get (x, 0);
+       double lambda = gsl_vector_get (x, 1);
+       double b = gsl_vector_get (x, 2);
+
+       size_t i;
+
+       for (i = 0; i < n; i++)
+         {
+           /* Model Yi = A * exp(-lambda * i) + b */
+           double t = i;
+           double Yi = A * exp (-lambda * t) + b;
+           gsl_vector_set (f, i, (Yi - y[i])/sigma[i]);
+         }
+
+       return GSL_SUCCESS;
+     }
+
+     int
+     expb_df (const gsl_vector * x, void *data,
+              gsl_matrix * J)
+     {
+       size_t n = ((struct data *)data)->n;
+       double *sigma = ((struct data *) data)->sigma;
+
+       double A = gsl_vector_get (x, 0);
+       double lambda = gsl_vector_get (x, 1);
+
+       size_t i;
+
+       for (i = 0; i < n; i++)
+         {
+           /* Jacobian matrix J(i,j) = dfi / dxj, */
+           /* where fi = (Yi - yi)/sigma[i],      */
+           /*       Yi = A * exp(-lambda * i) + b  */
+           /* and the xj are the parameters (A,lambda,b) */
+           double t = i;
+           double s = sigma[i];
+           double e = exp(-lambda * t);
+           gsl_matrix_set (J, i, 0, e/s);
+           gsl_matrix_set (J, i, 1, -t * A * e/s);
+           gsl_matrix_set (J, i, 2, 1/s);
+         }
+       return GSL_SUCCESS;
+     }
+
+     int
+     expb_fdf (const gsl_vector * x, void *data,
+               gsl_vector * f, gsl_matrix * J)
+     {
+       expb_f (x, data, f);
+       expb_df (x, data, J);
+
+       return GSL_SUCCESS;
+     }
+
+The main part of the program sets up a Levenberg-Marquardt solver and
+some simulated random data. The data uses the known parameters
+(1.0,5.0,0.1) combined with gaussian noise (standard deviation = 0.1)
+over a range of 40 timesteps. The initial guess for the parameters is
+chosen as (0.0, 1.0, 0.0).
+
+     #include <stdlib.h>
+     #include <stdio.h>
+     #include <gsl/gsl_rng.h>
+     #include <gsl/gsl_randist.h>
+     #include <gsl/gsl_vector.h>
+     #include <gsl/gsl_blas.h>
+     #include <gsl/gsl_multifit_nlin.h>
+
+     #include "expfit.c"
+
+     #define N 40
+
+     void print_state (size_t iter, gsl_multifit_fdfsolver * s);
+
+     int
+     main (void)
+     {
+       const gsl_multifit_fdfsolver_type *T;
+       gsl_multifit_fdfsolver *s;
+       int status;
+       unsigned int i, iter = 0;
+       const size_t n = N;
+       const size_t p = 3;
+
+       gsl_matrix *covar = gsl_matrix_alloc (p, p);
+       double y[N], sigma[N];
+       struct data d = { n, y, sigma};
+       gsl_multifit_function_fdf f;
+       double x_init[3] = { 1.0, 0.0, 0.0 };
+       gsl_vector_view x = gsl_vector_view_array (x_init, p);
+       const gsl_rng_type * type;
+       gsl_rng * r;
+
+       gsl_rng_env_setup();
+
+       type = gsl_rng_default;
+       r = gsl_rng_alloc (type);
+
+       f.f = &expb_f;
+       f.df = &expb_df;
+       f.fdf = &expb_fdf;
+       f.n = n;
+       f.p = p;
+       f.params = &d;
+
+       /* This is the data to be fitted */
+
+       for (i = 0; i < n; i++)
+         {
+           double t = i;
+           y[i] = 1.0 + 5 * exp (-0.1 * t)
+                      + gsl_ran_gaussian (r, 0.1);
+           sigma[i] = 0.1;
+           printf ("data: %u %g %g\n", i, y[i], sigma[i]);
+         };
+
+       T = gsl_multifit_fdfsolver_lmsder;
+       s = gsl_multifit_fdfsolver_alloc (T, n, p);
+       gsl_multifit_fdfsolver_set (s, &f, &x.vector);
+
+       print_state (iter, s);
+
+       do
+         {
+           iter++;
+           status = gsl_multifit_fdfsolver_iterate (s);
+
+           printf ("status = %s\n", gsl_strerror (status));
+
+           print_state (iter, s);
+
+           if (status)
+             break;
+
+           status = gsl_multifit_test_delta (s->dx, s->x,
+                                             1e-4, 1e-4);
+         }
+       while (status == GSL_CONTINUE && iter < 500);
+
+       gsl_multifit_covar (s->J, 0.0, covar);
+
+     #define FIT(i) gsl_vector_get(s->x, i)
+     #define ERR(i) sqrt(gsl_matrix_get(covar,i,i))
+
+       {
+         double chi = gsl_blas_dnrm2(s->f);
+         double dof = n - p;
+         double c = GSL_MAX_DBL(1, chi / sqrt(dof));
+
+         printf("chisq/dof = %g\n",  pow(chi, 2.0) / dof);
+
+         printf ("A      = %.5f +/- %.5f\n", FIT(0), c*ERR(0));
+         printf ("lambda = %.5f +/- %.5f\n", FIT(1), c*ERR(1));
+         printf ("b      = %.5f +/- %.5f\n", FIT(2), c*ERR(2));
+       }
+
+       printf ("status = %s\n", gsl_strerror (status));
+
+       gsl_multifit_fdfsolver_free (s);
+       gsl_matrix_free (covar);
+       gsl_rng_free (r);
+       return 0;
+     }
+
+     void
+     print_state (size_t iter, gsl_multifit_fdfsolver * s)
+     {
+       printf ("iter: %3u x = % 15.8f % 15.8f % 15.8f "
+               "|f(x)| = %g\n",
+               iter,
+               gsl_vector_get (s->x, 0),
+               gsl_vector_get (s->x, 1),
+               gsl_vector_get (s->x, 2),
+               gsl_blas_dnrm2 (s->f));
+     }
+
+The iteration terminates when the change in x is smaller than 0.0001, as
+both an absolute and relative change.  Here are the results of running
+the program:
+
+     iter: 0 x=1.00000000 0.00000000 0.00000000 |f(x)|=117.349
+     status=success
+     iter: 1 x=1.64659312 0.01814772 0.64659312 |f(x)|=76.4578
+     status=success
+     iter: 2 x=2.85876037 0.08092095 1.44796363 |f(x)|=37.6838
+     status=success
+     iter: 3 x=4.94899512 0.11942928 1.09457665 |f(x)|=9.58079
+     status=success
+     iter: 4 x=5.02175572 0.10287787 1.03388354 |f(x)|=5.63049
+     status=success
+     iter: 5 x=5.04520433 0.10405523 1.01941607 |f(x)|=5.44398
+     status=success
+     iter: 6 x=5.04535782 0.10404906 1.01924871 |f(x)|=5.44397
+     chisq/dof = 0.800996
+     A      = 5.04536 +/- 0.06028
+     lambda = 0.10405 +/- 0.00316
+     b      = 1.01925 +/- 0.03782
+     status = success
+
+The approximate values of the parameters are found correctly, and the
+chi-squared value indicates a good fit (the chi-squared per degree of
+freedom is approximately 1).  In this case the errors on the parameters
+can be estimated from the square roots of the diagonal elements of the
+covariance matrix.
+
+   If the chi-squared value shows a poor fit (i.e. chi^2/dof >> 1) then
+the error estimates obtained from the covariance matrix will be too
+small.  In the example program the error estimates are multiplied by
+\sqrt{\chi^2/dof} in this case, a common way of increasing the errors
+for a poor fit.  Note that a poor fit will result from the use an
+inappropriate model, and the scaled error estimates may then be outside
+the range of validity for gaussian errors.
+
+
+File: gsl-ref.info,  Node: References and Further Reading for Nonlinear Least-Squares Fitting,  Prev: Example programs for Nonlinear Least-Squares Fitting,  Up: Nonlinear Least-Squares Fitting
+
+37.10 References and Further Reading
+====================================
+
+The MINPACK algorithm is described in the following article,
+
+     J.J. More', `The Levenberg-Marquardt Algorithm: Implementation and
+     Theory', Lecture Notes in Mathematics, v630 (1978), ed G. Watson.
+
+The following paper is also relevant to the algorithms described in this
+section,
+
+     J.J. More', B.S. Garbow, K.E. Hillstrom, "Testing Unconstrained
+     Optimization Software", ACM Transactions on Mathematical Software,
+     Vol 7, No 1 (1981), p 17-41.
+
+
+File: gsl-ref.info,  Node: Basis Splines,  Next: Physical Constants,  Prev: Nonlinear Least-Squares Fitting,  Up: Top
+
+38 Basis Splines
+****************
+
+This chapter describes functions for the computation of smoothing basis
+splines (B-splines). The header file `gsl_bspline.h' contains
+prototypes for the bspline functions and related declarations.
+
+* Menu:
+
+* Overview of B-splines::
+* Initializing the B-splines solver::
+* Constructing the knots vector::
+* Evaluation of B-spline basis functions::
+* Example programs for B-splines::
+* References and Further Reading::
+
+
+File: gsl-ref.info,  Node: Overview of B-splines,  Next: Initializing the B-splines solver,  Up: Basis Splines
+
+38.1 Overview
+=============
+
+B-splines are commonly used as basis functions to fit smoothing curves
+to large data sets. To do this, the abscissa axis is broken up into
+some number of intervals, where the endpoints of each interval are
+called "breakpoints". These breakpoints are then converted to "knots"
+by imposing various continuity and smoothness conditions at each
+interface. Given a nondecreasing knot vector t = \t_0, t_1, \dots,
+t_n+k-1\, the n basis splines of order k are defined by
+
+     B_(i,1)(x) = (1, t_i <= x < t_(i+1)
+                  (0, else
+     B_(i,k)(x) = [(x - t_i)/(t_(i+k-1) - t_i)] B_(i,k-1)(x) + [(t_(i+k) - x)/(t_(i+k) - t_(i+1))] B_(i+1,k-1)(x)
+
+   for i = 0, \dots, n-1. The common case of cubic B-splines is given
+by k = 4. The above recurrence relation can be evaluated in a
+numerically stable way by the de Boor algorithm.
+
+   If we define appropriate knots on an interval [a,b] then the
+B-spline basis functions form a complete set on that interval.
+Therefore we can expand a smoothing function as
+
+     f(x) = \sum_i c_i B_(i,k)(x)
+
+   given enough (x_j, f(x_j)) data pairs. The c_i can be readily
+obtained from a least-squares fit.
+
+
+File: gsl-ref.info,  Node: Initializing the B-splines solver,  Next: Constructing the knots vector,  Prev: Overview of B-splines,  Up: Basis Splines
+
+38.2 Initializing the B-splines solver
+======================================
+
+ -- Function: gsl_bspline_workspace * gsl_bspline_alloc (const size_t
+          K, const size_t NBREAK)
+     This function allocates a workspace for computing B-splines of
+     order K. The number of breakpoints is given by NBREAK. This leads
+     to n = nbreak + k - 2 basis functions. Cubic B-splines are
+     specified by k = 4. The size of the workspace is O(5k + nbreak).
+
+ -- Function: void gsl_bspline_free (gsl_bspline_workspace * W)
+     This function frees the memory associated with the workspace W.
+
+
+File: gsl-ref.info,  Node: Constructing the knots vector,  Next: Evaluation of B-spline basis functions,  Prev: Initializing the B-splines solver,  Up: Basis Splines
+
+38.3 Constructing the knots vector
+==================================
+
+ -- Function: int gsl_bspline_knots (const gsl_vector * BREAKPTS,
+          gsl_bspline_workspace * W)
+     This function computes the knots associated with the given
+     breakpoints and stores them internally in `w->knots'.
+
+ -- Function: int gsl_bspline_knots_uniform (const double a, const
+          double b, gsl_bspline_workspace * W)
+     This function assumes uniformly spaced breakpoints on [a,b] and
+     constructs the corresponding knot vector using the previously
+     specified NBREAK parameter. The knots are stored in `w->knots'.
+
+
+File: gsl-ref.info,  Node: Evaluation of B-spline basis functions,  Next: Example programs for B-splines,  Prev: Constructing the knots vector,  Up: Basis Splines
+
+38.4 Evaluation of B-splines
+============================
+
+ -- Function: int gsl_bspline_eval (const double X, gsl_vector * B,
+          gsl_bspline_workspace * W)
+     This function evaluates all B-spline basis functions at the
+     position X and stores them in B, so that the ith element of B is
+     B_i(x). B must be of length n = nbreak + k - 2. This value is also
+     stored in `w->n'.  It is far more efficient to compute all of the
+     basis functions at once than to compute them individually, due to
+     the nature of the defining recurrence relation.
+
+
+File: gsl-ref.info,  Node: Example programs for B-splines,  Next: References and Further Reading,  Prev: Evaluation of B-spline basis functions,  Up: Basis Splines
+
+38.5 Example programs for B-splines
+===================================
+
+The following program computes a linear least squares fit to data using
+cubic B-spline basis functions with uniform breakpoints. The data is
+generated from the curve y(x) = \cos(x) \exp(-0.1 x) on [0, 15] with
+gaussian noise added.
+
+     #include <stdio.h>
+     #include <stdlib.h>
+     #include <math.h>
+     #include <gsl/gsl_bspline.h>
+     #include <gsl/gsl_multifit.h>
+     #include <gsl/gsl_rng.h>
+     #include <gsl/gsl_randist.h>
+
+     /* number of data points to fit */
+     #define N        200
+
+     /* number of fit coefficients */
+     #define NCOEFFS  8
+
+     /* nbreak = ncoeffs + 2 - k = ncoeffs - 2 since k = 4 */
+     #define NBREAK   (NCOEFFS - 2)
+
+     int
+     main (void)
+     {
+       const size_t n = N;
+       const size_t ncoeffs = NCOEFFS;
+       const size_t nbreak = NBREAK;
+       size_t i, j;
+       gsl_bspline_workspace *bw;
+       gsl_vector *B;
+       double dy;
+       gsl_rng *r;
+       gsl_vector *c, *w;
+       gsl_vector *x, *y;
+       gsl_matrix *X, *cov;
+       gsl_multifit_linear_workspace *mw;
+       double chisq;
+
+       gsl_rng_env_setup();
+       r = gsl_rng_alloc(gsl_rng_default);
+
+       /* allocate a cubic bspline workspace (k = 4) */
+       bw = gsl_bspline_alloc(4, nbreak);
+       B = gsl_vector_alloc(ncoeffs);
+
+       x = gsl_vector_alloc(n);
+       y = gsl_vector_alloc(n);
+       X = gsl_matrix_alloc(n, ncoeffs);
+       c = gsl_vector_alloc(ncoeffs);
+       w = gsl_vector_alloc(n);
+       cov = gsl_matrix_alloc(ncoeffs, ncoeffs);
+       mw = gsl_multifit_linear_alloc(n, ncoeffs);
+
+       printf("#m=0,S=0\n");
+       /* this is the data to be fitted */
+       for (i = 0; i < n; ++i)
+         {
+           double sigma;
+           double xi = (15.0/(N-1)) * i;
+           double yi = cos(xi) * exp(-0.1 * xi);
+
+           sigma = 0.1;
+           dy = gsl_ran_gaussian(r, sigma);
+           yi += dy;
+
+           gsl_vector_set(x, i, xi);
+           gsl_vector_set(y, i, yi);
+           gsl_vector_set(w, i, 1.0 / (sigma*sigma));
+
+           printf("%f %f\n", xi, yi);
+         }
+
+       /* use uniform breakpoints on [0, 15] */
+       gsl_bspline_knots_uniform(0.0, 15.0, bw);
+
+       /* construct the fit matrix X */
+       for (i = 0; i < n; ++i)
+         {
+           double xi = gsl_vector_get(x, i);
+
+           /* compute B_j(xi) for all j */
+           gsl_bspline_eval(xi, B, bw);
+
+           /* fill in row i of X */
+           for (j = 0; j < ncoeffs; ++j)
+             {
+               double Bj = gsl_vector_get(B, j);
+               gsl_matrix_set(X, i, j, Bj);
+             }
+         }
+
+       /* do the fit */
+       gsl_multifit_wlinear(X, w, y, c, cov, &chisq, mw);
+
+       /* output the smoothed curve */
+       {
+         double xi, yi, yerr;
+
+         printf("#m=1,S=0\n");
+         for (xi = 0.0; xi < 15.0; xi += 0.1)
+           {
+             gsl_bspline_eval(xi, B, bw);
+             gsl_multifit_linear_est(B, c, cov, &yi, &yerr);
+             printf("%f %f\n", xi, yi);
+           }
+       }
+
+       gsl_rng_free(r);
+       gsl_bspline_free(bw);
+       gsl_vector_free(B);
+       gsl_vector_free(x);
+       gsl_vector_free(y);
+       gsl_matrix_free(X);
+       gsl_vector_free(c);
+       gsl_vector_free(w);
+       gsl_matrix_free(cov);
+       gsl_multifit_linear_free(mw);
+
+       return 0;
+     } /* main() */
+
+   The output can be plotted with GNU `graph'.
+
+     $ ./a.out > bspline.dat
+     $ graph -T ps -X x -Y y -x 0 15 -y -1 1.3 < bspline.dat > bspline.ps
+
+
+File: gsl-ref.info,  Node: References and Further Reading,  Prev: Example programs for B-splines,  Up: Basis Splines
+
+38.6 References and Further Reading
+===================================
+
+Further information on the algorithms described in this section can be
+found in the following book,
+
+     C. de Boor, `A Practical Guide to Splines' (1978), Springer-Verlag,
+     ISBN 0-387-90356-9.
+
+A large collection of B-spline routines is available in the PPPACK
+library available at `http://www.netlib.org/pppack'.
+
+
+File: gsl-ref.info,  Node: Physical Constants,  Next: IEEE floating-point arithmetic,  Prev: Basis Splines,  Up: Top
+
+39 Physical Constants
+*********************
+
+This chapter describes macros for the values of physical constants, such
+as the speed of light, c, and gravitational constant, G.  The values
+are available in different unit systems, including the standard MKSA
+system (meters, kilograms, seconds, amperes) and the CGSM system
+(centimeters, grams, seconds, gauss), which is commonly used in
+Astronomy.
+
+   The definitions of constants in the MKSA system are available in the
+file `gsl_const_mksa.h'.  The constants in the CGSM system are defined
+in `gsl_const_cgsm.h'.  Dimensionless constants, such as the fine
+structure constant, which are pure numbers are defined in
+`gsl_const_num.h'.
+
+* Menu:
+
+* Fundamental Constants::
+* Astronomy and Astrophysics::
+* Atomic and Nuclear Physics::
+* Measurement of Time::
+* Imperial Units ::
+* Speed and Nautical Units::
+* Printers Units::
+* Volume Area and Length::
+* Mass and Weight ::
+* Thermal Energy and Power::
+* Pressure::
+* Viscosity::
+* Light and Illumination::
+* Radioactivity::
+* Force and Energy::
+* Prefixes::
+* Physical Constant Examples::
+* Physical Constant References and Further Reading::
+
+   The full list of constants is described briefly below.  Consult the
+header files themselves for the values of the constants used in the
+library.
+
+
+File: gsl-ref.info,  Node: Fundamental Constants,  Next: Astronomy and Astrophysics,  Up: Physical Constants
+
+39.1 Fundamental Constants
+==========================
+
+`GSL_CONST_MKSA_SPEED_OF_LIGHT'
+     The speed of light in vacuum, c.
+
+`GSL_CONST_MKSA_VACUUM_PERMEABILITY'
+     The permeability of free space, \mu_0. This constant is defined in
+     the MKSA system only.
+
+`GSL_CONST_MKSA_VACUUM_PERMITTIVITY'
+     The permittivity of free space, \epsilon_0.  This constant is
+     defined in the MKSA system only.
+
+`GSL_CONST_MKSA_PLANCKS_CONSTANT_H'
+     Planck's constant, h.
+
+`GSL_CONST_MKSA_PLANCKS_CONSTANT_HBAR'
+     Planck's constant divided by 2\pi, \hbar.
+
+`GSL_CONST_NUM_AVOGADRO'
+     Avogadro's number, N_a.
+
+`GSL_CONST_MKSA_FARADAY'
+     The molar charge of 1 Faraday.
+
+`GSL_CONST_MKSA_BOLTZMANN'
+     The Boltzmann constant, k.
+
+`GSL_CONST_MKSA_MOLAR_GAS'
+     The molar gas constant, R_0.
+
+`GSL_CONST_MKSA_STANDARD_GAS_VOLUME'
+     The standard gas volume, V_0.
+
+`GSL_CONST_MKSA_STEFAN_BOLTZMANN_CONSTANT'
+     The Stefan-Boltzmann radiation constant, \sigma.
+
+`GSL_CONST_MKSA_GAUSS'
+     The magnetic field of 1 Gauss.
+
+
+File: gsl-ref.info,  Node: Astronomy and Astrophysics,  Next: Atomic and Nuclear Physics,  Prev: Fundamental Constants,  Up: Physical Constants
+
+39.2 Astronomy and Astrophysics
+===============================
+
+`GSL_CONST_MKSA_ASTRONOMICAL_UNIT'
+     The length of 1 astronomical unit (mean earth-sun distance), au.
+
+`GSL_CONST_MKSA_GRAVITATIONAL_CONSTANT'
+     The gravitational constant, G.
+
+`GSL_CONST_MKSA_LIGHT_YEAR'
+     The distance of 1 light-year, ly.
+
+`GSL_CONST_MKSA_PARSEC'
+     The distance of 1 parsec, pc.
+
+`GSL_CONST_MKSA_GRAV_ACCEL'
+     The standard gravitational acceleration on Earth, g.
+
+`GSL_CONST_MKSA_SOLAR_MASS'
+     The mass of the Sun.
+
+
+File: gsl-ref.info,  Node: Atomic and Nuclear Physics,  Next: Measurement of Time,  Prev: Astronomy and Astrophysics,  Up: Physical Constants
+
+39.3 Atomic and Nuclear Physics
+===============================
+
+`GSL_CONST_MKSA_ELECTRON_CHARGE'
+     The charge of the electron, e.
+
+`GSL_CONST_MKSA_ELECTRON_VOLT'
+     The energy of 1 electron volt, eV.
+
+`GSL_CONST_MKSA_UNIFIED_ATOMIC_MASS'
+     The unified atomic mass, amu.
+
+`GSL_CONST_MKSA_MASS_ELECTRON'
+     The mass of the electron, m_e.
+
+`GSL_CONST_MKSA_MASS_MUON'
+     The mass of the muon, m_\mu.
+
+`GSL_CONST_MKSA_MASS_PROTON'
+     The mass of the proton, m_p.
+
+`GSL_CONST_MKSA_MASS_NEUTRON'
+     The mass of the neutron, m_n.
+
+`GSL_CONST_NUM_FINE_STRUCTURE'
+     The electromagnetic fine structure constant \alpha.
+
+`GSL_CONST_MKSA_RYDBERG'
+     The Rydberg constant, Ry, in units of energy.  This is related to
+     the Rydberg inverse wavelength R by Ry = h c R.
+
+`GSL_CONST_MKSA_BOHR_RADIUS'
+     The Bohr radius, a_0.
+
+`GSL_CONST_MKSA_ANGSTROM'
+     The length of 1 angstrom.
+
+`GSL_CONST_MKSA_BARN'
+     The area of 1 barn.
+
+`GSL_CONST_MKSA_BOHR_MAGNETON'
+     The Bohr Magneton, \mu_B.
+
+`GSL_CONST_MKSA_NUCLEAR_MAGNETON'
+     The Nuclear Magneton, \mu_N.
+
+`GSL_CONST_MKSA_ELECTRON_MAGNETIC_MOMENT'
+     The absolute value of the magnetic moment of the electron, \mu_e.
+     The physical magnetic moment of the electron is negative.
+
+`GSL_CONST_MKSA_PROTON_MAGNETIC_MOMENT'
+     The magnetic moment of the proton, \mu_p.
+
+`GSL_CONST_MKSA_THOMSON_CROSS_SECTION'
+     The Thomson cross section, \sigma_T.
+
+`GSL_CONST_MKSA_DEBYE'
+     The electric dipole moment of 1 Debye, D.
+
+
+File: gsl-ref.info,  Node: Measurement of Time,  Next: Imperial Units,  Prev: Atomic and Nuclear Physics,  Up: Physical Constants
+
+39.4 Measurement of Time
+========================
+
+`GSL_CONST_MKSA_MINUTE'
+     The number of seconds in 1 minute.
+
+`GSL_CONST_MKSA_HOUR'
+     The number of seconds in 1 hour.
+
+`GSL_CONST_MKSA_DAY'
+     The number of seconds in 1 day.
+
+`GSL_CONST_MKSA_WEEK'
+     The number of seconds in 1 week.
+
+
+File: gsl-ref.info,  Node: Imperial Units,  Next: Speed and Nautical Units,  Prev: Measurement of Time,  Up: Physical Constants
+
+39.5 Imperial Units
+===================
+
+`GSL_CONST_MKSA_INCH'
+     The length of 1 inch.
+
+`GSL_CONST_MKSA_FOOT'
+     The length of 1 foot.
+
+`GSL_CONST_MKSA_YARD'
+     The length of 1 yard.
+
+`GSL_CONST_MKSA_MILE'
+     The length of 1 mile.
+
+`GSL_CONST_MKSA_MIL'
+     The length of 1 mil (1/1000th of an inch).
+
+
+File: gsl-ref.info,  Node: Speed and Nautical Units,  Next: Printers Units,  Prev: Imperial Units,  Up: Physical Constants
+
+39.6 Speed and Nautical Units
+=============================
+
+`GSL_CONST_MKSA_KILOMETERS_PER_HOUR'
+     The speed of 1 kilometer per hour.
+
+`GSL_CONST_MKSA_MILES_PER_HOUR'
+     The speed of 1 mile per hour.
+
+`GSL_CONST_MKSA_NAUTICAL_MILE'
+     The length of 1 nautical mile.
+
+`GSL_CONST_MKSA_FATHOM'
+     The length of 1 fathom.
+
+`GSL_CONST_MKSA_KNOT'
+     The speed of 1 knot.
+
+
+File: gsl-ref.info,  Node: Printers Units,  Next: Volume Area and Length,  Prev: Speed and Nautical Units,  Up: Physical Constants
+
+39.7 Printers Units
+===================
+
+`GSL_CONST_MKSA_POINT'
+     The length of 1 printer's point (1/72 inch).
+
+`GSL_CONST_MKSA_TEXPOINT'
+     The length of 1 TeX point (1/72.27 inch).
+
+
+File: gsl-ref.info,  Node: Volume Area and Length,  Next: Mass and Weight,  Prev: Printers Units,  Up: Physical Constants
+
+39.8 Volume, Area and Length
+============================
+
+`GSL_CONST_MKSA_MICRON'
+     The length of 1 micron.
+
+`GSL_CONST_MKSA_HECTARE'
+     The area of 1 hectare.
+
+`GSL_CONST_MKSA_ACRE'
+     The area of 1 acre.
+
+`GSL_CONST_MKSA_LITER'
+     The volume of 1 liter.
+
+`GSL_CONST_MKSA_US_GALLON'
+     The volume of 1 US gallon.
+
+`GSL_CONST_MKSA_CANADIAN_GALLON'
+     The volume of 1 Canadian gallon.
+
+`GSL_CONST_MKSA_UK_GALLON'
+     The volume of 1 UK gallon.
+
+`GSL_CONST_MKSA_QUART'
+     The volume of 1 quart.
+
+`GSL_CONST_MKSA_PINT'
+     The volume of 1 pint.
+
+
+File: gsl-ref.info,  Node: Mass and Weight,  Next: Thermal Energy and Power,  Prev: Volume Area and Length,  Up: Physical Constants
+
+39.9 Mass and Weight
+====================
+
+`GSL_CONST_MKSA_POUND_MASS'
+     The mass of 1 pound.
+
+`GSL_CONST_MKSA_OUNCE_MASS'
+     The mass of 1 ounce.
+
+`GSL_CONST_MKSA_TON'
+     The mass of 1 ton.
+
+`GSL_CONST_MKSA_METRIC_TON'
+     The mass of 1 metric ton (1000 kg).
+
+`GSL_CONST_MKSA_UK_TON'
+     The mass of 1 UK ton.
+
+`GSL_CONST_MKSA_TROY_OUNCE'
+     The mass of 1 troy ounce.
+
+`GSL_CONST_MKSA_CARAT'
+     The mass of 1 carat.
+
+`GSL_CONST_MKSA_GRAM_FORCE'
+     The force of 1 gram weight.
+
+`GSL_CONST_MKSA_POUND_FORCE'
+     The force of 1 pound weight.
+
+`GSL_CONST_MKSA_KILOPOUND_FORCE'
+     The force of 1 kilopound weight.
+
+`GSL_CONST_MKSA_POUNDAL'
+     The force of 1 poundal.
+
+
+File: gsl-ref.info,  Node: Thermal Energy and Power,  Next: Pressure,  Prev: Mass and Weight,  Up: Physical Constants
+
+39.10 Thermal Energy and Power
+==============================
+
+`GSL_CONST_MKSA_CALORIE'
+     The energy of 1 calorie.
+
+`GSL_CONST_MKSA_BTU'
+     The energy of 1 British Thermal Unit, btu.
+
+`GSL_CONST_MKSA_THERM'
+     The energy of 1 Therm.
+
+`GSL_CONST_MKSA_HORSEPOWER'
+     The power of 1 horsepower.
+
+
+File: gsl-ref.info,  Node: Pressure,  Next: Viscosity,  Prev: Thermal Energy and Power,  Up: Physical Constants
+
+39.11 Pressure
+==============
+
+`GSL_CONST_MKSA_BAR'
+     The pressure of 1 bar.
+
+`GSL_CONST_MKSA_STD_ATMOSPHERE'
+     The pressure of 1 standard atmosphere.
+
+`GSL_CONST_MKSA_TORR'
+     The pressure of 1 torr.
+
+`GSL_CONST_MKSA_METER_OF_MERCURY'
+     The pressure of 1 meter of mercury.
+
+`GSL_CONST_MKSA_INCH_OF_MERCURY'
+     The pressure of 1 inch of mercury.
+
+`GSL_CONST_MKSA_INCH_OF_WATER'
+     The pressure of 1 inch of water.
+
+`GSL_CONST_MKSA_PSI'
+     The pressure of 1 pound per square inch.
+
+
+File: gsl-ref.info,  Node: Viscosity,  Next: Light and Illumination,  Prev: Pressure,  Up: Physical Constants
+
+39.12 Viscosity
+===============
+
+`GSL_CONST_MKSA_POISE'
+     The dynamic viscosity of 1 poise.
+
+`GSL_CONST_MKSA_STOKES'
+     The kinematic viscosity of 1 stokes.
+
+
+File: gsl-ref.info,  Node: Light and Illumination,  Next: Radioactivity,  Prev: Viscosity,  Up: Physical Constants
+
+39.13 Light and Illumination
+============================
+
+`GSL_CONST_MKSA_STILB'
+     The luminance of 1 stilb.
+
+`GSL_CONST_MKSA_LUMEN'
+     The luminous flux of 1 lumen.
+
+`GSL_CONST_MKSA_LUX'
+     The illuminance of 1 lux.
+
+`GSL_CONST_MKSA_PHOT'
+     The illuminance of 1 phot.
+
+`GSL_CONST_MKSA_FOOTCANDLE'
+     The illuminance of 1 footcandle.
+
+`GSL_CONST_MKSA_LAMBERT'
+     The luminance of 1 lambert.
+
+`GSL_CONST_MKSA_FOOTLAMBERT'
+     The luminance of 1 footlambert.
+
+
+File: gsl-ref.info,  Node: Radioactivity,  Next: Force and Energy,  Prev: Light and Illumination,  Up: Physical Constants
+
+39.14 Radioactivity
+===================
+
+`GSL_CONST_MKSA_CURIE'
+     The activity of 1 curie.
+
+`GSL_CONST_MKSA_ROENTGEN'
+     The exposure of 1 roentgen.
+
+`GSL_CONST_MKSA_RAD'
+     The absorbed dose of 1 rad.
+
+
+File: gsl-ref.info,  Node: Force and Energy,  Next: Prefixes,  Prev: Radioactivity,  Up: Physical Constants
+
+39.15 Force and Energy
+======================
+
+`GSL_CONST_MKSA_NEWTON'
+     The SI unit of force, 1 Newton.
+
+`GSL_CONST_MKSA_DYNE'
+     The force of 1 Dyne = 10^-5 Newton.
+
+`GSL_CONST_MKSA_JOULE'
+     The SI unit of energy, 1 Joule.
+
+`GSL_CONST_MKSA_ERG'
+     The energy 1 erg = 10^-7 Joule.
+
+
+File: gsl-ref.info,  Node: Prefixes,  Next: Physical Constant Examples,  Prev: Force and Energy,  Up: Physical Constants
+
+39.16 Prefixes
+==============
+
+These constants are dimensionless scaling factors.
+
+`GSL_CONST_NUM_YOTTA'
+     10^24
+
+`GSL_CONST_NUM_ZETTA'
+     10^21
+
+`GSL_CONST_NUM_EXA'
+     10^18
+
+`GSL_CONST_NUM_PETA'
+     10^15
+
+`GSL_CONST_NUM_TERA'
+     10^12
+
+`GSL_CONST_NUM_GIGA'
+     10^9
+
+`GSL_CONST_NUM_MEGA'
+     10^6
+
+`GSL_CONST_NUM_KILO'
+     10^3
+
+`GSL_CONST_NUM_MILLI'
+     10^-3
+
+`GSL_CONST_NUM_MICRO'
+     10^-6
+
+`GSL_CONST_NUM_NANO'
+     10^-9
+
+`GSL_CONST_NUM_PICO'
+     10^-12
+
+`GSL_CONST_NUM_FEMTO'
+     10^-15
+
+`GSL_CONST_NUM_ATTO'
+     10^-18
+
+`GSL_CONST_NUM_ZEPTO'
+     10^-21
+
+`GSL_CONST_NUM_YOCTO'
+     10^-24
+
+
+File: gsl-ref.info,  Node: Physical Constant Examples,  Next: Physical Constant References and Further Reading,  Prev: Prefixes,  Up: Physical Constants
+
+39.17 Examples
+==============
+
+The following program demonstrates the use of the physical constants in
+a calculation.  In this case, the goal is to calculate the range of
+light-travel times from Earth to Mars.
+
+   The required data is the average distance of each planet from the
+Sun in astronomical units (the eccentricities and inclinations of the
+orbits will be neglected for the purposes of this calculation).  The
+average radius of the orbit of Mars is 1.52 astronomical units, and for
+the orbit of Earth it is 1 astronomical unit (by definition).  These
+values are combined with the MKSA values of the constants for the speed
+of light and the length of an astronomical unit to produce a result for
+the shortest and longest light-travel times in seconds.  The figures are
+converted into minutes before being displayed.
+
+     #include <stdio.h>
+     #include <gsl/gsl_const_mksa.h>
+
+     int
+     main (void)
+     {
+       double c  = GSL_CONST_MKSA_SPEED_OF_LIGHT;
+       double au = GSL_CONST_MKSA_ASTRONOMICAL_UNIT;
+       double minutes = GSL_CONST_MKSA_MINUTE;
+
+       /* distance stored in meters */
+       double r_earth = 1.00 * au;
+       double r_mars  = 1.52 * au;
+
+       double t_min, t_max;
+
+       t_min = (r_mars - r_earth) / c;
+       t_max = (r_mars + r_earth) / c;
+
+       printf ("light travel time from Earth to Mars:\n");
+       printf ("minimum = %.1f minutes\n", t_min / minutes);
+       printf ("maximum = %.1f minutes\n", t_max / minutes);
+
+       return 0;
+     }
+
+Here is the output from the program,
+
+     light travel time from Earth to Mars:
+     minimum = 4.3 minutes
+     maximum = 21.0 minutes
+
+
+File: gsl-ref.info,  Node: Physical Constant References and Further Reading,  Prev: Physical Constant Examples,  Up: Physical Constants
+
+39.18 References and Further Reading
+====================================
+
+The authoritative sources for physical constants are the 2002 CODATA
+recommended values, published in the articles below. Further information
+on the values of physical constants is also available from the cited
+articles and the NIST website.
+
+     Journal of Physical and Chemical Reference Data, 28(6), 1713-1852,
+     1999
+
+     Reviews of Modern Physics, 72(2), 351-495, 2000
+
+     `http://www.physics.nist.gov/cuu/Constants/index.html'
+
+     `http://physics.nist.gov/Pubs/SP811/appenB9.html'
+
+
+File: gsl-ref.info,  Node: IEEE floating-point arithmetic,  Next: Debugging Numerical Programs,  Prev: Physical Constants,  Up: Top
+
+40 IEEE floating-point arithmetic
+*********************************
+
+This chapter describes functions for examining the representation of
+floating point numbers and controlling the floating point environment of
+your program.  The functions described in this chapter are declared in
+the header file `gsl_ieee_utils.h'.
+
+* Menu:
+
+* Representation of floating point numbers::
+* Setting up your IEEE environment::
+* IEEE References and Further Reading::
+
+
+File: gsl-ref.info,  Node: Representation of floating point numbers,  Next: Setting up your IEEE environment,  Up: IEEE floating-point arithmetic
+
+40.1 Representation of floating point numbers
+=============================================
+
+The IEEE Standard for Binary Floating-Point Arithmetic defines binary
+formats for single and double precision numbers.  Each number is
+composed of three parts: a "sign bit" (s), an "exponent" (E) and a
+"fraction" (f).  The numerical value of the combination (s,E,f) is
+given by the following formula,
+
+     (-1)^s (1.fffff...) 2^E
+
+The sign bit is either zero or one.  The exponent ranges from a minimum
+value E_min to a maximum value E_max depending on the precision.  The
+exponent is converted to an unsigned number e, known as the "biased
+exponent", for storage by adding a "bias" parameter, e = E + bias.  The
+sequence fffff... represents the digits of the binary fraction f.  The
+binary digits are stored in "normalized form", by adjusting the
+exponent to give a leading digit of 1.  Since the leading digit is
+always 1 for normalized numbers it is assumed implicitly and does not
+have to be stored.  Numbers smaller than 2^(E_min) are be stored in
+"denormalized form" with a leading zero,
+
+     (-1)^s (0.fffff...) 2^(E_min)
+
+This allows gradual underflow down to 2^(E_min - p) for p bits of
+precision.  A zero is encoded with the special exponent of 2^(E_min -
+1) and infinities with the exponent of 2^(E_max + 1).
+
+The format for single precision numbers uses 32 bits divided in the
+following way,
+
+     seeeeeeeefffffffffffffffffffffff
+
+     s = sign bit, 1 bit
+     e = exponent, 8 bits  (E_min=-126, E_max=127, bias=127)
+     f = fraction, 23 bits
+
+The format for double precision numbers uses 64 bits divided in the
+following way,
+
+     seeeeeeeeeeeffffffffffffffffffffffffffffffffffffffffffffffffffff
+
+     s = sign bit, 1 bit
+     e = exponent, 11 bits  (E_min=-1022, E_max=1023, bias=1023)
+     f = fraction, 52 bits
+
+It is often useful to be able to investigate the behavior of a
+calculation at the bit-level and the library provides functions for
+printing the IEEE representations in a human-readable form.
+
+ -- Function: void gsl_ieee_fprintf_float (FILE * STREAM, const float *
+          X)
+ -- Function: void gsl_ieee_fprintf_double (FILE * STREAM, const double
+          * X)
+     These functions output a formatted version of the IEEE
+     floating-point number pointed to by X to the stream STREAM. A
+     pointer is used to pass the number indirectly, to avoid any
+     undesired promotion from `float' to `double'.  The output takes
+     one of the following forms,
+
+    `NaN'
+          the Not-a-Number symbol
+
+    `Inf, -Inf'
+          positive or negative infinity
+
+    `1.fffff...*2^E, -1.fffff...*2^E'
+          a normalized floating point number
+
+    `0.fffff...*2^E, -0.fffff...*2^E'
+          a denormalized floating point number
+
+    `0, -0'
+          positive or negative zero
+
+
+     The output can be used directly in GNU Emacs Calc mode by
+     preceding it with `2#' to indicate binary.
+
+ -- Function: void gsl_ieee_printf_float (const float * X)
+ -- Function: void gsl_ieee_printf_double (const double * X)
+     These functions output a formatted version of the IEEE
+     floating-point number pointed to by X to the stream `stdout'.
+
+The following program demonstrates the use of the functions by printing
+the single and double precision representations of the fraction 1/3.
+For comparison the representation of the value promoted from single to
+double precision is also printed.
+
+     #include <stdio.h>
+     #include <gsl/gsl_ieee_utils.h>
+
+     int
+     main (void)
+     {
+       float f = 1.0/3.0;
+       double d = 1.0/3.0;
+
+       double fd = f; /* promote from float to double */
+
+       printf (" f="); gsl_ieee_printf_float(&f);
+       printf ("\n");
+
+       printf ("fd="); gsl_ieee_printf_double(&fd);
+       printf ("\n");
+
+       printf (" d="); gsl_ieee_printf_double(&d);
+       printf ("\n");
+
+       return 0;
+     }
+
+The binary representation of 1/3 is 0.01010101... .  The output below
+shows that the IEEE format normalizes this fraction to give a leading
+digit of 1,
+
+      f= 1.01010101010101010101011*2^-2
+     fd= 1.0101010101010101010101100000000000000000000000000000*2^-2
+      d= 1.0101010101010101010101010101010101010101010101010101*2^-2
+
+The output also shows that a single-precision number is promoted to
+double-precision by adding zeros in the binary representation.
+
+
+File: gsl-ref.info,  Node: Setting up your IEEE environment,  Next: IEEE References and Further Reading,  Prev: Representation of floating point numbers,  Up: IEEE floating-point arithmetic
+
+40.2 Setting up your IEEE environment
+=====================================
+
+The IEEE standard defines several "modes" for controlling the behavior
+of floating point operations.  These modes specify the important
+properties of computer arithmetic: the direction used for rounding (e.g.
+whether numbers should be rounded up, down or to the nearest number),
+the rounding precision and how the program should handle arithmetic
+exceptions, such as division by zero.
+
+   Many of these features can now be controlled via standard functions
+such as `fpsetround', which should be used whenever they are available.
+Unfortunately in the past there has been no universal API for
+controlling their behavior--each system has had its own low-level way
+of accessing them.  To help you write portable programs GSL allows you
+to specify modes in a platform-independent way using the environment
+variable `GSL_IEEE_MODE'.  The library then takes care of all the
+necessary machine-specific initializations for you when you call the
+function `gsl_ieee_env_setup'.
+
+ -- Function: void gsl_ieee_env_setup ()
+     This function reads the environment variable `GSL_IEEE_MODE' and
+     attempts to set up the corresponding specified IEEE modes.  The
+     environment variable should be a list of keywords, separated by
+     commas, like this,
+
+          `GSL_IEEE_MODE' = "KEYWORD,KEYWORD,..."
+
+     where KEYWORD is one of the following mode-names,
+
+          `single-precision'
+
+          `double-precision'
+
+          `extended-precision'
+
+          `round-to-nearest'
+
+          `round-down'
+
+          `round-up'
+
+          `round-to-zero'
+
+          `mask-all'
+
+          `mask-invalid'
+
+          `mask-denormalized'
+
+          `mask-division-by-zero'
+
+          `mask-overflow'
+
+          `mask-underflow'
+
+          `trap-inexact'
+
+          `trap-common'
+
+     If `GSL_IEEE_MODE' is empty or undefined then the function returns
+     immediately and no attempt is made to change the system's IEEE
+     mode.  When the modes from `GSL_IEEE_MODE' are turned on the
+     function prints a short message showing the new settings to remind
+     you that the results of the program will be affected.
+
+     If the requested modes are not supported by the platform being
+     used then the function calls the error handler and returns an
+     error code of `GSL_EUNSUP'.
+
+     When options are specified using this method, the resulting mode is
+     based on a default setting of the highest available precision
+     (double precision or extended precision, depending on the
+     platform) in round-to-nearest mode, with all exceptions enabled
+     apart from the INEXACT exception.  The INEXACT exception is
+     generated whenever rounding occurs, so it must generally be
+     disabled in typical scientific calculations.  All other
+     floating-point exceptions are enabled by default, including
+     underflows and the use of denormalized numbers, for safety.  They
+     can be disabled with the individual `mask-' settings or together
+     using `mask-all'.
+
+     The following adjusted combination of modes is convenient for many
+     purposes,
+
+          GSL_IEEE_MODE="double-precision,"\
+                          "mask-underflow,"\
+                            "mask-denormalized"
+
+     This choice ignores any errors relating to small numbers (either
+     denormalized, or underflowing to zero) but traps overflows,
+     division by zero and invalid operations.
+
+To demonstrate the effects of different rounding modes consider the
+following program which computes e, the base of natural logarithms, by
+summing a rapidly-decreasing series,
+
+     e = 1 + 1/2! + 1/3! + 1/4! + ...
+       = 2.71828182846...
+
+     #include <stdio.h>
+     #include <gsl/gsl_math.h>
+     #include <gsl/gsl_ieee_utils.h>
+
+     int
+     main (void)
+     {
+       double x = 1, oldsum = 0, sum = 0;
+       int i = 0;
+
+       gsl_ieee_env_setup (); /* read GSL_IEEE_MODE */
+
+       do
+         {
+           i++;
+
+           oldsum = sum;
+           sum += x;
+           x = x / i;
+
+           printf ("i=%2d sum=%.18f error=%g\n",
+                   i, sum, sum - M_E);
+
+           if (i > 30)
+              break;
+         }
+       while (sum != oldsum);
+
+       return 0;
+     }
+
+Here are the results of running the program in `round-to-nearest' mode.
+This is the IEEE default so it isn't really necessary to specify it
+here,
+
+     $ GSL_IEEE_MODE="round-to-nearest" ./a.out
+     i= 1 sum=1.000000000000000000 error=-1.71828
+     i= 2 sum=2.000000000000000000 error=-0.718282
+     ....
+     i=18 sum=2.718281828459045535 error=4.44089e-16
+     i=19 sum=2.718281828459045535 error=4.44089e-16
+
+After nineteen terms the sum converges to within 4 \times 10^-16 of the
+correct value.  If we now change the rounding mode to `round-down' the
+final result is less accurate,
+
+     $ GSL_IEEE_MODE="round-down" ./a.out
+     i= 1 sum=1.000000000000000000 error=-1.71828
+     ....
+     i=19 sum=2.718281828459041094 error=-3.9968e-15
+
+The result is about 4 \times 10^-15 below the correct value, an order
+of magnitude worse than the result obtained in the `round-to-nearest'
+mode.
+
+   If we change to rounding mode to `round-up' then the series no
+longer converges (the reason is that when we add each term to the sum
+the final result is always rounded up.  This is guaranteed to increase
+the sum by at least one tick on each iteration).  To avoid this problem
+we would need to use a safer converge criterion, such as `while
+(fabs(sum - oldsum) > epsilon)', with a suitably chosen value of
+epsilon.
+
+   Finally we can see the effect of computing the sum using
+single-precision rounding, in the default `round-to-nearest' mode.  In
+this case the program thinks it is still using double precision numbers
+but the CPU rounds the result of each floating point operation to
+single-precision accuracy.  This simulates the effect of writing the
+program using single-precision `float' variables instead of `double'
+variables.  The iteration stops after about half the number of
+iterations and the final result is much less accurate,
+
+     $ GSL_IEEE_MODE="single-precision" ./a.out
+     ....
+     i=12 sum=2.718281984329223633 error=1.5587e-07
+
+with an error of O(10^-7), which corresponds to single precision
+accuracy (about 1 part in 10^7).  Continuing the iterations further
+does not decrease the error because all the subsequent results are
+rounded to the same value.
+
+
+File: gsl-ref.info,  Node: IEEE References and Further Reading,  Prev: Setting up your IEEE environment,  Up: IEEE floating-point arithmetic
+
+40.3 References and Further Reading
+===================================
+
+The reference for the IEEE standard is,
+
+     ANSI/IEEE Std 754-1985, IEEE Standard for Binary Floating-Point
+     Arithmetic.
+
+A more pedagogical introduction to the standard can be found in the
+following paper,
+
+     David Goldberg: What Every Computer Scientist Should Know About
+     Floating-Point Arithmetic. `ACM Computing Surveys', Vol. 23, No. 1
+     (March 1991), pages 5-48.
+
+     Corrigendum: `ACM Computing Surveys', Vol. 23, No. 3 (September
+     1991), page 413. and see also the sections by B. A. Wichmann and
+     Charles B. Dunham in Surveyor's Forum: "What Every Computer
+     Scientist Should Know About Floating-Point Arithmetic". `ACM
+     Computing Surveys', Vol. 24, No. 3 (September 1992), page 319.
+
+A detailed textbook on IEEE arithmetic and its practical use is
+available from SIAM Press,
+
+     Michael L. Overton, `Numerical Computing with IEEE Floating Point
+     Arithmetic', SIAM Press, ISBN 0898715717.
+
+
+
+File: gsl-ref.info,  Node: Debugging Numerical Programs,  Next: Contributors to GSL,  Prev: IEEE floating-point arithmetic,  Up: Top
+
+Appendix A Debugging Numerical Programs
+***************************************
+
+This chapter describes some tips and tricks for debugging numerical
+programs which use GSL.
+
+* Menu:
+
+* Using gdb::
+* Examining floating point registers::
+* Handling floating point exceptions::
+* GCC warning options for numerical programs::
+* Debugging References::
+
+
+File: gsl-ref.info,  Node: Using gdb,  Next: Examining floating point registers,  Up: Debugging Numerical Programs
+
+A.1 Using gdb
+=============
+
+Any errors reported by the library are passed to the function
+`gsl_error'.  By running your programs under gdb and setting a
+breakpoint in this function you can automatically catch any library
+errors.  You can add a breakpoint for every session by putting
+
+     break gsl_error
+
+into your `.gdbinit' file in the directory where your program is
+started.
+
+   If the breakpoint catches an error then you can use a backtrace
+(`bt') to see the call-tree, and the arguments which possibly caused
+the error.  By moving up into the calling function you can investigate
+the values of variables at that point.  Here is an example from the
+program `fft/test_trap', which contains the following line,
+
+     status = gsl_fft_complex_wavetable_alloc (0, &complex_wavetable);
+
+The function `gsl_fft_complex_wavetable_alloc' takes the length of an
+FFT as its first argument.  When this line is executed an error will be
+generated because the length of an FFT is not allowed to be zero.
+
+   To debug this problem we start `gdb', using the file `.gdbinit' to
+define a breakpoint in `gsl_error',
+
+     $ gdb test_trap
+
+     GDB is free software and you are welcome to distribute copies
+     of it under certain conditions; type "show copying" to see
+     the conditions.  There is absolutely no warranty for GDB;
+     type "show warranty" for details.  GDB 4.16 (i586-debian-linux),
+     Copyright 1996 Free Software Foundation, Inc.
+
+     Breakpoint 1 at 0x8050b1e: file error.c, line 14.
+
+When we run the program this breakpoint catches the error and shows the
+reason for it.
+
+     (gdb) run
+     Starting program: test_trap
+
+     Breakpoint 1, gsl_error (reason=0x8052b0d
+         "length n must be positive integer",
+         file=0x8052b04 "c_init.c", line=108, gsl_errno=1)
+         at error.c:14
+     14        if (gsl_error_handler)
+
+The first argument of `gsl_error' is always a string describing the
+error.  Now we can look at the backtrace to see what caused the problem,
+
+     (gdb) bt
+     #0  gsl_error (reason=0x8052b0d
+         "length n must be positive integer",
+         file=0x8052b04 "c_init.c", line=108, gsl_errno=1)
+         at error.c:14
+     #1  0x8049376 in gsl_fft_complex_wavetable_alloc (n=0,
+         wavetable=0xbffff778) at c_init.c:108
+     #2  0x8048a00 in main (argc=1, argv=0xbffff9bc)
+         at test_trap.c:94
+     #3  0x80488be in ___crt_dummy__ ()
+
+We can see that the error was generated in the function
+`gsl_fft_complex_wavetable_alloc' when it was called with an argument
+of N=0.  The original call came from line 94 in the file `test_trap.c'.
+
+   By moving up to the level of the original call we can find the line
+that caused the error,
+
+     (gdb) up
+     #1  0x8049376 in gsl_fft_complex_wavetable_alloc (n=0,
+         wavetable=0xbffff778) at c_init.c:108
+     108   GSL_ERROR ("length n must be positive integer", GSL_EDOM);
+     (gdb) up
+     #2  0x8048a00 in main (argc=1, argv=0xbffff9bc)
+         at test_trap.c:94
+     94    status = gsl_fft_complex_wavetable_alloc (0,
+             &complex_wavetable);
+
+Thus we have found the line that caused the problem.  From this point we
+could also print out the values of other variables such as
+`complex_wavetable'.
+
+
+File: gsl-ref.info,  Node: Examining floating point registers,  Next: Handling floating point exceptions,  Prev: Using gdb,  Up: Debugging Numerical Programs
+
+A.2 Examining floating point registers
+======================================
+
+The contents of floating point registers can be examined using the
+command `info float' (on supported platforms).
+
+     (gdb) info float
+          st0: 0xc4018b895aa17a945000  Valid Normal -7.838871e+308
+          st1: 0x3ff9ea3f50e4d7275000  Valid Normal 0.0285946
+          st2: 0x3fe790c64ce27dad4800  Valid Normal 6.7415931e-08
+          st3: 0x3ffaa3ef0df6607d7800  Spec  Normal 0.0400229
+          st4: 0x3c028000000000000000  Valid Normal 4.4501477e-308
+          st5: 0x3ffef5412c22219d9000  Zero  Normal 0.9580257
+          st6: 0x3fff8000000000000000  Valid Normal 1
+          st7: 0xc4028b65a1f6d243c800  Valid Normal -1.566206e+309
+        fctrl: 0x0272 53 bit; NEAR; mask DENOR UNDER LOS;
+        fstat: 0xb9ba flags 0001; top 7; excep DENOR OVERF UNDER LOS
+         ftag: 0x3fff
+          fip: 0x08048b5c
+          fcs: 0x051a0023
+       fopoff: 0x08086820
+       fopsel: 0x002b
+
+Individual registers can be examined using the variables $REG, where
+REG is the register name.
+
+     (gdb) p $st1
+     $1 = 0.02859464454261210347719
+
+
+File: gsl-ref.info,  Node: Handling floating point exceptions,  Next: GCC warning options for numerical programs,  Prev: Examining floating point registers,  Up: Debugging Numerical Programs
+
+A.3 Handling floating point exceptions
+======================================
+
+It is possible to stop the program whenever a `SIGFPE' floating point
+exception occurs.  This can be useful for finding the cause of an
+unexpected infinity or `NaN'.  The current handler settings can be
+shown with the command `info signal SIGFPE'.
+
+     (gdb) info signal SIGFPE
+     Signal  Stop  Print  Pass to program Description
+     SIGFPE  Yes   Yes    Yes             Arithmetic exception
+
+Unless the program uses a signal handler the default setting should be
+changed so that SIGFPE is not passed to the program, as this would cause
+it to exit.  The command `handle SIGFPE stop nopass' prevents this.
+
+     (gdb) handle SIGFPE stop nopass
+     Signal  Stop  Print  Pass to program Description
+     SIGFPE  Yes   Yes    No              Arithmetic exception
+
+Depending on the platform it may be necessary to instruct the kernel to
+generate signals for floating point exceptions.  For programs using GSL
+this can be achieved using the `GSL_IEEE_MODE' environment variable in
+conjunction with the function `gsl_ieee_env_setup' as described in
+*note IEEE floating-point arithmetic::.
+
+     (gdb) set env GSL_IEEE_MODE=double-precision
+
+
+File: gsl-ref.info,  Node: GCC warning options for numerical programs,  Next: Debugging References,  Prev: Handling floating point exceptions,  Up: Debugging Numerical Programs
+
+A.4 GCC warning options for numerical programs
+==============================================
+
+Writing reliable numerical programs in C requires great care.  The
+following GCC warning options are recommended when compiling numerical
+programs:
+
+     gcc -ansi -pedantic -Werror -Wall -W
+       -Wmissing-prototypes -Wstrict-prototypes
+       -Wtraditional -Wconversion -Wshadow
+       -Wpointer-arith -Wcast-qual -Wcast-align
+       -Wwrite-strings -Wnested-externs
+       -fshort-enums -fno-common -Dinline= -g -O2
+
+For details of each option consult the manual `Using and Porting GCC'.
+The following table gives a brief explanation of what types of errors
+these options catch.
+
+`-ansi -pedantic'
+     Use ANSI C, and reject any non-ANSI extensions.  These flags help
+     in writing portable programs that will compile on other systems.
+
+`-Werror'
+     Consider warnings to be errors, so that compilation stops.  This
+     prevents warnings from scrolling off the top of the screen and
+     being lost.  You won't be able to compile the program until it is
+     completely warning-free.
+
+`-Wall'
+     This turns on a set of warnings for common programming problems.
+     You need `-Wall', but it is not enough on its own.
+
+`-O2'
+     Turn on optimization.  The warnings for uninitialized variables in
+     `-Wall' rely on the optimizer to analyze the code.  If there is no
+     optimization then these warnings aren't generated.
+
+`-W'
+     This turns on some extra warnings not included in `-Wall', such as
+     missing return values and comparisons between signed and unsigned
+     integers.
+
+`-Wmissing-prototypes -Wstrict-prototypes'
+     Warn if there are any missing or inconsistent prototypes.  Without
+     prototypes it is harder to detect problems with incorrect
+     arguments.
+
+`-Wtraditional'
+     This warns about certain constructs that behave differently in
+     traditional and ANSI C. Whether the traditional or ANSI
+     interpretation is used might be unpredictable on other compilers.
+
+`-Wconversion'
+     The main use of this option is to warn about conversions from
+     signed to unsigned integers.  For example, `unsigned int x = -1'.
+     If you need to perform such a conversion you can use an explicit
+     cast.
+
+`-Wshadow'
+     This warns whenever a local variable shadows another local
+     variable.  If two variables have the same name then it is a
+     potential source of confusion.
+
+`-Wpointer-arith -Wcast-qual -Wcast-align'
+     These options warn if you try to do pointer arithmetic for types
+     which don't have a size, such as `void', if you remove a `const'
+     cast from a pointer, or if you cast a pointer to a type which has a
+     different size, causing an invalid alignment.
+
+`-Wwrite-strings'
+     This option gives string constants a `const' qualifier so that it
+     will be a compile-time error to attempt to overwrite them.
+
+`-fshort-enums'
+     This option makes the type of `enum' as short as possible.
+     Normally this makes an `enum' different from an `int'.
+     Consequently any attempts to assign a pointer-to-int to a
+     pointer-to-enum will generate a cast-alignment warning.
+
+`-fno-common'
+     This option prevents global variables being simultaneously defined
+     in different object files (you get an error at link time).  Such a
+     variable should be defined in one file and referred to in other
+     files with an `extern' declaration.
+
+`-Wnested-externs'
+     This warns if an `extern' declaration is encountered within a
+     function.
+
+`-Dinline='
+     The `inline' keyword is not part of ANSI C. Thus if you want to use
+     `-ansi' with a program which uses inline functions you can use this
+     preprocessor definition to remove the `inline' keywords.
+
+`-g'
+     It always makes sense to put debugging symbols in the executable
+     so that you can debug it using `gdb'.  The only effect of
+     debugging symbols is to increase the size of the file, and you can
+     use the `strip' command to remove them later if necessary.
+
+
+File: gsl-ref.info,  Node: Debugging References,  Prev: GCC warning options for numerical programs,  Up: Debugging Numerical Programs
+
+A.5 References and Further Reading
+==================================
+
+The following books are essential reading for anyone writing and
+debugging numerical programs with GCC and GDB.
+
+     R.M. Stallman, `Using and Porting GNU CC', Free Software
+     Foundation, ISBN 1882114388
+
+     R.M. Stallman, R.H. Pesch, `Debugging with GDB: The GNU
+     Source-Level Debugger', Free Software Foundation, ISBN 1882114779
+
+For a tutorial introduction to the GNU C Compiler and related programs,
+see
+
+     B.J. Gough, `An Introduction to GCC', Network Theory Ltd, ISBN
+     0954161793
+
+
+File: gsl-ref.info,  Node: Contributors to GSL,  Next: Autoconf Macros,  Prev: Debugging Numerical Programs,  Up: Top
+
+Appendix B Contributors to GSL
+******************************
+
+(See the AUTHORS file in the distribution for up-to-date information.)
+
+*Mark Galassi*
+     Conceived GSL (with James Theiler) and wrote the design document.
+     Wrote the simulated annealing package and the relevant chapter in
+     the manual.
+
+*James Theiler*
+     Conceived GSL (with Mark Galassi).  Wrote the random number
+     generators and the relevant chapter in this manual.
+
+*Jim Davies*
+     Wrote the statistical routines and the relevant chapter in this
+     manual.
+
+*Brian Gough*
+     FFTs, numerical integration, random number generators and
+     distributions, root finding, minimization and fitting, polynomial
+     solvers, complex numbers, physical constants, permutations, vector
+     and matrix functions, histograms, statistics, ieee-utils, revised
+     CBLAS Level 2 & 3, matrix decompositions, eigensystems, cumulative
+     distribution functions, testing, documentation and releases.
+
+*Reid Priedhorsky*
+     Wrote and documented the initial version of the root finding
+     routines while at Los Alamos National Laboratory, Mathematical
+     Modeling and Analysis Group.
+
+*Gerard Jungman*
+     Special Functions, Series acceleration, ODEs, BLAS, Linear Algebra,
+     Eigensystems, Hankel Transforms.
+
+*Mike Booth*
+     Wrote the Monte Carlo library.
+
+*Jorma Olavi Ta"htinen*
+     Wrote the initial complex arithmetic functions.
+
+*Thomas Walter*
+     Wrote the initial heapsort routines and cholesky decomposition.
+
+*Fabrice Rossi*
+     Multidimensional minimization.
+
+*Carlo Perassi*
+     Implementation of the random number generators in Knuth's
+     `Seminumerical Algorithms', 3rd Ed.
+
+*Szymon Jaroszewicz*
+     Wrote the routines for generating combinations.
+
+*Nicolas Darnis*
+     Wrote the initial routines for canonical permutations.
+
+*Jason H. Stover*
+     Wrote the major cumulative distribution functions.
+
+*Ivo Alxneit*
+     Wrote the routines for wavelet transforms.
+
+*Tuomo Keskitalo*
+     Improved the implementation of the ODE solvers.
+
+*Lowell Johnson*
+     Implementation of the Mathieu functions.
+
+*Patrick Alken*
+     Implementation of non-symmetric eigensystems and B-splines.
+
+
+   Thanks to Nigel Lowry for help in proofreading the manual.
+
+   The non-symmetric eigensystems routines contain code based on the
+LAPACK linear algebra library.  LAPACK is distributed under the
+following license:
+
+
+     Copyright (c) 1992-2006 The University of Tennessee.  All rights
+     reserved.
+
+     Redistribution and use in source and binary forms, with or without
+     modification, are permitted provided that the following conditions
+     are met:
+
+     * Redistributions of source code must retain the above copyright
+     notice, this list of conditions and the following disclaimer.
+
+     * Redistributions in binary form must reproduce the above copyright
+     notice, this list of conditions and the following disclaimer listed
+     in this license in the documentation and/or other materials
+     provided with the distribution.
+
+     * Neither the name of the copyright holders nor the names of its
+     contributors may be used to endorse or promote products derived
+     from this software without specific prior written permission.
+
+     THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
+     "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
+     LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS
+     FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE
+     COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT,
+     INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES
+     (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
+     SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
+     HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
+     CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR
+     OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE,
+     EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
+
+
+File: gsl-ref.info,  Node: Autoconf Macros,  Next: GSL CBLAS Library,  Prev: Contributors to GSL,  Up: Top
+
+Appendix C Autoconf Macros
+**************************
+
+For applications using `autoconf' the standard macro `AC_CHECK_LIB' can
+be used to link with GSL automatically from a `configure' script.  The
+library itself depends on the presence of a CBLAS and math library as
+well, so these must also be located before linking with the main
+`libgsl' file.  The following commands should be placed in the
+`configure.ac' file to perform these tests,
+
+     AC_CHECK_LIB(m,main)
+     AC_CHECK_LIB(gslcblas,main)
+     AC_CHECK_LIB(gsl,main)
+
+It is important to check for `libm' and `libgslcblas' before `libgsl',
+otherwise the tests will fail.  Assuming the libraries are found the
+output during the configure stage looks like this,
+
+     checking for main in -lm... yes
+     checking for main in -lgslcblas... yes
+     checking for main in -lgsl... yes
+
+If the library is found then the tests will define the macros
+`HAVE_LIBGSL', `HAVE_LIBGSLCBLAS', `HAVE_LIBM' and add the options
+`-lgsl -lgslcblas -lm' to the variable `LIBS'.
+
+   The tests above will find any version of the library.  They are
+suitable for general use, where the versions of the functions are not
+important.  An alternative macro is available in the file `gsl.m4' to
+test for a specific version of the library.  To use this macro simply
+add the following line to your `configure.in' file instead of the tests
+above:
+
+     AM_PATH_GSL(GSL_VERSION,
+                [action-if-found],
+                [action-if-not-found])
+
+The argument `GSL_VERSION' should be the two or three digit MAJOR.MINOR
+or MAJOR.MINOR.MICRO version number of the release you require. A
+suitable choice for `action-if-not-found' is,
+
+     AC_MSG_ERROR(could not find required version of GSL)
+
+Then you can add the variables `GSL_LIBS' and `GSL_CFLAGS' to your
+Makefile.am files to obtain the correct compiler flags.  `GSL_LIBS' is
+equal to the output of the `gsl-config --libs' command and `GSL_CFLAGS'
+is equal to `gsl-config --cflags' command. For example,
+
+     libfoo_la_LDFLAGS = -lfoo $(GSL_LIBS) -lgslcblas
+
+Note that the macro `AM_PATH_GSL' needs to use the C compiler so it
+should appear in the `configure.in' file before the macro
+`AC_LANG_CPLUSPLUS' for programs that use C++.
+
+   To test for `inline' the following test should be placed in your
+`configure.in' file,
+
+     AC_C_INLINE
+
+     if test "$ac_cv_c_inline" != no ; then
+       AC_DEFINE(HAVE_INLINE,1)
+       AC_SUBST(HAVE_INLINE)
+     fi
+
+and the macro will then be defined in the compilation flags or by
+including the file `config.h' before any library headers.
+
+   The following autoconf test will check for `extern inline',
+
+     dnl Check for "extern inline", using a modified version
+     dnl of the test for AC_C_INLINE from acspecific.mt
+     dnl
+     AC_CACHE_CHECK([for extern inline], ac_cv_c_extern_inline,
+     [ac_cv_c_extern_inline=no
+     AC_TRY_COMPILE([extern $ac_cv_c_inline double foo(double x);
+     extern $ac_cv_c_inline double foo(double x) { return x+1.0; };
+     double foo (double x) { return x + 1.0; };],
+     [  foo(1.0)  ],
+     [ac_cv_c_extern_inline="yes"])
+     ])
+
+     if test "$ac_cv_c_extern_inline" != no ; then
+       AC_DEFINE(HAVE_INLINE,1)
+       AC_SUBST(HAVE_INLINE)
+     fi
+
+   The substitution of portability functions can be made automatically
+if you use `autoconf'. For example, to test whether the BSD function
+`hypot' is available you can include the following line in the
+configure file `configure.in' for your application,
+
+     AC_CHECK_FUNCS(hypot)
+
+and place the following macro definitions in the file `config.h.in',
+
+     /* Substitute gsl_hypot for missing system hypot */
+
+     #ifndef HAVE_HYPOT
+     #define hypot gsl_hypot
+     #endif
+
+The application source files can then use the include command `#include
+<config.h>' to substitute `gsl_hypot' for each occurrence of `hypot'
+when `hypot' is not available.
+
+
+File: gsl-ref.info,  Node: GSL CBLAS Library,  Next: Free Software Needs Free Documentation,  Prev: Autoconf Macros,  Up: Top
+
+Appendix D GSL CBLAS Library
+****************************
+
+The prototypes for the low-level CBLAS functions are declared in the
+file `gsl_cblas.h'.  For the definition of the functions consult the
+documentation available from Netlib (*note BLAS References and Further
+Reading::).
+
+* Menu:
+
+* Level 1 CBLAS Functions::
+* Level 2 CBLAS Functions::
+* Level 3 CBLAS Functions::
+* GSL CBLAS Examples::
+
+
+File: gsl-ref.info,  Node: Level 1 CBLAS Functions,  Next: Level 2 CBLAS Functions,  Up: GSL CBLAS Library
+
+D.1 Level 1
+===========
+
+ -- Function: float cblas_sdsdot (const int N, const float ALPHA, const
+          float * X, const int INCX, const float * Y, const int INCY)
+
+ -- Function: double cblas_dsdot (const int N, const float * X, const
+          int INCX, const float * Y, const int INCY)
+
+ -- Function: float cblas_sdot (const int N, const float * X, const int
+          INCX, const float * Y, const int INCY)
+
+ -- Function: double cblas_ddot (const int N, const double * X, const
+          int INCX, const double * Y, const int INCY)
+
+ -- Function: void cblas_cdotu_sub (const int N, const void * X, const
+          int INCX, const void * Y, const int INCY, void * DOTU)
+
+ -- Function: void cblas_cdotc_sub (const int N, const void * X, const
+          int INCX, const void * Y, const int INCY, void * DOTC)
+
+ -- Function: void cblas_zdotu_sub (const int N, const void * X, const
+          int INCX, const void * Y, const int INCY, void * DOTU)
+
+ -- Function: void cblas_zdotc_sub (const int N, const void * X, const
+          int INCX, const void * Y, const int INCY, void * DOTC)
+
+ -- Function: float cblas_snrm2 (const int N, const float * X, const
+          int INCX)
+
+ -- Function: float cblas_sasum (const int N, const float * X, const
+          int INCX)
+
+ -- Function: double cblas_dnrm2 (const int N, const double * X, const
+          int INCX)
+
+ -- Function: double cblas_dasum (const int N, const double * X, const
+          int INCX)
+
+ -- Function: float cblas_scnrm2 (const int N, const void * X, const
+          int INCX)
+
+ -- Function: float cblas_scasum (const int N, const void * X, const
+          int INCX)
+
+ -- Function: double cblas_dznrm2 (const int N, const void * X, const
+          int INCX)
+
+ -- Function: double cblas_dzasum (const int N, const void * X, const
+          int INCX)
+
+ -- Function: CBLAS_INDEX cblas_isamax (const int N, const float * X,
+          const int INCX)
+
+ -- Function: CBLAS_INDEX cblas_idamax (const int N, const double * X,
+          const int INCX)
+
+ -- Function: CBLAS_INDEX cblas_icamax (const int N, const void * X,
+          const int INCX)
+
+ -- Function: CBLAS_INDEX cblas_izamax (const int N, const void * X,
+          const int INCX)
+
+ -- Function: void cblas_sswap (const int N, float * X, const int INCX,
+          float * Y, const int INCY)
+
+ -- Function: void cblas_scopy (const int N, const float * X, const int
+          INCX, float * Y, const int INCY)
+
+ -- Function: void cblas_saxpy (const int N, const float ALPHA, const
+          float * X, const int INCX, float * Y, const int INCY)
+
+ -- Function: void cblas_dswap (const int N, double * X, const int
+          INCX, double * Y, const int INCY)
+
+ -- Function: void cblas_dcopy (const int N, const double * X, const
+          int INCX, double * Y, const int INCY)
+
+ -- Function: void cblas_daxpy (const int N, const double ALPHA, const
+          double * X, const int INCX, double * Y, const int INCY)
+
+ -- Function: void cblas_cswap (const int N, void * X, const int INCX,
+          void * Y, const int INCY)
+
+ -- Function: void cblas_ccopy (const int N, const void * X, const int
+          INCX, void * Y, const int INCY)
+
+ -- Function: void cblas_caxpy (const int N, const void * ALPHA, const
+          void * X, const int INCX, void * Y, const int INCY)
+
+ -- Function: void cblas_zswap (const int N, void * X, const int INCX,
+          void * Y, const int INCY)
+
+ -- Function: void cblas_zcopy (const int N, const void * X, const int
+          INCX, void * Y, const int INCY)
+
+ -- Function: void cblas_zaxpy (const int N, const void * ALPHA, const
+          void * X, const int INCX, void * Y, const int INCY)
+
+ -- Function: void cblas_srotg (float * A, float * B, float * C, float
+          * S)
+
+ -- Function: void cblas_srotmg (float * D1, float * D2, float * B1,
+          const float B2, float * P)
+
+ -- Function: void cblas_srot (const int N, float * X, const int INCX,
+          float * Y, const int INCY, const float C, const float S)
+
+ -- Function: void cblas_srotm (const int N, float * X, const int INCX,
+          float * Y, const int INCY, const float * P)
+
+ -- Function: void cblas_drotg (double * A, double * B, double * C,
+          double * S)
+
+ -- Function: void cblas_drotmg (double * D1, double * D2, double * B1,
+          const double B2, double * P)
+
+ -- Function: void cblas_drot (const int N, double * X, const int INCX,
+          double * Y, const int INCY, const double C, const double S)
+
+ -- Function: void cblas_drotm (const int N, double * X, const int
+          INCX, double * Y, const int INCY, const double * P)
+
+ -- Function: void cblas_sscal (const int N, const float ALPHA, float *
+          X, const int INCX)
+
+ -- Function: void cblas_dscal (const int N, const double ALPHA, double
+          * X, const int INCX)
+
+ -- Function: void cblas_cscal (const int N, const void * ALPHA, void *
+          X, const int INCX)
+
+ -- Function: void cblas_zscal (const int N, const void * ALPHA, void *
+          X, const int INCX)
+
+ -- Function: void cblas_csscal (const int N, const float ALPHA, void *
+          X, const int INCX)
+
+ -- Function: void cblas_zdscal (const int N, const double ALPHA, void
+          * X, const int INCX)
+
+
+File: gsl-ref.info,  Node: Level 2 CBLAS Functions,  Next: Level 3 CBLAS Functions,  Prev: Level 1 CBLAS Functions,  Up: GSL CBLAS Library
+
+D.2 Level 2
+===========
+
+ -- Function: void cblas_sgemv (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_TRANSPOSE TRANSA, const int M, const int N, const
+          float ALPHA, const float * A, const int LDA, const float * X,
+          const int INCX, const float BETA, float * Y, const int INCY)
+
+ -- Function: void cblas_sgbmv (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_TRANSPOSE TRANSA, const int M, const int N, const
+          int KL, const int KU, const float ALPHA, const float * A,
+          const int LDA, const float * X, const int INCX, const float
+          BETA, float * Y, const int INCY)
+
+ -- Function: void cblas_strmv (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA,
+          const enum CBLAS_DIAG DIAG, const int N, const float * A,
+          const int LDA, float * X, const int INCX)
+
+ -- Function: void cblas_stbmv (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA,
+          const enum CBLAS_DIAG DIAG, const int N, const int K, const
+          float * A, const int LDA, float * X, const int INCX)
+
+ -- Function: void cblas_stpmv (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA,
+          const enum CBLAS_DIAG DIAG, const int N, const float * AP,
+          float * X, const int INCX)
+
+ -- Function: void cblas_strsv (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA,
+          const enum CBLAS_DIAG DIAG, const int N, const float * A,
+          const int LDA, float * X, const int INCX)
+
+ -- Function: void cblas_stbsv (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA,
+          const enum CBLAS_DIAG DIAG, const int N, const int K, const
+          float * A, const int LDA, float * X, const int INCX)
+
+ -- Function: void cblas_stpsv (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA,
+          const enum CBLAS_DIAG DIAG, const int N, const float * AP,
+          float * X, const int INCX)
+
+ -- Function: void cblas_dgemv (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_TRANSPOSE TRANSA, const int M, const int N, const
+          double ALPHA, const double * A, const int LDA, const double *
+          X, const int INCX, const double BETA, double * Y, const int
+          INCY)
+
+ -- Function: void cblas_dgbmv (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_TRANSPOSE TRANSA, const int M, const int N, const
+          int KL, const int KU, const double ALPHA, const double * A,
+          const int LDA, const double * X, const int INCX, const double
+          BETA, double * Y, const int INCY)
+
+ -- Function: void cblas_dtrmv (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA,
+          const enum CBLAS_DIAG DIAG, const int N, const double * A,
+          const int LDA, double * X, const int INCX)
+
+ -- Function: void cblas_dtbmv (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA,
+          const enum CBLAS_DIAG DIAG, const int N, const int K, const
+          double * A, const int LDA, double * X, const int INCX)
+
+ -- Function: void cblas_dtpmv (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA,
+          const enum CBLAS_DIAG DIAG, const int N, const double * AP,
+          double * X, const int INCX)
+
+ -- Function: void cblas_dtrsv (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA,
+          const enum CBLAS_DIAG DIAG, const int N, const double * A,
+          const int LDA, double * X, const int INCX)
+
+ -- Function: void cblas_dtbsv (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA,
+          const enum CBLAS_DIAG DIAG, const int N, const int K, const
+          double * A, const int LDA, double * X, const int INCX)
+
+ -- Function: void cblas_dtpsv (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA,
+          const enum CBLAS_DIAG DIAG, const int N, const double * AP,
+          double * X, const int INCX)
+
+ -- Function: void cblas_cgemv (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_TRANSPOSE TRANSA, const int M, const int N, const
+          void * ALPHA, const void * A, const int LDA, const void * X,
+          const int INCX, const void * BETA, void * Y, const int INCY)
+
+ -- Function: void cblas_cgbmv (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_TRANSPOSE TRANSA, const int M, const int N, const
+          int KL, const int KU, const void * ALPHA, const void * A,
+          const int LDA, const void * X, const int INCX, const void *
+          BETA, void * Y, const int INCY)
+
+ -- Function: void cblas_ctrmv (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA,
+          const enum CBLAS_DIAG DIAG, const int N, const void * A,
+          const int LDA, void * X, const int INCX)
+
+ -- Function: void cblas_ctbmv (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA,
+          const enum CBLAS_DIAG DIAG, const int N, const int K, const
+          void * A, const int LDA, void * X, const int INCX)
+
+ -- Function: void cblas_ctpmv (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA,
+          const enum CBLAS_DIAG DIAG, const int N, const void * AP,
+          void * X, const int INCX)
+
+ -- Function: void cblas_ctrsv (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA,
+          const enum CBLAS_DIAG DIAG, const int N, const void * A,
+          const int LDA, void * X, const int INCX)
+
+ -- Function: void cblas_ctbsv (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA,
+          const enum CBLAS_DIAG DIAG, const int N, const int K, const
+          void * A, const int LDA, void * X, const int INCX)
+
+ -- Function: void cblas_ctpsv (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA,
+          const enum CBLAS_DIAG DIAG, const int N, const void * AP,
+          void * X, const int INCX)
+
+ -- Function: void cblas_zgemv (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_TRANSPOSE TRANSA, const int M, const int N, const
+          void * ALPHA, const void * A, const int LDA, const void * X,
+          const int INCX, const void * BETA, void * Y, const int INCY)
+
+ -- Function: void cblas_zgbmv (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_TRANSPOSE TRANSA, const int M, const int N, const
+          int KL, const int KU, const void * ALPHA, const void * A,
+          const int LDA, const void * X, const int INCX, const void *
+          BETA, void * Y, const int INCY)
+
+ -- Function: void cblas_ztrmv (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA,
+          const enum CBLAS_DIAG DIAG, const int N, const void * A,
+          const int LDA, void * X, const int INCX)
+
+ -- Function: void cblas_ztbmv (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA,
+          const enum CBLAS_DIAG DIAG, const int N, const int K, const
+          void * A, const int LDA, void * X, const int INCX)
+
+ -- Function: void cblas_ztpmv (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA,
+          const enum CBLAS_DIAG DIAG, const int N, const void * AP,
+          void * X, const int INCX)
+
+ -- Function: void cblas_ztrsv (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA,
+          const enum CBLAS_DIAG DIAG, const int N, const void * A,
+          const int LDA, void * X, const int INCX)
+
+ -- Function: void cblas_ztbsv (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA,
+          const enum CBLAS_DIAG DIAG, const int N, const int K, const
+          void * A, const int LDA, void * X, const int INCX)
+
+ -- Function: void cblas_ztpsv (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANSA,
+          const enum CBLAS_DIAG DIAG, const int N, const void * AP,
+          void * X, const int INCX)
+
+ -- Function: void cblas_ssymv (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const int N, const float ALPHA, const
+          float * A, const int LDA, const float * X, const int INCX,
+          const float BETA, float * Y, const int INCY)
+
+ -- Function: void cblas_ssbmv (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const int N, const int K, const float
+          ALPHA, const float * A, const int LDA, const float * X, const
+          int INCX, const float BETA, float * Y, const int INCY)
+
+ -- Function: void cblas_sspmv (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const int N, const float ALPHA, const
+          float * AP, const float * X, const int INCX, const float
+          BETA, float * Y, const int INCY)
+
+ -- Function: void cblas_sger (const enum CBLAS_ORDER ORDER, const int
+          M, const int N, const float ALPHA, const float * X, const int
+          INCX, const float * Y, const int INCY, float * A, const int
+          LDA)
+
+ -- Function: void cblas_ssyr (const enum CBLAS_ORDER ORDER, const enum
+          CBLAS_UPLO UPLO, const int N, const float ALPHA, const float
+          * X, const int INCX, float * A, const int LDA)
+
+ -- Function: void cblas_sspr (const enum CBLAS_ORDER ORDER, const enum
+          CBLAS_UPLO UPLO, const int N, const float ALPHA, const float
+          * X, const int INCX, float * AP)
+
+ -- Function: void cblas_ssyr2 (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const int N, const float ALPHA, const
+          float * X, const int INCX, const float * Y, const int INCY,
+          float * A, const int LDA)
+
+ -- Function: void cblas_sspr2 (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const int N, const float ALPHA, const
+          float * X, const int INCX, const float * Y, const int INCY,
+          float * A)
+
+ -- Function: void cblas_dsymv (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const int N, const double ALPHA, const
+          double * A, const int LDA, const double * X, const int INCX,
+          const double BETA, double * Y, const int INCY)
+
+ -- Function: void cblas_dsbmv (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const int N, const int K, const double
+          ALPHA, const double * A, const int LDA, const double * X,
+          const int INCX, const double BETA, double * Y, const int INCY)
+
+ -- Function: void cblas_dspmv (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const int N, const double ALPHA, const
+          double * AP, const double * X, const int INCX, const double
+          BETA, double * Y, const int INCY)
+
+ -- Function: void cblas_dger (const enum CBLAS_ORDER ORDER, const int
+          M, const int N, const double ALPHA, const double * X, const
+          int INCX, const double * Y, const int INCY, double * A, const
+          int LDA)
+
+ -- Function: void cblas_dsyr (const enum CBLAS_ORDER ORDER, const enum
+          CBLAS_UPLO UPLO, const int N, const double ALPHA, const
+          double * X, const int INCX, double * A, const int LDA)
+
+ -- Function: void cblas_dspr (const enum CBLAS_ORDER ORDER, const enum
+          CBLAS_UPLO UPLO, const int N, const double ALPHA, const
+          double * X, const int INCX, double * AP)
+
+ -- Function: void cblas_dsyr2 (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const int N, const double ALPHA, const
+          double * X, const int INCX, const double * Y, const int INCY,
+          double * A, const int LDA)
+
+ -- Function: void cblas_dspr2 (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const int N, const double ALPHA, const
+          double * X, const int INCX, const double * Y, const int INCY,
+          double * A)
+
+ -- Function: void cblas_chemv (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const int N, const void * ALPHA, const
+          void * A, const int LDA, const void * X, const int INCX,
+          const void * BETA, void * Y, const int INCY)
+
+ -- Function: void cblas_chbmv (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const int N, const int K, const void *
+          ALPHA, const void * A, const int LDA, const void * X, const
+          int INCX, const void * BETA, void * Y, const int INCY)
+
+ -- Function: void cblas_chpmv (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const int N, const void * ALPHA, const
+          void * AP, const void * X, const int INCX, const void * BETA,
+          void * Y, const int INCY)
+
+ -- Function: void cblas_cgeru (const enum CBLAS_ORDER ORDER, const int
+          M, const int N, const void * ALPHA, const void * X, const int
+          INCX, const void * Y, const int INCY, void * A, const int LDA)
+
+ -- Function: void cblas_cgerc (const enum CBLAS_ORDER ORDER, const int
+          M, const int N, const void * ALPHA, const void * X, const int
+          INCX, const void * Y, const int INCY, void * A, const int LDA)
+
+ -- Function: void cblas_cher (const enum CBLAS_ORDER ORDER, const enum
+          CBLAS_UPLO UPLO, const int N, const float ALPHA, const void *
+          X, const int INCX, void * A, const int LDA)
+
+ -- Function: void cblas_chpr (const enum CBLAS_ORDER ORDER, const enum
+          CBLAS_UPLO UPLO, const int N, const float ALPHA, const void *
+          X, const int INCX, void * A)
+
+ -- Function: void cblas_cher2 (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const int N, const void * ALPHA, const
+          void * X, const int INCX, const void * Y, const int INCY,
+          void * A, const int LDA)
+
+ -- Function: void cblas_chpr2 (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const int N, const void * ALPHA, const
+          void * X, const int INCX, const void * Y, const int INCY,
+          void * AP)
+
+ -- Function: void cblas_zhemv (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const int N, const void * ALPHA, const
+          void * A, const int LDA, const void * X, const int INCX,
+          const void * BETA, void * Y, const int INCY)
+
+ -- Function: void cblas_zhbmv (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const int N, const int K, const void *
+          ALPHA, const void * A, const int LDA, const void * X, const
+          int INCX, const void * BETA, void * Y, const int INCY)
+
+ -- Function: void cblas_zhpmv (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const int N, const void * ALPHA, const
+          void * AP, const void * X, const int INCX, const void * BETA,
+          void * Y, const int INCY)
+
+ -- Function: void cblas_zgeru (const enum CBLAS_ORDER ORDER, const int
+          M, const int N, const void * ALPHA, const void * X, const int
+          INCX, const void * Y, const int INCY, void * A, const int LDA)
+
+ -- Function: void cblas_zgerc (const enum CBLAS_ORDER ORDER, const int
+          M, const int N, const void * ALPHA, const void * X, const int
+          INCX, const void * Y, const int INCY, void * A, const int LDA)
+
+ -- Function: void cblas_zher (const enum CBLAS_ORDER ORDER, const enum
+          CBLAS_UPLO UPLO, const int N, const double ALPHA, const void
+          * X, const int INCX, void * A, const int LDA)
+
+ -- Function: void cblas_zhpr (const enum CBLAS_ORDER ORDER, const enum
+          CBLAS_UPLO UPLO, const int N, const double ALPHA, const void
+          * X, const int INCX, void * A)
+
+ -- Function: void cblas_zher2 (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const int N, const void * ALPHA, const
+          void * X, const int INCX, const void * Y, const int INCY,
+          void * A, const int LDA)
+
+ -- Function: void cblas_zhpr2 (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const int N, const void * ALPHA, const
+          void * X, const int INCX, const void * Y, const int INCY,
+          void * AP)
+
+
+File: gsl-ref.info,  Node: Level 3 CBLAS Functions,  Next: GSL CBLAS Examples,  Prev: Level 2 CBLAS Functions,  Up: GSL CBLAS Library
+
+D.3 Level 3
+===========
+
+ -- Function: void cblas_sgemm (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_TRANSPOSE TRANSA, const enum CBLAS_TRANSPOSE
+          TRANSB, const int M, const int N, const int K, const float
+          ALPHA, const float * A, const int LDA, const float * B, const
+          int LDB, const float BETA, float * C, const int LDC)
+
+ -- Function: void cblas_ssymm (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_SIDE SIDE, const enum CBLAS_UPLO UPLO, const int
+          M, const int N, const float ALPHA, const float * A, const int
+          LDA, const float * B, const int LDB, const float BETA, float
+          * C, const int LDC)
+
+ -- Function: void cblas_ssyrk (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANS, const
+          int N, const int K, const float ALPHA, const float * A, const
+          int LDA, const float BETA, float * C, const int LDC)
+
+ -- Function: void cblas_ssyr2k (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANS, const
+          int N, const int K, const float ALPHA, const float * A, const
+          int LDA, const float * B, const int LDB, const float BETA,
+          float * C, const int LDC)
+
+ -- Function: void cblas_strmm (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_SIDE SIDE, const enum CBLAS_UPLO UPLO, const enum
+          CBLAS_TRANSPOSE TRANSA, const enum CBLAS_DIAG DIAG, const int
+          M, const int N, const float ALPHA, const float * A, const int
+          LDA, float * B, const int LDB)
+
+ -- Function: void cblas_strsm (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_SIDE SIDE, const enum CBLAS_UPLO UPLO, const enum
+          CBLAS_TRANSPOSE TRANSA, const enum CBLAS_DIAG DIAG, const int
+          M, const int N, const float ALPHA, const float * A, const int
+          LDA, float * B, const int LDB)
+
+ -- Function: void cblas_dgemm (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_TRANSPOSE TRANSA, const enum CBLAS_TRANSPOSE
+          TRANSB, const int M, const int N, const int K, const double
+          ALPHA, const double * A, const int LDA, const double * B,
+          const int LDB, const double BETA, double * C, const int LDC)
+
+ -- Function: void cblas_dsymm (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_SIDE SIDE, const enum CBLAS_UPLO UPLO, const int
+          M, const int N, const double ALPHA, const double * A, const
+          int LDA, const double * B, const int LDB, const double BETA,
+          double * C, const int LDC)
+
+ -- Function: void cblas_dsyrk (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANS, const
+          int N, const int K, const double ALPHA, const double * A,
+          const int LDA, const double BETA, double * C, const int LDC)
+
+ -- Function: void cblas_dsyr2k (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANS, const
+          int N, const int K, const double ALPHA, const double * A,
+          const int LDA, const double * B, const int LDB, const double
+          BETA, double * C, const int LDC)
+
+ -- Function: void cblas_dtrmm (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_SIDE SIDE, const enum CBLAS_UPLO UPLO, const enum
+          CBLAS_TRANSPOSE TRANSA, const enum CBLAS_DIAG DIAG, const int
+          M, const int N, const double ALPHA, const double * A, const
+          int LDA, double * B, const int LDB)
+
+ -- Function: void cblas_dtrsm (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_SIDE SIDE, const enum CBLAS_UPLO UPLO, const enum
+          CBLAS_TRANSPOSE TRANSA, const enum CBLAS_DIAG DIAG, const int
+          M, const int N, const double ALPHA, const double * A, const
+          int LDA, double * B, const int LDB)
+
+ -- Function: void cblas_cgemm (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_TRANSPOSE TRANSA, const enum CBLAS_TRANSPOSE
+          TRANSB, const int M, const int N, const int K, const void *
+          ALPHA, const void * A, const int LDA, const void * B, const
+          int LDB, const void * BETA, void * C, const int LDC)
+
+ -- Function: void cblas_csymm (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_SIDE SIDE, const enum CBLAS_UPLO UPLO, const int
+          M, const int N, const void * ALPHA, const void * A, const int
+          LDA, const void * B, const int LDB, const void * BETA, void *
+          C, const int LDC)
+
+ -- Function: void cblas_csyrk (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANS, const
+          int N, const int K, const void * ALPHA, const void * A, const
+          int LDA, const void * BETA, void * C, const int LDC)
+
+ -- Function: void cblas_csyr2k (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANS, const
+          int N, const int K, const void * ALPHA, const void * A, const
+          int LDA, const void * B, const int LDB, const void * BETA,
+          void * C, const int LDC)
+
+ -- Function: void cblas_ctrmm (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_SIDE SIDE, const enum CBLAS_UPLO UPLO, const enum
+          CBLAS_TRANSPOSE TRANSA, const enum CBLAS_DIAG DIAG, const int
+          M, const int N, const void * ALPHA, const void * A, const int
+          LDA, void * B, const int LDB)
+
+ -- Function: void cblas_ctrsm (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_SIDE SIDE, const enum CBLAS_UPLO UPLO, const enum
+          CBLAS_TRANSPOSE TRANSA, const enum CBLAS_DIAG DIAG, const int
+          M, const int N, const void * ALPHA, const void * A, const int
+          LDA, void * B, const int LDB)
+
+ -- Function: void cblas_zgemm (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_TRANSPOSE TRANSA, const enum CBLAS_TRANSPOSE
+          TRANSB, const int M, const int N, const int K, const void *
+          ALPHA, const void * A, const int LDA, const void * B, const
+          int LDB, const void * BETA, void * C, const int LDC)
+
+ -- Function: void cblas_zsymm (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_SIDE SIDE, const enum CBLAS_UPLO UPLO, const int
+          M, const int N, const void * ALPHA, const void * A, const int
+          LDA, const void * B, const int LDB, const void * BETA, void *
+          C, const int LDC)
+
+ -- Function: void cblas_zsyrk (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANS, const
+          int N, const int K, const void * ALPHA, const void * A, const
+          int LDA, const void * BETA, void * C, const int LDC)
+
+ -- Function: void cblas_zsyr2k (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANS, const
+          int N, const int K, const void * ALPHA, const void * A, const
+          int LDA, const void * B, const int LDB, const void * BETA,
+          void * C, const int LDC)
+
+ -- Function: void cblas_ztrmm (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_SIDE SIDE, const enum CBLAS_UPLO UPLO, const enum
+          CBLAS_TRANSPOSE TRANSA, const enum CBLAS_DIAG DIAG, const int
+          M, const int N, const void * ALPHA, const void * A, const int
+          LDA, void * B, const int LDB)
+
+ -- Function: void cblas_ztrsm (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_SIDE SIDE, const enum CBLAS_UPLO UPLO, const enum
+          CBLAS_TRANSPOSE TRANSA, const enum CBLAS_DIAG DIAG, const int
+          M, const int N, const void * ALPHA, const void * A, const int
+          LDA, void * B, const int LDB)
+
+ -- Function: void cblas_chemm (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_SIDE SIDE, const enum CBLAS_UPLO UPLO, const int
+          M, const int N, const void * ALPHA, const void * A, const int
+          LDA, const void * B, const int LDB, const void * BETA, void *
+          C, const int LDC)
+
+ -- Function: void cblas_cherk (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANS, const
+          int N, const int K, const float ALPHA, const void * A, const
+          int LDA, const float BETA, void * C, const int LDC)
+
+ -- Function: void cblas_cher2k (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANS, const
+          int N, const int K, const void * ALPHA, const void * A, const
+          int LDA, const void * B, const int LDB, const float BETA,
+          void * C, const int LDC)
+
+ -- Function: void cblas_zhemm (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_SIDE SIDE, const enum CBLAS_UPLO UPLO, const int
+          M, const int N, const void * ALPHA, const void * A, const int
+          LDA, const void * B, const int LDB, const void * BETA, void *
+          C, const int LDC)
+
+ -- Function: void cblas_zherk (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANS, const
+          int N, const int K, const double ALPHA, const void * A, const
+          int LDA, const double BETA, void * C, const int LDC)
+
+ -- Function: void cblas_zher2k (const enum CBLAS_ORDER ORDER, const
+          enum CBLAS_UPLO UPLO, const enum CBLAS_TRANSPOSE TRANS, const
+          int N, const int K, const void * ALPHA, const void * A, const
+          int LDA, const void * B, const int LDB, const double BETA,
+          void * C, const int LDC)
+
+ -- Function: void cblas_xerbla (int P, const char * ROUT, const char *
+          FORM, ...)
+
+
+File: gsl-ref.info,  Node: GSL CBLAS Examples,  Prev: Level 3 CBLAS Functions,  Up: GSL CBLAS Library
+
+D.4 Examples
+============
+
+The following program computes the product of two matrices using the
+Level-3 BLAS function SGEMM,
+
+     [ 0.11 0.12 0.13 ]  [ 1011 1012 ]     [ 367.76 368.12 ]
+     [ 0.21 0.22 0.23 ]  [ 1021 1022 ]  =  [ 674.06 674.72 ]
+                         [ 1031 1032 ]
+
+The matrices are stored in row major order but could be stored in column
+major order if the first argument of the call to `cblas_sgemm' was
+changed to `CblasColMajor'.
+
+     #include <stdio.h>
+     #include <gsl/gsl_cblas.h>
+
+     int
+     main (void)
+     {
+       int lda = 3;
+
+       float A[] = { 0.11, 0.12, 0.13,
+                     0.21, 0.22, 0.23 };
+
+       int ldb = 2;
+
+       float B[] = { 1011, 1012,
+                     1021, 1022,
+                     1031, 1032 };
+
+       int ldc = 2;
+
+       float C[] = { 0.00, 0.00,
+                     0.00, 0.00 };
+
+       /* Compute C = A B */
+
+       cblas_sgemm (CblasRowMajor,
+                    CblasNoTrans, CblasNoTrans, 2, 2, 3,
+                    1.0, A, lda, B, ldb, 0.0, C, ldc);
+
+       printf ("[ %g, %g\n", C[0], C[1]);
+       printf ("  %g, %g ]\n", C[2], C[3]);
+
+       return 0;
+     }
+
+To compile the program use the following command line,
+
+     $ gcc -Wall demo.c -lgslcblas
+
+There is no need to link with the main library `-lgsl' in this case as
+the CBLAS library is an independent unit. Here is the output from the
+program,
+
+     $ ./a.out
+     [ 367.76, 368.12
+       674.06, 674.72 ]
+
+
+File: gsl-ref.info,  Node: Free Software Needs Free Documentation,  Next: GNU General Public License,  Prev: GSL CBLAS Library,  Up: Top
+
+Free Software Needs Free Documentation
+**************************************
+
+     The following article was written by Richard Stallman, founder of
+     the GNU Project.
+
+   The biggest deficiency in the free software community today is not in
+the software--it is the lack of good free documentation that we can
+include with the free software.  Many of our most important programs do
+not come with free reference manuals and free introductory texts.
+Documentation is an essential part of any software package; when an
+important free software package does not come with a free manual and a
+free tutorial, that is a major gap.  We have many such gaps today.
+
+   Consider Perl, for instance.  The tutorial manuals that people
+normally use are non-free.  How did this come about?  Because the
+authors of those manuals published them with restrictive terms--no
+copying, no modification, source files not available--which exclude
+them from the free software world.
+
+   That wasn't the first time this sort of thing happened, and it was
+far from the last.  Many times we have heard a GNU user eagerly
+describe a manual that he is writing, his intended contribution to the
+community, only to learn that he had ruined everything by signing a
+publication contract to make it non-free.
+
+   Free documentation, like free software, is a matter of freedom, not
+price.  The problem with the non-free manual is not that publishers
+charge a price for printed copies--that in itself is fine.  (The Free
+Software Foundation sells printed copies of manuals, too.)  The problem
+is the restrictions on the use of the manual.  Free manuals are
+available in source code form, and give you permission to copy and
+modify.  Non-free manuals do not allow this.
+
+   The criteria of freedom for a free manual are roughly the same as for
+free software.  Redistribution (including the normal kinds of
+commercial redistribution) must be permitted, so that the manual can
+accompany every copy of the program, both on-line and on paper.
+
+   Permission for modification of the technical content is crucial too.
+When people modify the software, adding or changing features, if they
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+accurate and clear documentation for the modified program.  A manual
+that leaves you no choice but to write a new manual to document a
+changed version of the program is not really available to our community.
+
+   Some kinds of limits on the way modification is handled are
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+author's copyright notice, the distribution terms, or the list of
+authors, are ok.  It is also no problem to require modified versions to
+include notice that they were modified.  Even entire sections that may
+not be deleted or changed are acceptable, as long as they deal with
+nontechnical topics (like this one).  These kinds of restrictions are
+acceptable because they don't obstruct the community's normal use of
+the manual.
+
+   However, it must be possible to modify all the _technical_ content
+of the manual, and then distribute the result in all the usual media,
+through all the usual channels.  Otherwise, the restrictions obstruct
+the use of the manual, it is not free, and we need another manual to
+replace it.
+
+   Please spread the word about this issue.  Our community continues to
+lose manuals to proprietary publishing.  If we spread the word that
+free software needs free reference manuals and free tutorials, perhaps
+the next person who wants to contribute by writing documentation will
+realize, before it is too late, that only free manuals contribute to
+the free software community.
+
+   If you are writing documentation, please insist on publishing it
+under the GNU Free Documentation License or another free documentation
+license.  Remember that this decision requires your approval--you don't
+have to let the publisher decide.  Some commercial publishers will use
+a free license if you insist, but they will not propose the option; it
+is up to you to raise the issue and say firmly that this is what you
+want.  If the publisher you are dealing with refuses, please try other
+publishers.  If you're not sure whether a proposed license is free,
+write to <licensing@gnu.org>.
+
+   You can encourage commercial publishers to sell more free, copylefted
+manuals and tutorials by buying them, and particularly by buying copies
+from the publishers that paid for their writing or for major
+improvements.  Meanwhile, try to avoid buying non-free documentation at
+all.  Check the distribution terms of a manual before you buy it, and
+insist that whoever seeks your business must respect your freedom.
+Check the history of the book, and try reward the publishers that have
+paid or pay the authors to work on it.
+
+   The Free Software Foundation maintains a list of free documentation
+published by other publishers:
+
+     `http://www.fsf.org/doc/other-free-books.html'
+
+
+File: gsl-ref.info,  Node: GNU General Public License,  Next: GNU Free Documentation License,  Prev: Free Software Needs Free Documentation,  Up: Top
+
+GNU General Public License
+**************************
+
+                         Version 2, June 1991
+     Copyright (C) 1989, 1991 Free Software Foundation, Inc.
+     59 Temple Place - Suite 330, Boston, MA  02111-1307, USA
+
+     Everyone is permitted to copy and distribute verbatim copies of this
+     license document, but changing it is not allowed.
+
+Preamble
+========
+
+The licenses for most software are designed to take away your freedom
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+Foundation's software and to any other program whose authors commit to
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+the GNU Library General Public License instead.)  You can apply it to
+your programs, too.
+
+   When we speak of free software, we are referring to freedom, not
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+
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+    TERMS AND CONDITIONS FOR COPYING, DISTRIBUTION AND MODIFICATION
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+  1. You may copy and distribute verbatim copies of the Program's
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+     this License along with the Program.
+
+     You may charge a fee for the physical act of transferring a copy,
+     and you may at your option offer warranty protection in exchange
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+
+  2. You may modify your copy or copies of the Program or any portion
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+     above, provided that you also meet all of these conditions:
+
+       a. You must cause the modified files to carry prominent notices
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+       b. You must cause any work that you distribute or publish, that
+          in whole or in part contains or is derived from the Program
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+          when run, you must cause it, when started running for such
+          interactive use in the most ordinary way, to print or display
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+
+     These requirements apply to the modified work as a whole.  If
+     identifiable sections of that work are not derived from the
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+  4. You may not copy, modify, sublicense, or distribute the Program
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+
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+     infringement or for any other reason (not limited to patent
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+
+     If any portion of this section is held invalid or unenforceable
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+
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+
+ 10. If you wish to incorporate parts of the Program into other free
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+
+                                NO WARRANTY
+ 11. BECAUSE THE PROGRAM IS LICENSED FREE OF CHARGE, THERE IS NO
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+
+ 12. IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN
+     WRITING WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MAY
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+
+                      END OF TERMS AND CONDITIONS
+Appendix: How to Apply These Terms to Your New Programs
+=======================================================
+
+If you develop a new program, and you want it to be of the greatest
+possible use to the public, the best way to achieve this is to make it
+free software which everyone can redistribute and change under these
+terms.
+
+   To do so, attach the following notices to the program.  It is safest
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+the "copyright" line and a pointer to where the full notice is found.
+
+     one line to give the program's name and a brief idea
+     of whAT IT DOES.
+     Copyright (C) YYYY  NAME OF AUTHOR
+
+     This program is free software; you can redistribute it
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+     version.
+
+     This program is distributed in the hope that it will be
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+     details.
+
+     You should have received a copy of the GNU General Public
+     License along with this program; if not, write to the Free
+     Software Foundation, Inc., 59 Temple Place - Suite 330,
+     Boston, MA 02111-1307, USA.
+
+   Also add information on how to contact you by electronic and paper
+mail.
+
+   If the program is interactive, make it output a short notice like
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+
+     Gnomovision version 69, Copyright (C) 19YY NAME OF AUTHOR
+     Gnomovision comes with ABSOLUTELY NO WARRANTY; for details
+     type `show w'.  This is free software, and you are welcome
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+     for details.
+
+   The hypothetical commands `show w' and `show c' should show the
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+
+   You should also get your employer (if you work as a programmer) or
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+
+     Yoyodyne, Inc., hereby disclaims all copyright interest in
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+     written by James Hacker.
+
+     SIGNATURE OF TY COON, 1 April 1989
+     Ty Coon, President of Vice
+
+   This General Public License does not permit incorporating your
+program into proprietary programs.  If your program is a subroutine
+library, you may consider it more useful to permit linking proprietary
+applications with the library.  If this is what you want to do, use the
+GNU Library General Public License instead of this License.
+
+
+File: gsl-ref.info,  Node: GNU Free Documentation License,  Next: Function Index,  Prev: GNU General Public License,  Up: Top
+
+GNU Free Documentation License
+******************************
+
+                      Version 1.2, November 2002
+     Copyright (C) 2000,2001,2002 Free Software Foundation, Inc.
+     51 Franklin St, Fifth Floor, Boston, MA  02110-1301, USA
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+
+     The "Invariant Sections" are certain Secondary Sections whose
+     titles are designated, as being those of Invariant Sections, in
+     the notice that says that the Document is released under this
+     License.  If a section does not fit the above definition of
+     Secondary then it is not allowed to be designated as Invariant.
+     The Document may contain zero Invariant Sections.  If the Document
+     does not identify any Invariant Sections then there are none.
+
+     The "Cover Texts" are certain short passages of text that are
+     listed, as Front-Cover Texts or Back-Cover Texts, in the notice
+     that says that the Document is released under this License.  A
+     Front-Cover Text may be at most 5 words, and a Back-Cover Text may
+     be at most 25 words.
+
+     A "Transparent" copy of the Document means a machine-readable copy,
+     represented in a format whose specification is available to the
+     general public, that is suitable for revising the document
+     straightforwardly with generic text editors or (for images
+     composed of pixels) generic paint programs or (for drawings) some
+     widely available drawing editor, and that is suitable for input to
+     text formatters or for automatic translation to a variety of
+     formats suitable for input to text formatters.  A copy made in an
+     otherwise Transparent file format whose markup, or absence of
+     markup, has been arranged to thwart or discourage subsequent
+     modification by readers is not Transparent.  An image format is
+     not Transparent if used for any substantial amount of text.  A
+     copy that is not "Transparent" is called "Opaque".
+
+     Examples of suitable formats for Transparent copies include plain
+     ASCII without markup, Texinfo input format, LaTeX input format,
+     SGML or XML using a publicly available DTD, and
+     standard-conforming simple HTML, PostScript or PDF designed for
+     human modification.  Examples of transparent image formats include
+     PNG, XCF and JPG.  Opaque formats include proprietary formats that
+     can be read and edited only by proprietary word processors, SGML or
+     XML for which the DTD and/or processing tools are not generally
+     available, and the machine-generated HTML, PostScript or PDF
+     produced by some word processors for output purposes only.
+
+     The "Title Page" means, for a printed book, the title page itself,
+     plus such following pages as are needed to hold, legibly, the
+     material this License requires to appear in the title page.  For
+     works in formats which do not have any title page as such, "Title
+     Page" means the text near the most prominent appearance of the
+     work's title, preceding the beginning of the body of the text.
+
+     A section "Entitled XYZ" means a named subunit of the Document
+     whose title either is precisely XYZ or contains XYZ in parentheses
+     following text that translates XYZ in another language.  (Here XYZ
+     stands for a specific section name mentioned below, such as
+     "Acknowledgements", "Dedications", "Endorsements", or "History".)
+     To "Preserve the Title" of such a section when you modify the
+     Document means that it remains a section "Entitled XYZ" according
+     to this definition.
+
+     The Document may include Warranty Disclaimers next to the notice
+     which states that this License applies to the Document.  These
+     Warranty Disclaimers are considered to be included by reference in
+     this License, but only as regards disclaiming warranties: any other
+     implication that these Warranty Disclaimers may have is void and
+     has no effect on the meaning of this License.
+
+  2. VERBATIM COPYING
+
+     You may copy and distribute the Document in any medium, either
+     commercially or noncommercially, provided that this License, the
+     copyright notices, and the license notice saying this License
+     applies to the Document are reproduced in all copies, and that you
+     add no other conditions whatsoever to those of this License.  You
+     may not use technical measures to obstruct or control the reading
+     or further copying of the copies you make or distribute.  However,
+     you may accept compensation in exchange for copies.  If you
+     distribute a large enough number of copies you must also follow
+     the conditions in section 3.
+
+     You may also lend copies, under the same conditions stated above,
+     and you may publicly display copies.
+
+  3. COPYING IN QUANTITY
+
+     If you publish printed copies (or copies in media that commonly
+     have printed covers) of the Document, numbering more than 100, and
+     the Document's license notice requires Cover Texts, you must
+     enclose the copies in covers that carry, clearly and legibly, all
+     these Cover Texts: Front-Cover Texts on the front cover, and
+     Back-Cover Texts on the back cover.  Both covers must also clearly
+     and legibly identify you as the publisher of these copies.  The
+     front cover must present the full title with all words of the
+     title equally prominent and visible.  You may add other material
+     on the covers in addition.  Copying with changes limited to the
+     covers, as long as they preserve the title of the Document and
+     satisfy these conditions, can be treated as verbatim copying in
+     other respects.
+
+     If the required texts for either cover are too voluminous to fit
+     legibly, you should put the first ones listed (as many as fit
+     reasonably) on the actual cover, and continue the rest onto
+     adjacent pages.
+
+     If you publish or distribute Opaque copies of the Document
+     numbering more than 100, you must either include a
+     machine-readable Transparent copy along with each Opaque copy, or
+     state in or with each Opaque copy a computer-network location from
+     which the general network-using public has access to download
+     using public-standard network protocols a complete Transparent
+     copy of the Document, free of added material.  If you use the
+     latter option, you must take reasonably prudent steps, when you
+     begin distribution of Opaque copies in quantity, to ensure that
+     this Transparent copy will remain thus accessible at the stated
+     location until at least one year after the last time you
+     distribute an Opaque copy (directly or through your agents or
+     retailers) of that edition to the public.
+
+     It is requested, but not required, that you contact the authors of
+     the Document well before redistributing any large number of
+     copies, to give them a chance to provide you with an updated
+     version of the Document.
+
+  4. MODIFICATIONS
+
+     You may copy and distribute a Modified Version of the Document
+     under the conditions of sections 2 and 3 above, provided that you
+     release the Modified Version under precisely this License, with
+     the Modified Version filling the role of the Document, thus
+     licensing distribution and modification of the Modified Version to
+     whoever possesses a copy of it.  In addition, you must do these
+     things in the Modified Version:
+
+       A. Use in the Title Page (and on the covers, if any) a title
+          distinct from that of the Document, and from those of
+          previous versions (which should, if there were any, be listed
+          in the History section of the Document).  You may use the
+          same title as a previous version if the original publisher of
+          that version gives permission.
+
+       B. List on the Title Page, as authors, one or more persons or
+          entities responsible for authorship of the modifications in
+          the Modified Version, together with at least five of the
+          principal authors of the Document (all of its principal
+          authors, if it has fewer than five), unless they release you
+          from this requirement.
+
+       C. State on the Title page the name of the publisher of the
+          Modified Version, as the publisher.
+
+       D. Preserve all the copyright notices of the Document.
+
+       E. Add an appropriate copyright notice for your modifications
+          adjacent to the other copyright notices.
+
+       F. Include, immediately after the copyright notices, a license
+          notice giving the public permission to use the Modified
+          Version under the terms of this License, in the form shown in
+          the Addendum below.
+
+       G. Preserve in that license notice the full lists of Invariant
+          Sections and required Cover Texts given in the Document's
+          license notice.
+
+       H. Include an unaltered copy of this License.
+
+       I. Preserve the section Entitled "History", Preserve its Title,
+          and add to it an item stating at least the title, year, new
+          authors, and publisher of the Modified Version as given on
+          the Title Page.  If there is no section Entitled "History" in
+          the Document, create one stating the title, year, authors,
+          and publisher of the Document as given on its Title Page,
+          then add an item describing the Modified Version as stated in
+          the previous sentence.
+
+       J. Preserve the network location, if any, given in the Document
+          for public access to a Transparent copy of the Document, and
+          likewise the network locations given in the Document for
+          previous versions it was based on.  These may be placed in
+          the "History" section.  You may omit a network location for a
+          work that was published at least four years before the
+          Document itself, or if the original publisher of the version
+          it refers to gives permission.
+
+       K. For any section Entitled "Acknowledgements" or "Dedications",
+          Preserve the Title of the section, and preserve in the
+          section all the substance and tone of each of the contributor
+          acknowledgements and/or dedications given therein.
+
+       L. Preserve all the Invariant Sections of the Document,
+          unaltered in their text and in their titles.  Section numbers
+          or the equivalent are not considered part of the section
+          titles.
+
+       M. Delete any section Entitled "Endorsements".  Such a section
+          may not be included in the Modified Version.
+
+       N. Do not retitle any existing section to be Entitled
+          "Endorsements" or to conflict in title with any Invariant
+          Section.
+
+       O. Preserve any Warranty Disclaimers.
+
+     If the Modified Version includes new front-matter sections or
+     appendices that qualify as Secondary Sections and contain no
+     material copied from the Document, you may at your option
+     designate some or all of these sections as invariant.  To do this,
+     add their titles to the list of Invariant Sections in the Modified
+     Version's license notice.  These titles must be distinct from any
+     other section titles.
+
+     You may add a section Entitled "Endorsements", provided it contains
+     nothing but endorsements of your Modified Version by various
+     parties--for example, statements of peer review or that the text
+     has been approved by an organization as the authoritative
+     definition of a standard.
+
+     You may add a passage of up to five words as a Front-Cover Text,
+     and a passage of up to 25 words as a Back-Cover Text, to the end
+     of the list of Cover Texts in the Modified Version.  Only one
+     passage of Front-Cover Text and one of Back-Cover Text may be
+     added by (or through arrangements made by) any one entity.  If the
+     Document already includes a cover text for the same cover,
+     previously added by you or by arrangement made by the same entity
+     you are acting on behalf of, you may not add another; but you may
+     replace the old one, on explicit permission from the previous
+     publisher that added the old one.
+
+     The author(s) and publisher(s) of the Document do not by this
+     License give permission to use their names for publicity for or to
+     assert or imply endorsement of any Modified Version.
+
+  5. COMBINING DOCUMENTS
+
+     You may combine the Document with other documents released under
+     this License, under the terms defined in section 4 above for
+     modified versions, provided that you include in the combination
+     all of the Invariant Sections of all of the original documents,
+     unmodified, and list them all as Invariant Sections of your
+     combined work in its license notice, and that you preserve all
+     their Warranty Disclaimers.
+
+     The combined work need only contain one copy of this License, and
+     multiple identical Invariant Sections may be replaced with a single
+     copy.  If there are multiple Invariant Sections with the same name
+     but different contents, make the title of each such section unique
+     by adding at the end of it, in parentheses, the name of the
+     original author or publisher of that section if known, or else a
+     unique number.  Make the same adjustment to the section titles in
+     the list of Invariant Sections in the license notice of the
+     combined work.
+
+     In the combination, you must combine any sections Entitled
+     "History" in the various original documents, forming one section
+     Entitled "History"; likewise combine any sections Entitled
+     "Acknowledgements", and any sections Entitled "Dedications".  You
+     must delete all sections Entitled "Endorsements."
+
+  6. COLLECTIONS OF DOCUMENTS
+
+     You may make a collection consisting of the Document and other
+     documents released under this License, and replace the individual
+     copies of this License in the various documents with a single copy
+     that is included in the collection, provided that you follow the
+     rules of this License for verbatim copying of each of the
+     documents in all other respects.
+
+     You may extract a single document from such a collection, and
+     distribute it individually under this License, provided you insert
+     a copy of this License into the extracted document, and follow
+     this License in all other respects regarding verbatim copying of
+     that document.
+
+  7. AGGREGATION WITH INDEPENDENT WORKS
+
+     A compilation of the Document or its derivatives with other
+     separate and independent documents or works, in or on a volume of
+     a storage or distribution medium, is called an "aggregate" if the
+     copyright resulting from the compilation is not used to limit the
+     legal rights of the compilation's users beyond what the individual
+     works permit.  When the Document is included in an aggregate, this
+     License does not apply to the other works in the aggregate which
+     are not themselves derivative works of the Document.
+
+     If the Cover Text requirement of section 3 is applicable to these
+     copies of the Document, then if the Document is less than one half
+     of the entire aggregate, the Document's Cover Texts may be placed
+     on covers that bracket the Document within the aggregate, or the
+     electronic equivalent of covers if the Document is in electronic
+     form.  Otherwise they must appear on printed covers that bracket
+     the whole aggregate.
+
+  8. TRANSLATION
+
+     Translation is considered a kind of modification, so you may
+     distribute translations of the Document under the terms of section
+     4.  Replacing Invariant Sections with translations requires special
+     permission from their copyright holders, but you may include
+     translations of some or all Invariant Sections in addition to the
+     original versions of these Invariant Sections.  You may include a
+     translation of this License, and all the license notices in the
+     Document, and any Warranty Disclaimers, provided that you also
+     include the original English version of this License and the
+     original versions of those notices and disclaimers.  In case of a
+     disagreement between the translation and the original version of
+     this License or a notice or disclaimer, the original version will
+     prevail.
+
+     If a section in the Document is Entitled "Acknowledgements",
+     "Dedications", or "History", the requirement (section 4) to
+     Preserve its Title (section 1) will typically require changing the
+     actual title.
+
+  9. TERMINATION
+
+     You may not copy, modify, sublicense, or distribute the Document
+     except as expressly provided for under this License.  Any other
+     attempt to copy, modify, sublicense or distribute the Document is
+     void, and will automatically terminate your rights under this
+     License.  However, parties who have received copies, or rights,
+     from you under this License will not have their licenses
+     terminated so long as such parties remain in full compliance.
+
+ 10. FUTURE REVISIONS OF THIS LICENSE
+
+     The Free Software Foundation may publish new, revised versions of
+     the GNU Free Documentation License from time to time.  Such new
+     versions will be similar in spirit to the present version, but may
+     differ in detail to address new problems or concerns.  See
+     `http://www.gnu.org/copyleft/'.
+
+     Each version of the License is given a distinguishing version
+     number.  If the Document specifies that a particular numbered
+     version of this License "or any later version" applies to it, you
+     have the option of following the terms and conditions either of
+     that specified version or of any later version that has been
+     published (not as a draft) by the Free Software Foundation.  If
+     the Document does not specify a version number of this License,
+     you may choose any version ever published (not as a draft) by the
+     Free Software Foundation.
+
+ADDENDUM: How to use this License for your documents
+====================================================
+
+To use this License in a document you have written, include a copy of
+the License in the document and put the following copyright and license
+notices just after the title page:
+
+       Copyright (C) YEAR YOUR NAME.
+       Permission is granted to copy, distribute and/or modify
+       this document under the terms of the GNU Free
+       Documentation License, Version 1.2 or any later version
+       published by the Free Software Foundation; with no
+       Invariant Sections, no Front-Cover Texts, and no
+       Back-Cover Texts.  A copy of the license is included in
+       the section entitled ``GNU Free Documentation License''.
+
+   If you have Invariant Sections, Front-Cover Texts and Back-Cover
+Texts, replace the "with...Texts." line with this:
+
+       with the Invariant Sections being LIST THEIR
+       TITLES, with the Front-Cover Texts being LIST, and
+       with the Back-Cover Texts being LIST.
+
+   If you have Invariant Sections without Cover Texts, or some other
+combination of the three, merge those two alternatives to suit the
+situation.
+
+   If your document contains nontrivial examples of program code, we
+recommend releasing these examples in parallel under your choice of
+free software license, such as the GNU General Public License, to
+permit their use in free software.
+
diff --git a/gsl-1.9/doc/gsl-ref.info-5 b/gsl-1.9/doc/gsl-ref.info-5
new file mode 100644
index 0000000..1b76f5e
--- /dev/null
+++ b/gsl-1.9/doc/gsl-ref.info-5
diff --git a/gsl-1.9/doc/gsl-ref.info-6 b/gsl-1.9/doc/gsl-ref.info-6
new file mode 100644
index 0000000..243ca4d
--- /dev/null
+++ b/gsl-1.9/doc/gsl-ref.info-6
diff --git a/gsl-1.9/doc/gsl-ref.texi b/gsl-1.9/doc/gsl-ref.texi
new file mode 100644
index 0000000..c2ba564
--- /dev/null
+++ b/gsl-1.9/doc/gsl-ref.texi
@@ -0,0 +1,647 @@
+\input texinfo @c -*-texinfo-*-
+@c This will be for the printing version of the manual
+@c @input config-local.texi
+@c %**start of header
+@setfilename gsl-ref.info
+@settitle GNU Scientific Library -- Reference Manual
+@finalout
+@set frontcontents
+@ifset publish
+@setchapternewpage odd
+@end ifset
+@c %**end of header
+
+@dircategory Scientific software
+@direntry
+* gsl-ref: (gsl-ref).                   GNU Scientific Library -- Reference
+@end direntry
+
+@c How to use the math macros
+@c ==========================
+@c 
+@c For simple expressions, simply use the @math{} command, e.g.
+@c
+@c     @math{\exp(x)/(1+x^2)}
+@c 
+@c but if the expression includes characters that need to be 'escaped'
+@c in texinfo, like '{' or '}', or needs different output for TeX and info,
+@c then use the following form,
+@c
+@c     blah blah blah @c{$y^{1+b} \le \pi$}
+@c     @math{y^@{1+b@} <= \pi}
+@c
+@c The first part using @c{} must appear at the end of a line (it reads
+@c up to the line end -- as far as texinfo is concerned it's actually
+@c a 'comment').  The comment command @c has been modified to capture
+@c a TeX expression which is output by the next @math.
+@c
+@c For ordinary comments use the @comment command.
+
+@tex
+% Mathematical macros taken from the GNU Calc Manual
+% ==================================================
+%
+% Some special kludges to make TeX formatting prettier.
+% Because makeinfo.c exists, we can't just define new commands.
+% So instead, we take over little-used existing commands.
+%
+% Redefine @cite{text} to act like $text$ in regular TeX.
+% Info will typeset this same as @samp{text}.
+\gdef\goodtex{\tex \let\rm\goodrm \let\t\ttfont \turnoffactive}
+\gdef\goodrm{\fam0\tenrm}
+\gdef\math{\goodtex$\mathxxx}
+\gdef\mathxxx#1{#1$\endgroup}
+\global\let\oldxrefX=\xrefX
+\gdef\xrefX[#1]{\begingroup\let\math=\dfn\oldxrefX[#1]\endgroup}
+%
+% Redefine @i{text} to be equivalent to @cite{text}, i.e., to use math mode.
+% This looks the same in TeX but omits the surrounding ` ' in Info.
+%\global\let\i=\cite
+%\global\let\math=\cite
+%
+% Redefine @c{tex-stuff} \n @whatever{info-stuff}.
+\gdef\c{\futurelet\next\mycxxx}
+\gdef\mycxxx{%
+  \ifx\next\bgroup \goodtex\let\next\mycxxy
+  \else\ifx\next\mindex \let\next\relax
+  \else\ifx\next\kindex \let\next\relax
+  \else\ifx\next\starindex \let\next\relax \else \let\next\comment
+  \fi\fi\fi\fi \next
+}
+\gdef\mycxxy#1#2{#1\endgroup\mycxxz}
+\gdef\mycxxz#1{}
+%
+% Define \Hat to take over from \hat as an accent
+\gdef\Hat{\mathaccent "705E}
+%
+%\gdef\beforedisplay{\vskip-10pt}
+%\gdef\afterdisplay{\vskip-5pt}
+\gdef\beforedisplay{}
+\gdef\afterdisplay{}
+{\globaldefs = 1
+\abovedisplayskip=7pt plus 2pt minus 1pt
+\belowdisplayskip=7pt plus 2pt minus 1pt
+\abovedisplayshortskip=7pt plus 2pt minus 1pt
+\belowdisplayshortskip=7pt plus 2pt minus 1pt}
+%\abovedisplayskip=12pt plus 3pt minus 3pt
+%\belowdisplayskip=12pt plus 3pt minus 3pt
+%\abovedisplayshortskip=7pt plus 1pt minus 1pt
+%\belowdisplayshortskip=7pt plus 1pt minus 1pt
+%\gdef\beforedisplayh{\vskip-25pt}
+%\gdef\afterdisplayh{\vskip-10pt}
+%
+\gdef\arcsec{\hbox{\rm arcsec}}
+\gdef\arccsc{\hbox{\rm arccsc}}
+\gdef\arccot{\hbox{\rm arccot}}
+\gdef\sech{\hbox{\rm sech}}
+\gdef\csch{\hbox{\rm csch}}
+\gdef\coth{\hbox{\rm coth}}
+\gdef\arcsinh{\hbox{\rm arcsinh}}
+\gdef\arccosh{\hbox{\rm arccosh}}
+\gdef\arctanh{\hbox{\rm arctanh}}
+\gdef\arcsech{\hbox{\rm arcsech}}
+\gdef\arccsch{\hbox{\rm arccsch}}
+\gdef\arccoth{\hbox{\rm arccoth}}
+%
+\gdef\Re{\hbox{\rm Re}}
+\gdef\Im{\hbox{\rm Im}}
+\gdef\Sin{\hbox{\rm Sin}}
+\gdef\Cos{\hbox{\rm Cos}}
+\gdef\Log{\hbox{\rm Log}}
+%
+\gdef\erf{\hbox{\rm erf}}
+\gdef\erfc{\hbox{\rm erfc}}
+\gdef\sinc{\hbox{\rm sinc}}
+\gdef\sgn{\hbox{\rm sgn}}
+\gdef\sign{\hbox{\rm sign}}
+\gdef\det{\hbox{\rm det}}
+\gdef\Var{\hbox{\rm Var}}
+\gdef\arg{\hbox{\rm arg}} % avoid temporary clobbering of arg in texinfo-4.8
+@end tex
+
+@include version-ref.texi
+@set GSL @i{GNU Scientific Library}
+
+@copying
+Copyright @copyright{} 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007 The GSL Team.
+
+Permission is granted to copy, distribute and/or modify this document
+under the terms of the GNU Free Documentation License, Version 1.2 or
+any later version published by the Free Software Foundation; with the
+Invariant Sections being ``GNU General Public License'' and ``Free Software
+Needs Free Documentation'', the Front-Cover text being ``A GNU Manual'',
+and with the Back-Cover Text being (a) (see below).  A copy of the
+license is included in the section entitled ``GNU Free Documentation
+License''.
+
+(a) The Back-Cover Text is: ``You have freedom to copy and modify this
+GNU Manual, like GNU software.''
+@end copying
+
+@titlepage
+@title GNU Scientific Library
+@subtitle Reference Manual
+@subtitle Edition @value{EDITION}, for GSL Version @value{VERSION}
+@subtitle @value{UPDATED}
+
+@author Mark Galassi 
+Los Alamos National Laboratory
+@comment @email{rosalia@@lanl.gov}
+@sp 1
+
+@author Jim Davies 
+Department of Computer Science, Georgia Institute of Technology
+@comment @email{jimmyd@@nis.lanl.gov}
+@sp 1
+
+@author James Theiler 
+Astrophysics and Radiation Measurements Group, Los Alamos National Laboratory
+@comment @email{jt@@nis.lanl.gov}
+@sp 1
+
+@author Brian Gough 
+Network Theory Limited
+@comment @email{bjg@@network-theory.co.uk}
+@sp 1
+
+@comment Asked to be moved to 'contributors' appendix
+@comment @author Reid Priedhorsky 
+@comment Mathematical Modeling and Analysis Group, Los Alamos National Laboratory
+@comment @email{rp@@lanl.gov}
+@comment @sp 1
+
+@author Gerard Jungman 
+Theoretical Astrophysics Group, Los Alamos National Laboratory
+@comment @email{jungman@@lanl.gov}
+@sp 1
+
+@author Michael Booth
+Department of Physics and Astronomy, The Johns Hopkins University
+@comment @email{booth@@planck.pha.jhu.edu} or @email{booth@@debian.org}
+@sp 1
+
+@author Fabrice Rossi
+University of Paris-Dauphine
+@comment @email{rossi@@ufrmd.dauphine.fr}
+@sp 1
+@page
+@vskip 0pt plus 1filll
+@insertcopying
+@ifclear publish
+Printed copies of this manual can be purchased from Network Theory Ltd
+at @uref{http://www.network-theory.co.uk/gsl/manual/}.
+
+The money raised from sales of the manual helps support the
+development of GSL.
+@end ifclear
+@end titlepage
+
+@iftex
+@ifset frontcontents
+@contents
+@end ifset
+@end iftex
+
+@ifnottex
+@node Top, Introduction, (dir), (dir)
+@top GSL
+
+This file documents the @value{GSL} (GSL), a collection of numerical
+routines for scientific computing.  It corresponds to release
+@value{VERSION} of the library.  Please report any errors in this
+manual to @email{bug-gsl@@gnu.org}.
+
+More information about GSL can be found at the project homepage,
+@uref{http://www.gnu.org/software/gsl/}.
+
+Printed copies of this manual can be purchased from Network Theory Ltd
+at @uref{http://www.network-theory.co.uk/gsl/manual/}. The money
+raised from sales of the manual helps support the development of GSL.
+
+A Japanese translation of this manual is available from the GSL
+project homepage thanks to Daisuke Tominaga.
+
+@insertcopying
+@sp 1
+@end ifnottex
+
+@menu
+* Introduction::                
+* Using the library::           
+* Error Handling::              
+* Mathematical Functions::      
+* Complex Numbers::             
+* Polynomials::                 
+* Special Functions::           
+* Vectors and Matrices::        
+* Permutations::                
+* Combinations::                
+* Sorting::                     
+* BLAS Support::                
+* Linear Algebra::              
+* Eigensystems::                
+* Fast Fourier Transforms::     
+* Numerical Integration::       
+* Random Number Generation::    
+* Quasi-Random Sequences::      
+* Random Number Distributions::  
+* Statistics::                  
+* Histograms::                  
+* N-tuples::                    
+* Monte Carlo Integration::     
+* Simulated Annealing::         
+* Ordinary Differential Equations::  
+* Interpolation::               
+* Numerical Differentiation::   
+* Chebyshev Approximations::    
+* Series Acceleration::         
+* Wavelet Transforms::          
+* Discrete Hankel Transforms::  
+* One dimensional Root-Finding::  
+* One dimensional Minimization::  
+* Multidimensional Root-Finding::  
+* Multidimensional Minimization::  
+* Least-Squares Fitting::       
+* Nonlinear Least-Squares Fitting::  
+* Basis Splines::
+* Physical Constants::          
+* IEEE floating-point arithmetic::  
+* Debugging Numerical Programs::  
+* Contributors to GSL::         
+* Autoconf Macros::             
+* GSL CBLAS Library::           
+* Free Software Needs Free Documentation::  
+* GNU General Public License::  
+* GNU Free Documentation License::  
+* Function Index::              
+* Variable Index::              
+* Type Index::                  
+* Concept Index::               
+@end menu
+
+@node  Introduction, Using the library, Top, Top
+@chapter Introduction
+@include intro.texi
+
+@node Using the library, Error Handling, Introduction, Top
+@chapter Using the library
+@cindex usage, compiling application programs
+@include usage.texi
+
+@node Error Handling, Mathematical Functions, Using the library, Top
+@chapter Error Handling
+@cindex Errors
+@include err.texi
+
+@node  Mathematical Functions, Complex Numbers, Error Handling, Top
+@chapter Mathematical Functions
+@include math.texi
+
+@node Complex Numbers, Polynomials, Mathematical Functions, Top
+@chapter Complex Numbers
+@include complex.texi
+
+@node Polynomials, Special Functions, Complex Numbers, Top
+@chapter Polynomials
+@include poly.texi
+
+@node Special Functions, Vectors and Matrices, Polynomials, Top
+@chapter Special Functions
+@include specfunc.texi
+
+@node Vectors and Matrices, Permutations, Special Functions, Top
+@chapter Vectors and Matrices
+@include vectors.texi
+
+@node   Permutations, Combinations, Vectors and Matrices, Top
+@chapter Permutations
+@include permutation.texi
+
+@node   Combinations, Sorting, Permutations, Top
+@chapter Combinations
+@include combination.texi
+
+@node  Sorting, BLAS Support, Combinations, Top
+@chapter Sorting
+@include sort.texi
+
+@node BLAS Support, Linear Algebra, Sorting, Top
+@chapter BLAS Support
+@include blas.texi
+
+@node Linear Algebra, Eigensystems, BLAS Support, Top
+@chapter Linear Algebra
+@include linalg.texi
+
+@node Eigensystems, Fast Fourier Transforms, Linear Algebra, Top
+@chapter Eigensystems
+@include eigen.texi
+
+@node Fast Fourier Transforms, Numerical Integration, Eigensystems, Top
+@chapter Fast Fourier Transforms (FFTs)
+@include fft.texi
+
+@node Numerical Integration, Random Number Generation, Fast Fourier Transforms, Top
+@chapter Numerical Integration
+@include integration.texi
+
+@node Random Number Generation, Quasi-Random Sequences, Numerical Integration, Top
+@chapter Random Number Generation
+@include rng.texi
+
+@node Quasi-Random Sequences, Random Number Distributions, Random Number Generation, Top
+@chapter Quasi-Random Sequences
+@include qrng.texi
+
+@node  Random Number Distributions, Statistics, Quasi-Random Sequences, Top
+@chapter Random Number Distributions
+@include randist.texi
+
+@node Statistics, Histograms, Random Number Distributions, Top
+@chapter Statistics
+@include statistics.texi
+
+@node Histograms, N-tuples, Statistics, Top
+@chapter Histograms
+@include histogram.texi
+
+@node  N-tuples, Monte Carlo Integration, Histograms, Top
+@chapter N-tuples
+@include ntuple.texi
+
+@node Monte Carlo Integration, Simulated Annealing, N-tuples, Top
+@chapter Monte Carlo Integration
+@include montecarlo.texi
+
+@node Simulated Annealing, Ordinary Differential Equations, Monte Carlo Integration, Top
+@chapter Simulated Annealing
+@include siman.texi
+
+@node Ordinary Differential Equations, Interpolation, Simulated Annealing, Top
+@chapter Ordinary Differential Equations
+@include ode-initval.texi
+
+@node   Interpolation, Numerical Differentiation, Ordinary Differential Equations, Top
+@chapter Interpolation
+@include interp.texi
+
+@node  Numerical Differentiation, Chebyshev Approximations, Interpolation, Top
+@chapter Numerical Differentiation
+@include diff.texi
+
+@node  Chebyshev Approximations, Series Acceleration, Numerical Differentiation, Top
+@chapter Chebyshev Approximations
+@include cheb.texi
+
+@node Series Acceleration, Wavelet Transforms, Chebyshev Approximations, Top
+@chapter Series Acceleration
+@include sum.texi
+
+@node  Wavelet Transforms, Discrete Hankel Transforms, Series Acceleration, Top
+@chapter Wavelet Transforms
+@include dwt.texi
+
+@node  Discrete Hankel Transforms, One dimensional Root-Finding, Wavelet Transforms, Top
+@chapter Discrete Hankel Transforms
+@include dht.texi
+
+@node  One dimensional Root-Finding, One dimensional Minimization, Discrete Hankel Transforms, Top
+@chapter One dimensional Root-Finding
+@include roots.texi
+
+@node   One dimensional Minimization, Multidimensional Root-Finding, One dimensional Root-Finding, Top
+@chapter One dimensional Minimization
+@include min.texi
+
+@node  Multidimensional Root-Finding, Multidimensional Minimization, One dimensional Minimization, Top
+@chapter Multidimensional Root-Finding
+@include multiroots.texi
+
+@node   Multidimensional Minimization, Least-Squares Fitting, Multidimensional Root-Finding, Top
+@chapter Multidimensional Minimization
+@include multimin.texi
+
+@node Least-Squares Fitting, Nonlinear Least-Squares Fitting, Multidimensional Minimization, Top
+@chapter Least-Squares Fitting
+@include fitting.texi
+
+@node Nonlinear Least-Squares Fitting, Basis Splines, Least-Squares Fitting, Top
+@chapter Nonlinear Least-Squares Fitting
+@include multifit.texi
+
+@node Basis Splines, Physical Constants, Nonlinear Least-Squares Fitting, Top
+@chapter Basis Splines
+@include bspline.texi
+
+@node  Physical Constants, IEEE floating-point arithmetic, Basis Splines, Top
+@chapter Physical Constants
+@include const.texi
+
+@node IEEE floating-point arithmetic, Debugging Numerical Programs, Physical Constants, Top
+@chapter IEEE floating-point arithmetic
+@include ieee754.texi
+
+@node Debugging Numerical Programs, Contributors to GSL, IEEE floating-point arithmetic, Top
+@appendix Debugging Numerical Programs
+@include debug.texi
+
+@node Contributors to GSL, Autoconf Macros, Debugging Numerical Programs, Top
+@appendix Contributors to GSL
+
+(See the AUTHORS file in the distribution for up-to-date information.)
+
+@table @strong
+@item Mark Galassi
+Conceived GSL (with James Theiler) and wrote the design document.  Wrote
+the simulated annealing package and the relevant chapter in the manual.
+
+@item James Theiler
+Conceived GSL (with Mark Galassi).  Wrote the random number generators
+and the relevant chapter in this manual.
+
+@item Jim Davies
+Wrote the statistical routines and the relevant chapter in this
+manual.
+
+@item Brian Gough
+FFTs, numerical integration, random number generators and distributions,
+root finding, minimization and fitting, polynomial solvers, complex
+numbers, physical constants, permutations, vector and matrix functions,
+histograms, statistics, ieee-utils, revised @sc{cblas} Level 2 & 3,
+matrix decompositions, eigensystems, cumulative distribution functions,
+testing, documentation and releases.
+
+@item Reid Priedhorsky 
+Wrote and documented the initial version of the root finding routines
+while at Los Alamos National Laboratory, Mathematical Modeling and
+Analysis Group.  
+@comment email: reid@reidster.net
+
+@item Gerard Jungman
+Special Functions, Series acceleration, ODEs, BLAS, Linear Algebra,
+Eigensystems, Hankel Transforms.
+
+@item Mike Booth
+Wrote the Monte Carlo library.
+
+@item Jorma Olavi T@"ahtinen 
+Wrote the initial complex arithmetic functions.
+
+@item  Thomas Walter 
+Wrote the initial heapsort routines and cholesky decomposition.
+
+@item  Fabrice Rossi
+Multidimensional minimization.
+
+@item Carlo Perassi
+Implementation of the random number generators in Knuth's
+@cite{Seminumerical Algorithms}, 3rd Ed.
+
+@item Szymon Jaroszewicz 
+@comment <sj@cs.umb.edu>
+Wrote the routines for generating combinations.
+
+@item Nicolas Darnis
+Wrote the initial routines for canonical permutations.
+
+@item Jason H. Stover
+@comment (jason@sakla.net) 
+Wrote the major cumulative distribution functions.
+
+@item Ivo Alxneit
+Wrote the routines for wavelet transforms.
+
+@item Tuomo Keskitalo
+Improved the implementation of the ODE solvers.
+
+@item Lowell Johnson
+Implementation of the Mathieu functions.
+
+@item Patrick Alken
+Implementation of non-symmetric eigensystems and B-splines.
+
+@end table
+
+Thanks to Nigel Lowry for help in proofreading the manual.
+
+The non-symmetric eigensystems routines contain code based on the
+LAPACK linear algebra library.  LAPACK is distributed under the
+following license:
+
+@iftex
+@smallerfonts @rm
+@end iftex
+@sp 1
+@quotation
+Copyright (c) 1992-2006 The University of Tennessee.  All rights reserved.
+
+Redistribution and use in source and binary forms, with or without
+modification, are permitted provided that the following conditions are
+met:
+
+@bullet{} Redistributions of source code must retain the above copyright
+notice, this list of conditions and the following disclaimer. 
+  
+@bullet{} Redistributions in binary form must reproduce the above copyright
+notice, this list of conditions and the following disclaimer listed
+in this license in the documentation and/or other materials
+provided with the distribution.
+  
+@bullet{} Neither the name of the copyright holders nor the names of its
+contributors may be used to endorse or promote products derived from
+this software without specific prior written permission.
+  
+THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
+"AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT  
+LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
+A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT 
+OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
+SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
+LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
+DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
+THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT  
+(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
+OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. 
+@end quotation
+
+@iftex
+@textfonts @rm
+@end iftex
+
+
+
+@node  Autoconf Macros, GSL CBLAS Library, Contributors to GSL, Top
+@appendix Autoconf Macros
+@include autoconf.texi
+
+@node GSL CBLAS Library, Free Software Needs Free Documentation, Autoconf Macros, Top
+@appendix GSL CBLAS Library
+@include cblas.texi
+
+@comment @node  Copyright, GNU General Public License, Contributors to GSL, Top
+@comment @unnumbered Copyright
+@comment @include science.texi
+
+@node  Free Software Needs Free Documentation, GNU General Public License, GSL CBLAS Library, Top
+@unnumbered Free Software Needs Free Documentation
+@include freemanuals.texi
+
+@node  GNU General Public License, GNU Free Documentation License, Free Software Needs Free Documentation, Top
+@unnumbered GNU General Public License
+@include gpl.texi
+
+@node GNU Free Documentation License, Function Index, GNU General Public License, Top
+@unnumbered GNU Free Documentation License
+@include fdl.texi
+
+@comment htmlhelp: @printindex fn
+@comment htmlhelp: @printindex vr
+@comment htmlhelp: @printindex tp
+@comment htmlhelp: @printindex cp
+@comment htmlhelp: @bye
+
+@iftex
+@normalbottom
+@end iftex
+
+@node Function Index, Variable Index, GNU Free Documentation License, Top
+@unnumbered Function Index
+
+@printindex fn
+
+@node Variable Index, Type Index, Function Index, Top
+@unnumbered Variable Index
+
+@printindex vr
+
+@node Type Index, Concept Index, Variable Index, Top
+@unnumbered Type Index
+
+@printindex tp
+
+@node Concept Index,  , Type Index, Top
+@unnumbered Concept Index
+
+@printindex cp
+
+@ifclear frontcontents
+@comment Use @setchapternewpage odd to ensure that the contents starts 
+@comment on an odd page so that it can always be moved to the front when 
+@comment printing two-up.
+@setchapternewpage odd
+@contents
+@end ifclear
+
+@ifset extrablankpages
+@comment final page must be blank for printed version
+@page
+@headings off
+@*
+@page
+@*
+@comment @page
+@comment @*
+@comment @page
+@comment @*
+@end ifset
+@bye
diff --git a/gsl-1.9/doc/gsl.3 b/gsl-1.9/doc/gsl.3
new file mode 100644
index 0000000..1f82292
--- /dev/null
+++ b/gsl-1.9/doc/gsl.3
@@ -0,0 +1,58 @@
+.TH GSL 3 "GNU Scientific Library" "GSL Team" \" -*- nroff -*-
+.SH NAME
+gsl - GNU Scientific Library
+.SH SYNOPSIS
+#include <gsl/...>
+.SH DESCRIPTION
+The GNU Scientific Library (GSL) is a collection of routines for
+numerical computing.  The routines are written from scratch by the GSL
+team in C, and present a modern Applications Programming Interface
+(API) for C programmers, allowing wrappers to be written for very high
+level languages.
+.PP
+The library covers the following areas,
+.TP
+.nf
+.BR
+Complex Numbers
+Roots of Polynomials
+Special Functions
+Vectors and Matrices
+Permutations
+Combinations
+Sorting
+BLAS Support
+Linear Algebra
+Eigensystems
+Fast Fourier Transforms
+Quadrature
+Random Numbers
+Quasi-Random Sequences
+Random Distributions
+Statistics
+Histograms
+N-Tuples
+Monte Carlo Integration
+Simulated Annealing
+Differential Equations
+Interpolation
+Numerical Differentiation
+Chebyshev Approximations
+Series Acceleration
+Discrete Hankel Transforms
+Root-Finding
+Minimization
+Least-Squares Fitting
+Physical Constants
+IEEE Floating-Point
+.fi
+.PP
+For more information please consult the GSL Reference Manual, which is
+available as an info file.  You can read it online using the shell
+command 
+.B info gsl-ref 
+(if the library is installed).
+.PP
+Please report any bugs to 
+.B bug-gsl@gnu.org.
+
diff --git a/gsl-1.9/doc/histogram.eps b/gsl-1.9/doc/histogram.eps
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diff --git a/gsl-1.9/doc/histogram.texi b/gsl-1.9/doc/histogram.texi
new file mode 100644
index 0000000..f2fe423
--- /dev/null
+++ b/gsl-1.9/doc/histogram.texi
@@ -0,0 +1,1173 @@
+@cindex histograms
+@cindex binning data
+This chapter describes functions for creating histograms.  Histograms
+provide a convenient way of summarizing the distribution of a set of
+data. A histogram consists of a set of @dfn{bins} which count the number
+of events falling into a given range of a continuous variable @math{x}.
+In GSL the bins of a histogram contain floating-point numbers, so they
+can be used to record both integer and non-integer distributions.  The
+bins can use arbitrary sets of ranges (uniformly spaced bins are the
+default).  Both one and two-dimensional histograms are supported.
+
+Once a histogram has been created it can also be converted into a
+probability distribution function.  The library provides efficient
+routines for selecting random samples from probability distributions.
+This can be useful for generating simulations based on real data.
+
+The functions are declared in the header files @file{gsl_histogram.h}
+and @file{gsl_histogram2d.h}.
+
+@menu
+* The histogram struct::        
+* Histogram allocation::        
+* Copying Histograms::          
+* Updating and accessing histogram elements::  
+* Searching histogram ranges::  
+* Histogram Statistics::        
+* Histogram Operations::        
+* Reading and writing histograms::  
+* Resampling from histograms::  
+* The histogram probability distribution struct::  
+* Example programs for histograms::  
+* Two dimensional histograms::  
+* The 2D histogram struct::     
+* 2D Histogram allocation::     
+* Copying 2D Histograms::       
+* Updating and accessing 2D histogram elements::  
+* Searching 2D histogram ranges::  
+* 2D Histogram Statistics::     
+* 2D Histogram Operations::     
+* Reading and writing 2D histograms::  
+* Resampling from 2D histograms::  
+* Example programs for 2D histograms::  
+@end menu
+
+@node The histogram struct
+@section The histogram struct
+
+A histogram is defined by the following struct,
+
+@deftp {Data Type} {gsl_histogram}
+@table @code
+@item size_t n
+This is the number of histogram bins
+@item double * range
+The ranges of the bins are stored in an array of @math{@var{n}+1} elements
+pointed to by @var{range}.
+@item double * bin
+The counts for each bin are stored in an array of @var{n} elements
+pointed to by @var{bin}.  The bins are floating-point numbers, so you can
+increment them by non-integer values if necessary.
+@end table
+@end deftp
+@comment 
+
+@noindent
+The range for @var{bin}[i] is given by @var{range}[i] to
+@var{range}[i+1].  For @math{n} bins there are @math{n+1} entries in the
+array @var{range}.  Each bin is inclusive at the lower end and exclusive
+at the upper end.  Mathematically this means that the bins are defined by
+the following inequality,
+@tex
+\beforedisplay
+$$
+\hbox{bin[i] corresponds to range[i]} \le x < \hbox{range[i+1]}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+@display
+bin[i] corresponds to range[i] <= x < range[i+1]
+@end display
+
+@end ifinfo
+@noindent
+Here is a diagram of the correspondence between ranges and bins on the
+number-line for @math{x},
+
+@smallexample
+
+     [ bin[0] )[ bin[1] )[ bin[2] )[ bin[3] )[ bin[4] )
+  ---|---------|---------|---------|---------|---------|---  x
+   r[0]      r[1]      r[2]      r[3]      r[4]      r[5]
+
+@end smallexample
+
+@noindent
+In this picture the values of the @var{range} array are denoted by
+@math{r}.  On the left-hand side of each bin the square bracket
+@samp{[} denotes an inclusive lower bound 
+(@c{$r \le x$}
+@math{r <= x}), and the round parentheses @samp{)} on the right-hand
+side denote an exclusive upper bound (@math{x < r}).  Thus any samples
+which fall on the upper end of the histogram are excluded.  If you want
+to include this value for the last bin you will need to add an extra bin
+to your histogram.
+
+The @code{gsl_histogram} struct and its associated functions are defined
+in the header file @file{gsl_histogram.h}.
+
+@node Histogram allocation
+@section Histogram allocation
+The functions for allocating memory to a histogram follow the style of
+@code{malloc} and @code{free}.  In addition they also perform their own
+error checking.  If there is insufficient memory available to allocate a
+histogram then the functions call the error handler (with an error
+number of @code{GSL_ENOMEM}) in addition to returning a null pointer.
+Thus if you use the library error handler to abort your program then it
+isn't necessary to check every histogram @code{alloc}.
+
+@deftypefun {gsl_histogram *} gsl_histogram_alloc (size_t @var{n})
+This function allocates memory for a histogram with @var{n} bins, and
+returns a pointer to a newly created @code{gsl_histogram} struct.  If
+insufficient memory is available a null pointer is returned and the
+error handler is invoked with an error code of @code{GSL_ENOMEM}. The
+bins and ranges are not initialized, and should be prepared using one of
+the range-setting functions below in order to make the histogram ready
+for use.
+@end deftypefun
+
+@comment @deftypefun {gsl_histogram *} gsl_histogram_calloc (size_t @var{n})
+@comment This function allocates memory for a histogram with @var{n} bins, and
+@comment returns a pointer to its newly initialized @code{gsl_histogram} struct.
+@comment The bins are uniformly spaced with a total range of 
+@comment @c{$0 \le  x < n$}
+@comment @math{0 <=  x < n},
+@comment as shown in the table below.
+
+@comment @tex
+@comment \beforedisplay
+@comment $$
+@comment \matrix{
+@comment \hbox{bin[0]}&\hbox{corresponds to}& 0 \le x < 1\cr
+@comment \hbox{bin[1]}&\hbox{corresponds to}& 1 \le x < 2\cr
+@comment \dots&\dots&\dots\cr
+@comment \hbox{bin[n-1]}&\hbox{corresponds to}&n-1 \le x < n}
+@comment $$
+@comment \afterdisplay
+@comment @end tex
+@comment @ifinfo
+@comment @display
+@comment bin[0] corresponds to 0 <= x < 1
+@comment bin[1] corresponds to 1 <= x < 2
+@comment @dots{}
+@comment bin[n-1] corresponds to n-1 <= x < n
+@comment @end display
+@comment @end ifinfo
+@comment @noindent
+@comment The bins are initialized to zero so the histogram is ready for use.
+
+@comment If insufficient memory is available a null pointer is returned and the
+@comment error handler is invoked with an error code of @code{GSL_ENOMEM}.
+@comment @end deftypefun
+
+@comment @deftypefun {gsl_histogram *} gsl_histogram_calloc_uniform (size_t @var{n}, double @var{xmin}, double @var{xmax})
+@comment This function allocates memory for a histogram with @var{n} uniformly
+@comment spaced bins from @var{xmin} to @var{xmax}, and returns a pointer to the
+@comment newly initialized @code{gsl_histogram} struct. 
+@comment If insufficient memory is available a null pointer is returned and the
+@comment error handler is invoked with an error code of @code{GSL_ENOMEM}.
+@comment @end deftypefun
+
+@comment @deftypefun {gsl_histogram *} gsl_histogram_calloc_range (size_t @var{n}, double * @var{range})
+@comment This function allocates a histogram of size @var{n} using the @math{n+1}
+@comment bin ranges specified by the array @var{range}.
+@comment @end deftypefun
+
+@deftypefun int gsl_histogram_set_ranges (gsl_histogram * @var{h}, const double @var{range}[], size_t @var{size})
+This function sets the ranges of the existing histogram @var{h} using
+the array @var{range} of size @var{size}.  The values of the histogram
+bins are reset to zero.  The @code{range} array should contain the
+desired bin limits.  The ranges can be arbitrary, subject to the
+restriction that they are monotonically increasing.
+
+The following example shows how to create a histogram with logarithmic
+bins with ranges [1,10), [10,100) and [100,1000).
+
+@example
+gsl_histogram * h = gsl_histogram_alloc (3);
+
+/* bin[0] covers the range 1 <= x < 10 */
+/* bin[1] covers the range 10 <= x < 100 */
+/* bin[2] covers the range 100 <= x < 1000 */
+
+double range[4] = @{ 1.0, 10.0, 100.0, 1000.0 @};
+
+gsl_histogram_set_ranges (h, range, 4);
+@end example
+
+@noindent
+Note that the size of the @var{range} array should be defined to be one
+element bigger than the number of bins.  The additional element is
+required for the upper value of the final bin.
+@end deftypefun
+
+@deftypefun int gsl_histogram_set_ranges_uniform (gsl_histogram * @var{h}, double @var{xmin}, double @var{xmax})
+This function sets the ranges of the existing histogram @var{h} to cover
+the range @var{xmin} to @var{xmax} uniformly.  The values of the
+histogram bins are reset to zero.  The bin ranges are shown in the table
+below,
+@tex
+\beforedisplay
+$$
+\matrix{\hbox{bin[0]}&\hbox{corresponds to}& xmin \le  x < xmin + d\cr
+\hbox{bin[1]} &\hbox{corresponds to}& xmin + d \le  x < xmin + 2 d\cr
+\dots&\dots&\dots\cr
+\hbox{bin[n-1]} & \hbox{corresponds to}& xmin + (n-1)d \le  x < xmax}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+@display
+bin[0] corresponds to xmin <= x < xmin + d
+bin[1] corresponds to xmin + d <= x < xmin + 2 d
+......
+bin[n-1] corresponds to xmin + (n-1)d <= x < xmax
+@end display
+
+@end ifinfo
+@noindent
+where @math{d} is the bin spacing, @math{d = (xmax-xmin)/n}.
+@end deftypefun
+
+@deftypefun void gsl_histogram_free (gsl_histogram * @var{h})
+This function frees the histogram @var{h} and all of the memory
+associated with it.
+@end deftypefun
+
+@node Copying Histograms
+@section Copying Histograms
+
+@deftypefun int gsl_histogram_memcpy (gsl_histogram * @var{dest}, const gsl_histogram * @var{src})
+This function copies the histogram @var{src} into the pre-existing
+histogram @var{dest}, making @var{dest} into an exact copy of @var{src}.
+The two histograms must be of the same size.
+@end deftypefun
+
+@deftypefun {gsl_histogram *} gsl_histogram_clone (const gsl_histogram * @var{src})
+This function returns a pointer to a newly created histogram which is an
+exact copy of the histogram @var{src}.
+@end deftypefun
+
+@node Updating and accessing histogram elements
+@section Updating and accessing histogram elements
+
+There are two ways to access histogram bins, either by specifying an
+@math{x} coordinate or by using the bin-index directly.  The functions
+for accessing the histogram through @math{x} coordinates use a binary
+search to identify the bin which covers the appropriate range.
+
+@deftypefun int gsl_histogram_increment (gsl_histogram * @var{h}, double @var{x})
+This function updates the histogram @var{h} by adding one (1.0) to the
+bin whose range contains the coordinate @var{x}. 
+
+If @var{x} lies in the valid range of the histogram then the function
+returns zero to indicate success.  If @var{x} is less than the lower
+limit of the histogram then the function returns @code{GSL_EDOM}, and
+none of bins are modified.  Similarly, if the value of @var{x} is greater
+than or equal to the upper limit of the histogram then the function
+returns @code{GSL_EDOM}, and none of the bins are modified.  The error
+handler is not called, however, since it is often necessary to compute
+histograms for a small range of a larger dataset, ignoring the values
+outside the range of interest.
+@end deftypefun
+
+@deftypefun int gsl_histogram_accumulate (gsl_histogram * @var{h}, double @var{x}, double @var{weight})
+This function is similar to @code{gsl_histogram_increment} but increases
+the value of the appropriate bin in the histogram @var{h} by the
+floating-point number @var{weight}.
+@end deftypefun
+
+@deftypefun double gsl_histogram_get (const gsl_histogram * @var{h}, size_t @var{i})
+This function returns the contents of the @var{i}-th bin of the histogram
+@var{h}.  If @var{i} lies outside the valid range of indices for the
+histogram then the error handler is called with an error code of
+@code{GSL_EDOM} and the function returns 0.
+@end deftypefun
+
+@deftypefun int gsl_histogram_get_range (const gsl_histogram * @var{h}, size_t @var{i}, double * @var{lower}, double * @var{upper})
+This function finds the upper and lower range limits of the @var{i}-th
+bin of the histogram @var{h}.  If the index @var{i} is valid then the
+corresponding range limits are stored in @var{lower} and @var{upper}.
+The lower limit is inclusive (i.e. events with this coordinate are
+included in the bin) and the upper limit is exclusive (i.e. events with
+the coordinate of the upper limit are excluded and fall in the
+neighboring higher bin, if it exists).  The function returns 0 to
+indicate success.  If @var{i} lies outside the valid range of indices for
+the histogram then the error handler is called and the function returns
+an error code of @code{GSL_EDOM}.
+@end deftypefun
+
+@deftypefun double gsl_histogram_max (const gsl_histogram * @var{h})
+@deftypefunx double gsl_histogram_min (const gsl_histogram * @var{h})
+@deftypefunx size_t gsl_histogram_bins (const gsl_histogram * @var{h})
+These functions return the maximum upper and minimum lower range limits
+and the number of bins of the histogram @var{h}.  They provide a way of
+determining these values without accessing the @code{gsl_histogram}
+struct directly.
+@end deftypefun
+
+@deftypefun void gsl_histogram_reset (gsl_histogram * @var{h})
+This function resets all the bins in the histogram @var{h} to zero.
+@end deftypefun
+
+@node Searching histogram ranges
+@section Searching histogram ranges
+
+The following functions are used by the access and update routines to
+locate the bin which corresponds to a given @math{x} coordinate.
+
+@deftypefun int gsl_histogram_find (const gsl_histogram * @var{h}, double @var{x}, size_t * @var{i})
+This function finds and sets the index @var{i} to the bin number which
+covers the coordinate @var{x} in the histogram @var{h}.  The bin is
+located using a binary search. The search includes an optimization for
+histograms with uniform range, and will return the correct bin
+immediately in this case.  If @var{x} is found in the range of the
+histogram then the function sets the index @var{i} and returns
+@code{GSL_SUCCESS}.  If @var{x} lies outside the valid range of the
+histogram then the function returns @code{GSL_EDOM} and the error
+handler is invoked.
+@end deftypefun
+
+@node Histogram Statistics
+@section Histogram Statistics
+@cindex histogram statistics
+@cindex statistics, from histogram
+@cindex maximum value, from histogram
+@cindex minimum value, from histogram
+@deftypefun double gsl_histogram_max_val (const gsl_histogram * @var{h})
+This function returns the maximum value contained in the histogram bins.
+@end deftypefun
+
+@deftypefun size_t gsl_histogram_max_bin (const gsl_histogram * @var{h})
+This function returns the index of the bin containing the maximum
+value. In the case where several bins contain the same maximum value the
+smallest index is returned.
+@end deftypefun
+
+@deftypefun double gsl_histogram_min_val (const gsl_histogram * @var{h})
+This function returns the minimum value contained in the histogram bins.
+@end deftypefun
+
+@deftypefun size_t gsl_histogram_min_bin (const gsl_histogram * @var{h})
+This function returns the index of the bin containing the minimum
+value. In the case where several bins contain the same maximum value the
+smallest index is returned.
+@end deftypefun
+
+@cindex mean value, from histogram
+@deftypefun double gsl_histogram_mean (const gsl_histogram * @var{h})
+This function returns the mean of the histogrammed variable, where the
+histogram is regarded as a probability distribution. Negative bin values
+are ignored for the purposes of this calculation.  The accuracy of the
+result is limited by the bin width.
+@end deftypefun
+
+@cindex standard deviation, from histogram
+@cindex variance, from histogram
+@deftypefun double gsl_histogram_sigma (const gsl_histogram * @var{h})
+This function returns the standard deviation of the histogrammed
+variable, where the histogram is regarded as a probability
+distribution. Negative bin values are ignored for the purposes of this
+calculation. The accuracy of the result is limited by the bin width.
+@end deftypefun
+
+@deftypefun double gsl_histogram_sum (const gsl_histogram * @var{h})
+This function returns the sum of all bin values. Negative bin values
+are included in the sum.
+@end deftypefun
+
+@node Histogram Operations
+@section Histogram Operations
+
+@deftypefun int gsl_histogram_equal_bins_p (const gsl_histogram * @var{h1}, const gsl_histogram * @var{h2})
+This function returns 1 if the all of the individual bin
+ranges of the two histograms are identical, and 0
+otherwise.
+@end deftypefun
+
+@deftypefun int gsl_histogram_add (gsl_histogram * @var{h1}, const gsl_histogram * @var{h2})
+This function adds the contents of the bins in histogram @var{h2} to the
+corresponding bins of histogram @var{h1},  i.e. @math{h'_1(i) = h_1(i) +
+h_2(i)}.  The two histograms must have identical bin ranges.
+@end deftypefun
+
+@deftypefun int gsl_histogram_sub (gsl_histogram * @var{h1}, const gsl_histogram * @var{h2})
+This function subtracts the contents of the bins in histogram @var{h2}
+from the corresponding bins of histogram @var{h1}, i.e. @math{h'_1(i) =
+h_1(i) - h_2(i)}.  The two histograms must have identical bin ranges.
+@end deftypefun
+
+@deftypefun int gsl_histogram_mul (gsl_histogram * @var{h1}, const gsl_histogram * @var{h2})
+This function multiplies the contents of the bins of histogram @var{h1}
+by the contents of the corresponding bins in histogram @var{h2},
+i.e. @math{h'_1(i) = h_1(i) * h_2(i)}.  The two histograms must have
+identical bin ranges.
+@end deftypefun
+
+@deftypefun int gsl_histogram_div (gsl_histogram * @var{h1}, const gsl_histogram * @var{h2})
+This function divides the contents of the bins of histogram @var{h1} by
+the contents of the corresponding bins in histogram @var{h2},
+i.e. @math{h'_1(i) = h_1(i) / h_2(i)}.  The two histograms must have
+identical bin ranges.
+@end deftypefun
+
+@deftypefun int gsl_histogram_scale (gsl_histogram * @var{h}, double @var{scale})
+This function multiplies the contents of the bins of histogram @var{h}
+by the constant @var{scale}, i.e. @c{$h'_1(i) = h_1(i) * \hbox{\it scale}$}
+@math{h'_1(i) = h_1(i) * scale}.
+@end deftypefun
+
+@deftypefun int gsl_histogram_shift (gsl_histogram * @var{h}, double @var{offset})
+This function shifts the contents of the bins of histogram @var{h} by
+the constant @var{offset}, i.e. @c{$h'_1(i) = h_1(i) + \hbox{\it offset}$}
+@math{h'_1(i) = h_1(i) + offset}.
+@end deftypefun
+
+@node Reading and writing histograms
+@section Reading and writing histograms
+
+The library provides functions for reading and writing histograms to a file
+as binary data or formatted text.
+
+@deftypefun int gsl_histogram_fwrite (FILE * @var{stream}, const gsl_histogram * @var{h})
+This function writes the ranges and bins of the histogram @var{h} to the
+stream @var{stream} in binary format.  The return value is 0 for success
+and @code{GSL_EFAILED} if there was a problem writing to the file.  Since
+the data is written in the native binary format it may not be portable
+between different architectures.
+@end deftypefun
+
+@deftypefun int gsl_histogram_fread (FILE * @var{stream}, gsl_histogram * @var{h})
+This function reads into the histogram @var{h} from the open stream
+@var{stream} in binary format.  The histogram @var{h} must be
+preallocated with the correct size since the function uses the number of
+bins in @var{h} to determine how many bytes to read.  The return value is
+0 for success and @code{GSL_EFAILED} if there was a problem reading from
+the file.  The data is assumed to have been written in the native binary
+format on the same architecture.
+@end deftypefun
+
+@deftypefun int gsl_histogram_fprintf (FILE * @var{stream}, const gsl_histogram * @var{h}, const char * @var{range_format}, const char * @var{bin_format})
+This function writes the ranges and bins of the histogram @var{h}
+line-by-line to the stream @var{stream} using the format specifiers
+@var{range_format} and @var{bin_format}.  These should be one of the
+@code{%g}, @code{%e} or @code{%f} formats for floating point
+numbers.  The function returns 0 for success and @code{GSL_EFAILED} if
+there was a problem writing to the file.  The histogram output is
+formatted in three columns, and the columns are separated by spaces,
+like this,
+
+@example
+range[0] range[1] bin[0]
+range[1] range[2] bin[1]
+range[2] range[3] bin[2]
+....
+range[n-1] range[n] bin[n-1]
+@end example
+
+@noindent
+The values of the ranges are formatted using @var{range_format} and the
+value of the bins are formatted using @var{bin_format}.  Each line
+contains the lower and upper limit of the range of the bins and the
+value of the bin itself.  Since the upper limit of one bin is the lower
+limit of the next there is duplication of these values between lines but
+this allows the histogram to be manipulated with line-oriented tools.
+@end deftypefun
+
+@deftypefun int gsl_histogram_fscanf (FILE * @var{stream}, gsl_histogram * @var{h})
+This function reads formatted data from the stream @var{stream} into the
+histogram @var{h}.  The data is assumed to be in the three-column format
+used by @code{gsl_histogram_fprintf}.  The histogram @var{h} must be
+preallocated with the correct length since the function uses the size of
+@var{h} to determine how many numbers to read.  The function returns 0
+for success and @code{GSL_EFAILED} if there was a problem reading from
+the file.
+@end deftypefun
+
+@node Resampling from histograms
+@section Resampling from histograms
+@cindex resampling from histograms
+@cindex sampling from histograms
+@cindex probability distributions, from histograms
+
+A histogram made by counting events can be regarded as a measurement of
+a probability distribution.  Allowing for statistical error, the height
+of each bin represents the probability of an event where the value of
+@math{x} falls in the range of that bin.  The probability distribution
+function has the one-dimensional form @math{p(x)dx} where,
+@tex
+\beforedisplay
+$$
+p(x) = n_i/ (N w_i)
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+p(x) = n_i/ (N w_i)
+@end example
+
+@end ifinfo
+@noindent
+In this equation @math{n_i} is the number of events in the bin which
+contains @math{x}, @math{w_i} is the width of the bin and @math{N} is
+the total number of events.  The distribution of events within each bin
+is assumed to be uniform.
+
+@node The histogram probability distribution struct
+@section The histogram probability distribution struct
+@cindex probability distribution, from histogram
+@cindex sampling from histograms
+@cindex random sampling from histograms
+@cindex histograms, random sampling from
+The probability distribution function for a histogram consists of a set
+of @dfn{bins} which measure the probability of an event falling into a
+given range of a continuous variable @math{x}. A probability
+distribution function is defined by the following struct, which actually
+stores the cumulative probability distribution function.  This is the
+natural quantity for generating samples via the inverse transform
+method, because there is a one-to-one mapping between the cumulative
+probability distribution and the range [0,1].  It can be shown that by
+taking a uniform random number in this range and finding its
+corresponding coordinate in the cumulative probability distribution we
+obtain samples with the desired probability distribution.
+
+@deftp {Data Type} {gsl_histogram_pdf}
+@table @code
+@item size_t n
+This is the number of bins used to approximate the probability
+distribution function. 
+@item double * range
+The ranges of the bins are stored in an array of @math{@var{n}+1} elements
+pointed to by @var{range}.
+@item double * sum
+The cumulative probability for the bins is stored in an array of
+@var{n} elements pointed to by @var{sum}.
+@end table
+@end deftp
+@comment 
+
+@noindent
+The following functions allow you to create a @code{gsl_histogram_pdf}
+struct which represents this probability distribution and generate
+random samples from it.
+
+@deftypefun {gsl_histogram_pdf *} gsl_histogram_pdf_alloc (size_t @var{n})
+This function allocates memory for a probability distribution with
+@var{n} bins and returns a pointer to a newly initialized
+@code{gsl_histogram_pdf} struct. If insufficient memory is available a
+null pointer is returned and the error handler is invoked with an error
+code of @code{GSL_ENOMEM}.
+@end deftypefun
+
+@deftypefun int gsl_histogram_pdf_init (gsl_histogram_pdf * @var{p}, const gsl_histogram * @var{h})
+This function initializes the probability distribution @var{p} with
+the contents of the histogram @var{h}. If any of the bins of @var{h} are
+negative then the error handler is invoked with an error code of
+@code{GSL_EDOM} because a probability distribution cannot contain
+negative values.
+@end deftypefun
+
+@deftypefun void gsl_histogram_pdf_free (gsl_histogram_pdf * @var{p})
+This function frees the probability distribution function @var{p} and
+all of the memory associated with it.
+@end deftypefun
+
+@deftypefun double gsl_histogram_pdf_sample (const gsl_histogram_pdf * @var{p}, double @var{r})
+This function uses @var{r}, a uniform random number between zero and
+one, to compute a single random sample from the probability distribution
+@var{p}.  The algorithm used to compute the sample @math{s} is given by
+the following formula,
+@tex
+\beforedisplay
+$$
+s = \hbox{range}[i] + \delta * (\hbox{range}[i+1] - \hbox{range}[i])
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+s = range[i] + delta * (range[i+1] - range[i])
+@end example
+
+@end ifinfo
+@noindent
+where @math{i} is the index which satisfies 
+@c{$sum[i] \le  r < sum[i+1]$}
+@math{sum[i] <=  r < sum[i+1]} and 
+@math{delta} is 
+@c{$(r - sum[i])/(sum[i+1] - sum[i])$}
+@math{(r - sum[i])/(sum[i+1] - sum[i])}.
+@end deftypefun
+
+@node Example programs for histograms
+@section Example programs for histograms
+
+The following program shows how to make a simple histogram of a column
+of numerical data supplied on @code{stdin}.  The program takes three
+arguments, specifying the upper and lower bounds of the histogram and
+the number of bins.  It then reads numbers from @code{stdin}, one line at
+a time, and adds them to the histogram.  When there is no more data to
+read it prints out the accumulated histogram using
+@code{gsl_histogram_fprintf}.
+
+@example
+@verbatiminclude examples/histogram.c
+@end example
+
+@noindent
+Here is an example of the program in use.  We generate 10000 random
+samples from a Cauchy distribution with a width of 30 and histogram
+them over the range -100 to 100, using 200 bins.
+
+@example
+$ gsl-randist 0 10000 cauchy 30 
+   | gsl-histogram -100 100 200 > histogram.dat
+@end example
+
+@noindent
+A plot of the resulting histogram shows the familiar shape of the
+Cauchy distribution and the fluctuations caused by the finite sample
+size.
+
+@example
+$ awk '@{print $1, $3 ; print $2, $3@}' histogram.dat 
+   | graph -T X
+@end example
+
+@iftex
+@sp 1
+@center @image{histogram,3.0in,2.8in}
+@end iftex
+
+@node Two dimensional histograms
+@section Two dimensional histograms
+@cindex two dimensional histograms
+@cindex 2D histograms
+
+A two dimensional histogram consists of a set of @dfn{bins} which count
+the number of events falling in a given area of the @math{(x,y)}
+plane.  The simplest way to use a two dimensional histogram is to record
+two-dimensional position information, @math{n(x,y)}.  Another possibility
+is to form a @dfn{joint distribution} by recording related
+variables.  For example a detector might record both the position of an
+event (@math{x}) and the amount of energy it deposited @math{E}.  These
+could be histogrammed as the joint distribution @math{n(x,E)}.
+
+@node The 2D histogram struct
+@section The 2D histogram struct
+
+Two dimensional histograms are defined by the following struct,
+
+@deftp {Data Type} {gsl_histogram2d}
+@table @code
+@item size_t nx, ny
+This is the number of histogram bins in the x and y directions.
+@item double * xrange
+The ranges of the bins in the x-direction are stored in an array of
+@math{@var{nx} + 1} elements pointed to by @var{xrange}.
+@item double * yrange
+The ranges of the bins in the y-direction are stored in an array of
+@math{@var{ny} + 1} elements pointed to by @var{yrange}.
+@item double * bin
+The counts for each bin are stored in an array pointed to by @var{bin}.
+The bins are floating-point numbers, so you can increment them by
+non-integer values if necessary.  The array @var{bin} stores the two
+dimensional array of bins in a single block of memory according to the
+mapping @code{bin(i,j)} = @code{bin[i * ny + j]}.
+@end table
+@end deftp
+@comment 
+
+@noindent
+The range for @code{bin(i,j)} is given by @code{xrange[i]} to
+@code{xrange[i+1]} in the x-direction and @code{yrange[j]} to
+@code{yrange[j+1]} in the y-direction.  Each bin is inclusive at the lower
+end and exclusive at the upper end.  Mathematically this means that the
+bins are defined by the following inequality,
+@tex
+\beforedisplay
+$$
+\matrix{
+\hbox{bin(i,j) corresponds to} & 
+          \hbox{\it xrange}[i] \le x < \hbox{\it xrange}[i+1] \cr
+   \hbox{and} & \hbox{\it yrange}[j] \le y < \hbox{\it yrange}[j+1]}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+@display
+bin(i,j) corresponds to xrange[i] <= x < xrange[i+1]
+                    and yrange[j] <= y < yrange[j+1]
+@end display
+
+@end ifinfo
+@noindent
+Note that any samples which fall on the upper sides of the histogram are
+excluded.  If you want to include these values for the side bins you will
+need to add an extra row or column to your histogram.
+
+The @code{gsl_histogram2d} struct and its associated functions are
+defined in the header file @file{gsl_histogram2d.h}.
+
+@node 2D Histogram allocation
+@section 2D Histogram allocation
+
+The functions for allocating memory to a 2D histogram follow the style
+of @code{malloc} and @code{free}.  In addition they also perform their
+own error checking.  If there is insufficient memory available to
+allocate a histogram then the functions call the error handler (with
+an error number of @code{GSL_ENOMEM}) in addition to returning a null
+pointer.  Thus if you use the library error handler to abort your program
+then it isn't necessary to check every 2D histogram @code{alloc}.
+
+@deftypefun {gsl_histogram2d *} gsl_histogram2d_alloc (size_t @var{nx}, size_t @var{ny})
+This function allocates memory for a two-dimensional histogram with
+@var{nx} bins in the x direction and @var{ny} bins in the y direction.
+The function returns a pointer to a newly created @code{gsl_histogram2d}
+struct. If insufficient memory is available a null pointer is returned
+and the error handler is invoked with an error code of
+@code{GSL_ENOMEM}. The bins and ranges must be initialized with one of
+the functions below before the histogram is ready for use.
+@end deftypefun
+
+@comment @deftypefun {gsl_histogram2d *} gsl_histogram2d_calloc (size_t @var{nx}, size_t @var{ny})
+@comment This function allocates memory for a two-dimensional histogram with
+@comment @var{nx} bins in the x direction and @var{ny} bins in the y
+@comment direction.  The function returns a pointer to a newly initialized
+@comment @code{gsl_histogram2d} struct.  The bins are uniformly spaced with a
+@comment total range of 
+@comment @c{$0 \le  x < nx$}
+@comment @math{0 <= x < nx} in the x-direction and 
+@comment @c{$0 \le  y < ny$} 
+@comment @math{0 <=  y < ny} in the y-direction, as shown in the table below.
+@comment 
+@comment The bins are initialized to zero so the histogram is ready for use.
+@comment 
+@comment If insufficient memory is available a null pointer is returned and the
+@comment error handler is invoked with an error code of @code{GSL_ENOMEM}.
+@comment @end deftypefun
+@comment 
+@comment @deftypefun {gsl_histogram2d *} gsl_histogram2d_calloc_uniform (size_t @var{nx}, size_t @var{ny}, double @var{xmin}, double @var{xmax}, double @var{ymin}, double @var{ymax})
+@comment This function allocates a histogram of size @var{nx}-by-@var{ny} which
+@comment uniformly covers the ranges @var{xmin} to @var{xmax} and @var{ymin} to
+@comment @var{ymax} in the @math{x} and @math{y} directions respectively.
+@comment @end deftypefun
+@comment 
+@comment @deftypefun {gsl_histogram2d *} gsl_histogram2d_calloc_range (size_t @var{nx}, size_t @var{ny}, double * @var{xrange}, double * @var{yrange})
+@comment This function allocates a histogram of size @var{nx}-by-@var{ny} using
+@comment the @math{nx+1} and @math{ny+1} bin ranges specified by the arrays
+@comment @var{xrange} and @var{xyrange}.
+@comment @end deftypefun
+
+@deftypefun int gsl_histogram2d_set_ranges (gsl_histogram2d * @var{h},  const double @var{xrange}[], size_t @var{xsize}, const double @var{yrange}[], size_t @var{ysize})
+This function sets the ranges of the existing histogram @var{h} using
+the arrays @var{xrange} and @var{yrange} of size @var{xsize} and
+@var{ysize} respectively.  The values of the histogram bins are reset to
+zero.
+@end deftypefun
+
+@deftypefun int gsl_histogram2d_set_ranges_uniform (gsl_histogram2d * @var{h}, double @var{xmin}, double @var{xmax}, double @var{ymin}, double @var{ymax})
+This function sets the ranges of the existing histogram @var{h} to cover
+the ranges @var{xmin} to @var{xmax} and @var{ymin} to @var{ymax}
+uniformly.  The values of the histogram bins are reset to zero.
+@end deftypefun
+
+@deftypefun void gsl_histogram2d_free (gsl_histogram2d * @var{h})
+This function frees the 2D histogram @var{h} and all of the memory
+associated with it.
+@end deftypefun
+
+@node Copying 2D Histograms
+@section Copying 2D Histograms
+
+@deftypefun int gsl_histogram2d_memcpy (gsl_histogram2d * @var{dest}, const gsl_histogram2d * @var{src})
+This function copies the histogram @var{src} into the pre-existing
+histogram @var{dest}, making @var{dest} into an exact copy of @var{src}.
+The two histograms must be of the same size.
+@end deftypefun
+
+@deftypefun {gsl_histogram2d *} gsl_histogram2d_clone (const gsl_histogram2d * @var{src})
+This function returns a pointer to a newly created histogram which is an
+exact copy of the histogram @var{src}.
+@end deftypefun
+
+@node Updating and accessing 2D histogram elements
+@section Updating and accessing 2D histogram elements
+
+You can access the bins of a two-dimensional histogram either by
+specifying a pair of @math{(x,y)} coordinates or by using the bin
+indices @math{(i,j)} directly.  The functions for accessing the histogram
+through @math{(x,y)} coordinates use binary searches in the x and y
+directions to identify the bin which covers the appropriate range.
+
+@deftypefun int gsl_histogram2d_increment (gsl_histogram2d * @var{h}, double @var{x}, double @var{y})
+This function updates the histogram @var{h} by adding one (1.0) to the
+bin whose x and y ranges contain the coordinates (@var{x},@var{y}).
+
+If the point @math{(x,y)} lies inside the valid ranges of the
+histogram then the function returns zero to indicate success.  If
+@math{(x,y)} lies outside the limits of the histogram then the
+function returns @code{GSL_EDOM}, and none of the bins are modified.  The
+error handler is not called, since it is often necessary to compute
+histograms for a small range of a larger dataset, ignoring any
+coordinates outside the range of interest.
+@end deftypefun
+
+@deftypefun int gsl_histogram2d_accumulate (gsl_histogram2d * @var{h}, double @var{x}, double @var{y}, double @var{weight})
+This function is similar to @code{gsl_histogram2d_increment} but increases
+the value of the appropriate bin in the histogram @var{h} by the
+floating-point number @var{weight}.
+@end deftypefun
+
+@deftypefun double gsl_histogram2d_get (const gsl_histogram2d * @var{h}, size_t @var{i}, size_t @var{j})
+This function returns the contents of the (@var{i},@var{j})-th bin of the
+histogram @var{h}.  If (@var{i},@var{j}) lies outside the valid range of
+indices for the histogram then the error handler is called with an error
+code of @code{GSL_EDOM} and the function returns 0.
+@end deftypefun
+
+@deftypefun int gsl_histogram2d_get_xrange (const gsl_histogram2d * @var{h}, size_t @var{i}, double * @var{xlower}, double * @var{xupper})
+@deftypefunx int gsl_histogram2d_get_yrange (const gsl_histogram2d * @var{h}, size_t @var{j}, double * @var{ylower}, double * @var{yupper})
+These functions find the upper and lower range limits of the @var{i}-th
+and @var{j}-th bins in the x and y directions of the histogram @var{h}.
+The range limits are stored in @var{xlower} and @var{xupper} or
+@var{ylower} and @var{yupper}.  The lower limits are inclusive
+(i.e. events with these coordinates are included in the bin) and the
+upper limits are exclusive (i.e. events with the value of the upper
+limit are not included and fall in the neighboring higher bin, if it
+exists).  The functions return 0 to indicate success.  If @var{i} or
+@var{j} lies outside the valid range of indices for the histogram then
+the error handler is called with an error code of @code{GSL_EDOM}.
+@end deftypefun
+
+@deftypefun double gsl_histogram2d_xmax (const gsl_histogram2d * @var{h})
+@deftypefunx double gsl_histogram2d_xmin (const gsl_histogram2d * @var{h})
+@deftypefunx size_t gsl_histogram2d_nx (const gsl_histogram2d * @var{h})
+@deftypefunx double gsl_histogram2d_ymax (const gsl_histogram2d * @var{h})
+@deftypefunx double gsl_histogram2d_ymin (const gsl_histogram2d * @var{h})
+@deftypefunx size_t gsl_histogram2d_ny (const gsl_histogram2d * @var{h})
+These functions return the maximum upper and minimum lower range limits
+and the number of bins for the x and y directions of the histogram
+@var{h}.  They provide a way of determining these values without
+accessing the @code{gsl_histogram2d} struct directly.
+@end deftypefun
+
+@deftypefun void gsl_histogram2d_reset (gsl_histogram2d * @var{h})
+This function resets all the bins of the histogram @var{h} to zero.
+@end deftypefun
+
+@node Searching 2D histogram ranges
+@section Searching 2D histogram ranges
+
+The following functions are used by the access and update routines to
+locate the bin which corresponds to a given @math{(x,y)} coordinate.
+
+@deftypefun int gsl_histogram2d_find (const gsl_histogram2d * @var{h}, double @var{x}, double @var{y}, size_t * @var{i}, size_t * @var{j})
+This function finds and sets the indices @var{i} and @var{j} to the to
+the bin which covers the coordinates (@var{x},@var{y}). The bin is
+located using a binary search.  The search includes an optimization for
+histograms with uniform ranges, and will return the correct bin immediately
+in this case. If @math{(x,y)} is found then the function sets the
+indices (@var{i},@var{j}) and returns @code{GSL_SUCCESS}.  If
+@math{(x,y)} lies outside the valid range of the histogram then the
+function returns @code{GSL_EDOM} and the error handler is invoked.
+@end deftypefun
+
+@node 2D Histogram Statistics
+@section 2D Histogram Statistics
+
+@deftypefun double gsl_histogram2d_max_val (const gsl_histogram2d * @var{h})
+This function returns the maximum value contained in the histogram bins.
+@end deftypefun
+
+@deftypefun void gsl_histogram2d_max_bin (const gsl_histogram2d * @var{h}, size_t * @var{i}, size_t * @var{j})
+This function finds the indices of the bin containing the maximum value
+in the histogram @var{h} and stores the result in (@var{i},@var{j}). In
+the case where several bins contain the same maximum value the first bin
+found is returned.
+@end deftypefun
+
+@deftypefun double gsl_histogram2d_min_val (const gsl_histogram2d * @var{h})
+This function returns the minimum value contained in the histogram bins.
+@end deftypefun
+
+@deftypefun void gsl_histogram2d_min_bin (const gsl_histogram2d * @var{h}, size_t * @var{i}, size_t * @var{j})
+This function finds the indices of the bin containing the minimum value
+in the histogram @var{h} and stores the result in (@var{i},@var{j}). In
+the case where several bins contain the same maximum value the first bin
+found is returned.
+@end deftypefun
+
+@deftypefun double gsl_histogram2d_xmean (const gsl_histogram2d * @var{h})
+This function returns the mean of the histogrammed x variable, where the
+histogram is regarded as a probability distribution. Negative bin values
+are ignored for the purposes of this calculation.
+@end deftypefun
+
+@deftypefun double gsl_histogram2d_ymean (const gsl_histogram2d * @var{h})
+This function returns the mean of the histogrammed y variable, where the
+histogram is regarded as a probability distribution. Negative bin values
+are ignored for the purposes of this calculation.
+@end deftypefun
+
+@deftypefun double gsl_histogram2d_xsigma (const gsl_histogram2d * @var{h})
+This function returns the standard deviation of the histogrammed
+x variable, where the histogram is regarded as a probability
+distribution. Negative bin values are ignored for the purposes of this
+calculation.
+@end deftypefun
+
+@deftypefun double gsl_histogram2d_ysigma (const gsl_histogram2d * @var{h})
+This function returns the standard deviation of the histogrammed
+y variable, where the histogram is regarded as a probability
+distribution. Negative bin values are ignored for the purposes of this
+calculation.
+@end deftypefun
+
+@deftypefun double gsl_histogram2d_cov (const gsl_histogram2d * @var{h})
+This function returns the covariance of the histogrammed x and y
+variables, where the histogram is regarded as a probability
+distribution. Negative bin values are ignored for the purposes of this
+calculation.
+@end deftypefun
+
+@deftypefun double gsl_histogram2d_sum (const gsl_histogram2d * @var{h})
+This function returns the sum of all bin values. Negative bin values
+are included in the sum.
+@end deftypefun
+
+@node 2D Histogram Operations
+@section 2D Histogram Operations
+
+@deftypefun int gsl_histogram2d_equal_bins_p (const gsl_histogram2d * @var{h1}, const gsl_histogram2d * @var{h2})
+This function returns 1 if all the individual bin ranges of the two
+histograms are identical, and 0 otherwise.
+@end deftypefun
+
+@deftypefun int gsl_histogram2d_add (gsl_histogram2d * @var{h1}, const gsl_histogram2d * @var{h2})
+This function adds the contents of the bins in histogram @var{h2} to the
+corresponding bins of histogram @var{h1},
+i.e. @math{h'_1(i,j) = h_1(i,j) + h_2(i,j)}.
+The two histograms must have identical bin ranges.
+@end deftypefun
+
+@deftypefun int gsl_histogram2d_sub (gsl_histogram2d * @var{h1}, const gsl_histogram2d * @var{h2})
+This function subtracts the contents of the bins in histogram @var{h2} from the
+corresponding bins of histogram @var{h1},
+i.e. @math{h'_1(i,j) = h_1(i,j) - h_2(i,j)}.
+The two histograms must have identical bin ranges.
+@end deftypefun
+
+@deftypefun int gsl_histogram2d_mul (gsl_histogram2d * @var{h1}, const gsl_histogram2d * @var{h2})
+This function multiplies the contents of the bins of histogram @var{h1}
+by the contents of the corresponding bins in histogram @var{h2},
+i.e. @math{h'_1(i,j) = h_1(i,j) * h_2(i,j)}.
+The two histograms must have identical bin ranges.
+@end deftypefun
+
+@deftypefun int gsl_histogram2d_div (gsl_histogram2d * @var{h1}, const gsl_histogram2d * @var{h2})
+This function divides the contents of the bins of histogram @var{h1}
+by the contents of the corresponding bins in histogram @var{h2},
+i.e. @math{h'_1(i,j) = h_1(i,j) / h_2(i,j)}.
+The two histograms must have identical bin ranges.
+@end deftypefun
+
+@deftypefun int gsl_histogram2d_scale (gsl_histogram2d * @var{h}, double @var{scale})
+This function multiplies the contents of the bins of histogram @var{h}
+by the constant @var{scale}, i.e. @c{$h'_1(i,j) = h_1(i,j) * \hbox{\it scale}$}
+@math{h'_1(i,j) = h_1(i,j) scale}.
+@end deftypefun
+
+@deftypefun int gsl_histogram2d_shift (gsl_histogram2d * @var{h}, double @var{offset})
+This function shifts the contents of the bins of histogram @var{h}
+by the constant @var{offset}, i.e. @c{$h'_1(i,j) = h_1(i,j) + \hbox{\it offset}$}
+@math{h'_1(i,j) = h_1(i,j) + offset}.
+@end deftypefun
+
+@node Reading and writing 2D histograms
+@section Reading and writing 2D histograms
+
+The library provides functions for reading and writing two dimensional
+histograms to a file as binary data or formatted text.
+
+@deftypefun int gsl_histogram2d_fwrite (FILE * @var{stream}, const gsl_histogram2d * @var{h})
+This function writes the ranges and bins of the histogram @var{h} to the
+stream @var{stream} in binary format.  The return value is 0 for success
+and @code{GSL_EFAILED} if there was a problem writing to the file.  Since
+the data is written in the native binary format it may not be portable
+between different architectures.
+@end deftypefun
+
+@deftypefun int gsl_histogram2d_fread (FILE * @var{stream}, gsl_histogram2d * @var{h})
+This function reads into the histogram @var{h} from the stream
+@var{stream} in binary format.  The histogram @var{h} must be
+preallocated with the correct size since the function uses the number of
+x and y bins in @var{h} to determine how many bytes to read.  The return
+value is 0 for success and @code{GSL_EFAILED} if there was a problem
+reading from the file.  The data is assumed to have been written in the
+native binary format on the same architecture.
+@end deftypefun
+
+@deftypefun int gsl_histogram2d_fprintf (FILE * @var{stream}, const gsl_histogram2d * @var{h}, const char * @var{range_format}, const char * @var{bin_format})
+This function writes the ranges and bins of the histogram @var{h}
+line-by-line to the stream @var{stream} using the format specifiers
+@var{range_format} and @var{bin_format}.  These should be one of the
+@code{%g}, @code{%e} or @code{%f} formats for floating point
+numbers.  The function returns 0 for success and @code{GSL_EFAILED} if
+there was a problem writing to the file.  The histogram output is
+formatted in five columns, and the columns are separated by spaces,
+like this,
+
+@smallexample
+xrange[0] xrange[1] yrange[0] yrange[1] bin(0,0)
+xrange[0] xrange[1] yrange[1] yrange[2] bin(0,1)
+xrange[0] xrange[1] yrange[2] yrange[3] bin(0,2)
+....
+xrange[0] xrange[1] yrange[ny-1] yrange[ny] bin(0,ny-1)
+
+xrange[1] xrange[2] yrange[0] yrange[1] bin(1,0)
+xrange[1] xrange[2] yrange[1] yrange[2] bin(1,1)
+xrange[1] xrange[2] yrange[1] yrange[2] bin(1,2)
+....
+xrange[1] xrange[2] yrange[ny-1] yrange[ny] bin(1,ny-1)
+
+....
+
+xrange[nx-1] xrange[nx] yrange[0] yrange[1] bin(nx-1,0)
+xrange[nx-1] xrange[nx] yrange[1] yrange[2] bin(nx-1,1)
+xrange[nx-1] xrange[nx] yrange[1] yrange[2] bin(nx-1,2)
+....
+xrange[nx-1] xrange[nx] yrange[ny-1] yrange[ny] bin(nx-1,ny-1)
+@end smallexample
+
+@noindent
+Each line contains the lower and upper limits of the bin and the
+contents of the bin.  Since the upper limits of the each bin are the
+lower limits of the neighboring bins there is duplication of these
+values but this allows the histogram to be manipulated with
+line-oriented tools.
+@end deftypefun
+
+@deftypefun int gsl_histogram2d_fscanf (FILE * @var{stream}, gsl_histogram2d * @var{h})
+This function reads formatted data from the stream @var{stream} into the
+histogram @var{h}.  The data is assumed to be in the five-column format
+used by @code{gsl_histogram_fprintf}.  The histogram @var{h} must be
+preallocated with the correct lengths since the function uses the sizes
+of @var{h} to determine how many numbers to read.  The function returns 0
+for success and @code{GSL_EFAILED} if there was a problem reading from
+the file.
+@end deftypefun
+
+@node Resampling from 2D histograms
+@section Resampling from 2D histograms
+
+As in the one-dimensional case, a two-dimensional histogram made by
+counting events can be regarded as a measurement of a probability
+distribution.  Allowing for statistical error, the height of each bin
+represents the probability of an event where (@math{x},@math{y}) falls in
+the range of that bin.  For a two-dimensional histogram the probability
+distribution takes the form @math{p(x,y) dx dy} where,
+@tex
+\beforedisplay
+$$
+p(x,y) = n_{ij}/ (N A_{ij})
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+p(x,y) = n_@{ij@}/ (N A_@{ij@})
+@end example
+
+@end ifinfo
+@noindent
+In this equation 
+@c{$n_{ij}$}
+@math{n_@{ij@}} is the number of events in the bin which
+contains @math{(x,y)}, 
+@c{$A_{ij}$}
+@math{A_@{ij@}} is the area of the bin and @math{N} is
+the total number of events.  The distribution of events within each bin
+is assumed to be uniform.
+
+@deftp {Data Type} {gsl_histogram2d_pdf}
+@table @code
+@item size_t nx, ny
+This is the number of histogram bins used to approximate the probability
+distribution function in the x and y directions.
+@item double * xrange
+The ranges of the bins in the x-direction are stored in an array of
+@math{@var{nx} + 1} elements pointed to by @var{xrange}.
+@item double * yrange
+The ranges of the bins in the y-direction are stored in an array of
+@math{@var{ny} + 1} pointed to by @var{yrange}.
+@item double * sum
+The cumulative probability for the bins is stored in an array of
+@var{nx}*@var{ny} elements pointed to by @var{sum}.
+@end table
+@end deftp
+@comment 
+
+@noindent
+The following functions allow you to create a @code{gsl_histogram2d_pdf}
+struct which represents a two dimensional probability distribution and
+generate random samples from it.
+
+@deftypefun {gsl_histogram2d_pdf *} gsl_histogram2d_pdf_alloc (size_t @var{nx}, size_t @var{ny})
+This function allocates memory for a two-dimensional probability
+distribution of size @var{nx}-by-@var{ny} and returns a pointer to a
+newly initialized @code{gsl_histogram2d_pdf} struct. If insufficient
+memory is available a null pointer is returned and the error handler is
+invoked with an error code of @code{GSL_ENOMEM}.
+@end deftypefun
+
+@deftypefun int gsl_histogram2d_pdf_init (gsl_histogram2d_pdf * @var{p}, const gsl_histogram2d * @var{h})
+This function initializes the two-dimensional probability distribution
+calculated @var{p} from the histogram @var{h}.  If any of the bins of
+@var{h} are negative then the error handler is invoked with an error
+code of @code{GSL_EDOM} because a probability distribution cannot
+contain negative values.
+@end deftypefun
+
+@deftypefun void gsl_histogram2d_pdf_free (gsl_histogram2d_pdf * @var{p})
+This function frees the two-dimensional probability distribution
+function @var{p} and all of the memory associated with it.
+@end deftypefun
+
+@deftypefun int gsl_histogram2d_pdf_sample (const gsl_histogram2d_pdf * @var{p}, double @var{r1}, double @var{r2}, double * @var{x}, double * @var{y})
+This function uses two uniform random numbers between zero and one,
+@var{r1} and @var{r2}, to compute a single random sample from the
+two-dimensional probability distribution @var{p}.
+@end deftypefun
+
+@page
+@node Example programs for 2D histograms
+@section Example programs for 2D histograms
+This program demonstrates two features of two-dimensional histograms.
+First a 10-by-10 two-dimensional histogram is created with x and y running
+from 0 to 1.  Then a few sample points are added to the histogram, at
+(0.3,0.3) with a height of 1, at (0.8,0.1) with a height of 5 and at
+(0.7,0.9) with a height of 0.5.  This histogram with three events is
+used to generate a random sample of 1000 simulated events, which are
+printed out.
+
+@example
+@verbatiminclude examples/histogram2d.c
+@end example
+
+@noindent
+@iftex
+The following plot shows the distribution of the simulated events.  Using
+a higher resolution grid we can see the original underlying histogram
+and also the statistical fluctuations caused by the events being
+uniformly distributed over the area of the original bins.
+
+@sp 1
+@center @image{histogram2d,3.4in}
+@end iftex
+
diff --git a/gsl-1.9/doc/histogram2d.eps b/gsl-1.9/doc/histogram2d.eps
new file mode 100644
index 0000000..a26062a
--- /dev/null
+++ b/gsl-1.9/doc/histogram2d.eps
@@ -0,0 +1,776 @@
+%!PS-Adobe-2.0 EPSF-2.0
+%%BoundingBox: 0 0 567 567
+%%Title: paw.eps
+%%Creator: HIGZ Version 1.23/07
+%%CreationDate: 98/05/04   19.25
+%%EndComments
+%%BeginProlog
+80 dict begin
+/s {stroke} def /l {lineto} def /m {moveto} def /t {translate} def
+/sw {stringwidth} def /r {rotate} def /rl {roll}  def /R {repeat} def
+/d {rlineto} def /rm {rmoveto} def /gr {grestore} def /f {eofill} def
+/c {setrgbcolor} def /lw {setlinewidth} def /sd {setdash} def
+/cl {closepath} def /sf {scalefont setfont} def /black {0 setgray} def
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+/NC{systemdict begin initclip end}def/C{NC box clip newpath}def
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+/mp {newpath /y exch def /x exch def} def
+/side {[w .77 mul w .23 mul] .385 w mul sd w 0 l currentpoint t -144 r} def
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+/reencdict 24 dict def /ReEncode {reencdict begin /nco&na exch def
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+/newfont basefontdict maxlength dict def basefontdict {exch dup /FID ne
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+/FontName nfnam put nco&na aload pop nco&na length 2 idiv {newfont
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+/accvec [ 176 /agrave 181 /Agrave 190 /acircumflex 192 /Acircumflex
+201 /adieresis 204 /Adieresis 209 /ccedilla 210 /Ccedilla 211 /eacute
+212 /Eacute 213 /egrave 214 /Egrave 215 /ecircumflex 216 /Ecircumflex
+217 /edieresis 218 /Edieresis 219 /icircumflex 220 /Icircumflex
+221 /idieresis 222 /Idieresis 223 /ntilde 224 /Ntilde 226 /ocircumflex
+228 /Ocircumflex 229 /odieresis 230 /Odieresis 231 /ucircumflex 236 /Ucircumflex
+237 /udieresis 238 /Udieresis 239 /aring 242 /Aring 243 /ydieresis
+244 /Ydieresis 246 /aacute 247 /Aacute 252 /ugrave 253 /Ugrave] def
+/Times-Roman /Times-Roman accvec ReEncode
+/Times-Italic /Times-Italic accvec ReEncode
+/Times-Bold /Times-Bold accvec ReEncode
+/Times-BoldItalic /Times-BoldItalic accvec ReEncode
+/Helvetica /Helvetica accvec ReEncode
+/Helvetica-Oblique /Helvetica-Oblique accvec ReEncode
+/Helvetica-Bold /Helvetica-Bold accvec ReEncode
+/Helvetica-BoldOblique /Helvetica-BoldOblique  accvec ReEncode
+/Courier /Courier accvec ReEncode
+/Courier-Oblique /Courier-Oblique accvec ReEncode
+/Courier-Bold /Courier-Bold accvec ReEncode
+/Courier-BoldOblique /Courier-BoldOblique accvec ReEncode
+/oshow {gsave [] 0 sd true charpath stroke gr} def
+/stwn { /fs exch def /fn exch def /text exch def fn findfont fs sf
+ text sw pop xs add /xs exch def} def
+/stwb { /fs exch def /fn exch def /nbas exch def /textf exch def
+textf length /tlen exch def nbas tlen gt {/nbas tlen def} if
+fn findfont fs sf textf dup length nbas sub nbas getinterval sw
+pop neg xs add /xs exch def} def
+/accspe [ 65 /plusminus 66 /bar 67 /existential 68 /universal
+69 /exclam 70 /numbersign 71 /greater 72 /question 73 /integral
+74 /colon 75 /semicolon 76 /less 77 /bracketleft 78 /bracketright
+79 /greaterequal 80 /braceleft 81 /braceright 82 /radical
+83 /spade 84 /heart 85 /diamond 86 /club 87 /lessequal
+88 /multiply 89 /percent 90 /infinity 48 /circlemultiply 49 /circleplus
+50 /emptyset 51 /lozenge 52 /bullet 53 /arrowright 54 /arrowup
+55 /arrowleft 56 /arrowdown 57 /arrowboth 48 /degree 44 /comma 43 /plus
+ 45 /angle 42 /angleleft 47 /divide 61 /notequal 40 /equivalence 41 /second
+ 97 /approxequal 98 /congruent 99 /perpendicular 100 /partialdiff 101 /florin
+ 102 /intersection 103 /union 104 /propersuperset 105 /reflexsuperset
+ 106 /notsubset 107 /propersubset 108 /reflexsubset 109 /element 110 /notelement
+ 111 /gradient 112 /logicaland 113 /logicalor 114 /arrowdblboth
+ 115 /arrowdblleft 116 /arrowdblup 117 /arrowdblright 118 /arrowdbldown
+ 119 /ampersand 120 /omega1 121 /similar 122 /aleph ] def
+/Symbol /Special accspe ReEncode
+%%EndProlog
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+ s 479 524 m 17 Y s 466 534 m 17 Y s 453 545 m 17 Y s 439 555 m 17 Y s 426 548 m
+ 34 Y s 413 575 m 17 Y s 399 585 m 17 Y s 386 595 m 17 Y s 373 605 m 17 Y s 360
+ 599 m 34 Y s 346 626 m 17 Y s 333 636 m 17 Y s 320 646 m 17 Y s 306 656 m 17 Y
+ s 293 649 m 34 Y s 280 676 m 17 Y s 267 687 m 17 Y s 253 697 m 17 Y s 240 707 m
+ 17 Y s 227 700 m 34 Y s 891 193 m 34 Y s 880 133 m -5 -2 d -3 -4 d -1 -8 d -4 Y
+ 1 -8 d 3 -5 d 5 -1 d 3 X 4 1 d 3 5 d 2 8 d 4 Y -2 8 d -3 4 d -4 2 d -3 X cl s
+ 771 183 m -4 -1 d -3 -5 d -2 -7 d -5 Y 2 -7 d 3 -5 d 4 -1 d 3 X 5 1 d 3 5 d 2 7
+ d 5 Y -2 7 d -3 5 d -5 1 d -3 X cl s 796 155 m -2 -2 d 2 -1 d 1 1 d -1 2 d cl s
+ 812 177 m 3 2 d 5 4 d -31 Y s 705 234 m -5 -1 d -3 -5 d -1 -7 d -5 Y 1 -8 d 3
+ -4 d 5 -2 d 3 X 5 2 d 3 4 d 1 8 d 5 Y -1 7 d -3 5 d -5 1 d -3 X cl s 729 205 m
+ -1 -1 d 1 -2 d 2 2 d -2 1 d cl s 743 227 m 1 Y 1 3 d 2 2 d 3 1 d 6 X 3 -1 d 1
+ -2 d 2 -3 d -3 Y -2 -3 d -3 -5 d -15 -15 d 21 X s 639 285 m -5 -2 d -3 -4 d -1
+ -8 d -4 Y 1 -8 d 3 -4 d 5 -2 d 3 X 4 2 d 3 4 d 2 8 d 4 Y -2 8 d -3 4 d -4 2 d
+ -3 X cl s 663 256 m -2 -1 d 2 -2 d 1 2 d -1 1 d cl s 678 285 m 17 X -10 -12 d 5
+ X 3 -2 d 2 -1 d 1 -5 d -3 Y -1 -4 d -3 -3 d -5 -2 d -5 X -4 2 d -2 1 d -1 3 d s
+ 572 336 m -4 -2 d -3 -4 d -2 -8 d -5 Y 2 -7 d 3 -5 d 4 -1 d 3 X 5 1 d 3 5 d 1 7
+ d 5 Y -1 8 d -3 4 d -5 2 d -3 X cl s 596 307 m -1 -2 d 1 -1 d 2 1 d -2 2 d cl s
+ 624 336 m -16 -22 d 23 X s 624 336 m -32 Y s 506 386 m -5 -1 d -3 -5 d -1 -7 d
+ -5 Y 1 -7 d 3 -5 d 5 -1 d 3 X 4 1 d 3 5 d 2 7 d 5 Y -2 7 d -3 5 d -4 1 d -3 X
+ cl s 530 358 m -2 -2 d 2 -1 d 1 1 d -1 2 d cl s 560 386 m -15 X -1 -13 d 1 1 d
+ 5 2 d 4 X 5 -2 d 3 -3 d 1 -4 d -3 Y -1 -5 d -3 -3 d -5 -1 d -4 X -5 1 d -1 2 d
+ -2 3 d s 439 437 m -4 -2 d -3 -4 d -2 -8 d -4 Y 2 -8 d 3 -4 d 4 -2 d 3 X 5 2 d
+ 3 4 d 1 8 d 4 Y -1 8 d -3 4 d -5 2 d -3 X cl s 464 408 m -2 -1 d 2 -2 d 1 2 d
+ -1 1 d cl s 495 432 m -1 3 d -5 2 d -3 X -4 -2 d -3 -4 d -2 -8 d -7 Y 2 -6 d 3
+ -3 d 4 -2 d 2 X 4 2 d 3 3 d 2 4 d 2 Y -2 4 d -3 3 d -4 2 d -2 X -4 -2 d -3 -3 d
+ -2 -4 d s 373 488 m -5 -2 d -3 -4 d -1 -8 d -4 Y 1 -8 d 3 -5 d 5 -1 d 3 X 4 1 d
+ 4 5 d 1 8 d 4 Y -1 8 d -4 4 d -4 2 d -3 X cl s 397 459 m -1 -2 d 1 -1 d 2 1 d
+ -2 2 d cl s 430 488 m -15 -32 d s 409 488 m 21 X s 307 538 m -5 -1 d -3 -5 d -2
+ -7 d -5 Y 2 -7 d 3 -5 d 5 -1 d 3 X 4 1 d 3 5 d 2 7 d 5 Y -2 7 d -3 5 d -4 1 d
+ -3 X cl s 331 510 m -2 -2 d 2 -1 d 1 1 d -1 2 d cl s 350 538 m -4 -1 d -2 -3 d
+ -3 Y 2 -3 d 3 -2 d 6 -1 d 4 -2 d 3 -3 d 2 -3 d -4 Y -2 -3 d -1 -2 d -5 -1 d -6
+ X -4 1 d -2 2 d -1 3 d 4 Y 1 3 d 3 3 d 5 2 d 6 1 d 3 2 d 1 3 d 3 Y -1 3 d -5 1
+ d -6 X cl s 240 589 m -4 -1 d -3 -5 d -2 -7 d -5 Y 2 -8 d 3 -4 d 4 -2 d 3 X 5 2
+ d 3 4 d 1 8 d 5 Y -1 7 d -3 5 d -5 1 d -3 X cl s 264 560 m -1 -1 d 1 -2 d 2 2 d
+ -2 1 d cl s 296 579 m -1 -5 d -3 -3 d -5 -2 d -2 X -4 2 d -3 3 d -2 5 d 1 Y 2 5
+ d 3 3 d 4 1 d 2 X 5 -1 d 3 -3 d 1 -6 d -8 Y -1 -8 d -3 -4 d -5 -2 d -3 X -5 2 d
+ -1 3 d s 211 634 m 3 1 d 5 5 d -32 Y s 227 734 m 1014 Y s 193 734 m 34 X s 210
+ 783 m 17 X s 210 832 m 17 X s 210 882 m 17 X s 193 931 m 34 X s 210 980 m 17 X
+ s 210 1029 m 17 X s 210 1079 m 17 X s 193 1128 m 34 X s 210 1177 m 17 X s 210
+ 1226 m 17 X s 210 1276 m 17 X s 193 1325 m 34 X s 210 1374 m 17 X s 210 1423 m
+ 17 X s 210 1473 m 17 X s 193 1522 m 34 X s 210 1571 m 17 X s 210 1620 m 17 X s
+ 210 1670 m 17 X s 193 1719 m 34 X s 193 1719 m 34 X s 153 750 m -4 -2 d -3 -4 d
+ -2 -8 d -4 Y 2 -8 d 3 -4 d 4 -2 d 3 X 5 2 d 3 4 d 2 8 d 4 Y -2 8 d -3 4 d -5 2
+ d -3 X cl s 116 939 m 2 Y 1 3 d 2 1 d 3 2 d 6 X 3 -2 d 1 -1 d 2 -3 d -3 Y -2 -3
+ d -3 -5 d -15 -15 d 21 X s 153 947 m -4 -2 d -3 -4 d -2 -8 d -4 Y 2 -8 d 3 -4 d
+ 4 -2 d 3 X 5 2 d 3 4 d 2 8 d 4 Y -2 8 d -3 4 d -5 2 d -3 X cl s 129 1144 m -15
+ -21 d 23 X s 129 1144 m -32 Y s 153 1144 m -4 -2 d -3 -4 d -2 -8 d -4 Y 2 -8 d
+ 3 -4 d 4 -2 d 3 X 5 2 d 3 4 d 2 8 d 4 Y -2 8 d -3 4 d -5 2 d -3 X cl s 134 1336
+ m -2 3 d -4 2 d -3 X -5 -2 d -3 -4 d -1 -8 d -7 Y 1 -6 d 3 -3 d 5 -2 d 1 X 5 2
+ d 3 3 d 1 4 d 2 Y -1 4 d -3 3 d -5 2 d -1 X -5 -2 d -3 -3 d -1 -4 d s 153 1341
+ m -4 -2 d -3 -4 d -2 -8 d -4 Y 2 -8 d 3 -4 d 4 -2 d 3 X 5 2 d 3 4 d 2 8 d 4 Y
+ -2 8 d -3 4 d -5 2 d -3 X cl s 122 1538 m -5 -2 d -1 -3 d -3 Y 1 -3 d 3 -1 d 6
+ -2 d 5 -1 d 3 -3 d 1 -3 d -5 Y -1 -3 d -2 -2 d -4 -1 d -6 X -5 1 d -1 2 d -2 3
+ d 5 Y 2 3 d 3 3 d 4 1 d 6 2 d 3 1 d 2 3 d 3 Y -2 3 d -4 2 d -6 X cl s 153 1538
+ m -4 -2 d -3 -4 d -2 -8 d -4 Y 2 -8 d 3 -5 d 4 -1 d 3 X 5 1 d 3 5 d 2 8 d 4 Y
+ -2 8 d -3 4 d -5 2 d -3 X cl s 88 1729 m 3 1 d 5 5 d -32 Y s 123 1735 m -4 -2 d
+ -3 -4 d -2 -8 d -4 Y 2 -8 d 3 -5 d 4 -1 d 3 X 5 1 d 3 5 d 1 8 d 4 Y -1 8 d -3 4
+ d -5 2 d -3 X cl s 153 1735 m -4 -2 d -3 -4 d -2 -8 d -4 Y 2 -8 d 3 -5 d 4 -1 d
+ 3 X 5 1 d 3 5 d 2 8 d 4 Y -2 8 d -3 4 d -5 2 d -3 X cl s 1041 71 m 18 X 3 Y -1
+ 3 d -2 2 d -3 1 d -4 X -3 -1 d -3 -3 d -2 -5 d -3 Y 2 -4 d 3 -4 d 3 -1 d 4 X 3
+ 1 d 3 4 d s 1068 80 m 17 -21 d s 1085 80 m -17 -21 d s 1112 80 m -21 Y s 1112
+ 76 m -3 3 d -3 1 d -5 X -3 -1 d -3 -3 d -1 -5 d -3 Y 1 -4 d 3 -4 d 3 -1 d 5 X 3
+ 1 d 3 4 d s 1124 80 m -21 Y s 1124 74 m 5 5 d 3 1 d 4 X 3 -1 d 2 -5 d -15 Y s
+ 1141 74 m 4 5 d 3 1 d 5 X 3 -1 d 1 -5 d -15 Y s 1170 80 m -32 Y s 1170 76 m 3 3
+ d 3 1 d 4 X 3 -1 d 3 -3 d 2 -5 d -3 Y -2 -4 d -3 -4 d -3 -1 d -4 X -3 1 d -3 4
+ d s 1198 91 m -32 Y s 1209 71 m 18 X 3 Y -2 3 d -1 2 d -3 1 d -5 X -3 -1 d -3
+ -3 d -1 -5 d -3 Y 1 -4 d 3 -4 d 3 -1 d 5 X 3 1 d 3 4 d s
+ gr  gr 
+showpage
+end
+%%EOF
diff --git a/gsl-1.9/doc/ieee754.texi b/gsl-1.9/doc/ieee754.texi
new file mode 100644
index 0000000..4cfecda
--- /dev/null
+++ b/gsl-1.9/doc/ieee754.texi
@@ -0,0 +1,461 @@
+@cindex IEEE floating point 
+
+This chapter describes functions for examining the representation of
+floating point numbers and controlling the floating point environment of
+your program.  The functions described in this chapter are declared in
+the header file @file{gsl_ieee_utils.h}.
+
+@menu
+* Representation of floating point numbers::  
+* Setting up your IEEE environment::  
+* IEEE References and Further Reading::  
+@end menu
+
+@node Representation of floating point numbers
+@section Representation of floating point numbers
+@cindex IEEE format for floating point numbers
+@cindex bias, IEEE format
+@cindex exponent, IEEE format
+@cindex sign bit, IEEE format
+@cindex mantissa, IEEE format
+The IEEE Standard for Binary Floating-Point Arithmetic defines binary
+formats for single and double precision numbers.  Each number is composed
+of three parts: a @dfn{sign bit} (@math{s}), an @dfn{exponent}
+(@math{E}) and a @dfn{fraction} (@math{f}).  The numerical value of the
+combination @math{(s,E,f)} is given by the following formula,
+@tex
+\beforedisplay
+$$
+(-1)^s (1 \cdot fffff\dots) 2^E
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+(-1)^s (1.fffff...) 2^E
+@end example
+
+@end ifinfo
+@noindent
+@cindex normalized form, IEEE format
+@cindex denormalized form, IEEE format
+The sign bit is either zero or one.  The exponent ranges from a minimum value
+@c{$E_{min}$}
+@math{E_min} 
+to a maximum value
+@c{$E_{max}$}
+@math{E_max} depending on the precision.  The exponent is converted to an 
+unsigned number
+@math{e}, known as the @dfn{biased exponent}, for storage by adding a
+@dfn{bias} parameter,
+@c{$e = E + \hbox{\it bias}$}
+@math{e = E + bias}.
+The sequence @math{fffff...} represents the digits of the binary
+fraction @math{f}.  The binary digits are stored in @dfn{normalized
+form}, by adjusting the exponent to give a leading digit of @math{1}. 
+Since the leading digit is always 1 for normalized numbers it is
+assumed implicitly and does not have to be stored.
+Numbers smaller than 
+@c{$2^{E_{min}}$}
+@math{2^(E_min)}
+are be stored in @dfn{denormalized form} with a leading zero,
+@tex
+\beforedisplay
+$$
+(-1)^s (0 \cdot fffff\dots) 2^{E_{min}}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+(-1)^s (0.fffff...) 2^(E_min)
+@end example
+
+@end ifinfo
+@noindent
+@cindex zero, IEEE format
+@cindex infinity, IEEE format
+This allows gradual underflow down to 
+@c{$2^{E_{min} - p}$}
+@math{2^(E_min - p)} for @math{p} bits of precision. 
+A zero is encoded with the special exponent of 
+@c{$2^{E_{min}-1}$}
+@math{2^(E_min - 1)} and infinities with the exponent of 
+@c{$2^{E_{max}+1}$}
+@math{2^(E_max + 1)}.
+
+@noindent
+@cindex single precision, IEEE format
+The format for single precision numbers uses 32 bits divided in the
+following way,
+
+@smallexample
+seeeeeeeefffffffffffffffffffffff
+    
+s = sign bit, 1 bit
+e = exponent, 8 bits  (E_min=-126, E_max=127, bias=127)
+f = fraction, 23 bits
+@end smallexample
+
+@noindent
+@cindex double precision, IEEE format
+The format for double precision numbers uses 64 bits divided in the
+following way,
+
+@smallexample
+seeeeeeeeeeeffffffffffffffffffffffffffffffffffffffffffffffffffff
+
+s = sign bit, 1 bit
+e = exponent, 11 bits  (E_min=-1022, E_max=1023, bias=1023)
+f = fraction, 52 bits
+@end smallexample
+
+@noindent
+It is often useful to be able to investigate the behavior of a
+calculation at the bit-level and the library provides functions for
+printing the IEEE representations in a human-readable form.
+
+@comment float vs double vs long double 
+@comment (how many digits are available for each)
+
+@deftypefun void gsl_ieee_fprintf_float (FILE * @var{stream}, const float * @var{x})
+@deftypefunx void gsl_ieee_fprintf_double (FILE * @var{stream}, const double * @var{x})
+These functions output a formatted version of the IEEE floating-point
+number pointed to by @var{x} to the stream @var{stream}. A pointer is
+used to pass the number indirectly, to avoid any undesired promotion
+from @code{float} to @code{double}.  The output takes one of the
+following forms,
+
+@table @code
+@item NaN
+the Not-a-Number symbol
+
+@item Inf, -Inf
+positive or negative infinity
+
+@item 1.fffff...*2^E, -1.fffff...*2^E 
+a normalized floating point number
+
+@item 0.fffff...*2^E, -0.fffff...*2^E 
+a denormalized floating point number
+
+@item 0, -0
+positive or negative zero
+
+@comment @item [non-standard IEEE float], [non-standard IEEE double]
+@comment an unrecognized encoding
+@end table
+
+The output can be used directly in GNU Emacs Calc mode by preceding it
+with @code{2#} to indicate binary.
+@end deftypefun
+
+@deftypefun void gsl_ieee_printf_float (const float * @var{x})
+@deftypefunx void gsl_ieee_printf_double (const double * @var{x})
+These functions output a formatted version of the IEEE floating-point
+number pointed to by @var{x} to the stream @code{stdout}.
+@end deftypefun
+
+@noindent
+The following program demonstrates the use of the functions by printing
+the single and double precision representations of the fraction
+@math{1/3}.  For comparison the representation of the value promoted from
+single to double precision is also printed.
+
+@example
+@verbatiminclude examples/ieee.c
+@end example
+
+@noindent
+The binary representation of @math{1/3} is @math{0.01010101... }.  The
+output below shows that the IEEE format normalizes this fraction to give
+a leading digit of 1,
+
+@smallexample
+ f= 1.01010101010101010101011*2^-2
+fd= 1.0101010101010101010101100000000000000000000000000000*2^-2
+ d= 1.0101010101010101010101010101010101010101010101010101*2^-2
+@end smallexample
+
+@noindent
+The output also shows that a single-precision number is promoted to
+double-precision by adding zeros in the binary representation.
+
+@comment importance of using 1.234L in long double calculations
+
+@comment @example
+@comment int main (void)
+@comment @{
+@comment   long double x = 1.0, y = 1.0;
+  
+@comment   x = x + 0.2;
+@comment   y = y + 0.2L;
+
+@comment   printf(" d %.20Lf\n",x);
+@comment   printf("ld %.20Lf\n",y);
+
+@comment   return 1;
+@comment @}
+
+@comment  d 1.20000000000000001110
+@comment ld 1.20000000000000000004
+@comment @end example
+
+
+@node Setting up your IEEE environment
+@section Setting up your IEEE environment
+@cindex IEEE exceptions
+@cindex precision, IEEE arithmetic
+@cindex rounding mode
+@cindex arithmetic exceptions
+@cindex exceptions, IEEE arithmetic
+@cindex division by zero, IEEE exceptions
+@cindex underflow, IEEE exceptions
+@cindex overflow, IEEE exceptions
+The IEEE standard defines several @dfn{modes} for controlling the
+behavior of floating point operations.  These modes specify the important
+properties of computer arithmetic: the direction used for rounding (e.g.
+whether numbers should be rounded up, down or to the nearest number),
+the rounding precision and how the program should handle arithmetic
+exceptions, such as division by zero.
+
+Many of these features can now be controlled via standard functions such
+as @code{fpsetround}, which should be used whenever they are available.
+Unfortunately in the past there has been no universal API for
+controlling their behavior---each system has had its own low-level way
+of accessing them.  To help you write portable programs GSL allows you
+to specify modes in a platform-independent way using the environment
+variable @code{GSL_IEEE_MODE}.  The library then takes care of all the
+necessary machine-specific initializations for you when you call the
+function @code{gsl_ieee_env_setup}.
+
+@deftypefun void gsl_ieee_env_setup ()
+This function reads the environment variable @code{GSL_IEEE_MODE} and
+attempts to set up the corresponding specified IEEE modes.  The
+environment variable should be a list of keywords, separated by
+commas, like this,
+
+@display
+@code{GSL_IEEE_MODE} = "@var{keyword},@var{keyword},..."
+@end display
+
+@noindent
+where @var{keyword} is one of the following mode-names,
+
+@itemize @asis
+@item 
+@code{single-precision}
+@item 
+@code{double-precision}
+@item 
+@code{extended-precision}
+@item 
+@code{round-to-nearest}
+@item 
+@code{round-down}
+@item 
+@code{round-up}
+@item 
+@code{round-to-zero}
+@item 
+@code{mask-all}
+@item 
+@code{mask-invalid}
+@item 
+@code{mask-denormalized}
+@item 
+@code{mask-division-by-zero}
+@item 
+@code{mask-overflow}
+@item 
+@code{mask-underflow}
+@item 
+@code{trap-inexact}
+@item 
+@code{trap-common}
+@end itemize
+
+If @code{GSL_IEEE_MODE} is empty or undefined then the function returns
+immediately and no attempt is made to change the system's IEEE
+mode.  When the modes from @code{GSL_IEEE_MODE} are turned on the
+function prints a short message showing the new settings to remind you
+that the results of the program will be affected.
+
+If the requested modes are not supported by the platform being used then
+the function calls the error handler and returns an error code of
+@code{GSL_EUNSUP}.
+
+When options are specified using this method, the resulting mode is
+based on a default setting of the highest available precision (double
+precision or extended precision, depending on the platform) in
+round-to-nearest mode, with all exceptions enabled apart from the
+@sc{inexact} exception.  The @sc{inexact} exception is generated
+whenever rounding occurs, so it must generally be disabled in typical
+scientific calculations.  All other floating-point exceptions are
+enabled by default, including underflows and the use of denormalized
+numbers, for safety.  They can be disabled with the individual
+@code{mask-} settings or together using @code{mask-all}.
+
+The following adjusted combination of modes is convenient for many
+purposes,
+
+@example
+GSL_IEEE_MODE="double-precision,"\
+                "mask-underflow,"\
+                  "mask-denormalized"
+@end example
+
+@noindent
+This choice ignores any errors relating to small numbers (either
+denormalized, or underflowing to zero) but traps overflows, division by
+zero and invalid operations.
+@end deftypefun
+
+@noindent
+To demonstrate the effects of different rounding modes consider the
+following program which computes @math{e}, the base of natural
+logarithms, by summing a rapidly-decreasing series,
+@tex
+\beforedisplay
+$$
+e = 1 + {1 \over 2!} + {1 \over 3!} + {1 \over 4!} + \dots 
+  = 2.71828182846...
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+e = 1 + 1/2! + 1/3! + 1/4! + ... 
+  = 2.71828182846...
+@end example
+@end ifinfo
+
+@example
+@verbatiminclude examples/ieeeround.c
+@end example
+
+@noindent
+Here are the results of running the program in @code{round-to-nearest}
+mode.  This is the IEEE default so it isn't really necessary to specify
+it here,
+
+@example
+$ GSL_IEEE_MODE="round-to-nearest" ./a.out 
+i= 1 sum=1.000000000000000000 error=-1.71828
+i= 2 sum=2.000000000000000000 error=-0.718282
+....
+i=18 sum=2.718281828459045535 error=4.44089e-16
+i=19 sum=2.718281828459045535 error=4.44089e-16
+@end example
+
+@noindent
+After nineteen terms the sum converges to within @c{$4 \times 10^{-16}$}
+@math{4 \times 10^-16} of the correct value.  
+If we now change the rounding mode to
+@code{round-down} the final result is less accurate,
+
+@example
+$ GSL_IEEE_MODE="round-down" ./a.out 
+i= 1 sum=1.000000000000000000 error=-1.71828
+....
+i=19 sum=2.718281828459041094 error=-3.9968e-15
+@end example
+
+@noindent
+The result is about 
+@c{$4 \times 10^{-15}$}
+@math{4 \times 10^-15} 
+below the correct value, an order of magnitude worse than the result
+obtained in the @code{round-to-nearest} mode.
+
+If we change to rounding mode to @code{round-up} then the series no
+longer converges (the reason is that when we add each term to the sum
+the final result is always rounded up.  This is guaranteed to increase the sum
+by at least one tick on each iteration).  To avoid this problem we would
+need to use a safer converge criterion, such as @code{while (fabs(sum -
+oldsum) > epsilon)}, with a suitably chosen value of epsilon.
+
+Finally we can see the effect of computing the sum using
+single-precision rounding, in the default @code{round-to-nearest}
+mode.  In this case the program thinks it is still using double precision
+numbers but the CPU rounds the result of each floating point operation
+to single-precision accuracy.  This simulates the effect of writing the
+program using single-precision @code{float} variables instead of
+@code{double} variables.  The iteration stops after about half the number
+of iterations and the final result is much less accurate,
+
+@example
+$ GSL_IEEE_MODE="single-precision" ./a.out 
+....
+i=12 sum=2.718281984329223633 error=1.5587e-07
+@end example
+
+@noindent
+with an error of 
+@c{$O(10^{-7})$}
+@math{O(10^-7)}, which corresponds to single
+precision accuracy (about 1 part in @math{10^7}).  Continuing the
+iterations further does not decrease the error because all the
+subsequent results are rounded to the same value.
+
+@node IEEE References and Further Reading
+@section References and Further Reading
+
+The reference for the IEEE standard is,
+
+@itemize @asis
+@item
+ANSI/IEEE Std 754-1985, IEEE Standard for Binary Floating-Point Arithmetic.
+@end itemize
+
+@noindent
+A more pedagogical introduction to the standard can be found in the
+following paper,
+
+@itemize @asis
+@item
+David Goldberg: What Every Computer Scientist Should Know About
+Floating-Point Arithmetic. @cite{ACM Computing Surveys}, Vol.@: 23, No.@: 1
+(March 1991), pages 5--48.
+
+Corrigendum: @cite{ACM Computing Surveys}, Vol.@: 23, No.@: 3 (September
+1991), page 413. and see also the sections by B. A. Wichmann and Charles
+B. Dunham in Surveyor's Forum: ``What Every Computer Scientist Should
+Know About Floating-Point Arithmetic''. @cite{ACM Computing Surveys},
+Vol.@: 24, No.@: 3 (September 1992), page 319.
+@end itemize
+
+@noindent
+
+A detailed textbook on IEEE arithmetic and its practical use is
+available from SIAM Press,
+
+@itemize @asis
+@item
+Michael L. Overton, @cite{Numerical Computing with IEEE Floating Point Arithmetic},
+SIAM Press, ISBN 0898715717.
+@end itemize
+
+@noindent
+
+@comment to turn on math exception handling use __setfpucw, see
+@comment /usr/include/i386/fpu_control.h
+@comment e.g.
+@comment #include <math.h>
+@comment #include <stdio.h>
+@comment #include <fpu_control.h>
+@comment double f (double x);
+@comment int main ()
+@comment {
+@comment   double a = 0;
+@comment   double y, z;
+@comment   __setfpucw(0x1372); 
+@comment mention extended vs double precision on Pentium, and how to get around
+@comment it (-ffloat-store, or selective use of volatile)
+
+@comment On the alpha the option -mieee is needed with gcc
+@comment In Digital's compiler the equivalent is -ieee, -ieee-with-no-inexact 
+@comment and -ieee-with-inexact,  or -fpe1 or -fpe2
diff --git a/gsl-1.9/doc/initial-route.eps b/gsl-1.9/doc/initial-route.eps
new file mode 100644
index 0000000..0f7e864
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diff --git a/gsl-1.9/doc/integration.texi b/gsl-1.9/doc/integration.texi
new file mode 100644
index 0000000..62afb3e
--- /dev/null
+++ b/gsl-1.9/doc/integration.texi
@@ -0,0 +1,854 @@
+@cindex quadrature
+@cindex numerical integration (quadrature)
+@cindex integration, numerical (quadrature)
+@cindex QUADPACK
+
+This chapter describes routines for performing numerical integration
+(quadrature) of a function in one dimension.  There are routines for
+adaptive and non-adaptive integration of general functions, with
+specialised routines for specific cases.  These include integration over
+infinite and semi-infinite ranges, singular integrals, including
+logarithmic singularities, computation of Cauchy principal values and
+oscillatory integrals.  The library reimplements the algorithms used in
+@sc{quadpack}, a numerical integration package written by Piessens,
+Doncker-Kapenga, Uberhuber and Kahaner.  Fortran code for @sc{quadpack} is
+available on Netlib.
+
+The functions described in this chapter are declared in the header file
+@file{gsl_integration.h}.
+
+@menu
+* Numerical Integration Introduction::  
+* QNG non-adaptive Gauss-Kronrod integration::  
+* QAG adaptive integration::    
+* QAGS adaptive integration with singularities::  
+* QAGP adaptive integration with known singular points::  
+* QAGI adaptive integration on infinite intervals::  
+* QAWC adaptive integration for Cauchy principal values::  
+* QAWS adaptive integration for singular functions::  
+* QAWO adaptive integration for oscillatory functions::  
+* QAWF adaptive integration for Fourier integrals::  
+* Numerical integration error codes::  
+* Numerical integration examples::  
+* Numerical integration References and Further Reading::  
+@end menu
+
+@node Numerical Integration Introduction
+@section Introduction
+
+Each algorithm computes an approximation to a definite integral of the
+form,
+@tex
+\beforedisplay
+$$
+I = \int_a^b f(x) w(x) \,dx
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+I = \int_a^b f(x) w(x) dx
+@end example
+
+@end ifinfo
+@noindent
+where @math{w(x)} is a weight function (for general integrands @math{w(x)=1}).
+The user provides absolute and relative error bounds 
+@c{$(\hbox{\it epsabs}, \hbox{\it epsrel}\,)$}
+@math{(epsabs, epsrel)} which specify the following accuracy requirement,
+@tex
+\beforedisplay
+$$
+|\hbox{\it RESULT} - I|  \leq \max(\hbox{\it epsabs}, \hbox{\it epsrel}\, |I|)
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+|RESULT - I|  <= max(epsabs, epsrel |I|)
+@end example
+
+@end ifinfo
+@noindent
+where 
+@c{$\hbox{\it RESULT}$}
+@math{RESULT} is the numerical approximation obtained by the
+algorithm.  The algorithms attempt to estimate the absolute error
+@c{$\hbox{\it ABSERR} = |\hbox{\it RESULT} - I|$}
+@math{ABSERR = |RESULT - I|} in such a way that the following inequality
+holds,
+@tex
+\beforedisplay
+$$
+|\hbox{\it RESULT} - I| \leq \hbox{\it ABSERR} \leq \max(\hbox{\it epsabs}, \hbox{\it epsrel}\,|I|)
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+|RESULT - I| <= ABSERR <= max(epsabs, epsrel |I|)
+@end example
+
+@end ifinfo
+@noindent
+The routines will fail to converge if the error bounds are too
+stringent, but always return the best approximation obtained up to that
+stage.
+
+The algorithms in @sc{quadpack} use a naming convention based on the
+following letters,
+
+@display 
+@code{Q} - quadrature routine
+
+@code{N} - non-adaptive integrator
+@code{A} - adaptive integrator
+
+@code{G} - general integrand (user-defined)
+@code{W} - weight function with integrand
+
+@code{S} - singularities can be more readily integrated
+@code{P} - points of special difficulty can be supplied
+@code{I} - infinite range of integration
+@code{O} - oscillatory weight function, cos or sin
+@code{F} - Fourier integral
+@code{C} - Cauchy principal value
+@end display
+
+@noindent
+The algorithms are built on pairs of quadrature rules, a higher order
+rule and a lower order rule.  The higher order rule is used to compute
+the best approximation to an integral over a small range.  The
+difference between the results of the higher order rule and the lower
+order rule gives an estimate of the error in the approximation.
+
+@menu
+* Integrands without weight functions::  
+* Integrands with weight functions::  
+* Integrands with singular weight functions::  
+@end menu
+
+@node Integrands without weight functions
+@subsection Integrands without weight functions
+@cindex Gauss-Kronrod quadrature
+The algorithms for general functions (without a weight function) are
+based on Gauss-Kronrod rules. 
+
+A Gauss-Kronrod rule begins with a classical Gaussian quadrature rule of
+order @math{m}.  This is extended with additional points between each of
+the abscissae to give a higher order Kronrod rule of order @math{2m+1}.
+The Kronrod rule is efficient because it reuses existing function
+evaluations from the Gaussian rule.  
+
+The higher order Kronrod rule is used as the best approximation to the
+integral, and the difference between the two rules is used as an
+estimate of the error in the approximation.
+
+@node Integrands with weight functions
+@subsection Integrands with weight functions
+@cindex Clenshaw-Curtis quadrature
+@cindex Modified Clenshaw-Curtis quadrature
+For integrands with weight functions the algorithms use Clenshaw-Curtis
+quadrature rules.  
+
+A Clenshaw-Curtis rule begins with an @math{n}-th order Chebyshev
+polynomial approximation to the integrand.  This polynomial can be
+integrated exactly to give an approximation to the integral of the
+original function.  The Chebyshev expansion can be extended to higher
+orders to improve the approximation and provide an estimate of the
+error.
+
+@node Integrands with singular weight functions
+@subsection Integrands with singular weight functions
+
+The presence of singularities (or other behavior) in the integrand can
+cause slow convergence in the Chebyshev approximation.  The modified
+Clenshaw-Curtis rules used in @sc{quadpack} separate out several common
+weight functions which cause slow convergence.  
+
+These weight functions are integrated analytically against the Chebyshev
+polynomials to precompute @dfn{modified Chebyshev moments}.  Combining
+the moments with the Chebyshev approximation to the function gives the
+desired integral.  The use of analytic integration for the singular part
+of the function allows exact cancellations and substantially improves
+the overall convergence behavior of the integration.
+
+
+@node QNG non-adaptive Gauss-Kronrod integration
+@section QNG non-adaptive Gauss-Kronrod integration
+@cindex QNG quadrature algorithm
+
+The QNG algorithm is a non-adaptive procedure which uses fixed
+Gauss-Kronrod abscissae to sample the integrand at a maximum of 87
+points.  It is provided for fast integration of smooth functions.
+
+@deftypefun int gsl_integration_qng (const gsl_function * @var{f}, double @var{a}, double @var{b}, double @var{epsabs}, double @var{epsrel}, double * @var{result}, double * @var{abserr}, size_t * @var{neval})
+
+This function applies the Gauss-Kronrod 10-point, 21-point, 43-point and
+87-point integration rules in succession until an estimate of the
+integral of @math{f} over @math{(a,b)} is achieved within the desired
+absolute and relative error limits, @var{epsabs} and @var{epsrel}.  The
+function returns the final approximation, @var{result}, an estimate of
+the absolute error, @var{abserr} and the number of function evaluations
+used, @var{neval}.  The Gauss-Kronrod rules are designed in such a way
+that each rule uses all the results of its predecessors, in order to
+minimize the total number of function evaluations.
+@end deftypefun
+
+
+@node QAG adaptive integration
+@section QAG adaptive integration
+@cindex QAG quadrature algorithm
+
+The QAG algorithm is a simple adaptive integration procedure.  The
+integration region is divided into subintervals, and on each iteration
+the subinterval with the largest estimated error is bisected.  This
+reduces the overall error rapidly, as the subintervals become
+concentrated around local difficulties in the integrand.  These
+subintervals are managed by a @code{gsl_integration_workspace} struct,
+which handles the memory for the subinterval ranges, results and error
+estimates.
+
+@deftypefun {gsl_integration_workspace *} gsl_integration_workspace_alloc (size_t @var{n}) 
+This function allocates a workspace sufficient to hold @var{n} double
+precision intervals, their integration results and error estimates.
+@end deftypefun
+
+@deftypefun void gsl_integration_workspace_free (gsl_integration_workspace * @var{w})
+This function frees the memory associated with the workspace @var{w}.
+@end deftypefun
+
+@deftypefun int gsl_integration_qag (const gsl_function * @var{f}, double @var{a}, double @var{b}, double @var{epsabs}, double @var{epsrel}, size_t @var{limit}, int @var{key}, gsl_integration_workspace * @var{workspace},  double * @var{result}, double * @var{abserr})
+
+This function applies an integration rule adaptively until an estimate
+of the integral of @math{f} over @math{(a,b)} is achieved within the
+desired absolute and relative error limits, @var{epsabs} and
+@var{epsrel}.  The function returns the final approximation,
+@var{result}, and an estimate of the absolute error, @var{abserr}.  The
+integration rule is determined by the value of @var{key}, which should
+be chosen from the following symbolic names,
+
+@example
+GSL_INTEG_GAUSS15  (key = 1)
+GSL_INTEG_GAUSS21  (key = 2)
+GSL_INTEG_GAUSS31  (key = 3)
+GSL_INTEG_GAUSS41  (key = 4)
+GSL_INTEG_GAUSS51  (key = 5)
+GSL_INTEG_GAUSS61  (key = 6)
+@end example
+
+@noindent
+corresponding to the 15, 21, 31, 41, 51 and 61 point Gauss-Kronrod
+rules.  The higher-order rules give better accuracy for smooth functions,
+while lower-order rules save time when the function contains local
+difficulties, such as discontinuities.
+
+On each iteration the adaptive integration strategy bisects the interval
+with the largest error estimate.  The subintervals and their results are
+stored in the memory provided by @var{workspace}.  The maximum number of
+subintervals is given by @var{limit}, which may not exceed the allocated
+size of the workspace.
+@end deftypefun
+
+
+@node QAGS adaptive integration with singularities
+@section QAGS adaptive integration with singularities
+@cindex QAGS quadrature algorithm
+
+The presence of an integrable singularity in the integration region
+causes an adaptive routine to concentrate new subintervals around the
+singularity.  As the subintervals decrease in size the successive
+approximations to the integral converge in a limiting fashion.  This
+approach to the limit can be accelerated using an extrapolation
+procedure.  The QAGS algorithm combines adaptive bisection with the Wynn
+epsilon-algorithm to speed up the integration of many types of
+integrable singularities.
+
+@deftypefun int gsl_integration_qags (const gsl_function * @var{f}, double @var{a}, double @var{b}, double @var{epsabs}, double @var{epsrel}, size_t @var{limit}, gsl_integration_workspace * @var{workspace}, double * @var{result}, double * @var{abserr})
+
+This function applies the Gauss-Kronrod 21-point integration rule
+adaptively until an estimate of the integral of @math{f} over
+@math{(a,b)} is achieved within the desired absolute and relative error
+limits, @var{epsabs} and @var{epsrel}.  The results are extrapolated
+using the epsilon-algorithm, which accelerates the convergence of the
+integral in the presence of discontinuities and integrable
+singularities.  The function returns the final approximation from the
+extrapolation, @var{result}, and an estimate of the absolute error,
+@var{abserr}.  The subintervals and their results are stored in the
+memory provided by @var{workspace}.  The maximum number of subintervals
+is given by @var{limit}, which may not exceed the allocated size of the
+workspace.
+
+@end deftypefun
+
+@node QAGP adaptive integration with known singular points
+@section QAGP adaptive integration with known singular points
+@cindex QAGP quadrature algorithm
+@cindex singular points, specifying positions in quadrature
+@deftypefun int gsl_integration_qagp (const gsl_function * @var{f}, double * @var{pts}, size_t @var{npts}, double @var{epsabs}, double @var{epsrel}, size_t @var{limit}, gsl_integration_workspace * @var{workspace}, double * @var{result}, double * @var{abserr})
+
+This function applies the adaptive integration algorithm QAGS taking
+account of the user-supplied locations of singular points.  The array
+@var{pts} of length @var{npts} should contain the endpoints of the
+integration ranges defined by the integration region and locations of
+the singularities.  For example, to integrate over the region
+@math{(a,b)} with break-points at @math{x_1, x_2, x_3} (where 
+@math{a < x_1 < x_2 < x_3 < b}) the following @var{pts} array should be used
+
+@example
+pts[0] = a
+pts[1] = x_1
+pts[2] = x_2
+pts[3] = x_3
+pts[4] = b
+@end example
+
+@noindent
+with @var{npts} = 5.
+
+@noindent
+If you know the locations of the singular points in the integration
+region then this routine will be faster than @code{QAGS}.
+
+@end deftypefun
+
+@node QAGI adaptive integration on infinite intervals
+@section QAGI adaptive integration on infinite intervals
+@cindex QAGI quadrature algorithm
+
+@deftypefun int gsl_integration_qagi (gsl_function * @var{f}, double @var{epsabs}, double @var{epsrel}, size_t @var{limit}, gsl_integration_workspace * @var{workspace}, double * @var{result}, double * @var{abserr})
+
+This function computes the integral of the function @var{f} over the
+infinite interval @math{(-\infty,+\infty)}.  The integral is mapped onto the
+semi-open interval @math{(0,1]} using the transformation @math{x = (1-t)/t},
+@tex
+\beforedisplay
+$$
+\int_{-\infty}^{+\infty} dx \, f(x) 
+  = \int_0^1 dt \, (f((1-t)/t) + f(-(1-t)/t))/t^2.
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+\int_@{-\infty@}^@{+\infty@} dx f(x) = 
+     \int_0^1 dt (f((1-t)/t) + f((-1+t)/t))/t^2.
+@end example
+
+@end ifinfo
+@noindent
+It is then integrated using the QAGS algorithm.  The normal 21-point
+Gauss-Kronrod rule of QAGS is replaced by a 15-point rule, because the
+transformation can generate an integrable singularity at the origin.  In
+this case a lower-order rule is more efficient.
+@end deftypefun
+
+@deftypefun int gsl_integration_qagiu (gsl_function * @var{f}, double @var{a}, double @var{epsabs}, double @var{epsrel}, size_t @var{limit}, gsl_integration_workspace * @var{workspace}, double * @var{result}, double * @var{abserr})
+
+This function computes the integral of the function @var{f} over the
+semi-infinite interval @math{(a,+\infty)}.  The integral is mapped onto the
+semi-open interval @math{(0,1]} using the transformation @math{x = a + (1-t)/t},
+@tex
+\beforedisplay
+$$
+\int_{a}^{+\infty} dx \, f(x) 
+  = \int_0^1 dt \, f(a + (1-t)/t)/t^2
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+\int_@{a@}^@{+\infty@} dx f(x) = 
+     \int_0^1 dt f(a + (1-t)/t)/t^2
+@end example
+
+@end ifinfo
+@noindent
+and then integrated using the QAGS algorithm.
+@end deftypefun
+
+@deftypefun int gsl_integration_qagil (gsl_function * @var{f}, double @var{b}, double @var{epsabs}, double @var{epsrel}, size_t @var{limit}, gsl_integration_workspace * @var{workspace}, double * @var{result}, double * @var{abserr})
+This function computes the integral of the function @var{f} over the
+semi-infinite interval @math{(-\infty,b)}.  The integral is mapped onto the
+semi-open interval @math{(0,1]} using the transformation @math{x = b - (1-t)/t},
+@tex
+\beforedisplay
+$$
+\int_{-\infty}^{b} dx \, f(x) 
+  = \int_0^1 dt \, f(b - (1-t)/t)/t^2
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+\int_@{+\infty@}^@{b@} dx f(x) = 
+     \int_0^1 dt f(b - (1-t)/t)/t^2
+@end example
+
+@end ifinfo
+@noindent
+and then integrated using the QAGS algorithm.
+@end deftypefun
+
+@node  QAWC adaptive integration for Cauchy principal values
+@section QAWC adaptive integration for Cauchy principal values
+@cindex QAWC quadrature algorithm
+@cindex Cauchy principal value, by numerical quadrature
+@deftypefun int gsl_integration_qawc (gsl_function * @var{f}, double @var{a}, double @var{b}, double @var{c}, double @var{epsabs}, double @var{epsrel}, size_t @var{limit}, gsl_integration_workspace * @var{workspace}, double * @var{result}, double * @var{abserr})
+
+This function computes the Cauchy principal value of the integral of
+@math{f} over @math{(a,b)}, with a singularity at @var{c},
+@tex
+\beforedisplay
+$$
+I = \int_a^b dx\, {f(x) \over x - c}
+  = \lim_{\epsilon \to 0} 
+\left\{
+\int_a^{c-\epsilon} dx\, {f(x) \over x - c}
++
+\int_{c+\epsilon}^b dx\, {f(x) \over x - c}
+\right\}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+I = \int_a^b dx f(x) / (x - c)
+@end example
+
+@end ifinfo
+@noindent
+The adaptive bisection algorithm of QAG is used, with modifications to
+ensure that subdivisions do not occur at the singular point @math{x = c}.
+When a subinterval contains the point @math{x = c} or is close to
+it then a special 25-point modified Clenshaw-Curtis rule is used to control
+the singularity.  Further away from the
+singularity the algorithm uses an ordinary 15-point Gauss-Kronrod
+integration rule.
+
+@end deftypefun
+
+@node  QAWS adaptive integration for singular functions
+@section QAWS adaptive integration for singular functions
+@cindex QAWS quadrature algorithm
+@cindex singular functions, numerical integration of
+The QAWS algorithm is designed for integrands with algebraic-logarithmic
+singularities at the end-points of an integration region.  In order to
+work efficiently the algorithm requires a precomputed table of
+Chebyshev moments.
+
+@deftypefun {gsl_integration_qaws_table *} gsl_integration_qaws_table_alloc (double @var{alpha}, double @var{beta}, int @var{mu}, int @var{nu})
+
+This function allocates space for a @code{gsl_integration_qaws_table}
+struct describing a singular weight function
+@math{W(x)} with the parameters @math{(\alpha, \beta, \mu, \nu)},
+@tex
+\beforedisplay
+$$
+W(x) = (x - a)^\alpha (b - x)^\beta \log^\mu (x - a) \log^\nu (b - x)
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+W(x) = (x-a)^alpha (b-x)^beta log^mu (x-a) log^nu (b-x)
+@end example
+
+@end ifinfo
+@noindent
+where @math{\alpha > -1}, @math{\beta > -1}, and @math{\mu = 0, 1},
+@math{\nu = 0, 1}.  The weight function can take four different forms
+depending on the values of @math{\mu} and @math{\nu},
+@tex
+\beforedisplay
+$$
+\matrix{
+W(x) = (x - a)^\alpha (b - x)^\beta  
+                                                \hfill~ (\mu = 0, \nu = 0) \cr
+W(x) = (x - a)^\alpha (b - x)^\beta \log(x - a) 
+                                                \hfill~ (\mu = 1, \nu = 0) \cr
+W(x) = (x - a)^\alpha (b - x)^\beta \log(b - x) 
+                                                \hfill~ (\mu = 0, \nu = 1) \cr
+W(x) = (x - a)^\alpha (b - x)^\beta \log(x - a) \log(b - x) 
+                                                \hfill~ (\mu = 1, \nu = 1)
+}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+W(x) = (x-a)^alpha (b-x)^beta                   (mu = 0, nu = 0)
+W(x) = (x-a)^alpha (b-x)^beta log(x-a)          (mu = 1, nu = 0)
+W(x) = (x-a)^alpha (b-x)^beta log(b-x)          (mu = 0, nu = 1)
+W(x) = (x-a)^alpha (b-x)^beta log(x-a) log(b-x) (mu = 1, nu = 1)
+@end example
+
+@end ifinfo
+@noindent
+The singular points @math{(a,b)} do not have to be specified until the
+integral is computed, where they are the endpoints of the integration
+range.
+
+The function returns a pointer to the newly allocated table
+@code{gsl_integration_qaws_table} if no errors were detected, and 0 in
+the case of error.
+@end deftypefun
+
+@deftypefun int gsl_integration_qaws_table_set (gsl_integration_qaws_table * @var{t}, double @var{alpha}, double @var{beta}, int @var{mu}, int @var{nu})
+This function modifies the parameters @math{(\alpha, \beta, \mu, \nu)} of
+an existing @code{gsl_integration_qaws_table} struct @var{t}.
+@end deftypefun
+
+@deftypefun void gsl_integration_qaws_table_free (gsl_integration_qaws_table * @var{t})
+This function frees all the memory associated with the
+@code{gsl_integration_qaws_table} struct @var{t}.
+@end deftypefun
+
+@deftypefun int gsl_integration_qaws (gsl_function * @var{f}, const double @var{a}, const double @var{b}, gsl_integration_qaws_table * @var{t}, const double @var{epsabs}, const double @var{epsrel}, const size_t @var{limit}, gsl_integration_workspace * @var{workspace}, double * @var{result}, double * @var{abserr})
+
+This function computes the integral of the function @math{f(x)} over the
+interval @math{(a,b)} with the singular weight function
+@math{(x-a)^\alpha (b-x)^\beta \log^\mu (x-a) \log^\nu (b-x)}.  The parameters 
+of the weight function @math{(\alpha, \beta, \mu, \nu)} are taken from the
+table @var{t}.  The integral is,
+@tex
+\beforedisplay
+$$
+I = \int_a^b dx\, f(x) (x - a)^\alpha (b - x)^\beta 
+        \log^\mu (x - a) \log^\nu (b - x).
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+I = \int_a^b dx f(x) (x-a)^alpha (b-x)^beta log^mu (x-a) log^nu (b-x).
+@end example
+
+@end ifinfo
+@noindent
+The adaptive bisection algorithm of QAG is used.  When a subinterval
+contains one of the endpoints then a special 25-point modified
+Clenshaw-Curtis rule is used to control the singularities.  For
+subintervals which do not include the endpoints an ordinary 15-point
+Gauss-Kronrod integration rule is used.
+
+@end deftypefun
+
+@node  QAWO adaptive integration for oscillatory functions
+@section QAWO adaptive integration for oscillatory functions
+@cindex QAWO quadrature algorithm
+@cindex oscillatory functions, numerical integration of
+The QAWO algorithm is designed for integrands with an oscillatory
+factor, @math{\sin(\omega x)} or @math{\cos(\omega x)}.  In order to
+work efficiently the algorithm requires a table of Chebyshev moments
+which must be pre-computed with calls to the functions below.
+
+@deftypefun {gsl_integration_qawo_table *} gsl_integration_qawo_table_alloc (double @var{omega}, double @var{L}, enum gsl_integration_qawo_enum @var{sine}, size_t @var{n})
+
+This function allocates space for a @code{gsl_integration_qawo_table}
+struct and its associated workspace describing a sine or cosine weight
+function @math{W(x)} with the parameters @math{(\omega, L)},
+@tex
+\beforedisplay
+$$
+\eqalign{
+W(x) & = \left\{\matrix{\sin(\omega x) \cr \cos(\omega x)} \right\}
+}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+W(x) = sin(omega x)
+W(x) = cos(omega x)
+@end example
+
+@end ifinfo
+@noindent
+The parameter @var{L} must be the length of the interval over which the
+function will be integrated @math{L = b - a}.  The choice of sine or
+cosine is made with the parameter @var{sine} which should be chosen from
+one of the two following symbolic values:
+
+@example
+GSL_INTEG_COSINE
+GSL_INTEG_SINE
+@end example
+
+@noindent
+The @code{gsl_integration_qawo_table} is a table of the trigonometric
+coefficients required in the integration process.  The parameter @var{n}
+determines the number of levels of coefficients that are computed.  Each
+level corresponds to one bisection of the interval @math{L}, so that
+@var{n} levels are sufficient for subintervals down to the length
+@math{L/2^n}.  The integration routine @code{gsl_integration_qawo}
+returns the error @code{GSL_ETABLE} if the number of levels is
+insufficient for the requested accuracy.
+
+@end deftypefun
+
+@deftypefun int gsl_integration_qawo_table_set (gsl_integration_qawo_table * @var{t}, double @var{omega}, double @var{L}, enum gsl_integration_qawo_enum @var{sine})
+This function changes the parameters @var{omega}, @var{L} and @var{sine}
+of the existing workspace @var{t}.
+@end deftypefun
+
+@deftypefun int gsl_integration_qawo_table_set_length (gsl_integration_qawo_table * @var{t}, double @var{L})
+This function allows the length parameter @var{L} of the workspace
+@var{t} to be changed.
+@end deftypefun
+
+@deftypefun void gsl_integration_qawo_table_free (gsl_integration_qawo_table * @var{t})
+This function frees all the memory associated with the workspace @var{t}.
+@end deftypefun
+
+@deftypefun int gsl_integration_qawo (gsl_function * @var{f}, const double @var{a}, const double @var{epsabs}, const double @var{epsrel}, const size_t @var{limit}, gsl_integration_workspace * @var{workspace}, gsl_integration_qawo_table * @var{wf}, double * @var{result}, double * @var{abserr})
+
+This function uses an adaptive algorithm to compute the integral of
+@math{f} over @math{(a,b)} with the weight function 
+@math{\sin(\omega x)} or @math{\cos(\omega x)} defined 
+by the table @var{wf},
+@tex
+\beforedisplay
+$$
+\eqalign{
+I & = \int_a^b dx\, f(x) \left\{ \matrix{\sin(\omega x) \cr \cos(\omega x)}\right\}
+}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+I = \int_a^b dx f(x) sin(omega x)
+I = \int_a^b dx f(x) cos(omega x)
+@end example
+
+@end ifinfo
+@noindent
+The results are extrapolated using the epsilon-algorithm to accelerate
+the convergence of the integral.  The function returns the final
+approximation from the extrapolation, @var{result}, and an estimate of
+the absolute error, @var{abserr}.  The subintervals and their results are
+stored in the memory provided by @var{workspace}.  The maximum number of
+subintervals is given by @var{limit}, which may not exceed the allocated
+size of the workspace.
+
+Those subintervals with ``large'' widths @math{d} where @math{d\omega > 4} are
+computed using a 25-point Clenshaw-Curtis integration rule, which handles the
+oscillatory behavior.  Subintervals with a ``small'' widths where
+@math{d\omega < 4} are computed using a 15-point Gauss-Kronrod integration.
+
+@end deftypefun
+
+@node  QAWF adaptive integration for Fourier integrals
+@section QAWF adaptive integration for Fourier integrals
+@cindex QAWF quadrature algorithm
+@cindex Fourier integrals, numerical
+
+@deftypefun int gsl_integration_qawf (gsl_function * @var{f}, const double @var{a}, const double @var{epsabs}, const size_t @var{limit}, gsl_integration_workspace * @var{workspace}, gsl_integration_workspace * @var{cycle_workspace}, gsl_integration_qawo_table * @var{wf}, double * @var{result}, double * @var{abserr})
+
+This function attempts to compute a Fourier integral of the function
+@var{f} over the semi-infinite interval @math{[a,+\infty)}.
+@tex
+\beforedisplay
+$$
+\eqalign{
+I & = \int_a^{+\infty} dx\, f(x) \left\{ \matrix{ \sin(\omega x) \cr
+                                                 \cos(\omega x) } \right\}
+}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+I = \int_a^@{+\infty@} dx f(x) sin(omega x)
+I = \int_a^@{+\infty@} dx f(x) cos(omega x)
+@end example
+@end ifinfo
+
+The parameter @math{\omega} and choice of @math{\sin} or @math{\cos} is
+taken from the table @var{wf} (the length @var{L} can take any value,
+since it is overridden by this function to a value appropriate for the
+fourier integration).  The integral is computed using the QAWO algorithm
+over each of the subintervals,
+@tex
+\beforedisplay
+$$
+\eqalign{
+C_1 & = [a, a + c] \cr
+C_2 & = [a + c, a + 2c] \cr
+\dots & = \dots \cr
+C_k & = [a + (k-1) c, a + k c]
+}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+C_1 = [a, a + c]
+C_2 = [a + c, a + 2 c]
+... = ...
+C_k = [a + (k-1) c, a + k c]
+@end example
+
+@end ifinfo
+@noindent
+where 
+@c{$c = (2 \,\hbox{floor}(|\omega|) + 1) \pi/|\omega|$}
+@math{c = (2 floor(|\omega|) + 1) \pi/|\omega|}.  The width @math{c} is
+chosen to cover an odd number of periods so that the contributions from
+the intervals alternate in sign and are monotonically decreasing when
+@var{f} is positive and monotonically decreasing.  The sum of this
+sequence of contributions is accelerated using the epsilon-algorithm.
+
+This function works to an overall absolute tolerance of
+@var{abserr}.  The following strategy is used: on each interval
+@math{C_k} the algorithm tries to achieve the tolerance
+@tex
+\beforedisplay
+$$
+TOL_k = u_k \hbox{\it abserr}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+TOL_k = u_k abserr
+@end example
+
+@end ifinfo
+@noindent
+where 
+@c{$u_k = (1 - p)p^{k-1}$}
+@math{u_k = (1 - p)p^@{k-1@}} and @math{p = 9/10}.  
+The sum of the geometric series of contributions from each interval
+gives an overall tolerance of @var{abserr}.
+
+If the integration of a subinterval leads to difficulties then the
+accuracy requirement for subsequent intervals is relaxed,
+@tex
+\beforedisplay
+$$
+TOL_k = u_k \max(\hbox{\it abserr}, \max_{i<k}\{E_i\})
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+TOL_k = u_k max(abserr, max_@{i<k@}@{E_i@})
+@end example
+
+@end ifinfo
+@noindent
+where @math{E_k} is the estimated error on the interval @math{C_k}.
+
+The subintervals and their results are stored in the memory provided by
+@var{workspace}.  The maximum number of subintervals is given by
+@var{limit}, which may not exceed the allocated size of the workspace.
+The integration over each subinterval uses the memory provided by
+@var{cycle_workspace} as workspace for the QAWO algorithm.
+
+@end deftypefun
+
+@node Numerical integration error codes
+@section Error codes
+
+In addition to the standard error codes for invalid arguments the
+functions can return the following values,
+
+@table @code
+@item GSL_EMAXITER
+the maximum number of subdivisions was exceeded.
+@item GSL_EROUND
+cannot reach tolerance because of roundoff error,
+or roundoff error was detected in the extrapolation table.
+@item GSL_ESING  
+a non-integrable singularity or other bad integrand behavior was found
+in the integration interval.
+@item GSL_EDIVERGE
+the integral is divergent, or too slowly convergent to be integrated
+numerically.
+@end table
+
+@node Numerical integration examples
+@section Examples
+
+The integrator @code{QAGS} will handle a large class of definite
+integrals.  For example, consider the following integral, which has a
+algebraic-logarithmic singularity at the origin,
+@tex
+\beforedisplay
+$$
+\int_0^1 x^{-1/2} \log(x) \,dx = -4
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+\int_0^1 x^@{-1/2@} log(x) dx = -4
+@end example
+
+@end ifinfo
+@noindent
+The program below computes this integral to a relative accuracy bound of
+@code{1e-7}.
+
+@example
+@verbatiminclude examples/integration.c
+@end example
+
+@noindent
+The results below show that the desired accuracy is achieved after 8
+subdivisions. 
+
+@example
+$ ./a.out 
+@verbatiminclude examples/integration.out
+@end example
+
+@noindent
+In fact, the extrapolation procedure used by @code{QAGS} produces an
+accuracy of almost twice as many digits.  The error estimate returned by
+the extrapolation procedure is larger than the actual error, giving a
+margin of safety of one order of magnitude.
+
+
+@node Numerical integration References and Further Reading
+@section References and Further Reading
+
+The following book is the definitive reference for @sc{quadpack}, and was
+written by the original authors.  It provides descriptions of the
+algorithms, program listings, test programs and examples.  It also
+includes useful advice on numerical integration and many references to
+the numerical integration literature used in developing @sc{quadpack}.
+
+@itemize @asis
+@item
+R. Piessens, E. de Doncker-Kapenga, C.W. Uberhuber, D.K. Kahaner.
+@cite{@sc{quadpack} A subroutine package for automatic integration}
+Springer Verlag, 1983.
+@end itemize
+
+@noindent
+
+
+
+
+
+
diff --git a/gsl-1.9/doc/interp.texi b/gsl-1.9/doc/interp.texi
new file mode 100644
index 0000000..e1e6b1c
--- /dev/null
+++ b/gsl-1.9/doc/interp.texi
@@ -0,0 +1,318 @@
+@cindex interpolation
+@cindex spline
+
+This chapter describes functions for performing interpolation.  The
+library provides a variety of interpolation methods, including Cubic
+splines and Akima splines.  The interpolation types are interchangeable,
+allowing different methods to be used without recompiling.
+Interpolations can be defined for both normal and periodic boundary
+conditions.  Additional functions are available for computing
+derivatives and integrals of interpolating functions.
+
+The functions described in this section are declared in the header files
+@file{gsl_interp.h} and @file{gsl_spline.h}.
+
+@menu
+* Introduction to Interpolation::  
+* Interpolation Functions::     
+* Interpolation Types::         
+* Index Look-up and Acceleration::  
+* Evaluation of Interpolating Functions::  
+* Higher-level Interface::      
+* Interpolation Example programs::  
+* Interpolation References and Further Reading::  
+@end menu
+
+@node Introduction to Interpolation
+@section Introduction
+
+Given a set of data points @math{(x_1, y_1) \dots (x_n, y_n)} the
+routines described in this section compute a continuous interpolating
+function @math{y(x)} such that @math{y(x_i) = y_i}.  The interpolation
+is piecewise smooth, and its behavior at the end-points is determined by
+the type of interpolation used.
+
+@node Interpolation Functions
+@section Interpolation Functions
+
+The interpolation function for a given dataset is stored in a
+@code{gsl_interp} object.  These are created by the following functions.
+
+@deftypefun {gsl_interp *} gsl_interp_alloc (const gsl_interp_type * @var{T}, size_t @var{size})
+This function returns a pointer to a newly allocated interpolation
+object of type @var{T} for @var{size} data-points.
+@end deftypefun
+
+@deftypefun int gsl_interp_init (gsl_interp * @var{interp}, const double @var{xa}[], const double @var{ya}[], size_t @var{size})
+This function initializes the interpolation object @var{interp} for the
+data (@var{xa},@var{ya}) where @var{xa} and @var{ya} are arrays of size
+@var{size}.  The interpolation object (@code{gsl_interp}) does not save
+the data arrays @var{xa} and @var{ya} and only stores the static state
+computed from the data.  The @var{xa} data array is always assumed to be
+strictly ordered; the behavior for other arrangements is not defined.
+@end deftypefun
+
+@deftypefun void gsl_interp_free (gsl_interp * @var{interp})
+This function frees the interpolation object @var{interp}.
+@end deftypefun
+
+@node Interpolation Types
+@section Interpolation Types
+
+The interpolation library provides five interpolation types:
+
+@deffn {Interpolation Type} gsl_interp_linear
+@cindex linear interpolation
+Linear interpolation.  This interpolation method does not require any
+additional memory.
+@end deffn
+
+@deffn {Interpolation Type} gsl_interp_polynomial
+@cindex polynomial interpolation
+Polynomial interpolation.  This method should only be used for
+interpolating small numbers of points because polynomial interpolation
+introduces large oscillations, even for well-behaved datasets.  The
+number of terms in the interpolating polynomial is equal to the number
+of points.
+@end deffn
+
+@deffn {Interpolation Type} gsl_interp_cspline
+@cindex cubic splines
+Cubic spline with natural boundary conditions.  The resulting curve is
+piecewise cubic on each interval, with matching first and second
+derivatives at the supplied data-points.  The second derivative is
+chosen to be zero at the first point and last point.
+@end deffn
+
+@deffn {Interpolation Type} gsl_interp_cspline_periodic
+Cubic spline with periodic boundary conditions.  The resulting curve
+is piecewise cubic on each interval, with matching first and second
+derivatives at the supplied data-points.  The derivatives at the first
+and last points are also matched.  Note that the last point in the
+data must have the same y-value as the first point, otherwise the
+resulting periodic interpolation will have a discontinuity at the
+boundary.
+
+@end deffn
+
+@deffn {Interpolation Type} gsl_interp_akima
+@cindex Akima splines
+Non-rounded Akima spline with natural boundary conditions.  This method
+uses the non-rounded corner algorithm of Wodicka.
+@end deffn
+
+@deffn {Interpolation Type} gsl_interp_akima_periodic
+Non-rounded Akima spline with periodic boundary conditions.  This method
+uses the non-rounded corner algorithm of Wodicka.
+@end deffn
+
+The following related functions are available:
+
+@deftypefun {const char *} gsl_interp_name (const gsl_interp * @var{interp})
+This function returns the name of the interpolation type used by @var{interp}.
+For example,
+
+@example
+printf ("interp uses '%s' interpolation.\n", 
+        gsl_interp_name (interp));
+@end example
+
+@noindent
+would print something like,
+
+@example
+interp uses 'cspline' interpolation.
+@end example
+@end deftypefun
+
+@deftypefun {unsigned int} gsl_interp_min_size (const gsl_interp * @var{interp})
+This function returns the minimum number of points required by the
+interpolation type of @var{interp}.  For example, Akima spline interpolation
+requires a minimum of 5 points.
+@end deftypefun
+
+@node Index Look-up and Acceleration
+@section Index Look-up and Acceleration
+
+The state of searches can be stored in a @code{gsl_interp_accel} object,
+which is a kind of iterator for interpolation lookups.  It caches the
+previous value of an index lookup.  When the subsequent interpolation
+point falls in the same interval its index value can be returned
+immediately.
+
+@deftypefun size_t gsl_interp_bsearch (const double @var{x_array}[], double @var{x}, size_t @var{index_lo}, size_t @var{index_hi})
+This function returns the index @math{i} of the array @var{x_array} such
+that @code{x_array[i] <= x < x_array[i+1]}.  The index is searched for
+in the range [@var{index_lo},@var{index_hi}].
+@end deftypefun
+
+@deftypefun {gsl_interp_accel *} gsl_interp_accel_alloc (void)
+This function returns a pointer to an accelerator object, which is a
+kind of iterator for interpolation lookups.  It tracks the state of
+lookups, thus allowing for application of various acceleration
+strategies.
+@end deftypefun
+
+@deftypefun size_t gsl_interp_accel_find (gsl_interp_accel * @var{a}, const double @var{x_array}[], size_t @var{size}, double @var{x})
+This function performs a lookup action on the data array @var{x_array}
+of size @var{size}, using the given accelerator @var{a}.  This is how
+lookups are performed during evaluation of an interpolation.  The
+function returns an index @math{i} such that @code{x_array[i] <= x <
+x_array[i+1]}.
+@end deftypefun
+
+@deftypefun void gsl_interp_accel_free (gsl_interp_accel* @var{acc})
+This function frees the accelerator object @var{acc}.
+@end deftypefun
+
+@node Evaluation of Interpolating Functions
+@section Evaluation of Interpolating Functions
+
+@deftypefun  double gsl_interp_eval (const gsl_interp * @var{interp}, const double @var{xa}[], const double @var{ya}[], double @var{x}, gsl_interp_accel * @var{acc})
+@deftypefunx int gsl_interp_eval_e (const gsl_interp * @var{interp}, const double @var{xa}[], const double @var{ya}[], double @var{x}, gsl_interp_accel * @var{acc}, double * @var{y})
+These functions return the interpolated value of @var{y} for a given
+point @var{x}, using the interpolation object @var{interp}, data arrays
+@var{xa} and @var{ya} and the accelerator @var{acc}.
+@end deftypefun
+
+@deftypefun double gsl_interp_eval_deriv (const gsl_interp * @var{interp}, const double @var{xa}[], const double @var{ya}[], double @var{x}, gsl_interp_accel * @var{acc})
+@deftypefunx int gsl_interp_eval_deriv_e (const gsl_interp * @var{interp}, const double @var{xa}[], const double @var{ya}[], double @var{x}, gsl_interp_accel * @var{acc}, double * @var{d})
+These functions return the derivative @var{d} of an interpolated
+function for a given point @var{x}, using the interpolation object
+@var{interp}, data arrays @var{xa} and @var{ya} and the accelerator
+@var{acc}. 
+@end deftypefun
+
+@deftypefun double gsl_interp_eval_deriv2 (const gsl_interp * @var{interp}, const double @var{xa}[], const double @var{ya}[], double @var{x}, gsl_interp_accel * @var{acc})
+@deftypefunx int gsl_interp_eval_deriv2_e (const gsl_interp * @var{interp}, const double @var{xa}[], const double @var{ya}[], double @var{x}, gsl_interp_accel * @var{acc}, double * @var{d2})
+These functions return the second derivative @var{d2} of an interpolated
+function for a given point @var{x}, using the interpolation object
+@var{interp}, data arrays @var{xa} and @var{ya} and the accelerator
+@var{acc}. 
+@end deftypefun
+
+@deftypefun double gsl_interp_eval_integ (const gsl_interp * @var{interp}, const double @var{xa}[], const double @var{ya}[], double @var{a}, double @var{b}, gsl_interp_accel * @var{acc})
+@deftypefunx int gsl_interp_eval_integ_e (const gsl_interp * @var{interp}, const double @var{xa}[], const double @var{ya}[], double @var{a}, double @var{b}, gsl_interp_accel * @var{acc}, double * @var{result})
+These functions return the numerical integral @var{result} of an
+interpolated function over the range [@var{a}, @var{b}], using the
+interpolation object @var{interp}, data arrays @var{xa} and @var{ya} and
+the accelerator @var{acc}.
+@end deftypefun
+
+@node Higher-level Interface
+@section Higher-level Interface
+
+The functions described in the previous sections required the user to
+supply pointers to the @math{x} and @math{y} arrays on each call.  The
+following functions are equivalent to the corresponding
+@code{gsl_interp} functions but maintain a copy of this data in the
+@code{gsl_spline} object.  This removes the need to pass both @var{xa}
+and @var{ya} as arguments on each evaluation. These functions are
+defined in the header file @file{gsl_spline.h}.
+
+@deftypefun {gsl_spline *} gsl_spline_alloc (const gsl_interp_type * @var{T}, size_t @var{size})
+@end deftypefun
+
+@deftypefun int gsl_spline_init (gsl_spline * @var{spline}, const double @var{xa}[], const double @var{ya}[], size_t @var{size})
+@end deftypefun
+
+@deftypefun void gsl_spline_free (gsl_spline * @var{spline})
+@end deftypefun
+
+@deftypefun {const char *} gsl_spline_name (const gsl_spline * @var{spline})
+@end deftypefun
+
+@deftypefun {unsigned int} gsl_spline_min_size (const gsl_spline * @var{spline})
+@end deftypefun
+
+@deftypefun double gsl_spline_eval (const gsl_spline * @var{spline}, double @var{x}, gsl_interp_accel * @var{acc})
+@deftypefunx int gsl_spline_eval_e (const gsl_spline * @var{spline}, double @var{x}, gsl_interp_accel * @var{acc}, double * @var{y})
+@end deftypefun
+
+@deftypefun double gsl_spline_eval_deriv (const gsl_spline * @var{spline}, double @var{x}, gsl_interp_accel * @var{acc})
+@deftypefunx int gsl_spline_eval_deriv_e (const gsl_spline * @var{spline}, double @var{x}, gsl_interp_accel * @var{acc}, double * @var{d})
+@end deftypefun
+
+@deftypefun double gsl_spline_eval_deriv2 (const gsl_spline * @var{spline}, double @var{x}, gsl_interp_accel * @var{acc})
+@deftypefunx int gsl_spline_eval_deriv2_e (const gsl_spline * @var{spline}, double @var{x}, gsl_interp_accel * @var{acc}, double * @var{d2})
+@end deftypefun
+
+@deftypefun double gsl_spline_eval_integ (const gsl_spline * @var{spline}, double @var{a}, double @var{b}, gsl_interp_accel * @var{acc})
+@deftypefunx int gsl_spline_eval_integ_e (const gsl_spline * @var{spline}, double @var{a}, double @var{b}, gsl_interp_accel * @var{acc}, double * @var{result})
+@end deftypefun
+
+@node Interpolation Example programs
+@section Examples
+
+The following program demonstrates the use of the interpolation and
+spline functions.  It computes a cubic spline interpolation of the
+10-point dataset @math{(x_i, y_i)} where @math{x_i = i + \sin(i)/2} and
+@math{y_i = i + \cos(i^2)} for @math{i = 0 \dots 9}.
+
+@example
+@verbatiminclude examples/interp.c
+@end example
+
+@noindent
+The output is designed to be used with the @sc{gnu} plotutils
+@code{graph} program,
+
+@example
+$ ./a.out > interp.dat
+$ graph -T ps < interp.dat > interp.ps
+@end example
+
+@iftex
+@sp 1
+@center @image{interp2,3.4in}
+@end iftex
+
+@noindent
+The result shows a smooth interpolation of the original points.  The
+interpolation method can changed simply by varying the first argument of
+@code{gsl_spline_alloc}.
+
+The next program demonstrates a periodic cubic spline with 4 data
+points.  Note that the first and last points must be supplied with 
+the same y-value for a periodic spline.
+
+@example
+@verbatiminclude examples/interpp.c
+@end example
+
+@noindent
+
+The output can be plotted with @sc{gnu} @code{graph}.
+
+@example
+$ ./a.out > interp.dat
+$ graph -T ps < interp.dat > interp.ps
+@end example
+
+@iftex
+@sp 1
+@center @image{interpp2,3.4in}
+@end iftex
+
+@noindent
+The result shows a periodic interpolation of the original points. The
+slope of the fitted curve is the same at the beginning and end of the
+data, and the second derivative is also.
+
+@node Interpolation References and Further Reading
+@section References and Further Reading
+
+Descriptions of the interpolation algorithms and further references can
+be found in the following books:
+
+@itemize @asis
+@item C.W. Ueberhuber,
+@cite{Numerical Computation (Volume 1), Chapter 9 ``Interpolation''},
+Springer (1997), ISBN 3-540-62058-3.
+
+@item D.M. Young, R.T. Gregory
+@cite{A Survey of Numerical Mathematics (Volume 1), Chapter 6.8},
+Dover (1988), ISBN 0-486-65691-8.
+@end itemize
+
+@noindent
diff --git a/gsl-1.9/doc/interp2.eps b/gsl-1.9/doc/interp2.eps
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diff --git a/gsl-1.9/doc/intro.texi b/gsl-1.9/doc/intro.texi
new file mode 100644
index 0000000..dca7be9
--- /dev/null
+++ b/gsl-1.9/doc/intro.texi
@@ -0,0 +1,247 @@
+@cindex license of GSL
+@cindex GNU General Public License
+The GNU Scientific Library (GSL) is a collection of routines for
+numerical computing.  The routines have been written from scratch in C,
+and present a modern Applications Programming Interface
+(API) for C programmers, allowing wrappers to be written for very
+high level languages.  The source code is distributed under the GNU
+General Public License.
+
+
+@menu
+* Routines available in GSL::   
+* GSL is Free Software::        
+* Obtaining GSL::               
+* No Warranty::                 
+* Reporting Bugs::              
+* Further Information::         
+* Conventions used in this manual::  
+@end menu
+
+@node Routines available in GSL
+@section Routines available in GSL
+
+The library covers a wide range of topics in numerical computing.
+Routines are available for the following areas,
+
+@iftex
+@sp 1
+@end iftex
+@multitable @columnfractions 0.05 0.45 0.45 0.05
+@item @tab Complex Numbers            @tab Roots of Polynomials        
+@item @tab Special Functions          @tab Vectors and Matrices        
+@item @tab Permutations               @tab Combinations              
+@item @tab Sorting                    @tab BLAS Support               
+@item @tab Linear Algebra             @tab CBLAS Library
+@item @tab Fast Fourier Transforms    @tab Eigensystems               
+@item @tab Random Numbers             @tab Quadrature                 
+@item @tab Random Distributions       @tab Quasi-Random Sequences     
+@item @tab Histograms                 @tab Statistics                 
+@item @tab Monte Carlo Integration    @tab N-Tuples                   
+@item @tab Differential Equations     @tab Simulated Annealing        
+@item @tab Numerical Differentiation  @tab Interpolation              
+@item @tab Series Acceleration        @tab Chebyshev Approximations   
+@item @tab Root-Finding               @tab Discrete Hankel Transforms 
+@item @tab Least-Squares Fitting      @tab Minimization               
+@item @tab IEEE Floating-Point        @tab Physical Constants         
+@item @tab Wavelets
+@end multitable
+@iftex
+@sp 1
+@end iftex
+
+@noindent
+The use of these routines is described in this manual.  Each chapter
+provides detailed definitions of the functions, followed by example
+programs and references to the articles on which the algorithms are
+based.
+
+Where possible the routines have been based on reliable public-domain
+packages such as FFTPACK and QUADPACK, which the developers of GSL
+have reimplemented in C with modern coding conventions.
+
+@node GSL is Free Software
+@section GSL is Free Software
+@cindex free software, explanation of
+The subroutines in the GNU Scientific Library are ``free software'';
+this means that everyone is free to use them, and to redistribute them
+in other free programs.  The library is not in the public domain; it is
+copyrighted and there are conditions on its distribution.  These
+conditions are designed to permit everything that a good cooperating
+citizen would want to do.  What is not allowed is to try to prevent
+others from further sharing any version of the software that they might
+get from you.
+
+Specifically, we want to make sure that you have the right to share
+copies of programs that you are given which use the GNU Scientific
+Library, that you receive their source code or else can get it if you
+want it, that you can change these programs or use pieces of them in new
+free programs, and that you know you can do these things.
+
+To make sure that everyone has such rights, we have to forbid you to
+deprive anyone else of these rights.  For example, if you distribute
+copies of any code which uses the GNU Scientific Library, you must give
+the recipients all the rights that you have received.  You must make
+sure that they, too, receive or can get the source code, both to the
+library and the code which uses it.  And you must tell them their
+rights.  This means that the library should not be redistributed in
+proprietary programs.
+
+Also, for our own protection, we must make certain that everyone finds
+out that there is no warranty for the GNU Scientific Library.  If these
+programs are modified by someone else and passed on, we want their
+recipients to know that what they have is not what we distributed, so
+that any problems introduced by others will not reflect on our
+reputation.
+
+The precise conditions for the distribution of software related to the
+GNU Scientific Library are found in the GNU General Public License
+(@pxref{GNU General Public License}).  Further information about this
+license is available from the GNU Project webpage @cite{Frequently Asked
+Questions about the GNU GPL},
+
+@itemize @asis
+@item 
+@uref{http://www.gnu.org/copyleft/gpl-faq.html}
+@end itemize
+
+@noindent
+The Free Software Foundation also operates a license consulting
+service for commercial users (contact details available from
+@uref{http://www.fsf.org/}).
+
+@node Obtaining GSL
+@section Obtaining GSL
+@cindex obtaining GSL
+@cindex downloading GSL
+@cindex mailing list for GSL announcements
+@cindex info-gsl mailing list
+The source code for the library can be obtained in different ways, by
+copying it from a friend, purchasing it on @sc{cdrom} or downloading it
+from the internet. A list of public ftp servers which carry the source
+code can be found on the GNU website,
+
+@itemize @asis
+@item 
+@uref{http://www.gnu.org/software/gsl/}
+@end itemize
+
+@noindent
+The preferred platform for the library is a GNU system, which allows it
+to take advantage of additional features in the GNU C compiler and GNU C
+library.  However, the library is fully portable and should compile on
+most systems with a C compiler.  Precompiled versions of the library can be purchased from
+commercial redistributors listed on the website above.
+
+Announcements of new releases, updates and other relevant events are
+made on the @code{info-gsl@@gnu.org} mailing list.  To subscribe to this
+low-volume list, send an email of the following form:
+
+@example
+To: info-gsl-request@@gnu.org 
+Subject: subscribe
+@end example
+
+@noindent
+You will receive a response asking you to reply in order to confirm
+your subscription.
+
+@node  No Warranty
+@section No Warranty
+@cindex warranty (none)
+The software described in this manual has no warranty, it is provided
+``as is''.  It is your responsibility to validate the behavior of the
+routines and their accuracy using the source code provided, or to
+purchase support and warranties from commercial redistributors.  Consult
+the GNU General Public license for further details (@pxref{GNU General
+Public License}).
+
+@node  Reporting Bugs
+@section Reporting Bugs
+@cindex reporting bugs in GSL
+@cindex bugs, how to report
+@cindex bug-gsl mailing list
+@cindex mailing list, bug-gsl
+A list of known bugs can be found in the @file{BUGS} file included in
+the GSL distribution.  Details of compilation problems can be found in
+the @file{INSTALL} file.
+
+If you find a bug which is not listed in these files, please report it to
+@email{bug-gsl@@gnu.org}.
+
+All bug reports should include:
+
+@itemize @bullet
+@item
+The version number of GSL
+@item
+The hardware and operating system
+@item
+The compiler used, including version number and compilation options
+@item
+A description of the bug behavior
+@item
+A short program which exercises the bug
+@end itemize
+
+@noindent
+It is useful if you can check whether the same problem occurs when the
+library is compiled without optimization.  Thank you.
+
+Any errors or omissions in this manual can also be reported to the
+same address.
+
+@node  Further Information
+@section Further Information
+@cindex mailing list archives
+@cindex website, developer information
+@cindex contacting the GSL developers
+Additional information, including online copies of this manual, links to
+related projects, and mailing list archives are available from the
+website mentioned above.  
+
+Any questions about the use and installation of the library can be asked
+on the mailing list @code{help-gsl@@gnu.org}.  To subscribe to this
+list, send an email of the following form:
+
+@example
+To: help-gsl-request@@gnu.org
+Subject: subscribe
+@end example
+
+@noindent
+This mailing list can be used to ask questions not covered by this
+manual, and to contact the developers of the library.
+
+If you would like to refer to the GNU Scientific Library in a journal
+article, the recommended way is to cite this reference manual,
+e.g. @cite{M. Galassi et al, GNU Scientific Library Reference Manual (2nd
+Ed.), ISBN 0954161734}.
+
+If you want to give a url, use ``@uref{http://www.gnu.org/software/gsl/}''.
+
+@node Conventions used in this manual
+@section Conventions used in this manual
+@cindex conventions, used in manual
+@cindex examples, conventions used in
+@cindex shell prompt
+@cindex @code{$}, shell prompt
+This manual contains many examples which can be typed at the keyboard.
+A command entered at the terminal is shown like this,
+
+@example
+$ @i{command}
+@end example
+
+@noindent
+@cindex dollar sign @code{$}, shell prompt
+The first character on the line is the terminal prompt, and should not
+be typed.  The dollar sign @samp{$} is used as the standard prompt in
+this manual, although some systems may use a different character.
+
+The examples assume the use of the GNU operating system.  There may be
+minor differences in the output on other systems.  The commands for
+setting environment variables use the Bourne shell syntax of the
+standard GNU shell (@code{bash}).
+
+
diff --git a/gsl-1.9/doc/landau.dat b/gsl-1.9/doc/landau.dat
new file mode 100644
index 0000000..0c7d337
--- /dev/null
+++ b/gsl-1.9/doc/landau.dat
@@ -0,0 +1,201 @@
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diff --git a/gsl-1.9/doc/linalg.texi b/gsl-1.9/doc/linalg.texi
new file mode 100644
index 0000000..839f4d5
--- /dev/null
+++ b/gsl-1.9/doc/linalg.texi
@@ -0,0 +1,1117 @@
+@cindex linear algebra
+@cindex solution of linear systems, Ax=b
+@cindex matrix factorization
+@cindex factorization of matrices
+
+This chapter describes functions for solving linear systems.  The
+library provides linear algebra operations which operate directly on
+the @code{gsl_vector} and @code{gsl_matrix} objects.  These routines
+use the standard algorithms from Golub & Van Loan's @cite{Matrix
+Computations}.
+
+@cindex LAPACK, recommended for linear algebra
+When dealing with very large systems the routines found in @sc{lapack}
+should be considered.  These support specialized data representations
+and other optimizations.
+
+The functions described in this chapter are declared in the header file
+@file{gsl_linalg.h}.
+
+
+@menu
+* LU Decomposition::            
+* QR Decomposition::            
+* QR Decomposition with Column Pivoting::  
+* Singular Value Decomposition::  
+* Cholesky Decomposition::      
+* Tridiagonal Decomposition of Real Symmetric Matrices::  
+* Tridiagonal Decomposition of Hermitian Matrices::  
+* Hessenberg Decomposition of Real Matrices::
+* Bidiagonalization::           
+* Householder Transformations::  
+* Householder solver for linear systems::  
+* Tridiagonal Systems::         
+* Balancing::
+* Linear Algebra Examples::     
+* Linear Algebra References and Further Reading::  
+@end menu
+
+@node LU Decomposition
+@section LU Decomposition
+@cindex LU decomposition
+
+A general square matrix @math{A} has an @math{LU} decomposition into
+upper and lower triangular matrices,
+@tex
+\beforedisplay
+$$
+P A = L U
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+P A = L U
+@end example
+
+@end ifinfo
+@noindent
+where @math{P} is a permutation matrix, @math{L} is unit lower
+triangular matrix and @math{U} is upper triangular matrix. For square
+matrices this decomposition can be used to convert the linear system
+@math{A x = b} into a pair of triangular systems (@math{L y = P b},
+@math{U x = y}), which can be solved by forward and back-substitution.
+Note that the @math{LU} decomposition is valid for singular matrices.
+
+@deftypefun int gsl_linalg_LU_decomp (gsl_matrix * @var{A}, gsl_permutation * @var{p}, int * @var{signum})
+@deftypefunx int gsl_linalg_complex_LU_decomp (gsl_matrix_complex * @var{A}, gsl_permutation * @var{p}, int * @var{signum})
+These functions factorize the square matrix @var{A} into the @math{LU}
+decomposition @math{PA = LU}.  On output the diagonal and upper
+triangular part of the input matrix @var{A} contain the matrix
+@math{U}. The lower triangular part of the input matrix (excluding the
+diagonal) contains @math{L}.  The diagonal elements of @math{L} are
+unity, and are not stored.
+
+The permutation matrix @math{P} is encoded in the permutation
+@var{p}. The @math{j}-th column of the matrix @math{P} is given by the
+@math{k}-th column of the identity matrix, where @math{k = p_j} the
+@math{j}-th element of the permutation vector. The sign of the
+permutation is given by @var{signum}. It has the value @math{(-1)^n},
+where @math{n} is the number of interchanges in the permutation.
+
+The algorithm used in the decomposition is Gaussian Elimination with
+partial pivoting (Golub & Van Loan, @cite{Matrix Computations},
+Algorithm 3.4.1).
+@end deftypefun
+
+@cindex linear systems, solution of
+@deftypefun int gsl_linalg_LU_solve (const gsl_matrix * @var{LU}, const gsl_permutation * @var{p}, const gsl_vector * @var{b}, gsl_vector * @var{x})
+@deftypefunx int gsl_linalg_complex_LU_solve (const gsl_matrix_complex * @var{LU}, const gsl_permutation * @var{p}, const gsl_vector_complex * @var{b}, gsl_vector_complex * @var{x})
+These functions solve the square system @math{A x = b} using the @math{LU}
+decomposition of @math{A} into (@var{LU}, @var{p}) given by
+@code{gsl_linalg_LU_decomp} or @code{gsl_linalg_complex_LU_decomp}.
+@end deftypefun
+
+@deftypefun int gsl_linalg_LU_svx (const gsl_matrix * @var{LU}, const gsl_permutation * @var{p}, gsl_vector * @var{x})
+@deftypefunx int gsl_linalg_complex_LU_svx (const gsl_matrix_complex * @var{LU}, const gsl_permutation * @var{p}, gsl_vector_complex * @var{x})
+These functions solve the square system @math{A x = b} in-place using the
+@math{LU} decomposition of @math{A} into (@var{LU},@var{p}). On input
+@var{x} should contain the right-hand side @math{b}, which is replaced
+by the solution on output.
+@end deftypefun
+
+@cindex refinement of solutions in linear systems
+@cindex iterative refinement of solutions in linear systems
+@cindex linear systems, refinement of solutions
+@deftypefun int gsl_linalg_LU_refine (const gsl_matrix * @var{A}, const gsl_matrix * @var{LU}, const gsl_permutation * @var{p}, const gsl_vector * @var{b}, gsl_vector * @var{x}, gsl_vector * @var{residual})
+@deftypefunx int gsl_linalg_complex_LU_refine (const gsl_matrix_complex * @var{A}, const gsl_matrix_complex * @var{LU}, const gsl_permutation * @var{p}, const gsl_vector_complex * @var{b}, gsl_vector_complex * @var{x}, gsl_vector_complex * @var{residual})
+These functions apply an iterative improvement to @var{x}, the solution
+of @math{A x = b}, using the @math{LU} decomposition of @math{A} into
+(@var{LU},@var{p}). The initial residual @math{r = A x - b} is also
+computed and stored in @var{residual}.
+@end deftypefun
+
+@cindex inverse of a matrix, by LU decomposition
+@cindex matrix inverse
+@deftypefun int gsl_linalg_LU_invert (const gsl_matrix * @var{LU}, const gsl_permutation * @var{p}, gsl_matrix * @var{inverse})
+@deftypefunx int gsl_linalg_complex_LU_invert (const gsl_matrix_complex * @var{LU}, const gsl_permutation * @var{p}, gsl_matrix_complex * @var{inverse})
+These functions compute the inverse of a matrix @math{A} from its
+@math{LU} decomposition (@var{LU},@var{p}), storing the result in the
+matrix @var{inverse}. The inverse is computed by solving the system
+@math{A x = b} for each column of the identity matrix.  It is preferable
+to avoid direct use of the inverse whenever possible, as the linear
+solver functions can obtain the same result more efficiently and
+reliably (consult any introductory textbook on numerical linear algebra
+for details).
+@end deftypefun
+
+@cindex determinant of a matrix, by LU decomposition
+@cindex matrix determinant
+@deftypefun double gsl_linalg_LU_det (gsl_matrix * @var{LU}, int @var{signum})
+@deftypefunx gsl_complex gsl_linalg_complex_LU_det (gsl_matrix_complex * @var{LU}, int @var{signum})
+These functions compute the determinant of a matrix @math{A} from its
+@math{LU} decomposition, @var{LU}. The determinant is computed as the
+product of the diagonal elements of @math{U} and the sign of the row
+permutation @var{signum}.
+@end deftypefun
+
+@cindex logarithm of the determinant of a matrix
+@deftypefun double gsl_linalg_LU_lndet (gsl_matrix * @var{LU})
+@deftypefunx double gsl_linalg_complex_LU_lndet (gsl_matrix_complex * @var{LU})
+These functions compute the logarithm of the absolute value of the
+determinant of a matrix @math{A}, @math{\ln|\det(A)|}, from its @math{LU}
+decomposition, @var{LU}.  This function may be useful if the direct
+computation of the determinant would overflow or underflow.
+@end deftypefun
+
+@cindex sign of the determinant of a matrix
+@deftypefun int gsl_linalg_LU_sgndet (gsl_matrix * @var{LU}, int @var{signum})
+@deftypefunx gsl_complex gsl_linalg_complex_LU_sgndet (gsl_matrix_complex * @var{LU}, int @var{signum})
+These functions compute the sign or phase factor of the determinant of a
+matrix @math{A}, @math{\det(A)/|\det(A)|}, from its @math{LU} decomposition,
+@var{LU}.
+@end deftypefun
+
+@node QR Decomposition
+@section QR Decomposition
+@cindex QR decomposition
+
+A general rectangular @math{M}-by-@math{N} matrix @math{A} has a
+@math{QR} decomposition into the product of an orthogonal
+@math{M}-by-@math{M} square matrix @math{Q} (where @math{Q^T Q = I}) and
+an @math{M}-by-@math{N} right-triangular matrix @math{R},
+@tex
+\beforedisplay
+$$
+A = Q R
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+A = Q R
+@end example
+
+@end ifinfo
+@noindent
+This decomposition can be used to convert the linear system @math{A x =
+b} into the triangular system @math{R x = Q^T b}, which can be solved by
+back-substitution. Another use of the @math{QR} decomposition is to
+compute an orthonormal basis for a set of vectors. The first @math{N}
+columns of @math{Q} form an orthonormal basis for the range of @math{A},
+@math{ran(A)}, when @math{A} has full column rank.
+
+@deftypefun int gsl_linalg_QR_decomp (gsl_matrix * @var{A}, gsl_vector * @var{tau})
+This function factorizes the @math{M}-by-@math{N} matrix @var{A} into
+the @math{QR} decomposition @math{A = Q R}.  On output the diagonal and
+upper triangular part of the input matrix contain the matrix
+@math{R}. The vector @var{tau} and the columns of the lower triangular
+part of the matrix @var{A} contain the Householder coefficients and
+Householder vectors which encode the orthogonal matrix @var{Q}.  The
+vector @var{tau} must be of length @math{k=\min(M,N)}. The matrix
+@math{Q} is related to these components by, @math{Q = Q_k ... Q_2 Q_1}
+where @math{Q_i = I - \tau_i v_i v_i^T} and @math{v_i} is the
+Householder vector @math{v_i =
+(0,...,1,A(i+1,i),A(i+2,i),...,A(m,i))}. This is the same storage scheme
+as used by @sc{lapack}.
+
+The algorithm used to perform the decomposition is Householder QR (Golub
+& Van Loan, @cite{Matrix Computations}, Algorithm 5.2.1).
+@end deftypefun
+
+@deftypefun int gsl_linalg_QR_solve (const gsl_matrix * @var{QR}, const gsl_vector * @var{tau}, const gsl_vector * @var{b}, gsl_vector * @var{x})
+This function solves the square system @math{A x = b} using the @math{QR}
+decomposition of @math{A} into (@var{QR}, @var{tau}) given by
+@code{gsl_linalg_QR_decomp}. The least-squares solution for rectangular systems can
+be found using @code{gsl_linalg_QR_lssolve}.
+@end deftypefun
+
+@deftypefun int gsl_linalg_QR_svx (const gsl_matrix * @var{QR}, const gsl_vector * @var{tau}, gsl_vector * @var{x})
+This function solves the square system @math{A x = b} in-place using the
+@math{QR} decomposition of @math{A} into (@var{QR},@var{tau}) given by
+@code{gsl_linalg_QR_decomp}. On input @var{x} should contain the
+right-hand side @math{b}, which is replaced by the solution on output.
+@end deftypefun
+
+@deftypefun int gsl_linalg_QR_lssolve (const gsl_matrix * @var{QR}, const gsl_vector * @var{tau}, const gsl_vector * @var{b}, gsl_vector * @var{x}, gsl_vector * @var{residual})
+This function finds the least squares solution to the overdetermined
+system @math{A x = b} where the matrix @var{A} has more rows than
+columns.  The least squares solution minimizes the Euclidean norm of the
+residual, @math{||Ax - b||}.The routine uses the @math{QR} decomposition
+of @math{A} into (@var{QR}, @var{tau}) given by
+@code{gsl_linalg_QR_decomp}.  The solution is returned in @var{x}.  The
+residual is computed as a by-product and stored in @var{residual}.
+@end deftypefun
+
+@deftypefun int gsl_linalg_QR_QTvec (const gsl_matrix * @var{QR}, const gsl_vector * @var{tau}, gsl_vector * @var{v})
+This function applies the matrix @math{Q^T} encoded in the decomposition
+(@var{QR},@var{tau}) to the vector @var{v}, storing the result @math{Q^T
+v} in @var{v}.  The matrix multiplication is carried out directly using
+the encoding of the Householder vectors without needing to form the full
+matrix @math{Q^T}.
+@end deftypefun
+
+@deftypefun int gsl_linalg_QR_Qvec (const gsl_matrix * @var{QR}, const gsl_vector * @var{tau}, gsl_vector * @var{v})
+This function applies the matrix @math{Q} encoded in the decomposition
+(@var{QR},@var{tau}) to the vector @var{v}, storing the result @math{Q
+v} in @var{v}.  The matrix multiplication is carried out directly using
+the encoding of the Householder vectors without needing to form the full
+matrix @math{Q}.
+@end deftypefun
+
+@deftypefun int gsl_linalg_QR_Rsolve (const gsl_matrix * @var{QR}, const gsl_vector * @var{b}, gsl_vector * @var{x})
+This function solves the triangular system @math{R x = b} for
+@var{x}. It may be useful if the product @math{b' = Q^T b} has already
+been computed using @code{gsl_linalg_QR_QTvec}.
+@end deftypefun
+
+@deftypefun int gsl_linalg_QR_Rsvx (const gsl_matrix * @var{QR}, gsl_vector * @var{x})
+This function solves the triangular system @math{R x = b} for @var{x}
+in-place. On input @var{x} should contain the right-hand side @math{b}
+and is replaced by the solution on output. This function may be useful if
+the product @math{b' = Q^T b} has already been computed using
+@code{gsl_linalg_QR_QTvec}.
+@end deftypefun
+
+@deftypefun int gsl_linalg_QR_unpack (const gsl_matrix * @var{QR}, const gsl_vector * @var{tau}, gsl_matrix * @var{Q}, gsl_matrix * @var{R})
+This function unpacks the encoded @math{QR} decomposition
+(@var{QR},@var{tau}) into the matrices @var{Q} and @var{R}, where
+@var{Q} is @math{M}-by-@math{M} and @var{R} is @math{M}-by-@math{N}. 
+@end deftypefun
+
+@deftypefun int gsl_linalg_QR_QRsolve (gsl_matrix * @var{Q}, gsl_matrix * @var{R}, const gsl_vector * @var{b}, gsl_vector * @var{x})
+This function solves the system @math{R x = Q^T b} for @var{x}. It can
+be used when the @math{QR} decomposition of a matrix is available in
+unpacked form as (@var{Q}, @var{R}).
+@end deftypefun
+
+@deftypefun int gsl_linalg_QR_update (gsl_matrix * @var{Q}, gsl_matrix * @var{R}, gsl_vector * @var{w}, const gsl_vector * @var{v})
+This function performs a rank-1 update @math{w v^T} of the @math{QR}
+decomposition (@var{Q}, @var{R}). The update is given by @math{Q'R' = Q
+R + w v^T} where the output matrices @math{Q'} and @math{R'} are also
+orthogonal and right triangular. Note that @var{w} is destroyed by the
+update.
+@end deftypefun
+
+@deftypefun int gsl_linalg_R_solve (const gsl_matrix * @var{R}, const gsl_vector * @var{b}, gsl_vector * @var{x})
+This function solves the triangular system @math{R x = b} for the
+@math{N}-by-@math{N} matrix @var{R}.
+@end deftypefun
+
+@deftypefun int gsl_linalg_R_svx (const gsl_matrix * @var{R}, gsl_vector * @var{x})
+This function solves the triangular system @math{R x = b} in-place. On
+input @var{x} should contain the right-hand side @math{b}, which is
+replaced by the solution on output.
+@end deftypefun
+
+@node QR Decomposition with Column Pivoting
+@section QR Decomposition with Column Pivoting
+@cindex QR decomposition with column pivoting
+
+The @math{QR} decomposition can be extended to the rank deficient case
+by introducing a column permutation @math{P},
+@tex
+\beforedisplay
+$$
+A P = Q R
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+A P = Q R
+@end example
+
+@end ifinfo
+@noindent
+The first @math{r} columns of @math{Q} form an orthonormal basis
+for the range of @math{A} for a matrix with column rank @math{r}.  This
+decomposition can also be used to convert the linear system @math{A x =
+b} into the triangular system @math{R y = Q^T b, x = P y}, which can be
+solved by back-substitution and permutation.  We denote the @math{QR}
+decomposition with column pivoting by @math{QRP^T} since @math{A = Q R
+P^T}.
+
+@deftypefun int gsl_linalg_QRPT_decomp (gsl_matrix * @var{A}, gsl_vector * @var{tau}, gsl_permutation * @var{p}, int * @var{signum}, gsl_vector * @var{norm})
+This function factorizes the @math{M}-by-@math{N} matrix @var{A} into
+the @math{QRP^T} decomposition @math{A = Q R P^T}.  On output the
+diagonal and upper triangular part of the input matrix contain the
+matrix @math{R}. The permutation matrix @math{P} is stored in the
+permutation @var{p}.  The sign of the permutation is given by
+@var{signum}. It has the value @math{(-1)^n}, where @math{n} is the
+number of interchanges in the permutation. The vector @var{tau} and the
+columns of the lower triangular part of the matrix @var{A} contain the
+Householder coefficients and vectors which encode the orthogonal matrix
+@var{Q}.  The vector @var{tau} must be of length @math{k=\min(M,N)}. The
+matrix @math{Q} is related to these components by, @math{Q = Q_k ... Q_2
+Q_1} where @math{Q_i = I - \tau_i v_i v_i^T} and @math{v_i} is the
+Householder vector @math{v_i =
+(0,...,1,A(i+1,i),A(i+2,i),...,A(m,i))}. This is the same storage scheme
+as used by @sc{lapack}.  The vector @var{norm} is a workspace of length
+@var{N} used for column pivoting.
+
+The algorithm used to perform the decomposition is Householder QR with
+column pivoting (Golub & Van Loan, @cite{Matrix Computations}, Algorithm
+5.4.1).
+@end deftypefun
+
+@deftypefun int gsl_linalg_QRPT_decomp2 (const gsl_matrix * @var{A}, gsl_matrix * @var{q}, gsl_matrix * @var{r}, gsl_vector * @var{tau}, gsl_permutation * @var{p}, int * @var{signum}, gsl_vector * @var{norm})
+This function factorizes the matrix @var{A} into the decomposition
+@math{A = Q R P^T} without modifying @var{A} itself and storing the
+output in the separate matrices @var{q} and @var{r}.
+@end deftypefun
+
+@deftypefun int gsl_linalg_QRPT_solve (const gsl_matrix * @var{QR}, const gsl_vector * @var{tau}, const gsl_permutation * @var{p}, const gsl_vector * @var{b}, gsl_vector * @var{x})
+This function solves the square system @math{A x = b} using the @math{QRP^T}
+decomposition of @math{A} into (@var{QR}, @var{tau}, @var{p}) given by
+@code{gsl_linalg_QRPT_decomp}.
+@end deftypefun
+
+@deftypefun int gsl_linalg_QRPT_svx (const gsl_matrix * @var{QR}, const gsl_vector * @var{tau}, const gsl_permutation * @var{p}, gsl_vector * @var{x})
+This function solves the square system @math{A x = b} in-place using the
+@math{QRP^T} decomposition of @math{A} into
+(@var{QR},@var{tau},@var{p}). On input @var{x} should contain the
+right-hand side @math{b}, which is replaced by the solution on output.
+@end deftypefun
+
+@deftypefun int gsl_linalg_QRPT_QRsolve (const gsl_matrix * @var{Q}, const gsl_matrix * @var{R}, const gsl_permutation * @var{p}, const gsl_vector * @var{b}, gsl_vector * @var{x})
+This function solves the square system @math{R P^T x = Q^T b} for
+@var{x}. It can be used when the @math{QR} decomposition of a matrix is
+available in unpacked form as (@var{Q}, @var{R}).
+@end deftypefun
+
+@deftypefun int gsl_linalg_QRPT_update (gsl_matrix * @var{Q}, gsl_matrix * @var{R}, const gsl_permutation * @var{p}, gsl_vector * @var{u}, const gsl_vector * @var{v})
+This function performs a rank-1 update @math{w v^T} of the @math{QRP^T}
+decomposition (@var{Q}, @var{R}, @var{p}). The update is given by
+@math{Q'R' = Q R + w v^T} where the output matrices @math{Q'} and
+@math{R'} are also orthogonal and right triangular. Note that @var{w} is
+destroyed by the update. The permutation @var{p} is not changed.
+@end deftypefun
+
+@deftypefun int gsl_linalg_QRPT_Rsolve (const gsl_matrix * @var{QR}, const gsl_permutation * @var{p}, const gsl_vector * @var{b}, gsl_vector * @var{x})
+This function solves the triangular system @math{R P^T x = b} for the
+@math{N}-by-@math{N} matrix @math{R} contained in @var{QR}.
+@end deftypefun
+
+@deftypefun int gsl_linalg_QRPT_Rsvx (const gsl_matrix * @var{QR}, const gsl_permutation * @var{p}, gsl_vector * @var{x})
+This function solves the triangular system @math{R P^T x = b} in-place
+for the @math{N}-by-@math{N} matrix @math{R} contained in @var{QR}. On
+input @var{x} should contain the right-hand side @math{b}, which is
+replaced by the solution on output.
+@end deftypefun
+
+@node Singular Value Decomposition
+@section Singular Value Decomposition
+@cindex SVD
+@cindex singular value decomposition
+
+A general rectangular @math{M}-by-@math{N} matrix @math{A} has a
+singular value decomposition (@sc{svd}) into the product of an
+@math{M}-by-@math{N} orthogonal matrix @math{U}, an @math{N}-by-@math{N}
+diagonal matrix of singular values @math{S} and the transpose of an
+@math{N}-by-@math{N} orthogonal square matrix @math{V},
+@tex
+\beforedisplay
+$$
+A = U S V^T
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+A = U S V^T
+@end example
+
+@end ifinfo
+@noindent
+The singular values
+@c{$\sigma_i = S_{ii}$}
+@math{\sigma_i = S_@{ii@}} are all non-negative and are
+generally chosen to form a non-increasing sequence 
+@c{$\sigma_1 \ge \sigma_2 \ge ... \ge \sigma_N \ge 0$}
+@math{\sigma_1 >= \sigma_2 >= ... >= \sigma_N >= 0}.
+
+The singular value decomposition of a matrix has many practical uses.
+The condition number of the matrix is given by the ratio of the largest
+singular value to the smallest singular value. The presence of a zero
+singular value indicates that the matrix is singular. The number of
+non-zero singular values indicates the rank of the matrix.  In practice
+singular value decomposition of a rank-deficient matrix will not produce
+exact zeroes for singular values, due to finite numerical
+precision.  Small singular values should be edited by choosing a suitable
+tolerance.
+
+For a rank-deficient matrix, the null space of @math{A} is given by
+the columns of @math{V} corresponding to the zero singular values.
+Similarly, the range of @math{A} is given by columns of @math{U}
+corresponding to the non-zero singular values.
+
+@deftypefun int gsl_linalg_SV_decomp (gsl_matrix * @var{A}, gsl_matrix * @var{V}, gsl_vector * @var{S}, gsl_vector * @var{work})
+This function factorizes the @math{M}-by-@math{N} matrix @var{A} into
+the singular value decomposition @math{A = U S V^T} for @c{$M \ge N$}
+@math{M >= N}.  On output the matrix @var{A} is replaced by
+@math{U}. The diagonal elements of the singular value matrix @math{S}
+are stored in the vector @var{S}. The singular values are non-negative
+and form a non-increasing sequence from @math{S_1} to @math{S_N}. The
+matrix @var{V} contains the elements of @math{V} in untransposed
+form. To form the product @math{U S V^T} it is necessary to take the
+transpose of @var{V}.  A workspace of length @var{N} is required in
+@var{work}.
+
+This routine uses the Golub-Reinsch SVD algorithm.
+@end deftypefun
+
+@deftypefun int gsl_linalg_SV_decomp_mod (gsl_matrix * @var{A}, gsl_matrix * @var{X}, gsl_matrix * @var{V}, gsl_vector * @var{S}, gsl_vector * @var{work})
+This function computes the SVD using the modified Golub-Reinsch
+algorithm, which is faster for @c{$M \gg N$}
+@math{M>>N}.  It requires the vector @var{work} of length @var{N} and the
+@math{N}-by-@math{N} matrix @var{X} as additional working space.
+@end deftypefun
+
+@deftypefun int gsl_linalg_SV_decomp_jacobi (gsl_matrix * @var{A}, gsl_matrix * @var{V}, gsl_vector * @var{S})
+This function computes the SVD of the @math{M}-by-@math{N} matrix @var{A}
+using one-sided Jacobi orthogonalization for @c{$M \ge N$} 
+@math{M >= N}.  The Jacobi method can compute singular values to higher
+relative accuracy than Golub-Reinsch algorithms (see references for
+details).
+@end deftypefun
+
+@deftypefun int gsl_linalg_SV_solve (gsl_matrix * @var{U}, gsl_matrix * @var{V}, gsl_vector * @var{S}, const gsl_vector * @var{b}, gsl_vector * @var{x})
+This function solves the system @math{A x = b} using the singular value
+decomposition (@var{U}, @var{S}, @var{V}) of @math{A} given by
+@code{gsl_linalg_SV_decomp}.
+
+Only non-zero singular values are used in computing the solution. The
+parts of the solution corresponding to singular values of zero are
+ignored.  Other singular values can be edited out by setting them to
+zero before calling this function. 
+
+In the over-determined case where @var{A} has more rows than columns the
+system is solved in the least squares sense, returning the solution
+@var{x} which minimizes @math{||A x - b||_2}.
+@end deftypefun
+
+@node Cholesky Decomposition
+@section Cholesky Decomposition
+@cindex Cholesky decomposition
+@cindex square root of a matrix, Cholesky decomposition
+@cindex matrix square root, Cholesky decomposition
+
+A symmetric, positive definite square matrix @math{A} has a Cholesky
+decomposition into a product of a lower triangular matrix @math{L} and
+its transpose @math{L^T},
+@tex
+\beforedisplay
+$$
+A = L L^T
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+A = L L^T
+@end example
+
+@end ifinfo
+@noindent
+This is sometimes referred to as taking the square-root of a matrix. The
+Cholesky decomposition can only be carried out when all the eigenvalues
+of the matrix are positive.  This decomposition can be used to convert
+the linear system @math{A x = b} into a pair of triangular systems
+(@math{L y = b}, @math{L^T x = y}), which can be solved by forward and
+back-substitution.
+
+@deftypefun int gsl_linalg_cholesky_decomp (gsl_matrix * @var{A})
+This function factorizes the positive-definite symmetric square matrix
+@var{A} into the Cholesky decomposition @math{A = L L^T}. On input
+only the diagonal and lower-triangular part of the matrix @var{A} are
+needed.  On output the diagonal and lower triangular part of the input
+matrix @var{A} contain the matrix @math{L}. The upper triangular part
+of the input matrix contains @math{L^T}, the diagonal terms being
+identical for both @math{L} and @math{L^T}.  If the matrix is not
+positive-definite then the decomposition will fail, returning the
+error code @code{GSL_EDOM}.
+@end deftypefun
+
+@deftypefun int gsl_linalg_cholesky_solve (const gsl_matrix * @var{cholesky}, const gsl_vector * @var{b}, gsl_vector * @var{x})
+This function solves the system @math{A x = b} using the Cholesky
+decomposition of @math{A} into the matrix @var{cholesky} given by
+@code{gsl_linalg_cholesky_decomp}.
+@end deftypefun
+
+@deftypefun int gsl_linalg_cholesky_svx (const gsl_matrix * @var{cholesky}, gsl_vector * @var{x})
+This function solves the system @math{A x = b} in-place using the
+Cholesky decomposition of @math{A} into the matrix @var{cholesky} given
+by @code{gsl_linalg_cholesky_decomp}. On input @var{x} should contain
+the right-hand side @math{b}, which is replaced by the solution on
+output.
+@end deftypefun
+
+@node Tridiagonal Decomposition of Real Symmetric Matrices
+@section Tridiagonal Decomposition of Real Symmetric Matrices
+@cindex  tridiagonal decomposition
+
+A symmetric matrix @math{A} can be factorized by similarity
+transformations into the form,
+@tex
+\beforedisplay
+$$
+A = Q T Q^T
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+A = Q T Q^T
+@end example
+
+@end ifinfo
+@noindent
+where @math{Q} is an orthogonal matrix and @math{T} is a symmetric
+tridiagonal matrix.
+
+@deftypefun int gsl_linalg_symmtd_decomp (gsl_matrix * @var{A}, gsl_vector * @var{tau})
+This function factorizes the symmetric square matrix @var{A} into the
+symmetric tridiagonal decomposition @math{Q T Q^T}.  On output the
+diagonal and subdiagonal part of the input matrix @var{A} contain the
+tridiagonal matrix @math{T}.  The remaining lower triangular part of the
+input matrix contains the Householder vectors which, together with the
+Householder coefficients @var{tau}, encode the orthogonal matrix
+@math{Q}. This storage scheme is the same as used by @sc{lapack}.  The
+upper triangular part of @var{A} is not referenced.
+@end deftypefun
+
+@deftypefun int gsl_linalg_symmtd_unpack (const gsl_matrix * @var{A}, const gsl_vector * @var{tau}, gsl_matrix * @var{Q}, gsl_vector * @var{diag}, gsl_vector * @var{subdiag})
+This function unpacks the encoded symmetric tridiagonal decomposition
+(@var{A}, @var{tau}) obtained from @code{gsl_linalg_symmtd_decomp} into
+the orthogonal matrix @var{Q}, the vector of diagonal elements @var{diag}
+and the vector of subdiagonal elements @var{subdiag}.  
+@end deftypefun
+
+@deftypefun int gsl_linalg_symmtd_unpack_T (const gsl_matrix * @var{A}, gsl_vector * @var{diag}, gsl_vector * @var{subdiag})
+This function unpacks the diagonal and subdiagonal of the encoded
+symmetric tridiagonal decomposition (@var{A}, @var{tau}) obtained from
+@code{gsl_linalg_symmtd_decomp} into the vectors @var{diag} and @var{subdiag}.
+@end deftypefun
+
+@node Tridiagonal Decomposition of Hermitian Matrices
+@section Tridiagonal Decomposition of Hermitian Matrices
+@cindex tridiagonal decomposition
+
+A hermitian matrix @math{A} can be factorized by similarity
+transformations into the form,
+@tex
+\beforedisplay
+$$
+A = U T U^T
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+A = U T U^T
+@end example
+
+@end ifinfo
+@noindent
+where @math{U} is a unitary matrix and @math{T} is a real symmetric
+tridiagonal matrix.
+
+
+@deftypefun int gsl_linalg_hermtd_decomp (gsl_matrix_complex * @var{A}, gsl_vector_complex * @var{tau})
+This function factorizes the hermitian matrix @var{A} into the symmetric
+tridiagonal decomposition @math{U T U^T}.  On output the real parts of
+the diagonal and subdiagonal part of the input matrix @var{A} contain
+the tridiagonal matrix @math{T}.  The remaining lower triangular part of
+the input matrix contains the Householder vectors which, together with
+the Householder coefficients @var{tau}, encode the orthogonal matrix
+@math{Q}. This storage scheme is the same as used by @sc{lapack}.  The
+upper triangular part of @var{A} and imaginary parts of the diagonal are
+not referenced.
+@end deftypefun
+
+@deftypefun int gsl_linalg_hermtd_unpack (const gsl_matrix_complex * @var{A}, const gsl_vector_complex * @var{tau}, gsl_matrix_complex * @var{Q}, gsl_vector * @var{diag}, gsl_vector * @var{subdiag})
+This function unpacks the encoded tridiagonal decomposition (@var{A},
+@var{tau}) obtained from @code{gsl_linalg_hermtd_decomp} into the
+unitary matrix @var{U}, the real vector of diagonal elements @var{diag} and
+the real vector of subdiagonal elements @var{subdiag}. 
+@end deftypefun
+
+@deftypefun int gsl_linalg_hermtd_unpack_T (const gsl_matrix_complex * @var{A}, gsl_vector * @var{diag}, gsl_vector * @var{subdiag})
+This function unpacks the diagonal and subdiagonal of the encoded
+tridiagonal decomposition (@var{A}, @var{tau}) obtained from the
+@code{gsl_linalg_hermtd_decomp} into the real vectors
+@var{diag} and @var{subdiag}.
+@end deftypefun
+
+@node Hessenberg Decomposition of Real Matrices
+@section Hessenberg Decomposition of Real Matrices
+@cindex hessenberg decomposition
+
+A general matrix @math{A} can be decomposed by orthogonal
+similarity transformations into the form
+@tex
+\beforedisplay
+$$
+A = U H U^T
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+A = U H U^T
+@end example
+
+@end ifinfo
+where @math{U} is orthogonal and @math{H} is an upper Hessenberg matrix,
+meaning that it has zeros below the first subdiagonal. The
+Hessenberg reduction is the first step in the Schur decomposition
+for the nonsymmetric eigenvalue problem, but has applications in
+other areas as well.
+
+@deftypefun int gsl_linalg_hessenberg (gsl_matrix * @var{A}, gsl_vector * @var{tau})
+This function computes the Hessenberg decomposition of the matrix
+@var{A} by applying the similarity transformation @math{H = U^T A U}.
+On output, @math{H} is stored in the upper portion of @var{A}. The
+information required to construct the matrix @math{U} is stored in
+the lower triangular portion of @var{A}. @math{U} is a product
+of @math{N - 2} Householder matrices. The Householder vectors
+are stored in the lower portion of @var{A} (below the subdiagonal)
+and the Householder coefficients are stored in the vector @var{tau}.
+@var{tau} must be of length @var{N}.
+@end deftypefun
+
+@deftypefun int gsl_linalg_hessenberg_unpack (gsl_matrix * @var{H}, gsl_vector * @var{tau}, gsl_matrix * @var{U})
+This function constructs the orthogonal matrix @math{U} from the
+information stored in the Hessenberg matrix @var{H} along with the
+vector @var{tau}. @var{H} and @var{tau} are outputs from
+@code{gsl_linalg_hessenberg}.
+@end deftypefun
+
+@deftypefun int gsl_linalg_hessenberg_unpack_accum (gsl_matrix * @var{H}, gsl_vector * @var{tau}, gsl_matrix * @var{V})
+This function is similar to @code{gsl_linalg_hessenberg_unpack}, except
+it accumulates the matrix @var{U} into @var{V}, so that @math{V' = VU}.
+The matrix @var{V} must be initialized prior to calling this function.
+Setting @var{V} to the identity matrix provides the same result as
+@code{gsl_linalg_hessenberg_unpack}. If @var{H} is order @var{N}, then
+@var{V} must have @var{N} columns but may have any number of rows.
+@end deftypefun
+
+@deftypefun void gsl_linalg_hessenberg_set_zero (gsl_matrix * @var{H})
+This function sets the lower triangular portion of @var{H}, below
+the subdiagonal, to zero. It is useful for clearing out the
+Householder vectors after calling @code{gsl_linalg_hessenberg}.
+@end deftypefun
+
+@node Bidiagonalization
+@section Bidiagonalization 
+@cindex bidiagonalization of real matrices
+
+A general matrix @math{A} can be factorized by similarity
+transformations into the form,
+@tex
+\beforedisplay
+$$
+A = U B V^T
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+A = U B V^T
+@end example
+
+@end ifinfo
+@noindent
+where @math{U} and @math{V} are orthogonal matrices and @math{B} is a
+@math{N}-by-@math{N} bidiagonal matrix with non-zero entries only on the
+diagonal and superdiagonal.  The size of @var{U} is @math{M}-by-@math{N}
+and the size of @var{V} is @math{N}-by-@math{N}.
+
+@deftypefun int gsl_linalg_bidiag_decomp (gsl_matrix * @var{A}, gsl_vector * @var{tau_U}, gsl_vector * @var{tau_V})
+This function factorizes the @math{M}-by-@math{N} matrix @var{A} into
+bidiagonal form @math{U B V^T}.  The diagonal and superdiagonal of the
+matrix @math{B} are stored in the diagonal and superdiagonal of @var{A}.
+The orthogonal matrices @math{U} and @var{V} are stored as compressed
+Householder vectors in the remaining elements of @var{A}.  The
+Householder coefficients are stored in the vectors @var{tau_U} and
+@var{tau_V}.  The length of @var{tau_U} must equal the number of
+elements in the diagonal of @var{A} and the length of @var{tau_V} should
+be one element shorter.
+@end deftypefun
+
+@deftypefun int gsl_linalg_bidiag_unpack (const gsl_matrix * @var{A}, const gsl_vector * @var{tau_U}, gsl_matrix * @var{U}, const gsl_vector * @var{tau_V}, gsl_matrix * @var{V}, gsl_vector * @var{diag}, gsl_vector * @var{superdiag})
+This function unpacks the bidiagonal decomposition of @var{A} given by
+@code{gsl_linalg_bidiag_decomp}, (@var{A}, @var{tau_U}, @var{tau_V})
+into the separate orthogonal matrices @var{U}, @var{V} and the diagonal
+vector @var{diag} and superdiagonal @var{superdiag}.  Note that @var{U}
+is stored as a compact @math{M}-by-@math{N} orthogonal matrix satisfying
+@math{U^T U = I} for efficiency.
+@end deftypefun
+
+@deftypefun int gsl_linalg_bidiag_unpack2 (gsl_matrix * @var{A}, gsl_vector * @var{tau_U}, gsl_vector * @var{tau_V}, gsl_matrix * @var{V})
+This function unpacks the bidiagonal decomposition of @var{A} given by
+@code{gsl_linalg_bidiag_decomp}, (@var{A}, @var{tau_U}, @var{tau_V})
+into the separate orthogonal matrices @var{U}, @var{V} and the diagonal
+vector @var{diag} and superdiagonal @var{superdiag}.  The matrix @var{U}
+is stored in-place in @var{A}.
+@end deftypefun
+
+@deftypefun int gsl_linalg_bidiag_unpack_B (const gsl_matrix * @var{A}, gsl_vector * @var{diag}, gsl_vector * @var{superdiag})
+This function unpacks the diagonal and superdiagonal of the bidiagonal
+decomposition of @var{A} given by @code{gsl_linalg_bidiag_decomp}, into
+the diagonal vector @var{diag} and superdiagonal vector @var{superdiag}.
+@end deftypefun
+
+@node Householder Transformations
+@section Householder Transformations
+@cindex Householder matrix
+@cindex Householder transformation
+@cindex transformation, Householder
+
+A Householder transformation is a rank-1 modification of the identity
+matrix which can be used to zero out selected elements of a vector.  A
+Householder matrix @math{P} takes the form,
+@tex
+\beforedisplay
+$$
+P = I - \tau v v^T
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+P = I - \tau v v^T
+@end example
+
+@end ifinfo
+@noindent
+where @math{v} is a vector (called the @dfn{Householder vector}) and
+@math{\tau = 2/(v^T v)}.  The functions described in this section use the
+rank-1 structure of the Householder matrix to create and apply
+Householder transformations efficiently.
+
+@deftypefun double gsl_linalg_householder_transform (gsl_vector * @var{v})
+This function prepares a Householder transformation @math{P = I - \tau v
+v^T} which can be used to zero all the elements of the input vector except
+the first.  On output the transformation is stored in the vector @var{v}
+and the scalar @math{\tau} is returned.
+@end deftypefun
+
+@deftypefun int gsl_linalg_householder_hm (double tau, const gsl_vector * v, gsl_matrix * A)
+This function applies the Householder matrix @math{P} defined by the
+scalar @var{tau} and the vector @var{v} to the left-hand side of the
+matrix @var{A}. On output the result @math{P A} is stored in @var{A}.
+@end deftypefun
+
+@deftypefun int gsl_linalg_householder_mh (double tau, const gsl_vector * v, gsl_matrix * A)
+This function applies the Householder matrix @math{P} defined by the
+scalar @var{tau} and the vector @var{v} to the right-hand side of the
+matrix @var{A}. On output the result @math{A P} is stored in @var{A}.
+@end deftypefun
+
+@deftypefun int gsl_linalg_householder_hv (double tau, const gsl_vector * v, gsl_vector * w)
+This function applies the Householder transformation @math{P} defined by
+the scalar @var{tau} and the vector @var{v} to the vector @var{w}.  On
+output the result @math{P w} is stored in @var{w}.
+@end deftypefun
+
+@comment @deftypefun int gsl_linalg_householder_hm1 (double tau, gsl_matrix * A)
+@comment This function applies the Householder transform, defined by the scalar
+@comment @var{tau} and the vector @var{v}, to a matrix being build up from the
+@comment identity matrix, using the first column of @var{A} as a householder vector.
+@comment @end deftypefun
+
+@node Householder solver for linear systems
+@section Householder solver for linear systems
+@cindex solution of linear system by Householder transformations
+@cindex Householder linear solver
+
+@deftypefun int gsl_linalg_HH_solve (gsl_matrix * @var{A}, const gsl_vector * @var{b}, gsl_vector * @var{x})
+This function solves the system @math{A x = b} directly using
+Householder transformations. On output the solution is stored in @var{x}
+and @var{b} is not modified. The matrix @var{A} is destroyed by the
+Householder transformations.
+@end deftypefun
+
+@deftypefun int gsl_linalg_HH_svx (gsl_matrix * @var{A}, gsl_vector * @var{x})
+This function solves the system @math{A x = b} in-place using
+Householder transformations.  On input @var{x} should contain the
+right-hand side @math{b}, which is replaced by the solution on output.  The
+matrix @var{A} is destroyed by the Householder transformations.
+@end deftypefun
+
+@node Tridiagonal Systems
+@section Tridiagonal Systems
+@cindex tridiagonal systems
+
+@deftypefun int gsl_linalg_solve_tridiag (const gsl_vector * @var{diag}, const gsl_vector * @var{e}, const gsl_vector * @var{f}, const gsl_vector * @var{b}, gsl_vector * @var{x})
+This function solves the general @math{N}-by-@math{N} system @math{A x =
+b} where @var{A} is tridiagonal (@c{$N\geq 2$}
+@math{N >= 2}). The super-diagonal and
+sub-diagonal vectors @var{e} and @var{f} must be one element shorter
+than the diagonal vector @var{diag}.  The form of @var{A} for the 4-by-4
+case is shown below,
+@tex
+\beforedisplay
+$$
+A = \pmatrix{d_0&e_0&  0& 0\cr
+             f_0&d_1&e_1& 0\cr
+             0  &f_1&d_2&e_2\cr 
+             0  &0  &f_2&d_3\cr}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+A = ( d_0 e_0  0   0  )
+    ( f_0 d_1 e_1  0  )
+    (  0  f_1 d_2 e_2 )
+    (  0   0  f_2 d_3 )
+@end example
+@end ifinfo
+@end deftypefun
+
+@deftypefun int gsl_linalg_solve_symm_tridiag (const gsl_vector * @var{diag}, const gsl_vector * @var{e}, const gsl_vector * @var{b}, gsl_vector * @var{x})
+This function solves the general @math{N}-by-@math{N} system @math{A x =
+b} where @var{A} is symmetric tridiagonal (@c{$N\geq 2$}
+@math{N >= 2}).  The off-diagonal vector
+@var{e} must be one element shorter than the diagonal vector @var{diag}.
+The form of @var{A} for the 4-by-4 case is shown below,
+@tex
+\beforedisplay
+$$
+A = \pmatrix{d_0&e_0&  0& 0\cr
+             e_0&d_1&e_1& 0\cr
+             0  &e_1&d_2&e_2\cr 
+             0  &0  &e_2&d_3\cr}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+A = ( d_0 e_0  0   0  )
+    ( e_0 d_1 e_1  0  )
+    (  0  e_1 d_2 e_2 )
+    (  0   0  e_2 d_3 )
+@end example
+@end ifinfo
+The current implementation uses a variant of Cholesky decomposition
+which can cause division by zero if the matrix is not positive definite.
+@end deftypefun
+
+@deftypefun int gsl_linalg_solve_cyc_tridiag (const gsl_vector * @var{diag}, const gsl_vector * @var{e}, const gsl_vector * @var{f}, const gsl_vector * @var{b}, gsl_vector * @var{x})
+This function solves the general @math{N}-by-@math{N} system @math{A x =
+b} where @var{A} is cyclic tridiagonal (@c{$N\geq 3$}
+@math{N >= 3}).  The cyclic super-diagonal and
+sub-diagonal vectors @var{e} and @var{f} must have the same number of
+elements as the diagonal vector @var{diag}.  The form of @var{A} for the
+4-by-4 case is shown below,
+@tex
+\beforedisplay
+$$
+A = \pmatrix{d_0&e_0& 0 &f_3\cr
+             f_0&d_1&e_1& 0 \cr
+              0 &f_1&d_2&e_2\cr 
+             e_3& 0 &f_2&d_3\cr}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+A = ( d_0 e_0  0  f_3 )
+    ( f_0 d_1 e_1  0  )
+    (  0  f_1 d_2 e_2 )
+    ( e_3  0  f_2 d_3 )
+@end example
+@end ifinfo
+@end deftypefun
+
+
+@deftypefun int gsl_linalg_solve_symm_cyc_tridiag (const gsl_vector * @var{diag}, const gsl_vector * @var{e}, const gsl_vector * @var{b}, gsl_vector * @var{x})
+This function solves the general @math{N}-by-@math{N} system @math{A x =
+b} where @var{A} is symmetric cyclic tridiagonal (@c{$N\geq 3$}
+@math{N >= 3}).  The cyclic
+off-diagonal vector @var{e} must have the same number of elements as the
+diagonal vector @var{diag}.  The form of @var{A} for the 4-by-4 case is
+shown below,
+@tex
+\beforedisplay
+$$
+A = \pmatrix{d_0&e_0& 0 &e_3\cr
+             e_0&d_1&e_1& 0 \cr
+              0 &e_1&d_2&e_2\cr 
+             e_3& 0 &e_2&d_3\cr}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+A = ( d_0 e_0  0  e_3 )
+    ( e_0 d_1 e_1  0  )
+    (  0  e_1 d_2 e_2 )
+    ( e_3  0  e_2 d_3 )
+@end example
+@end ifinfo
+@end deftypefun
+
+@node Balancing
+@section Balancing
+@cindex balancing matrices
+
+The process of balancing a matrix applies similarity transformations
+to make the rows and columns have comparable norms. This is
+useful, for example, to reduce roundoff errors in the solution
+of eigenvalue problems. Balancing a matrix @math{A} consists
+of replacing @math{A} with a similar matrix
+@tex
+\beforedisplay
+$$
+A' = D^{-1} A D
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+A' = D^(-1) A D
+@end example
+
+@end ifinfo
+where @math{D} is a diagonal matrix whose entries are powers
+of the floating point radix.
+
+@deftypefun int gsl_linalg_balance_matrix (gsl_matrix * @var{A}, gsl_vector * @var{D})
+This function replaces the matrix @var{A} with its balanced counterpart
+and stores the diagonal elements of the similarity transformation
+into the vector @var{D}.
+@end deftypefun
+
+@node Linear Algebra Examples
+@section Examples
+
+The following program solves the linear system @math{A x = b}. The
+system to be solved is,
+@tex
+\beforedisplay
+$$
+\left(
+\matrix{0.18& 0.60& 0.57& 0.96\cr
+0.41& 0.24& 0.99& 0.58\cr
+0.14& 0.30& 0.97& 0.66\cr
+0.51& 0.13& 0.19& 0.85}
+\right)
+\left(
+\matrix{x_0\cr
+x_1\cr
+x_2\cr
+x_3}
+\right)
+=
+\left(
+\matrix{1.0\cr
+2.0\cr
+3.0\cr
+4.0}
+\right)
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+[ 0.18 0.60 0.57 0.96 ] [x0]   [1.0]
+[ 0.41 0.24 0.99 0.58 ] [x1] = [2.0]
+[ 0.14 0.30 0.97 0.66 ] [x2]   [3.0]
+[ 0.51 0.13 0.19 0.85 ] [x3]   [4.0]
+@end example
+
+@end ifinfo
+@noindent
+and the solution is found using LU decomposition of the matrix @math{A}.
+
+@example
+@verbatiminclude examples/linalglu.c
+@end example
+
+@noindent
+Here is the output from the program,
+
+@example
+@verbatiminclude examples/linalglu.out
+@end example
+
+@noindent
+This can be verified by multiplying the solution @math{x} by the
+original matrix @math{A} using @sc{gnu octave},
+
+@example
+octave> A = [ 0.18, 0.60, 0.57, 0.96;
+              0.41, 0.24, 0.99, 0.58; 
+              0.14, 0.30, 0.97, 0.66; 
+              0.51, 0.13, 0.19, 0.85 ];
+
+octave> x = [ -4.05205; -12.6056; 1.66091; 8.69377];
+
+octave> A * x
+ans =
+  1.0000
+  2.0000
+  3.0000
+  4.0000
+@end example
+
+@noindent
+This reproduces the original right-hand side vector, @math{b}, in
+accordance with the equation @math{A x = b}.
+
+@node Linear Algebra References and Further Reading
+@section References and Further Reading
+
+Further information on the algorithms described in this section can be
+found in the following book,
+
+@itemize @asis
+@item
+G. H. Golub, C. F. Van Loan, @cite{Matrix Computations} (3rd Ed, 1996),
+Johns Hopkins University Press, ISBN 0-8018-5414-8.
+@end itemize
+
+@noindent
+The @sc{lapack} library is described in the following manual,
+
+@itemize @asis
+@item
+@cite{LAPACK Users' Guide} (Third Edition, 1999), Published by SIAM,
+ISBN 0-89871-447-8.
+
+@uref{http://www.netlib.org/lapack} 
+@end itemize
+
+@noindent
+The @sc{lapack} source code can be found at the website above, along
+with an online copy of the users guide.
+
+@noindent
+The Modified Golub-Reinsch algorithm is described in the following paper,
+
+@itemize @asis
+@item
+T.F. Chan, ``An Improved Algorithm for Computing the Singular Value
+Decomposition'', @cite{ACM Transactions on Mathematical Software}, 8
+(1982), pp 72--83.
+@end itemize
+
+@noindent
+The Jacobi algorithm for singular value decomposition is described in
+the following papers,
+
+@itemize @asis
+@item
+J.C. Nash, ``A one-sided transformation method for the singular value
+decomposition and algebraic eigenproblem'', @cite{Computer Journal},
+Volume 18, Number 1 (1973), p 74--76
+
+@item
+James Demmel, Kresimir Veselic, ``Jacobi's Method is more accurate than
+QR'', @cite{Lapack Working Note 15} (LAWN-15), October 1989. Available
+from netlib, @uref{http://www.netlib.org/lapack/} in the @code{lawns} or
+@code{lawnspdf} directories.
+@end itemize
+
+
+
diff --git a/gsl-1.9/doc/math.texi b/gsl-1.9/doc/math.texi
new file mode 100644
index 0000000..5f28d71
--- /dev/null
+++ b/gsl-1.9/doc/math.texi
@@ -0,0 +1,339 @@
+@cindex elementary functions
+@cindex mathematical functions, elementary
+
+This chapter describes basic mathematical functions.  Some of these
+functions are present in system libraries, but the alternative versions
+given here can be used as a substitute when the system functions are not
+available.
+
+The functions and macros described in this chapter are defined in the
+header file @file{gsl_math.h}.
+
+@menu
+* Mathematical Constants::      
+* Infinities and Not-a-number::  
+* Elementary Functions::        
+* Small integer powers::        
+* Testing the Sign of Numbers::  
+* Testing for Odd and Even Numbers::  
+* Maximum and Minimum functions::  
+* Approximate Comparison of Floating Point Numbers::  
+@end menu
+
+@node Mathematical Constants
+@section Mathematical Constants
+@cindex mathematical constants, defined as macros
+@cindex numerical constants, defined as macros
+@cindex constants, mathematical---defined as macros
+@cindex macros for mathematical constants
+The library ensures that the standard @sc{bsd} mathematical constants
+are defined. For reference, here is a list of the constants:
+
+@table @code
+@item M_E
+@cindex e, defined as a macro
+The base of exponentials, @math{e}
+
+@item M_LOG2E
+The base-2 logarithm of @math{e}, @math{\log_2 (e)}
+
+@item M_LOG10E
+The base-10 logarithm of @math{e}, @c{$\log_{10}(e)$}
+@math{\log_10 (e)}
+
+@item M_SQRT2
+The square root of two, @math{\sqrt 2}
+
+@item M_SQRT1_2
+The square root of one-half, @c{$\sqrt{1/2}$}
+@math{\sqrt@{1/2@}}
+
+@item M_SQRT3
+The square root of three, @math{\sqrt 3}
+
+@item M_PI
+@cindex pi, defined as a macro
+The constant pi, @math{\pi}
+
+@item M_PI_2
+Pi divided by two, @math{\pi/2}
+
+@item M_PI_4
+Pi divided by four, @math{\pi/4}
+
+@item M_SQRTPI
+The square root of pi, @math{\sqrt\pi}
+
+@item M_2_SQRTPI
+Two divided by the square root of pi, @math{2/\sqrt\pi}
+
+@item M_1_PI
+The reciprocal of pi, @math{1/\pi}
+
+@item M_2_PI
+Twice the reciprocal of pi, @math{2/\pi}
+
+@item M_LN10
+The natural logarithm of ten, @math{\ln(10)}
+
+@item M_LN2
+The natural logarithm of two, @math{\ln(2)}
+
+@item M_LNPI
+The natural logarithm of pi, @math{\ln(\pi)}
+
+@item M_EULER
+@cindex Euler's constant, defined as a macro
+Euler's constant, @math{\gamma}
+
+@end table
+
+@node Infinities and Not-a-number
+@section Infinities and Not-a-number
+
+@cindex infinity, defined as a macro
+@cindex IEEE infinity, defined as a macro
+@cindex NaN, defined as a macro
+@cindex Not-a-number, defined as a macro
+@cindex IEEE NaN, defined as a macro
+
+@defmac GSL_POSINF
+This macro contains the IEEE representation of positive infinity,
+@math{+\infty}. It is computed from the expression @code{+1.0/0.0}.
+@end defmac
+
+@defmac GSL_NEGINF
+This macro contains the IEEE representation of negative infinity,
+@math{-\infty}. It is computed from the expression @code{-1.0/0.0}.
+@end defmac
+
+@defmac GSL_NAN
+This macro contains the IEEE representation of the Not-a-Number symbol,
+@code{NaN}. It is computed from the ratio @code{0.0/0.0}.
+@end defmac
+
+@deftypefun int gsl_isnan (const double @var{x})
+This function returns 1 if @var{x} is not-a-number.
+@end deftypefun
+
+@deftypefun int gsl_isinf (const double @var{x})
+This function returns @math{+1} if @var{x} is positive infinity,
+@math{-1} if @var{x} is negative infinity and 0 otherwise.
+@end deftypefun
+
+@deftypefun int gsl_finite (const double @var{x})
+This function returns 1 if @var{x} is a real number, and 0 if it is
+infinite or not-a-number.
+@end deftypefun
+
+
+@node Elementary Functions
+@section Elementary Functions
+
+The following routines provide portable implementations of functions
+found in the BSD math library.  When native versions are not available
+the functions described here can be used instead.  The substitution can
+be made automatically if you use @code{autoconf} to compile your
+application (@pxref{Portability functions}).
+
+@deftypefun double gsl_log1p (const double @var{x})
+@cindex log1p
+@cindex logarithm, computed accurately near 1
+This function computes the value of @math{\log(1+x)} in a way that is
+accurate for small @var{x}. It provides an alternative to the BSD math
+function @code{log1p(x)}.
+@end deftypefun
+
+@deftypefun double gsl_expm1 (const double @var{x})
+@cindex expm1
+@cindex exponential, difference from 1 computed accurately
+This function computes the value of @math{\exp(x)-1} in a way that is
+accurate for small @var{x}. It provides an alternative to the BSD math
+function @code{expm1(x)}.
+@end deftypefun
+
+@deftypefun double gsl_hypot (const double @var{x}, const double @var{y})
+@cindex hypot
+@cindex euclidean distance function, hypot
+@cindex length, computed accurately using hypot
+This function computes the value of
+@c{$\sqrt{x^2 + y^2}$}
+@math{\sqrt@{x^2 + y^2@}} in a way that avoids overflow. It provides an
+alternative to the BSD math function @code{hypot(x,y)}.
+@end deftypefun
+
+@deftypefun double gsl_acosh (const double @var{x})
+@cindex acosh
+@cindex hyperbolic cosine, inverse
+@cindex inverse hyperbolic cosine
+This function computes the value of @math{\arccosh(x)}. It provides an
+alternative to the standard math function @code{acosh(x)}.
+@end deftypefun
+
+@deftypefun double gsl_asinh (const double @var{x})
+@cindex asinh
+@cindex hyperbolic sine, inverse
+@cindex inverse hyperbolic sine
+This function computes the value of @math{\arcsinh(x)}. It provides an
+alternative to the standard math function @code{asinh(x)}.
+@end deftypefun
+
+@deftypefun double gsl_atanh (const double @var{x})
+@cindex atanh
+@cindex hyperbolic tangent, inverse
+@cindex inverse hyperbolic tangent
+This function computes the value of @math{\arctanh(x)}. It provides an
+alternative to the standard math function @code{atanh(x)}.
+@end deftypefun
+
+
+@deftypefun double gsl_ldexp (double @var{x}, int @var{e})
+@cindex ldexp
+This function computes the value of @math{x * 2^e}. It provides an
+alternative to the standard math function @code{ldexp(x,e)}.
+@end deftypefun
+
+@deftypefun double gsl_frexp (double @var{x}, int * @var{e})
+@cindex frexp
+This function splits the number @math{x} into its normalized fraction
+@math{f} and exponent @math{e}, such that @math{x = f * 2^e} and
+@c{$0.5 \le f < 1$}
+@math{0.5 <= f < 1}. The function returns @math{f} and stores the
+exponent in @math{e}. If @math{x} is zero, both @math{f} and @math{e}
+are set to zero. This function provides an alternative to the standard
+math function @code{frexp(x, e)}.
+@end deftypefun
+
+@node Small integer powers
+@section Small integer powers
+
+A common complaint about the standard C library is its lack of a
+function for calculating (small) integer powers.  GSL provides some simple
+functions to fill this gap.  For reasons of efficiency, these functions
+do not check for overflow or underflow conditions. 
+
+@deftypefun double gsl_pow_int (double @var{x}, int @var{n})
+This routine computes the power @math{x^n} for integer @var{n}.  The
+power is computed efficiently---for example, @math{x^8} is computed as
+@math{((x^2)^2)^2}, requiring only 3 multiplications.  A version of this
+function which also computes the numerical error in the result is
+available as @code{gsl_sf_pow_int_e}.
+@end deftypefun
+
+@deftypefun double gsl_pow_2 (const double @var{x})
+@deftypefunx double gsl_pow_3 (const double @var{x})
+@deftypefunx double gsl_pow_4 (const double @var{x})
+@deftypefunx double gsl_pow_5 (const double @var{x})
+@deftypefunx double gsl_pow_6 (const double @var{x})
+@deftypefunx double gsl_pow_7 (const double @var{x})
+@deftypefunx double gsl_pow_8 (const double @var{x})
+@deftypefunx double gsl_pow_9 (const double @var{x})
+These functions can be used to compute small integer powers @math{x^2},
+@math{x^3}, etc. efficiently. The functions will be inlined when
+possible so that use of these functions should be as efficient as
+explicitly writing the corresponding product expression.
+@end deftypefun
+
+@example
+#include <gsl/gsl_math.h>
+double y = gsl_pow_4 (3.141)  /* compute 3.141**4 */
+@end example
+
+@node Testing the Sign of Numbers
+@section Testing the Sign of Numbers
+
+@defmac GSL_SIGN (x)
+This macro returns the sign of @var{x}. It is defined as @code{((x) >= 0
+? 1 : -1)}. Note that with this definition the sign of zero is positive
+(regardless of its @sc{ieee} sign bit).
+@end defmac
+
+@node Testing for Odd and Even Numbers
+@section Testing for Odd and Even Numbers
+
+@defmac GSL_IS_ODD (n)
+This macro evaluates to 1 if @var{n} is odd and 0 if @var{n} is
+even. The argument @var{n} must be of integer type.
+@end defmac
+
+@defmac GSL_IS_EVEN (n)
+This macro is the opposite of @code{GSL_IS_ODD(n)}. It evaluates to 1 if
+@var{n} is even and 0 if @var{n} is odd. The argument @var{n} must be of
+integer type.
+@end defmac
+
+@node Maximum and Minimum functions
+@section Maximum and Minimum functions
+
+@defmac GSL_MAX (a, b)
+@cindex maximum of two numbers
+This macro returns the maximum of @var{a} and @var{b}. It is defined as 
+@code{((a) > (b) ? (a):(b))}.
+@end defmac
+
+@defmac GSL_MIN (a, b)
+@cindex minimum of two numbers
+This macro returns the minimum of @var{a} and @var{b}. It is defined as 
+@code{((a) < (b) ? (a):(b))}.
+@end defmac
+
+@deftypefun {extern inline double} GSL_MAX_DBL (double @var{a}, double @var{b})
+This function returns the maximum of the double precision numbers
+@var{a} and @var{b} using an inline function. The use of a function
+allows for type checking of the arguments as an extra safety feature. On
+platforms where inline functions are not available the macro
+@code{GSL_MAX} will be automatically substituted.
+@end deftypefun
+
+@deftypefun {extern inline double} GSL_MIN_DBL (double @var{a}, double @var{b})
+This function returns the minimum of the double precision numbers
+@var{a} and @var{b} using an inline function. The use of a function
+allows for type checking of the arguments as an extra safety feature. On
+platforms where inline functions are not available the macro
+@code{GSL_MIN} will be automatically substituted.
+@end deftypefun
+
+@deftypefun {extern inline int} GSL_MAX_INT (int @var{a}, int @var{b})
+@deftypefunx {extern inline int} GSL_MIN_INT (int @var{a}, int @var{b})
+These functions return the maximum or minimum of the integers @var{a}
+and @var{b} using an inline function.  On platforms where inline
+functions are not available the macros @code{GSL_MAX} or @code{GSL_MIN}
+will be automatically substituted.
+@end deftypefun
+
+@deftypefun {extern inline long double} GSL_MAX_LDBL (long double @var{a}, long double @var{b})
+@deftypefunx {extern inline long double} GSL_MIN_LDBL (long double @var{a}, long double @var{b})
+These functions return the maximum or minimum of the long doubles @var{a}
+and @var{b} using an inline function.  On platforms where inline
+functions are not available the macros @code{GSL_MAX} or @code{GSL_MIN}
+will be automatically substituted.
+@end deftypefun
+
+@node Approximate Comparison of Floating Point Numbers
+@section Approximate Comparison of Floating Point Numbers
+
+It is sometimes useful to be able to compare two floating point numbers
+approximately, to allow for rounding and truncation errors.  The following
+function implements the approximate floating-point comparison algorithm
+proposed by D.E. Knuth in Section 4.2.2 of @cite{Seminumerical
+Algorithms} (3rd edition).
+
+@deftypefun int gsl_fcmp (double @var{x}, double @var{y}, double @var{epsilon})
+@cindex approximate comparison of floating point numbers
+@cindex safe comparison of floating point numbers
+@cindex floating point numbers, approximate comparison
+This function determines whether @math{x} and @math{y} are approximately
+equal to a relative accuracy @var{epsilon}.
+
+The relative accuracy is measured using an interval of size @math{2
+\delta}, where @math{\delta = 2^k \epsilon} and @math{k} is the
+maximum base-2 exponent of @math{x} and @math{y} as computed by the
+function @code{frexp}.  
+
+If @math{x} and @math{y} lie within this interval, they are considered
+approximately equal and the function returns 0. Otherwise if @math{x <
+y}, the function returns @math{-1}, or if @math{x > y}, the function returns
+@math{+1}.
+
+The implementation is based on the package @code{fcmp} by T.C. Belding.
+@end deftypefun
diff --git a/gsl-1.9/doc/mdate-sh b/gsl-1.9/doc/mdate-sh
new file mode 100755
index 0000000..4b2efdf
--- /dev/null
+++ b/gsl-1.9/doc/mdate-sh
@@ -0,0 +1,193 @@
+#!/bin/sh
+# Get modification time of a file or directory and pretty-print it.
+
+scriptversion=2005-02-07.09
+
+# Copyright (C) 1995, 1996, 1997, 2003, 2004, 2005 Free Software
+# Foundation, Inc.
+# written by Ulrich Drepper <drepper@gnu.ai.mit.edu>, June 1995
+#
+# This program is free software; you can redistribute it and/or modify
+# it under the terms of the GNU General Public License as published by
+# the Free Software Foundation; either version 2, or (at your option)
+# any later version.
+#
+# This program is distributed in the hope that it will be useful,
+# but WITHOUT ANY WARRANTY; without even the implied warranty of
+# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+# GNU General Public License for more details.
+#
+# You should have received a copy of the GNU General Public License
+# along with this program; if not, write to the Free Software Foundation,
+# Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA.
+
+# As a special exception to the GNU General Public License, if you
+# distribute this file as part of a program that contains a
+# configuration script generated by Autoconf, you may include it under
+# the same distribution terms that you use for the rest of that program.
+
+# This file is maintained in Automake, please report
+# bugs to <bug-automake@gnu.org> or send patches to
+# <automake-patches@gnu.org>.
+
+case $1 in
+  '')
+     echo "$0: No file.  Try \`$0 --help' for more information." 1>&2
+     exit 1;
+     ;;
+  -h | --h*)
+    cat <<\EOF
+Usage: mdate-sh [--help] [--version] FILE
+
+Pretty-print the modification time of FILE.
+
+Report bugs to <bug-automake@gnu.org>.
+EOF
+    exit $?
+    ;;
+  -v | --v*)
+    echo "mdate-sh $scriptversion"
+    exit $?
+    ;;
+esac
+
+# Prevent date giving response in another language.
+LANG=C
+export LANG
+LC_ALL=C
+export LC_ALL
+LC_TIME=C
+export LC_TIME
+
+save_arg1="$1"
+
+# Find out how to get the extended ls output of a file or directory.
+if ls -L /dev/null 1>/dev/null 2>&1; then
+  ls_command='ls -L -l -d'
+else
+  ls_command='ls -l -d'
+fi
+
+# A `ls -l' line looks as follows on OS/2.
+#  drwxrwx---        0 Aug 11  2001 foo
+# This differs from Unix, which adds ownership information.
+#  drwxrwx---   2 root  root      4096 Aug 11  2001 foo
+#
+# To find the date, we split the line on spaces and iterate on words
+# until we find a month.  This cannot work with files whose owner is a
+# user named `Jan', or `Feb', etc.  However, it's unlikely that `/'
+# will be owned by a user whose name is a month.  So we first look at
+# the extended ls output of the root directory to decide how many
+# words should be skipped to get the date.
+
+# On HPUX /bin/sh, "set" interprets "-rw-r--r--" as options, so the "x" below.
+set x`ls -l -d /`
+
+# Find which argument is the month.
+month=
+command=
+until test $month
+do
+  shift
+  # Add another shift to the command.
+  command="$command shift;"
+  case $1 in
+    Jan) month=January; nummonth=1;;
+    Feb) month=February; nummonth=2;;
+    Mar) month=March; nummonth=3;;
+    Apr) month=April; nummonth=4;;
+    May) month=May; nummonth=5;;
+    Jun) month=June; nummonth=6;;
+    Jul) month=July; nummonth=7;;
+    Aug) month=August; nummonth=8;;
+    Sep) month=September; nummonth=9;;
+    Oct) month=October; nummonth=10;;
+    Nov) month=November; nummonth=11;;
+    Dec) month=December; nummonth=12;;
+  esac
+done
+
+# Get the extended ls output of the file or directory.
+set dummy x`eval "$ls_command \"\$save_arg1\""`
+
+# Remove all preceding arguments
+eval $command
+
+# Because of the dummy argument above, month is in $2.
+#
+# On a POSIX system, we should have
+#
+# $# = 5
+# $1 = file size
+# $2 = month
+# $3 = day
+# $4 = year or time
+# $5 = filename
+#
+# On Darwin 7.7.0 and 7.6.0, we have
+#
+# $# = 4
+# $1 = day
+# $2 = month
+# $3 = year or time
+# $4 = filename
+
+# Get the month.
+case $2 in
+  Jan) month=January; nummonth=1;;
+  Feb) month=February; nummonth=2;;
+  Mar) month=March; nummonth=3;;
+  Apr) month=April; nummonth=4;;
+  May) month=May; nummonth=5;;
+  Jun) month=June; nummonth=6;;
+  Jul) month=July; nummonth=7;;
+  Aug) month=August; nummonth=8;;
+  Sep) month=September; nummonth=9;;
+  Oct) month=October; nummonth=10;;
+  Nov) month=November; nummonth=11;;
+  Dec) month=December; nummonth=12;;
+esac
+
+case $3 in
+  ???*) day=$1;;
+  *) day=$3; shift;;
+esac
+
+# Here we have to deal with the problem that the ls output gives either
+# the time of day or the year.
+case $3 in
+  *:*) set `date`; eval year=\$$#
+       case $2 in
+	 Jan) nummonthtod=1;;
+	 Feb) nummonthtod=2;;
+	 Mar) nummonthtod=3;;
+	 Apr) nummonthtod=4;;
+	 May) nummonthtod=5;;
+	 Jun) nummonthtod=6;;
+	 Jul) nummonthtod=7;;
+	 Aug) nummonthtod=8;;
+	 Sep) nummonthtod=9;;
+	 Oct) nummonthtod=10;;
+	 Nov) nummonthtod=11;;
+	 Dec) nummonthtod=12;;
+       esac
+       # For the first six month of the year the time notation can also
+       # be used for files modified in the last year.
+       if (expr $nummonth \> $nummonthtod) > /dev/null;
+       then
+	 year=`expr $year - 1`
+       fi;;
+  *) year=$3;;
+esac
+
+# The result.
+echo $day $month $year
+
+# Local Variables:
+# mode: shell-script
+# sh-indentation: 2
+# eval: (add-hook 'write-file-hooks 'time-stamp)
+# time-stamp-start: "scriptversion="
+# time-stamp-format: "%:y-%02m-%02d.%02H"
+# time-stamp-end: "$"
+# End:
diff --git a/gsl-1.9/doc/min-interval.eps b/gsl-1.9/doc/min-interval.eps
new file mode 100644
index 0000000..96b7913
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+++ b/gsl-1.9/doc/min-interval.eps
@@ -0,0 +1,330 @@
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diff --git a/gsl-1.9/doc/min.texi b/gsl-1.9/doc/min.texi
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+@cindex optimization, see minimization
+@cindex maximization, see minimization
+@cindex minimization, one-dimensional
+@cindex finding minima
+@cindex nonlinear functions, minimization
+
+This chapter describes routines for finding minima of arbitrary
+one-dimensional functions.  The library provides low level components
+for a variety of iterative minimizers and convergence tests.  These can be
+combined by the user to achieve the desired solution, with full access
+to the intermediate steps of the algorithms.  Each class of methods uses
+the same framework, so that you can switch between minimizers at runtime
+without needing to recompile your program.  Each instance of a minimizer
+keeps track of its own state, allowing the minimizers to be used in
+multi-threaded programs.
+
+The header file @file{gsl_min.h} contains prototypes for the
+minimization functions and related declarations.  To use the minimization
+algorithms to find the maximum of a function simply invert its sign.
+
+@menu
+* Minimization Overview::       
+* Minimization Caveats::        
+* Initializing the Minimizer::  
+* Providing the function to minimize::  
+* Minimization Iteration::      
+* Minimization Stopping Parameters::  
+* Minimization Algorithms::     
+* Minimization Examples::       
+* Minimization References and Further Reading::  
+@end menu
+
+@node Minimization Overview
+@section Overview
+@cindex minimization, overview
+
+The minimization algorithms begin with a bounded region known to contain
+a minimum.  The region is described by a lower bound @math{a} and an
+upper bound @math{b}, with an estimate of the location of the minimum
+@math{x}.
+
+@iftex
+@sp 1
+@center @image{min-interval,3.4in}
+@end iftex
+
+@noindent
+The value of the function at @math{x} must be less than the value of the
+function at the ends of the interval,
+@tex
+$$f(a) > f(x) < f(b)$$
+@end tex
+@ifinfo
+
+@example
+f(a) > f(x) < f(b)
+@end example
+
+@end ifinfo
+@noindent
+This condition guarantees that a minimum is contained somewhere within
+the interval.  On each iteration a new point @math{x'} is selected using
+one of the available algorithms.  If the new point is a better estimate
+of the minimum, i.e.@: where @math{f(x') < f(x)}, then the current
+estimate of the minimum @math{x} is updated.  The new point also allows
+the size of the bounded interval to be reduced, by choosing the most
+compact set of points which satisfies the constraint @math{f(a) > f(x) <
+f(b)}.  The interval is reduced until it encloses the true minimum to a
+desired tolerance.  This provides a best estimate of the location of the
+minimum and a rigorous error estimate.
+
+Several bracketing algorithms are available within a single framework.
+The user provides a high-level driver for the algorithm, and the
+library provides the individual functions necessary for each of the
+steps.  There are three main phases of the iteration.  The steps are,
+
+@itemize @bullet
+@item
+initialize minimizer state, @var{s}, for algorithm @var{T}
+
+@item
+update @var{s} using the iteration @var{T}
+
+@item
+test @var{s} for convergence, and repeat iteration if necessary
+@end itemize
+
+@noindent
+The state for the minimizers is held in a @code{gsl_min_fminimizer}
+struct.  The updating procedure uses only function evaluations (not
+derivatives).
+
+@node Minimization Caveats
+@section Caveats
+@cindex minimization, caveats
+
+Note that minimization functions can only search for one minimum at a
+time.  When there are several minima in the search area, the first
+minimum to be found will be returned; however it is difficult to predict
+which of the minima this will be. @emph{In most cases, no error will be
+reported if you try to find a minimum in an area where there is more
+than one.}
+
+With all minimization algorithms it can be difficult to determine the
+location of the minimum to full numerical precision.  The behavior of the
+function in the region of the minimum @math{x^*} can be approximated by
+a Taylor expansion,
+@tex
+$$
+y = f(x^*) + {1 \over 2} f''(x^*) (x - x^*)^2
+$$
+@end tex
+@ifinfo
+
+@example
+y = f(x^*) + (1/2) f''(x^*) (x - x^*)^2
+@end example
+
+@end ifinfo
+@noindent
+and the second term of this expansion can be lost when added to the
+first term at finite precision.  This magnifies the error in locating
+@math{x^*}, making it proportional to @math{\sqrt \epsilon} (where
+@math{\epsilon} is the relative accuracy of the floating point numbers).
+For functions with higher order minima, such as @math{x^4}, the
+magnification of the error is correspondingly worse.  The best that can
+be achieved is to converge to the limit of numerical accuracy in the
+function values, rather than the location of the minimum itself.
+
+@node Initializing the Minimizer
+@section Initializing the Minimizer
+
+@deftypefun {gsl_min_fminimizer *} gsl_min_fminimizer_alloc (const gsl_min_fminimizer_type * @var{T})
+This function returns a pointer to a newly allocated instance of a
+minimizer of type @var{T}.  For example, the following code
+creates an instance of a golden section minimizer,
+
+@example
+const gsl_min_fminimizer_type * T 
+  = gsl_min_fminimizer_goldensection;
+gsl_min_fminimizer * s 
+  = gsl_min_fminimizer_alloc (T);
+@end example
+
+If there is insufficient memory to create the minimizer then the function
+returns a null pointer and the error handler is invoked with an error
+code of @code{GSL_ENOMEM}.
+@end deftypefun
+
+@deftypefun int gsl_min_fminimizer_set (gsl_min_fminimizer * @var{s}, gsl_function * @var{f}, double @var{x_minimum}, double @var{x_lower}, double @var{x_upper})
+This function sets, or resets, an existing minimizer @var{s} to use the
+function @var{f} and the initial search interval [@var{x_lower},
+@var{x_upper}], with a guess for the location of the minimum
+@var{x_minimum}.
+
+If the interval given does not contain a minimum, then the function
+returns an error code of @code{GSL_EINVAL}.
+@end deftypefun
+
+@deftypefun int gsl_min_fminimizer_set_with_values (gsl_min_fminimizer * @var{s}, gsl_function * @var{f}, double @var{x_minimum}, double @var{f_minimum}, double @var{x_lower}, double @var{f_lower}, double @var{x_upper}, double @var{f_upper})
+This function is equivalent to @code{gsl_min_fminimizer_set} but uses
+the values @var{f_minimum}, @var{f_lower} and @var{f_upper} instead of
+computing @code{f(x_minimum)}, @code{f(x_lower)} and @code{f(x_upper)}.
+@end deftypefun
+
+
+@deftypefun void gsl_min_fminimizer_free (gsl_min_fminimizer * @var{s})
+This function frees all the memory associated with the minimizer
+@var{s}.
+@end deftypefun
+
+@deftypefun {const char *} gsl_min_fminimizer_name (const gsl_min_fminimizer * @var{s})
+This function returns a pointer to the name of the minimizer.  For example,
+
+@example
+printf ("s is a '%s' minimizer\n",
+        gsl_min_fminimizer_name (s));
+@end example
+
+@noindent
+would print something like @code{s is a 'brent' minimizer}.
+@end deftypefun
+
+@node Providing the function to minimize
+@section Providing the function to minimize
+@cindex minimization, providing a function to minimize
+
+You must provide a continuous function of one variable for the
+minimizers to operate on.  In order to allow for general parameters the
+functions are defined by a @code{gsl_function} data type
+(@pxref{Providing the function to solve}).
+
+@node Minimization Iteration
+@section Iteration
+
+The following functions drive the iteration of each algorithm.  Each
+function performs one iteration to update the state of any minimizer of the
+corresponding type.  The same functions work for all minimizers so that
+different methods can be substituted at runtime without modifications to
+the code.
+
+@deftypefun int gsl_min_fminimizer_iterate (gsl_min_fminimizer * @var{s})
+This function performs a single iteration of the minimizer @var{s}.  If the
+iteration encounters an unexpected problem then an error code will be
+returned,
+
+@table @code
+@item GSL_EBADFUNC
+the iteration encountered a singular point where the function evaluated
+to @code{Inf} or @code{NaN}.
+
+@item GSL_FAILURE
+the algorithm could not improve the current best approximation or
+bounding interval.
+@end table
+@end deftypefun
+
+The minimizer maintains a current best estimate of the position of the
+minimum at all times, and the current interval bounding the minimum.
+This information can be accessed with the following auxiliary functions,
+
+@deftypefun double gsl_min_fminimizer_x_minimum (const gsl_min_fminimizer * @var{s})
+This function returns the current estimate of the position of the
+minimum for the minimizer @var{s}.
+@end deftypefun
+
+@deftypefun double gsl_min_fminimizer_x_upper (const gsl_min_fminimizer * @var{s})
+@deftypefunx double gsl_min_fminimizer_x_lower (const gsl_min_fminimizer * @var{s})
+These functions return the current upper and lower bound of the interval
+for the minimizer @var{s}.
+@end deftypefun
+
+@deftypefun double gsl_min_fminimizer_f_minimum (const gsl_min_fminimizer * @var{s})
+@deftypefunx double gsl_min_fminimizer_f_upper (const gsl_min_fminimizer * @var{s})
+@deftypefunx double gsl_min_fminimizer_f_lower (const gsl_min_fminimizer * @var{s})
+These functions return the value of the function at the current estimate
+of the minimum and at the upper and lower bounds of the interval for the
+minimizer @var{s}.
+@end deftypefun
+
+@node Minimization Stopping Parameters
+@section Stopping Parameters
+@cindex minimization, stopping parameters
+
+A minimization procedure should stop when one of the following
+conditions is true:
+
+@itemize @bullet
+@item
+A minimum has been found to within the user-specified precision.
+
+@item
+A user-specified maximum number of iterations has been reached.
+
+@item
+An error has occurred.
+@end itemize
+
+@noindent
+The handling of these conditions is under user control.  The function
+below allows the user to test the precision of the current result.
+
+@deftypefun int gsl_min_test_interval (double @var{x_lower}, double @var{x_upper},  double @var{epsabs}, double @var{epsrel})
+This function tests for the convergence of the interval [@var{x_lower},
+@var{x_upper}] with absolute error @var{epsabs} and relative error
+@var{epsrel}.  The test returns @code{GSL_SUCCESS} if the following
+condition is achieved,
+@tex
+\beforedisplay
+$$
+|a - b| < \hbox{\it epsabs} + \hbox{\it epsrel\/}\, \min(|a|,|b|)
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+|a - b| < epsabs + epsrel min(|a|,|b|) 
+@end example
+
+@end ifinfo
+@noindent
+when the interval @math{x = [a,b]} does not include the origin.  If the
+interval includes the origin then @math{\min(|a|,|b|)} is replaced by
+zero (which is the minimum value of @math{|x|} over the interval).  This
+ensures that the relative error is accurately estimated for minima close
+to the origin.
+
+This condition on the interval also implies that any estimate of the
+minimum @math{x_m} in the interval satisfies the same condition with respect
+to the true minimum @math{x_m^*},
+@tex
+\beforedisplay
+$$
+|x_m - x_m^*| < \hbox{\it epsabs} + \hbox{\it epsrel\/}\, x_m^*
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+|x_m - x_m^*| < epsabs + epsrel x_m^*
+@end example
+
+@end ifinfo
+@noindent
+assuming that the true minimum @math{x_m^*} is contained within the interval.
+@end deftypefun
+
+@comment ============================================================
+
+@node Minimization Algorithms
+@section Minimization Algorithms
+
+The minimization algorithms described in this section require an initial
+interval which is guaranteed to contain a minimum---if @math{a} and
+@math{b} are the endpoints of the interval and @math{x} is an estimate
+of the minimum then @math{f(a) > f(x) < f(b)}.  This ensures that the
+function has at least one minimum somewhere in the interval.  If a valid
+initial interval is used then these algorithm cannot fail, provided the
+function is well-behaved.
+
+@deffn {Minimizer} gsl_min_fminimizer_goldensection
+
+@cindex golden section algorithm for finding minima
+@cindex minimum finding, golden section algorithm
+
+The @dfn{golden section algorithm} is the simplest method of bracketing
+the minimum of a function.  It is the slowest algorithm provided by the
+library, with linear convergence.
+
+On each iteration, the algorithm first compares the subintervals from
+the endpoints to the current minimum.  The larger subinterval is divided
+in a golden section (using the famous ratio @math{(3-\sqrt 5)/2 =
+0.3189660}@dots{}) and the value of the function at this new point is
+calculated.  The new value is used with the constraint @math{f(a') >
+f(x') < f(b')} to a select new interval containing the minimum, by
+discarding the least useful point.  This procedure can be continued
+indefinitely until the interval is sufficiently small.  Choosing the
+golden section as the bisection ratio can be shown to provide the
+fastest convergence for this type of algorithm.
+
+@end deffn
+
+@comment ============================================================
+
+@deffn {Minimizer} gsl_min_fminimizer_brent
+@cindex Brent's method for finding minima
+@cindex minimum finding, Brent's method
+
+The @dfn{Brent minimization algorithm} combines a parabolic
+interpolation with the golden section algorithm.  This produces a fast
+algorithm which is still robust.
+
+The outline of the algorithm can be summarized as follows: on each
+iteration Brent's method approximates the function using an
+interpolating parabola through three existing points.  The minimum of the
+parabola is taken as a guess for the minimum.  If it lies within the
+bounds of the current interval then the interpolating point is accepted,
+and used to generate a smaller interval.  If the interpolating point is
+not accepted then the algorithm falls back to an ordinary golden section
+step.  The full details of Brent's method include some additional checks
+to improve convergence.
+@end deffn
+
+@comment ============================================================
+
+@node Minimization Examples
+@section Examples
+
+The following program uses the Brent algorithm to find the minimum of
+the function @math{f(x) = \cos(x) + 1}, which occurs at @math{x = \pi}.
+The starting interval is @math{(0,6)}, with an initial guess for the
+minimum of @math{2}.
+
+@example
+@verbatiminclude examples/min.c
+@end example
+
+@noindent
+Here are the results of the minimization procedure.
+
+@smallexample
+$ ./a.out 
+@verbatiminclude examples/min.out
+@end smallexample
+
+@node Minimization References and Further Reading
+@section References and Further Reading
+
+Further information on Brent's algorithm is available in the following
+book,
+
+@itemize @asis
+@item
+Richard Brent, @cite{Algorithms for minimization without derivatives},
+Prentice-Hall (1973), republished by Dover in paperback (2002), ISBN
+0-486-41998-3.
+@end itemize
+
diff --git a/gsl-1.9/doc/montecarlo.texi b/gsl-1.9/doc/montecarlo.texi
new file mode 100644
index 0000000..0c3aabe
--- /dev/null
+++ b/gsl-1.9/doc/montecarlo.texi
@@ -0,0 +1,718 @@
+@cindex Monte Carlo integration
+@cindex stratified sampling in monte carlo integration
+This chapter describes routines for multidimensional Monte Carlo
+integration.  These include the traditional Monte Carlo method and
+adaptive algorithms such as @sc{vegas} and @sc{miser} which use
+importance sampling and stratified sampling techniques. Each algorithm
+computes an estimate of a multidimensional definite integral of the
+form,
+@tex
+\beforedisplay
+$$
+I = \int_{x_l}^{x_u} dx\,\int_{y_l}^{y_u}dy\,... f(x,y,...)
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+I = \int_xl^xu dx \int_yl^yu  dy ...  f(x, y, ...)
+@end example
+
+@end ifinfo
+@noindent
+over a hypercubic region @math{((x_l,x_u)}, @math{(y_l,y_u), ...)} using
+a fixed number of function calls.  The routines also provide a
+statistical estimate of the error on the result.  This error estimate
+should be taken as a guide rather than as a strict error bound---random 
+sampling of the region may not uncover all the important features
+of the function, resulting in an underestimate of the error.
+
+The functions are defined in separate header files for each routine,
+@code{gsl_monte_plain.h}, @file{gsl_monte_miser.h} and
+@file{gsl_monte_vegas.h}.
+
+@menu
+* Monte Carlo Interface::       
+* PLAIN Monte Carlo::  
+* MISER::                       
+* VEGAS::                       
+* Monte Carlo Examples::        
+* Monte Carlo Integration References and Further Reading::  
+@end menu
+
+@node Monte Carlo Interface
+@section Interface
+All of the Monte Carlo integration routines use the same general form of
+interface.  There is an allocator to allocate memory for control
+variables and workspace, a routine to initialize those control
+variables, the integrator itself, and a function to free the space when
+done.
+
+Each integration function requires a random number generator to be
+supplied, and returns an estimate of the integral and its standard
+deviation.  The accuracy of the result is determined by the number of
+function calls specified by the user.  If a known level of accuracy is
+required this can be achieved by calling the integrator several times
+and averaging the individual results until the desired accuracy is
+obtained.  
+
+Random sample points used within the Monte Carlo routines are always
+chosen strictly within the integration region, so that endpoint
+singularities are automatically avoided.
+
+The function to be integrated has its own datatype, defined in the
+header file @file{gsl_monte.h}.
+
+@deftp {Data Type} gsl_monte_function 
+
+This data type defines a general function with parameters for Monte
+Carlo integration.
+
+@table @code
+@item double (* f) (double * @var{x}, size_t @var{dim}, void * @var{params})
+this function should return the value
+@c{$f(x,\hbox{\it params})$}
+@math{f(x,params)} for the argument @var{x} and parameters @var{params},
+where @var{x} is an array of size @var{dim} giving the coordinates of
+the point where the function is to be evaluated.
+
+@item size_t dim
+the number of dimensions for @var{x}.
+
+@item void * params
+a pointer to the parameters of the function.
+@end table
+@end deftp
+
+@noindent
+Here is an example for a quadratic function in two dimensions,
+@tex
+\beforedisplay
+$$
+f(x,y) = a x^2 + b x y + c y^2
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+f(x,y) = a x^2 + b x y + c y^2
+@end example
+
+@end ifinfo
+@noindent
+with @math{a = 3}, @math{b = 2}, @math{c = 1}.  The following code
+defines a @code{gsl_monte_function} @code{F} which you could pass to an
+integrator:
+
+@example
+struct my_f_params @{ double a; double b; double c; @};
+
+double
+my_f (double x[], size_t dim, void * p) @{
+   struct my_f_params * fp = (struct my_f_params *)p;
+
+   if (dim != 2)
+      @{
+        fprintf (stderr, "error: dim != 2");
+        abort ();
+      @}
+
+   return  fp->a * x[0] * x[0] 
+             + fp->b * x[0] * x[1] 
+               + fp->c * x[1] * x[1];
+@}
+
+gsl_monte_function F;
+struct my_f_params params = @{ 3.0, 2.0, 1.0 @};
+
+F.f = &my_f;
+F.dim = 2;
+F.params = &params;
+@end example
+
+@noindent
+The function @math{f(x)} can be evaluated using the following macro,
+
+@example
+#define GSL_MONTE_FN_EVAL(F,x) 
+    (*((F)->f))(x,(F)->dim,(F)->params)
+@end example
+
+@node PLAIN Monte Carlo
+@section PLAIN Monte Carlo
+@cindex plain monte carlo
+The plain Monte Carlo algorithm samples points randomly from the
+integration region to estimate the integral and its error.  Using this
+algorithm the estimate of the integral @math{E(f; N)} for @math{N}
+randomly distributed points @math{x_i} is given by,
+@tex
+\beforedisplay
+$$
+E(f; N) =  V \langle f \rangle = {V \over N} \sum_i^N f(x_i)
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+E(f; N) = =  V <f> = (V / N) \sum_i^N f(x_i)
+@end example
+
+@end ifinfo
+@noindent
+where @math{V} is the volume of the integration region.  The error on
+this estimate @math{\sigma(E;N)} is calculated from the estimated
+variance of the mean,
+@tex
+\beforedisplay
+$$
+\sigma^2 (E; N) = {V \over N } \sum_i^N (f(x_i) - \langle f \rangle)^2.
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+\sigma^2 (E; N) = (V / N) \sum_i^N (f(x_i) -  <f>)^2.
+@end example
+
+@end ifinfo
+@noindent
+For large @math{N} this variance decreases asymptotically as
+@math{\Var(f)/N}, where @math{\Var(f)} is the true variance of the
+function over the integration region.  The error estimate itself should
+decrease as @c{$\sigma(f)/\sqrt{N}$}
+@math{\sigma(f)/\sqrt@{N@}}.  The familiar law of errors
+decreasing as @c{$1/\sqrt{N}$}
+@math{1/\sqrt@{N@}} applies---to reduce the error by a
+factor of 10 requires a 100-fold increase in the number of sample
+points.
+
+The functions described in this section are declared in the header file
+@file{gsl_monte_plain.h}.
+
+@deftypefun {gsl_monte_plain_state *} gsl_monte_plain_alloc (size_t @var{dim})
+This function allocates and initializes a workspace for Monte Carlo
+integration in @var{dim} dimensions.  
+@end deftypefun
+
+@deftypefun int gsl_monte_plain_init (gsl_monte_plain_state* @var{s})
+This function initializes a previously allocated integration state.
+This allows an existing workspace to be reused for different
+integrations.
+@end deftypefun
+
+@deftypefun int gsl_monte_plain_integrate (gsl_monte_function * @var{f}, double * @var{xl}, double * @var{xu}, size_t @var{dim}, size_t @var{calls}, gsl_rng * @var{r}, gsl_monte_plain_state * @var{s}, double * @var{result}, double * @var{abserr})
+This routines uses the plain Monte Carlo algorithm to integrate the
+function @var{f} over the @var{dim}-dimensional hypercubic region
+defined by the lower and upper limits in the arrays @var{xl} and
+@var{xu}, each of size @var{dim}.  The integration uses a fixed number
+of function calls @var{calls}, and obtains random sampling points using
+the random number generator @var{r}. A previously allocated workspace
+@var{s} must be supplied.  The result of the integration is returned in
+@var{result}, with an estimated absolute error @var{abserr}.
+@end deftypefun
+
+@deftypefun void gsl_monte_plain_free (gsl_monte_plain_state * @var{s})
+This function frees the memory associated with the integrator state
+@var{s}.
+@end deftypefun
+
+@node MISER
+@section MISER
+@cindex MISER monte carlo integration
+@cindex recursive stratified sampling, MISER
+
+The @sc{miser} algorithm of Press and Farrar is based on recursive
+stratified sampling.  This technique aims to reduce the overall
+integration error by concentrating integration points in the regions of
+highest variance.
+
+The idea of stratified sampling begins with the observation that for two
+disjoint regions @math{a} and @math{b} with Monte Carlo estimates of the
+integral @math{E_a(f)} and @math{E_b(f)} and variances
+@math{\sigma_a^2(f)} and @math{\sigma_b^2(f)}, the variance
+@math{\Var(f)} of the combined estimate 
+@c{$E(f) = {1\over 2} (E_a(f) + E_b(f))$} 
+@math{E(f) = (1/2) (E_a(f) + E_b(f))} 
+is given by,
+@tex
+\beforedisplay
+$$
+\Var(f) = {\sigma_a^2(f) \over 4 N_a} + {\sigma_b^2(f) \over 4 N_b}.
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+\Var(f) = (\sigma_a^2(f) / 4 N_a) + (\sigma_b^2(f) / 4 N_b).
+@end example
+
+@end ifinfo
+@noindent
+It can be shown that this variance is minimized by distributing the
+points such that,
+@tex
+\beforedisplay
+$$
+{N_a \over N_a+N_b} = {\sigma_a \over \sigma_a + \sigma_b}.
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+N_a / (N_a + N_b) = \sigma_a / (\sigma_a + \sigma_b).
+@end example
+
+@end ifinfo
+@noindent
+Hence the smallest error estimate is obtained by allocating sample
+points in proportion to the standard deviation of the function in each
+sub-region.
+
+The @sc{miser} algorithm proceeds by bisecting the integration region
+along one coordinate axis to give two sub-regions at each step.  The
+direction is chosen by examining all @math{d} possible bisections and
+selecting the one which will minimize the combined variance of the two
+sub-regions.  The variance in the sub-regions is estimated by sampling
+with a fraction of the total number of points available to the current
+step.  The same procedure is then repeated recursively for each of the
+two half-spaces from the best bisection. The remaining sample points are
+allocated to the sub-regions using the formula for @math{N_a} and
+@math{N_b}.  This recursive allocation of integration points continues
+down to a user-specified depth where each sub-region is integrated using
+a plain Monte Carlo estimate.  These individual values and their error
+estimates are then combined upwards to give an overall result and an
+estimate of its error.
+
+The functions described in this section are declared in the header file
+@file{gsl_monte_miser.h}.
+
+@deftypefun {gsl_monte_miser_state *} gsl_monte_miser_alloc (size_t @var{dim})
+This function allocates and initializes a workspace for Monte Carlo
+integration in @var{dim} dimensions.  The workspace is used to maintain
+the state of the integration.
+@end deftypefun
+
+@deftypefun int gsl_monte_miser_init (gsl_monte_miser_state* @var{s})
+This function initializes a previously allocated integration state.
+This allows an existing workspace to be reused for different
+integrations.
+@end deftypefun
+
+@deftypefun int gsl_monte_miser_integrate (gsl_monte_function * @var{f}, double * @var{xl}, double * @var{xu}, size_t @var{dim}, size_t @var{calls}, gsl_rng * @var{r}, gsl_monte_miser_state * @var{s}, double * @var{result}, double * @var{abserr})
+This routines uses the @sc{miser} Monte Carlo algorithm to integrate the
+function @var{f} over the @var{dim}-dimensional hypercubic region
+defined by the lower and upper limits in the arrays @var{xl} and
+@var{xu}, each of size @var{dim}.  The integration uses a fixed number
+of function calls @var{calls}, and obtains random sampling points using
+the random number generator @var{r}. A previously allocated workspace
+@var{s} must be supplied.  The result of the integration is returned in
+@var{result}, with an estimated absolute error @var{abserr}.
+@end deftypefun
+
+@deftypefun void gsl_monte_miser_free (gsl_monte_miser_state * @var{s}) 
+This function frees the memory associated with the integrator state
+@var{s}.
+@end deftypefun
+
+The @sc{miser} algorithm has several configurable parameters. The
+following variables can be accessed through the
+@code{gsl_monte_miser_state} struct,
+
+@deftypevar double estimate_frac
+This parameter specifies the fraction of the currently available number of
+function calls which are allocated to estimating the variance at each
+recursive step. The default value is 0.1.
+@end deftypevar
+
+@deftypevar size_t min_calls
+This parameter specifies the minimum number of function calls required
+for each estimate of the variance. If the number of function calls
+allocated to the estimate using @var{estimate_frac} falls below
+@var{min_calls} then @var{min_calls} are used instead.  This ensures
+that each estimate maintains a reasonable level of accuracy.  The
+default value of @var{min_calls} is @code{16 * dim}.
+@end deftypevar
+
+@deftypevar size_t min_calls_per_bisection
+This parameter specifies the minimum number of function calls required
+to proceed with a bisection step.  When a recursive step has fewer calls
+available than @var{min_calls_per_bisection} it performs a plain Monte
+Carlo estimate of the current sub-region and terminates its branch of
+the recursion.  The default value of this parameter is @code{32 *
+min_calls}.
+@end deftypevar
+
+@deftypevar double alpha
+This parameter controls how the estimated variances for the two
+sub-regions of a bisection are combined when allocating points.  With
+recursive sampling the overall variance should scale better than
+@math{1/N}, since the values from the sub-regions will be obtained using
+a procedure which explicitly minimizes their variance.  To accommodate
+this behavior the @sc{miser} algorithm allows the total variance to
+depend on a scaling parameter @math{\alpha},
+@tex
+\beforedisplay
+$$
+\Var(f) = {\sigma_a \over N_a^\alpha} + {\sigma_b \over N_b^\alpha}.
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+\Var(f) = @{\sigma_a \over N_a^\alpha@} + @{\sigma_b \over N_b^\alpha@}.
+@end example
+
+@end ifinfo
+@noindent
+The authors of the original paper describing @sc{miser} recommend the
+value @math{\alpha = 2} as a good choice, obtained from numerical
+experiments, and this is used as the default value in this
+implementation.
+@end deftypevar
+
+@deftypevar double dither
+This parameter introduces a random fractional variation of size
+@var{dither} into each bisection, which can be used to break the
+symmetry of integrands which are concentrated near the exact center of
+the hypercubic integration region.  The default value of dither is zero,
+so no variation is introduced. If needed, a typical value of
+@var{dither} is 0.1.
+@end deftypevar
+
+@node VEGAS
+@section VEGAS
+@cindex VEGAS monte carlo integration
+@cindex importance sampling, VEGAS
+
+The @sc{vegas} algorithm of Lepage is based on importance sampling.  It
+samples points from the probability distribution described by the
+function @math{|f|}, so that the points are concentrated in the regions
+that make the largest contribution to the integral.
+
+In general, if the Monte Carlo integral of @math{f} is sampled with
+points distributed according to a probability distribution described by
+the function @math{g}, we obtain an estimate @math{E_g(f; N)},
+@tex
+\beforedisplay
+$$
+E_g(f; N) = E(f/g; N)
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+E_g(f; N) = E(f/g; N)
+@end example
+
+@end ifinfo
+@noindent
+with a corresponding variance,
+@tex
+\beforedisplay
+$$
+\Var_g(f; N) = \Var(f/g; N).
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+\Var_g(f; N) = \Var(f/g; N).
+@end example
+
+@end ifinfo
+@noindent
+If the probability distribution is chosen as @math{g = |f|/I(|f|)} then
+it can be shown that the variance @math{V_g(f; N)} vanishes, and the
+error in the estimate will be zero.  In practice it is not possible to
+sample from the exact distribution @math{g} for an arbitrary function, so
+importance sampling algorithms aim to produce efficient approximations
+to the desired distribution.
+
+The @sc{vegas} algorithm approximates the exact distribution by making a
+number of passes over the integration region while histogramming the
+function @math{f}. Each histogram is used to define a sampling
+distribution for the next pass.  Asymptotically this procedure converges
+to the desired distribution. In order
+to avoid the number of histogram bins growing like @math{K^d} the
+probability distribution is approximated by a separable function:
+@c{$g(x_1, x_2,\ldots) = g_1(x_1) g_2(x_2)\ldots$} 
+@math{g(x_1, x_2, ...) = g_1(x_1) g_2(x_2) ...}  
+so that the number of bins required is only @math{Kd}.     
+This is equivalent to locating the peaks of the function from the
+projections of the integrand onto the coordinate axes.  The efficiency
+of @sc{vegas} depends on the validity of this assumption.  It is most
+efficient when the peaks of the integrand are well-localized.  If an
+integrand can be rewritten in a form which is approximately separable
+this will increase the efficiency of integration with @sc{vegas}.
+
+@sc{vegas} incorporates a number of additional features, and combines both
+stratified sampling and importance sampling.  The integration region is
+divided into a number of ``boxes'', with each box getting a fixed
+number of points (the goal is 2).  Each box can then have a fractional
+number of bins, but if the ratio of bins-per-box is less than two, Vegas switches to a
+kind variance reduction (rather than importance sampling).
+
+
+@deftypefun {gsl_monte_vegas_state *} gsl_monte_vegas_alloc (size_t @var{dim})
+This function allocates and initializes a workspace for Monte Carlo
+integration in @var{dim} dimensions.  The workspace is used to maintain
+the state of the integration.
+@end deftypefun
+
+@deftypefun int gsl_monte_vegas_init (gsl_monte_vegas_state* @var{s})
+This function initializes a previously allocated integration state.
+This allows an existing workspace to be reused for different
+integrations.
+@end deftypefun
+
+@deftypefun int gsl_monte_vegas_integrate (gsl_monte_function * @var{f}, double * @var{xl}, double * @var{xu}, size_t @var{dim}, size_t @var{calls}, gsl_rng * @var{r}, gsl_monte_vegas_state * @var{s}, double * @var{result}, double * @var{abserr})
+This routines uses the @sc{vegas} Monte Carlo algorithm to integrate the
+function @var{f} over the @var{dim}-dimensional hypercubic region
+defined by the lower and upper limits in the arrays @var{xl} and
+@var{xu}, each of size @var{dim}.  The integration uses a fixed number
+of function calls @var{calls}, and obtains random sampling points using
+the random number generator @var{r}. A previously allocated workspace
+@var{s} must be supplied.  The result of the integration is returned in
+@var{result}, with an estimated absolute error @var{abserr}.  The result
+and its error estimate are based on a weighted average of independent
+samples. The chi-squared per degree of freedom for the weighted average
+is returned via the state struct component, @var{s->chisq}, and must be
+consistent with 1 for the weighted average to be reliable.
+@end deftypefun
+
+@deftypefun void gsl_monte_vegas_free (gsl_monte_vegas_state * @var{s})
+This function frees the memory associated with the integrator state
+@var{s}.
+@end deftypefun
+
+The @sc{vegas} algorithm computes a number of independent estimates of the
+integral internally, according to the @code{iterations} parameter
+described below, and returns their weighted average.  Random sampling of
+the integrand can occasionally produce an estimate where the error is
+zero, particularly if the function is constant in some regions. An
+estimate with zero error causes the weighted average to break down and
+must be handled separately. In the original Fortran implementations of
+@sc{vegas} the error estimate is made non-zero by substituting a small
+value (typically @code{1e-30}).  The implementation in GSL differs from
+this and avoids the use of an arbitrary constant---it either assigns
+the value a weight which is the average weight of the preceding
+estimates or discards it according to the following procedure,
+
+@table @asis
+@item current estimate has zero error, weighted average has finite error
+
+The current estimate is assigned a weight which is the average weight of
+the preceding estimates.
+
+@item current estimate has finite error, previous estimates had zero error
+
+The previous estimates are discarded and the weighted averaging
+procedure begins with the current estimate.
+
+@item current estimate has zero error, previous estimates had zero error
+
+The estimates are averaged using the arithmetic mean, but no error is computed.
+@end table
+
+The @sc{vegas} algorithm is highly configurable. The following variables
+can be accessed through the @code{gsl_monte_vegas_state} struct,
+
+@deftypevar double result
+@deftypevarx double sigma
+These parameters contain the raw value of the integral @var{result} and
+its error @var{sigma} from the last iteration of the algorithm.
+@end deftypevar
+
+@deftypevar double chisq
+This parameter gives the chi-squared per degree of freedom for the
+weighted estimate of the integral.  The value of @var{chisq} should be
+close to 1.  A value of @var{chisq} which differs significantly from 1
+indicates that the values from different iterations are inconsistent.
+In this case the weighted error will be under-estimated, and further
+iterations of the algorithm are needed to obtain reliable results.
+@end deftypevar
+
+@deftypevar double alpha
+The parameter @code{alpha} controls the stiffness of the rebinning
+algorithm.  It is typically set between one and two. A value of zero
+prevents rebinning of the grid.  The default value is 1.5.
+@end deftypevar
+
+@deftypevar size_t iterations
+The number of iterations to perform for each call to the routine. The
+default value is 5 iterations.
+@end deftypevar
+
+@deftypevar int stage
+Setting this determines the @dfn{stage} of the calculation.  Normally,
+@code{stage = 0} which begins with a new uniform grid and empty weighted
+average.  Calling vegas with @code{stage = 1} retains the grid from the
+previous run but discards the weighted average, so that one can ``tune''
+the grid using a relatively small number of points and then do a large
+run with @code{stage = 1} on the optimized grid.  Setting @code{stage =
+2} keeps the grid and the weighted average from the previous run, but
+may increase (or decrease) the number of histogram bins in the grid
+depending on the number of calls available.  Choosing @code{stage = 3}
+enters at the main loop, so that nothing is changed, and is equivalent
+to performing additional iterations in a previous call.
+@end deftypevar
+
+@deftypevar int mode
+The possible choices are @code{GSL_VEGAS_MODE_IMPORTANCE},
+@code{GSL_VEGAS_MODE_STRATIFIED}, @code{GSL_VEGAS_MODE_IMPORTANCE_ONLY}.
+This determines whether @sc{vegas} will use importance sampling or
+stratified sampling, or whether it can pick on its own.  In low
+dimensions @sc{vegas} uses strict stratified sampling (more precisely,
+stratified sampling is chosen if there are fewer than 2 bins per box).
+@end deftypevar
+
+@deftypevar int verbose
+@deftypevarx {FILE *} ostream
+These parameters set the level of information printed by @sc{vegas}. All
+information is written to the stream @var{ostream}.  The default setting
+of @var{verbose} is @code{-1}, which turns off all output.  A
+@var{verbose} value of @code{0} prints summary information about the
+weighted average and final result, while a value of @code{1} also
+displays the grid coordinates.  A value of @code{2} prints information
+from the rebinning procedure for each iteration.
+@end deftypevar
+
+@node Monte Carlo Examples
+@section Examples
+
+The example program below uses the Monte Carlo routines to estimate the
+value of the following 3-dimensional integral from the theory of random
+walks,
+@tex
+\beforedisplay
+$$
+I = \int_{-\pi}^{+\pi} {dk_x \over 2\pi} 
+    \int_{-\pi}^{+\pi} {dk_y \over 2\pi} 
+    \int_{-\pi}^{+\pi} {dk_z \over 2\pi} 
+     { 1 \over (1 - \cos(k_x)\cos(k_y)\cos(k_z))}.
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+I = \int_@{-pi@}^@{+pi@} @{dk_x/(2 pi)@} 
+    \int_@{-pi@}^@{+pi@} @{dk_y/(2 pi)@} 
+    \int_@{-pi@}^@{+pi@} @{dk_z/(2 pi)@} 
+     1 / (1 - cos(k_x)cos(k_y)cos(k_z)).
+@end example
+
+@end ifinfo
+@noindent
+The analytic value of this integral can be shown to be @math{I =
+\Gamma(1/4)^4/(4 \pi^3) = 1.393203929685676859...}.  The integral gives
+the mean time spent at the origin by a random walk on a body-centered
+cubic lattice in three dimensions.
+
+For simplicity we will compute the integral over the region
+@math{(0,0,0)} to @math{(\pi,\pi,\pi)} and multiply by 8 to obtain the
+full result.  The integral is slowly varying in the middle of the region
+but has integrable singularities at the corners @math{(0,0,0)},
+@math{(0,\pi,\pi)}, @math{(\pi,0,\pi)} and @math{(\pi,\pi,0)}.  The
+Monte Carlo routines only select points which are strictly within the
+integration region and so no special measures are needed to avoid these
+singularities.
+
+@smallexample
+@verbatiminclude examples/monte.c
+@end smallexample
+
+@noindent
+With 500,000 function calls the plain Monte Carlo algorithm achieves a
+fractional error of 0.6%.  The estimated error @code{sigma} is
+consistent with the actual error, and the computed result differs from
+the true result by about one standard deviation,
+
+@example
+plain ==================
+result =  1.385867
+sigma  =  0.007938
+exact  =  1.393204
+error  = -0.007337 = 0.9 sigma
+@end example
+
+@noindent
+The @sc{miser} algorithm reduces the error by a factor of two, and also
+correctly estimates the error,
+
+@example
+miser ==================
+result =  1.390656
+sigma  =  0.003743
+exact  =  1.393204
+error  = -0.002548 = 0.7 sigma
+@end example
+
+@noindent
+In the case of the @sc{vegas} algorithm the program uses an initial
+warm-up run of 10,000 function calls to prepare, or ``warm up'', the grid.
+This is followed by a main run with five iterations of 100,000 function
+calls. The chi-squared per degree of freedom for the five iterations are
+checked for consistency with 1, and the run is repeated if the results
+have not converged. In this case the estimates are consistent on the
+first pass.
+
+@example
+vegas warm-up ==================
+result =  1.386925
+sigma  =  0.002651
+exact  =  1.393204
+error  = -0.006278 = 2 sigma
+converging...
+result =  1.392957 sigma =  0.000452 chisq/dof = 1.1
+vegas final ==================
+result =  1.392957
+sigma  =  0.000452
+exact  =  1.393204
+error  = -0.000247 = 0.5 sigma
+@end example
+
+@noindent
+If the value of @code{chisq} had differed significantly from 1 it would
+indicate inconsistent results, with a correspondingly underestimated
+error.  The final estimate from @sc{vegas} (using a similar number of
+function calls) is significantly more accurate than the other two
+algorithms.
+
+@node Monte Carlo Integration References and Further Reading
+@section References and Further Reading
+
+The @sc{miser} algorithm is described in the following article by Press
+and Farrar,
+
+@itemize @asis
+@item
+W.H. Press, G.R. Farrar, @cite{Recursive Stratified Sampling for
+Multidimensional Monte Carlo Integration},
+Computers in Physics, v4 (1990), pp190--195.
+@end itemize
+
+@noindent
+The @sc{vegas} algorithm is described in the following papers,
+
+@itemize @asis
+@item
+G.P. Lepage,
+@cite{A New Algorithm for Adaptive Multidimensional Integration},
+Journal of Computational Physics 27, 192--203, (1978)
+
+@item
+G.P. Lepage,
+@cite{VEGAS: An Adaptive Multi-dimensional Integration Program},
+Cornell preprint CLNS 80-447, March 1980
+@end itemize
+
diff --git a/gsl-1.9/doc/multifit.texi b/gsl-1.9/doc/multifit.texi
new file mode 100644
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+@cindex nonlinear least squares fitting
+@cindex least squares fitting, nonlinear
+
+This chapter describes functions for multidimensional nonlinear
+least-squares fitting.  The library provides low level components for a
+variety of iterative solvers and convergence tests.  These can be
+combined by the user to achieve the desired solution, with full access
+to the intermediate steps of the iteration.  Each class of methods uses
+the same framework, so that you can switch between solvers at runtime
+without needing to recompile your program.  Each instance of a solver
+keeps track of its own state, allowing the solvers to be used in
+multi-threaded programs.
+
+The header file @file{gsl_multifit_nlin.h} contains prototypes for the
+multidimensional nonlinear fitting functions and related declarations.
+
+@menu
+* Overview of Nonlinear Least-Squares Fitting::  
+* Initializing the Nonlinear Least-Squares Solver::  
+* Providing the Function to be Minimized::  
+* Iteration of the Minimization Algorithm::  
+* Search Stopping Parameters for Minimization Algorithms::  
+* Minimization Algorithms using Derivatives::  
+* Minimization Algorithms without Derivatives::  
+* Computing the covariance matrix of best fit parameters::  
+* Example programs for Nonlinear Least-Squares Fitting::  
+* References and Further Reading for Nonlinear Least-Squares Fitting::  
+@end menu
+
+@node Overview of Nonlinear Least-Squares Fitting
+@section Overview
+@cindex nonlinear least squares fitting, overview
+
+The problem of multidimensional nonlinear least-squares fitting requires
+the minimization of the squared residuals of @math{n} functions,
+@math{f_i}, in @math{p} parameters, @math{x_i},
+@tex
+\beforedisplay
+$$
+\Phi(x)  = {1 \over 2} || F(x) ||^2
+         = {1 \over 2} \sum_{i=1}^{n} f_i (x_1, \dots, x_p)^2
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+\Phi(x) = (1/2) || F(x) ||^2
+        = (1/2) \sum_@{i=1@}^@{n@} f_i(x_1, ..., x_p)^2 
+@end example
+
+@end ifinfo
+@noindent
+All algorithms proceed from an initial guess using the linearization,
+@tex
+\beforedisplay
+$$
+\psi(p) = || F(x+p) || \approx || F(x) + J p\, ||
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+\psi(p) = || F(x+p) || ~=~ || F(x) + J p ||
+@end example
+
+@end ifinfo
+@noindent
+where @math{x} is the initial point, @math{p} is the proposed step
+and @math{J} is the
+Jacobian matrix @c{$J_{ij} = \partial f_i / \partial x_j$}
+@math{J_@{ij@} = d f_i / d x_j}.  
+Additional strategies are used to enlarge the region of convergence.
+These include requiring a decrease in the norm @math{||F||} on each
+step or using a trust region to avoid steps which fall outside the linear 
+regime.
+
+To perform a weighted least-squares fit of a nonlinear model
+@math{Y(x,t)} to data (@math{t_i}, @math{y_i}) with independent gaussian
+errors @math{\sigma_i}, use function components of the following form,
+@tex
+\beforedisplay
+$$
+f_i = {(Y(x, t_i) - y_i) \over \sigma_i}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+f_i = (Y(x, t_i) - y_i) / \sigma_i
+@end example
+
+@end ifinfo
+@noindent
+Note that the model parameters are denoted by @math{x} in this chapter
+since the non-linear least-squares algorithms are described
+geometrically (i.e. finding the minimum of a surface).  The
+independent variable of any data to be fitted is denoted by @math{t}.
+
+With the definition above the Jacobian is 
+@c{$J_{ij} = (1 / \sigma_i) \partial Y_i / \partial x_j$}
+@math{J_@{ij@} =(1 / \sigma_i)  d Y_i / d x_j}, where @math{Y_i = Y(x,t_i)}.  
+
+
+@node Initializing the Nonlinear Least-Squares Solver
+@section Initializing the Solver
+
+@deftypefun {gsl_multifit_fsolver *} gsl_multifit_fsolver_alloc (const gsl_multifit_fsolver_type * @var{T}, size_t @var{n}, size_t @var{p})
+This function returns a pointer to a newly allocated instance of a
+solver of type @var{T} for @var{n} observations and @var{p} parameters.
+The number of observations @var{n} must be greater than or equal to
+parameters @var{p}. 
+
+If there is insufficient memory to create the solver then the function
+returns a null pointer and the error handler is invoked with an error
+code of @code{GSL_ENOMEM}.
+@end deftypefun
+
+@deftypefun {gsl_multifit_fdfsolver *} gsl_multifit_fdfsolver_alloc (const gsl_multifit_fdfsolver_type * @var{T}, size_t @var{n}, size_t @var{p})
+This function returns a pointer to a newly allocated instance of a
+derivative solver of type @var{T} for @var{n} observations and @var{p}
+parameters.  For example, the following code creates an instance of a
+Levenberg-Marquardt solver for 100 data points and 3 parameters,
+
+@example
+const gsl_multifit_fdfsolver_type * T 
+    = gsl_multifit_fdfsolver_lmder;
+gsl_multifit_fdfsolver * s 
+    = gsl_multifit_fdfsolver_alloc (T, 100, 3);
+@end example
+
+@noindent
+The number of observations @var{n} must be greater than or equal to
+parameters @var{p}.
+
+If there is insufficient memory to create the solver then the function
+returns a null pointer and the error handler is invoked with an error
+code of @code{GSL_ENOMEM}.
+@end deftypefun
+
+@deftypefun int gsl_multifit_fsolver_set (gsl_multifit_fsolver * @var{s}, gsl_multifit_function * @var{f}, gsl_vector * @var{x})
+This function initializes, or reinitializes, an existing solver @var{s}
+to use the function @var{f} and the initial guess @var{x}.
+@end deftypefun
+
+@deftypefun int gsl_multifit_fdfsolver_set (gsl_multifit_fdfsolver * @var{s}, gsl_multifit_function_fdf * @var{fdf}, gsl_vector * @var{x})
+This function initializes, or reinitializes, an existing solver @var{s}
+to use the function and derivative @var{fdf} and the initial guess
+@var{x}.
+@end deftypefun
+
+@deftypefun void gsl_multifit_fsolver_free (gsl_multifit_fsolver * @var{s})
+@deftypefunx void gsl_multifit_fdfsolver_free (gsl_multifit_fdfsolver * @var{s})
+These functions free all the memory associated with the solver @var{s}.
+@end deftypefun
+
+@deftypefun {const char *} gsl_multifit_fsolver_name (const gsl_multifit_fsolver * @var{s})
+@deftypefunx {const char *} gsl_multifit_fdfsolver_name (const gsl_multifit_fdfsolver * @var{s})
+These functions return a pointer to the name of the solver.  For example,
+
+@example
+printf ("s is a '%s' solver\n", 
+        gsl_multifit_fdfsolver_name (s));
+@end example
+
+@noindent
+would print something like @code{s is a 'lmder' solver}.
+@end deftypefun
+
+@node Providing the Function to be Minimized
+@section Providing the Function to be Minimized
+
+You must provide @math{n} functions of @math{p} variables for the
+minimization algorithms to operate on.  In order to allow for
+arbitrary parameters the functions are defined by the following data
+types:
+
+@deftp {Data Type} gsl_multifit_function 
+This data type defines a general system of functions with arbitrary parameters.  
+
+@table @code
+@item int (* f) (const gsl_vector * @var{x}, void * @var{params}, gsl_vector * @var{f})
+this function should store the vector result
+@c{$f(x,\hbox{\it params})$}
+@math{f(x,params)} in @var{f} for argument @var{x} and arbitrary parameters @var{params},
+returning an appropriate error code if the function cannot be computed.
+
+@item size_t n
+the number of functions, i.e. the number of components of the
+vector @var{f}.
+
+@item size_t p
+the number of independent variables, i.e. the number of components of
+the vector @var{x}.
+
+@item void * params
+a pointer to the arbitrary parameters of the function.
+@end table
+@end deftp
+
+@deftp {Data Type} gsl_multifit_function_fdf
+This data type defines a general system of functions with arbitrary parameters and
+the corresponding Jacobian matrix of derivatives,
+
+@table @code
+@item int (* f) (const gsl_vector * @var{x}, void * @var{params}, gsl_vector * @var{f})
+this function should store the vector result
+@c{$f(x,\hbox{\it params})$}
+@math{f(x,params)} in @var{f} for argument @var{x} and arbitrary parameters @var{params},
+returning an appropriate error code if the function cannot be computed.
+
+@item int (* df) (const gsl_vector * @var{x}, void * @var{params}, gsl_matrix * @var{J})
+this function should store the @var{n}-by-@var{p} matrix result
+@c{$J_{ij} = \partial f_i(x,\hbox{\it params}) / \partial x_j$}
+@math{J_ij = d f_i(x,params) / d x_j} in @var{J} for argument @var{x} 
+and arbitrary parameters @var{params}, returning an appropriate error code if the
+function cannot be computed.
+
+@item int (* fdf) (const gsl_vector * @var{x}, void * @var{params}, gsl_vector * @var{f}, gsl_matrix * @var{J})
+This function should set the values of the @var{f} and @var{J} as above,
+for arguments @var{x} and arbitrary parameters @var{params}.  This function
+provides an optimization of the separate functions for @math{f(x)} and
+@math{J(x)}---it is always faster to compute the function and its
+derivative at the same time.
+
+@item size_t n
+the number of functions, i.e. the number of components of the
+vector @var{f}.
+
+@item size_t p
+the number of independent variables, i.e. the number of components of
+the vector @var{x}.
+
+@item void * params
+a pointer to the arbitrary parameters of the function.
+@end table
+@end deftp
+
+Note that when fitting a non-linear model against experimental data,
+the data is passed to the functions above using the
+@var{params} argument and the trial best-fit parameters through the
+@var{x} argument.
+
+@node Iteration of the Minimization Algorithm
+@section Iteration
+
+The following functions drive the iteration of each algorithm.  Each
+function performs one iteration to update the state of any solver of the
+corresponding type.  The same functions work for all solvers so that
+different methods can be substituted at runtime without modifications to
+the code.
+
+@deftypefun int gsl_multifit_fsolver_iterate (gsl_multifit_fsolver * @var{s})
+@deftypefunx int gsl_multifit_fdfsolver_iterate (gsl_multifit_fdfsolver * @var{s})
+These functions perform a single iteration of the solver @var{s}.  If
+the iteration encounters an unexpected problem then an error code will
+be returned.  The solver maintains a current estimate of the best-fit
+parameters at all times. 
+@end deftypefun
+
+The solver struct @var{s} contains the following entries, which can
+be used to track the progress of the solution:
+
+@table @code
+@item gsl_vector * x
+The current position.
+
+@item gsl_vector * f
+The function value at the current position.
+
+@item gsl_vector * dx
+The difference between the current position and the previous position,
+i.e. the last step, taken as a vector.
+
+@item gsl_matrix * J
+The Jacobian matrix at the current position (for the
+@code{gsl_multifit_fdfsolver} struct only)
+@end table
+
+The best-fit information also can be accessed with the following
+auxiliary functions,
+
+@deftypefun {gsl_vector *} gsl_multifit_fsolver_position (const gsl_multifit_fsolver * @var{s})
+@deftypefunx {gsl_vector *} gsl_multifit_fdfsolver_position (const gsl_multifit_fdfsolver * @var{s})
+These functions return the current position (i.e. best-fit parameters)
+@code{s->x} of the solver @var{s}.
+@end deftypefun
+
+@node Search Stopping Parameters for Minimization Algorithms
+@section Search Stopping Parameters
+@cindex nonlinear fitting, stopping parameters
+
+A minimization procedure should stop when one of the following conditions is
+true:
+
+@itemize @bullet
+@item
+A minimum has been found to within the user-specified precision.
+
+@item
+A user-specified maximum number of iterations has been reached.
+
+@item
+An error has occurred.
+@end itemize
+
+@noindent
+The handling of these conditions is under user control.  The functions
+below allow the user to test the current estimate of the best-fit
+parameters in several standard ways.
+
+@deftypefun int gsl_multifit_test_delta (const gsl_vector * @var{dx}, const gsl_vector * @var{x}, double @var{epsabs}, double @var{epsrel})
+
+This function tests for the convergence of the sequence by comparing the
+last step @var{dx} with the absolute error @var{epsabs} and relative
+error @var{epsrel} to the current position @var{x}.  The test returns
+@code{GSL_SUCCESS} if the following condition is achieved,
+@tex
+\beforedisplay
+$$
+|dx_i| < \hbox{\it epsabs} + \hbox{\it epsrel\/}\, |x_i|
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+|dx_i| < epsabs + epsrel |x_i|
+@end example
+
+@end ifinfo
+@noindent
+for each component of @var{x} and returns @code{GSL_CONTINUE} otherwise.
+@end deftypefun
+
+@cindex residual, in nonlinear systems of equations
+@deftypefun int gsl_multifit_test_gradient (const gsl_vector * @var{g}, double @var{epsabs})
+This function tests the residual gradient @var{g} against the absolute
+error bound @var{epsabs}.  Mathematically, the gradient should be
+exactly zero at the minimum. The test returns @code{GSL_SUCCESS} if the
+following condition is achieved,
+@tex
+\beforedisplay
+$$
+\sum_i |g_i| < \hbox{\it epsabs}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+\sum_i |g_i| < epsabs
+@end example
+
+@end ifinfo
+@noindent
+and returns @code{GSL_CONTINUE} otherwise.  This criterion is suitable
+for situations where the precise location of the minimum, @math{x},
+is unimportant provided a value can be found where the gradient is small
+enough.
+@end deftypefun
+
+
+@deftypefun int gsl_multifit_gradient (const gsl_matrix * @var{J}, const gsl_vector * @var{f}, gsl_vector * @var{g})
+This function computes the gradient @var{g} of @math{\Phi(x) = (1/2)
+||F(x)||^2} from the Jacobian matrix @math{J} and the function values
+@var{f}, using the formula @math{g = J^T f}.
+@end deftypefun
+
+@node Minimization Algorithms using Derivatives
+@section Minimization Algorithms using Derivatives
+
+The minimization algorithms described in this section make use of both
+the function and its derivative.  They require an initial guess for the
+location of the minimum. There is no absolute guarantee of
+convergence---the function must be suitable for this technique and the
+initial guess must be sufficiently close to the minimum for it to work.
+
+@comment ============================================================
+@cindex Levenberg-Marquardt algorithms
+@deffn {Derivative Solver} gsl_multifit_fdfsolver_lmsder
+@cindex LMDER algorithm
+@cindex MINPACK, minimization algorithms
+This is a robust and efficient version of the Levenberg-Marquardt
+algorithm as implemented in the scaled @sc{lmder} routine in
+@sc{minpack}.  Minpack was written by Jorge J. Mor@'e, Burton S. Garbow
+and Kenneth E. Hillstrom.
+
+The algorithm uses a generalized trust region to keep each step under
+control.  In order to be accepted a proposed new position @math{x'} must
+satisfy the condition @math{|D (x' - x)| < \delta}, where @math{D} is a
+diagonal scaling matrix and @math{\delta} is the size of the trust
+region.  The components of @math{D} are computed internally, using the
+column norms of the Jacobian to estimate the sensitivity of the residual
+to each component of @math{x}.  This improves the behavior of the
+algorithm for badly scaled functions.
+
+On each iteration the algorithm attempts to minimize the linear system
+@math{|F + J p|} subject to the constraint @math{|D p| < \Delta}.  The
+solution to this constrained linear system is found using the
+Levenberg-Marquardt method.
+
+The proposed step is now tested by evaluating the function at the
+resulting point, @math{x'}.  If the step reduces the norm of the
+function sufficiently, and follows the predicted behavior of the
+function within the trust region, then it is accepted and the size of the
+trust region is increased.  If the proposed step fails to improve the
+solution, or differs significantly from the expected behavior within
+the trust region, then the size of the trust region is decreased and
+another trial step is computed.
+
+The algorithm also monitors the progress of the solution and returns an
+error if the changes in the solution are smaller than the machine
+precision.  The possible error codes are,
+
+@table @code
+@item GSL_ETOLF
+the decrease in the function falls below machine precision
+
+@item GSL_ETOLX
+the change in the position vector falls below machine precision
+
+@item GSL_ETOLG
+the norm of the gradient, relative to the norm of the function, falls
+below machine precision
+@end table
+
+@noindent
+These error codes indicate that further iterations will be unlikely to
+change the solution from its current value.
+
+@end deffn
+
+@deffn {Derivative Solver} gsl_multifit_fdfsolver_lmder
+This is an unscaled version of the @sc{lmder} algorithm.  The elements of the
+diagonal scaling matrix @math{D} are set to 1.  This algorithm may be
+useful in circumstances where the scaled version of @sc{lmder} converges too
+slowly, or the function is already scaled appropriately.
+@end deffn
+
+@node Minimization Algorithms without Derivatives
+@section Minimization Algorithms without Derivatives
+
+There are no algorithms implemented in this section at the moment.
+
+@node Computing the covariance matrix of best fit parameters
+@section Computing the covariance matrix of best fit parameters
+@cindex best-fit parameters, covariance
+@cindex least squares, covariance of best-fit parameters
+@cindex covariance matrix, nonlinear fits
+
+@deftypefun int gsl_multifit_covar (const gsl_matrix * @var{J}, double @var{epsrel}, gsl_matrix * @var{covar})
+This function uses the Jacobian matrix @var{J} to compute the covariance
+matrix of the best-fit parameters, @var{covar}.  The parameter
+@var{epsrel} is used to remove linear-dependent columns when @var{J} is
+rank deficient.
+
+The covariance matrix is given by,
+@tex
+\beforedisplay
+$$
+C = (J^T J)^{-1}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+covar = (J^T J)^@{-1@}
+@end example
+
+@end ifinfo
+@noindent
+and is computed by QR decomposition of J with column-pivoting.  Any
+columns of @math{R} which satisfy 
+@tex
+\beforedisplay
+$$
+|R_{kk}| \leq epsrel |R_{11}|
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+|R_@{kk@}| <= epsrel |R_@{11@}|
+@end example
+
+@end ifinfo
+@noindent
+are considered linearly-dependent and are excluded from the covariance
+matrix (the corresponding rows and columns of the covariance matrix are
+set to zero).
+
+If the minimisation uses the weighted least-squares function
+@math{f_i = (Y(x, t_i) - y_i) / \sigma_i} then the covariance
+matrix above gives the statistical error on the best-fit parameters
+resulting from the gaussian errors @math{\sigma_i} on 
+the underlying data @math{y_i}.  This can be verified from the relation 
+@math{\delta f = J \delta c} and the fact that the fluctuations in @math{f}
+from the data @math{y_i} are normalised by @math{\sigma_i} and 
+so satisfy @c{$\langle \delta f \delta f^T \rangle = I$}
+@math{<\delta f \delta f^T> = I}.
+
+For an unweighted least-squares function @math{f_i = (Y(x, t_i) -
+y_i)} the covariance matrix above should be multiplied by the variance
+of the residuals about the best-fit @math{\sigma^2 = \sum (y_i - Y(x,t_i))^2 / (n-p)}
+to give the variance-covariance
+matrix @math{\sigma^2 C}.  This estimates the statistical error on the
+best-fit parameters from the scatter of the underlying data.
+
+For more information about covariance matrices see @ref{Fitting Overview}.
+@end deftypefun
+
+@comment ============================================================
+
+@node Example programs for Nonlinear Least-Squares Fitting
+@section Examples
+
+The following example program fits a weighted exponential model with
+background to experimental data, @math{Y = A \exp(-\lambda t) + b}. The
+first part of the program sets up the functions @code{expb_f} and
+@code{expb_df} to calculate the model and its Jacobian.  The appropriate
+fitting function is given by,
+@tex
+\beforedisplay
+$$
+f_i = ((A \exp(-\lambda t_i) + b) - y_i)/\sigma_i
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+f_i = ((A \exp(-\lambda t_i) + b) - y_i)/\sigma_i
+@end example
+
+@end ifinfo
+@noindent
+where we have chosen @math{t_i = i}.  The Jacobian matrix @math{J} is
+the derivative of these functions with respect to the three parameters
+(@math{A}, @math{\lambda}, @math{b}).  It is given by,
+@tex
+\beforedisplay
+$$
+J_{ij} = {\partial f_i \over \partial x_j}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+J_@{ij@} = d f_i / d x_j
+@end example
+
+@end ifinfo
+@noindent
+where @math{x_0 = A}, @math{x_1 = \lambda} and @math{x_2 = b}.
+
+@example
+@verbatiminclude examples/expfit.c
+@end example
+
+@noindent
+The main part of the program sets up a Levenberg-Marquardt solver and
+some simulated random data. The data uses the known parameters
+(1.0,5.0,0.1) combined with gaussian noise (standard deviation = 0.1)
+over a range of 40 timesteps. The initial guess for the parameters is
+chosen as (0.0, 1.0, 0.0).
+
+@example
+@verbatiminclude examples/nlfit.c
+@end example
+
+@noindent
+The iteration terminates when the change in x is smaller than 0.0001, as
+both an absolute and relative change.  Here are the results of running
+the program:
+
+@smallexample
+iter: 0 x=1.00000000 0.00000000 0.00000000 |f(x)|=117.349
+status=success
+iter: 1 x=1.64659312 0.01814772 0.64659312 |f(x)|=76.4578
+status=success
+iter: 2 x=2.85876037 0.08092095 1.44796363 |f(x)|=37.6838
+status=success
+iter: 3 x=4.94899512 0.11942928 1.09457665 |f(x)|=9.58079
+status=success
+iter: 4 x=5.02175572 0.10287787 1.03388354 |f(x)|=5.63049
+status=success
+iter: 5 x=5.04520433 0.10405523 1.01941607 |f(x)|=5.44398
+status=success
+iter: 6 x=5.04535782 0.10404906 1.01924871 |f(x)|=5.44397
+chisq/dof = 0.800996
+A      = 5.04536 +/- 0.06028
+lambda = 0.10405 +/- 0.00316
+b      = 1.01925 +/- 0.03782
+status = success
+@end smallexample
+
+@noindent
+The approximate values of the parameters are found correctly, and the
+chi-squared value indicates a good fit (the chi-squared per degree of
+freedom is approximately 1).  In this case the errors on the parameters
+can be estimated from the square roots of the diagonal elements of the
+covariance matrix.  
+
+If the chi-squared value shows a poor fit (i.e. @c{$\chi^2/(n-p) \gg 1$}
+@math{chi^2/dof >> 1}) then the error estimates obtained from the
+covariance matrix will be too small.  In the example program the error estimates
+are multiplied by @c{$\sqrt{\chi^2/(n-p)}$}
+@math{\sqrt@{\chi^2/dof@}} in this case, a common way of increasing the
+errors for a poor fit.  Note that a poor fit will result from the use
+an inappropriate model, and the scaled error estimates may then
+be outside the range of validity for gaussian errors.
+
+@iftex
+@sp 1
+@center @image{fit-exp,3.4in}
+@end iftex
+
+@node References and Further Reading for Nonlinear Least-Squares Fitting
+@section References and Further Reading
+
+The @sc{minpack} algorithm is described in the following article,
+
+@itemize @asis
+@item
+J.J. Mor@'e, @cite{The Levenberg-Marquardt Algorithm: Implementation and
+Theory}, Lecture Notes in Mathematics, v630 (1978), ed G. Watson.
+@end itemize
+
+@noindent
+The following paper is also relevant to the algorithms described in this
+section,
+
+@itemize @asis
+@item 
+J.J. Mor@'e, B.S. Garbow, K.E. Hillstrom, ``Testing Unconstrained
+Optimization Software'', ACM Transactions on Mathematical Software, Vol
+7, No 1 (1981), p 17--41.
+@end itemize
+
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diff --git a/gsl-1.9/doc/multimin.texi b/gsl-1.9/doc/multimin.texi
new file mode 100644
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+@cindex minimization, multidimensional
+
+This chapter describes routines for finding minima of arbitrary
+multidimensional functions.  The library provides low level components
+for a variety of iterative minimizers and convergence tests.  These can
+be combined by the user to achieve the desired solution, while providing
+full access to the intermediate steps of the algorithms.  Each class of
+methods uses the same framework, so that you can switch between
+minimizers at runtime without needing to recompile your program.  Each
+instance of a minimizer keeps track of its own state, allowing the
+minimizers to be used in multi-threaded programs. The minimization
+algorithms can be used to maximize a function by inverting its sign.
+
+The header file @file{gsl_multimin.h} contains prototypes for the
+minimization functions and related declarations.  
+
+@menu
+* Multimin Overview::       
+* Multimin Caveats::        
+* Initializing the Multidimensional Minimizer::  
+* Providing a function to minimize::  
+* Multimin Iteration::      
+* Multimin Stopping Criteria::  
+* Multimin Algorithms::     
+* Multimin Examples::       
+* Multimin References and Further Reading::  
+@end menu
+
+@node Multimin Overview
+@section Overview
+
+The problem of multidimensional minimization requires finding a point
+@math{x} such that the scalar function,
+@tex
+\beforedisplay
+$$
+f(x_1, \dots, x_n)
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+f(x_1, @dots{}, x_n)
+@end example
+
+@end ifinfo
+@noindent
+takes a value which is lower than at any neighboring point. For smooth
+functions the gradient @math{g = \nabla f} vanishes at the minimum. In
+general there are no bracketing methods available for the
+minimization of @math{n}-dimensional functions.  The algorithms
+proceed from an initial guess using a search algorithm which attempts
+to move in a downhill direction. 
+
+Algorithms making use of the gradient of the function perform a
+one-dimensional line minimisation along this direction until the lowest
+point is found to a suitable tolerance.  The search direction is then
+updated with local information from the function and its derivatives,
+and the whole process repeated until the true @math{n}-dimensional
+minimum is found.
+
+The Nelder-Mead Simplex algorithm applies a different strategy.  It
+maintains @math{n+1} trial parameter vectors as the vertices of a
+@math{n}-dimensional simplex.  In each iteration step it tries to
+improve the worst vertex by a simple geometrical transformation until
+the size of the simplex falls below a given tolerance.
+
+Both types of algorithms use a standard framework. The user provides a
+high-level driver for the algorithms, and the library provides the
+individual functions necessary for each of the steps.  There are three
+main phases of the iteration.  The steps are,
+
+@itemize @bullet
+@item
+initialize minimizer state, @var{s}, for algorithm @var{T}
+
+@item
+update @var{s} using the iteration @var{T}
+
+@item
+test @var{s} for convergence, and repeat iteration if necessary
+@end itemize
+
+@noindent
+Each iteration step consists either of an improvement to the
+line-minimisation in the current direction or an update to the search
+direction itself.  The state for the minimizers is held in a
+@code{gsl_multimin_fdfminimizer} struct or a
+@code{gsl_multimin_fminimizer} struct.
+
+@node Multimin Caveats
+@section Caveats
+@cindex Multimin, caveats
+
+Note that the minimization algorithms can only search for one local
+minimum at a time.  When there are several local minima in the search
+area, the first minimum to be found will be returned; however it is
+difficult to predict which of the minima this will be.  In most cases,
+no error will be reported if you try to find a local minimum in an area
+where there is more than one.
+
+It is also important to note that the minimization algorithms find local
+minima; there is no way to determine whether a minimum is a global
+minimum of the function in question.
+
+@node Initializing the Multidimensional Minimizer
+@section Initializing the Multidimensional Minimizer
+
+The following function initializes a multidimensional minimizer.  The
+minimizer itself depends only on the dimension of the problem and the
+algorithm and can be reused for different problems.
+
+@deftypefun {gsl_multimin_fdfminimizer *} gsl_multimin_fdfminimizer_alloc (const gsl_multimin_fdfminimizer_type * @var{T}, size_t @var{n})
+@deftypefunx {gsl_multimin_fminimizer *} gsl_multimin_fminimizer_alloc (const gsl_multimin_fminimizer_type * @var{T}, size_t @var{n})
+This function returns a pointer to a newly allocated instance of a
+minimizer of type @var{T} for an @var{n}-dimension function.  If there
+is insufficient memory to create the minimizer then the function returns
+a null pointer and the error handler is invoked with an error code of
+@code{GSL_ENOMEM}.
+@end deftypefun
+
+@deftypefun int gsl_multimin_fdfminimizer_set (gsl_multimin_fdfminimizer * @var{s}, gsl_multimin_function_fdf * @var{fdf}, const gsl_vector * @var{x}, double @var{step_size}, double @var{tol})
+This function initializes the minimizer @var{s} to minimize the function
+@var{fdf} starting from the initial point @var{x}.  The size of the
+first trial step is given by @var{step_size}.  The accuracy of the line
+minimization is specified by @var{tol}.  The precise meaning of this
+parameter depends on the method used.  Typically the line minimization
+is considered successful if the gradient of the function @math{g} is
+orthogonal to the current search direction @math{p} to a relative
+accuracy of @var{tol}, where @c{$p\cdot g < tol |p| |g|$} 
+@math{dot(p,g) < tol |p| |g|}.  A @var{tol} value of 0.1 is 
+suitable for most purposes, since line minimization only needs to
+be carried out approximately.    Note that setting @var{tol} to zero will
+force the use of ``exact'' line-searches, which are extremely expensive.
+
+@deftypefunx int gsl_multimin_fminimizer_set (gsl_multimin_fminimizer * @var{s}, gsl_multimin_function * @var{f}, const gsl_vector * @var{x}, const gsl_vector * @var{step_size})
+This function initializes the minimizer @var{s} to minimize the function
+@var{f}, starting from the initial point
+@var{x}. The size of the initial trial steps is given in vector
+@var{step_size}. The precise meaning of this parameter depends on the
+method used. 
+@end deftypefun
+
+@deftypefun void gsl_multimin_fdfminimizer_free (gsl_multimin_fdfminimizer * @var{s})
+@deftypefunx void gsl_multimin_fminimizer_free (gsl_multimin_fminimizer * @var{s})
+This function frees all the memory associated with the minimizer
+@var{s}.
+@end deftypefun
+
+@deftypefun {const char *} gsl_multimin_fdfminimizer_name (const gsl_multimin_fdfminimizer * @var{s})
+@deftypefunx {const char *} gsl_multimin_fminimizer_name (const gsl_multimin_fminimizer * @var{s})
+This function returns a pointer to the name of the minimizer.  For example,
+
+@example
+printf ("s is a '%s' minimizer\n", 
+        gsl_multimin_fdfminimizer_name (s));
+@end example
+
+@noindent
+would print something like @code{s is a 'conjugate_pr' minimizer}.
+@end deftypefun
+
+@node Providing a function to minimize
+@section Providing a function to minimize
+
+You must provide a parametric function of @math{n} variables for the
+minimizers to operate on.  You may also need to provide a routine which
+calculates the gradient of the function and a third routine which
+calculates both the function value and the gradient together.  In order
+to allow for general parameters the functions are defined by the
+following data types:
+
+@deftp {Data Type} gsl_multimin_function_fdf
+This data type defines a general function of @math{n} variables with
+parameters and the corresponding gradient vector of derivatives,
+
+@table @code
+@item double (* f) (const gsl_vector * @var{x}, void * @var{params})
+this function should return the result
+@c{$f(x,\hbox{\it params})$}
+@math{f(x,params)} for argument @var{x} and parameters @var{params}.
+
+@item void (* df) (const gsl_vector * @var{x}, void * @var{params}, gsl_vector * @var{g})
+this function should store the @var{n}-dimensional gradient
+@c{$g_i = \partial f(x,\hbox{\it params}) / \partial x_i$}
+@math{g_i = d f(x,params) / d x_i} in the vector @var{g} for argument @var{x} 
+and parameters @var{params}, returning an appropriate error code if the
+function cannot be computed.
+
+@item void (* fdf) (const gsl_vector * @var{x}, void * @var{params}, double * f, gsl_vector * @var{g})
+This function should set the values of the @var{f} and @var{g} as above,
+for arguments @var{x} and parameters @var{params}.  This function
+provides an optimization of the separate functions for @math{f(x)} and
+@math{g(x)}---it is always faster to compute the function and its
+derivative at the same time.
+
+@item size_t n
+the dimension of the system, i.e. the number of components of the
+vectors @var{x}.
+
+@item void * params
+a pointer to the parameters of the function.
+@end table
+@end deftp
+@deftp {Data Type} gsl_multimin_function
+This data type defines a general function of @math{n} variables with
+parameters,
+
+@table @code
+@item double (* f) (const gsl_vector * @var{x}, void * @var{params})
+this function should return the result
+@c{$f(x,\hbox{\it params})$}
+@math{f(x,params)} for argument @var{x} and parameters @var{params}.
+
+@item size_t n
+the dimension of the system, i.e. the number of components of the
+vectors @var{x}.
+
+@item void * params
+a pointer to the parameters of the function.
+@end table
+@end deftp
+
+@noindent
+The following example function defines a simple paraboloid with two
+parameters,
+
+@example
+/* Paraboloid centered on (dp[0],dp[1]) */
+
+double
+my_f (const gsl_vector *v, void *params)
+@{
+  double x, y;
+  double *dp = (double *)params;
+  
+  x = gsl_vector_get(v, 0);
+  y = gsl_vector_get(v, 1);
+ 
+  return 10.0 * (x - dp[0]) * (x - dp[0]) +
+           20.0 * (y - dp[1]) * (y - dp[1]) + 30.0; 
+@}
+
+/* The gradient of f, df = (df/dx, df/dy). */
+void 
+my_df (const gsl_vector *v, void *params, 
+       gsl_vector *df)
+@{
+  double x, y;
+  double *dp = (double *)params;
+  
+  x = gsl_vector_get(v, 0);
+  y = gsl_vector_get(v, 1);
+ 
+  gsl_vector_set(df, 0, 20.0 * (x - dp[0]));
+  gsl_vector_set(df, 1, 40.0 * (y - dp[1]));
+@}
+
+/* Compute both f and df together. */
+void 
+my_fdf (const gsl_vector *x, void *params, 
+        double *f, gsl_vector *df) 
+@{
+  *f = my_f(x, params); 
+  my_df(x, params, df);
+@}
+@end example
+
+@noindent
+The function can be initialized using the following code,
+
+@example
+gsl_multimin_function_fdf my_func;
+
+double p[2] = @{ 1.0, 2.0 @}; /* center at (1,2) */
+
+my_func.f = &my_f;
+my_func.df = &my_df;
+my_func.fdf = &my_fdf;
+my_func.n = 2;
+my_func.params = (void *)p;
+@end example
+
+@node Multimin Iteration
+@section Iteration
+
+The following function drives the iteration of each algorithm.  The
+function performs one iteration to update the state of the minimizer.
+The same function works for all minimizers so that different methods can
+be substituted at runtime without modifications to the code.
+
+@deftypefun int gsl_multimin_fdfminimizer_iterate (gsl_multimin_fdfminimizer * @var{s})
+@deftypefunx int gsl_multimin_fminimizer_iterate (gsl_multimin_fminimizer * @var{s})
+These functions perform a single iteration of the minimizer @var{s}.  If
+the iteration encounters an unexpected problem then an error code will
+be returned.
+@end deftypefun
+
+@noindent
+The minimizer maintains a current best estimate of the minimum at all
+times.  This information can be accessed with the following auxiliary
+functions,
+
+@deftypefun {gsl_vector *} gsl_multimin_fdfminimizer_x (const gsl_multimin_fdfminimizer * @var{s})
+@deftypefunx {gsl_vector *} gsl_multimin_fminimizer_x (const gsl_multimin_fminimizer * @var{s})
+@deftypefunx double gsl_multimin_fdfminimizer_minimum (const gsl_multimin_fdfminimizer * @var{s})
+@deftypefunx double gsl_multimin_fminimizer_minimum (const gsl_multimin_fminimizer * @var{s})
+@deftypefunx {gsl_vector *} gsl_multimin_fdfminimizer_gradient (const gsl_multimin_fdfminimizer * @var{s})
+@deftypefunx double gsl_multimin_fminimizer_size (const gsl_multimin_fminimizer * @var{s})
+These functions return the current best estimate of the location of the
+minimum, the value of the function at that point, its gradient, 
+and minimizer specific characteristic size for the minimizer @var{s}.
+@end deftypefun
+
+@deftypefun int gsl_multimin_fdfminimizer_restart (gsl_multimin_fdfminimizer * @var{s})
+This function resets the minimizer @var{s} to use the current point as a
+new starting point.
+@end deftypefun
+
+@node Multimin Stopping Criteria
+@section Stopping Criteria
+
+A minimization procedure should stop when one of the following
+conditions is true:
+
+@itemize @bullet
+@item
+A minimum has been found to within the user-specified precision.
+
+@item
+A user-specified maximum number of iterations has been reached.
+
+@item
+An error has occurred.
+@end itemize
+
+@noindent
+The handling of these conditions is under user control.  The functions
+below allow the user to test the precision of the current result.
+
+@deftypefun int gsl_multimin_test_gradient (const gsl_vector * @var{g}, double @var{epsabs})
+This function tests the norm of the gradient @var{g} against the
+absolute tolerance @var{epsabs}. The gradient of a multidimensional
+function goes to zero at a minimum. The test returns @code{GSL_SUCCESS}
+if the following condition is achieved,
+@tex
+\beforedisplay
+$$
+|g| < \hbox{\it epsabs}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+|g| < epsabs
+@end example
+
+@end ifinfo
+@noindent
+and returns @code{GSL_CONTINUE} otherwise.  A suitable choice of
+@var{epsabs} can be made from the desired accuracy in the function for
+small variations in @math{x}.  The relationship between these quantities
+is given by @c{$\delta{f} = g\,\delta{x}$}
+@math{\delta f = g \delta x}.
+@end deftypefun
+
+@deftypefun int gsl_multimin_test_size (const double @var{size}, double @var{epsabs})
+This function tests the minimizer specific characteristic
+size (if applicable to the used minimizer) against absolute tolerance @var{epsabs}. 
+The test returns @code{GSL_SUCCESS} if the size is smaller than tolerance,
+otherwise @code{GSL_CONTINUE} is returned.
+@end deftypefun
+
+@node Multimin Algorithms
+@section Algorithms
+
+There are several minimization methods available. The best choice of
+algorithm depends on the problem.  All of the algorithms use the value
+of the function and its gradient at each evaluation point, except for
+the Simplex algorithm which uses function values only.
+
+@deffn {Minimizer} gsl_multimin_fdfminimizer_conjugate_fr
+@cindex Fletcher-Reeves conjugate gradient algorithm, minimization
+@cindex Conjugate gradient algorithm, minimization
+@cindex minimization, conjugate gradient algorithm
+This is the Fletcher-Reeves conjugate gradient algorithm. The conjugate
+gradient algorithm proceeds as a succession of line minimizations. The
+sequence of search directions is used to build up an approximation to the
+curvature of the function in the neighborhood of the minimum.  
+
+An initial search direction @var{p} is chosen using the gradient, and line
+minimization is carried out in that direction.  The accuracy of the line
+minimization is specified by the parameter @var{tol}.  The minimum
+along this line occurs when the function gradient @var{g} and the search direction
+@var{p} are orthogonal.  The line minimization terminates when
+@c{$p\cdot g < tol |p| |g|$} 
+@math{dot(p,g) < tol |p| |g|}.  The
+search direction is updated  using the Fletcher-Reeves formula
+@math{p' = g' - \beta g} where @math{\beta=-|g'|^2/|g|^2}, and
+the line minimization is then repeated for the new search
+direction.
+@end deffn
+
+@deffn {Minimizer} gsl_multimin_fdfminimizer_conjugate_pr
+@cindex Polak-Ribiere algorithm, minimization
+@cindex minimization, Polak-Ribiere algorithm
+This is the Polak-Ribiere conjugate gradient algorithm.  It is similar
+to the Fletcher-Reeves method, differing only in the choice of the
+coefficient @math{\beta}. Both methods work well when the evaluation
+point is close enough to the minimum of the objective function that it
+is well approximated by a quadratic hypersurface.
+@end deffn
+
+@deffn {Minimizer} gsl_multimin_fdfminimizer_vector_bfgs2
+@deffnx {Minimizer} gsl_multimin_fdfminimizer_vector_bfgs
+@cindex BFGS algorithm, minimization
+@cindex minimization, BFGS algorithm
+These methods use the vector Broyden-Fletcher-Goldfarb-Shanno (BFGS)
+algorithm.  This is a quasi-Newton method which builds up an approximation
+to the second derivatives of the function @math{f} using the difference
+between successive gradient vectors.  By combining the first and second
+derivatives the algorithm is able to take Newton-type steps towards the
+function minimum, assuming quadratic behavior in that region.
+
+The @code{bfgs2} version of this minimizer is the most efficient
+version available, and is a faithful implementation of the line
+minimization scheme described in Fletcher's @cite{Practical Methods of
+Optimization}, Algorithms 2.6.2 and 2.6.4.  It supercedes the original
+@code{bfgs} routine and requires substantially fewer function and
+gradient evaluations.  The user-supplied tolerance @var{tol}
+corresponds to the parameter @math{\sigma} used by Fletcher.  A value
+of 0.1 is recommended for typical use (larger values correspond to
+less accurate line searches).
+
+@end deffn
+
+@deffn {Minimizer} gsl_multimin_fdfminimizer_steepest_descent
+@cindex steepest descent algorithm, minimization
+@cindex minimization, steepest descent algorithm
+The steepest descent algorithm follows the downhill gradient of the
+function at each step. When a downhill step is successful the step-size
+is increased by a factor of two.  If the downhill step leads to a higher
+function value then the algorithm backtracks and the step size is
+decreased using the parameter @var{tol}.  A suitable value of @var{tol}
+for most applications is 0.1.  The steepest descent method is
+inefficient and is included only for demonstration purposes.
+@end deffn
+
+@deffn {Minimizer} gsl_multimin_fminimizer_nmsimplex
+@cindex Nelder-Mead simplex algorithm for minimization
+@cindex simplex algorithm, minimization
+@cindex minimization, simplex algorithm
+This is the Simplex algorithm of Nelder and Mead. It constructs 
+@math{n} vectors @math{p_i} from the
+starting vector @var{x} and the vector @var{step_size} as follows:
+@tex
+\beforedisplay
+$$
+\eqalign{
+p_0 & = (x_0, x_1, \cdots , x_n) \cr
+p_1 & = (x_0 + step\_size_0, x_1, \cdots , x_n) \cr
+p_2 & = (x_0, x_1 + step\_size_1, \cdots , x_n) \cr
+\dots &= \dots \cr
+p_n & = (x_0, x_1, \cdots , x_n+step\_size_n) \cr
+}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+p_0 = (x_0, x_1, ... , x_n) 
+p_1 = (x_0 + step_size_0, x_1, ... , x_n) 
+p_2 = (x_0, x_1 + step_size_1, ... , x_n) 
+... = ...
+p_n = (x_0, x_1, ... , x_n+step_size_n)
+@end example
+
+@end ifinfo
+@noindent
+These vectors form the @math{n+1} vertices of a simplex in @math{n}
+dimensions.  On each iteration the algorithm tries to improve
+the parameter vector @math{p_i} corresponding to the highest
+function value by simple geometrical transformations.  These
+are reflection, reflection followed by expansion, contraction and multiple
+contraction. Using these transformations the simplex moves through 
+the parameter space towards the minimum, where it contracts itself.  
+
+After each iteration, the best vertex is returned.  Note, that due to
+the nature of the algorithm not every step improves the current
+best parameter vector.  Usually several iterations are required.
+
+The routine calculates the minimizer specific characteristic size as the
+average distance from the geometrical center of the simplex to all its
+vertices.  This size can be used as a stopping criteria, as the simplex
+contracts itself near the minimum. The size is returned by the function
+@code{gsl_multimin_fminimizer_size}.
+@end deffn
+
+@node Multimin Examples
+@section Examples
+
+This example program finds the minimum of the paraboloid function
+defined earlier.  The location of the minimum is offset from the origin
+in @math{x} and @math{y}, and the function value at the minimum is
+non-zero. The main program is given below, it requires the example
+function given earlier in this chapter.
+
+@smallexample
+int
+main (void)
+@{
+  size_t iter = 0;
+  int status;
+
+  const gsl_multimin_fdfminimizer_type *T;
+  gsl_multimin_fdfminimizer *s;
+
+  /* Position of the minimum (1,2). */
+  double par[2] = @{ 1.0, 2.0 @};
+
+  gsl_vector *x;
+  gsl_multimin_function_fdf my_func;
+
+  my_func.f = &my_f;
+  my_func.df = &my_df;
+  my_func.fdf = &my_fdf;
+  my_func.n = 2;
+  my_func.params = &par;
+
+  /* Starting point, x = (5,7) */
+  x = gsl_vector_alloc (2);
+  gsl_vector_set (x, 0, 5.0);
+  gsl_vector_set (x, 1, 7.0);
+
+  T = gsl_multimin_fdfminimizer_conjugate_fr;
+  s = gsl_multimin_fdfminimizer_alloc (T, 2);
+
+  gsl_multimin_fdfminimizer_set (s, &my_func, x, 0.01, 1e-4);
+
+  do
+    @{
+      iter++;
+      status = gsl_multimin_fdfminimizer_iterate (s);
+
+      if (status)
+        break;
+
+      status = gsl_multimin_test_gradient (s->gradient, 1e-3);
+
+      if (status == GSL_SUCCESS)
+        printf ("Minimum found at:\n");
+
+      printf ("%5d %.5f %.5f %10.5f\n", iter,
+              gsl_vector_get (s->x, 0), 
+              gsl_vector_get (s->x, 1), 
+              s->f);
+
+    @}
+  while (status == GSL_CONTINUE && iter < 100);
+
+  gsl_multimin_fdfminimizer_free (s);
+  gsl_vector_free (x);
+
+  return 0;
+@}
+@end smallexample  
+
+@noindent
+The initial step-size is chosen as 0.01, a conservative estimate in this
+case, and the line minimization parameter is set at 0.0001.  The program
+terminates when the norm of the gradient has been reduced below
+0.001. The output of the program is shown below,
+
+@example
+         x       y         f
+    1 4.99629 6.99072  687.84780
+    2 4.98886 6.97215  683.55456
+    3 4.97400 6.93501  675.01278
+    4 4.94429 6.86073  658.10798
+    5 4.88487 6.71217  625.01340
+    6 4.76602 6.41506  561.68440
+    7 4.52833 5.82083  446.46694
+    8 4.05295 4.63238  261.79422
+    9 3.10219 2.25548   75.49762
+   10 2.85185 1.62963   67.03704
+   11 2.19088 1.76182   45.31640
+   12 0.86892 2.02622   30.18555
+Minimum found at:
+   13 1.00000 2.00000   30.00000
+@end example
+
+@noindent
+Note that the algorithm gradually increases the step size as it
+successfully moves downhill, as can be seen by plotting the successive
+points.
+
+@iftex
+@sp 1
+@center @image{multimin,3.4in}
+@end iftex
+
+@noindent
+The conjugate gradient algorithm finds the minimum on its second
+direction because the function is purely quadratic. Additional
+iterations would be needed for a more complicated function.
+
+Here is another example using the Nelder-Mead Simplex algorithm to
+minimize the same example object function, as above.
+
+@smallexample
+int 
+main(void)
+@{
+  size_t np = 2;
+  double par[2] = @{1.0, 2.0@};
+
+  const gsl_multimin_fminimizer_type *T = 
+    gsl_multimin_fminimizer_nmsimplex;
+  gsl_multimin_fminimizer *s = NULL;
+  gsl_vector *ss, *x;
+  gsl_multimin_function minex_func;
+
+  size_t iter = 0, i;
+  int status;
+  double size;
+
+  /* Initial vertex size vector */
+  ss = gsl_vector_alloc (np);
+
+  /* Set all step sizes to 1 */
+  gsl_vector_set_all (ss, 1.0);
+
+  /* Starting point */
+  x = gsl_vector_alloc (np);
+
+  gsl_vector_set (x, 0, 5.0);
+  gsl_vector_set (x, 1, 7.0);
+
+  /* Initialize method and iterate */
+  minex_func.f = &my_f;
+  minex_func.n = np;
+  minex_func.params = (void *)&par;
+
+  s = gsl_multimin_fminimizer_alloc (T, np);
+  gsl_multimin_fminimizer_set (s, &minex_func, x, ss);
+
+  do
+    @{
+      iter++;
+      status = gsl_multimin_fminimizer_iterate(s);
+      
+      if (status) 
+        break;
+
+      size = gsl_multimin_fminimizer_size (s);
+      status = gsl_multimin_test_size (size, 1e-2);
+
+      if (status == GSL_SUCCESS)
+        @{
+          printf ("converged to minimum at\n");
+        @}
+
+      printf ("%5d ", iter);
+      for (i = 0; i < np; i++)
+        @{
+          printf ("%10.3e ", gsl_vector_get (s->x, i));
+        @}
+      printf ("f() = %7.3f size = %.3f\n", s->fval, size);
+    @}
+  while (status == GSL_CONTINUE && iter < 100);
+  
+  gsl_vector_free(x);
+  gsl_vector_free(ss);
+  gsl_multimin_fminimizer_free (s);
+
+  return status;
+@}
+@end smallexample
+
+@noindent
+The minimum search stops when the Simplex size drops to 0.01. The output is
+shown below.
+
+@example
+    1  6.500e+00  5.000e+00 f() = 512.500 size = 1.082
+    2  5.250e+00  4.000e+00 f() = 290.625 size = 1.372
+    3  5.250e+00  4.000e+00 f() = 290.625 size = 1.372
+    4  5.500e+00  1.000e+00 f() = 252.500 size = 1.372
+    5  2.625e+00  3.500e+00 f() = 101.406 size = 1.823
+    6  3.469e+00  1.375e+00 f() = 98.760  size = 1.526
+    7  1.820e+00  3.156e+00 f() = 63.467  size = 1.105
+    8  1.820e+00  3.156e+00 f() = 63.467  size = 1.105
+    9  1.016e+00  2.812e+00 f() = 43.206  size = 1.105
+   10  2.041e+00  2.008e+00 f() = 40.838  size = 0.645
+   11  1.236e+00  1.664e+00 f() = 32.816  size = 0.645
+   12  1.236e+00  1.664e+00 f() = 32.816  size = 0.447
+   13  5.225e-01  1.980e+00 f() = 32.288  size = 0.447
+   14  1.103e+00  2.073e+00 f() = 30.214  size = 0.345
+   15  1.103e+00  2.073e+00 f() = 30.214  size = 0.264
+   16  1.103e+00  2.073e+00 f() = 30.214  size = 0.160
+   17  9.864e-01  1.934e+00 f() = 30.090  size = 0.132
+   18  9.190e-01  1.987e+00 f() = 30.069  size = 0.092
+   19  1.028e+00  2.017e+00 f() = 30.013  size = 0.056
+   20  1.028e+00  2.017e+00 f() = 30.013  size = 0.046
+   21  1.028e+00  2.017e+00 f() = 30.013  size = 0.033
+   22  9.874e-01  1.985e+00 f() = 30.006  size = 0.028
+   23  9.846e-01  1.995e+00 f() = 30.003  size = 0.023
+   24  1.007e+00  2.003e+00 f() = 30.001  size = 0.012
+converged to minimum at                  
+   25  1.007e+00  2.003e+00 f() = 30.001  size = 0.010
+@end example
+
+@noindent
+The simplex size first increases, while the simplex moves towards the
+minimum. After a while the size begins to decrease as the simplex
+contracts around the minimum.
+
+@node Multimin References and Further Reading
+@section References and Further Reading
+
+The conjugate gradient and BFGS methods are described in detail in the
+following book,
+
+@itemize @asis
+@item R. Fletcher,
+@cite{Practical Methods of Optimization (Second Edition)} Wiley
+(1987), ISBN 0471915475.
+@end itemize
+
+A brief description of multidimensional minimization algorithms and
+more recent references can be found in,
+
+@itemize @asis
+@item C.W. Ueberhuber,
+@cite{Numerical Computation (Volume 2)}, Chapter 14, Section 4.4
+``Minimization Methods'', p.@: 325--335, Springer (1997), ISBN
+3-540-62057-5.
+@end itemize
+
+@noindent
+The simplex algorithm is described in the following paper, 
+
+@itemize @asis
+@item J.A. Nelder and R. Mead,
+@cite{A simplex method for function minimization}, Computer Journal
+vol.@: 7 (1965), 308--315.
+@end itemize
+
+@noindent
diff --git a/gsl-1.9/doc/multiroots.texi b/gsl-1.9/doc/multiroots.texi
new file mode 100644
index 0000000..b0dc49c
--- /dev/null
+++ b/gsl-1.9/doc/multiroots.texi
@@ -0,0 +1,1083 @@
+@cindex solving nonlinear systems of equations
+@cindex nonlinear systems of equations, solution of
+@cindex systems of equations, nonlinear
+
+This chapter describes functions for multidimensional root-finding
+(solving nonlinear systems with @math{n} equations in @math{n}
+unknowns).  The library provides low level components for a variety of
+iterative solvers and convergence tests.  These can be combined by the
+user to achieve the desired solution, with full access to the
+intermediate steps of the iteration.  Each class of methods uses the
+same framework, so that you can switch between solvers at runtime
+without needing to recompile your program.  Each instance of a solver
+keeps track of its own state, allowing the solvers to be used in
+multi-threaded programs.  The solvers are based on the original Fortran
+library @sc{minpack}.
+
+The header file @file{gsl_multiroots.h} contains prototypes for the
+multidimensional root finding functions and related declarations.
+
+@menu
+* Overview of Multidimensional Root Finding::  
+* Initializing the Multidimensional Solver::  
+* Providing the multidimensional system of equations to solve::  
+* Iteration of the multidimensional solver::  
+* Search Stopping Parameters for the multidimensional solver::  
+* Algorithms using Derivatives::  
+* Algorithms without Derivatives::  
+* Example programs for Multidimensional Root finding::  
+* References and Further Reading for Multidimensional Root Finding::  
+@end menu
+
+@node Overview of Multidimensional Root Finding
+@section Overview
+@cindex multidimensional root finding, overview
+
+The problem of multidimensional root finding requires the simultaneous
+solution of @math{n} equations, @math{f_i}, in @math{n} variables,
+@math{x_i},
+@tex
+\beforedisplay
+$$
+f_i (x_1, \dots, x_n) = 0 \qquad\hbox{for}~i = 1 \dots n.
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+f_i (x_1, ..., x_n) = 0    for i = 1 ... n.
+@end example
+
+@end ifinfo
+@noindent
+In general there are no bracketing methods available for @math{n}
+dimensional systems, and no way of knowing whether any solutions
+exist.  All algorithms proceed from an initial guess using a variant of
+the Newton iteration,
+@tex
+\beforedisplay
+$$
+x \to x' = x - J^{-1} f(x)
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+x -> x' = x - J^@{-1@} f(x)
+@end example
+
+@end ifinfo
+@noindent
+where @math{x}, @math{f} are vector quantities and @math{J} is the
+Jacobian matrix @c{$J_{ij} = \partial f_i / \partial x_j$} 
+@math{J_@{ij@} = d f_i / d x_j}.
+Additional strategies can be used to enlarge the region of
+convergence.  These include requiring a decrease in the norm @math{|f|} on
+each step proposed by Newton's method, or taking steepest-descent steps in
+the direction of the negative gradient of @math{|f|}.
+
+Several root-finding algorithms are available within a single framework.
+The user provides a high-level driver for the algorithms, and the
+library provides the individual functions necessary for each of the
+steps.  There are three main phases of the iteration.  The steps are,
+
+@itemize @bullet
+@item
+initialize solver state, @var{s}, for algorithm @var{T}
+
+@item
+update @var{s} using the iteration @var{T}
+
+@item
+test @var{s} for convergence, and repeat iteration if necessary
+@end itemize
+
+@noindent
+The evaluation of the Jacobian matrix can be problematic, either because
+programming the derivatives is intractable or because computation of the
+@math{n^2} terms of the matrix becomes too expensive.  For these reasons
+the algorithms provided by the library are divided into two classes according
+to whether the derivatives are available or not.
+
+The state for solvers with an analytic Jacobian matrix is held in a
+@code{gsl_multiroot_fdfsolver} struct.  The updating procedure requires
+both the function and its derivatives to be supplied by the user.
+
+The state for solvers which do not use an analytic Jacobian matrix is
+held in a @code{gsl_multiroot_fsolver} struct.  The updating procedure
+uses only function evaluations (not derivatives).  The algorithms
+estimate the matrix @math{J} or @c{$J^{-1}$} 
+@math{J^@{-1@}} by approximate methods.
+
+@node Initializing the Multidimensional Solver
+@section Initializing the Solver
+
+The following functions initialize a multidimensional solver, either
+with or without derivatives.  The solver itself depends only on the
+dimension of the problem and the algorithm and can be reused for
+different problems.
+
+@deftypefun {gsl_multiroot_fsolver *} gsl_multiroot_fsolver_alloc (const gsl_multiroot_fsolver_type * @var{T}, size_t @var{n})
+This function returns a pointer to a newly allocated instance of a
+solver of type @var{T} for a system of @var{n} dimensions.
+For example, the following code creates an instance of a hybrid solver, 
+to solve a 3-dimensional system of equations.
+
+@example
+const gsl_multiroot_fsolver_type * T 
+    = gsl_multiroot_fsolver_hybrid;
+gsl_multiroot_fsolver * s 
+    = gsl_multiroot_fsolver_alloc (T, 3);
+@end example
+
+@noindent
+If there is insufficient memory to create the solver then the function
+returns a null pointer and the error handler is invoked with an error
+code of @code{GSL_ENOMEM}.
+@end deftypefun
+
+@deftypefun {gsl_multiroot_fdfsolver *} gsl_multiroot_fdfsolver_alloc (const gsl_multiroot_fdfsolver_type * @var{T}, size_t @var{n})
+This function returns a pointer to a newly allocated instance of a
+derivative solver of type @var{T} for a system of @var{n} dimensions.
+For example, the following code creates an instance of a Newton-Raphson solver,
+for a 2-dimensional system of equations.
+
+@example
+const gsl_multiroot_fdfsolver_type * T 
+    = gsl_multiroot_fdfsolver_newton;
+gsl_multiroot_fdfsolver * s = 
+    gsl_multiroot_fdfsolver_alloc (T, 2);
+@end example
+
+@noindent
+If there is insufficient memory to create the solver then the function
+returns a null pointer and the error handler is invoked with an error
+code of @code{GSL_ENOMEM}.
+@end deftypefun
+
+@deftypefun int gsl_multiroot_fsolver_set (gsl_multiroot_fsolver * @var{s}, gsl_multiroot_function * @var{f}, gsl_vector * @var{x})
+This function sets, or resets, an existing solver @var{s} to use the
+function @var{f} and the initial guess @var{x}.
+@end deftypefun
+
+@deftypefun int gsl_multiroot_fdfsolver_set (gsl_multiroot_fdfsolver * @var{s}, gsl_multiroot_function_fdf * @var{fdf}, gsl_vector * @var{x})
+This function sets, or resets, an existing solver @var{s} to use the
+function and derivative @var{fdf} and the initial guess @var{x}.
+@end deftypefun
+
+@deftypefun void gsl_multiroot_fsolver_free (gsl_multiroot_fsolver * @var{s})
+@deftypefunx void gsl_multiroot_fdfsolver_free (gsl_multiroot_fdfsolver * @var{s})
+These functions free all the memory associated with the solver @var{s}.
+@end deftypefun
+
+@deftypefun {const char *} gsl_multiroot_fsolver_name (const gsl_multiroot_fsolver * @var{s})
+@deftypefunx {const char *} gsl_multiroot_fdfsolver_name (const gsl_multiroot_fdfsolver * @var{s})
+These functions return a pointer to the name of the solver.  For example,
+
+@example
+printf ("s is a '%s' solver\n", 
+        gsl_multiroot_fdfsolver_name (s));
+@end example
+
+@noindent
+would print something like @code{s is a 'newton' solver}.
+@end deftypefun
+
+@node Providing the multidimensional system of equations to solve
+@section Providing the function to solve
+@cindex multidimensional root finding, providing a function to solve
+
+You must provide @math{n} functions of @math{n} variables for the root
+finders to operate on.  In order to allow for general parameters the
+functions are defined by the following data types:
+
+@deftp {Data Type} gsl_multiroot_function 
+This data type defines a general system of functions with parameters.
+
+@table @code
+@item int (* f) (const gsl_vector * @var{x}, void * @var{params}, gsl_vector * @var{f})
+this function should store the vector result
+@c{$f(x,\hbox{\it params})$}
+@math{f(x,params)} in @var{f} for argument @var{x} and parameters @var{params},
+returning an appropriate error code if the function cannot be computed.
+
+@item size_t n
+the dimension of the system, i.e. the number of components of the
+vectors @var{x} and @var{f}.
+
+@item void * params
+a pointer to the parameters of the function.
+@end table
+@end deftp
+
+@noindent
+Here is an example using Powell's test function,
+@tex
+\beforedisplay
+$$
+f_1(x) = A x_0 x_1 - 1,
+f_2(x) = \exp(-x_0) + \exp(-x_1) - (1 + 1/A)
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+f_1(x) = A x_0 x_1 - 1,
+f_2(x) = exp(-x_0) + exp(-x_1) - (1 + 1/A)
+@end example
+
+@end ifinfo
+@noindent
+with @math{A = 10^4}.  The following code defines a
+@code{gsl_multiroot_function} system @code{F} which you could pass to a
+solver:
+
+@example
+struct powell_params @{ double A; @};
+
+int
+powell (gsl_vector * x, void * p, gsl_vector * f) @{
+   struct powell_params * params 
+     = *(struct powell_params *)p;
+   const double A = (params->A);
+   const double x0 = gsl_vector_get(x,0);
+   const double x1 = gsl_vector_get(x,1);
+
+   gsl_vector_set (f, 0, A * x0 * x1 - 1);
+   gsl_vector_set (f, 1, (exp(-x0) + exp(-x1) 
+                          - (1.0 + 1.0/A)));
+   return GSL_SUCCESS
+@}
+
+gsl_multiroot_function F;
+struct powell_params params = @{ 10000.0 @};
+
+F.f = &powell;
+F.n = 2;
+F.params = &params;
+@end example
+
+@deftp {Data Type} gsl_multiroot_function_fdf
+This data type defines a general system of functions with parameters and
+the corresponding Jacobian matrix of derivatives,
+
+@table @code
+@item int (* f) (const gsl_vector * @var{x}, void * @var{params}, gsl_vector * @var{f})
+this function should store the vector result
+@c{$f(x,\hbox{\it params})$}
+@math{f(x,params)} in @var{f} for argument @var{x} and parameters @var{params},
+returning an appropriate error code if the function cannot be computed.
+
+@item int (* df) (const gsl_vector * @var{x}, void * @var{params}, gsl_matrix * @var{J})
+this function should store the @var{n}-by-@var{n} matrix result
+@c{$J_{ij} = \partial f_i(x,\hbox{\it params}) / \partial x_j$}
+@math{J_ij = d f_i(x,params) / d x_j} in @var{J} for argument @var{x} 
+and parameters @var{params}, returning an appropriate error code if the
+function cannot be computed.
+
+@item int (* fdf) (const gsl_vector * @var{x}, void * @var{params}, gsl_vector * @var{f}, gsl_matrix * @var{J})
+This function should set the values of the @var{f} and @var{J} as above,
+for arguments @var{x} and parameters @var{params}.  This function
+provides an optimization of the separate functions for @math{f(x)} and
+@math{J(x)}---it is always faster to compute the function and its
+derivative at the same time.
+
+@item size_t n
+the dimension of the system, i.e. the number of components of the
+vectors @var{x} and @var{f}.
+
+@item void * params
+a pointer to the parameters of the function.
+@end table
+@end deftp
+
+@noindent
+The example of Powell's test function defined above can be extended to
+include analytic derivatives using the following code,
+
+@example
+int
+powell_df (gsl_vector * x, void * p, gsl_matrix * J) 
+@{
+   struct powell_params * params 
+     = *(struct powell_params *)p;
+   const double A = (params->A);
+   const double x0 = gsl_vector_get(x,0);
+   const double x1 = gsl_vector_get(x,1);
+   gsl_matrix_set (J, 0, 0, A * x1);
+   gsl_matrix_set (J, 0, 1, A * x0);
+   gsl_matrix_set (J, 1, 0, -exp(-x0));
+   gsl_matrix_set (J, 1, 1, -exp(-x1));
+   return GSL_SUCCESS
+@}
+
+int
+powell_fdf (gsl_vector * x, void * p, 
+            gsl_matrix * f, gsl_matrix * J) @{
+   struct powell_params * params 
+     = *(struct powell_params *)p;
+   const double A = (params->A);
+   const double x0 = gsl_vector_get(x,0);
+   const double x1 = gsl_vector_get(x,1);
+
+   const double u0 = exp(-x0);
+   const double u1 = exp(-x1);
+
+   gsl_vector_set (f, 0, A * x0 * x1 - 1);
+   gsl_vector_set (f, 1, u0 + u1 - (1 + 1/A));
+
+   gsl_matrix_set (J, 0, 0, A * x1);
+   gsl_matrix_set (J, 0, 1, A * x0);
+   gsl_matrix_set (J, 1, 0, -u0);
+   gsl_matrix_set (J, 1, 1, -u1);
+   return GSL_SUCCESS
+@}
+
+gsl_multiroot_function_fdf FDF;
+
+FDF.f = &powell_f;
+FDF.df = &powell_df;
+FDF.fdf = &powell_fdf;
+FDF.n = 2;
+FDF.params = 0;
+@end example
+
+@noindent
+Note that the function @code{powell_fdf} is able to reuse existing terms
+from the function when calculating the Jacobian, thus saving time.
+
+@node Iteration of the multidimensional solver
+@section Iteration
+
+The following functions drive the iteration of each algorithm.  Each
+function performs one iteration to update the state of any solver of the
+corresponding type.  The same functions work for all solvers so that
+different methods can be substituted at runtime without modifications to
+the code.
+
+@deftypefun int gsl_multiroot_fsolver_iterate (gsl_multiroot_fsolver * @var{s})
+@deftypefunx int gsl_multiroot_fdfsolver_iterate (gsl_multiroot_fdfsolver * @var{s})
+These functions perform a single iteration of the solver @var{s}.  If the
+iteration encounters an unexpected problem then an error code will be
+returned,
+
+@table @code
+@item GSL_EBADFUNC
+the iteration encountered a singular point where the function or its
+derivative evaluated to @code{Inf} or @code{NaN}.
+
+@item GSL_ENOPROG
+the iteration is not making any progress, preventing the algorithm from
+continuing.
+@end table
+@end deftypefun
+
+The solver maintains a current best estimate of the root at all times.
+This information can be accessed with the following auxiliary functions,
+
+@deftypefun {gsl_vector *} gsl_multiroot_fsolver_root (const gsl_multiroot_fsolver * @var{s})
+@deftypefunx {gsl_vector *} gsl_multiroot_fdfsolver_root (const gsl_multiroot_fdfsolver * @var{s})
+These functions return the current estimate of the root for the solver @var{s}.
+@end deftypefun
+
+@deftypefun {gsl_vector *} gsl_multiroot_fsolver_f (const gsl_multiroot_fsolver * @var{s})
+@deftypefunx {gsl_vector *} gsl_multiroot_fdfsolver_f (const gsl_multiroot_fdfsolver * @var{s})
+These functions return the function value @math{f(x)} at the current
+estimate of the root for the solver @var{s}.
+@end deftypefun
+
+@deftypefun {gsl_vector *} gsl_multiroot_fsolver_dx (const gsl_multiroot_fsolver * @var{s})
+@deftypefunx {gsl_vector *} gsl_multiroot_fdfsolver_dx (const gsl_multiroot_fdfsolver * @var{s})
+These functions return the last step @math{dx} taken by the solver
+@var{s}.
+@end deftypefun
+
+@node Search Stopping Parameters for the multidimensional solver
+@section Search Stopping Parameters
+@cindex root finding, stopping parameters
+
+A root finding procedure should stop when one of the following conditions is
+true:
+
+@itemize @bullet
+@item
+A multidimensional root has been found to within the user-specified precision.
+
+@item
+A user-specified maximum number of iterations has been reached.
+
+@item
+An error has occurred.
+@end itemize
+
+@noindent
+The handling of these conditions is under user control.  The functions
+below allow the user to test the precision of the current result in
+several standard ways.
+
+@deftypefun int gsl_multiroot_test_delta (const gsl_vector * @var{dx}, const gsl_vector * @var{x}, double @var{epsabs}, double @var{epsrel})
+
+This function tests for the convergence of the sequence by comparing the
+last step @var{dx} with the absolute error @var{epsabs} and relative
+error @var{epsrel} to the current position @var{x}.  The test returns
+@code{GSL_SUCCESS} if the following condition is achieved,
+@tex
+\beforedisplay
+$$
+|dx_i| < \hbox{\it epsabs} + \hbox{\it epsrel\/}\, |x_i|
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+|dx_i| < epsabs + epsrel |x_i|
+@end example
+
+@end ifinfo
+@noindent
+for each component of @var{x} and returns @code{GSL_CONTINUE} otherwise.
+@end deftypefun
+
+@cindex residual, in nonlinear systems of equations
+@deftypefun int gsl_multiroot_test_residual (const gsl_vector * @var{f}, double @var{epsabs})
+This function tests the residual value @var{f} against the absolute
+error bound @var{epsabs}.  The test returns @code{GSL_SUCCESS} if the
+following condition is achieved,
+@tex
+\beforedisplay
+$$
+\sum_i |f_i| < \hbox{\it epsabs}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+\sum_i |f_i| < epsabs
+@end example
+
+@end ifinfo
+@noindent
+and returns @code{GSL_CONTINUE} otherwise.  This criterion is suitable
+for situations where the precise location of the root, @math{x}, is
+unimportant provided a value can be found where the residual is small
+enough.
+@end deftypefun
+
+@comment ============================================================
+
+@node Algorithms using Derivatives
+@section Algorithms using Derivatives
+
+The root finding algorithms described in this section make use of both
+the function and its derivative.  They require an initial guess for the
+location of the root, but there is no absolute guarantee of
+convergence---the function must be suitable for this technique and the
+initial guess must be sufficiently close to the root for it to work.
+When the conditions are satisfied then convergence is quadratic.
+
+
+@comment ============================================================
+@cindex HYBRID algorithms for nonlinear systems
+@deffn {Derivative Solver} gsl_multiroot_fdfsolver_hybridsj
+@cindex HYBRIDSJ algorithm
+@cindex MINPACK, minimization algorithms
+This is a modified version of Powell's Hybrid method as implemented in
+the @sc{hybrj} algorithm in @sc{minpack}.  Minpack was written by Jorge
+J. Mor@'e, Burton S. Garbow and Kenneth E. Hillstrom.  The Hybrid
+algorithm retains the fast convergence of Newton's method but will also
+reduce the residual when Newton's method is unreliable. 
+
+The algorithm uses a generalized trust region to keep each step under
+control.  In order to be accepted a proposed new position @math{x'} must
+satisfy the condition @math{|D (x' - x)| < \delta}, where @math{D} is a
+diagonal scaling matrix and @math{\delta} is the size of the trust
+region.  The components of @math{D} are computed internally, using the
+column norms of the Jacobian to estimate the sensitivity of the residual
+to each component of @math{x}.  This improves the behavior of the
+algorithm for badly scaled functions.
+
+On each iteration the algorithm first determines the standard Newton
+step by solving the system @math{J dx = - f}.  If this step falls inside
+the trust region it is used as a trial step in the next stage.  If not,
+the algorithm uses the linear combination of the Newton and gradient
+directions which is predicted to minimize the norm of the function while
+staying inside the trust region,
+@tex
+\beforedisplay
+$$
+dx = - \alpha J^{-1} f(x) - \beta \nabla |f(x)|^2.
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+dx = - \alpha J^@{-1@} f(x) - \beta \nabla |f(x)|^2.
+@end example
+
+@end ifinfo
+@noindent
+This combination of Newton and gradient directions is referred to as a
+@dfn{dogleg step}.
+
+The proposed step is now tested by evaluating the function at the
+resulting point, @math{x'}.  If the step reduces the norm of the function
+sufficiently then it is accepted and size of the trust region is
+increased.  If the proposed step fails to improve the solution then the
+size of the trust region is decreased and another trial step is
+computed.
+
+The speed of the algorithm is increased by computing the changes to the
+Jacobian approximately, using a rank-1 update.  If two successive
+attempts fail to reduce the residual then the full Jacobian is
+recomputed.  The algorithm also monitors the progress of the solution
+and returns an error if several steps fail to make any improvement,
+
+@table @code
+@item GSL_ENOPROG
+the iteration is not making any progress, preventing the algorithm from
+continuing.
+
+@item GSL_ENOPROGJ
+re-evaluations of the Jacobian indicate that the iteration is not
+making any progress, preventing the algorithm from continuing.
+@end table
+
+@end deffn
+
+@deffn {Derivative Solver} gsl_multiroot_fdfsolver_hybridj
+@cindex HYBRIDJ algorithm
+This algorithm is an unscaled version of @code{hybridsj}.  The steps are
+controlled by a spherical trust region @math{|x' - x| < \delta}, instead
+of a generalized region.  This can be useful if the generalized region
+estimated by @code{hybridsj} is inappropriate.
+@end deffn
+
+
+@deffn {Derivative Solver} gsl_multiroot_fdfsolver_newton
+@cindex Newton's method for systems of nonlinear equations
+
+Newton's Method is the standard root-polishing algorithm.  The algorithm
+begins with an initial guess for the location of the solution.  On each
+iteration a linear approximation to the function @math{F} is used to
+estimate the step which will zero all the components of the residual.
+The iteration is defined by the following sequence,
+@tex
+\beforedisplay
+$$
+x \to x' = x - J^{-1} f(x)
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+x -> x' = x - J^@{-1@} f(x)
+@end example
+
+@end ifinfo
+@noindent
+where the Jacobian matrix @math{J} is computed from the derivative
+functions provided by @var{f}.  The step @math{dx} is obtained by solving
+the linear system,
+@tex
+\beforedisplay
+$$
+J \,dx = - f(x)
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+J dx = - f(x)
+@end example
+
+@end ifinfo
+@noindent
+using LU decomposition.
+@end deffn
+
+@comment ============================================================
+
+@deffn {Derivative Solver} gsl_multiroot_fdfsolver_gnewton
+@cindex Modified Newton's method for nonlinear systems
+@cindex Newton algorithm, globally convergent
+This is a modified version of Newton's method which attempts to improve
+global convergence by requiring every step to reduce the Euclidean norm
+of the residual, @math{|f(x)|}.  If the Newton step leads to an increase
+in the norm then a reduced step of relative size,
+@tex
+\beforedisplay
+$$
+t = (\sqrt{1 + 6 r} - 1) / (3 r)
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+t = (\sqrt(1 + 6 r) - 1) / (3 r)
+@end example
+
+@end ifinfo
+@noindent
+is proposed, with @math{r} being the ratio of norms
+@math{|f(x')|^2/|f(x)|^2}.  This procedure is repeated until a suitable step
+size is found. 
+@end deffn
+
+@comment ============================================================
+
+@node Algorithms without Derivatives
+@section Algorithms without Derivatives
+
+The algorithms described in this section do not require any derivative
+information to be supplied by the user.  Any derivatives needed are
+approximated by finite differences.  Note that if the
+finite-differencing step size chosen by these routines is inappropriate,
+an explicit user-supplied numerical derivative can always be used with
+the algorithms described in the previous section.
+
+@deffn {Solver} gsl_multiroot_fsolver_hybrids
+@cindex HYBRIDS algorithm, scaled without derivatives
+This is a version of the Hybrid algorithm which replaces calls to the
+Jacobian function by its finite difference approximation.  The finite
+difference approximation is computed using @code{gsl_multiroots_fdjac}
+with a relative step size of @code{GSL_SQRT_DBL_EPSILON}.  Note that
+this step size will not be suitable for all problems.
+@end deffn
+
+@deffn {Solver} gsl_multiroot_fsolver_hybrid
+@cindex HYBRID algorithm, unscaled without derivatives
+This is a finite difference version of the Hybrid algorithm without
+internal scaling.
+@end deffn
+
+@comment ============================================================
+
+@deffn {Solver} gsl_multiroot_fsolver_dnewton
+
+@cindex Discrete Newton algorithm for multidimensional roots
+@cindex Newton algorithm, discrete
+
+The @dfn{discrete Newton algorithm} is the simplest method of solving a
+multidimensional system.  It uses the Newton iteration
+@tex
+\beforedisplay
+$$
+x \to x - J^{-1} f(x)
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+x -> x - J^@{-1@} f(x)
+@end example
+
+@end ifinfo
+@noindent
+where the Jacobian matrix @math{J} is approximated by taking finite
+differences of the function @var{f}.  The approximation scheme used by
+this implementation is,
+@tex
+\beforedisplay
+$$
+J_{ij} = (f_i(x + \delta_j) - f_i(x)) /  \delta_j
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+J_@{ij@} = (f_i(x + \delta_j) - f_i(x)) /  \delta_j
+@end example
+
+@end ifinfo
+@noindent
+where @math{\delta_j} is a step of size @math{\sqrt\epsilon |x_j|} with
+@math{\epsilon} being the machine precision 
+(@c{$\epsilon \approx 2.22 \times 10^{-16}$}
+@math{\epsilon \approx 2.22 \times 10^-16}).
+The order of convergence of Newton's algorithm is quadratic, but the
+finite differences require @math{n^2} function evaluations on each
+iteration.  The algorithm may become unstable if the finite differences
+are not a good approximation to the true derivatives.
+@end deffn
+
+@comment ============================================================
+
+@deffn {Solver} gsl_multiroot_fsolver_broyden
+@cindex Broyden algorithm for multidimensional roots
+@cindex multidimensional root finding, Broyden algorithm
+
+The @dfn{Broyden algorithm} is a version of the discrete Newton
+algorithm which attempts to avoids the expensive update of the Jacobian
+matrix on each iteration.  The changes to the Jacobian are also
+approximated, using a rank-1 update,
+@tex
+\beforedisplay
+$$
+J^{-1} \to J^{-1} - (J^{-1} df - dx) dx^T J^{-1} / dx^T J^{-1} df
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+J^@{-1@} \to J^@{-1@} - (J^@{-1@} df - dx) dx^T J^@{-1@} / dx^T J^@{-1@} df
+@end example
+
+@end ifinfo
+@noindent
+where the vectors @math{dx} and @math{df} are the changes in @math{x}
+and @math{f}.  On the first iteration the inverse Jacobian is estimated
+using finite differences, as in the discrete Newton algorithm.
+ 
+This approximation gives a fast update but is unreliable if the changes
+are not small, and the estimate of the inverse Jacobian becomes worse as
+time passes.  The algorithm has a tendency to become unstable unless it
+starts close to the root.  The Jacobian is refreshed if this instability
+is detected (consult the source for details).
+
+This algorithm is included only for demonstration purposes, and is not
+recommended for serious use.
+@end deffn
+
+@comment ============================================================
+
+
+@node Example programs for Multidimensional Root finding
+@section Examples
+
+The multidimensional solvers are used in a similar way to the
+one-dimensional root finding algorithms.  This first example
+demonstrates the @code{hybrids} scaled-hybrid algorithm, which does not
+require derivatives. The program solves the Rosenbrock system of equations,
+@tex
+\beforedisplay
+$$
+f_1 (x, y) = a (1 - x),~
+f_2 (x, y) = b (y - x^2)
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+f_1 (x, y) = a (1 - x)
+f_2 (x, y) = b (y - x^2)
+@end example
+
+@end ifinfo
+@noindent
+with @math{a = 1, b = 10}. The solution of this system lies at
+@math{(x,y) = (1,1)} in a narrow valley.
+
+The first stage of the program is to define the system of equations,
+
+@example
+#include <stdlib.h>
+#include <stdio.h>
+#include <gsl/gsl_vector.h>
+#include <gsl/gsl_multiroots.h>
+
+struct rparams
+  @{
+    double a;
+    double b;
+  @};
+
+int
+rosenbrock_f (const gsl_vector * x, void *params, 
+              gsl_vector * f)
+@{
+  double a = ((struct rparams *) params)->a;
+  double b = ((struct rparams *) params)->b;
+
+  const double x0 = gsl_vector_get (x, 0);
+  const double x1 = gsl_vector_get (x, 1);
+
+  const double y0 = a * (1 - x0);
+  const double y1 = b * (x1 - x0 * x0);
+
+  gsl_vector_set (f, 0, y0);
+  gsl_vector_set (f, 1, y1);
+
+  return GSL_SUCCESS;
+@}
+@end example
+
+@noindent
+The main program begins by creating the function object @code{f}, with
+the arguments @code{(x,y)} and parameters @code{(a,b)}. The solver
+@code{s} is initialized to use this function, with the @code{hybrids}
+method.
+
+@example
+int
+main (void)
+@{
+  const gsl_multiroot_fsolver_type *T;
+  gsl_multiroot_fsolver *s;
+
+  int status;
+  size_t i, iter = 0;
+
+  const size_t n = 2;
+  struct rparams p = @{1.0, 10.0@};
+  gsl_multiroot_function f = @{&rosenbrock_f, n, &p@};
+
+  double x_init[2] = @{-10.0, -5.0@};
+  gsl_vector *x = gsl_vector_alloc (n);
+
+  gsl_vector_set (x, 0, x_init[0]);
+  gsl_vector_set (x, 1, x_init[1]);
+
+  T = gsl_multiroot_fsolver_hybrids;
+  s = gsl_multiroot_fsolver_alloc (T, 2);
+  gsl_multiroot_fsolver_set (s, &f, x);
+
+  print_state (iter, s);
+
+  do
+    @{
+      iter++;
+      status = gsl_multiroot_fsolver_iterate (s);
+
+      print_state (iter, s);
+
+      if (status)   /* check if solver is stuck */
+        break;
+
+      status = 
+        gsl_multiroot_test_residual (s->f, 1e-7);
+    @}
+  while (status == GSL_CONTINUE && iter < 1000);
+
+  printf ("status = %s\n", gsl_strerror (status));
+
+  gsl_multiroot_fsolver_free (s);
+  gsl_vector_free (x);
+  return 0;
+@}
+@end example
+
+@noindent
+Note that it is important to check the return status of each solver
+step, in case the algorithm becomes stuck.  If an error condition is
+detected, indicating that the algorithm cannot proceed, then the error
+can be reported to the user, a new starting point chosen or a different
+algorithm used.
+
+The intermediate state of the solution is displayed by the following
+function.  The solver state contains the vector @code{s->x} which is the
+current position, and the vector @code{s->f} with corresponding function
+values.
+
+@example
+int
+print_state (size_t iter, gsl_multiroot_fsolver * s)
+@{
+  printf ("iter = %3u x = % .3f % .3f "
+          "f(x) = % .3e % .3e\n",
+          iter,
+          gsl_vector_get (s->x, 0), 
+          gsl_vector_get (s->x, 1),
+          gsl_vector_get (s->f, 0), 
+          gsl_vector_get (s->f, 1));
+@}
+@end example
+
+@noindent
+Here are the results of running the program. The algorithm is started at
+@math{(-10,-5)} far from the solution.  Since the solution is hidden in
+a narrow valley the earliest steps follow the gradient of the function
+downhill, in an attempt to reduce the large value of the residual. Once
+the root has been approximately located, on iteration 8, the Newton
+behavior takes over and convergence is very rapid.
+
+@smallexample
+iter =  0 x = -10.000  -5.000  f(x) = 1.100e+01 -1.050e+03
+iter =  1 x = -10.000  -5.000  f(x) = 1.100e+01 -1.050e+03
+iter =  2 x =  -3.976  24.827  f(x) = 4.976e+00  9.020e+01
+iter =  3 x =  -3.976  24.827  f(x) = 4.976e+00  9.020e+01
+iter =  4 x =  -3.976  24.827  f(x) = 4.976e+00  9.020e+01
+iter =  5 x =  -1.274  -5.680  f(x) = 2.274e+00 -7.302e+01
+iter =  6 x =  -1.274  -5.680  f(x) = 2.274e+00 -7.302e+01
+iter =  7 x =   0.249   0.298  f(x) = 7.511e-01  2.359e+00
+iter =  8 x =   0.249   0.298  f(x) = 7.511e-01  2.359e+00
+iter =  9 x =   1.000   0.878  f(x) = 1.268e-10 -1.218e+00
+iter = 10 x =   1.000   0.989  f(x) = 1.124e-11 -1.080e-01
+iter = 11 x =   1.000   1.000  f(x) = 0.000e+00  0.000e+00
+status = success
+@end smallexample
+
+@noindent
+Note that the algorithm does not update the location on every
+iteration. Some iterations are used to adjust the trust-region
+parameter, after trying a step which was found to be divergent, or to
+recompute the Jacobian, when poor convergence behavior is detected.
+
+The next example program adds derivative information, in order to
+accelerate the solution. There are two derivative functions
+@code{rosenbrock_df} and @code{rosenbrock_fdf}. The latter computes both
+the function and its derivative simultaneously. This allows the
+optimization of any common terms.  For simplicity we substitute calls to
+the separate @code{f} and @code{df} functions at this point in the code
+below.
+
+@example
+int
+rosenbrock_df (const gsl_vector * x, void *params, 
+               gsl_matrix * J)
+@{
+  const double a = ((struct rparams *) params)->a;
+  const double b = ((struct rparams *) params)->b;
+
+  const double x0 = gsl_vector_get (x, 0);
+
+  const double df00 = -a;
+  const double df01 = 0;
+  const double df10 = -2 * b  * x0;
+  const double df11 = b;
+
+  gsl_matrix_set (J, 0, 0, df00);
+  gsl_matrix_set (J, 0, 1, df01);
+  gsl_matrix_set (J, 1, 0, df10);
+  gsl_matrix_set (J, 1, 1, df11);
+
+  return GSL_SUCCESS;
+@}
+
+int
+rosenbrock_fdf (const gsl_vector * x, void *params,
+                gsl_vector * f, gsl_matrix * J)
+@{
+  rosenbrock_f (x, params, f);
+  rosenbrock_df (x, params, J);
+
+  return GSL_SUCCESS;
+@}
+@end example
+
+@noindent
+The main program now makes calls to the corresponding @code{fdfsolver}
+versions of the functions,
+
+@example
+int
+main (void)
+@{
+  const gsl_multiroot_fdfsolver_type *T;
+  gsl_multiroot_fdfsolver *s;
+
+  int status;
+  size_t i, iter = 0;
+
+  const size_t n = 2;
+  struct rparams p = @{1.0, 10.0@};
+  gsl_multiroot_function_fdf f = @{&rosenbrock_f, 
+                                  &rosenbrock_df, 
+                                  &rosenbrock_fdf, 
+                                  n, &p@};
+
+  double x_init[2] = @{-10.0, -5.0@};
+  gsl_vector *x = gsl_vector_alloc (n);
+
+  gsl_vector_set (x, 0, x_init[0]);
+  gsl_vector_set (x, 1, x_init[1]);
+
+  T = gsl_multiroot_fdfsolver_gnewton;
+  s = gsl_multiroot_fdfsolver_alloc (T, n);
+  gsl_multiroot_fdfsolver_set (s, &f, x);
+
+  print_state (iter, s);
+
+  do
+    @{
+      iter++;
+
+      status = gsl_multiroot_fdfsolver_iterate (s);
+
+      print_state (iter, s);
+
+      if (status)
+        break;
+
+      status = gsl_multiroot_test_residual (s->f, 1e-7);
+    @}
+  while (status == GSL_CONTINUE && iter < 1000);
+
+  printf ("status = %s\n", gsl_strerror (status));
+
+  gsl_multiroot_fdfsolver_free (s);
+  gsl_vector_free (x);
+  return 0;
+@}
+@end example
+
+@noindent
+The addition of derivative information to the @code{hybrids} solver does
+not make any significant difference to its behavior, since it able to
+approximate the Jacobian numerically with sufficient accuracy.  To
+illustrate the behavior of a different derivative solver we switch to
+@code{gnewton}. This is a traditional Newton solver with the constraint
+that it scales back its step if the full step would lead ``uphill''. Here
+is the output for the @code{gnewton} algorithm,
+
+@smallexample
+iter = 0 x = -10.000  -5.000 f(x) =  1.100e+01 -1.050e+03
+iter = 1 x =  -4.231 -65.317 f(x) =  5.231e+00 -8.321e+02
+iter = 2 x =   1.000 -26.358 f(x) = -8.882e-16 -2.736e+02
+iter = 3 x =   1.000   1.000 f(x) = -2.220e-16 -4.441e-15
+status = success
+@end smallexample
+
+@noindent
+The convergence is much more rapid, but takes a wide excursion out to
+the point @math{(-4.23,-65.3)}. This could cause the algorithm to go
+astray in a realistic application.  The hybrid algorithm follows the
+downhill path to the solution more reliably.
+
+@node References and Further Reading for Multidimensional Root Finding
+@section References and Further Reading
+
+The original version of the Hybrid method is described in the following
+articles by Powell,
+
+@itemize @asis
+@item
+M.J.D. Powell, ``A Hybrid Method for Nonlinear Equations'' (Chap 6, p
+87--114) and ``A Fortran Subroutine for Solving systems of Nonlinear
+Algebraic Equations'' (Chap 7, p 115--161), in @cite{Numerical Methods for
+Nonlinear Algebraic Equations}, P. Rabinowitz, editor.  Gordon and
+Breach, 1970.
+@end itemize
+
+@noindent
+The following papers are also relevant to the algorithms described in
+this section,
+
+@itemize @asis
+@item
+J.J. Mor@'e, M.Y. Cosnard, ``Numerical Solution of Nonlinear Equations'',
+@cite{ACM Transactions on Mathematical Software}, Vol 5, No 1, (1979), p 64--85
+
+@item
+C.G. Broyden, ``A Class of Methods for Solving Nonlinear
+Simultaneous Equations'', @cite{Mathematics of Computation}, Vol 19 (1965),
+p 577--593
+
+@item 
+J.J. Mor@'e, B.S. Garbow, K.E. Hillstrom, ``Testing Unconstrained
+Optimization Software'', ACM Transactions on Mathematical Software, Vol
+7, No 1 (1981), p 17--41
+@end itemize
+
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diff --git a/gsl-1.9/doc/ntuple.texi b/gsl-1.9/doc/ntuple.texi
new file mode 100644
index 0000000..ccbd156
--- /dev/null
+++ b/gsl-1.9/doc/ntuple.texi
@@ -0,0 +1,194 @@
+@cindex ntuples
+
+This chapter describes functions for creating and manipulating
+@dfn{ntuples}, sets of values associated with events.  The ntuples
+are stored in files. Their values can be extracted in any combination
+and @dfn{booked} in a histogram using a selection function.
+
+The values to be stored are held in a user-defined data structure, and
+an ntuple is created associating this data structure with a file.  The
+values are then written to the file (normally inside a loop) using
+the ntuple functions described below.
+
+A histogram can be created from ntuple data by providing a selection
+function and a value function.  The selection function specifies whether
+an event should be included in the subset to be analyzed or not. The value
+function computes the entry to be added to the histogram for each
+event.
+
+All the ntuple functions are defined in the header file
+@file{gsl_ntuple.h}
+
+@menu
+* The ntuple struct::           
+* Creating ntuples::            
+* Opening an existing ntuple file::  
+* Writing ntuples::             
+* Reading ntuples ::            
+* Closing an ntuple file::      
+* Histogramming ntuple values::  
+* Example ntuple programs::     
+* Ntuple References and Further Reading::  
+@end menu
+
+@node The ntuple struct
+@section The ntuple struct
+
+Ntuples are manipulated using the @code{gsl_ntuple} struct. This struct
+contains information on the file where the ntuple data is stored, a
+pointer to the current ntuple data row and the size of the user-defined
+ntuple data struct.
+
+@example
+typedef struct @{
+    FILE * file;
+    void * ntuple_data;
+    size_t size;
+@} gsl_ntuple;
+@end example
+
+@node Creating ntuples
+@section Creating ntuples
+
+@deftypefun {gsl_ntuple *} gsl_ntuple_create (char * @var{filename}, void * @var{ntuple_data}, size_t @var{size})
+This function creates a new write-only ntuple file @var{filename} for
+ntuples of size @var{size} and returns a pointer to the newly created
+ntuple struct.  Any existing file with the same name is truncated to
+zero length and overwritten.  A pointer to memory for the current ntuple
+row @var{ntuple_data} must be supplied---this is used to copy ntuples
+in and out of the file.
+@end deftypefun
+
+@node Opening an existing ntuple file
+@section Opening an existing ntuple file
+
+@deftypefun {gsl_ntuple *} gsl_ntuple_open (char * @var{filename}, void * @var{ntuple_data}, size_t @var{size})
+This function opens an existing ntuple file @var{filename} for reading
+and returns a pointer to a corresponding ntuple struct. The ntuples in
+the file must have size @var{size}.  A pointer to memory for the current
+ntuple row @var{ntuple_data} must be supplied---this is used to copy
+ntuples in and out of the file.
+@end deftypefun
+
+@node Writing ntuples
+@section Writing ntuples
+
+@deftypefun int gsl_ntuple_write (gsl_ntuple * @var{ntuple})
+This function writes the current ntuple @var{ntuple->ntuple_data} of
+size @var{ntuple->size} to the corresponding file.
+@end deftypefun
+
+@deftypefun int gsl_ntuple_bookdata (gsl_ntuple * @var{ntuple})
+This function is a synonym for @code{gsl_ntuple_write}.
+@end deftypefun
+
+@node Reading ntuples 
+@section Reading ntuples
+
+@deftypefun int gsl_ntuple_read (gsl_ntuple * @var{ntuple})
+This function reads the current row of the ntuple file for @var{ntuple}
+and stores the values in @var{ntuple->data}.
+@end deftypefun
+
+@node Closing an ntuple file
+@section Closing an ntuple file
+
+@deftypefun int gsl_ntuple_close (gsl_ntuple * @var{ntuple})
+This function closes the ntuple file @var{ntuple} and frees its
+associated allocated memory.
+@end deftypefun
+
+@node Histogramming ntuple values
+@section Histogramming ntuple values
+
+Once an ntuple has been created its contents can be histogrammed in
+various ways using the function @code{gsl_ntuple_project}.  Two
+user-defined functions must be provided, a function to select events and
+a function to compute scalar values. The selection function and the
+value function both accept the ntuple row as a first argument and other
+parameters as a second argument.
+
+@cindex selection function, ntuples
+The @dfn{selection function} determines which ntuple rows are selected
+for histogramming.  It is defined by the following struct,
+@smallexample
+typedef struct @{
+  int (* function) (void * ntuple_data, void * params);
+  void * params;
+@} gsl_ntuple_select_fn;
+@end smallexample
+
+@noindent
+The struct component @var{function} should return a non-zero value for
+each ntuple row that is to be included in the histogram.
+
+@cindex value function, ntuples
+The @dfn{value function} computes scalar values for those ntuple rows
+selected by the selection function,
+@smallexample
+typedef struct @{
+  double (* function) (void * ntuple_data, void * params);
+  void * params;
+@} gsl_ntuple_value_fn;
+@end smallexample
+
+@noindent
+In this case the struct component @var{function} should return the value
+to be added to the histogram for the ntuple row.  
+
+@cindex histogram, from ntuple
+@cindex projection of ntuples
+@deftypefun int gsl_ntuple_project (gsl_histogram * @var{h}, gsl_ntuple * @var{ntuple}, gsl_ntuple_value_fn * @var{value_func}, gsl_ntuple_select_fn * @var{select_func})
+This function updates the histogram @var{h} from the ntuple @var{ntuple}
+using the functions @var{value_func} and @var{select_func}. For each
+ntuple row where the selection function @var{select_func} is non-zero the
+corresponding value of that row is computed using the function
+@var{value_func} and added to the histogram.  Those ntuple rows where
+@var{select_func} returns zero are ignored.  New entries are added to
+the histogram, so subsequent calls can be used to accumulate further
+data in the same histogram.
+@end deftypefun
+
+@node Example ntuple programs
+@section Examples
+
+The following example programs demonstrate the use of ntuples in
+managing a large dataset.  The first program creates a set of 10,000
+simulated ``events'', each with 3 associated values @math{(x,y,z)}.  These
+are generated from a gaussian distribution with unit variance, for
+demonstration purposes, and written to the ntuple file @file{test.dat}.
+
+@example
+@verbatiminclude examples/ntuplew.c
+@end example
+
+@noindent
+The next program analyses the ntuple data in the file @file{test.dat}.
+The analysis procedure is to compute the squared-magnitude of each
+event, @math{E^2=x^2+y^2+z^2}, and select only those which exceed a
+lower limit of 1.5.  The selected events are then histogrammed using
+their @math{E^2} values.
+
+@example
+@verbatiminclude examples/ntupler.c
+@end example
+
+@need 3000
+The following plot shows the distribution of the selected events.
+Note the cut-off at the lower bound.
+
+@iftex
+@sp 1
+@center @image{ntuple,3.4in}
+@end iftex
+
+@node Ntuple References and Further Reading
+@section References and Further Reading
+@cindex PAW
+@cindex HBOOK
+
+Further information on the use of ntuples can be found in the
+documentation for the @sc{cern} packages @sc{paw} and @sc{hbook}
+(available online).
+
+
diff --git a/gsl-1.9/doc/ode-initval.texi b/gsl-1.9/doc/ode-initval.texi
new file mode 100644
index 0000000..ae8e68b
--- /dev/null
+++ b/gsl-1.9/doc/ode-initval.texi
@@ -0,0 +1,556 @@
+@cindex differential equations, initial value problems
+@cindex initial value problems, differential equations
+@cindex ordinary differential equations, initial value problem
+@cindex ODEs, initial value problems
+This chapter describes functions for solving ordinary differential
+equation (ODE) initial value problems.  The library provides a variety
+of low-level methods, such as Runge-Kutta and Bulirsch-Stoer routines,
+and higher-level components for adaptive step-size control.  The
+components can be combined by the user to achieve the desired solution,
+with full access to any intermediate steps.
+
+These functions are declared in the header file @file{gsl_odeiv.h}.
+
+@menu
+* Defining the ODE System::     
+* Stepping Functions::          
+* Adaptive Step-size Control::  
+* Evolution::                   
+* ODE Example programs::        
+* ODE References and Further Reading::  
+@end menu
+
+@node Defining the ODE System
+@section Defining the ODE System
+
+The routines solve the general @math{n}-dimensional first-order system,
+@tex
+\beforedisplay
+$$
+{dy_i(t) \over dt} = f_i (t, y_1(t), \dots y_n(t))
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+dy_i(t)/dt = f_i(t, y_1(t), ..., y_n(t))
+@end example
+
+@end ifinfo
+@noindent
+for @math{i = 1, \dots, n}.  The stepping functions rely on the vector
+of derivatives @math{f_i} and the Jacobian matrix, 
+@c{$J_{ij} = \partial f_i(t, y(t)) / \partial y_j$}
+@math{J_@{ij@} = df_i(t,y(t)) / dy_j}. 
+A system of equations is defined using the @code{gsl_odeiv_system}
+datatype.
+
+@deftp {Data Type} gsl_odeiv_system
+This data type defines a general ODE system with arbitrary parameters.
+
+@table @code
+@item int (* function) (double t, const double y[], double dydt[], void * params)
+This function should store the vector elements
+@c{$f_i(t,y,\hbox{\it params})$}
+@math{f_i(t,y,params)} in the array @var{dydt},
+for arguments (@var{t},@var{y}) and parameters @var{params}.
+The function should return @code{GSL_SUCCESS} if the calculation 
+was completed successfully. Any other return value indicates
+an error.
+
+@item  int (* jacobian) (double t, const double y[], double * dfdy, double dfdt[], void * params);
+This function should store the vector of derivative elements 
+@c{$\partial f_i(t,y,params) / \partial t$}
+@math{df_i(t,y,params)/dt} in the array @var{dfdt} and the 
+Jacobian matrix @c{$J_{ij}$}
+@math{J_@{ij@}} in the array
+@var{dfdy}, regarded as a row-ordered matrix @code{J(i,j) = dfdy[i * dimension + j]}
+where @code{dimension} is the dimension of the system.
+The function should return @code{GSL_SUCCESS} if the calculation 
+was completed successfully.  Any other return value indicates
+an error.
+
+Some of the simpler solver algorithms do not make use of the Jacobian
+matrix, so it is not always strictly necessary to provide it (the
+@code{jacobian} element of the struct can be replaced by a null pointer
+for those algorithms).  However, it is useful to provide the Jacobian to allow
+the solver algorithms to be interchanged---the best algorithms make use
+of the Jacobian.
+
+@item size_t dimension;
+This is the dimension of the system of equations.
+
+@item void * params
+This is a pointer to the arbitrary parameters of the system.
+@end table
+@end deftp
+
+@node Stepping Functions
+@section Stepping Functions
+
+The lowest level components are the @dfn{stepping functions} which
+advance a solution from time @math{t} to @math{t+h} for a fixed
+step-size @math{h} and estimate the resulting local error.
+
+@deftypefun {gsl_odeiv_step *} gsl_odeiv_step_alloc (const gsl_odeiv_step_type * @var{T}, size_t @var{dim})
+This function returns a pointer to a newly allocated instance of a
+stepping function of type @var{T} for a system of @var{dim} dimensions.
+@end deftypefun
+
+@deftypefun int gsl_odeiv_step_reset (gsl_odeiv_step * @var{s})
+This function resets the stepping function @var{s}.  It should be used
+whenever the next use of @var{s} will not be a continuation of a
+previous step.
+@end deftypefun
+
+@deftypefun void gsl_odeiv_step_free (gsl_odeiv_step * @var{s})
+This function frees all the memory associated with the stepping function
+@var{s}.
+@end deftypefun
+
+@deftypefun {const char *} gsl_odeiv_step_name (const gsl_odeiv_step * @var{s})
+This function returns a pointer to the name of the stepping function.
+For example,
+
+@example
+printf ("step method is '%s'\n",
+         gsl_odeiv_step_name (s));
+@end example
+
+@noindent
+would print something like @code{step method is 'rk4'}.
+@end deftypefun
+
+@deftypefun {unsigned int} gsl_odeiv_step_order (const gsl_odeiv_step * @var{s})
+This function returns the order of the stepping function on the previous
+step.  This order can vary if the stepping function itself is adaptive.
+@end deftypefun
+
+@deftypefun int gsl_odeiv_step_apply (gsl_odeiv_step * @var{s}, double @var{t}, double @var{h}, double @var{y}[], double @var{yerr}[], const double @var{dydt_in}[], double @var{dydt_out}[], const gsl_odeiv_system * @var{dydt})
+This function applies the stepping function @var{s} to the system of
+equations defined by @var{dydt}, using the step size @var{h} to advance
+the system from time @var{t} and state @var{y} to time @var{t}+@var{h}.
+The new state of the system is stored in @var{y} on output, with an
+estimate of the absolute error in each component stored in @var{yerr}.
+If the argument @var{dydt_in} is not null it should point an array
+containing the derivatives for the system at time @var{t} on input. This
+is optional as the derivatives will be computed internally if they are
+not provided, but allows the reuse of existing derivative information.
+On output the new derivatives of the system at time @var{t}+@var{h} will
+be stored in @var{dydt_out} if it is not null.
+
+If the user-supplied functions defined in the system @var{dydt} return a
+status other than @code{GSL_SUCCESS} the step will be aborted.  In this
+case, the elements of @var{y} will be restored to their pre-step values
+and the error code from the user-supplied function will be returned. 
+The step-size @var{h} will be set to the step-size which caused the error.  
+If the function is called again with a smaller step-size, e.g. @math{@var{h}/10},  
+it should be possible to get closer to any singularity.
+To distinguish between error codes from the user-supplied functions and
+those from @code{gsl_odeiv_step_apply} itself, any user-defined return
+values should be distinct from the standard GSL error codes.
+@end deftypefun
+
+The following algorithms are available,
+
+@deffn {Step Type} gsl_odeiv_step_rk2
+@cindex RK2, Runge-Kutta method
+@cindex Runge-Kutta methods, ordinary differential equations
+Embedded Runge-Kutta (2, 3) method.
+@end deffn
+
+@deffn {Step Type} gsl_odeiv_step_rk4
+@cindex RK4, Runge-Kutta method
+4th order (classical) Runge-Kutta.
+@end deffn
+
+@deffn {Step Type} gsl_odeiv_step_rkf45
+@cindex Fehlberg method, differential equations
+@cindex RKF45, Runge-Kutta-Fehlberg method
+Embedded Runge-Kutta-Fehlberg (4, 5) method.  This method is a good
+general-purpose integrator.
+@end deffn
+
+@deffn {Step Type} gsl_odeiv_step_rkck 
+@cindex Runge-Kutta Cash-Karp method
+@cindex Cash-Karp, Runge-Kutta method
+Embedded Runge-Kutta Cash-Karp (4, 5) method.
+@end deffn
+
+@deffn {Step Type} gsl_odeiv_step_rk8pd  
+@cindex Runge-Kutta Prince-Dormand method
+@cindex Prince-Dormand, Runge-Kutta method
+Embedded Runge-Kutta Prince-Dormand (8,9) method.
+@end deffn
+
+@deffn {Step Type} gsl_odeiv_step_rk2imp  
+Implicit 2nd order Runge-Kutta at Gaussian points.
+@end deffn
+
+@deffn {Step Type} gsl_odeiv_step_rk4imp  
+Implicit 4th order Runge-Kutta at Gaussian points.
+@end deffn
+
+@deffn {Step Type} gsl_odeiv_step_bsimp   
+@cindex Bulirsch-Stoer method
+@cindex Bader and Deuflhard, Bulirsch-Stoer method.
+@cindex Deuflhard and Bader, Bulirsch-Stoer method.
+Implicit Bulirsch-Stoer method of Bader and Deuflhard.  This algorithm
+requires the Jacobian.
+@end deffn
+
+@deffn {Step Type} gsl_odeiv_step_gear1 
+@cindex Gear method, differential equations
+M=1 implicit Gear method.
+@end deffn
+
+@deffn {Step Type} gsl_odeiv_step_gear2 
+M=2 implicit Gear method.
+@end deffn
+
+@node Adaptive Step-size Control
+@section Adaptive Step-size Control
+@cindex Adaptive step-size control, differential equations
+
+The control function examines the proposed change to the solution
+produced by a stepping function and attempts to determine the optimal
+step-size for a user-specified level of error.
+
+@deftypefun {gsl_odeiv_control *} gsl_odeiv_control_standard_new (double @var{eps_abs}, double @var{eps_rel}, double @var{a_y}, double @var{a_dydt})
+The standard control object is a four parameter heuristic based on
+absolute and relative errors @var{eps_abs} and @var{eps_rel}, and
+scaling factors @var{a_y} and @var{a_dydt} for the system state
+@math{y(t)} and derivatives @math{y'(t)} respectively.
+
+The step-size adjustment procedure for this method begins by computing
+the desired error level @math{D_i} for each component,
+@tex
+\beforedisplay
+$$
+D_i = \epsilon_{abs} + \epsilon_{rel} * (a_{y} |y_i| + a_{dydt} h |y'_i|)
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+D_i = eps_abs + eps_rel * (a_y |y_i| + a_dydt h |y'_i|)
+@end example
+
+@end ifinfo
+@noindent
+and comparing it with the observed error @math{E_i = |yerr_i|}.  If the
+observed error @var{E} exceeds the desired error level @var{D} by more
+than 10% for any component then the method reduces the step-size by an
+appropriate factor,
+@tex
+\beforedisplay
+$$
+h_{new} = h_{old} * S * (E/D)^{-1/q}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+h_new = h_old * S * (E/D)^(-1/q)
+@end example
+
+@end ifinfo
+@noindent
+where @math{q} is the consistency order of the method (e.g. @math{q=4} for
+4(5) embedded RK), and @math{S} is a safety factor of 0.9. The ratio
+@math{E/D} is taken to be the maximum of the ratios
+@math{E_i/D_i}. 
+
+If the observed error @math{E} is less than 50% of the desired error
+level @var{D} for the maximum ratio @math{E_i/D_i} then the algorithm
+takes the opportunity to increase the step-size to bring the error in
+line with the desired level,
+@tex
+\beforedisplay
+$$
+h_{new} = h_{old} * S * (E/D)^{-1/(q+1)}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+h_new = h_old * S * (E/D)^(-1/(q+1))
+@end example
+
+@end ifinfo
+@noindent
+This encompasses all the standard error scaling methods. To avoid
+uncontrolled changes in the stepsize, the overall scaling factor is
+limited to the range @math{1/5} to 5.
+@end deftypefun
+
+@deftypefun {gsl_odeiv_control *} gsl_odeiv_control_y_new (double @var{eps_abs}, double @var{eps_rel})
+This function creates a new control object which will keep the local
+error on each step within an absolute error of @var{eps_abs} and
+relative error of @var{eps_rel} with respect to the solution @math{y_i(t)}.
+This is equivalent to the standard control object with @var{a_y}=1 and
+@var{a_dydt}=0.
+@end deftypefun
+
+@deftypefun {gsl_odeiv_control *} gsl_odeiv_control_yp_new (double @var{eps_abs}, double @var{eps_rel})
+This function creates a new control object which will keep the local
+error on each step within an absolute error of @var{eps_abs} and
+relative error of @var{eps_rel} with respect to the derivatives of the
+solution @math{y'_i(t)}.  This is equivalent to the standard control
+object with @var{a_y}=0 and @var{a_dydt}=1.
+@end deftypefun
+
+
+@deftypefun {gsl_odeiv_control *} gsl_odeiv_control_scaled_new (double @var{eps_abs}, double @var{eps_rel}, double @var{a_y}, double @var{a_dydt}, const double @var{scale_abs}[], size_t @var{dim})
+This function creates a new control object which uses the same algorithm
+as @code{gsl_odeiv_control_standard_new} but with an absolute error
+which is scaled for each component by the array @var{scale_abs}.
+The formula for @math{D_i} for this control object is,
+@tex
+\beforedisplay
+$$
+D_i = \epsilon_{abs} s_i + \epsilon_{rel} * (a_{y} |y_i| + a_{dydt} h |y'_i|)
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+D_i = eps_abs * s_i + eps_rel * (a_y |y_i| + a_dydt h |y'_i|)
+@end example
+
+@end ifinfo
+@noindent
+where @math{s_i} is the @math{i}-th component of the array @var{scale_abs}.
+The same error control heuristic is used by the Matlab @sc{ode} suite. 
+@end deftypefun
+
+@deftypefun {gsl_odeiv_control *} gsl_odeiv_control_alloc (const gsl_odeiv_control_type * @var{T})
+This function returns a pointer to a newly allocated instance of a
+control function of type @var{T}.  This function is only needed for
+defining new types of control functions.  For most purposes the standard
+control functions described above should be sufficient. 
+@end deftypefun
+
+@deftypefun int gsl_odeiv_control_init (gsl_odeiv_control * @var{c}, double @var{eps_abs}, double @var{eps_rel}, double @var{a_y}, double @var{a_dydt})
+This function initializes the control function @var{c} with the
+parameters @var{eps_abs} (absolute error), @var{eps_rel} (relative
+error), @var{a_y} (scaling factor for y) and @var{a_dydt} (scaling
+factor for derivatives).
+@end deftypefun
+
+@deftypefun void gsl_odeiv_control_free (gsl_odeiv_control * @var{c})
+This function frees all the memory associated with the control function
+@var{c}.
+@end deftypefun
+
+@deftypefun int gsl_odeiv_control_hadjust (gsl_odeiv_control * @var{c}, gsl_odeiv_step * @var{s}, const double @var{y}[], const double @var{yerr}[], const double @var{dydt}[], double * @var{h})
+This function adjusts the step-size @var{h} using the control function
+@var{c}, and the current values of @var{y}, @var{yerr} and @var{dydt}.
+The stepping function @var{step} is also needed to determine the order
+of the method.  If the error in the y-values @var{yerr} is found to be
+too large then the step-size @var{h} is reduced and the function returns
+@code{GSL_ODEIV_HADJ_DEC}.  If the error is sufficiently small then
+@var{h} may be increased and @code{GSL_ODEIV_HADJ_INC} is returned.  The
+function returns @code{GSL_ODEIV_HADJ_NIL} if the step-size is
+unchanged.  The goal of the function is to estimate the largest
+step-size which satisfies the user-specified accuracy requirements for
+the current point.
+@end deftypefun
+
+@deftypefun {const char *} gsl_odeiv_control_name (const gsl_odeiv_control * @var{c})
+This function returns a pointer to the name of the control function.
+For example,
+
+@example
+printf ("control method is '%s'\n", 
+        gsl_odeiv_control_name (c));
+@end example
+
+@noindent
+would print something like @code{control method is 'standard'}
+@end deftypefun
+
+
+@node Evolution
+@section Evolution
+
+The highest level of the system is the evolution function which combines
+the results of a stepping function and control function to reliably
+advance the solution forward over an interval @math{(t_0, t_1)}.  If the
+control function signals that the step-size should be decreased the
+evolution function backs out of the current step and tries the proposed
+smaller step-size.  This process is continued until an acceptable
+step-size is found.
+
+@deftypefun {gsl_odeiv_evolve *} gsl_odeiv_evolve_alloc (size_t @var{dim})
+This function returns a pointer to a newly allocated instance of an
+evolution function for a system of @var{dim} dimensions.
+@end deftypefun
+
+@deftypefun int gsl_odeiv_evolve_apply (gsl_odeiv_evolve * @var{e}, gsl_odeiv_control * @var{con}, gsl_odeiv_step * @var{step}, const gsl_odeiv_system * @var{dydt}, double * @var{t}, double @var{t1}, double * @var{h}, double @var{y}[])
+This function advances the system (@var{e}, @var{dydt}) from time
+@var{t} and position @var{y} using the stepping function @var{step}.
+The new time and position are stored in @var{t} and @var{y} on output.
+The initial step-size is taken as @var{h}, but this will be modified
+using the control function @var{c} to achieve the appropriate error
+bound if necessary.  The routine may make several calls to @var{step} in
+order to determine the optimum step-size. An estimate of 
+the local error for the step can be obtained from the components of the array @code{@var{e}->yerr[]}. 
+If the step-size has been changed the value of @var{h} will be modified on output. 
+The maximum time @var{t1} is guaranteed not to be exceeded by the time-step.  On the
+final time-step the value of @var{t} will be set to @var{t1} exactly.
+
+If the user-supplied functions defined in the system @var{dydt} return a
+status other than @code{GSL_SUCCESS} the step will be aborted.  In this
+case,  @var{t} and @var{y} will be restored to their pre-step values
+and the error code from the user-supplied function will be returned.  To
+distinguish between error codes from the user-supplied functions and
+those from @code{gsl_odeiv_evolve_apply} itself, any user-defined return
+values should be distinct from the standard GSL error codes.
+@end deftypefun
+
+@deftypefun int gsl_odeiv_evolve_reset (gsl_odeiv_evolve * @var{e})
+This function resets the evolution function @var{e}.  It should be used
+whenever the next use of @var{e} will not be a continuation of a
+previous step.
+@end deftypefun
+
+@deftypefun void gsl_odeiv_evolve_free (gsl_odeiv_evolve * @var{e})
+This function frees all the memory associated with the evolution function
+@var{e}.
+@end deftypefun
+
+@node ODE Example programs
+@section Examples
+@cindex Van der Pol oscillator, example
+The following program solves the second-order nonlinear Van der Pol
+oscillator equation,
+@tex
+\beforedisplay
+$$
+x''(t) + \mu x'(t) (x(t)^2 - 1) + x(t) = 0
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+x''(t) + \mu x'(t) (x(t)^2 - 1) + x(t) = 0
+@end example
+
+@end ifinfo
+@noindent
+This can be converted into a first order system suitable for use with
+the routines described in this chapter by introducing a separate
+variable for the velocity, @math{y = x'(t)},
+@tex
+\beforedisplay
+$$
+\eqalign{
+x' &= y\cr
+y' &= -x + \mu y (1-x^2)
+}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+x' = y
+y' = -x + \mu y (1-x^2)
+@end example
+
+@end ifinfo
+@noindent
+The program begins by defining functions for these derivatives and
+their Jacobian,
+
+@example
+@verbatiminclude examples/ode-initval.c
+@end example
+
+@noindent
+For functions with multiple parameters, the appropriate information
+can be passed in through the @var{params} argument using a pointer to
+a struct.
+
+The main loop of the program evolves the solution from @math{(y, y') =
+(1, 0)} at @math{t=0} to @math{t=100}.  The step-size @math{h} is
+automatically adjusted by the controller to maintain an absolute
+accuracy of @c{$10^{-6}$} 
+@math{10^@{-6@}} in the function values @var{y}.
+
+@iftex
+@sp 1
+@center @image{vdp,3.4in}
+@center Numerical solution of the Van der Pol oscillator equation 
+@center using Prince-Dormand 8th order Runge-Kutta.
+@end iftex
+
+@noindent
+To obtain the values at regular intervals, rather than the variable
+spacings chosen by the control function, the main loop can be modified
+to advance the solution from one point to the next.  For example, the
+following main loop prints the solution at the fixed points @math{t = 0,
+1, 2, \dots, 100},
+
+@example
+  for (i = 1; i <= 100; i++)
+    @{
+      double ti = i * t1 / 100.0;
+
+      while (t < ti)
+        @{
+          gsl_odeiv_evolve_apply (e, c, s, 
+                                  &sys, 
+                                  &t, ti, &h,
+                                  y);
+        @}
+ 
+      printf ("%.5e %.5e %.5e\n", t, y[0], y[1]);
+    @}
+@end example
+
+@noindent
+It is also possible to work with a non-adaptive integrator, using only
+the stepping function itself.  The following program uses the @code{rk4}
+fourth-order Runge-Kutta stepping function with a fixed stepsize of
+0.01,
+
+@example
+@verbatiminclude examples/odefixed.c
+@end example
+
+@noindent
+The derivatives must be initialized for the starting point @math{t=0}
+before the first step is taken.  Subsequent steps use the output
+derivatives @var{dydt_out} as inputs to the next step by copying their
+values into @var{dydt_in}.
+
+@node ODE References and Further Reading
+@section References and Further Reading
+
+Many of the basic Runge-Kutta formulas can be found in the Handbook of
+Mathematical Functions,
+
+@itemize @asis
+@item
+Abramowitz & Stegun (eds.), @cite{Handbook of Mathematical Functions},
+Section 25.5.
+@end itemize
+
+@noindent
+The implicit Bulirsch-Stoer algorithm @code{bsimp} is described in the
+following paper,
+
+@itemize @asis
+@item
+G. Bader and P. Deuflhard, ``A Semi-Implicit Mid-Point Rule for Stiff
+Systems of Ordinary Differential Equations.'', Numer.@: Math.@: 41, 373--398,
+1983.
+@end itemize
diff --git a/gsl-1.9/doc/permutation.texi b/gsl-1.9/doc/permutation.texi
new file mode 100644
index 0000000..4ae54b9
--- /dev/null
+++ b/gsl-1.9/doc/permutation.texi
@@ -0,0 +1,377 @@
+@cindex permutations
+
+This chapter describes functions for creating and manipulating
+permutations. A permutation @math{p} is represented by an array of
+@math{n} integers in the range 0 to @math{n-1}, where each value
+@math{p_i} occurs once and only once.  The application of a permutation
+@math{p} to a vector @math{v} yields a new vector @math{v'} where
+@c{$v'_i = v_{p_i}$}
+@math{v'_i = v_@{p_i@}}. 
+For example, the array @math{(0,1,3,2)} represents a permutation
+which exchanges the last two elements of a four element vector.
+The corresponding identity permutation is @math{(0,1,2,3)}.   
+
+Note that the permutations produced by the linear algebra routines
+correspond to the exchange of matrix columns, and so should be considered
+as applying to row-vectors in the form @math{v' = v P} rather than
+column-vectors, when permuting the elements of a vector.
+
+The functions described in this chapter are defined in the header file
+@file{gsl_permutation.h}.
+
+@menu
+* The Permutation struct::      
+* Permutation allocation::      
+* Accessing permutation elements::  
+* Permutation properties::      
+* Permutation functions::       
+* Applying Permutations::       
+* Reading and writing permutations::  
+* Permutations in cyclic form::  
+* Permutation Examples::        
+* Permutation References and Further Reading::  
+@end menu
+
+@node The Permutation struct
+@section The Permutation struct
+
+A permutation is defined by a structure containing two components, the size
+of the permutation and a pointer to the permutation array.  The elements
+of the permutation array are all of type @code{size_t}.  The
+@code{gsl_permutation} structure looks like this,
+
+@example
+typedef struct
+@{
+  size_t size;
+  size_t * data;
+@} gsl_permutation;
+@end example
+@comment
+
+@noindent
+
+@node Permutation allocation
+@section Permutation allocation
+
+@deftypefun {gsl_permutation *} gsl_permutation_alloc (size_t @var{n})
+This function allocates memory for a new permutation of size @var{n}.
+The permutation is not initialized and its elements are undefined.  Use
+the function @code{gsl_permutation_calloc} if you want to create a
+permutation which is initialized to the identity. A null pointer is
+returned if insufficient memory is available to create the permutation.
+@end deftypefun
+
+@deftypefun {gsl_permutation *} gsl_permutation_calloc (size_t @var{n})
+This function allocates memory for a new permutation of size @var{n} and
+initializes it to the identity. A null pointer is returned if
+insufficient memory is available to create the permutation.
+@end deftypefun
+
+@deftypefun void gsl_permutation_init (gsl_permutation * @var{p})
+@cindex identity permutation
+This function initializes the permutation @var{p} to the identity, i.e.
+@math{(0,1,2,@dots{},n-1)}.
+@end deftypefun
+
+@deftypefun void gsl_permutation_free (gsl_permutation * @var{p})
+This function frees all the memory used by the permutation @var{p}.
+@end deftypefun
+
+@deftypefun int gsl_permutation_memcpy (gsl_permutation * @var{dest}, const gsl_permutation * @var{src})
+This function copies the elements of the permutation @var{src} into the
+permutation @var{dest}.  The two permutations must have the same size.
+@end deftypefun
+
+@node Accessing permutation elements
+@section Accessing permutation elements
+
+The following functions can be used to access and manipulate
+permutations.
+
+@deftypefun size_t gsl_permutation_get (const gsl_permutation * @var{p}, const size_t @var{i})
+This function returns the value of the @var{i}-th element of the
+permutation @var{p}.  If @var{i} lies outside the allowed range of 0 to
+@math{@var{n}-1} then the error handler is invoked and 0 is returned.
+@end deftypefun
+
+@deftypefun int gsl_permutation_swap (gsl_permutation * @var{p}, const size_t @var{i}, const size_t @var{j})
+@cindex exchanging permutation elements
+@cindex swapping permutation elements
+This function exchanges the @var{i}-th and @var{j}-th elements of the
+permutation @var{p}.
+@end deftypefun
+
+@node Permutation properties
+@section Permutation properties
+
+@deftypefun size_t gsl_permutation_size (const gsl_permutation * @var{p})
+This function returns the size of the permutation @var{p}.
+@end deftypefun
+
+@deftypefun {size_t *} gsl_permutation_data (const gsl_permutation * @var{p})
+This function returns a pointer to the array of elements in the
+permutation @var{p}.
+@end deftypefun
+
+@deftypefun int gsl_permutation_valid (gsl_permutation * @var{p})
+@cindex checking permutation for validity
+@cindex testing permutation for validity
+This function checks that the permutation @var{p} is valid.  The @var{n}
+elements should contain each of the numbers 0 to @math{@var{n}-1} once and only
+once.
+@end deftypefun
+
+@node Permutation functions
+@section Permutation functions
+
+@deftypefun void gsl_permutation_reverse (gsl_permutation * @var{p})
+@cindex reversing a permutation
+This function reverses the elements of the permutation @var{p}.
+@end deftypefun
+
+@deftypefun int gsl_permutation_inverse (gsl_permutation * @var{inv}, const gsl_permutation * @var{p})
+@cindex inverting a permutation
+This function computes the inverse of the permutation @var{p}, storing
+the result in @var{inv}.
+@end deftypefun
+
+@deftypefun int gsl_permutation_next (gsl_permutation * @var{p})
+@cindex iterating through permutations
+This function advances the permutation @var{p} to the next permutation
+in lexicographic order and returns @code{GSL_SUCCESS}.  If no further
+permutations are available it returns @code{GSL_FAILURE} and leaves
+@var{p} unmodified.  Starting with the identity permutation and
+repeatedly applying this function will iterate through all possible
+permutations of a given order.
+@end deftypefun
+
+@deftypefun int gsl_permutation_prev (gsl_permutation * @var{p})
+This function steps backwards from the permutation @var{p} to the
+previous permutation in lexicographic order, returning
+@code{GSL_SUCCESS}.  If no previous permutation is available it returns
+@code{GSL_FAILURE} and leaves @var{p} unmodified.
+@end deftypefun
+
+@node Applying Permutations
+@section Applying Permutations
+
+@deftypefun int gsl_permute (const size_t * @var{p}, double * @var{data}, size_t @var{stride}, size_t @var{n})
+This function applies the permutation @var{p} to the array @var{data} of
+size @var{n} with stride @var{stride}.
+@end deftypefun
+
+@deftypefun int gsl_permute_inverse (const size_t * @var{p}, double * @var{data}, size_t @var{stride}, size_t @var{n})
+This function applies the inverse of the permutation @var{p} to the
+array @var{data} of size @var{n} with stride @var{stride}.
+@end deftypefun
+
+@deftypefun int gsl_permute_vector (const gsl_permutation * @var{p}, gsl_vector * @var{v})
+This function applies the permutation @var{p} to the elements of the
+vector @var{v}, considered as a row-vector acted on by a permutation
+matrix from the right, @math{v' = v P}.  The @math{j}-th column of the
+permutation matrix @math{P} is given by the @math{p_j}-th column of the
+identity matrix. The permutation @var{p} and the vector @var{v} must
+have the same length.
+@end deftypefun
+
+@deftypefun int gsl_permute_vector_inverse (const gsl_permutation * @var{p}, gsl_vector * @var{v})
+This function applies the inverse of the permutation @var{p} to the
+elements of the vector @var{v}, considered as a row-vector acted on by
+an inverse permutation matrix from the right, @math{v' = v P^T}.  Note
+that for permutation matrices the inverse is the same as the transpose.
+The @math{j}-th column of the permutation matrix @math{P} is given by
+the @math{p_j}-th column of the identity matrix. The permutation @var{p}
+and the vector @var{v} must have the same length.
+@end deftypefun
+
+@deftypefun int gsl_permutation_mul (gsl_permutation * @var{p}, const gsl_permutation * @var{pa}, const gsl_permutation * @var{pb})
+This function combines the two permutations @var{pa} and @var{pb} into a
+single permutation @var{p}, where @math{p = pa . pb}. The permutation
+@var{p} is equivalent to applying @math{pb} first and then @var{pa}.
+@end deftypefun
+
+@node Reading and writing permutations
+@section Reading and writing permutations
+
+The library provides functions for reading and writing permutations to a
+file as binary data or formatted text.
+
+@deftypefun int gsl_permutation_fwrite (FILE * @var{stream}, const gsl_permutation * @var{p})
+This function writes the elements of the permutation @var{p} to the
+stream @var{stream} in binary format.  The function returns
+@code{GSL_EFAILED} if there was a problem writing to the file.  Since the
+data is written in the native binary format it may not be portable
+between different architectures.
+@end deftypefun
+
+@deftypefun int gsl_permutation_fread (FILE * @var{stream}, gsl_permutation * @var{p})
+This function reads into the permutation @var{p} from the open stream
+@var{stream} in binary format.  The permutation @var{p} must be
+preallocated with the correct length since the function uses the size of
+@var{p} to determine how many bytes to read.  The function returns
+@code{GSL_EFAILED} if there was a problem reading from the file.  The
+data is assumed to have been written in the native binary format on the
+same architecture.
+@end deftypefun
+
+@deftypefun int gsl_permutation_fprintf (FILE * @var{stream}, const gsl_permutation * @var{p}, const char * @var{format})
+This function writes the elements of the permutation @var{p}
+line-by-line to the stream @var{stream} using the format specifier
+@var{format}, which should be suitable for a type of @var{size_t}.  On a
+GNU system the type modifier @code{Z} represents @code{size_t}, so
+@code{"%Zu\n"} is a suitable format.  The function returns
+@code{GSL_EFAILED} if there was a problem writing to the file.
+@end deftypefun
+
+@deftypefun int gsl_permutation_fscanf (FILE * @var{stream}, gsl_permutation * @var{p})
+This function reads formatted data from the stream @var{stream} into the
+permutation @var{p}.  The permutation @var{p} must be preallocated with
+the correct length since the function uses the size of @var{p} to
+determine how many numbers to read.  The function returns
+@code{GSL_EFAILED} if there was a problem reading from the file.
+@end deftypefun
+
+@node Permutations in cyclic form
+@section Permutations in cyclic form
+
+A permutation can be represented in both @dfn{linear} and @dfn{cyclic}
+notations.  The functions described in this section convert between the
+two forms.  The linear notation is an index mapping, and has already
+been described above.  The cyclic notation expresses a permutation as a
+series of circular rearrangements of groups of elements, or
+@dfn{cycles}.
+
+For example, under the cycle (1 2 3), 1 is replaced by 2, 2 is replaced
+by 3 and 3 is replaced by 1 in a circular fashion. Cycles of different
+sets of elements can be combined independently, for example (1 2 3) (4
+5) combines the cycle (1 2 3) with the cycle (4 5), which is an exchange
+of elements 4 and 5.  A cycle of length one represents an element which
+is unchanged by the permutation and is referred to as a @dfn{singleton}.
+
+It can be shown that every permutation can be decomposed into
+combinations of cycles.  The decomposition is not unique, but can always
+be rearranged into a standard @dfn{canonical form} by a reordering of
+elements.  The library uses the canonical form defined in Knuth's
+@cite{Art of Computer Programming} (Vol 1, 3rd Ed, 1997) Section 1.3.3,
+p.178.
+
+The procedure for obtaining the canonical form given by Knuth is,
+
+@enumerate
+@item Write all singleton cycles explicitly
+@item Within each cycle, put the smallest number first
+@item Order the cycles in decreasing order of the first number in the cycle.
+@end enumerate
+
+@noindent
+For example, the linear representation (2 4 3 0 1) is represented as (1
+4) (0 2 3) in canonical form. The permutation corresponds to an
+exchange of elements 1 and 4, and rotation of elements 0, 2 and 3.
+
+The important property of the canonical form is that it can be
+reconstructed from the contents of each cycle without the brackets. In
+addition, by removing the brackets it can be considered as a linear
+representation of a different permutation. In the example given above
+the permutation (2 4 3 0 1) would become (1 4 0 2 3).  This mapping has
+many applications in the theory of permutations.
+
+@deftypefun int gsl_permutation_linear_to_canonical (gsl_permutation * @var{q}, const gsl_permutation * @var{p})
+This function computes the canonical form of the permutation @var{p} and
+stores it in the output argument @var{q}.
+@end deftypefun
+
+@deftypefun int gsl_permutation_canonical_to_linear (gsl_permutation * @var{p}, const gsl_permutation * @var{q})
+This function converts a permutation @var{q} in canonical form back into
+linear form storing it in the output argument @var{p}.
+@end deftypefun
+
+@deftypefun size_t gsl_permutation_inversions (const gsl_permutation * @var{p})
+This function counts the number of inversions in the permutation
+@var{p}.  An inversion is any pair of elements that are not in order.
+For example, the permutation 2031 has three inversions, corresponding to
+the pairs (2,0) (2,1) and (3,1).  The identity permutation has no
+inversions.
+@end deftypefun
+
+@deftypefun size_t gsl_permutation_linear_cycles (const gsl_permutation * @var{p})
+This function counts the number of cycles in the permutation @var{p}, given in linear form.
+@end deftypefun
+
+@deftypefun size_t gsl_permutation_canonical_cycles (const gsl_permutation * @var{q})
+This function counts the number of cycles in the permutation @var{q}, given in canonical form.
+@end deftypefun
+
+
+@node Permutation Examples
+@section Examples
+The example program below creates a random permutation (by shuffling the
+elements of the identity) and finds its inverse.
+
+@example
+@verbatiminclude examples/permshuffle.c
+@end example
+
+@noindent
+Here is the output from the program,
+
+@example
+$ ./a.out 
+initial permutation: 0 1 2 3 4 5 6 7 8 9
+ random permutation: 1 3 5 2 7 6 0 4 9 8
+inverse permutation: 6 0 3 1 7 2 5 4 9 8
+@end example
+
+@noindent
+The random permutation @code{p[i]} and its inverse @code{q[i]} are
+related through the identity @code{p[q[i]] = i}, which can be verified
+from the output.
+
+The next example program steps forwards through all possible third order
+permutations, starting from the identity,
+
+@example
+@verbatiminclude examples/permseq.c
+@end example
+
+@noindent
+Here is the output from the program,
+
+@example
+$ ./a.out 
+ 0 1 2
+ 0 2 1
+ 1 0 2
+ 1 2 0
+ 2 0 1
+ 2 1 0
+@end example
+
+@noindent
+The permutations are generated in lexicographic order.  To reverse the
+sequence, begin with the final permutation (which is the reverse of the
+identity) and replace @code{gsl_permutation_next} with
+@code{gsl_permutation_prev}.
+
+@node Permutation References and Further Reading
+@section References and Further Reading
+
+The subject of permutations is covered extensively in Knuth's
+@cite{Sorting and Searching},
+
+@itemize @asis
+@item
+Donald E. Knuth, @cite{The Art of Computer Programming: Sorting and
+Searching} (Vol 3, 3rd Ed, 1997), Addison-Wesley, ISBN 0201896850.
+@end itemize
+
+@noindent
+For the definition of the @dfn{canonical form} see,
+
+@itemize @asis
+@item
+Donald E. Knuth, @cite{The Art of Computer Programming: Fundamental
+Algorithms} (Vol 1, 3rd Ed, 1997), Addison-Wesley, ISBN 0201896850.
+Section 1.3.3, @cite{An Unusual Correspondence}, p.178--179.
+@end itemize
+
diff --git a/gsl-1.9/doc/poly.texi b/gsl-1.9/doc/poly.texi
new file mode 100644
index 0000000..531f77a
--- /dev/null
+++ b/gsl-1.9/doc/poly.texi
@@ -0,0 +1,302 @@
+@cindex polynomials, roots of 
+
+This chapter describes functions for evaluating and solving polynomials.
+There are routines for finding real and complex roots of quadratic and
+cubic equations using analytic methods.  An iterative polynomial solver
+is also available for finding the roots of general polynomials with real
+coefficients (of any order).  The functions are declared in the header
+file @code{gsl_poly.h}.
+
+@menu
+* Polynomial Evaluation::       
+* Divided Difference Representation of Polynomials::  
+* Quadratic Equations::         
+* Cubic Equations::             
+* General Polynomial Equations::  
+* Roots of Polynomials Examples::  
+* Roots of Polynomials References and Further Reading::  
+@end menu
+
+@node Polynomial Evaluation
+@section Polynomial Evaluation
+@cindex polynomial evaluation
+@cindex evaluation of polynomials
+
+@deftypefun double gsl_poly_eval (const double @var{c}[], const int @var{len}, const double @var{x})
+This function evaluates the polynomial 
+@c{$c[0] + c[1] x + c[2] x^2 + \dots + c[len-1] x^{len-1}$}
+@math{c[0] + c[1] x + c[2] x^2 + \dots + c[len-1] x^@{len-1@}} using
+Horner's method for stability.  The function is inlined when possible.
+@end deftypefun
+
+@node Divided Difference Representation of Polynomials
+@section Divided Difference Representation of Polynomials
+@cindex divided differences, polynomials
+@cindex evaluation of polynomials, in divided difference form
+
+The functions described here manipulate polynomials stored in Newton's
+divided-difference representation.  The use of divided-differences is
+described in Abramowitz & Stegun sections 25.1.4 and 25.2.26.
+
+@deftypefun int gsl_poly_dd_init (double @var{dd}[], const double @var{xa}[], const double @var{ya}[], size_t @var{size})
+This function computes a divided-difference representation of the
+interpolating polynomial for the points (@var{xa}, @var{ya}) stored in
+the arrays @var{xa} and @var{ya} of length @var{size}.  On output the
+divided-differences of (@var{xa},@var{ya}) are stored in the array
+@var{dd}, also of length @var{size}.
+@end deftypefun
+
+@deftypefun double gsl_poly_dd_eval (const double @var{dd}[], const double @var{xa}[], const size_t @var{size}, const double @var{x})
+This function evaluates the polynomial stored in divided-difference form
+in the arrays @var{dd} and @var{xa} of length @var{size} at the point
+@var{x}.
+@end deftypefun
+
+@deftypefun int gsl_poly_dd_taylor (double @var{c}[], double @var{xp}, const double @var{dd}[], const double @var{xa}[], size_t @var{size}, double @var{w}[])
+This function converts the divided-difference representation of a
+polynomial to a Taylor expansion.  The divided-difference representation
+is supplied in the arrays @var{dd} and @var{xa} of length @var{size}.
+On output the Taylor coefficients of the polynomial expanded about the
+point @var{xp} are stored in the array @var{c} also of length
+@var{size}.  A workspace of length @var{size} must be provided in the
+array @var{w}.
+@end deftypefun
+
+@node  Quadratic Equations
+@section Quadratic Equations
+@cindex quadratic equation, solving
+
+@deftypefun int gsl_poly_solve_quadratic (double @var{a}, double @var{b}, double @var{c}, double * @var{x0}, double * @var{x1})
+This function finds the real roots of the quadratic equation,
+@tex
+\beforedisplay
+$$
+a x^2 + b x + c = 0
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+a x^2 + b x + c = 0
+@end example
+
+@end ifinfo
+@noindent
+The number of real roots (either zero, one or two) is returned, and
+their locations are stored in @var{x0} and @var{x1}.  If no real roots
+are found then @var{x0} and @var{x1} are not modified.  If one real root
+is found (i.e. if @math{a=0}) then it is stored in @var{x0}.  When two
+real roots are found they are stored in @var{x0} and @var{x1} in
+ascending order.  The case of coincident roots is not considered
+special.  For example @math{(x-1)^2=0} will have two roots, which happen
+to have exactly equal values.
+
+The number of roots found depends on the sign of the discriminant
+@math{b^2 - 4 a c}.  This will be subject to rounding and cancellation
+errors when computed in double precision, and will also be subject to
+errors if the coefficients of the polynomial are inexact.  These errors
+may cause a discrete change in the number of roots.  However, for
+polynomials with small integer coefficients the discriminant can always
+be computed exactly.
+
+@end deftypefun
+
+@deftypefun int gsl_poly_complex_solve_quadratic (double @var{a}, double @var{b}, double @var{c}, gsl_complex * @var{z0}, gsl_complex * @var{z1})
+
+This function finds the complex roots of the quadratic equation,
+@tex
+\beforedisplay
+$$
+a z^2 + b z + c = 0
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+a z^2 + b z + c = 0
+@end example
+
+@end ifinfo
+@noindent
+The number of complex roots is returned (either one or two) and the
+locations of the roots are stored in @var{z0} and @var{z1}.  The roots
+are returned in ascending order, sorted first by their real components
+and then by their imaginary components.  If only one real root is found
+(i.e. if @math{a=0}) then it is stored in @var{z0}.
+
+@end deftypefun
+
+
+@node Cubic Equations
+@section Cubic Equations
+@cindex cubic equation, solving
+
+@deftypefun int gsl_poly_solve_cubic (double @var{a}, double @var{b}, double @var{c}, double * @var{x0}, double * @var{x1}, double * @var{x2})
+
+This function finds the real roots of the cubic equation,
+@tex
+\beforedisplay
+$$
+x^3 + a x^2 + b x + c = 0
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+x^3 + a x^2 + b x + c = 0
+@end example
+
+@end ifinfo
+@noindent
+with a leading coefficient of unity.  The number of real roots (either
+one or three) is returned, and their locations are stored in @var{x0},
+@var{x1} and @var{x2}.  If one real root is found then only @var{x0} is
+modified.  When three real roots are found they are stored in @var{x0},
+@var{x1} and @var{x2} in ascending order.  The case of coincident roots
+is not considered special.  For example, the equation @math{(x-1)^3=0}
+will have three roots with exactly equal values.
+
+@end deftypefun
+
+@deftypefun int gsl_poly_complex_solve_cubic (double @var{a}, double @var{b}, double @var{c}, gsl_complex * @var{z0}, gsl_complex * @var{z1}, gsl_complex * @var{z2})
+
+This function finds the complex roots of the cubic equation,
+@tex
+\beforedisplay
+$$
+z^3 + a z^2 + b z + c = 0
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+z^3 + a z^2 + b z + c = 0
+@end example
+
+@end ifinfo
+@noindent
+The number of complex roots is returned (always three) and the locations
+of the roots are stored in @var{z0}, @var{z1} and @var{z2}.  The roots
+are returned in ascending order, sorted first by their real components
+and then by their imaginary components.
+
+@end deftypefun
+
+
+@node General Polynomial Equations
+@section General Polynomial Equations
+@cindex general polynomial equations, solving
+
+The roots of polynomial equations cannot be found analytically beyond
+the special cases of the quadratic, cubic and quartic equation.  The
+algorithm described in this section uses an iterative method to find the
+approximate locations of roots of higher order polynomials.
+
+@deftypefun {gsl_poly_complex_workspace *} gsl_poly_complex_workspace_alloc (size_t @var{n})
+This function allocates space for a @code{gsl_poly_complex_workspace}
+struct and a workspace suitable for solving a polynomial with @var{n}
+coefficients using the routine @code{gsl_poly_complex_solve}.
+
+The function returns a pointer to the newly allocated
+@code{gsl_poly_complex_workspace} if no errors were detected, and a null
+pointer in the case of error.
+@end deftypefun
+
+@deftypefun void gsl_poly_complex_workspace_free (gsl_poly_complex_workspace * @var{w})
+This function frees all the memory associated with the workspace
+@var{w}.
+@end deftypefun
+
+@deftypefun int gsl_poly_complex_solve (const double * @var{a}, size_t @var{n}, gsl_poly_complex_workspace * @var{w}, gsl_complex_packed_ptr @var{z})
+This function computes the roots of the general polynomial 
+@c{$P(x) = a_0 + a_1 x + a_2 x^2 + ... + a_{n-1} x^{n-1}$} 
+@math{P(x) = a_0 + a_1 x + a_2 x^2 + ... + a_@{n-1@} x^@{n-1@}} using 
+balanced-QR reduction of the companion matrix.  The parameter @var{n}
+specifies the length of the coefficient array.  The coefficient of the
+highest order term must be non-zero.  The function requires a workspace
+@var{w} of the appropriate size.  The @math{n-1} roots are returned in
+the packed complex array @var{z} of length @math{2(n-1)}, alternating
+real and imaginary parts.
+
+The function returns @code{GSL_SUCCESS} if all the roots are found. If
+the QR reduction does not converge, the error handler is invoked with
+an error code of @code{GSL_EFAILED}.  Note that due to finite precision,
+roots of higher multiplicity are returned as a cluster of simple roots
+with reduced accuracy.  The solution of polynomials with higher-order
+roots requires specialized algorithms that take the multiplicity
+structure into account (see e.g. Z. Zeng, Algorithm 835, ACM
+Transactions on Mathematical Software, Volume 30, Issue 2 (2004), pp
+218--236).
+@end deftypefun
+
+@node Roots of Polynomials Examples
+@section Examples
+
+To demonstrate the use of the general polynomial solver we will take the
+polynomial @math{P(x) = x^5 - 1} which has the following roots,
+@tex
+\beforedisplay
+$$
+1, e^{2\pi i /5}, e^{4\pi i /5}, e^{6\pi i /5}, e^{8\pi i /5}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+1, e^@{2\pi i /5@}, e^@{4\pi i /5@}, e^@{6\pi i /5@}, e^@{8\pi i /5@}
+@end example
+
+@end ifinfo
+@noindent
+The following program will find these roots.
+
+@example
+@verbatiminclude examples/polyroots.c
+@end example
+
+@noindent
+The output of the program is,
+
+@example
+$ ./a.out 
+@verbatiminclude examples/polyroots.out
+@end example
+
+@noindent
+which agrees with the analytic result, @math{z_n = \exp(2 \pi n i/5)}.
+
+@node Roots of Polynomials References and Further Reading
+@section References and Further Reading
+
+The balanced-QR method and its error analysis are described in the
+following papers,
+
+@itemize @asis
+@item
+R.S. Martin, G. Peters and J.H. Wilkinson, ``The QR Algorithm for Real
+Hessenberg Matrices'', @cite{Numerische Mathematik}, 14 (1970), 219--231.
+
+@item
+B.N. Parlett and C. Reinsch, ``Balancing a Matrix for Calculation of
+Eigenvalues and Eigenvectors'', @cite{Numerische Mathematik}, 13 (1969),
+293--304.
+
+@item
+A. Edelman and H. Murakami, ``Polynomial roots from companion matrix
+eigenvalues'', @cite{Mathematics of Computation}, Vol.@: 64, No.@: 210
+(1995), 763--776.
+@end itemize
+
+@noindent
+The formulas for divided differences are given in Abramowitz and Stegun,
+
+@itemize @asis
+@item
+Abramowitz and Stegun, @cite{Handbook of Mathematical Functions},
+Sections 25.1.4 and 25.2.26.
+@end itemize
diff --git a/gsl-1.9/doc/qrng.eps b/gsl-1.9/doc/qrng.eps
new file mode 100644
index 0000000..2fa00f1
--- /dev/null
+++ b/gsl-1.9/doc/qrng.eps
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+stroke
+grestore
+end
+showpage
+%%Trailer
+%%DocumentFonts: Helvetica
diff --git a/gsl-1.9/doc/qrng.texi b/gsl-1.9/doc/qrng.texi
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+@cindex quasi-random sequences
+@cindex low discrepancy sequences
+@cindex Sobol sequence
+@cindex Niederreiter sequence
+This chapter describes functions for generating quasi-random sequences
+in arbitrary dimensions.  A quasi-random sequence progressively covers a
+@math{d}-dimensional space with a set of points that are uniformly
+distributed.  Quasi-random sequences are also known as low-discrepancy
+sequences.  The quasi-random sequence generators use an interface that
+is similar to the interface for random number generators, except that
+seeding is not required---each generator produces a single sequence.
+
+The functions described in this section are declared in the header file
+@file{gsl_qrng.h}.
+
+@menu
+* Quasi-random number generator initialization::  
+* Sampling from a quasi-random number generator::  
+* Auxiliary quasi-random number generator functions::  
+* Saving and resorting quasi-random number generator state::  
+* Quasi-random number generator algorithms::  
+* Quasi-random number generator examples::  
+* Quasi-random number references::  
+@end menu
+
+@node Quasi-random number generator initialization
+@section Quasi-random number generator initialization
+
+@deftypefun {gsl_qrng *} gsl_qrng_alloc (const gsl_qrng_type * @var{T}, unsigned int @var{d})
+This function returns a pointer to a newly-created instance of a
+quasi-random sequence generator of type @var{T} and dimension @var{d}.
+If there is insufficient memory to create the generator then the
+function returns a null pointer and the error handler is invoked with an
+error code of @code{GSL_ENOMEM}.
+@end deftypefun
+
+@deftypefun void gsl_qrng_free (gsl_qrng * @var{q})
+This function frees all the memory associated with the generator
+@var{q}.
+@end deftypefun
+
+@deftypefun void gsl_qrng_init (gsl_qrng * @var{q})
+This function reinitializes the generator @var{q} to its starting point.
+Note that quasi-random sequences do not use a seed and always produce
+the same set of values.
+@end deftypefun
+
+@node Sampling from a quasi-random number generator
+@section Sampling from a quasi-random number generator
+
+@deftypefun int gsl_qrng_get (const gsl_qrng * @var{q}, double @var{x}[])
+This function stores the next point from the sequence generator @var{q}
+in the array @var{x}.  The space available for @var{x} must match the
+dimension of the generator.  The point @var{x} will lie in the range
+@math{0 < x_i < 1} for each @math{x_i}.
+@end deftypefun
+
+@node Auxiliary quasi-random number generator functions
+@section Auxiliary quasi-random number generator functions
+
+@deftypefun {const char *} gsl_qrng_name (const gsl_qrng * @var{q})
+This function returns a pointer to the name of the generator.
+@end deftypefun 
+
+@deftypefun size_t gsl_qrng_size (const gsl_qrng * @var{q})
+@deftypefunx {void *} gsl_qrng_state (const gsl_qrng * @var{q})
+These functions return a pointer to the state of generator @var{r} and
+its size.  You can use this information to access the state directly.  For
+example, the following code will write the state of a generator to a
+stream,
+
+@example
+void * state = gsl_qrng_state (q);
+size_t n = gsl_qrng_size (q);
+fwrite (state, n, 1, stream);
+@end example
+@end deftypefun
+
+
+@node Saving and resorting quasi-random number generator state
+@section Saving and resorting quasi-random number generator state
+
+@deftypefun int gsl_qrng_memcpy (gsl_qrng * @var{dest}, const gsl_qrng * @var{src})
+This function copies the quasi-random sequence generator @var{src} into the
+pre-existing generator @var{dest}, making @var{dest} into an exact copy
+of @var{src}.  The two generators must be of the same type.
+@end deftypefun
+
+@deftypefun {gsl_qrng *} gsl_qrng_clone (const gsl_qrng * @var{q})
+This function returns a pointer to a newly created generator which is an
+exact copy of the generator @var{q}.
+@end deftypefun
+
+@node Quasi-random number generator algorithms
+@section Quasi-random number generator algorithms
+
+The following quasi-random sequence algorithms are available,
+
+@deffn {Generator} gsl_qrng_niederreiter_2
+This generator uses the algorithm described in Bratley, Fox,
+Niederreiter, @cite{ACM Trans. Model. Comp. Sim.} 2, 195 (1992). It is
+valid up to 12 dimensions.
+@end deffn
+
+@deffn {Generator} gsl_qrng_sobol
+This generator uses the Sobol sequence described in Antonov, Saleev,
+@cite{USSR Comput. Maths. Math. Phys.} 19, 252 (1980). It is valid up to
+40 dimensions.
+@end deffn
+
+@node Quasi-random number generator examples
+@section Examples
+
+The following program prints the first 1024 points of the 2-dimensional
+Sobol sequence.
+
+@example
+@verbatiminclude examples/qrng.c
+@end example
+
+@noindent
+Here is the output from the program,
+
+@example
+$ ./a.out
+0.50000 0.50000
+0.75000 0.25000
+0.25000 0.75000
+0.37500 0.37500
+0.87500 0.87500
+0.62500 0.12500
+0.12500 0.62500
+....
+@end example
+
+@noindent
+It can be seen that successive points progressively fill-in the spaces
+between previous points. 
+
+@iftex
+@need 4000
+The following plot shows the distribution in the x-y plane of the first
+1024 points from the Sobol sequence,
+@sp 1
+@center @image{qrng,3.4in}
+@sp 1
+@center Distribution of the first 1024 points 
+@center from the quasi-random Sobol sequence
+@end iftex
+
+@node Quasi-random number references
+@section References
+
+The implementations of the quasi-random sequence routines are based on
+the algorithms described in the following paper,
+
+@itemize @asis
+P. Bratley and B.L. Fox and H. Niederreiter, ``Algorithm 738: Programs
+to Generate Niederreiter's Low-discrepancy Sequences'', @cite{ACM
+Transactions on Mathematical Software}, Vol.@: 20, No.@: 4, December, 1994,
+p.@: 494--495.
+@end itemize
+
diff --git a/gsl-1.9/doc/rand-bernoulli.tex b/gsl-1.9/doc/rand-bernoulli.tex
new file mode 100644
index 0000000..6e086e7
--- /dev/null
+++ b/gsl-1.9/doc/rand-bernoulli.tex
@@ -0,0 +1,695 @@
+% GNUPLOT: plain TeX with Postscript
+\begingroup
+  \catcode`\@=11\relax
+  \def\GNUPLOTspecial{%
+    \def\do##1{\catcode`##1=12\relax}\dospecials
+    \catcode`\{=1\catcode`\}=2\catcode\%=14\relax\special}%
+%
+\expandafter\ifx\csname GNUPLOTpicture\endcsname\relax
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+   gsave 1 setlinecap M 0 0 V stroke grestore } def
+/Dia { stroke [] 0 setdash 2 copy vpt add M
+  hpt neg vpt neg V hpt vpt neg V
+  hpt vpt V hpt neg vpt V closepath stroke
+  Pnt } def
+/Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V
+  currentpoint stroke M
+  hpt neg vpt neg R hpt2 0 V stroke
+  } def
+/Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M
+  0 vpt2 neg V hpt2 0 V 0 vpt2 V
+  hpt2 neg 0 V closepath stroke
+  Pnt } def
+/Crs { stroke [] 0 setdash exch hpt sub exch vpt add M
+  hpt2 vpt2 neg V currentpoint stroke M
+  hpt2 neg 0 R hpt2 vpt2 V stroke } def
+/TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M
+  hpt neg vpt -1.62 mul V
+  hpt 2 mul 0 V
+  hpt neg vpt 1.62 mul V closepath stroke
+  Pnt  } def
+/Star { 2 copy Pls Crs } def
+/BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M
+  0 vpt2 neg V  hpt2 0 V  0 vpt2 V
+  hpt2 neg 0 V  closepath fill } def
+/TriUF { stroke [] 0 setdash vpt 1.12 mul add M
+  hpt neg vpt -1.62 mul V
+  hpt 2 mul 0 V
+  hpt neg vpt 1.62 mul V closepath fill } def
+/TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M
+  hpt neg vpt 1.62 mul V
+  hpt 2 mul 0 V
+  hpt neg vpt -1.62 mul V closepath stroke
+  Pnt  } def
+/TriDF { stroke [] 0 setdash vpt 1.12 mul sub M
+  hpt neg vpt 1.62 mul V
+  hpt 2 mul 0 V
+  hpt neg vpt -1.62 mul V closepath fill} def
+/DiaF { stroke [] 0 setdash vpt add M
+  hpt neg vpt neg V hpt vpt neg V
+  hpt vpt V hpt neg vpt V closepath fill } def
+/Pent { stroke [] 0 setdash 2 copy gsave
+  translate 0 hpt M 4 {72 rotate 0 hpt L} repeat
+  closepath stroke grestore Pnt } def
+/PentF { stroke [] 0 setdash gsave
+  translate 0 hpt M 4 {72 rotate 0 hpt L} repeat
+  closepath fill grestore } def
+/Circle { stroke [] 0 setdash 2 copy
+  hpt 0 360 arc stroke Pnt } def
+/CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def
+/C0 { BL [] 0 setdash 2 copy moveto vpt 90 450  arc } bind def
+/C1 { BL [] 0 setdash 2 copy        moveto
+       2 copy  vpt 0 90 arc closepath fill
+               vpt 0 360 arc closepath } bind def
+/C2 { BL [] 0 setdash 2 copy moveto
+       2 copy  vpt 90 180 arc closepath fill
+               vpt 0 360 arc closepath } bind def
+/C3 { BL [] 0 setdash 2 copy moveto
+       2 copy  vpt 0 180 arc closepath fill
+               vpt 0 360 arc closepath } bind def
+/C4 { BL [] 0 setdash 2 copy moveto
+       2 copy  vpt 180 270 arc closepath fill
+               vpt 0 360 arc closepath } bind def
+/C5 { BL [] 0 setdash 2 copy moveto
+       2 copy  vpt 0 90 arc
+       2 copy moveto
+       2 copy  vpt 180 270 arc closepath fill
+               vpt 0 360 arc } bind def
+/C6 { BL [] 0 setdash 2 copy moveto
+      2 copy  vpt 90 270 arc closepath fill
+              vpt 0 360 arc closepath } bind def
+/C7 { BL [] 0 setdash 2 copy moveto
+      2 copy  vpt 0 270 arc closepath fill
+              vpt 0 360 arc closepath } bind def
+/C8 { BL [] 0 setdash 2 copy moveto
+      2 copy vpt 270 360 arc closepath fill
+              vpt 0 360 arc closepath } bind def
+/C9 { BL [] 0 setdash 2 copy moveto
+      2 copy  vpt 270 450 arc closepath fill
+              vpt 0 360 arc closepath } bind def
+/C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill
+       2 copy moveto
+       2 copy vpt 90 180 arc closepath fill
+               vpt 0 360 arc closepath } bind def
+/C11 { BL [] 0 setdash 2 copy moveto
+       2 copy  vpt 0 180 arc closepath fill
+       2 copy moveto
+       2 copy  vpt 270 360 arc closepath fill
+               vpt 0 360 arc closepath } bind def
+/C12 { BL [] 0 setdash 2 copy moveto
+       2 copy  vpt 180 360 arc closepath fill
+               vpt 0 360 arc closepath } bind def
+/C13 { BL [] 0 setdash  2 copy moveto
+       2 copy  vpt 0 90 arc closepath fill
+       2 copy moveto
+       2 copy  vpt 180 360 arc closepath fill
+               vpt 0 360 arc closepath } bind def
+/C14 { BL [] 0 setdash 2 copy moveto
+       2 copy  vpt 90 360 arc closepath fill
+               vpt 0 360 arc } bind def
+/C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill
+               vpt 0 360 arc closepath } bind def
+/Rec   { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto
+       neg 0 rlineto closepath } bind def
+/Square { dup Rec } bind def
+/Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def
+/S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def
+/S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def
+/S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def
+/S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def
+/S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def
+/S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill
+       exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def
+/S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def
+/S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill
+       2 copy vpt Square fill
+       Bsquare } bind def
+/S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def
+/S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def
+/S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill
+       Bsquare } bind def
+/S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill
+       Bsquare } bind def
+/S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def
+/S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill
+       2 copy vpt Square fill Bsquare } bind def
+/S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill
+       2 copy exch vpt sub exch vpt Square fill Bsquare } bind def
+/S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def
+/D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def
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+/D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def
+/D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def
+/D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def
+/D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def
+/D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def
+/D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def
+/D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def
+/D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def
+/D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def
+/D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def
+/D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def
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+  hpt neg vpt neg V hpt vpt neg V
+  hpt vpt V hpt neg vpt V closepath stroke } def
+/BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M
+  0 vpt2 neg V hpt2 0 V 0 vpt2 V
+  hpt2 neg 0 V closepath stroke } def
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+  hpt 2 mul 0 V
+  hpt neg vpt 1.62 mul V closepath stroke } def
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+  hpt 2 mul 0 V
+  hpt neg vpt -1.62 mul V closepath stroke } def
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+  translate 0 hpt M 4 {72 rotate 0 hpt L} repeat
+  closepath stroke grestore } def
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+  hpt 0 360 arc stroke } def
+/Opaque { gsave closepath 1 setgray fill grestore 0 setgray closepath } def
+/DiaW { stroke [] 0 setdash vpt add M
+  hpt neg vpt neg V hpt vpt neg V
+  hpt vpt V hpt neg vpt V Opaque stroke } def
+/BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M
+  0 vpt2 neg V hpt2 0 V 0 vpt2 V
+  hpt2 neg 0 V Opaque stroke } def
+/TriUW { stroke [] 0 setdash vpt 1.12 mul add M
+  hpt neg vpt -1.62 mul V
+  hpt 2 mul 0 V
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+/TriDW { stroke [] 0 setdash vpt 1.12 mul sub M
+  hpt neg vpt 1.62 mul V
+  hpt 2 mul 0 V
+  hpt neg vpt -1.62 mul V Opaque stroke } def
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+  translate 0 hpt M 4 {72 rotate 0 hpt L} repeat
+  Opaque stroke grestore } def
+/CircW { stroke [] 0 setdash 
+  hpt 0 360 arc Opaque stroke } def
+/BoxFill { gsave Rec 1 setgray fill grestore } def
+/BoxColFill {
+  gsave Rec
+  /Fillden exch def
+  currentrgbcolor
+  /ColB exch def /ColG exch def /ColR exch def
+  /ColR ColR Fillden mul Fillden sub 1 add def
+  /ColG ColG Fillden mul Fillden sub 1 add def
+  /ColB ColB Fillden mul Fillden sub 1 add def
+  ColR ColG ColB setrgbcolor
+  fill grestore } def
+%
+% PostScript Level 1 Pattern Fill routine
+% Usage: x y w h s a XX PatternFill
+%	x,y = lower left corner of box to be filled
+%	w,h = width and height of box
+%	  a = angle in degrees between lines and x-axis
+%	 XX = 0/1 for no/yes cross-hatch
+%
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+    PFa 0 get PFa 2 get 2 div add PFa 1 get PFa 3 get 2 div add translate
+    PFa 2 get -2 div PFa 3 get -2 div PFa 2 get PFa 3 get Rec
+    gsave 1 setgray fill grestore clip
+    currentlinewidth 0.5 mul setlinewidth
+    /PFs PFa 2 get dup mul PFa 3 get dup mul add sqrt def
+    0 0 M PFa 5 get rotate PFs -2 div dup translate
+	0 1 PFs PFa 4 get div 1 add floor cvi
+	{ PFa 4 get mul 0 M 0 PFs V } for
+    0 PFa 6 get ne {
+	0 1 PFs PFa 4 get div 1 add floor cvi
+	{ PFa 4 get mul 0 2 1 roll M PFs 0 V } for
+    } if
+    stroke grestore } def
+%
+/Symbol-Oblique /Symbol findfont [1 0 .167 1 0 0] makefont
+dup length dict begin {1 index /FID eq {pop pop} {def} ifelse} forall
+currentdict end definefont pop
+end
+%%EndProlog
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+/S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def
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+/D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def
+/D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def
+/D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def
+/D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def
+/D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def
+/D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def
+/D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def
+/D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def
+/D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def
+/D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def
+/DiaE { stroke [] 0 setdash vpt add M
+  hpt neg vpt neg V hpt vpt neg V
+  hpt vpt V hpt neg vpt V closepath stroke } def
+/BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M
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+  hpt2 neg 0 V closepath stroke } def
+/TriUE { stroke [] 0 setdash vpt 1.12 mul add M
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+/TriDE { stroke [] 0 setdash vpt 1.12 mul sub M
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+/PentE { stroke [] 0 setdash gsave
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+  closepath stroke grestore } def
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+/DiaW { stroke [] 0 setdash vpt add M
+  hpt neg vpt neg V hpt vpt neg V
+  hpt vpt V hpt neg vpt V Opaque stroke } def
+/BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M
+  0 vpt2 neg V hpt2 0 V 0 vpt2 V
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+/TriUW { stroke [] 0 setdash vpt 1.12 mul add M
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+/TriDW { stroke [] 0 setdash vpt 1.12 mul sub M
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+/PentW { stroke [] 0 setdash gsave
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+  Opaque stroke grestore } def
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+/BoxFill { gsave Rec 1 setgray fill grestore } def
+/BoxColFill {
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+  /Fillden exch def
+  currentrgbcolor
+  /ColB exch def /ColG exch def /ColR exch def
+  /ColR ColR Fillden mul Fillden sub 1 add def
+  /ColG ColG Fillden mul Fillden sub 1 add def
+  /ColB ColB Fillden mul Fillden sub 1 add def
+  ColR ColG ColB setrgbcolor
+  fill grestore } def
+%
+% PostScript Level 1 Pattern Fill routine
+% Usage: x y w h s a XX PatternFill
+%	x,y = lower left corner of box to be filled
+%	w,h = width and height of box
+%	  a = angle in degrees between lines and x-axis
+%	 XX = 0/1 for no/yes cross-hatch
+%
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+    PFa 2 get -2 div PFa 3 get -2 div PFa 2 get PFa 3 get Rec
+    gsave 1 setgray fill grestore clip
+    currentlinewidth 0.5 mul setlinewidth
+    /PFs PFa 2 get dup mul PFa 3 get dup mul add sqrt def
+    0 0 M PFa 5 get rotate PFs -2 div dup translate
+	0 1 PFs PFa 4 get div 1 add floor cvi
+	{ PFa 4 get mul 0 M 0 PFs V } for
+    0 PFa 6 get ne {
+	0 1 PFs PFa 4 get div 1 add floor cvi
+	{ PFa 4 get mul 0 2 1 roll M PFs 0 V } for
+    } if
+    stroke grestore } def
+%
+/Symbol-Oblique /Symbol findfont [1 0 .167 1 0 0] makefont
+dup length dict begin {1 index /FID eq {pop pop} {def} ifelse} forall
+currentdict end definefont pop
+end
+%%EndProlog
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diff --git a/gsl-1.9/doc/rand-gamma.tex b/gsl-1.9/doc/rand-gamma.tex
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diff --git a/gsl-1.9/doc/rand-landau.tex b/gsl-1.9/doc/rand-landau.tex
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+  /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def
+/DL { Color {setrgbcolor Solid {pop []} if 0 setdash }
+ {pop pop pop 0 setgray Solid {pop []} if 0 setdash} ifelse } def
+/BL { stroke userlinewidth 2 mul setlinewidth
+      Rounded { 1 setlinejoin 1 setlinecap } if } def
+/AL { stroke userlinewidth 2 div setlinewidth
+      Rounded { 1 setlinejoin 1 setlinecap } if } def
+/UL { dup gnulinewidth mul /userlinewidth exch def
+      dup 1 lt {pop 1} if 10 mul /udl exch def } def
+/PL { stroke userlinewidth setlinewidth
+      Rounded { 1 setlinejoin 1 setlinecap } if } def
+/LTw { PL [] 1 setgray } def
+/LTb { BL [] 0 0 0 DL } def
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+/LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def
+/Pnt { stroke [] 0 setdash
+   gsave 1 setlinecap M 0 0 V stroke grestore } def
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+  hpt neg vpt neg V hpt vpt neg V
+  hpt vpt V hpt neg vpt V closepath stroke
+  Pnt } def
+/Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V
+  currentpoint stroke M
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+  } def
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+  0 vpt2 neg V hpt2 0 V 0 vpt2 V
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+  Pnt } def
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+  hpt2 vpt2 neg V currentpoint stroke M
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+  Pnt  } def
+/Star { 2 copy Pls Crs } def
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+  hpt neg vpt -1.62 mul V closepath stroke
+  Pnt  } def
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+  translate 0 hpt M 4 {72 rotate 0 hpt L} repeat
+  closepath stroke grestore Pnt } def
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+  translate 0 hpt M 4 {72 rotate 0 hpt L} repeat
+  closepath fill grestore } def
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+       2 copy  vpt 0 90 arc closepath fill
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+       2 copy moveto
+       2 copy  vpt 180 270 arc closepath fill
+               vpt 0 360 arc } bind def
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+      2 copy vpt 270 360 arc closepath fill
+              vpt 0 360 arc closepath } bind def
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+      2 copy  vpt 270 450 arc closepath fill
+              vpt 0 360 arc closepath } bind def
+/C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill
+       2 copy moveto
+       2 copy vpt 90 180 arc closepath fill
+               vpt 0 360 arc closepath } bind def
+/C11 { BL [] 0 setdash 2 copy moveto
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+       2 copy moveto
+       2 copy  vpt 270 360 arc closepath fill
+               vpt 0 360 arc closepath } bind def
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+               vpt 0 360 arc closepath } bind def
+/C13 { BL [] 0 setdash  2 copy moveto
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+       2 copy moveto
+       2 copy  vpt 180 360 arc closepath fill
+               vpt 0 360 arc closepath } bind def
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+               vpt 0 360 arc } bind def
+/C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill
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+       neg 0 rlineto closepath } bind def
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+/S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def
+/S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def
+/S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill
+       exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def
+/S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def
+/S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill
+       2 copy vpt Square fill
+       Bsquare } bind def
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+/S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill
+       Bsquare } bind def
+/S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill
+       Bsquare } bind def
+/S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def
+/S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill
+       2 copy vpt Square fill Bsquare } bind def
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+       2 copy exch vpt sub exch vpt Square fill Bsquare } bind def
+/S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def
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+/D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def
+/D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def
+/D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def
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+/D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def
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+  hpt2 neg 0 V closepath stroke } def
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+  translate 0 hpt M 4 {72 rotate 0 hpt L} repeat
+  closepath stroke grestore } def
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+  hpt neg vpt neg V hpt vpt neg V
+  hpt vpt V hpt neg vpt V Opaque stroke } def
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+  translate 0 hpt M 4 {72 rotate 0 hpt L} repeat
+  Opaque stroke grestore } def
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+  hpt 0 360 arc Opaque stroke } def
+/BoxFill { gsave Rec 1 setgray fill grestore } def
+/BoxColFill {
+  gsave Rec
+  /Fillden exch def
+  currentrgbcolor
+  /ColB exch def /ColG exch def /ColR exch def
+  /ColR ColR Fillden mul Fillden sub 1 add def
+  /ColG ColG Fillden mul Fillden sub 1 add def
+  /ColB ColB Fillden mul Fillden sub 1 add def
+  ColR ColG ColB setrgbcolor
+  fill grestore } def
+%
+% PostScript Level 1 Pattern Fill routine
+% Usage: x y w h s a XX PatternFill
+%	x,y = lower left corner of box to be filled
+%	w,h = width and height of box
+%	  a = angle in degrees between lines and x-axis
+%	 XX = 0/1 for no/yes cross-hatch
+%
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+    PFa 2 get -2 div PFa 3 get -2 div PFa 2 get PFa 3 get Rec
+    gsave 1 setgray fill grestore clip
+    currentlinewidth 0.5 mul setlinewidth
+    /PFs PFa 2 get dup mul PFa 3 get dup mul add sqrt def
+    0 0 M PFa 5 get rotate PFs -2 div dup translate
+	0 1 PFs PFa 4 get div 1 add floor cvi
+	{ PFa 4 get mul 0 M 0 PFs V } for
+    0 PFa 6 get ne {
+	0 1 PFs PFa 4 get div 1 add floor cvi
+	{ PFa 4 get mul 0 2 1 roll M PFs 0 V } for
+    } if
+    stroke grestore } def
+%
+/Symbol-Oblique /Symbol findfont [1 0 .167 1 0 0] makefont
+dup length dict begin {1 index /FID eq {pop pop} {def} ifelse} forall
+currentdict end definefont pop
+end
+%%EndProlog
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+/C14 { BL [] 0 setdash 2 copy moveto
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+               vpt 0 360 arc } bind def
+/C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill
+               vpt 0 360 arc closepath } bind def
+/Rec   { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto
+       neg 0 rlineto closepath } bind def
+/Square { dup Rec } bind def
+/Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def
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+/S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def
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+/S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def
+/S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def
+/S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill
+       exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def
+/S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def
+/S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill
+       2 copy vpt Square fill
+       Bsquare } bind def
+/S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def
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+/S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill
+       Bsquare } bind def
+/S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill
+       Bsquare } bind def
+/S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def
+/S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill
+       2 copy vpt Square fill Bsquare } bind def
+/S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill
+       2 copy exch vpt sub exch vpt Square fill Bsquare } bind def
+/S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def
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+/D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def
+/D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def
+/D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def
+/D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def
+/D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def
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+  hpt vpt V hpt neg vpt V closepath stroke } def
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+/PentE { stroke [] 0 setdash gsave
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+  closepath stroke grestore } def
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+/DiaW { stroke [] 0 setdash vpt add M
+  hpt neg vpt neg V hpt vpt neg V
+  hpt vpt V hpt neg vpt V Opaque stroke } def
+/BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M
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+/TriUW { stroke [] 0 setdash vpt 1.12 mul add M
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+/TriDW { stroke [] 0 setdash vpt 1.12 mul sub M
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+  Opaque stroke grestore } def
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+/BoxFill { gsave Rec 1 setgray fill grestore } def
+/BoxColFill {
+  gsave Rec
+  /Fillden exch def
+  currentrgbcolor
+  /ColB exch def /ColG exch def /ColR exch def
+  /ColR ColR Fillden mul Fillden sub 1 add def
+  /ColG ColG Fillden mul Fillden sub 1 add def
+  /ColB ColB Fillden mul Fillden sub 1 add def
+  ColR ColG ColB setrgbcolor
+  fill grestore } def
+%
+% PostScript Level 1 Pattern Fill routine
+% Usage: x y w h s a XX PatternFill
+%	x,y = lower left corner of box to be filled
+%	w,h = width and height of box
+%	  a = angle in degrees between lines and x-axis
+%	 XX = 0/1 for no/yes cross-hatch
+%
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+    PFa 2 get -2 div PFa 3 get -2 div PFa 2 get PFa 3 get Rec
+    gsave 1 setgray fill grestore clip
+    currentlinewidth 0.5 mul setlinewidth
+    /PFs PFa 2 get dup mul PFa 3 get dup mul add sqrt def
+    0 0 M PFa 5 get rotate PFs -2 div dup translate
+	0 1 PFs PFa 4 get div 1 add floor cvi
+	{ PFa 4 get mul 0 M 0 PFs V } for
+    0 PFa 6 get ne {
+	0 1 PFs PFa 4 get div 1 add floor cvi
+	{ PFa 4 get mul 0 2 1 roll M PFs 0 V } for
+    } if
+    stroke grestore } def
+%
+/Symbol-Oblique /Symbol findfont [1 0 .167 1 0 0] makefont
+dup length dict begin {1 index /FID eq {pop pop} {def} ifelse} forall
+currentdict end definefont pop
+end
+%%EndProlog
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diff --git a/gsl-1.9/doc/randist.texi b/gsl-1.9/doc/randist.texi
new file mode 100644
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@@ -0,0 +1,2296 @@
+@cindex random number distributions
+@cindex cumulative distribution functions (CDFs)
+@cindex CDFs, cumulative distribution functions
+@cindex inverse cumulative distribution functions
+@cindex quantile functions
+This chapter describes functions for generating random variates and
+computing their probability distributions.  Samples from the
+distributions described in this chapter can be obtained using any of the
+random number generators in the library as an underlying source of
+randomness.  
+
+In the simplest cases a non-uniform distribution can be obtained
+analytically from the uniform distribution of a random number generator
+by applying an appropriate transformation.  This method uses one call to
+the random number generator.  More complicated distributions are created
+by the @dfn{acceptance-rejection} method, which compares the desired
+distribution against a distribution which is similar and known
+analytically.  This usually requires several samples from the generator.
+
+The library also provides cumulative distribution functions and inverse
+cumulative distribution functions, sometimes referred to as quantile
+functions.  The cumulative distribution functions and their inverses are
+computed separately for the upper and lower tails of the distribution,
+allowing full accuracy to be retained for small results.
+
+The functions for random variates and probability density functions
+described in this section are declared in @file{gsl_randist.h}.  The
+corresponding cumulative distribution functions are declared in
+@file{gsl_cdf.h}.
+
+Note that the discrete random variate functions always
+return a value of type @code{unsigned int}, and on most platforms this
+has a maximum value of @c{$2^{32}-1 \approx 4.29\times10^9$} 
+@math{2^32-1 ~=~ 4.29e9}. They should only be called with
+a safe range of parameters (where there is a negligible probability of
+a variate exceeding this limit) to prevent incorrect results due to
+overflow.
+
+@menu
+* Random Number Distribution Introduction::  
+* The Gaussian Distribution::   
+* The Gaussian Tail Distribution::  
+* The Bivariate Gaussian Distribution::  
+* The Exponential Distribution::  
+* The Laplace Distribution::    
+* The Exponential Power Distribution::  
+* The Cauchy Distribution::     
+* The Rayleigh Distribution::   
+* The Rayleigh Tail Distribution::  
+* The Landau Distribution::     
+* The Levy alpha-Stable Distributions::  
+* The Levy skew alpha-Stable Distribution::  
+* The Gamma Distribution::      
+* The Flat (Uniform) Distribution::  
+* The Lognormal Distribution::  
+* The Chi-squared Distribution::  
+* The F-distribution::          
+* The t-distribution::          
+* The Beta Distribution::       
+* The Logistic Distribution::   
+* The Pareto Distribution::     
+* Spherical Vector Distributions::  
+* The Weibull Distribution::    
+* The Type-1 Gumbel Distribution::  
+* The Type-2 Gumbel Distribution::  
+* The Dirichlet Distribution::  
+* General Discrete Distributions::  
+* The Poisson Distribution::    
+* The Bernoulli Distribution::  
+* The Binomial Distribution::   
+* The Multinomial Distribution::  
+* The Negative Binomial Distribution::  
+* The Pascal Distribution::     
+* The Geometric Distribution::  
+* The Hypergeometric Distribution::  
+* The Logarithmic Distribution::  
+* Shuffling and Sampling::      
+* Random Number Distribution Examples::  
+* Random Number Distribution References and Further Reading::  
+@end menu
+
+@node Random Number Distribution Introduction
+@section Introduction
+
+Continuous random number distributions are defined by a probability
+density function, @math{p(x)}, such that the probability of @math{x}
+occurring in the infinitesimal range @math{x} to @math{x+dx} is @c{$p\,dx$}
+@math{p dx}.
+
+The cumulative distribution function for the lower tail @math{P(x)} is
+defined by the integral,
+@tex
+\beforedisplay
+$$
+P(x) = \int_{-\infty}^{x} dx' p(x')
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+P(x) = \int_@{-\infty@}^@{x@} dx' p(x')
+@end example
+
+@end ifinfo
+@noindent
+and gives the probability of a variate taking a value less than @math{x}.
+
+The cumulative distribution function for the upper tail @math{Q(x)} is
+defined by the integral,
+@tex
+\beforedisplay
+$$
+Q(x) = \int_{x}^{+\infty} dx' p(x')
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+Q(x) = \int_@{x@}^@{+\infty@} dx' p(x')
+@end example
+
+@end ifinfo
+@noindent
+and gives the probability of a variate taking a value greater than @math{x}.
+
+The upper and lower cumulative distribution functions are related by
+@math{P(x) + Q(x) = 1} and satisfy @c{$0 \le P(x) \le 1$}
+@math{0 <= P(x) <= 1}, @c{$0 \le Q(x) \le 1$}
+@math{0 <= Q(x) <= 1}.
+
+The inverse cumulative distributions, @c{$x=P^{-1}(P)$}
+@math{x=P^@{-1@}(P)} and @c{$x=Q^{-1}(Q)$}
+@math{x=Q^@{-1@}(Q)} give the values of @math{x}
+which correspond to a specific value of @math{P} or @math{Q}.  
+They can be used to find confidence limits from probability values.
+
+For discrete distributions the probability of sampling the integer
+value @math{k} is given by @math{p(k)}, where @math{\sum_k p(k) = 1}.
+The cumulative distribution for the lower tail @math{P(k)} of a
+discrete distribution is defined as,
+@tex
+\beforedisplay
+$$
+P(k) = \sum_{i \le k} p(i) 
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+P(k) = \sum_@{i <= k@} p(i)
+@end example
+
+@end ifinfo
+@noindent
+where the sum is over the allowed range of the distribution less than
+or equal to @math{k}.  
+
+The cumulative distribution for the upper tail of a discrete
+distribution @math{Q(k)} is defined as
+@tex
+\beforedisplay
+$$
+Q(k) = \sum_{i > k} p(i) 
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+Q(k) = \sum_@{i > k@} p(i)
+@end example
+
+@end ifinfo
+@noindent
+giving the sum of probabilities for all values greater than @math{k}.
+These two definitions satisfy the identity @math{P(k)+Q(k)=1}.
+
+If the range of the distribution is 1 to @math{n} inclusive then
+@math{P(n)=1}, @math{Q(n)=0} while @math{P(1) = p(1)},
+@math{Q(1)=1-p(1)}.
+
+@page
+@node The Gaussian Distribution
+@section The Gaussian Distribution
+@deftypefun double gsl_ran_gaussian (const gsl_rng * @var{r}, double @var{sigma})
+@cindex Gaussian distribution
+This function returns a Gaussian random variate, with mean zero and
+standard deviation @var{sigma}.  The probability distribution for
+Gaussian random variates is,
+@tex
+\beforedisplay
+$$
+p(x) dx = {1 \over \sqrt{2 \pi \sigma^2}} \exp (-x^2 / 2\sigma^2) dx
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+p(x) dx = @{1 \over \sqrt@{2 \pi \sigma^2@}@} \exp (-x^2 / 2\sigma^2) dx
+@end example
+
+@end ifinfo
+@noindent
+for @math{x} in the range @math{-\infty} to @math{+\infty}.  Use the
+transformation @math{z = \mu + x} on the numbers returned by
+@code{gsl_ran_gaussian} to obtain a Gaussian distribution with mean
+@math{\mu}.  This function uses the Box-Mueller algorithm which requires two
+calls to the random number generator @var{r}.
+@end deftypefun
+
+@deftypefun double gsl_ran_gaussian_pdf (double @var{x}, double @var{sigma})
+This function computes the probability density @math{p(x)} at @var{x}
+for a Gaussian distribution with standard deviation @var{sigma}, using
+the formula given above.
+@end deftypefun
+
+@sp 1
+@tex
+\centerline{\input rand-gaussian.tex}
+@end tex
+
+@deftypefun double gsl_ran_gaussian_ziggurat (const gsl_rng * @var{r}, double @var{sigma})
+@deftypefunx double gsl_ran_gaussian_ratio_method (const gsl_rng * @var{r}, double @var{sigma})
+This function computes a Gaussian random variate using the alternative
+Marsaglia-Tsang ziggurat and Kinderman-Monahan-Leva ratio methods.  The
+Ziggurat algorithm is the fastest available algorithm in most cases.
+@end deftypefun
+
+@deftypefun double gsl_ran_ugaussian (const gsl_rng * @var{r})
+@deftypefunx double gsl_ran_ugaussian_pdf (double @var{x})
+@deftypefunx double gsl_ran_ugaussian_ratio_method (const gsl_rng * @var{r})
+These functions compute results for the unit Gaussian distribution.  They
+are equivalent to the functions above with a standard deviation of one,
+@var{sigma} = 1.
+@end deftypefun
+
+@deftypefun double gsl_cdf_gaussian_P (double @var{x}, double @var{sigma})
+@deftypefunx double gsl_cdf_gaussian_Q (double @var{x}, double @var{sigma})
+@deftypefunx double gsl_cdf_gaussian_Pinv (double @var{P}, double @var{sigma})
+@deftypefunx double gsl_cdf_gaussian_Qinv (double @var{Q}, double @var{sigma})
+These functions compute the cumulative distribution functions
+@math{P(x)}, @math{Q(x)} and their inverses for the Gaussian
+distribution with standard deviation @var{sigma}.
+@end deftypefun
+
+@deftypefun double gsl_cdf_ugaussian_P (double @var{x})
+@deftypefunx double gsl_cdf_ugaussian_Q (double @var{x})
+@deftypefunx double gsl_cdf_ugaussian_Pinv (double @var{P})
+@deftypefunx double gsl_cdf_ugaussian_Qinv (double @var{Q})
+These functions compute the cumulative distribution functions
+@math{P(x)}, @math{Q(x)} and their inverses for the unit Gaussian
+distribution.
+@end deftypefun
+
+@page
+@node The Gaussian Tail Distribution
+@section The Gaussian Tail Distribution
+@deftypefun double gsl_ran_gaussian_tail (const gsl_rng * @var{r}, double @var{a}, double @var{sigma})
+@cindex Gaussian Tail distribution
+This function provides random variates from the upper tail of a Gaussian
+distribution with standard deviation @var{sigma}.  The values returned
+are larger than the lower limit @var{a}, which must be positive.  The
+method is based on Marsaglia's famous rectangle-wedge-tail algorithm (Ann. 
+Math. Stat. 32, 894--899 (1961)), with this aspect explained in Knuth, v2,
+3rd ed, p139,586 (exercise 11).
+
+The probability distribution for Gaussian tail random variates is,
+@tex
+\beforedisplay
+$$
+p(x) dx = {1 \over N(a;\sigma) \sqrt{2 \pi \sigma^2}} \exp (- x^2 / 2\sigma^2) dx
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+p(x) dx = @{1 \over N(a;\sigma) \sqrt@{2 \pi \sigma^2@}@} \exp (- x^2/(2 \sigma^2)) dx
+@end example
+
+@end ifinfo
+@noindent
+for @math{x > a} where @math{N(a;\sigma)} is the normalization constant,
+@tex
+\beforedisplay
+$$
+N(a;\sigma) = {1 \over 2} \hbox{erfc}\left({a \over \sqrt{2 \sigma^2}}\right).
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+N(a;\sigma) = (1/2) erfc(a / sqrt(2 sigma^2)).
+@end example
+@end ifinfo
+
+@end deftypefun
+
+@deftypefun double gsl_ran_gaussian_tail_pdf (double @var{x}, double @var{a}, double @var{sigma})
+This function computes the probability density @math{p(x)} at @var{x}
+for a Gaussian tail distribution with standard deviation @var{sigma} and
+lower limit @var{a}, using the formula given above.
+@end deftypefun
+
+@sp 1
+@tex
+\centerline{\input rand-gaussian-tail.tex}
+@end tex
+
+@deftypefun double gsl_ran_ugaussian_tail (const gsl_rng * @var{r}, double @var{a})
+@deftypefunx double gsl_ran_ugaussian_tail_pdf (double @var{x}, double @var{a})
+These functions compute results for the tail of a unit Gaussian
+distribution.  They are equivalent to the functions above with a standard
+deviation of one, @var{sigma} = 1.
+@end deftypefun
+
+
+@page
+@node The Bivariate Gaussian Distribution
+@section The Bivariate Gaussian Distribution
+
+@deftypefun void gsl_ran_bivariate_gaussian (const gsl_rng * @var{r}, double @var{sigma_x}, double @var{sigma_y}, double @var{rho}, double * @var{x}, double * @var{y})
+@cindex Bivariate Gaussian distribution
+@cindex two dimensional Gaussian distribution
+@cindex Gaussian distribution, bivariate
+This function generates a pair of correlated Gaussian variates, with
+mean zero, correlation coefficient @var{rho} and standard deviations
+@var{sigma_x} and @var{sigma_y} in the @math{x} and @math{y} directions.
+The probability distribution for bivariate Gaussian random variates is,
+@tex
+\beforedisplay
+$$
+p(x,y) dx dy = {1 \over 2 \pi \sigma_x \sigma_y \sqrt{1-\rho^2}} \exp \left(-{(x^2/\sigma_x^2 + y^2/\sigma_y^2 - 2 \rho x y/(\sigma_x\sigma_y)) \over 2(1-\rho^2)}\right) dx dy
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+p(x,y) dx dy = @{1 \over 2 \pi \sigma_x \sigma_y \sqrt@{1-\rho^2@}@} \exp (-(x^2/\sigma_x^2 + y^2/\sigma_y^2 - 2 \rho x y/(\sigma_x\sigma_y))/2(1-\rho^2)) dx dy
+@end example
+
+@end ifinfo
+@noindent
+for @math{x,y} in the range @math{-\infty} to @math{+\infty}.  The
+correlation coefficient @var{rho} should lie between @math{1} and
+@math{-1}.
+@end deftypefun
+
+@deftypefun double gsl_ran_bivariate_gaussian_pdf (double @var{x}, double @var{y}, double @var{sigma_x}, double @var{sigma_y}, double @var{rho})
+This function computes the probability density @math{p(x,y)} at
+(@var{x},@var{y}) for a bivariate Gaussian distribution with standard
+deviations @var{sigma_x}, @var{sigma_y} and correlation coefficient
+@var{rho}, using the formula given above.
+@end deftypefun
+
+@sp 1
+@tex
+\centerline{\input rand-bivariate-gaussian.tex}
+@end tex
+
+@page
+@node The Exponential Distribution
+@section The Exponential Distribution
+@deftypefun double gsl_ran_exponential (const gsl_rng * @var{r}, double @var{mu})
+@cindex Exponential distribution
+This function returns a random variate from the exponential distribution
+with mean @var{mu}. The distribution is,
+@tex
+\beforedisplay
+$$
+p(x) dx = {1 \over \mu} \exp(-x/\mu) dx
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+p(x) dx = @{1 \over \mu@} \exp(-x/\mu) dx
+@end example
+
+@end ifinfo
+@noindent
+for @c{$x \ge 0$}
+@math{x >= 0}. 
+@end deftypefun
+
+@deftypefun double gsl_ran_exponential_pdf (double @var{x}, double @var{mu})
+This function computes the probability density @math{p(x)} at @var{x}
+for an exponential distribution with mean @var{mu}, using the formula
+given above.
+@end deftypefun
+
+@sp 1
+@tex
+\centerline{\input rand-exponential.tex}
+@end tex
+
+@deftypefun double gsl_cdf_exponential_P (double @var{x}, double @var{mu})
+@deftypefunx double gsl_cdf_exponential_Q (double @var{x}, double @var{mu})
+@deftypefunx double gsl_cdf_exponential_Pinv (double @var{P}, double @var{mu})
+@deftypefunx double gsl_cdf_exponential_Qinv (double @var{Q}, double @var{mu})
+These functions compute the cumulative distribution functions
+@math{P(x)}, @math{Q(x)} and their inverses for the exponential
+distribution with mean @var{mu}.
+@end deftypefun
+
+@page
+@node The Laplace Distribution
+@section The Laplace Distribution
+@deftypefun double gsl_ran_laplace (const gsl_rng * @var{r}, double @var{a})
+@cindex two-sided exponential distribution
+@cindex Laplace distribution
+This function returns a random variate from the Laplace distribution
+with width @var{a}.  The distribution is,
+@tex
+\beforedisplay
+$$
+p(x) dx = {1 \over 2 a}  \exp(-|x/a|) dx
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+p(x) dx = @{1 \over 2 a@}  \exp(-|x/a|) dx
+@end example
+
+@end ifinfo
+@noindent
+for @math{-\infty < x < \infty}.
+@end deftypefun
+
+@deftypefun double gsl_ran_laplace_pdf (double @var{x}, double @var{a})
+This function computes the probability density @math{p(x)} at @var{x}
+for a Laplace distribution with width @var{a}, using the formula
+given above.
+@end deftypefun
+
+@sp 1
+@tex
+\centerline{\input rand-laplace.tex}
+@end tex
+
+@deftypefun double gsl_cdf_laplace_P (double @var{x}, double @var{a})
+@deftypefunx double gsl_cdf_laplace_Q (double @var{x}, double @var{a})
+@deftypefunx double gsl_cdf_laplace_Pinv (double @var{P}, double @var{a})
+@deftypefunx double gsl_cdf_laplace_Qinv (double @var{Q}, double @var{a})
+These functions compute the cumulative distribution functions
+@math{P(x)}, @math{Q(x)} and their inverses for the Laplace
+distribution with width @var{a}.
+@end deftypefun
+
+
+@page
+@node The Exponential Power Distribution
+@section The Exponential Power Distribution
+@deftypefun double gsl_ran_exppow (const gsl_rng * @var{r}, double @var{a}, double @var{b})
+@cindex Exponential power distribution
+This function returns a random variate from the exponential power distribution
+with scale parameter @var{a} and exponent @var{b}.  The distribution is,
+@tex
+\beforedisplay
+$$
+p(x) dx = {1 \over 2 a \Gamma(1+1/b)} \exp(-|x/a|^b) dx
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+p(x) dx = @{1 \over 2 a \Gamma(1+1/b)@} \exp(-|x/a|^b) dx
+@end example
+
+@end ifinfo
+@noindent
+for @c{$x \ge 0$}
+@math{x >= 0}.  For @math{b = 1} this reduces to the Laplace
+distribution.  For @math{b = 2} it has the same form as a gaussian
+distribution, but with @c{$a = \sqrt{2} \sigma$}
+@math{a = \sqrt@{2@} \sigma}.
+@end deftypefun
+
+@deftypefun double gsl_ran_exppow_pdf (double @var{x}, double @var{a}, double @var{b})
+This function computes the probability density @math{p(x)} at @var{x}
+for an exponential power distribution with scale parameter @var{a}
+and exponent @var{b}, using the formula given above.
+@end deftypefun
+
+@sp 1
+@tex
+\centerline{\input rand-exppow.tex}
+@end tex
+
+@deftypefun double gsl_cdf_exppow_P (double @var{x}, double @var{a}, double @var{b})
+@deftypefunx double gsl_cdf_exppow_Q (double @var{x}, double @var{a}, double @var{b})
+These functions compute the cumulative distribution functions
+@math{P(x)}, @math{Q(x)} for the exponential power distribution with
+parameters @var{a} and @var{b}.
+@end deftypefun
+
+
+@page
+@node The Cauchy Distribution
+@section The Cauchy Distribution
+@deftypefun double gsl_ran_cauchy (const gsl_rng * @var{r}, double @var{a})
+@cindex Cauchy distribution
+This function returns a random variate from the Cauchy distribution with
+scale parameter @var{a}.  The probability distribution for Cauchy
+random variates is,
+@tex
+\beforedisplay
+$$
+p(x) dx = {1 \over a\pi (1 + (x/a)^2) } dx
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+p(x) dx = @{1 \over a\pi (1 + (x/a)^2) @} dx
+@end example
+
+@end ifinfo
+@noindent
+for @math{x} in the range @math{-\infty} to @math{+\infty}.  The Cauchy
+distribution is also known as the Lorentz distribution.
+@end deftypefun
+
+@deftypefun double gsl_ran_cauchy_pdf (double @var{x}, double @var{a})
+This function computes the probability density @math{p(x)} at @var{x}
+for a Cauchy distribution with scale parameter @var{a}, using the formula
+given above.
+@end deftypefun
+
+@sp 1
+@tex
+\centerline{\input rand-cauchy.tex}
+@end tex
+
+@deftypefun double gsl_cdf_cauchy_P (double @var{x}, double @var{a})
+@deftypefunx double gsl_cdf_cauchy_Q (double @var{x}, double @var{a})
+@deftypefunx double gsl_cdf_cauchy_Pinv (double @var{P}, double @var{a})
+@deftypefunx double gsl_cdf_cauchy_Qinv (double @var{Q}, double @var{a})
+These functions compute the cumulative distribution functions
+@math{P(x)}, @math{Q(x)} and their inverses for the Cauchy
+distribution with scale parameter @var{a}.
+@end deftypefun
+
+
+@page
+@node The Rayleigh Distribution
+@section The Rayleigh Distribution
+@deftypefun double gsl_ran_rayleigh (const gsl_rng * @var{r}, double @var{sigma})
+@cindex Rayleigh distribution
+This function returns a random variate from the Rayleigh distribution with
+scale parameter @var{sigma}.  The distribution is,
+@tex
+\beforedisplay
+$$
+p(x) dx = {x \over \sigma^2} \exp(- x^2/(2 \sigma^2)) dx
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+p(x) dx = @{x \over \sigma^2@} \exp(- x^2/(2 \sigma^2)) dx
+@end example
+
+@end ifinfo
+@noindent
+for @math{x > 0}.
+@end deftypefun
+
+@deftypefun double gsl_ran_rayleigh_pdf (double @var{x}, double @var{sigma})
+This function computes the probability density @math{p(x)} at @var{x}
+for a Rayleigh distribution with scale parameter @var{sigma}, using the
+formula given above.
+@end deftypefun
+
+@sp 1
+@tex
+\centerline{\input rand-rayleigh.tex}
+@end tex
+
+@deftypefun double gsl_cdf_rayleigh_P (double @var{x}, double @var{sigma})
+@deftypefunx double gsl_cdf_rayleigh_Q (double @var{x}, double @var{sigma})
+@deftypefunx double gsl_cdf_rayleigh_Pinv (double @var{P}, double @var{sigma})
+@deftypefunx double gsl_cdf_rayleigh_Qinv (double @var{Q}, double @var{sigma})
+These functions compute the cumulative distribution functions
+@math{P(x)}, @math{Q(x)} and their inverses for the Rayleigh
+distribution with scale parameter @var{sigma}.
+@end deftypefun
+
+
+@page
+@node The Rayleigh Tail Distribution
+@section The Rayleigh Tail Distribution
+@deftypefun double gsl_ran_rayleigh_tail (const gsl_rng * @var{r}, double @var{a}, double @var{sigma})
+@cindex Rayleigh Tail distribution
+This function returns a random variate from the tail of the Rayleigh
+distribution with scale parameter @var{sigma} and a lower limit of
+@var{a}.  The distribution is,
+@tex
+\beforedisplay
+$$
+p(x) dx = {x \over \sigma^2} \exp ((a^2 - x^2) /(2 \sigma^2)) dx
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+p(x) dx = @{x \over \sigma^2@} \exp ((a^2 - x^2) /(2 \sigma^2)) dx
+@end example
+
+@end ifinfo
+@noindent
+for @math{x > a}.
+@end deftypefun
+
+@deftypefun double gsl_ran_rayleigh_tail_pdf (double @var{x}, double @var{a}, double @var{sigma})
+This function computes the probability density @math{p(x)} at @var{x}
+for a Rayleigh tail distribution with scale parameter @var{sigma} and
+lower limit @var{a}, using the formula given above.
+@end deftypefun
+
+@sp 1
+@tex
+\centerline{\input rand-rayleigh-tail.tex}
+@end tex
+
+@page
+@node The Landau Distribution
+@section The Landau Distribution
+@deftypefun double gsl_ran_landau (const gsl_rng * @var{r})
+@cindex Landau distribution
+This function returns a random variate from the Landau distribution.  The
+probability distribution for Landau random variates is defined
+analytically by the complex integral,
+@tex
+\beforedisplay
+$$
+p(x) = 
+{1 \over {2 \pi i}} \int_{c-i\infty}^{c+i\infty} ds\, \exp(s \log(s) + x s) 
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+p(x) = (1/(2 \pi i)) \int_@{c-i\infty@}^@{c+i\infty@} ds exp(s log(s) + x s) 
+@end example
+@end ifinfo
+For numerical purposes it is more convenient to use the following
+equivalent form of the integral,
+@tex
+\beforedisplay
+$$
+p(x) = (1/\pi) \int_0^\infty dt \exp(-t \log(t) - x t) \sin(\pi t).
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+p(x) = (1/\pi) \int_0^\infty dt \exp(-t \log(t) - x t) \sin(\pi t).
+@end example
+@end ifinfo
+@end deftypefun
+
+@deftypefun double gsl_ran_landau_pdf (double @var{x})
+This function computes the probability density @math{p(x)} at @var{x}
+for the Landau distribution using an approximation to the formula given
+above.
+@end deftypefun
+
+@sp 1
+@tex
+\centerline{\input rand-landau.tex}
+@end tex
+
+@page
+@node The Levy alpha-Stable Distributions
+@section The Levy alpha-Stable Distributions
+@deftypefun double gsl_ran_levy (const gsl_rng * @var{r}, double @var{c}, double @var{alpha})
+@cindex Levy distribution
+This function returns a random variate from the Levy symmetric stable
+distribution with scale @var{c} and exponent @var{alpha}.  The symmetric
+stable probability distribution is defined by a fourier transform,
+@tex
+\beforedisplay
+$$
+p(x) = {1 \over 2 \pi} \int_{-\infty}^{+\infty} dt \exp(-it x - |c t|^\alpha)
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+p(x) = @{1 \over 2 \pi@} \int_@{-\infty@}^@{+\infty@} dt \exp(-it x - |c t|^alpha)
+@end example
+
+@end ifinfo
+@noindent
+There is no explicit solution for the form of @math{p(x)} and the
+library does not define a corresponding @code{pdf} function.  For
+@math{\alpha = 1} the distribution reduces to the Cauchy distribution.  For
+@math{\alpha = 2} it is a Gaussian distribution with @c{$\sigma = \sqrt{2} c$} 
+@math{\sigma = \sqrt@{2@} c}.  For @math{\alpha < 1} the tails of the
+distribution become extremely wide.
+
+The algorithm only works for @c{$0 < \alpha \le 2$}
+@math{0 < alpha <= 2}.
+@end deftypefun
+
+@sp 1
+@tex
+\centerline{\input rand-levy.tex}
+@end tex
+
+@page
+@node The Levy skew alpha-Stable Distribution
+@section The Levy skew alpha-Stable Distribution
+
+@deftypefun double gsl_ran_levy_skew (const gsl_rng * @var{r}, double @var{c}, double @var{alpha}, double @var{beta})
+@cindex Levy distribution, skew
+@cindex Skew Levy distribution
+This function returns a random variate from the Levy skew stable
+distribution with scale @var{c}, exponent @var{alpha} and skewness
+parameter @var{beta}.  The skewness parameter must lie in the range
+@math{[-1,1]}.  The Levy skew stable probability distribution is defined
+by a fourier transform,
+@tex
+\beforedisplay
+$$
+p(x) = {1 \over 2 \pi} \int_{-\infty}^{+\infty} dt \exp(-it x - |c t|^\alpha (1-i \beta \sign(t) \tan(\pi\alpha/2)))
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+p(x) = @{1 \over 2 \pi@} \int_@{-\infty@}^@{+\infty@} dt \exp(-it x - |c t|^alpha (1-i beta sign(t) tan(pi alpha/2)))
+@end example
+
+@end ifinfo
+@noindent
+When @math{\alpha = 1} the term @math{\tan(\pi \alpha/2)} is replaced by
+@math{-(2/\pi)\log|t|}.  There is no explicit solution for the form of
+@math{p(x)} and the library does not define a corresponding @code{pdf}
+function.  For @math{\alpha = 2} the distribution reduces to a Gaussian
+distribution with @c{$\sigma = \sqrt{2} c$} 
+@math{\sigma = \sqrt@{2@} c} and the skewness parameter has no effect.  
+For @math{\alpha < 1} the tails of the distribution become extremely
+wide.  The symmetric distribution corresponds to @math{\beta =
+0}.
+
+The algorithm only works for @c{$0 < \alpha \le 2$}
+@math{0 < alpha <= 2}.
+@end deftypefun
+
+The Levy alpha-stable distributions have the property that if @math{N}
+alpha-stable variates are drawn from the distribution @math{p(c, \alpha,
+\beta)} then the sum @math{Y = X_1 + X_2 + \dots + X_N} will also be
+distributed as an alpha-stable variate,
+@c{$p(N^{1/\alpha} c, \alpha, \beta)$}  
+@math{p(N^(1/\alpha) c, \alpha, \beta)}.
+
+@comment PDF not available because there is no analytic expression for it
+@comment
+@comment @deftypefun double gsl_ran_levy_pdf (double @var{x}, double @var{mu})
+@comment This function computes the probability density @math{p(x)} at @var{x}
+@comment for a symmetric Levy distribution with scale parameter @var{mu} and
+@comment exponent @var{a}, using the formula given above.
+@comment @end deftypefun
+
+@sp 1
+@tex
+\centerline{\input rand-levyskew.tex}
+@end tex
+
+@page
+@node The Gamma Distribution
+@section The Gamma Distribution
+@deftypefun double gsl_ran_gamma (const gsl_rng * @var{r}, double @var{a}, double @var{b})
+@cindex Gamma distribution
+This function returns a random variate from the gamma
+distribution.  The distribution function is,
+@tex
+\beforedisplay
+$$
+p(x) dx = {1 \over \Gamma(a) b^a} x^{a-1} e^{-x/b} dx
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+p(x) dx = @{1 \over \Gamma(a) b^a@} x^@{a-1@} e^@{-x/b@} dx
+@end example
+
+@end ifinfo
+@noindent
+for @math{x > 0}.
+@comment If @xmath{X} and @xmath{Y} are independent gamma-distributed random
+@comment variables of order @xmath{a} and @xmath{b}, then @xmath{X+Y} has a gamma
+@comment distribution of order @xmath{a+b}.
+
+@cindex Erlang distribution
+The gamma distribution with an integer parameter @var{a} is known as the Erlang distribution.
+
+The variates are computed using the Marsaglia-Tsang fast gamma method.
+This function for this method was previously called
+@code{gsl_ran_gamma_mt} and can still be accessed using this name.
+@end deftypefun
+
+@deftypefun double gsl_ran_gamma_knuth (const gsl_rng * @var{r}, double @var{a}, double @var{b})
+This function returns a gamma variate using the algorithms from Knuth (vol 2).
+@end deftypefun
+
+@deftypefun double gsl_ran_gamma_pdf (double @var{x}, double @var{a}, double @var{b})
+This function computes the probability density @math{p(x)} at @var{x}
+for a gamma distribution with parameters @var{a} and @var{b}, using the
+formula given above.
+@end deftypefun
+
+@sp 1
+@tex
+\centerline{\input rand-gamma.tex}
+@end tex
+
+@deftypefun double gsl_cdf_gamma_P (double @var{x}, double @var{a}, double @var{b})
+@deftypefunx double gsl_cdf_gamma_Q (double @var{x}, double @var{a}, double @var{b})
+@deftypefunx double gsl_cdf_gamma_Pinv (double @var{P}, double @var{a}, double @var{b})
+@deftypefunx double gsl_cdf_gamma_Qinv (double @var{Q}, double @var{a}, double @var{b})
+These functions compute the cumulative distribution functions
+@math{P(x)}, @math{Q(x)} and their inverses for the gamma
+distribution with parameters @var{a} and @var{b}.
+@end deftypefun
+
+@page
+@node The Flat (Uniform) Distribution
+@section The Flat (Uniform) Distribution
+@deftypefun double gsl_ran_flat (const gsl_rng * @var{r}, double @var{a}, double @var{b})
+@cindex flat distribution
+@cindex uniform distribution
+This function returns a random variate from the flat (uniform)
+distribution from @var{a} to @var{b}. The distribution is,
+@tex
+\beforedisplay
+$$
+p(x) dx = {1 \over (b-a)} dx
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+p(x) dx = @{1 \over (b-a)@} dx
+@end example
+
+@end ifinfo
+@noindent
+if @c{$a \le x < b$}
+@math{a <= x < b} and 0 otherwise.
+@end deftypefun
+
+@deftypefun double gsl_ran_flat_pdf (double @var{x}, double @var{a}, double @var{b})
+This function computes the probability density @math{p(x)} at @var{x}
+for a uniform distribution from @var{a} to @var{b}, using the formula
+given above.
+@end deftypefun
+
+@sp 1
+@tex
+\centerline{\input rand-flat.tex}
+@end tex
+
+@deftypefun double gsl_cdf_flat_P (double @var{x}, double @var{a}, double @var{b})
+@deftypefunx double gsl_cdf_flat_Q (double @var{x}, double @var{a}, double @var{b})
+@deftypefunx double gsl_cdf_flat_Pinv (double @var{P}, double @var{a}, double @var{b})
+@deftypefunx double gsl_cdf_flat_Qinv (double @var{Q}, double @var{a}, double @var{b})
+These functions compute the cumulative distribution functions
+@math{P(x)}, @math{Q(x)} and their inverses for a uniform distribution
+from @var{a} to @var{b}.
+@end deftypefun
+
+
+@page
+@node The Lognormal Distribution
+@section The Lognormal Distribution
+@deftypefun double gsl_ran_lognormal (const gsl_rng * @var{r}, double @var{zeta}, double @var{sigma})
+@cindex Lognormal distribution
+This function returns a random variate from the lognormal
+distribution.  The distribution function is,
+@tex
+\beforedisplay
+$$
+p(x) dx = {1 \over x \sqrt{2 \pi \sigma^2}} \exp(-(\ln(x) - \zeta)^2/2 \sigma^2) dx
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+p(x) dx = @{1 \over x \sqrt@{2 \pi \sigma^2@} @} \exp(-(\ln(x) - \zeta)^2/2 \sigma^2) dx
+@end example
+
+@end ifinfo
+@noindent
+for @math{x > 0}.
+@end deftypefun
+
+@deftypefun double gsl_ran_lognormal_pdf (double @var{x}, double @var{zeta}, double @var{sigma})
+This function computes the probability density @math{p(x)} at @var{x}
+for a lognormal distribution with parameters @var{zeta} and @var{sigma},
+using the formula given above.
+@end deftypefun
+
+@sp 1
+@tex
+\centerline{\input rand-lognormal.tex}
+@end tex
+
+@deftypefun double gsl_cdf_lognormal_P (double @var{x}, double @var{zeta}, double @var{sigma})
+@deftypefunx double gsl_cdf_lognormal_Q (double @var{x}, double @var{zeta}, double @var{sigma})
+@deftypefunx double gsl_cdf_lognormal_Pinv (double @var{P}, double @var{zeta}, double @var{sigma})
+@deftypefunx double gsl_cdf_lognormal_Qinv (double @var{Q}, double @var{zeta}, double @var{sigma})
+These functions compute the cumulative distribution functions
+@math{P(x)}, @math{Q(x)} and their inverses for the lognormal
+distribution with parameters @var{zeta} and @var{sigma}.
+@end deftypefun
+
+
+@page
+@node The Chi-squared Distribution
+@section The Chi-squared Distribution
+The chi-squared distribution arises in statistics.  If @math{Y_i} are
+@math{n} independent gaussian random variates with unit variance then the
+sum-of-squares,
+@tex
+\beforedisplay
+$$
+X_i = \sum_i Y_i^2
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+X_i = \sum_i Y_i^2
+@end example
+
+@end ifinfo
+@noindent
+has a chi-squared distribution with @math{n} degrees of freedom.
+
+@deftypefun double gsl_ran_chisq (const gsl_rng * @var{r}, double @var{nu})
+@cindex Chi-squared distribution
+This function returns a random variate from the chi-squared distribution
+with @var{nu} degrees of freedom. The distribution function is,
+@tex
+\beforedisplay
+$$
+p(x) dx = {1 \over 2 \Gamma(\nu/2) } (x/2)^{\nu/2 - 1} \exp(-x/2) dx
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+p(x) dx = @{1 \over 2 \Gamma(\nu/2) @} (x/2)^@{\nu/2 - 1@} \exp(-x/2) dx
+@end example
+
+@end ifinfo
+@noindent
+for @c{$x \ge 0$}
+@math{x >= 0}. 
+@end deftypefun
+
+@deftypefun double gsl_ran_chisq_pdf (double @var{x}, double @var{nu})
+This function computes the probability density @math{p(x)} at @var{x}
+for a chi-squared distribution with @var{nu} degrees of freedom, using
+the formula given above.
+@end deftypefun
+
+@sp 1
+@tex
+\centerline{\input rand-chisq.tex}
+@end tex
+
+@deftypefun double gsl_cdf_chisq_P (double @var{x}, double @var{nu})
+@deftypefunx double gsl_cdf_chisq_Q (double @var{x}, double @var{nu})
+@deftypefunx double gsl_cdf_chisq_Pinv (double @var{P}, double @var{nu})
+@deftypefunx double gsl_cdf_chisq_Qinv (double @var{Q}, double @var{nu})
+These functions compute the cumulative distribution functions
+@math{P(x)}, @math{Q(x)} and their inverses for the chi-squared
+distribution with @var{nu} degrees of freedom.
+@end deftypefun
+
+
+
+@page
+@node The F-distribution
+@section The F-distribution
+The F-distribution arises in statistics.  If @math{Y_1} and @math{Y_2}
+are chi-squared deviates with @math{\nu_1} and @math{\nu_2} degrees of
+freedom then the ratio,
+@tex
+\beforedisplay
+$$
+X = { (Y_1 / \nu_1) \over (Y_2 / \nu_2) }
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+X = @{ (Y_1 / \nu_1) \over (Y_2 / \nu_2) @}
+@end example
+
+@end ifinfo
+@noindent
+has an F-distribution @math{F(x;\nu_1,\nu_2)}.
+
+@deftypefun double gsl_ran_fdist (const gsl_rng * @var{r}, double @var{nu1}, double @var{nu2})
+@cindex F-distribution
+This function returns a random variate from the F-distribution with degrees of freedom @var{nu1} and @var{nu2}. The distribution function is,
+@tex
+\beforedisplay
+$$
+p(x) dx = 
+   { \Gamma((\nu_1 + \nu_2)/2)
+        \over \Gamma(\nu_1/2) \Gamma(\nu_2/2) } 
+   \nu_1^{\nu_1/2} \nu_2^{\nu_2/2} 
+   x^{\nu_1/2 - 1} (\nu_2 + \nu_1 x)^{-\nu_1/2 -\nu_2/2}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+p(x) dx = 
+   @{ \Gamma((\nu_1 + \nu_2)/2)
+        \over \Gamma(\nu_1/2) \Gamma(\nu_2/2) @} 
+   \nu_1^@{\nu_1/2@} \nu_2^@{\nu_2/2@} 
+   x^@{\nu_1/2 - 1@} (\nu_2 + \nu_1 x)^@{-\nu_1/2 -\nu_2/2@}
+@end example
+
+@end ifinfo
+@noindent
+for @c{$x \ge 0$}
+@math{x >= 0}. 
+@end deftypefun
+
+@deftypefun double gsl_ran_fdist_pdf (double @var{x}, double @var{nu1}, double @var{nu2})
+This function computes the probability density @math{p(x)} at @var{x}
+for an F-distribution with @var{nu1} and @var{nu2} degrees of freedom,
+using the formula given above.
+@end deftypefun
+
+@sp 1
+@tex
+\centerline{\input rand-fdist.tex}
+@end tex
+
+@deftypefun double gsl_cdf_fdist_P (double @var{x}, double @var{nu1}, double @var{nu2})
+@deftypefunx double gsl_cdf_fdist_Q (double @var{x}, double @var{nu1}, double @var{nu2})
+@deftypefunx double gsl_cdf_fdist_Pinv (double @var{P}, double @var{nu1}, double @var{nu2})
+@deftypefunx double gsl_cdf_fdist_Qinv (double @var{Q}, double @var{nu1}, double @var{nu2})
+These functions compute the cumulative distribution functions
+@math{P(x)}, @math{Q(x)} and their inverses for the F-distribution
+with @var{nu1} and @var{nu2} degrees of freedom.
+@end deftypefun
+
+@page
+@node The t-distribution
+@section The t-distribution
+The t-distribution arises in statistics.  If @math{Y_1} has a normal
+distribution and @math{Y_2} has a chi-squared distribution with
+@math{\nu} degrees of freedom then the ratio,
+@tex
+\beforedisplay
+$$
+X = { Y_1 \over \sqrt{Y_2 / \nu} }
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+X = @{ Y_1 \over \sqrt@{Y_2 / \nu@} @}
+@end example
+
+@end ifinfo
+@noindent
+has a t-distribution @math{t(x;\nu)} with @math{\nu} degrees of freedom.
+
+@deftypefun double gsl_ran_tdist (const gsl_rng * @var{r}, double @var{nu})
+@cindex t-distribution
+@cindex Student t-distribution
+This function returns a random variate from the t-distribution.  The
+distribution function is,
+@tex
+\beforedisplay
+$$
+p(x) dx = {\Gamma((\nu + 1)/2) \over \sqrt{\pi \nu} \Gamma(\nu/2)}
+   (1 + x^2/\nu)^{-(\nu + 1)/2} dx
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+p(x) dx = @{\Gamma((\nu + 1)/2) \over \sqrt@{\pi \nu@} \Gamma(\nu/2)@}
+   (1 + x^2/\nu)^@{-(\nu + 1)/2@} dx
+@end example
+
+@end ifinfo
+@noindent
+for @math{-\infty < x < +\infty}.
+@end deftypefun
+
+@deftypefun double gsl_ran_tdist_pdf (double @var{x}, double @var{nu})
+This function computes the probability density @math{p(x)} at @var{x}
+for a t-distribution with @var{nu} degrees of freedom, using the formula
+given above.
+@end deftypefun
+
+@sp 1
+@tex
+\centerline{\input rand-tdist.tex}
+@end tex
+
+@deftypefun double gsl_cdf_tdist_P (double @var{x}, double @var{nu})
+@deftypefunx double gsl_cdf_tdist_Q (double @var{x}, double @var{nu})
+@deftypefunx double gsl_cdf_tdist_Pinv (double @var{P}, double @var{nu})
+@deftypefunx double gsl_cdf_tdist_Qinv (double @var{Q}, double @var{nu})
+These functions compute the cumulative distribution functions
+@math{P(x)}, @math{Q(x)} and their inverses for the t-distribution
+with @var{nu} degrees of freedom.
+@end deftypefun
+
+@page
+@node The Beta Distribution
+@section The Beta Distribution
+@deftypefun double gsl_ran_beta (const gsl_rng * @var{r}, double @var{a}, double @var{b})
+@cindex Beta distribution
+This function returns a random variate from the beta
+distribution.  The distribution function is,
+@tex
+\beforedisplay
+$$
+p(x) dx = {\Gamma(a+b) \over \Gamma(a) \Gamma(b)} x^{a-1} (1-x)^{b-1} dx
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+p(x) dx = @{\Gamma(a+b) \over \Gamma(a) \Gamma(b)@} x^@{a-1@} (1-x)^@{b-1@} dx
+@end example
+
+@end ifinfo
+@noindent
+for @c{$0 \le x \le 1$}
+@math{0 <= x <= 1}.
+@end deftypefun
+
+@deftypefun double gsl_ran_beta_pdf (double @var{x}, double @var{a}, double @var{b})
+This function computes the probability density @math{p(x)} at @var{x}
+for a beta distribution with parameters @var{a} and @var{b}, using the
+formula given above.
+@end deftypefun
+
+@sp 1
+@tex
+\centerline{\input rand-beta.tex}
+@end tex
+
+@deftypefun double gsl_cdf_beta_P (double @var{x}, double @var{a}, double @var{b})
+@deftypefunx double gsl_cdf_beta_Q (double @var{x}, double @var{a}, double @var{b})
+@deftypefunx double gsl_cdf_beta_Pinv (double @var{P}, double @var{a}, double @var{b})
+@deftypefunx double gsl_cdf_beta_Qinv (double @var{Q}, double @var{a}, double @var{b})
+These functions compute the cumulative distribution functions
+@math{P(x)}, @math{Q(x)} and their inverses for the beta
+distribution with parameters @var{a} and @var{b}.
+@end deftypefun
+
+@page
+@node The Logistic Distribution
+@section The Logistic Distribution
+
+@deftypefun double gsl_ran_logistic (const gsl_rng * @var{r}, double @var{a})
+@cindex Logistic distribution
+This function returns a random variate from the logistic
+distribution.  The distribution function is,
+@tex
+\beforedisplay
+$$
+p(x) dx = { \exp(-x/a) \over a (1 + \exp(-x/a))^2 } dx
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+p(x) dx = @{ \exp(-x/a) \over a (1 + \exp(-x/a))^2 @} dx
+@end example
+
+@end ifinfo
+@noindent
+for @math{-\infty < x < +\infty}.
+@end deftypefun
+
+@deftypefun double gsl_ran_logistic_pdf (double @var{x}, double @var{a})
+This function computes the probability density @math{p(x)} at @var{x}
+for a logistic distribution with scale parameter @var{a}, using the
+formula given above.
+@end deftypefun
+
+@sp 1
+@tex
+\centerline{\input rand-logistic.tex}
+@end tex
+
+@deftypefun double gsl_cdf_logistic_P (double @var{x}, double @var{a})
+@deftypefunx double gsl_cdf_logistic_Q (double @var{x}, double @var{a})
+@deftypefunx double gsl_cdf_logistic_Pinv (double @var{P}, double @var{a})
+@deftypefunx double gsl_cdf_logistic_Qinv (double @var{Q}, double @var{a})
+These functions compute the cumulative distribution functions
+@math{P(x)}, @math{Q(x)} and their inverses for the logistic
+distribution with scale parameter @var{a}.
+@end deftypefun
+
+@page
+@node The Pareto Distribution
+@section The Pareto Distribution
+@deftypefun double gsl_ran_pareto (const gsl_rng * @var{r}, double @var{a}, double @var{b})
+@cindex Pareto distribution
+This function returns a random variate from the Pareto distribution of
+order @var{a}.  The distribution function is,
+@tex
+\beforedisplay
+$$
+p(x) dx = (a/b) / (x/b)^{a+1} dx
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+p(x) dx = (a/b) / (x/b)^@{a+1@} dx
+@end example
+
+@end ifinfo
+@noindent
+for @c{$x \ge b$}
+@math{x >= b}.
+@end deftypefun
+
+@deftypefun double gsl_ran_pareto_pdf (double @var{x}, double @var{a}, double @var{b})
+This function computes the probability density @math{p(x)} at @var{x}
+for a Pareto distribution with exponent @var{a} and scale @var{b}, using
+the formula given above.
+@end deftypefun
+
+@sp 1
+@tex
+\centerline{\input rand-pareto.tex}
+@end tex
+
+@deftypefun double gsl_cdf_pareto_P (double @var{x}, double @var{a}, double @var{b})
+@deftypefunx double gsl_cdf_pareto_Q (double @var{x}, double @var{a}, double @var{b})
+@deftypefunx double gsl_cdf_pareto_Pinv (double @var{P}, double @var{a}, double @var{b})
+@deftypefunx double gsl_cdf_pareto_Qinv (double @var{Q}, double @var{a}, double @var{b})
+These functions compute the cumulative distribution functions
+@math{P(x)}, @math{Q(x)} and their inverses for the Pareto
+distribution with exponent @var{a} and scale @var{b}.
+@end deftypefun
+
+@page
+@node Spherical Vector Distributions
+@section Spherical Vector Distributions
+
+The spherical distributions generate random vectors, located on a
+spherical surface.  They can be used as random directions, for example in
+the steps of a random walk.
+
+@deftypefun void gsl_ran_dir_2d (const gsl_rng * @var{r}, double * @var{x}, double * @var{y})
+@deftypefunx void gsl_ran_dir_2d_trig_method (const gsl_rng * @var{r}, double * @var{x}, double * @var{y})
+@cindex 2D random direction vector
+@cindex direction vector, random 2D
+@cindex spherical random variates, 2D
+This function returns a random direction vector @math{v} =
+(@var{x},@var{y}) in two dimensions.  The vector is normalized such that
+@math{|v|^2 = x^2 + y^2 = 1}.  The obvious way to do this is to take a
+uniform random number between 0 and @math{2\pi} and let @var{x} and
+@var{y} be the sine and cosine respectively.  Two trig functions would
+have been expensive in the old days, but with modern hardware
+implementations, this is sometimes the fastest way to go.  This is the
+case for the Pentium (but not the case for the Sun Sparcstation).
+One can avoid the trig evaluations by choosing @var{x} and
+@var{y} in the interior of a unit circle (choose them at random from the
+interior of the enclosing square, and then reject those that are outside
+the unit circle), and then dividing by @c{$\sqrt{x^2 + y^2}$}
+@math{\sqrt@{x^2 + y^2@}}.
+A much cleverer approach, attributed to von Neumann (See Knuth, v2, 3rd
+ed, p140, exercise 23), requires neither trig nor a square root.  In
+this approach, @var{u} and @var{v} are chosen at random from the
+interior of a unit circle, and then @math{x=(u^2-v^2)/(u^2+v^2)} and
+@math{y=2uv/(u^2+v^2)}.
+@end deftypefun
+
+@deftypefun void gsl_ran_dir_3d (const gsl_rng * @var{r}, double * @var{x}, double * @var{y}, double * @var{z})
+@cindex 3D random direction vector
+@cindex direction vector, random 3D
+@cindex spherical random variates, 3D
+This function returns a random direction vector @math{v} =
+(@var{x},@var{y},@var{z}) in three dimensions.  The vector is normalized
+such that @math{|v|^2 = x^2 + y^2 + z^2 = 1}.  The method employed is
+due to Robert E. Knop (CACM 13, 326 (1970)), and explained in Knuth, v2,
+3rd ed, p136.  It uses the surprising fact that the distribution
+projected along any axis is actually uniform (this is only true for 3
+dimensions).
+@end deftypefun
+
+@deftypefun void gsl_ran_dir_nd (const gsl_rng * @var{r}, size_t @var{n}, double * @var{x})
+@cindex N-dimensional random direction vector
+@cindex direction vector, random N-dimensional
+@cindex spherical random variates, N-dimensional
+
+This function returns a random direction vector
+@c{$v = (x_1,x_2,\ldots,x_n)$}
+@math{v = (x_1,x_2,...,x_n)} in @var{n} dimensions.  The vector is normalized
+such that 
+@c{$|v|^2 = x_1^2 + x_2^2 + \cdots + x_n^2 = 1$}
+@math{|v|^2 = x_1^2 + x_2^2 + ... + x_n^2 = 1}.  The method
+uses the fact that a multivariate gaussian distribution is spherically
+symmetric.  Each component is generated to have a gaussian distribution,
+and then the components are normalized.  The method is described by
+Knuth, v2, 3rd ed, p135--136, and attributed to G. W. Brown, Modern
+Mathematics for the Engineer (1956).
+@end deftypefun
+
+@page
+@node The Weibull Distribution
+@section The Weibull Distribution
+@deftypefun double gsl_ran_weibull (const gsl_rng * @var{r}, double @var{a}, double @var{b})
+@cindex Weibull distribution
+This function returns a random variate from the Weibull distribution.  The
+distribution function is,
+@tex
+\beforedisplay
+$$
+p(x) dx = {b \over a^b} x^{b-1}  \exp(-(x/a)^b) dx
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+p(x) dx = @{b \over a^b@} x^@{b-1@}  \exp(-(x/a)^b) dx
+@end example
+
+@end ifinfo
+@noindent
+for @c{$x \ge 0$}
+@math{x >= 0}.
+@end deftypefun
+
+@deftypefun double gsl_ran_weibull_pdf (double @var{x}, double @var{a}, double @var{b})
+This function computes the probability density @math{p(x)} at @var{x}
+for a Weibull distribution with scale @var{a} and exponent @var{b},
+using the formula given above.
+@end deftypefun
+
+@sp 1
+@tex
+\centerline{\input rand-weibull.tex}
+@end tex
+
+@deftypefun double gsl_cdf_weibull_P (double @var{x}, double @var{a}, double @var{b})
+@deftypefunx double gsl_cdf_weibull_Q (double @var{x}, double @var{a}, double @var{b})
+@deftypefunx double gsl_cdf_weibull_Pinv (double @var{P}, double @var{a}, double @var{b})
+@deftypefunx double gsl_cdf_weibull_Qinv (double @var{Q}, double @var{a}, double @var{b})
+These functions compute the cumulative distribution functions
+@math{P(x)}, @math{Q(x)} and their inverses for the Weibull
+distribution with scale @var{a} and exponent @var{b}.
+@end deftypefun
+
+
+@page
+@node The Type-1 Gumbel Distribution
+@section  The Type-1 Gumbel Distribution
+@deftypefun double gsl_ran_gumbel1 (const gsl_rng * @var{r}, double @var{a}, double @var{b})
+@cindex Gumbel distribution (Type 1)
+@cindex Type 1 Gumbel distribution, random variates
+This function returns  a random variate from the Type-1 Gumbel
+distribution.  The Type-1 Gumbel distribution function is,
+@tex
+\beforedisplay
+$$
+p(x) dx = a b \exp(-(b \exp(-ax) + ax)) dx
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+p(x) dx = a b \exp(-(b \exp(-ax) + ax)) dx
+@end example
+
+@end ifinfo
+@noindent
+for @math{-\infty < x < \infty}. 
+@end deftypefun
+
+@deftypefun double gsl_ran_gumbel1_pdf (double @var{x}, double @var{a}, double @var{b})
+This function computes the probability density @math{p(x)} at @var{x}
+for a Type-1 Gumbel distribution with parameters @var{a} and @var{b},
+using the formula given above.
+@end deftypefun
+
+@sp 1
+@tex
+\centerline{\input rand-gumbel1.tex}
+@end tex
+
+@deftypefun double gsl_cdf_gumbel1_P (double @var{x}, double @var{a}, double @var{b})
+@deftypefunx double gsl_cdf_gumbel1_Q (double @var{x}, double @var{a}, double @var{b})
+@deftypefunx double gsl_cdf_gumbel1_Pinv (double @var{P}, double @var{a}, double @var{b})
+@deftypefunx double gsl_cdf_gumbel1_Qinv (double @var{Q}, double @var{a}, double @var{b})
+These functions compute the cumulative distribution functions
+@math{P(x)}, @math{Q(x)} and their inverses for the Type-1 Gumbel
+distribution with parameters @var{a} and @var{b}.
+@end deftypefun
+
+
+@page
+@node The Type-2 Gumbel Distribution
+@section  The Type-2 Gumbel Distribution
+@deftypefun double gsl_ran_gumbel2 (const gsl_rng * @var{r}, double @var{a}, double @var{b})
+@cindex Gumbel distribution (Type 2)
+@cindex Type 2 Gumbel distribution
+This function returns a random variate from the Type-2 Gumbel
+distribution.  The Type-2 Gumbel distribution function is,
+@tex
+\beforedisplay
+$$
+p(x) dx = a b x^{-a-1} \exp(-b x^{-a}) dx
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+p(x) dx = a b x^@{-a-1@} \exp(-b x^@{-a@}) dx
+@end example
+
+@end ifinfo
+@noindent
+for @math{0 < x < \infty}.
+@end deftypefun
+
+@deftypefun double gsl_ran_gumbel2_pdf (double @var{x}, double @var{a}, double @var{b})
+This function computes the probability density @math{p(x)} at @var{x}
+for a Type-2 Gumbel distribution with parameters @var{a} and @var{b},
+using the formula given above.
+@end deftypefun
+
+@sp 1
+@tex
+\centerline{\input rand-gumbel2.tex}
+@end tex
+
+@deftypefun double gsl_cdf_gumbel2_P (double @var{x}, double @var{a}, double @var{b})
+@deftypefunx double gsl_cdf_gumbel2_Q (double @var{x}, double @var{a}, double @var{b})
+@deftypefunx double gsl_cdf_gumbel2_Pinv (double @var{P}, double @var{a}, double @var{b})
+@deftypefunx double gsl_cdf_gumbel2_Qinv (double @var{Q}, double @var{a}, double @var{b})
+These functions compute the cumulative distribution functions
+@math{P(x)}, @math{Q(x)} and their inverses for the Type-2 Gumbel
+distribution with parameters @var{a} and @var{b}.
+@end deftypefun
+
+
+@page
+@node The Dirichlet Distribution
+@section The Dirichlet Distribution
+@deftypefun void gsl_ran_dirichlet (const gsl_rng * @var{r}, size_t @var{K}, const double @var{alpha}[], double @var{theta}[])
+@cindex Dirichlet distribution 
+This function returns an array of @var{K} random variates from a Dirichlet
+distribution of order @var{K}-1. The distribution function is
+@tex
+\beforedisplay
+$$
+p(\theta_1,\ldots,\theta_K) \, d\theta_1 \cdots d\theta_K = 
+        {1 \over Z} \prod_{i=1}^{K} \theta_i^{\alpha_i - 1} 
+          \; \delta(1 -\sum_{i=1}^K \theta_i) d\theta_1 \cdots d\theta_K
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+p(\theta_1, ..., \theta_K) d\theta_1 ... d\theta_K = 
+  (1/Z) \prod_@{i=1@}^K \theta_i^@{\alpha_i - 1@} \delta(1 -\sum_@{i=1@}^K \theta_i) d\theta_1 ... d\theta_K
+@end example
+
+@end ifinfo
+@noindent
+for @c{$\theta_i \ge 0$} 
+@math{theta_i >= 0}
+and @c{$\alpha_i \ge 0$} 
+@math{alpha_i >= 0}.  The delta function ensures that @math{\sum \theta_i = 1}.
+The normalization factor @math{Z} is
+@tex
+\beforedisplay
+$$
+Z = {\prod_{i=1}^K \Gamma(\alpha_i) \over \Gamma( \sum_{i=1}^K \alpha_i)}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+Z = @{\prod_@{i=1@}^K \Gamma(\alpha_i)@} / @{\Gamma( \sum_@{i=1@}^K \alpha_i)@}
+@end example
+@end ifinfo
+
+The random variates are generated by sampling @var{K} values 
+from gamma distributions with parameters 
+@c{$a=\alpha_i$, $b=1$} 
+@math{a=alpha_i, b=1}, 
+and renormalizing. 
+See A.M. Law, W.D. Kelton, @cite{Simulation Modeling and Analysis} (1991).
+@end deftypefun
+
+@deftypefun double gsl_ran_dirichlet_pdf (size_t @var{K}, const double @var{alpha}[], const double @var{theta}[]) 
+This function computes the probability density 
+@c{$p(\theta_1, \ldots , \theta_K)$}
+@math{p(\theta_1, ... , \theta_K)}
+at @var{theta}[@var{K}] for a Dirichlet distribution with parameters 
+@var{alpha}[@var{K}], using the formula given above.
+@end deftypefun
+
+@deftypefun double gsl_ran_dirichlet_lnpdf (size_t @var{K}, const double @var{alpha}[], const double @var{theta}[]) 
+This function computes the logarithm of the probability density 
+@c{$p(\theta_1, \ldots , \theta_K)$}
+@math{p(\theta_1, ... , \theta_K)}
+for a Dirichlet distribution with parameters 
+@var{alpha}[@var{K}].
+@end deftypefun
+
+@page
+@node General Discrete Distributions
+@section General Discrete Distributions
+
+Given @math{K} discrete events with different probabilities @math{P[k]},
+produce a random value @math{k} consistent with its probability.
+
+The obvious way to do this is to preprocess the probability list by
+generating a cumulative probability array with @math{K+1} elements:
+@tex
+\beforedisplay
+$$
+\eqalign{
+C[0] & = 0 \cr
+C[k+1] &= C[k]+P[k].
+}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+  C[0] = 0 
+C[k+1] = C[k]+P[k].
+@end example
+
+@end ifinfo
+@noindent
+Note that this construction produces @math{C[K]=1}.  Now choose a
+uniform deviate @math{u} between 0 and 1, and find the value of @math{k}
+such that @c{$C[k] \le u < C[k+1]$}
+@math{C[k] <= u < C[k+1]}.  
+Although this in principle requires of order @math{\log K} steps per
+random number generation, they are fast steps, and if you use something
+like @math{\lfloor uK \rfloor} as a starting point, you can often do
+pretty well.
+
+But faster methods have been devised.  Again, the idea is to preprocess
+the probability list, and save the result in some form of lookup table;
+then the individual calls for a random discrete event can go rapidly.
+An approach invented by G. Marsaglia (Generating discrete random numbers
+in a computer, Comm ACM 6, 37--38 (1963)) is very clever, and readers
+interested in examples of good algorithm design are directed to this
+short and well-written paper.  Unfortunately, for large @math{K},
+Marsaglia's lookup table can be quite large.  
+
+A much better approach is due to Alastair J. Walker (An efficient method
+for generating discrete random variables with general distributions, ACM
+Trans on Mathematical Software 3, 253--256 (1977); see also Knuth, v2,
+3rd ed, p120--121,139).  This requires two lookup tables, one floating
+point and one integer, but both only of size @math{K}.  After
+preprocessing, the random numbers are generated in O(1) time, even for
+large @math{K}.  The preprocessing suggested by Walker requires
+@math{O(K^2)} effort, but that is not actually necessary, and the
+implementation provided here only takes @math{O(K)} effort.  In general,
+more preprocessing leads to faster generation of the individual random
+numbers, but a diminishing return is reached pretty early.  Knuth points
+out that the optimal preprocessing is combinatorially difficult for
+large @math{K}.
+
+This method can be used to speed up some of the discrete random number
+generators below, such as the binomial distribution.  To use it for
+something like the Poisson Distribution, a modification would have to
+be made, since it only takes a finite set of @math{K} outcomes.
+
+@deftypefun {gsl_ran_discrete_t *} gsl_ran_discrete_preproc (size_t @var{K}, const double * @var{P})
+@cindex Discrete random numbers
+@cindex Discrete random numbers, preprocessing
+This function returns a pointer to a structure that contains the lookup
+table for the discrete random number generator.  The array @var{P}[] contains
+the probabilities of the discrete events; these array elements must all be 
+positive, but they needn't add up to one (so you can think of them more
+generally as ``weights'')---the preprocessor will normalize appropriately.
+This return value is used
+as an argument for the @code{gsl_ran_discrete} function below.
+@end deftypefun
+
+@deftypefun {size_t} gsl_ran_discrete (const gsl_rng * @var{r}, const gsl_ran_discrete_t * @var{g})
+@cindex Discrete random numbers
+After the preprocessor, above, has been called, you use this function to
+get the discrete random numbers.
+@end deftypefun
+
+@deftypefun {double} gsl_ran_discrete_pdf (size_t @var{k}, const gsl_ran_discrete_t * @var{g})
+@cindex Discrete random numbers
+Returns the probability @math{P[k]} of observing the variable @var{k}.
+Since @math{P[k]} is not stored as part of the lookup table, it must be
+recomputed; this computation takes @math{O(K)}, so if @var{K} is large
+and you care about the original array @math{P[k]} used to create the
+lookup table, then you should just keep this original array @math{P[k]}
+around.
+@end deftypefun
+
+@deftypefun {void} gsl_ran_discrete_free (gsl_ran_discrete_t * @var{g})
+@cindex Discrete random numbers
+De-allocates the lookup table pointed to by @var{g}.
+@end deftypefun
+
+@page
+@node The Poisson Distribution
+@section The Poisson Distribution
+@deftypefun {unsigned int} gsl_ran_poisson (const gsl_rng * @var{r}, double @var{mu})
+@cindex Poisson random numbers
+This function returns a random integer from the Poisson distribution
+with mean @var{mu}.  The probability distribution for Poisson variates is,
+@tex
+\beforedisplay
+$$
+p(k) = {\mu^k \over k!} \exp(-\mu)
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+p(k) = @{\mu^k \over k!@} \exp(-\mu)
+@end example
+
+@end ifinfo
+@noindent
+for @c{$k \ge 0$}
+@math{k >= 0}.
+@end deftypefun
+
+@deftypefun double gsl_ran_poisson_pdf (unsigned int @var{k}, double @var{mu})
+This function computes the probability @math{p(k)} of obtaining  @var{k}
+from a Poisson distribution with mean @var{mu}, using the formula
+given above.
+@end deftypefun
+
+@sp 1
+@tex
+\centerline{\input rand-poisson.tex}
+@end tex
+
+@deftypefun double gsl_cdf_poisson_P (unsigned int @var{k}, double @var{mu})
+@deftypefunx double gsl_cdf_poisson_Q (unsigned int @var{k}, double @var{mu})
+These functions compute the cumulative distribution functions
+@math{P(k)}, @math{Q(k)} for the Poisson distribution with parameter
+@var{mu}.
+@end deftypefun
+
+
+@page
+@node The Bernoulli Distribution
+@section The Bernoulli Distribution
+@deftypefun {unsigned int} gsl_ran_bernoulli (const gsl_rng * @var{r}, double @var{p})
+@cindex Bernoulli trial, random variates
+This function returns either 0 or 1, the result of a Bernoulli trial
+with probability @var{p}.  The probability distribution for a Bernoulli
+trial is,
+@tex
+\beforedisplay
+$$
+\eqalign{
+p(0) & = 1 - p \cr
+p(1) & = p
+}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+p(0) = 1 - p
+p(1) = p
+@end example
+@end ifinfo
+
+@end deftypefun
+
+@deftypefun double gsl_ran_bernoulli_pdf (unsigned int @var{k}, double @var{p})
+This function computes the probability @math{p(k)} of obtaining
+@var{k} from a Bernoulli distribution with probability parameter
+@var{p}, using the formula given above.
+@end deftypefun
+
+@sp 1
+@tex
+\centerline{\input rand-bernoulli.tex}
+@end tex
+
+@page
+@node The Binomial Distribution
+@section The Binomial Distribution
+@deftypefun {unsigned int} gsl_ran_binomial (const gsl_rng * @var{r}, double @var{p}, unsigned int @var{n})
+@cindex Binomial random variates
+This function returns a random integer from the binomial distribution,
+the number of successes in @var{n} independent trials with probability
+@var{p}.  The probability distribution for binomial variates is,
+@tex
+\beforedisplay
+$$
+p(k) = {n! \over k! (n-k)!} p^k (1-p)^{n-k}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+p(k) = @{n! \over k! (n-k)! @} p^k (1-p)^@{n-k@}
+@end example
+
+@end ifinfo
+@noindent
+for @c{$0 \le k \le n$}
+@math{0 <= k <= n}.
+@end deftypefun
+
+@deftypefun double gsl_ran_binomial_pdf (unsigned int @var{k}, double @var{p}, unsigned int @var{n})
+This function computes the probability @math{p(k)} of obtaining @var{k}
+from a binomial distribution with parameters @var{p} and @var{n}, using
+the formula given above.
+@end deftypefun
+
+@sp 1
+@tex
+\centerline{\input rand-binomial.tex}
+@end tex
+
+@deftypefun double gsl_cdf_binomial_P (unsigned int @var{k}, double @var{p}, unsigned int @var{n})
+@deftypefunx double gsl_cdf_binomial_Q (unsigned int @var{k}, double @var{p}, unsigned int @var{n})
+These functions compute the cumulative distribution functions
+@math{P(k)}, @math{Q(k)}  for the binomial
+distribution with parameters @var{p} and @var{n}.
+@end deftypefun
+
+
+@page
+@node The Multinomial Distribution
+@section The Multinomial Distribution
+@deftypefun void gsl_ran_multinomial (const gsl_rng * @var{r}, size_t @var{K}, unsigned int @var{N}, const double @var{p}[], unsigned int @var{n}[])
+@cindex Multinomial distribution 
+
+This function computes a random sample @var{n}[] from the multinomial
+distribution formed by @var{N} trials from an underlying distribution
+@var{p}[@var{K}]. The distribution function for @var{n}[] is,
+@tex
+\beforedisplay
+$$
+P(n_1, n_2,\cdots, n_K) = {{ N!}\over{n_1 ! n_2 ! \cdots n_K !}} \,
+  p_1^{n_1} p_2^{n_2} \cdots p_K^{n_K}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+P(n_1, n_2, ..., n_K) = 
+  (N!/(n_1! n_2! ... n_K!)) p_1^n_1 p_2^n_2 ... p_K^n_K
+@end example
+
+@end ifinfo
+@noindent
+where @c{($n_1$, $n_2$, $\ldots$, $n_K$)}
+@math{(n_1, n_2, ..., n_K)} 
+are nonnegative integers with 
+@c{$\sum_{k=1}^{K} n_k =N$} 
+@math{sum_@{k=1@}^K n_k = N},
+and
+@c{$(p_1, p_2, \ldots, p_K)$} 
+@math{(p_1, p_2, ..., p_K)}
+is a probability distribution with @math{\sum p_i = 1}.  
+If the array @var{p}[@var{K}] is not normalized then its entries will be
+treated as weights and normalized appropriately.  The arrays @var{n}[]
+and @var{p}[] must both be of length @var{K}.
+
+Random variates are generated using the conditional binomial method (see
+C.S. David, @cite{The computer generation of multinomial random
+variates}, Comp. Stat. Data Anal. 16 (1993) 205--217 for details).
+@end deftypefun
+
+@deftypefun double gsl_ran_multinomial_pdf (size_t @var{K}, const double @var{p}[], const unsigned int @var{n}[]) 
+This function computes the probability 
+@c{$P(n_1, n_2, \ldots, n_K)$}
+@math{P(n_1, n_2, ..., n_K)}
+of sampling @var{n}[@var{K}] from a multinomial distribution 
+with parameters @var{p}[@var{K}], using the formula given above.
+@end deftypefun
+
+@deftypefun double gsl_ran_multinomial_lnpdf (size_t @var{K}, const double @var{p}[], const unsigned int @var{n}[]) 
+This function returns the logarithm of the probability for the
+multinomial distribution @c{$P(n_1, n_2, \ldots, n_K)$}
+@math{P(n_1, n_2, ..., n_K)} with parameters @var{p}[@var{K}].
+@end deftypefun
+
+@page
+@node The Negative Binomial Distribution
+@section The Negative Binomial Distribution
+@deftypefun {unsigned int} gsl_ran_negative_binomial (const gsl_rng * @var{r}, double @var{p}, double @var{n})
+@cindex Negative Binomial distribution, random variates
+This function returns a random integer from the negative binomial
+distribution, the number of failures occurring before @var{n} successes
+in independent trials with probability @var{p} of success.  The
+probability distribution for negative binomial variates is,
+@tex
+\beforedisplay
+$$
+p(k) = {\Gamma(n + k) \over \Gamma(k+1) \Gamma(n) } p^n (1-p)^k
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+p(k) = @{\Gamma(n + k) \over \Gamma(k+1) \Gamma(n) @} p^n (1-p)^k
+@end example
+
+@end ifinfo
+@noindent
+Note that @math{n} is not required to be an integer.
+@end deftypefun
+
+@deftypefun double gsl_ran_negative_binomial_pdf (unsigned int @var{k}, double @var{p}, double  @var{n})
+This function computes the probability @math{p(k)} of obtaining @var{k}
+from a negative binomial distribution with parameters @var{p} and
+@var{n}, using the formula given above.
+@end deftypefun
+
+@sp 1
+@tex
+\centerline{\input rand-nbinomial.tex}
+@end tex
+
+@deftypefun double gsl_cdf_negative_binomial_P (unsigned int @var{k}, double @var{p}, double @var{n})
+@deftypefunx double gsl_cdf_negative_binomial_Q (unsigned int @var{k}, double @var{p}, double @var{n})
+These functions compute the cumulative distribution functions
+@math{P(k)}, @math{Q(k)} for the negative binomial distribution with
+parameters @var{p} and @var{n}.
+@end deftypefun
+
+@page
+@node The Pascal Distribution
+@section The Pascal Distribution
+
+@deftypefun {unsigned int} gsl_ran_pascal (const gsl_rng * @var{r}, double @var{p}, unsigned int @var{n})
+This function returns a random integer from the Pascal distribution.  The
+Pascal distribution is simply a negative binomial distribution with an
+integer value of @math{n}.
+@tex
+\beforedisplay
+$$
+p(k) = {(n + k - 1)! \over k! (n - 1)! } p^n (1-p)^k
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+p(k) = @{(n + k - 1)! \over k! (n - 1)! @} p^n (1-p)^k
+@end example
+
+@end ifinfo
+@noindent
+for @c{$k \ge 0$}
+@math{k >= 0}
+@end deftypefun
+
+@deftypefun double gsl_ran_pascal_pdf (unsigned int @var{k}, double @var{p}, unsigned int @var{n})
+This function computes the probability @math{p(k)} of obtaining @var{k}
+from a Pascal distribution with parameters @var{p} and
+@var{n}, using the formula given above.
+@end deftypefun
+
+@sp 1
+@tex
+\centerline{\input rand-pascal.tex}
+@end tex
+
+@deftypefun double gsl_cdf_pascal_P (unsigned int @var{k}, double @var{p}, unsigned int @var{n})
+@deftypefunx double gsl_cdf_pascal_Q (unsigned int @var{k}, double @var{p}, unsigned int @var{n})
+These functions compute the cumulative distribution functions
+@math{P(k)}, @math{Q(k)} for the Pascal distribution with
+parameters @var{p} and @var{n}.
+@end deftypefun
+
+@page
+@node The Geometric Distribution
+@section The Geometric Distribution
+@deftypefun {unsigned int} gsl_ran_geometric (const gsl_rng * @var{r}, double @var{p})
+@cindex Geometric random variates
+This function returns a random integer from the geometric distribution,
+the number of independent trials with probability @var{p} until the
+first success.  The probability distribution for geometric variates
+is,
+@tex
+\beforedisplay
+$$
+p(k) =  p (1-p)^{k-1}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+p(k) =  p (1-p)^(k-1)
+@end example
+
+@end ifinfo
+@noindent
+for @c{$k \ge 1$}
+@math{k >= 1}.  Note that the distribution begins with @math{k=1} with this
+definition.  There is another convention in which the exponent @math{k-1} 
+is replaced by @math{k}.
+@end deftypefun
+
+@deftypefun double gsl_ran_geometric_pdf (unsigned int @var{k}, double @var{p})
+This function computes the probability @math{p(k)} of obtaining @var{k}
+from a geometric distribution with probability parameter @var{p}, using
+the formula given above.
+@end deftypefun
+
+@sp 1
+@tex
+\centerline{\input rand-geometric.tex}
+@end tex
+
+@deftypefun double gsl_cdf_geometric_P (unsigned int @var{k}, double @var{p})
+@deftypefunx double gsl_cdf_geometric_Q (unsigned int @var{k}, double @var{p})
+These functions compute the cumulative distribution functions
+@math{P(k)}, @math{Q(k)} for the geometric distribution with parameter
+@var{p}.
+@end deftypefun
+
+
+@page
+@node The Hypergeometric Distribution
+@section The Hypergeometric Distribution
+@cindex hypergeometric random variates
+@deftypefun {unsigned int} gsl_ran_hypergeometric (const gsl_rng * @var{r}, unsigned int @var{n1}, unsigned int @var{n2}, unsigned int @var{t})
+@cindex Geometric random variates
+This function returns a random integer from the hypergeometric
+distribution.  The probability distribution for hypergeometric
+random variates is,
+@tex
+\beforedisplay
+$$
+p(k) =  C(n_1, k) C(n_2, t - k) / C(n_1 + n_2, t)
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+p(k) =  C(n_1, k) C(n_2, t - k) / C(n_1 + n_2, t)
+@end example
+
+@end ifinfo
+@noindent
+where @math{C(a,b) = a!/(b!(a-b)!)} and 
+@c{$t \leq n_1 + n_2$}
+@math{t <= n_1 + n_2}.  The domain of @math{k} is 
+@c{$\hbox{max}(0,t-n_2), \ldots, \hbox{min}(t,n_1)$} 
+@math{max(0,t-n_2), ..., min(t,n_1)}.
+
+If a population contains @math{n_1} elements of ``type 1'' and
+@math{n_2} elements of ``type 2'' then the hypergeometric
+distribution gives the probability of obtaining @math{k} elements of
+``type 1'' in @math{t} samples from the population without
+replacement.
+@end deftypefun
+
+@deftypefun double gsl_ran_hypergeometric_pdf (unsigned int @var{k}, unsigned int @var{n1}, unsigned int @var{n2}, unsigned int @var{t})
+This function computes the probability @math{p(k)} of obtaining @var{k}
+from a hypergeometric distribution with parameters @var{n1}, @var{n2},
+@var{t}, using the formula given above.
+@end deftypefun
+
+@sp 1
+@tex
+\centerline{\input rand-hypergeometric.tex}
+@end tex
+
+@deftypefun double gsl_cdf_hypergeometric_P (unsigned int @var{k}, unsigned int @var{n1}, unsigned int @var{n2}, unsigned int @var{t})
+@deftypefunx double gsl_cdf_hypergeometric_Q (unsigned int @var{k}, unsigned int @var{n1}, unsigned int @var{n2}, unsigned int @var{t})
+These functions compute the cumulative distribution functions
+@math{P(k)}, @math{Q(k)} for the hypergeometric distribution with
+parameters @var{n1}, @var{n2} and @var{t}.
+@end deftypefun
+
+
+@page
+@node The Logarithmic Distribution
+@section The Logarithmic Distribution
+@deftypefun {unsigned int} gsl_ran_logarithmic (const gsl_rng * @var{r}, double @var{p})
+@cindex Logarithmic random variates
+This function returns a random integer from the logarithmic
+distribution.  The probability distribution for logarithmic random variates
+is,
+@tex
+\beforedisplay
+$$
+p(k) = {-1 \over \log(1-p)} {\left( p^k \over k \right)}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+p(k) = @{-1 \over \log(1-p)@} @{(p^k \over k)@}
+@end example
+
+@end ifinfo
+@noindent
+for @c{$k \ge 1$}
+@math{k >= 1}.
+@end deftypefun
+
+@deftypefun double gsl_ran_logarithmic_pdf (unsigned int @var{k}, double @var{p})
+This function computes the probability @math{p(k)} of obtaining @var{k}
+from a logarithmic distribution with probability parameter @var{p},
+using the formula given above.
+@end deftypefun
+
+@sp 1
+@tex
+\centerline{\input rand-logarithmic.tex}
+@end tex
+
+@page
+@node Shuffling and Sampling
+@section Shuffling and Sampling
+
+The following functions allow the shuffling and sampling of a set of
+objects.  The algorithms rely on a random number generator as a source of
+randomness and a poor quality generator can lead to correlations in the
+output.  In particular it is important to avoid generators with a short
+period.  For more information see Knuth, v2, 3rd ed, Section 3.4.2,
+``Random Sampling and Shuffling''.
+
+@deftypefun void gsl_ran_shuffle (const gsl_rng * @var{r}, void * @var{base}, size_t @var{n}, size_t @var{size})
+
+This function randomly shuffles the order of @var{n} objects, each of
+size @var{size}, stored in the array @var{base}[0..@var{n}-1].  The
+output of the random number generator @var{r} is used to produce the
+permutation.  The algorithm generates all possible @math{n!}
+permutations with equal probability, assuming a perfect source of random
+numbers.
+
+The following code shows how to shuffle the numbers from 0 to 51,
+
+@example
+int a[52];
+
+for (i = 0; i < 52; i++)
+  @{
+    a[i] = i;
+  @}
+
+gsl_ran_shuffle (r, a, 52, sizeof (int));
+@end example
+
+@end deftypefun
+
+@deftypefun int gsl_ran_choose (const gsl_rng * @var{r}, void * @var{dest}, size_t @var{k}, void * @var{src}, size_t @var{n}, size_t @var{size})
+This function fills the array @var{dest}[k] with @var{k} objects taken
+randomly from the @var{n} elements of the array
+@var{src}[0..@var{n}-1].  The objects are each of size @var{size}.  The
+output of the random number generator @var{r} is used to make the
+selection.  The algorithm ensures all possible samples are equally
+likely, assuming a perfect source of randomness.
+
+The objects are sampled @emph{without} replacement, thus each object can
+only appear once in @var{dest}[k].  It is required that @var{k} be less
+than or equal to @code{n}.  The objects in @var{dest} will be in the
+same relative order as those in @var{src}.  You will need to call
+@code{gsl_ran_shuffle(r, dest, n, size)} if you want to randomize the
+order.
+
+The following code shows how to select a random sample of three unique
+numbers from the set 0 to 99,
+
+@example
+double a[3], b[100];
+
+for (i = 0; i < 100; i++)
+  @{
+    b[i] = (double) i;
+  @}
+
+gsl_ran_choose (r, a, 3, b, 100, sizeof (double));
+@end example
+
+@end deftypefun
+
+@deftypefun void gsl_ran_sample (const gsl_rng * @var{r}, void * @var{dest}, size_t @var{k}, void * @var{src}, size_t @var{n}, size_t @var{size})
+This function is like @code{gsl_ran_choose} but samples @var{k} items
+from the original array of @var{n} items @var{src} with replacement, so
+the same object can appear more than once in the output sequence
+@var{dest}.  There is no requirement that @var{k} be less than @var{n}
+in this case.
+@end deftypefun
+
+
+@node Random Number Distribution Examples
+@section Examples
+
+The following program demonstrates the use of a random number generator
+to produce variates from a distribution.  It prints 10 samples from the
+Poisson distribution with a mean of 3.
+
+@example
+@verbatiminclude examples/randpoisson.c
+@end example
+
+@noindent
+If the library and header files are installed under @file{/usr/local}
+(the default location) then the program can be compiled with these
+options,
+
+@example
+$ gcc -Wall demo.c -lgsl -lgslcblas -lm
+@end example
+
+@noindent
+Here is the output of the program,
+
+@example
+$ ./a.out 
+@verbatiminclude examples/randpoisson.out
+@end example
+
+@noindent
+The variates depend on the seed used by the generator.  The seed for the
+default generator type @code{gsl_rng_default} can be changed with the
+@code{GSL_RNG_SEED} environment variable to produce a different stream
+of variates,
+
+@example
+$ GSL_RNG_SEED=123 ./a.out 
+@verbatiminclude examples/randpoisson.2.out
+@end example
+
+@noindent
+The following program generates a random walk in two dimensions.
+
+@example
+@verbatiminclude examples/randwalk.c
+@end example
+
+@noindent
+Here is the output from the program, three 10-step random walks from the origin,
+
+@tex
+\centerline{\input random-walk.tex}
+@end tex
+
+The following program computes the upper and lower cumulative
+distribution functions for the standard normal distribution at
+@math{x=2}.
+
+@example
+@verbatiminclude examples/cdf.c
+@end example
+
+@noindent
+Here is the output of the program,
+
+@example
+@verbatiminclude examples/cdf.out
+@end example
+
+@node Random Number Distribution References and Further Reading
+@section References and Further Reading
+
+For an encyclopaedic coverage of the subject readers are advised to
+consult the book @cite{Non-Uniform Random Variate Generation} by Luc
+Devroye.  It covers every imaginable distribution and provides hundreds
+of algorithms.
+
+@itemize @asis
+@item
+Luc Devroye, @cite{Non-Uniform Random Variate Generation},
+Springer-Verlag, ISBN 0-387-96305-7.
+@end itemize
+
+@noindent
+The subject of random variate generation is also reviewed by Knuth, who
+describes algorithms for all the major distributions.
+
+@itemize @asis
+@item
+Donald E. Knuth, @cite{The Art of Computer Programming: Seminumerical
+Algorithms} (Vol 2, 3rd Ed, 1997), Addison-Wesley, ISBN 0201896842.
+@end itemize
+
+@noindent
+The Particle Data Group provides a short review of techniques for
+generating distributions of random numbers in the ``Monte Carlo''
+section of its Annual Review of Particle Physics.
+
+@itemize @asis
+@item
+@cite{Review of Particle Properties}
+R.M. Barnett et al., Physical Review D54, 1 (1996)
+@uref{http://pdg.lbl.gov/}.
+@end itemize
+
+@noindent
+The Review of Particle Physics is available online in postscript and pdf
+format.
+
+@noindent
+An overview of methods used to compute cumulative distribution functions
+can be found in @cite{Statistical Computing} by W.J. Kennedy and
+J.E. Gentle. Another general reference is @cite{Elements of Statistical
+Computing} by R.A. Thisted.
+
+@itemize @asis
+@item
+William E. Kennedy and James E. Gentle, @cite{Statistical Computing} (1980),
+Marcel Dekker, ISBN 0-8247-6898-1.
+@end itemize
+
+@itemize @asis
+@item
+Ronald A. Thisted, @cite{Elements of Statistical Computing} (1988), 
+Chapman & Hall, ISBN 0-412-01371-1.
+@end itemize
+
+@noindent
+The cumulative distribution functions for the Gaussian distribution
+are based on the following papers,
+
+@itemize @asis
+@item
+@cite{Rational Chebyshev Approximations Using Linear Equations},
+W.J. Cody, W. Fraser, J.F. Hart. Numerische Mathematik 12, 242--251 (1968).
+@end itemize
+
+@itemize @asis
+@item
+@cite{Rational Chebyshev Approximations for the Error Function},
+W.J. Cody. Mathematics of Computation 23, n107, 631--637 (July 1969).
+@end itemize
diff --git a/gsl-1.9/doc/random-walk.tex b/gsl-1.9/doc/random-walk.tex
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diff --git a/gsl-1.9/doc/randplots.gnp b/gsl-1.9/doc/randplots.gnp
new file mode 100644
index 0000000..88221b5
--- /dev/null
+++ b/gsl-1.9/doc/randplots.gnp
@@ -0,0 +1,374 @@
+#set term postscript eps enhanced
+set term pstex monochrome
+set samples 300
+#set size 0.8,0.88  # paperback 6x9  #CHANGE THESE LATER IN FILE ALSO 
+set size 1,1.1      # us letter      #IF MODIFIED 
+set border 31 lw 0.5
+set tics out
+set ticscale 0.5 0.25
+set xtics border nomirror norotate
+set ytics border nomirror norotate
+set format x "%g"
+
+# Continuous distributions
+
+set xlabel "$x$"
+set ylabel "$p(x)$"
+
+set xrange [-5:5]
+set yrange [0:0.5]
+set ytics 0.1
+set xtics 1
+set output "rand-gaussian.tex"
+set title "Gaussian Distribution"
+p(x)=1/sqrt(2*pi*sigma**2)*exp(-x**2/(2*sigma**2))
+plot sigma=1.0, p(x) title "$\\sigma=1$", \
+	sigma=2.0, p(x) title "$\\sigma=2$"
+
+set xrange [0:5]
+set yrange [0:2]
+set ytics 0.5
+set xtics 1
+set output "rand-gaussian-tail.tex"
+set title "Gaussian Tail Distribution"
+p(x)= x > a ? 1/(0.5*erfc(a/sqrt(2*sigma**2))*sqrt(2*pi*sigma**2))*exp(-x**2/(2*sigma**2)) : 0
+plot a=1.5, sigma=1.0, p(x) title "$\\sigma=1,a=1.5$"
+
+set xrange [0:5]
+set yrange [0:1.09]
+set ytics 0.5
+set xtics 1
+set output "rand-rayleigh-tail.tex"
+set title "Rayleigh Tail Distribution"
+p(x)= x > a ? (x/sigma**2)*exp((a**2 - x**2)/(2*sigma**2)) : 0
+plot a=1.0, sigma=1.0, p(x) title "$a=1,\\sigma=1$", \
+	a=0.5, sigma=2.0, p(x) title "$a=0.5,\\sigma=2$"
+
+set size ratio -1 0.8,1.143
+set xlabel "$x$"
+set ylabel "$y$"
+set xrange [-2:2]
+set yrange [-2:2]
+set ytics 1
+set xtics 1
+set output "rand-bivariate-gaussian.tex"
+set title "Bivariate Gaussian Distribution"
+set isosamples 30
+p(x,y)=1/sqrt(2*pi*sqrt(1-rho**2))*exp(-(x**2 + y**2 - 2*rho*x*y)/(2*(1-rho**2)))
+#set cntrparam cspline
+set noclabel
+set contour
+set view 0,0
+set nosurface
+splot rho=0.9, p(x,y) title "$\\sigma_x=1, \\sigma_y=1, \\rho=0.9$"
+set size noratio 1,1.1  # us letter
+
+set xlabel "$x$"
+set ylabel "$p(x)$"
+set ytics 0.1
+set xrange [-5:5]
+set yrange [0:0.4]
+set output "rand-cauchy.tex"
+set title "Cauchy Distribution"
+p(x)=1/(pi*a*(1+(x/a)**2))
+plot a=1.0, p(x) title "$a=1$", \
+	a=2.0, p(x) title "$a=2$"
+
+set ytics 0.5
+set yrange [0:1]
+set xrange [0:3]
+set output "rand-exponential.tex"
+set title "Exponential Distribution"
+p(x)=exp(-x/mu)/mu
+plot mu=1.0, p(x) title "$\\mu=1$", \
+	mu=2.0, p(x) title "$\\mu=2$"
+
+set yrange [0:1]
+set xrange [0:3]
+set output "rand-chisq.tex"
+set title "Chi-squared Distribution"
+p(x)=(x/2)**(nu/2.0-1.0) *exp(-x/2)/ (2*gamma(nu/2.0))
+plot nu=1.0, p(x) title "$\\nu=1$", \
+      nu=2.0, p(x) title "$\\nu=2$", \
+	nu=3.0, p(x) title "$\\nu=3$"
+
+set yrange [0:1]
+set xrange [0:5]
+set output "rand-erlang.tex"
+set title "Erlang Distribution"
+p(x)=x**(n-1) *exp(-x/a)/ (gamma(n)*a**n)
+plot n=1.0, a=1, p(x) title "$n=1$", \
+      n=2.0, a=1.0, p(x) title "$n=2$", \
+	n=3.0, a=1.0, p(x) title "$n=3$"
+
+set xrange [0:2]
+set xtics 1
+set output "rand-fdist.tex"
+set title "F-Distribution"
+p(x)=gamma((v1+v2)/2.0)*v1**(v1/2.0)*v2**(v2/2.0)\
+	/gamma(v1/2.0)/gamma(v2/2.0) * \
+	x**(v1/2.0-1)*(v2+v1*x)**-(v1/2.0+v2/2.0)
+plot v1=1.0, v2=1.0, p(x) title "$\\nu_1=1, \\nu_2=1$", \
+      v1=3.0, v2=1.0, p(x) title "$\\nu_1=3, \\nu_2=2$"
+
+set xrange [0:5]
+set output "rand-flat.tex"
+set title "Flat Distribution"
+p(x)= (x<b && x>a) ? 1/(b-a) : 0
+plot a=0.5, b=2.5, p(x) title "$a=0.5,b=2.5$" with step, \
+      a=1.2, b=4.8, p(x) title "$a=1.2,b=4.8$" with step
+
+set xrange [0:5]
+set yrange [0:1]
+set ytics 0.5
+set output "rand-gamma.tex"
+set title "Gamma Distribution"
+p(x)= x**(a-1)* exp(-x) / gamma(a)
+plot a=1.0, p(x) title "$a=1$", \
+      a=2.0, p(x) title "$a=2$", \
+      a=3.0, p(x) title "$a=3$"
+
+set xrange [0:1]
+set yrange [0:4]
+set xtics 0.25
+set ytics 1
+set output "rand-beta.tex"
+set title "Beta Distribution"
+p(x)= gamma(a+b)/(gamma(a)*gamma(b))*x**(a-1)*(1-x)**(b-1)
+plot a=2.0, b=2.0, p(x) title "$a=2,b=2$", \
+      a=4.0, b=1.0, p(x) title "$a=4,b=1$", \
+      a=1.0, b=4.0, p(x) title "$a=1,b=4$"
+
+
+set xrange [-5:5]
+set yrange [0:0.3]
+set ytics 0.1
+set xtics 1
+set output "rand-logistic.tex"
+set title "Logistic Distribution"
+p(x)= exp(-x/a)/a/(1+exp(-x/a))**2
+plot a=1.0, p(x) title "$a=1$", \
+	a=2.0, p(x) title "$a=2$"
+
+set xrange [0:3]
+set yrange [0:0.7]
+set xtics 1
+set ytics 0.5
+set output "rand-lognormal.tex"
+set title "Lognormal Distribution"
+p(x)= exp(-(log(x)-zeta)**2/2/sigma**2)/sqrt(2*pi*sigma**2)/x
+plot zeta=0.0, sigma=1.0, p(x) title "$\\zeta=0, \\sigma=1$", \
+	zeta=1.0, sigma=1.0, p(x) title "$\\zeta=1, \\sigma=1$"
+
+set xrange [0:5]
+set yrange [0:2]
+set ytics 0.5
+set xtics 1
+set output "rand-pareto.tex"
+set title "Pareto Distribution"
+p(x)= x>b ? a*b**a/x**(a+1) : 0
+plot a=1.0, b=1, p(x) title "$a=1, b=1$", \
+	a=3.0, b=2, p(x) title "$a=3, b=2 $"
+
+set xrange [0:5]
+set yrange [0:0.7]
+set ytics 0.1
+set xtics 1
+set output "rand-rayleigh.tex"
+set title "Rayleigh Distribution"
+p(x)= (x/sigma**2)*exp(-x**2/(2*sigma**2))
+plot sigma=1.0, p(x) title "$\\sigma=1$", \
+	sigma=2.0, p(x) title "$\\sigma=2$"
+
+set xrange [0:5]
+set yrange [0:1.09]
+set ytics 0.5
+set xtics 1
+set output "rand-rayleigh-tail.tex"
+set title "Rayleigh Tail Distribution"
+p(x)= x > a ? (x/sigma**2)*exp((a**2 - x**2)/(2*sigma**2)) : 0
+plot a=1.0, sigma=1.0, p(x) title "$a=1,\\sigma=1$", \
+	a=0.5, sigma=2.0, p(x) title "$a=0.5,\\sigma=2$"
+
+set xrange [-4:4]
+set yrange [0:0.5]
+set ytics 0.1
+set output "rand-tdist.tex"
+set title "Student's t distribution"
+p(x)=gamma((v+1.0)/2)/sqrt(pi*v)/gamma(v/2)*(1+(x**2)/v)**-((v+1.0)/2)
+plot v=1.0, p(x) title "$\\nu_1=1$", \
+      v=5.0, p(x) title "$\\nu_1=5$"
+
+set xrange [-5:5]
+set yrange [0:0.55]
+set ytics 0.1
+set output "rand-laplace.tex"
+set title "Laplace Distribution (Two-sided Exponential)"
+p(x)=exp(-abs(x)/a)/2/a
+plot a=1.0, p(x) title "$a=1$", \
+	a=2.0, p(x) title "$a=2$"
+
+set xrange [-5:5]
+set yrange [0:0.8]
+set ytics 0.2
+set output "rand-exppow.tex"
+set title "Exponential Power Distribution"
+p(x)=exp(-(abs(x/a))**b)/2/a/gamma(1.0+1.0/b)
+plot a=1.0, b=2.5, p(x) title "$a=1, b=2.5$", \
+	a=1.0, b=0.5, p(x) title "$a=1, b=0.5$"
+
+set xrange [-5:10]
+set yrange [0:0.2]
+set ytics 0.1
+set output "rand-landau.tex"
+set title "Landau Distribution"
+plot 'landau.dat' notitle w lines
+
+set xrange [-5:5]
+set yrange [0:0.45]
+set ytics 0.1
+set output "rand-levy.tex"
+set title "Levy Distribution"
+p1(x)=1/(pi*mu*(1+(x/mu)**2))
+p2(x)=1/sqrt(2*pi*2*mu**2)*exp(-x**2/(4*mu**2)) 
+plot mu=1.0, a=1, p1(x) title "$c=1, \\alpha=1.0$", \
+	mu=1.0, a=2, p2(x) title "$c=1, \\alpha=2.0$"
+
+set xrange [-5:5]
+set yrange [0:0.09]
+set ytics 0.05
+set output "rand-levyskew.tex"
+set title "Levy Skew Distribution"
+logp(x)= 1.0990e+01+x*(-3.3388e-01+x*(-4.2372e-01+x*(2.1749e-01+x*(-4.4910e-02 + x*(4.1801e-03 - 1.4456e-04*x)))))
+p1(x)=exp(logp(x))/1000000
+plot mu=1.0, a=1, p1(x) title "$c=1, \\alpha=1.0, \\beta=1.0$"
+
+
+set xrange [0:2]
+set yrange [0:1.5]
+set ytics 0.5
+set xtics 0.5
+set output "rand-weibull.tex"
+set title "Weibull Distribution"
+p(x)=(b/a) * (x/a)**(b-1) * exp(-(x/a)**b)
+plot a=1.0, b=1.0, p(x) title "$a=1,b=1$", \
+       a=1.0, b=2.0, p(x) title "$a=1,b=2$", \
+	 a=2.0, b=3.0, p(x) title "$a=2,b=3$"
+
+set xrange [-2:5]
+set yrange [0:0.5]
+set ytics 0.1
+set xtics 1.0
+set output "rand-gumbel1.tex"
+set title "Type 1 Gumbel Distribution"
+p1(x)=a*b*exp(-(b*exp(-a*x)+a*x))
+plot a=1.0, b=1.0, p1(x) title "Type 1, $a=1,b=1$"
+
+set xrange [0:2]
+set yrange [0:0.7]
+set ytics 0.1
+set xtics 0.5
+set output "rand-gumbel2.tex"
+set title "Type 2 Gumbel Distribution"
+p2(x)=x > 0 ? a*b*x**(-a-1)*exp(-b*x**-a) : 0
+plot a=1.0, b=1.0, p2(x) title "Type 2, $a=1,b=1$"
+
+# Discrete distributions
+
+set xlabel "$k$"
+set ylabel "$p(k)$"
+set ticscale 0.5 1
+set mxtics 2
+
+set xrange [-0.5:10.5]
+set yrange [0:0.3]
+set ytics 0.1
+set xtics 1
+set function style step
+set output "rand-poisson.tex"
+set title "Poisson Distribution"
+p(x)=mu**int(x)/(int(x)!) * exp(-mu)
+plot mu=2.5, p(x+0.5) title "$\\mu=2.5$"
+
+set xrange [:1.5]
+set yrange [0:1]
+set ytics 0.5
+set xtics 1
+set function style step
+set output "rand-bernoulli.tex"
+set title "Bernoulli Trial"
+p(x)= (int(x) == 0) ? 1-p : p
+plot p = 0.7, p(x+0.5) title "$p=0.7$"
+
+set xrange [:10.5]
+set yrange [0:0.3]
+set xtics 1
+set ytics 0.1
+set output "rand-binomial.tex"
+set title "Binomial Distribution"
+p(x)= P**int(x)*(1-P)**(n-int(x))*gamma(n+1.0)/gamma(int(x)+1.0)/gamma(n-int(x)+1.0) + 1e-5
+plot P=0.5,n=9.99999999, p(x+0.5) title "$p=0.5,n=9$"
+
+set xrange [:5.5]
+set yrange [0:0.7]
+set ytics 0.1
+set xtics 1
+set output "rand-geometric.tex"
+set title "Geometric Distribution"
+p(x)=int(x) ? P*(1-P)**int(x-1) : 0
+plot P=0.5, p(x+0.5) title "$p=0.5$"
+
+set xrange [:10.5]
+set yrange [0:0.7]
+set ytics 0.1
+set xtics 1
+set output "rand-logarithmic.tex"
+set title "Logarithmic Distribution"
+p(x)=int(x) ? (-1/log(1-P))*(P**int(x))/int(x) : 0
+plot P=0.7, p(x+0.5) title "$p=0.7$"
+
+set xrange [:10.5]
+set yrange [0:0.7]
+set ytics 0.1
+set xtics 1
+set output "rand-hypergeometric.tex"
+set title "Hypergeometric Distribution"
+choose(a,b)=gamma(a+1)/(gamma(b+1)*gamma(a-b+1))
+p(x)=choose(n1,int(x))*choose(n2,t-int(x))/choose(n1+n2,t)
+plot n1=5, n2=20, t=3, p(x+0.5) title "$n1=5, n2=20, t=3$"
+
+set xrange [:10.5]
+set yrange [0:0.3]
+set xtics 1
+set ytics 0.1
+set output "rand-nbinomial.tex"
+set title "Negative Binomial Distribution"
+p(x)= (P**n)*((1-P)**(int(x)))*gamma(n+int(x))/gamma(n)/gamma(int(x)+1.0) + 1e-5
+plot P=0.5,n=3.5, p(x+0.5) title "$p=0.5,n=3.5$"
+
+set xrange [:10.5]
+set yrange [0:0.3]
+set xtics 1
+set ytics 0.1
+set output "rand-pascal.tex"
+set title "Pascal Distribution"
+p(x)= (P**n)*((1-P)**(int(x)))*gamma(n+int(x))/gamma(n)/gamma(int(x)+1.0) + 1e-5
+plot P=0.5,n=3, p(x+0.5) title "$p=0.5,n=3$"
+
+######################################################################
+
+set ticscale 1 0
+set xrange [-5:5]
+set yrange [-5:5]
+set xtics 1
+set ytics 1
+set size 0.8,1.143
+set size square
+set output "random-walk.tex"
+set title "Random walk"
+set xzeroaxis
+set yzeroaxis
+set nokey
+plot "rand-walk.dat" using 1:2 with linesp
+
+
diff --git a/gsl-1.9/doc/rng.texi b/gsl-1.9/doc/rng.texi
new file mode 100644
index 0000000..a947f31
--- /dev/null
+++ b/gsl-1.9/doc/rng.texi
@@ -0,0 +1,1456 @@
+@cindex random number generators
+
+The library provides a large collection of random number generators
+which can be accessed through a uniform interface.  Environment
+variables allow you to select different generators and seeds at runtime,
+so that you can easily switch between generators without needing to
+recompile your program.  Each instance of a generator keeps track of its
+own state, allowing the generators to be used in multi-threaded
+programs.  Additional functions are available for transforming uniform
+random numbers into samples from continuous or discrete probability
+distributions such as the Gaussian, log-normal or Poisson distributions.
+
+These functions are declared in the header file @file{gsl_rng.h}.
+
+@comment Need to explain the difference between SERIAL and PARALLEL random 
+@comment number generators here
+
+@menu
+* General comments on random numbers::  
+* The Random Number Generator Interface::  
+* Random number generator initialization::  
+* Sampling from a random number generator::  
+* Auxiliary random number generator functions::  
+* Random number environment variables::  
+* Copying random number generator state::   
+* Reading and writing random number generator state::   
+* Random number generator algorithms::  
+* Unix random number generators::  
+* Other random number generators::  
+* Random Number Generator Performance::  
+* Random Number Generator Examples::  
+* Random Number References and Further Reading::  
+* Random Number Acknowledgements::  
+@end menu
+
+@node General comments on random numbers
+@section General comments on random numbers
+
+In 1988, Park and Miller wrote a paper entitled ``Random number
+generators: good ones are hard to find.'' [Commun.@: ACM, 31, 1192--1201].
+Fortunately, some excellent random number generators are available,
+though poor ones are still in common use.  You may be happy with the
+system-supplied random number generator on your computer, but you should
+be aware that as computers get faster, requirements on random number
+generators increase.  Nowadays, a simulation that calls a random number
+generator millions of times can often finish before you can make it down
+the hall to the coffee machine and back.
+
+A very nice review of random number generators was written by Pierre
+L'Ecuyer, as Chapter 4 of the book: Handbook on Simulation, Jerry Banks,
+ed. (Wiley, 1997).  The chapter is available in postscript from
+L'Ecuyer's ftp site (see references).  Knuth's volume on Seminumerical
+Algorithms (originally published in 1968) devotes 170 pages to random
+number generators, and has recently been updated in its 3rd edition
+(1997).
+@comment is only now starting to show its age.
+@comment Nonetheless, 
+It is brilliant, a classic.  If you don't own it, you should stop reading
+right now, run to the nearest bookstore, and buy it.
+
+A good random number generator will satisfy both theoretical and
+statistical properties.  Theoretical properties are often hard to obtain
+(they require real math!), but one prefers a random number generator
+with a long period, low serial correlation, and a tendency @emph{not} to
+``fall mainly on the planes.''  Statistical tests are performed with
+numerical simulations.  Generally, a random number generator is used to
+estimate some quantity for which the theory of probability provides an
+exact answer.  Comparison to this exact answer provides a measure of
+``randomness''.
+
+@node The Random Number Generator Interface
+@section The Random Number Generator Interface
+
+It is important to remember that a random number generator is not a
+``real'' function like sine or cosine.  Unlike real functions, successive
+calls to a random number generator yield different return values.  Of
+course that is just what you want for a random number generator, but to
+achieve this effect, the generator must keep track of some kind of
+``state'' variable.  Sometimes this state is just an integer (sometimes
+just the value of the previously generated random number), but often it
+is more complicated than that and may involve a whole array of numbers,
+possibly with some indices thrown in.  To use the random number
+generators, you do not need to know the details of what comprises the
+state, and besides that varies from algorithm to algorithm.
+
+The random number generator library uses two special structs,
+@code{gsl_rng_type} which holds static information about each type of
+generator and @code{gsl_rng} which describes an instance of a generator
+created from a given @code{gsl_rng_type}.
+
+The functions described in this section are declared in the header file
+@file{gsl_rng.h}.
+
+@node Random number generator initialization
+@section Random number generator initialization
+
+@deftypefun {gsl_rng *} gsl_rng_alloc (const gsl_rng_type * @var{T})
+This function returns a pointer to a newly-created
+instance of a random number generator of type @var{T}.
+For example, the following code creates an instance of the Tausworthe
+generator,
+
+@example
+gsl_rng * r = gsl_rng_alloc (gsl_rng_taus);
+@end example
+
+If there is insufficient memory to create the generator then the
+function returns a null pointer and the error handler is invoked with an
+error code of @code{GSL_ENOMEM}.
+
+The generator is automatically initialized with the default seed,
+@code{gsl_rng_default_seed}.  This is zero by default but can be changed
+either directly or by using the environment variable @code{GSL_RNG_SEED}
+(@pxref{Random number environment variables}).
+
+The details of the available generator types are
+described later in this chapter.
+@end deftypefun
+
+@deftypefun void gsl_rng_set (const gsl_rng * @var{r}, unsigned long int @var{s})
+This function initializes (or `seeds') the random number generator.  If
+the generator is seeded with the same value of @var{s} on two different
+runs, the same stream of random numbers will be generated by successive
+calls to the routines below.  If different values of @var{s} are
+supplied, then the generated streams of random numbers should be
+completely different.  If the seed @var{s} is zero then the standard seed
+from the original implementation is used instead.  For example, the
+original Fortran source code for the @code{ranlux} generator used a seed
+of 314159265, and so choosing @var{s} equal to zero reproduces this when
+using @code{gsl_rng_ranlux}.
+@end deftypefun
+
+@deftypefun void gsl_rng_free (gsl_rng * @var{r})
+This function frees all the memory associated with the generator
+@var{r}.
+@end deftypefun
+
+@node  Sampling from a random number generator
+@section Sampling from a random number generator
+
+The following functions return uniformly distributed random numbers,
+either as integers or double precision floating point numbers.  To obtain
+non-uniform distributions @pxref{Random Number Distributions}.
+
+@deftypefun {unsigned long int} gsl_rng_get (const gsl_rng * @var{r})
+This function returns a random integer from the generator @var{r}.  The
+minimum and maximum values depend on the algorithm used, but all
+integers in the range [@var{min},@var{max}] are equally likely.  The
+values of @var{min} and @var{max} can determined using the auxiliary
+functions @code{gsl_rng_max (r)} and @code{gsl_rng_min (r)}.
+@end deftypefun
+
+@deftypefun double gsl_rng_uniform (const gsl_rng * @var{r})
+This function returns a double precision floating point number uniformly
+distributed in the range [0,1).  The range includes 0.0 but excludes 1.0.
+The value is typically obtained by dividing the result of
+@code{gsl_rng_get(r)} by @code{gsl_rng_max(r) + 1.0} in double
+precision.  Some generators compute this ratio internally so that they
+can provide floating point numbers with more than 32 bits of randomness
+(the maximum number of bits that can be portably represented in a single
+@code{unsigned long int}).
+@end deftypefun
+
+@deftypefun double gsl_rng_uniform_pos (const gsl_rng * @var{r})
+This function returns a positive double precision floating point number
+uniformly distributed in the range (0,1), excluding both 0.0 and 1.0.
+The number is obtained by sampling the generator with the algorithm of
+@code{gsl_rng_uniform} until a non-zero value is obtained.  You can use
+this function if you need to avoid a singularity at 0.0.
+@end deftypefun
+
+@deftypefun {unsigned long int} gsl_rng_uniform_int (const gsl_rng * @var{r}, unsigned long int @var{n})
+This function returns a random integer from 0 to @math{n-1} inclusive
+by scaling down and/or discarding samples from the generator @var{r}.
+All integers in the range @math{[0,n-1]} are produced with equal
+probability.  For generators with a non-zero minimum value an offset
+is applied so that zero is returned with the correct probability.
+
+Note that this function is designed for sampling from ranges smaller
+than the range of the underlying generator.  The parameter @var{n}
+must be less than or equal to the range of the generator @var{r}.
+If @var{n} is larger than the range of the generator then the function
+calls the error handler with an error code of @code{GSL_EINVAL} and
+returns zero.
+
+In particular, this function is not intended for generating the full range of
+unsigned integer values @c{$[0,2^{32}-1]$} 
+@math{[0,2^32-1]}. Instead
+choose a generator with the maximal integer range and zero mimimum
+value, such as @code{gsl_rng_ranlxd1}, @code{gsl_rng_mt19937} or
+@code{gsl_rng_taus}, and sample it directly using
+@code{gsl_rng_get}.  The range of each generator can be found using
+the auxiliary functions described in the next section.
+@end deftypefun
+
+@node Auxiliary random number generator functions
+@section Auxiliary random number generator functions
+The following functions provide information about an existing
+generator.  You should use them in preference to hard-coding the generator
+parameters into your own code.
+
+@deftypefun {const char *} gsl_rng_name (const gsl_rng * @var{r})
+This function returns a pointer to the name of the generator.
+For example,
+
+@example
+printf ("r is a '%s' generator\n", 
+        gsl_rng_name (r));
+@end example
+
+@noindent
+would print something like @code{r is a 'taus' generator}.
+@end deftypefun
+
+@deftypefun {unsigned long int} gsl_rng_max (const gsl_rng * @var{r})
+@code{gsl_rng_max} returns the largest value that @code{gsl_rng_get}
+can return.
+@end deftypefun
+
+@deftypefun {unsigned long int} gsl_rng_min (const gsl_rng * @var{r})
+@code{gsl_rng_min} returns the smallest value that @code{gsl_rng_get}
+can return.  Usually this value is zero.  There are some generators with
+algorithms that cannot return zero, and for these generators the minimum
+value is 1.
+@end deftypefun
+
+@deftypefun {void *} gsl_rng_state (const gsl_rng * @var{r})
+@deftypefunx size_t gsl_rng_size (const gsl_rng * @var{r})
+These functions return a pointer to the state of generator @var{r} and
+its size.  You can use this information to access the state directly.  For
+example, the following code will write the state of a generator to a
+stream,
+
+@example
+void * state = gsl_rng_state (r);
+size_t n = gsl_rng_size (r);
+fwrite (state, n, 1, stream);
+@end example
+@end deftypefun
+
+@deftypefun {const gsl_rng_type **} gsl_rng_types_setup (void)
+This function returns a pointer to an array of all the available
+generator types, terminated by a null pointer. The function should be
+called once at the start of the program, if needed.  The following code
+fragment shows how to iterate over the array of generator types to print
+the names of the available algorithms,
+
+@example
+const gsl_rng_type **t, **t0;
+
+t0 = gsl_rng_types_setup ();
+
+printf ("Available generators:\n");
+
+for (t = t0; *t != 0; t++)
+  @{
+    printf ("%s\n", (*t)->name);
+  @}
+@end example
+@end deftypefun
+
+@node Random number environment variables
+@section Random number environment variables
+
+The library allows you to choose a default generator and seed from the
+environment variables @code{GSL_RNG_TYPE} and @code{GSL_RNG_SEED} and
+the function @code{gsl_rng_env_setup}.  This makes it easy try out
+different generators and seeds without having to recompile your program.
+
+@deftypefun {const gsl_rng_type *} gsl_rng_env_setup (void)
+This function reads the environment variables @code{GSL_RNG_TYPE} and
+@code{GSL_RNG_SEED} and uses their values to set the corresponding
+library variables @code{gsl_rng_default} and
+@code{gsl_rng_default_seed}.  These global variables are defined as
+follows,
+
+@example
+extern const gsl_rng_type *gsl_rng_default
+extern unsigned long int gsl_rng_default_seed
+@end example
+
+The environment variable @code{GSL_RNG_TYPE} should be the name of a
+generator, such as @code{taus} or @code{mt19937}.  The environment
+variable @code{GSL_RNG_SEED} should contain the desired seed value.  It
+is converted to an @code{unsigned long int} using the C library function
+@code{strtoul}.
+
+If you don't specify a generator for @code{GSL_RNG_TYPE} then
+@code{gsl_rng_mt19937} is used as the default.  The initial value of
+@code{gsl_rng_default_seed} is zero.
+
+@end deftypefun
+
+@noindent
+@need 2000
+Here is a short program which shows how to create a global
+generator using the environment variables @code{GSL_RNG_TYPE} and
+@code{GSL_RNG_SEED},
+
+@example
+@verbatiminclude examples/rng.c
+@end example
+
+@noindent
+Running the program without any environment variables uses the initial
+defaults, an @code{mt19937} generator with a seed of 0,
+
+@example
+$ ./a.out 
+@verbatiminclude examples/rng.out
+@end example
+
+@noindent
+By setting the two variables on the command line we can
+change the default generator and the seed,
+
+@example
+$ GSL_RNG_TYPE="taus" GSL_RNG_SEED=123 ./a.out 
+GSL_RNG_TYPE=taus
+GSL_RNG_SEED=123
+generator type: taus
+seed = 123
+first value = 2720986350
+@end example
+
+@node Copying random number generator state
+@section Copying random number generator state
+
+The above methods do not expose the random number `state' which changes
+from call to call.  It is often useful to be able to save and restore
+the state.  To permit these practices, a few somewhat more advanced
+functions are supplied.  These include:
+
+@deftypefun int gsl_rng_memcpy (gsl_rng * @var{dest}, const gsl_rng * @var{src})
+This function copies the random number generator @var{src} into the
+pre-existing generator @var{dest}, making @var{dest} into an exact copy
+of @var{src}.  The two generators must be of the same type.
+@end deftypefun
+
+@deftypefun {gsl_rng *} gsl_rng_clone (const gsl_rng * @var{r})
+This function returns a pointer to a newly created generator which is an
+exact copy of the generator @var{r}.
+@end deftypefun
+
+@node Reading and writing random number generator state 
+@section Reading and writing random number generator state
+
+The library provides functions for reading and writing the random
+number state to a file as binary data or formatted text.
+
+@deftypefun int gsl_rng_fwrite (FILE * @var{stream}, const gsl_rng * @var{r})
+This function writes the random number state of the random number
+generator @var{r} to the stream @var{stream} in binary format.  The
+return value is 0 for success and @code{GSL_EFAILED} if there was a
+problem writing to the file.  Since the data is written in the native
+binary format it may not be portable between different architectures.
+@end deftypefun
+
+@deftypefun int gsl_rng_fread (FILE * @var{stream}, gsl_rng * @var{r})
+This function reads the random number state into the random number
+generator @var{r} from the open stream @var{stream} in binary format.
+The random number generator @var{r} must be preinitialized with the
+correct random number generator type since type information is not
+saved.  The return value is 0 for success and @code{GSL_EFAILED} if
+there was a problem reading from the file.  The data is assumed to
+have been written in the native binary format on the same
+architecture.
+@end deftypefun
+
+@node Random number generator algorithms
+@section Random number generator algorithms
+
+The functions described above make no reference to the actual algorithm
+used.  This is deliberate so that you can switch algorithms without
+having to change any of your application source code.  The library
+provides a large number of generators of different types, including
+simulation quality generators, generators provided for compatibility
+with other libraries and historical generators from the past.
+
+The following generators are recommended for use in simulation.  They
+have extremely long periods, low correlation and pass most statistical
+tests.  For the most reliable source of uncorrelated numbers, the
+second-generation @sc{ranlux} generators have the strongest proof of
+randomness.
+
+@deffn {Generator} gsl_rng_mt19937
+@cindex MT19937 random number generator
+The MT19937 generator of Makoto Matsumoto and Takuji Nishimura is a
+variant of the twisted generalized feedback shift-register algorithm,
+and is known as the ``Mersenne Twister'' generator.  It has a Mersenne
+prime period of 
+@comment
+@c{$2^{19937} - 1$} 
+@math{2^19937 - 1} (about 
+@c{$10^{6000}$}
+@math{10^6000}) and is
+equi-distributed in 623 dimensions.  It has passed the @sc{diehard}
+statistical tests.  It uses 624 words of state per generator and is
+comparable in speed to the other generators.  The original generator used
+a default seed of 4357 and choosing @var{s} equal to zero in
+@code{gsl_rng_set} reproduces this.  Later versions switched to 5489
+as the default seed, you can choose this explicitly via @code{gsl_rng_set} 
+instead if you require it.
+
+For more information see,
+@itemize @asis
+@item
+Makoto Matsumoto and Takuji Nishimura, ``Mersenne Twister: A
+623-dimensionally equidistributed uniform pseudorandom number
+generator''. @cite{ACM Transactions on Modeling and Computer
+Simulation}, Vol.@: 8, No.@: 1 (Jan. 1998), Pages 3--30
+@end itemize
+
+@noindent
+The generator @code{gsl_rng_mt19937} uses the second revision of the
+seeding procedure published by the two authors above in 2002.  The
+original seeding procedures could cause spurious artifacts for some seed
+values. They are still available through the alternative generators
+@code{gsl_rng_mt19937_1999} and @code{gsl_rng_mt19937_1998}.
+@end deffn
+
+@deffn {Generator} gsl_rng_ranlxs0
+@deffnx {Generator} gsl_rng_ranlxs1
+@deffnx {Generator} gsl_rng_ranlxs2
+@cindex RANLXS random number generator
+
+The generator @code{ranlxs0} is a second-generation version of the
+@sc{ranlux} algorithm of L@"uscher, which produces ``luxury random
+numbers''.  This generator provides single precision output (24 bits) at
+three luxury levels @code{ranlxs0}, @code{ranlxs1} and @code{ranlxs2}, 
+in increasing order of strength.
+It uses double-precision floating point arithmetic internally and can be
+significantly faster than the integer version of @code{ranlux},
+particularly on 64-bit architectures.  The period of the generator is
+about @c{$10^{171}$} 
+@math{10^171}.  The algorithm has mathematically proven properties and
+can provide truly decorrelated numbers at a known level of randomness.
+The higher luxury levels provide increased decorrelation between samples
+as an additional safety margin.
+@end deffn
+
+@deffn {Generator} gsl_rng_ranlxd1
+@deffnx {Generator} gsl_rng_ranlxd2
+@cindex RANLXD random number generator
+
+These generators produce double precision output (48 bits) from the
+@sc{ranlxs} generator.  The library provides two luxury levels
+@code{ranlxd1} and @code{ranlxd2}, in increasing order of strength.
+@end deffn
+
+
+@deffn {Generator} gsl_rng_ranlux
+@deffnx {Generator} gsl_rng_ranlux389
+
+@cindex RANLUX random number generator
+The @code{ranlux} generator is an implementation of the original
+algorithm developed by L@"uscher.  It uses a
+lagged-fibonacci-with-skipping algorithm to produce ``luxury random
+numbers''.  It is a 24-bit generator, originally designed for
+single-precision IEEE floating point numbers.  This implementation is
+based on integer arithmetic, while the second-generation versions
+@sc{ranlxs} and @sc{ranlxd} described above provide floating-point
+implementations which will be faster on many platforms.
+The period of the generator is about @c{$10^{171}$} 
+@math{10^171}.  The algorithm has mathematically proven properties and
+it can provide truly decorrelated numbers at a known level of
+randomness.  The default level of decorrelation recommended by L@"uscher
+is provided by @code{gsl_rng_ranlux}, while @code{gsl_rng_ranlux389}
+gives the highest level of randomness, with all 24 bits decorrelated.
+Both types of generator use 24 words of state per generator.
+
+For more information see,
+@itemize @asis
+@item
+M. L@"uscher, ``A portable high-quality random number generator for
+lattice field theory calculations'', @cite{Computer Physics
+Communications}, 79 (1994) 100--110.
+@item
+F. James, ``RANLUX: A Fortran implementation of the high-quality
+pseudo-random number generator of L@"uscher'', @cite{Computer Physics
+Communications}, 79 (1994) 111--114
+@end itemize
+@end deffn
+
+
+@deffn {Generator} gsl_rng_cmrg
+@cindex CMRG, combined multiple recursive random number generator
+This is a combined multiple recursive generator by L'Ecuyer. 
+Its sequence is,
+@tex
+\beforedisplay
+$$
+z_n = (x_n - y_n) \,\hbox{mod}\, m_1
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+z_n = (x_n - y_n) mod m_1
+@end example
+
+@end ifinfo
+@noindent
+where the two underlying generators @math{x_n} and @math{y_n} are,
+@tex
+\beforedisplay
+$$
+\eqalign{ 
+x_n & = (a_1 x_{n-1} + a_2 x_{n-2} + a_3 x_{n-3}) \,\hbox{mod}\, m_1 \cr
+y_n & = (b_1 y_{n-1} + b_2 y_{n-2} + b_3 y_{n-3}) \,\hbox{mod}\, m_2
+}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+x_n = (a_1 x_@{n-1@} + a_2 x_@{n-2@} + a_3 x_@{n-3@}) mod m_1
+y_n = (b_1 y_@{n-1@} + b_2 y_@{n-2@} + b_3 y_@{n-3@}) mod m_2
+@end example
+
+@end ifinfo
+@noindent
+with coefficients 
+@math{a_1 = 0}, 
+@math{a_2 = 63308}, 
+@math{a_3 = -183326},
+@math{b_1 = 86098}, 
+@math{b_2 = 0},
+@math{b_3 = -539608},
+and moduli 
+@c{$m_1 = 2^{31} - 1 = 2147483647$} 
+@math{m_1 = 2^31 - 1 = 2147483647}
+and 
+@c{$m_2 = 2145483479$}
+@math{m_2 = 2145483479}.
+
+The period of this generator is  
+@c{$\hbox{lcm}(m_1^3-1, m_2^3-1)$}
+@math{lcm(m_1^3-1, m_2^3-1)},
+which is approximately
+@c{$2^{185}$}
+@math{2^185} 
+(about 
+@c{$10^{56}$}
+@math{10^56}).  It uses
+6 words of state per generator.  For more information see,
+
+@itemize @asis
+@item
+P. L'Ecuyer, ``Combined Multiple Recursive Random Number
+Generators'', @cite{Operations Research}, 44, 5 (1996), 816--822.
+@end itemize
+@end deffn
+
+@deffn {Generator} gsl_rng_mrg
+@cindex MRG, multiple recursive random number generator
+This is a fifth-order multiple recursive generator by L'Ecuyer, Blouin
+and Coutre.  Its sequence is,
+@tex
+\beforedisplay
+$$
+x_n = (a_1 x_{n-1} + a_5 x_{n-5}) \,\hbox{mod}\, m
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+x_n = (a_1 x_@{n-1@} + a_5 x_@{n-5@}) mod m
+@end example
+
+@end ifinfo
+@noindent
+with 
+@math{a_1 = 107374182}, 
+@math{a_2 = a_3 = a_4 = 0}, 
+@math{a_5 = 104480}
+and 
+@c{$m = 2^{31}-1$}
+@math{m = 2^31 - 1}.
+
+The period of this generator is about 
+@c{$10^{46}$}
+@math{10^46}.  It uses 5 words
+of state per generator.  More information can be found in the following
+paper,
+@itemize @asis
+@item
+P. L'Ecuyer, F. Blouin, and R. Coutre, ``A search for good multiple
+recursive random number generators'', @cite{ACM Transactions on Modeling and
+Computer Simulation} 3, 87--98 (1993).
+@end itemize
+@end deffn
+
+@deffn {Generator} gsl_rng_taus
+@deffnx {Generator} gsl_rng_taus2
+@cindex Tausworthe random number generator
+This is a maximally equidistributed combined Tausworthe generator by
+L'Ecuyer.  The sequence is,
+@tex
+\beforedisplay
+$$
+x_n = (s^1_n \oplus s^2_n \oplus s^3_n) 
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+x_n = (s1_n ^^ s2_n ^^ s3_n) 
+@end example
+
+@end ifinfo
+@noindent
+where,
+@tex
+\beforedisplay
+$$
+\eqalign{
+s^1_{n+1} &= (((s^1_n \& 4294967294)\ll 12) \oplus (((s^1_n\ll 13) \oplus s^1_n)\gg 19)) \cr
+s^2_{n+1} &= (((s^2_n \& 4294967288)\ll 4) \oplus (((s^2_n\ll 2) \oplus s^2_n)\gg 25)) \cr
+s^3_{n+1} &= (((s^3_n \& 4294967280)\ll 17) \oplus (((s^3_n\ll 3) \oplus s^3_n)\gg 11))
+}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+s1_@{n+1@} = (((s1_n&4294967294)<<12)^^(((s1_n<<13)^^s1_n)>>19))
+s2_@{n+1@} = (((s2_n&4294967288)<< 4)^^(((s2_n<< 2)^^s2_n)>>25))
+s3_@{n+1@} = (((s3_n&4294967280)<<17)^^(((s3_n<< 3)^^s3_n)>>11))
+@end example
+
+@end ifinfo
+@noindent
+computed modulo 
+@c{$2^{32}$}
+@math{2^32}.  In the formulas above 
+@c{$\oplus$}
+@math{^^}
+denotes ``exclusive-or''.  Note that the algorithm relies on the properties
+of 32-bit unsigned integers and has been implemented using a bitmask
+of @code{0xFFFFFFFF} to make it work on 64 bit machines.
+
+The period of this generator is @c{$2^{88}$}
+@math{2^88} (about
+@c{$10^{26}$}
+@math{10^26}).  It uses 3 words of state per generator.  For more
+information see,
+
+@itemize @asis
+@item
+P. L'Ecuyer, ``Maximally Equidistributed Combined Tausworthe
+Generators'', @cite{Mathematics of Computation}, 65, 213 (1996), 203--213.
+@end itemize
+
+@noindent
+The generator @code{gsl_rng_taus2} uses the same algorithm as
+@code{gsl_rng_taus} but with an improved seeding procedure described in
+the paper,
+
+@itemize @asis
+@item
+P. L'Ecuyer, ``Tables of Maximally Equidistributed Combined LFSR
+Generators'', @cite{Mathematics of Computation}, 68, 225 (1999), 261--269
+@end itemize
+
+@noindent
+The generator @code{gsl_rng_taus2} should now be used in preference to
+@code{gsl_rng_taus}.
+@end deffn
+
+@deffn {Generator} gsl_rng_gfsr4
+@cindex Four-tap Generalized Feedback Shift Register
+The @code{gfsr4} generator is like a lagged-fibonacci generator, and 
+produces each number as an @code{xor}'d sum of four previous values.
+@tex
+\beforedisplay
+$$
+r_n = r_{n-A} \oplus r_{n-B} \oplus r_{n-C} \oplus r_{n-D}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+r_n = r_@{n-A@} ^^ r_@{n-B@} ^^ r_@{n-C@} ^^ r_@{n-D@}
+@end example
+@end ifinfo
+
+Ziff (ref below) notes that ``it is now widely known'' that two-tap
+registers (such as R250, which is described below)
+have serious flaws, the most obvious one being the three-point
+correlation that comes from the definition of the generator.  Nice
+mathematical properties can be derived for GFSR's, and numerics bears
+out the claim that 4-tap GFSR's with appropriately chosen offsets are as
+random as can be measured, using the author's test.
+
+This implementation uses the values suggested the example on p392 of
+Ziff's article: @math{A=471}, @math{B=1586}, @math{C=6988}, @math{D=9689}.
+
+
+If the offsets are appropriately chosen (such as the one ones in this
+implementation), then the sequence is said to be maximal; that means
+that the period is @math{2^D - 1}, where @math{D} is the longest lag.
+(It is one less than @math{2^D} because it is not permitted to have all
+zeros in the @code{ra[]} array.)  For this implementation with
+@math{D=9689} that works out to about @c{$10^{2917}$}
+@math{10^2917}.
+
+Note that the implementation of this generator using a 32-bit
+integer amounts to 32 parallel implementations of one-bit
+generators.  One consequence of this is that the period of this
+32-bit generator is the same as for the one-bit generator.
+Moreover, this independence means that all 32-bit patterns are
+equally likely, and in particular that 0 is an allowed random
+value.  (We are grateful to Heiko Bauke for clarifying for us these
+properties of GFSR random number generators.)
+
+For more information see,
+@itemize @asis
+@item
+Robert M. Ziff, ``Four-tap shift-register-sequence random-number 
+generators'', @cite{Computers in Physics}, 12(4), Jul/Aug
+1998, pp 385--392.
+@end itemize
+@end deffn
+
+@node Unix random number generators
+@section Unix random number generators
+
+The standard Unix random number generators @code{rand}, @code{random}
+and @code{rand48} are provided as part of GSL. Although these
+generators are widely available individually often they aren't all
+available on the same platform.  This makes it difficult to write
+portable code using them and so we have included the complete set of
+Unix generators in GSL for convenience.  Note that these generators
+don't produce high-quality randomness and aren't suitable for work
+requiring accurate statistics.  However, if you won't be measuring
+statistical quantities and just want to introduce some variation into
+your program then these generators are quite acceptable.
+
+@cindex rand, BSD random number generator
+@cindex Unix random number generators, rand
+@cindex Unix random number generators, rand48
+
+@deffn {Generator} gsl_rng_rand
+@cindex BSD random number generator
+This is the BSD @code{rand} generator.  Its sequence is
+@tex
+\beforedisplay
+$$
+x_{n+1} = (a x_n + c) \,\hbox{mod}\, m
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+x_@{n+1@} = (a x_n + c) mod m
+@end example
+
+@end ifinfo
+@noindent
+with 
+@math{a = 1103515245}, 
+@math{c = 12345} and 
+@c{$m = 2^{31}$}
+@math{m = 2^31}.
+The seed specifies the initial value, 
+@math{x_1}.  The period of this
+generator is 
+@c{$2^{31}$}
+@math{2^31}, and it uses 1 word of storage per
+generator.
+@end deffn
+
+@deffn {Generator} gsl_rng_random_bsd
+@deffnx {Generator} gsl_rng_random_libc5
+@deffnx {Generator} gsl_rng_random_glibc2
+These generators implement the @code{random} family of functions, a
+set of linear feedback shift register generators originally used in BSD
+Unix.  There are several versions of @code{random} in use today: the
+original BSD version (e.g. on SunOS4), a libc5 version (found on
+older GNU/Linux systems) and a glibc2 version.  Each version uses a
+different seeding procedure, and thus produces different sequences.
+
+The original BSD routines accepted a variable length buffer for the
+generator state, with longer buffers providing higher-quality
+randomness.  The @code{random} function implemented algorithms for
+buffer lengths of 8, 32, 64, 128 and 256 bytes, and the algorithm with
+the largest length that would fit into the user-supplied buffer was
+used.  To support these algorithms additional generators are available
+with the following names,
+
+@example
+gsl_rng_random8_bsd
+gsl_rng_random32_bsd
+gsl_rng_random64_bsd
+gsl_rng_random128_bsd
+gsl_rng_random256_bsd
+@end example
+
+@noindent
+where the numeric suffix indicates the buffer length.  The original BSD
+@code{random} function used a 128-byte default buffer and so
+@code{gsl_rng_random_bsd} has been made equivalent to
+@code{gsl_rng_random128_bsd}.  Corresponding versions of the @code{libc5}
+and @code{glibc2} generators are also available, with the names
+@code{gsl_rng_random8_libc5}, @code{gsl_rng_random8_glibc2}, etc.
+@end deffn
+
+@deffn {Generator} gsl_rng_rand48
+@cindex rand48 random number generator
+This is the Unix @code{rand48} generator.  Its sequence is
+@tex
+\beforedisplay
+$$
+x_{n+1} = (a x_n + c) \,\hbox{mod}\, m
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+x_@{n+1@} = (a x_n + c) mod m
+@end example
+
+@end ifinfo
+@noindent
+defined on 48-bit unsigned integers with 
+@math{a = 25214903917}, 
+@math{c = 11} and 
+@c{$m = 2^{48}$} 
+@math{m = 2^48}. 
+The seed specifies the upper 32 bits of the initial value, @math{x_1},
+with the lower 16 bits set to @code{0x330E}.  The function
+@code{gsl_rng_get} returns the upper 32 bits from each term of the
+sequence.  This does not have a direct parallel in the original
+@code{rand48} functions, but forcing the result to type @code{long int}
+reproduces the output of @code{mrand48}.  The function
+@code{gsl_rng_uniform} uses the full 48 bits of internal state to return
+the double precision number @math{x_n/m}, which is equivalent to the
+function @code{drand48}.  Note that some versions of the GNU C Library
+contained a bug in @code{mrand48} function which caused it to produce
+different results (only the lower 16-bits of the return value were set).
+@end deffn
+
+@node Other random number generators
+@section Other random number generators
+
+The generators in this section are provided for compatibility with
+existing libraries.  If you are converting an existing program to use GSL
+then you can select these generators to check your new implementation
+against the original one, using the same random number generator.  After
+verifying that your new program reproduces the original results you can
+then switch to a higher-quality generator.
+
+Note that most of the generators in this section are based on single
+linear congruence relations, which are the least sophisticated type of
+generator.  In particular, linear congruences have poor properties when
+used with a non-prime modulus, as several of these routines do (e.g.
+with a power of two modulus, 
+@c{$2^{31}$}
+@math{2^31} or 
+@c{$2^{32}$}
+@math{2^32}).  This
+leads to periodicity in the least significant bits of each number,
+with only the higher bits having any randomness.  Thus if you want to
+produce a random bitstream it is best to avoid using the least
+significant bits.
+
+@deffn {Generator} gsl_rng_ranf
+@cindex RANF random number generator
+@cindex CRAY random number generator, RANF
+This is the CRAY random number generator @code{RANF}.  Its sequence is
+@tex
+\beforedisplay
+$$
+x_{n+1} = (a x_n) \,\hbox{mod}\, m
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+x_@{n+1@} = (a x_n) mod m
+@end example
+
+@end ifinfo
+@noindent
+defined on 48-bit unsigned integers with @math{a = 44485709377909} and
+@c{$m = 2^{48}$}
+@math{m = 2^48}.  The seed specifies the lower
+32 bits of the initial value, 
+@math{x_1}, with the lowest bit set to
+prevent the seed taking an even value.  The upper 16 bits of 
+@math{x_1}
+are set to 0. A consequence of this procedure is that the pairs of seeds
+2 and 3, 4 and 5, etc produce the same sequences.
+
+The generator compatible with the CRAY MATHLIB routine RANF. It
+produces double precision floating point numbers which should be
+identical to those from the original RANF.
+
+There is a subtlety in the implementation of the seeding.  The initial
+state is reversed through one step, by multiplying by the modular
+inverse of @math{a} mod @math{m}.  This is done for compatibility with
+the original CRAY implementation.
+
+Note that you can only seed the generator with integers up to
+@c{$2^{32}$}
+@math{2^32}, while the original CRAY implementation uses
+non-portable wide integers which can cover all 
+@c{$2^{48}$}
+@math{2^48} states of the generator.
+
+The function @code{gsl_rng_get} returns the upper 32 bits from each term
+of the sequence.  The function @code{gsl_rng_uniform} uses the full 48
+bits to return the double precision number @math{x_n/m}.
+
+The period of this generator is @c{$2^{46}$}
+@math{2^46}.
+@end deffn
+
+@deffn {Generator} gsl_rng_ranmar
+@cindex RANMAR random number generator
+This is the RANMAR lagged-fibonacci generator of Marsaglia, Zaman and
+Tsang.  It is a 24-bit generator, originally designed for
+single-precision IEEE floating point numbers.  It was included in the
+CERNLIB high-energy physics library.
+@end deffn
+
+@deffn {Generator} gsl_rng_r250
+@cindex shift-register random number generator
+@cindex R250 shift-register random number generator
+This is the shift-register generator of Kirkpatrick and Stoll.  The
+sequence is based on the recurrence
+@tex
+\beforedisplay
+$$ 
+x_n = x_{n-103} \oplus x_{n-250}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+x_n = x_@{n-103@} ^^ x_@{n-250@}
+@end example
+
+@end ifinfo
+@noindent
+where 
+@c{$\oplus$}
+@math{^^} denotes ``exclusive-or'', defined on
+32-bit words.  The period of this generator is about @c{$2^{250}$}
+@math{2^250} and it
+uses 250 words of state per generator.
+
+For more information see,
+@itemize @asis
+@item
+S. Kirkpatrick and E. Stoll, ``A very fast shift-register sequence random
+number generator'', @cite{Journal of Computational Physics}, 40, 517--526
+(1981)
+@end itemize
+@end deffn
+
+@deffn {Generator} gsl_rng_tt800
+@cindex TT800 random number generator
+This is an earlier version of the twisted generalized feedback
+shift-register generator, and has been superseded by the development of
+MT19937.  However, it is still an acceptable generator in its own
+right.  It has a period of 
+@c{$2^{800}$}
+@math{2^800} and uses 33 words of storage
+per generator.
+
+For more information see,
+@itemize @asis
+@item
+Makoto Matsumoto and Yoshiharu Kurita, ``Twisted GFSR Generators
+II'', @cite{ACM Transactions on Modelling and Computer Simulation},
+Vol.@: 4, No.@: 3, 1994, pages 254--266.
+@end itemize
+@end deffn
+
+@comment The following generators are included only for historical reasons, so
+@comment that you can reproduce results from old programs which might have used
+@comment them.  These generators should not be used for real simulations since
+@comment they have poor statistical properties by modern standards.
+
+@deffn {Generator} gsl_rng_vax
+@cindex VAX random number generator
+This is the VAX generator @code{MTH$RANDOM}.  Its sequence is,
+@tex
+\beforedisplay
+$$
+x_{n+1} = (a x_n + c) \,\hbox{mod}\, m
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+x_@{n+1@} = (a x_n + c) mod m
+@end example
+
+@end ifinfo
+@noindent
+with 
+@math{a = 69069}, @math{c = 1} and 
+@c{$m = 2^{32}$}
+@math{m = 2^32}.  The seed specifies the initial value, 
+@math{x_1}.  The
+period of this generator is 
+@c{$2^{32}$}
+@math{2^32} and it uses 1 word of storage per
+generator.
+@end deffn
+
+@deffn {Generator} gsl_rng_transputer
+This is the random number generator from the INMOS Transputer
+Development system.  Its sequence is,
+@tex
+\beforedisplay
+$$
+x_{n+1} = (a x_n) \,\hbox{mod}\, m
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+x_@{n+1@} = (a x_n) mod m
+@end example
+
+@end ifinfo
+@noindent
+with @math{a = 1664525} and 
+@c{$m = 2^{32}$}
+@math{m = 2^32}.
+The seed specifies the initial value, 
+@c{$x_1$}
+@math{x_1}.
+@end deffn
+
+@deffn {Generator} gsl_rng_randu
+@cindex RANDU random number generator
+This is the IBM @code{RANDU} generator.  Its sequence is
+@tex
+\beforedisplay
+$$
+x_{n+1} = (a x_n) \,\hbox{mod}\, m
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+x_@{n+1@} = (a x_n) mod m
+@end example
+
+@end ifinfo
+@noindent
+with @math{a = 65539} and 
+@c{$m = 2^{31}$}
+@math{m = 2^31}.  The
+seed specifies the initial value, 
+@math{x_1}.  The period of this
+generator was only 
+@c{$2^{29}$}
+@math{2^29}.  It has become a textbook example of a
+poor generator.
+@end deffn
+
+@deffn {Generator} gsl_rng_minstd
+@cindex RANMAR random number generator
+This is Park and Miller's ``minimal standard'' @sc{minstd} generator, a
+simple linear congruence which takes care to avoid the major pitfalls of
+such algorithms.  Its sequence is,
+@tex
+\beforedisplay
+$$
+x_{n+1} = (a x_n) \,\hbox{mod}\, m
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+x_@{n+1@} = (a x_n) mod m
+@end example
+
+@end ifinfo
+@noindent
+with @math{a = 16807} and 
+@c{$m = 2^{31} - 1 = 2147483647$}
+@math{m = 2^31 - 1 = 2147483647}. 
+The seed specifies the initial value, 
+@c{$x_1$}
+@math{x_1}.  The period of this
+generator is about 
+@c{$2^{31}$}
+@math{2^31}.
+
+This generator is used in the IMSL Library (subroutine RNUN) and in
+MATLAB (the RAND function).  It is also sometimes known by the acronym
+``GGL'' (I'm not sure what that stands for).
+
+For more information see,
+@itemize @asis
+@item
+Park and Miller, ``Random Number Generators: Good ones are hard to find'',
+@cite{Communications of the ACM}, October 1988, Volume 31, No 10, pages
+1192--1201.
+@end itemize
+@end deffn
+
+@deffn {Generator} gsl_rng_uni
+@deffnx {Generator} gsl_rng_uni32
+This is a reimplementation of the 16-bit SLATEC random number generator
+RUNIF. A generalization of the generator to 32 bits is provided by
+@code{gsl_rng_uni32}.  The original source code is available from NETLIB.
+@end deffn
+
+@deffn {Generator} gsl_rng_slatec
+This is the SLATEC random number generator RAND. It is ancient.  The
+original source code is available from NETLIB.
+@end deffn
+
+
+@deffn {Generator} gsl_rng_zuf
+This is the ZUFALL lagged Fibonacci series generator of Peterson.  Its
+sequence is,
+@tex
+\beforedisplay
+$$ 
+\eqalign{
+t &= u_{n-273} + u_{n-607} \cr
+u_n  &= t - \hbox{floor}(t)
+}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+t = u_@{n-273@} + u_@{n-607@}
+u_n  = t - floor(t)
+@end example
+@end ifinfo
+
+The original source code is available from NETLIB.  For more information
+see,
+@itemize @asis
+@item
+W. Petersen, ``Lagged Fibonacci Random Number Generators for the NEC
+SX-3'', @cite{International Journal of High Speed Computing} (1994).
+@end itemize
+@end deffn
+
+@deffn {Generator} gsl_rng_knuthran2
+This is a second-order multiple recursive generator described by Knuth
+in @cite{Seminumerical Algorithms}, 3rd Ed., page 108.  Its sequence is,
+@tex
+\beforedisplay
+$$
+x_n = (a_1 x_{n-1} + a_2 x_{n-2}) \,\hbox{mod}\, m
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+x_n = (a_1 x_@{n-1@} + a_2 x_@{n-2@}) mod m
+@end example
+
+@end ifinfo
+@noindent
+with 
+@math{a_1 = 271828183}, 
+@math{a_2 = 314159269}, 
+and 
+@c{$m = 2^{31}-1$}
+@math{m = 2^31 - 1}.
+@end deffn
+
+@deffn {Generator} gsl_rng_knuthran2002
+@deffnx {Generator} gsl_rng_knuthran
+This is a second-order multiple recursive generator described by Knuth
+in @cite{Seminumerical Algorithms}, 3rd Ed., Section 3.6.  Knuth
+provides its C code.  The updated routine @code{gsl_rng_knuthran2002}
+is from the revised 9th printing and corrects some weaknesses in the
+earlier version, which is implemented as @code{gsl_rng_knuthran}.
+@end deffn
+
+@deffn {Generator} gsl_rng_borosh13
+@deffnx {Generator} gsl_rng_fishman18
+@deffnx {Generator} gsl_rng_fishman20
+@deffnx {Generator} gsl_rng_lecuyer21
+@deffnx {Generator} gsl_rng_waterman14
+These multiplicative generators are taken from Knuth's
+@cite{Seminumerical Algorithms}, 3rd Ed., pages 106--108. Their sequence
+is,
+@tex
+\beforedisplay
+$$
+x_{n+1} = (a x_n) \,\hbox{mod}\, m
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+x_@{n+1@} = (a x_n) mod m
+@end example
+
+@end ifinfo
+@noindent
+where the seed specifies the initial value, @c{$x_1$}
+@math{x_1}.
+The parameters @math{a} and @math{m} are as follows,
+Borosh-Niederreiter: 
+@math{a = 1812433253}, @c{$m = 2^{32}$}
+@math{m = 2^32},
+Fishman18:
+@math{a = 62089911},
+@c{$m = 2^{31}-1$}
+@math{m = 2^31 - 1},
+Fishman20:
+@math{a = 48271},
+@c{$m = 2^{31}-1$}
+@math{m = 2^31 - 1},
+L'Ecuyer:
+@math{a = 40692},
+@c{$m = 2^{31}-249$}
+@math{m = 2^31 - 249},
+Waterman:
+@math{a = 1566083941},
+@c{$m = 2^{32}$}
+@math{m = 2^32}.
+@end deffn
+
+@deffn {Generator} gsl_rng_fishman2x
+This is the L'Ecuyer--Fishman random number generator. It is taken from
+Knuth's @cite{Seminumerical Algorithms}, 3rd Ed., page 108. Its sequence
+is,
+@tex
+\beforedisplay
+$$
+z_{n+1} = (x_n - y_n) \,\hbox{mod}\, m
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+z_@{n+1@} = (x_n - y_n) mod m
+@end example
+
+@end ifinfo
+@noindent
+with @c{$m = 2^{31}-1$}
+@math{m = 2^31 - 1}.
+@math{x_n} and @math{y_n} are given by the @code{fishman20} 
+and @code{lecuyer21} algorithms.
+The seed specifies the initial value, 
+@c{$x_1$}
+@math{x_1}.
+
+@end deffn
+
+
+@deffn {Generator} gsl_rng_coveyou
+This is the Coveyou random number generator. It is taken from Knuth's
+@cite{Seminumerical Algorithms}, 3rd Ed., Section 3.2.2. Its sequence
+is,
+@tex
+\beforedisplay
+$$
+x_{n+1} = (x_n (x_n + 1)) \,\hbox{mod}\, m
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+x_@{n+1@} = (x_n (x_n + 1)) mod m
+@end example
+
+@end ifinfo
+@noindent
+with @c{$m = 2^{32}$}
+@math{m = 2^32}.
+The seed specifies the initial value, 
+@c{$x_1$}
+@math{x_1}.
+@end deffn
+
+
+
+
+
+@node Random Number Generator Performance
+@section Performance
+
+@comment
+@comment I made the original plot like this
+@comment ./benchmark > tmp; cat tmp | perl -n -e '($n,$s) = split(" ",$_); printf("%17s ",$n); print "-" x ($s/1e5), "\n";'
+@comment
+
+The following table shows the relative performance of a selection the
+available random number generators.  The fastest simulation quality
+generators are @code{taus}, @code{gfsr4} and @code{mt19937}.  The
+generators which offer the best mathematically-proven quality are those
+based on the @sc{ranlux} algorithm.
+
+@comment The large number of generators based on single linear congruences are
+@comment represented by the @code{random} generator below.  These generators are
+@comment fast but have the lowest statistical quality.
+
+@example
+1754 k ints/sec,    870 k doubles/sec, taus
+1613 k ints/sec,    855 k doubles/sec, gfsr4
+1370 k ints/sec,    769 k doubles/sec, mt19937
+ 565 k ints/sec,    571 k doubles/sec, ranlxs0
+ 400 k ints/sec,    405 k doubles/sec, ranlxs1
+ 490 k ints/sec,    389 k doubles/sec, mrg
+ 407 k ints/sec,    297 k doubles/sec, ranlux
+ 243 k ints/sec,    254 k doubles/sec, ranlxd1
+ 251 k ints/sec,    253 k doubles/sec, ranlxs2
+ 238 k ints/sec,    215 k doubles/sec, cmrg
+ 247 k ints/sec,    198 k doubles/sec, ranlux389
+ 141 k ints/sec,    140 k doubles/sec, ranlxd2
+
+1852 k ints/sec,    935 k doubles/sec, ran3
+ 813 k ints/sec,    575 k doubles/sec, ran0
+ 787 k ints/sec,    476 k doubles/sec, ran1
+ 379 k ints/sec,    292 k doubles/sec, ran2
+@end example
+
+@node Random Number Generator Examples
+@section Examples
+
+The following program demonstrates the use of a random number generator
+to produce uniform random numbers in the range [0.0, 1.0),
+
+@example
+@verbatiminclude examples/rngunif.c
+@end example
+
+@noindent
+Here is the output of the program,
+
+@example
+$ ./a.out 
+@verbatiminclude examples/rngunif.out
+@end example
+
+@noindent
+The numbers depend on the seed used by the generator.  The default seed
+can be changed with the @code{GSL_RNG_SEED} environment variable to
+produce a different stream of numbers.  The generator itself can be
+changed using the environment variable @code{GSL_RNG_TYPE}.  Here is the
+output of the program using a seed value of 123 and the
+multiple-recursive generator @code{mrg},
+
+@example
+$ GSL_RNG_SEED=123 GSL_RNG_TYPE=mrg ./a.out 
+@verbatiminclude examples/rngunif.2.out
+@end example
+
+@node Random Number References and Further Reading
+@section References and Further Reading
+
+The subject of random number generation and testing is reviewed
+extensively in Knuth's @cite{Seminumerical Algorithms}.
+
+@itemize @asis
+@item
+Donald E. Knuth, @cite{The Art of Computer Programming: Seminumerical
+Algorithms} (Vol 2, 3rd Ed, 1997), Addison-Wesley, ISBN 0201896842.
+@end itemize
+
+@noindent
+Further information is available in the review paper written by Pierre
+L'Ecuyer,
+
+@itemize @asis
+P. L'Ecuyer, ``Random Number Generation'', Chapter 4 of the
+Handbook on Simulation, Jerry Banks Ed., Wiley, 1998, 93--137.
+
+@uref{http://www.iro.umontreal.ca/~lecuyer/papers.html}
+in the file @file{handsim.ps}.
+@end itemize
+
+@noindent
+The source code for the @sc{diehard} random number generator tests is also
+available online,
+
+@itemize @asis
+@item
+@cite{DIEHARD source code} G. Marsaglia,
+@item
+@uref{http://stat.fsu.edu/pub/diehard/}
+@end itemize
+
+@noindent
+A comprehensive set of random number generator tests is available from
+@sc{nist},
+
+@itemize @asis
+@item
+NIST Special Publication 800-22, ``A Statistical Test Suite for the
+Validation of Random Number Generators and Pseudo Random Number
+Generators for Cryptographic Applications''.
+@item
+@uref{http://csrc.nist.gov/rng/}
+@end itemize
+
+@node Random Number Acknowledgements
+@section Acknowledgements
+
+Thanks to Makoto Matsumoto, Takuji Nishimura and Yoshiharu Kurita for
+making the source code to their generators (MT19937, MM&TN; TT800,
+MM&YK) available under the GNU General Public License.  Thanks to Martin
+L@"uscher for providing notes and source code for the @sc{ranlxs} and
+@sc{ranlxd} generators.
+
+@comment lcg
+@comment [ LCG(n) := n * 69069 mod (2^32) ]
+@comment First 6: [69069, 475559465, 2801775573, 1790562961, 3104832285, 4238970681]
+@comment %2^31-1   69069, 475559465, 654291926, 1790562961, 957348638, 2091487034
+@comment mrg
+@comment [q([x1, x2, x3, x4, x5]) := [107374182 mod 2147483647 * x1 + 104480 mod 2147483647 * x5, x1, x2, x3, x4]]
+@comment
+@comment cmrg
+@comment [q1([x1,x2,x3]) := [63308 mod 2147483647 * x2 -183326 mod 2147483647 * x3, x1, x2],
+@comment  q2([x1,x2,x3]) := [86098 mod 2145483479 * x1 -539608 mod 2145483479 * x3, x1, x2] ]
+@comment  initial for q1 is [69069, 475559465, 654291926]
+@comment  initial for q2 is  [1790562961, 959348806, 2093487202]
+
+@comment tausworthe
+@comment    [ b1(x) := rsh(xor(lsh(x, 13), x), 19),
+@comment      q1(x) := xor(lsh(and(x, 4294967294), 12), b1(x)),
+@comment      b2(x) := rsh(xor(lsh(x, 2), x), 25),
+@comment      q2(x) := xor(lsh(and(x, 4294967288), 4), b2(x)),
+@comment      b3(x) := rsh(xor(lsh(x, 3), x), 11),
+@comment      q3(x) := xor(lsh(and(x, 4294967280), 17), b3(x)) ]
+@comment      [s1, s2, s3] = [600098857, 1131373026, 1223067536] 
+@comment [2948905028, 441213979, 394017882]
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diff --git a/gsl-1.9/doc/roots.texi b/gsl-1.9/doc/roots.texi
new file mode 100644
index 0000000..fdcefe1
--- /dev/null
+++ b/gsl-1.9/doc/roots.texi
@@ -0,0 +1,928 @@
+@cindex root finding
+@cindex zero finding
+@cindex finding roots
+@cindex finding zeros
+@cindex roots
+@cindex solving a nonlinear equation
+@cindex nonlinear equation, solutions of
+
+This chapter describes routines for finding roots of arbitrary
+one-dimensional functions.  The library provides low level components
+for a variety of iterative solvers and convergence tests.  These can be
+combined by the user to achieve the desired solution, with full access
+to the intermediate steps of the iteration.  Each class of methods uses
+the same framework, so that you can switch between solvers at runtime
+without needing to recompile your program.  Each instance of a solver
+keeps track of its own state, allowing the solvers to be used in
+multi-threaded programs.
+
+The header file @file{gsl_roots.h} contains prototypes for the root
+finding functions and related declarations.
+
+@menu
+* Root Finding Overview::       
+* Root Finding Caveats::        
+* Initializing the Solver::     
+* Providing the function to solve::  
+* Search Bounds and Guesses::   
+* Root Finding Iteration::      
+* Search Stopping Parameters::  
+* Root Bracketing Algorithms::  
+* Root Finding Algorithms using Derivatives::  
+* Root Finding Examples::       
+* Root Finding References and Further Reading::  
+@end menu
+
+@node Root Finding Overview
+@section Overview
+@cindex root finding, overview
+
+One-dimensional root finding algorithms can be divided into two classes,
+@dfn{root bracketing} and @dfn{root polishing}.  Algorithms which proceed
+by bracketing a root are guaranteed to converge.  Bracketing algorithms
+begin with a bounded region known to contain a root.  The size of this
+bounded region is reduced, iteratively, until it encloses the root to a
+desired tolerance.  This provides a rigorous error estimate for the
+location of the root.
+
+The technique of @dfn{root polishing} attempts to improve an initial
+guess to the root.  These algorithms converge only if started ``close
+enough'' to a root, and sacrifice a rigorous error bound for speed.  By
+approximating the behavior of a function in the vicinity of a root they
+attempt to find a higher order improvement of an initial guess.  When the
+behavior of the function is compatible with the algorithm and a good
+initial guess is available a polishing algorithm can provide rapid
+convergence.
+
+In GSL both types of algorithm are available in similar frameworks.  The
+user provides a high-level driver for the algorithms, and the library
+provides the individual functions necessary for each of the steps.
+There are three main phases of the iteration.  The steps are,
+
+@itemize @bullet
+@item
+initialize solver state, @var{s}, for algorithm @var{T}
+
+@item
+update @var{s} using the iteration @var{T}
+
+@item
+test @var{s} for convergence, and repeat iteration if necessary
+@end itemize
+
+@noindent
+The state for bracketing solvers is held in a @code{gsl_root_fsolver}
+struct.  The updating procedure uses only function evaluations (not
+derivatives).  The state for root polishing solvers is held in a
+@code{gsl_root_fdfsolver} struct.  The updates require both the function
+and its derivative (hence the name @code{fdf}) to be supplied by the
+user.
+
+@node Root Finding Caveats
+@section Caveats
+@cindex root finding, caveats
+
+Note that root finding functions can only search for one root at a time.
+When there are several roots in the search area, the first root to be
+found will be returned; however it is difficult to predict which of the
+roots this will be. @emph{In most cases, no error will be reported if
+you try to find a root in an area where there is more than one.}
+
+Care must be taken when a function may have a multiple root (such as 
+@c{$f(x) = (x-x_0)^2$}
+@math{f(x) = (x-x_0)^2} or 
+@c{$f(x) = (x-x_0)^3$}
+@math{f(x) = (x-x_0)^3}).  
+It is not possible to use root-bracketing algorithms on
+even-multiplicity roots.  For these algorithms the initial interval must
+contain a zero-crossing, where the function is negative at one end of
+the interval and positive at the other end.  Roots with even-multiplicity
+do not cross zero, but only touch it instantaneously.  Algorithms based
+on root bracketing will still work for odd-multiplicity roots
+(e.g. cubic, quintic, @dots{}). 
+Root polishing algorithms generally work with higher multiplicity roots,
+but at a reduced rate of convergence.  In these cases the @dfn{Steffenson
+algorithm} can be used to accelerate the convergence of multiple roots.
+
+While it is not absolutely required that @math{f} have a root within the
+search region, numerical root finding functions should not be used
+haphazardly to check for the @emph{existence} of roots.  There are better
+ways to do this.  Because it is easy to create situations where numerical
+root finders can fail, it is a bad idea to throw a root finder at a
+function you do not know much about.  In general it is best to examine
+the function visually by plotting before searching for a root.
+
+@node Initializing the Solver
+@section Initializing the Solver
+
+@deftypefun {gsl_root_fsolver *} gsl_root_fsolver_alloc (const gsl_root_fsolver_type * @var{T})
+This function returns a pointer to a newly allocated instance of a
+solver of type @var{T}.  For example, the following code creates an
+instance of a bisection solver,
+
+@example
+const gsl_root_fsolver_type * T 
+  = gsl_root_fsolver_bisection;
+gsl_root_fsolver * s 
+  = gsl_root_fsolver_alloc (T);
+@end example
+
+If there is insufficient memory to create the solver then the function
+returns a null pointer and the error handler is invoked with an error
+code of @code{GSL_ENOMEM}.
+@end deftypefun
+
+@deftypefun {gsl_root_fdfsolver *} gsl_root_fdfsolver_alloc (const gsl_root_fdfsolver_type * @var{T})
+This function returns a pointer to a newly allocated instance of a
+derivative-based solver of type @var{T}.  For example, the following
+code creates an instance of a Newton-Raphson solver,
+
+@example
+const gsl_root_fdfsolver_type * T 
+  = gsl_root_fdfsolver_newton;
+gsl_root_fdfsolver * s 
+  = gsl_root_fdfsolver_alloc (T);
+@end example
+
+If there is insufficient memory to create the solver then the function
+returns a null pointer and the error handler is invoked with an error
+code of @code{GSL_ENOMEM}.
+@end deftypefun
+
+
+@deftypefun int gsl_root_fsolver_set (gsl_root_fsolver * @var{s}, gsl_function * @var{f}, double @var{x_lower}, double @var{x_upper})
+This function initializes, or reinitializes, an existing solver @var{s}
+to use the function @var{f} and the initial search interval
+[@var{x_lower}, @var{x_upper}].
+@end deftypefun
+
+@deftypefun int gsl_root_fdfsolver_set (gsl_root_fdfsolver * @var{s}, gsl_function_fdf * @var{fdf}, double @var{root})
+This function initializes, or reinitializes, an existing solver @var{s}
+to use the function and derivative @var{fdf} and the initial guess
+@var{root}.
+@end deftypefun
+
+@deftypefun void gsl_root_fsolver_free (gsl_root_fsolver * @var{s})
+@deftypefunx void gsl_root_fdfsolver_free (gsl_root_fdfsolver * @var{s})
+These functions free all the memory associated with the solver @var{s}.
+@end deftypefun
+
+@deftypefun {const char *} gsl_root_fsolver_name (const gsl_root_fsolver * @var{s})
+@deftypefunx {const char *} gsl_root_fdfsolver_name (const gsl_root_fdfsolver * @var{s})
+These functions return a pointer to the name of the solver.  For example,
+
+@example
+printf ("s is a '%s' solver\n",
+        gsl_root_fsolver_name (s));
+@end example
+
+@noindent
+would print something like @code{s is a 'bisection' solver}.
+@end deftypefun
+
+@node Providing the function to solve
+@section Providing the function to solve
+@cindex root finding, providing a function to solve
+
+You must provide a continuous function of one variable for the root
+finders to operate on, and, sometimes, its first derivative.  In order
+to allow for general parameters the functions are defined by the
+following data types:
+
+@deftp {Data Type} gsl_function 
+This data type defines a general function with parameters. 
+
+@table @code
+@item double (* function) (double @var{x}, void * @var{params})
+this function should return the value
+@c{$f(x,\hbox{\it params})$}
+@math{f(x,params)} for argument @var{x} and parameters @var{params}
+
+@item void * params
+a pointer to the parameters of the function
+@end table
+@end deftp
+
+Here is an example for the general quadratic function,
+@tex
+\beforedisplay
+$$
+f(x) = a x^2 + b x + c
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+f(x) = a x^2 + b x + c
+@end example
+
+@end ifinfo
+@noindent
+with @math{a = 3}, @math{b = 2}, @math{c = 1}.  The following code
+defines a @code{gsl_function} @code{F} which you could pass to a root
+finder:
+
+@example
+struct my_f_params @{ double a; double b; double c; @};
+
+double
+my_f (double x, void * p) @{
+   struct my_f_params * params 
+     = (struct my_f_params *)p;
+   double a = (params->a);
+   double b = (params->b);
+   double c = (params->c);
+
+   return  (a * x + b) * x + c;
+@}
+
+gsl_function F;
+struct my_f_params params = @{ 3.0, 2.0, 1.0 @};
+
+F.function = &my_f;
+F.params = &params;
+@end example
+
+@noindent
+The function @math{f(x)} can be evaluated using the following macro,
+
+@example
+#define GSL_FN_EVAL(F,x) 
+    (*((F)->function))(x,(F)->params)
+@end example
+
+@deftp {Data Type} gsl_function_fdf
+This data type defines a general function with parameters and its first
+derivative.
+
+@table @code
+@item double (* f) (double @var{x}, void * @var{params})
+this function should return the value of
+@c{$f(x,\hbox{\it params})$}
+@math{f(x,params)} for argument @var{x} and parameters @var{params}
+
+@item double (* df) (double @var{x}, void * @var{params})
+this function should return the value of the derivative of @var{f} with
+respect to @var{x},
+@c{$f'(x,\hbox{\it params})$}
+@math{f'(x,params)}, for argument @var{x} and parameters @var{params}
+
+@item void (* fdf) (double @var{x}, void * @var{params}, double * @var{f}, double * @var{d}f)
+this function should set the values of the function @var{f} to 
+@c{$f(x,\hbox{\it params})$}
+@math{f(x,params)}
+and its derivative @var{df} to
+@c{$f'(x,\hbox{\it params})$}
+@math{f'(x,params)} 
+for argument @var{x} and parameters @var{params}.  This function
+provides an optimization of the separate functions for @math{f(x)} and
+@math{f'(x)}---it is always faster to compute the function and its
+derivative at the same time.
+
+@item void * params
+a pointer to the parameters of the function
+@end table
+@end deftp
+
+Here is an example where 
+@c{$f(x) = \exp(2x)$}
+@math{f(x) = 2\exp(2x)}:
+
+@example
+double
+my_f (double x, void * params)
+@{
+   return exp (2 * x);
+@}
+
+double
+my_df (double x, void * params)
+@{
+   return 2 * exp (2 * x);
+@}
+
+void
+my_fdf (double x, void * params, 
+        double * f, double * df)
+@{
+   double t = exp (2 * x);
+
+   *f = t;
+   *df = 2 * t;   /* uses existing value */
+@}
+
+gsl_function_fdf FDF;
+
+FDF.f = &my_f;
+FDF.df = &my_df;
+FDF.fdf = &my_fdf;
+FDF.params = 0;
+@end example
+
+@noindent
+The function @math{f(x)} can be evaluated using the following macro,
+
+@example
+#define GSL_FN_FDF_EVAL_F(FDF,x) 
+     (*((FDF)->f))(x,(FDF)->params)
+@end example
+
+@noindent
+The derivative @math{f'(x)} can be evaluated using the following macro,
+
+@example
+#define GSL_FN_FDF_EVAL_DF(FDF,x) 
+     (*((FDF)->df))(x,(FDF)->params)
+@end example
+
+@noindent
+and both the function @math{y = f(x)} and its derivative @math{dy = f'(x)}
+can be evaluated at the same time using the following macro,
+
+@example
+#define GSL_FN_FDF_EVAL_F_DF(FDF,x,y,dy) 
+     (*((FDF)->fdf))(x,(FDF)->params,(y),(dy))
+@end example
+
+@noindent
+The macro stores @math{f(x)} in its @var{y} argument and @math{f'(x)} in
+its @var{dy} argument---both of these should be pointers to
+@code{double}.
+
+@node Search Bounds and Guesses
+@section Search Bounds and Guesses
+@cindex root finding, search bounds
+@cindex root finding, initial guess
+
+You provide either search bounds or an initial guess; this section
+explains how search bounds and guesses work and how function arguments
+control them.
+
+A guess is simply an @math{x} value which is iterated until it is within
+the desired precision of a root.  It takes the form of a @code{double}.
+
+Search bounds are the endpoints of a interval which is iterated until
+the length of the interval is smaller than the requested precision.  The
+interval is defined by two values, the lower limit and the upper limit.
+Whether the endpoints are intended to be included in the interval or not
+depends on the context in which the interval is used.
+
+@node Root Finding Iteration
+@section Iteration
+
+The following functions drive the iteration of each algorithm.  Each
+function performs one iteration to update the state of any solver of the
+corresponding type.  The same functions work for all solvers so that
+different methods can be substituted at runtime without modifications to
+the code.
+
+@deftypefun int gsl_root_fsolver_iterate (gsl_root_fsolver * @var{s})
+@deftypefunx int gsl_root_fdfsolver_iterate (gsl_root_fdfsolver * @var{s})
+These functions perform a single iteration of the solver @var{s}.  If the
+iteration encounters an unexpected problem then an error code will be
+returned,
+
+@table @code
+@item GSL_EBADFUNC
+the iteration encountered a singular point where the function or its
+derivative evaluated to @code{Inf} or @code{NaN}.
+
+@item GSL_EZERODIV
+the derivative of the function vanished at the iteration point,
+preventing the algorithm from continuing without a division by zero.
+@end table
+@end deftypefun
+
+The solver maintains a current best estimate of the root at all
+times.  The bracketing solvers also keep track of the current best
+interval bounding the root.  This information can be accessed with the
+following auxiliary functions,
+
+@deftypefun double gsl_root_fsolver_root (const gsl_root_fsolver * @var{s})
+@deftypefunx double gsl_root_fdfsolver_root (const gsl_root_fdfsolver * @var{s})
+These functions return the current estimate of the root for the solver @var{s}.
+@end deftypefun
+
+@deftypefun double gsl_root_fsolver_x_lower (const gsl_root_fsolver * @var{s})
+@deftypefunx double gsl_root_fsolver_x_upper (const gsl_root_fsolver * @var{s})
+These functions return the current bracketing interval for the solver @var{s}.
+@end deftypefun
+
+@node Search Stopping Parameters
+@section Search Stopping Parameters
+@cindex root finding, stopping parameters
+
+A root finding procedure should stop when one of the following conditions is
+true:
+
+@itemize @bullet
+@item
+A root has been found to within the user-specified precision.
+
+@item
+A user-specified maximum number of iterations has been reached.
+
+@item
+An error has occurred.
+@end itemize
+
+@noindent
+The handling of these conditions is under user control.  The functions
+below allow the user to test the precision of the current result in
+several standard ways.
+
+@deftypefun int gsl_root_test_interval (double @var{x_lower}, double @var{x_upper}, double @var{epsabs}, double @var{epsrel})
+This function tests for the convergence of the interval [@var{x_lower},
+@var{x_upper}] with absolute error @var{epsabs} and relative error
+@var{epsrel}.  The test returns @code{GSL_SUCCESS} if the following
+condition is achieved,
+@tex
+\beforedisplay
+$$
+|a - b| < \hbox{\it epsabs} + \hbox{\it epsrel\/}\, \min(|a|,|b|)
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+|a - b| < epsabs + epsrel min(|a|,|b|) 
+@end example
+
+@end ifinfo
+@noindent
+when the interval @math{x = [a,b]} does not include the origin.  If the
+interval includes the origin then @math{\min(|a|,|b|)} is replaced by
+zero (which is the minimum value of @math{|x|} over the interval).  This
+ensures that the relative error is accurately estimated for roots close
+to the origin.
+
+This condition on the interval also implies that any estimate of the
+root @math{r} in the interval satisfies the same condition with respect
+to the true root @math{r^*},
+@tex
+\beforedisplay
+$$
+|r - r^*| < \hbox{\it epsabs} + \hbox{\it epsrel\/}\, r^*
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+|r - r^*| < epsabs + epsrel r^*
+@end example
+
+@end ifinfo
+@noindent
+assuming that the true root @math{r^*} is contained within the interval.
+@end deftypefun
+
+@deftypefun int gsl_root_test_delta (double @var{x1}, double @var{x0}, double @var{epsabs}, double @var{epsrel})
+
+This function tests for the convergence of the sequence @dots{}, @var{x0},
+@var{x1} with absolute error @var{epsabs} and relative error
+@var{epsrel}.  The test returns @code{GSL_SUCCESS} if the following
+condition is achieved,
+@tex
+\beforedisplay
+$$
+|x_1 - x_0| < \hbox{\it epsabs} + \hbox{\it epsrel\/}\, |x_1|
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+|x_1 - x_0| < epsabs + epsrel |x_1|
+@end example
+
+@end ifinfo
+@noindent
+and returns @code{GSL_CONTINUE} otherwise.
+@end deftypefun
+
+
+@deftypefun int gsl_root_test_residual (double @var{f}, double @var{epsabs})
+This function tests the residual value @var{f} against the absolute
+error bound @var{epsabs}.  The test returns @code{GSL_SUCCESS} if the
+following condition is achieved,
+@tex
+\beforedisplay
+$$
+|f| < \hbox{\it epsabs}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+|f| < epsabs
+@end example
+
+@end ifinfo
+@noindent
+and returns @code{GSL_CONTINUE} otherwise.  This criterion is suitable
+for situations where the precise location of the root, @math{x}, is
+unimportant provided a value can be found where the residual,
+@math{|f(x)|}, is small enough.
+@end deftypefun
+
+@comment ============================================================
+
+@node Root Bracketing Algorithms
+@section Root Bracketing Algorithms
+
+The root bracketing algorithms described in this section require an
+initial interval which is guaranteed to contain a root---if @math{a}
+and @math{b} are the endpoints of the interval then @math{f(a)} must
+differ in sign from @math{f(b)}.  This ensures that the function crosses
+zero at least once in the interval.  If a valid initial interval is used
+then these algorithm cannot fail, provided the function is well-behaved.
+
+Note that a bracketing algorithm cannot find roots of even degree, since
+these do not cross the @math{x}-axis.
+
+@deffn {Solver} gsl_root_fsolver_bisection
+
+@cindex bisection algorithm for finding roots
+@cindex root finding, bisection algorithm
+
+The @dfn{bisection algorithm} is the simplest method of bracketing the
+roots of a function.   It is the slowest algorithm provided by
+the library, with linear convergence.
+
+On each iteration, the interval is bisected and the value of the
+function at the midpoint is calculated.  The sign of this value is used
+to determine which half of the interval does not contain a root.  That
+half is discarded to give a new, smaller interval containing the
+root.  This procedure can be continued indefinitely until the interval is
+sufficiently small.
+
+At any time the current estimate of the root is taken as the midpoint of
+the interval.
+
+@comment eps file "roots-bisection.eps"
+@comment @iftex
+@comment @sp 1
+@comment @center @image{roots-bisection,3.4in}
+
+@comment @quotation
+@comment Four iterations of bisection, where @math{a_n} is @math{n}th position of
+@comment the beginning of the interval and @math{b_n} is the @math{n}th position
+@comment of the end.  The midpoint of each interval is also indicated.
+@comment @end quotation
+@comment @end iftex
+@end deffn
+
+@comment ============================================================
+
+@deffn {Solver} gsl_root_fsolver_falsepos
+@cindex false position algorithm for finding roots
+@cindex root finding, false position algorithm
+
+The @dfn{false position algorithm} is a method of finding roots based on
+linear interpolation.  Its convergence is linear, but it is usually
+faster than bisection.
+
+On each iteration a line is drawn between the endpoints @math{(a,f(a))}
+and @math{(b,f(b))} and the point where this line crosses the
+@math{x}-axis taken as a ``midpoint''.  The value of the function at
+this point is calculated and its sign is used to determine which side of
+the interval does not contain a root.  That side is discarded to give a
+new, smaller interval containing the root.  This procedure can be
+continued indefinitely until the interval is sufficiently small.
+
+The best estimate of the root is taken from the linear interpolation of
+the interval on the current iteration.
+
+@comment eps file "roots-false-position.eps"
+@comment @iftex
+@comment @image{roots-false-position,4in}
+@comment @quotation
+@comment Several iterations of false position, where @math{a_n} is @math{n}th
+@comment position of the beginning of the interval and @math{b_n} is the
+@comment @math{n}th position of the end.
+@comment @end quotation
+@comment @end iftex
+@end deffn
+
+@comment ============================================================
+
+@deffn {Solver} gsl_root_fsolver_brent
+@cindex Brent's method for finding roots
+@cindex root finding, Brent's method
+
+The @dfn{Brent-Dekker method} (referred to here as @dfn{Brent's method})
+combines an interpolation strategy with the bisection algorithm.  This
+produces a fast algorithm which is still robust.
+
+On each iteration Brent's method approximates the function using an
+interpolating curve.  On the first iteration this is a linear
+interpolation of the two endpoints.  For subsequent iterations the
+algorithm uses an inverse quadratic fit to the last three points, for
+higher accuracy.  The intercept of the interpolating curve with the
+@math{x}-axis is taken as a guess for the root.  If it lies within the
+bounds of the current interval then the interpolating point is accepted,
+and used to generate a smaller interval.  If the interpolating point is
+not accepted then the algorithm falls back to an ordinary bisection
+step.
+
+The best estimate of the root is taken from the most recent
+interpolation or bisection.
+@end deffn
+
+@comment ============================================================
+
+@node Root Finding Algorithms using Derivatives
+@section Root Finding Algorithms using Derivatives
+
+The root polishing algorithms described in this section require an
+initial guess for the location of the root.  There is no absolute
+guarantee of convergence---the function must be suitable for this
+technique and the initial guess must be sufficiently close to the root
+for it to work.  When these conditions are satisfied then convergence is
+quadratic.
+
+These algorithms make use of both the function and its derivative. 
+
+@deffn {Derivative Solver} gsl_root_fdfsolver_newton
+@cindex Newton's method for finding roots
+@cindex root finding, Newton's method
+
+Newton's Method is the standard root-polishing algorithm.  The algorithm
+begins with an initial guess for the location of the root.  On each
+iteration, a line tangent to the function @math{f} is drawn at that
+position.  The point where this line crosses the @math{x}-axis becomes
+the new guess.  The iteration is defined by the following sequence,
+@tex
+\beforedisplay
+$$
+x_{i+1} = x_i - {f(x_i) \over f'(x_i)}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+x_@{i+1@} = x_i - f(x_i)/f'(x_i)
+@end example
+
+@end ifinfo
+@noindent
+Newton's method converges quadratically for single roots, and linearly
+for multiple roots.
+
+@comment eps file "roots-newtons-method.eps"
+@comment @iftex
+@comment @sp 1
+@comment @center @image{roots-newtons-method,3.4in}
+
+@comment @quotation
+@comment Several iterations of Newton's Method, where @math{g_n} is the
+@comment @math{n}th guess.
+@comment @end quotation
+@comment @end iftex
+@end deffn
+
+@comment ============================================================
+
+@deffn {Derivative Solver} gsl_root_fdfsolver_secant
+@cindex secant method for finding roots
+@cindex root finding, secant method
+
+The @dfn{secant method} is a simplified version of Newton's method which does
+not require the computation of the derivative on every step.
+
+On its first iteration the algorithm begins with Newton's method, using
+the derivative to compute a first step,
+@tex
+\beforedisplay
+$$
+x_1 = x_0 - {f(x_0) \over f'(x_0)}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+x_1 = x_0 - f(x_0)/f'(x_0)
+@end example
+
+@end ifinfo
+@noindent
+Subsequent iterations avoid the evaluation of the derivative by
+replacing it with a numerical estimate, the slope of the line through
+the previous two points,
+@tex
+\beforedisplay
+$$
+x_{i+1} = x_i - {f(x_i) \over f'_{est}}
+ ~\hbox{where}~
+ f'_{est} =  {f(x_{i}) - f(x_{i-1}) \over x_i - x_{i-1}}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+x_@{i+1@} = x_i f(x_i) / f'_@{est@} where
+ f'_@{est@} = (f(x_i) - f(x_@{i-1@})/(x_i - x_@{i-1@})
+@end example
+
+@end ifinfo
+@noindent
+When the derivative does not change significantly in the vicinity of the
+root the secant method gives a useful saving.  Asymptotically the secant
+method is faster than Newton's method whenever the cost of evaluating
+the derivative is more than 0.44 times the cost of evaluating the
+function itself.  As with all methods of computing a numerical
+derivative the estimate can suffer from cancellation errors if the
+separation of the points becomes too small.
+
+On single roots, the method has a convergence of order @math{(1 + \sqrt
+5)/2} (approximately @math{1.62}).  It converges linearly for multiple
+roots.  
+
+@comment eps file "roots-secant-method.eps"
+@comment @iftex
+@comment @tex
+@comment \input epsf
+@comment \medskip
+@comment \centerline{\epsfxsize=5in\epsfbox{roots-secant-method.eps}}
+@comment @end tex
+@comment @quotation
+@comment Several iterations of Secant Method, where @math{g_n} is the @math{n}th
+@comment guess.
+@comment @end quotation
+@comment @end iftex
+@end deffn
+
+@comment ============================================================
+
+@deffn {Derivative Solver} gsl_root_fdfsolver_steffenson
+@cindex Steffenson's method for finding roots
+@cindex root finding, Steffenson's method
+
+The @dfn{Steffenson Method} provides the fastest convergence of all the
+routines.  It combines the basic Newton algorithm with an Aitken
+``delta-squared'' acceleration.  If the Newton iterates are @math{x_i}
+then the acceleration procedure generates a new sequence @math{R_i},
+@tex
+\beforedisplay
+$$
+R_i = x_i - {(x_{i+1} - x_i)^2 \over (x_{i+2} - 2 x_{i+1} + x_i)}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+R_i = x_i - (x_@{i+1@} - x_i)^2 / (x_@{i+2@} - 2 x_@{i+1@} + x_@{i@})
+@end example
+
+@end ifinfo
+@noindent
+which converges faster than the original sequence under reasonable
+conditions.  The new sequence requires three terms before it can produce
+its first value so the method returns accelerated values on the second
+and subsequent iterations.  On the first iteration it returns the
+ordinary Newton estimate.  The Newton iterate is also returned if the
+denominator of the acceleration term ever becomes zero.
+
+As with all acceleration procedures this method can become unstable if
+the function is not well-behaved. 
+@end deffn
+
+@node Root Finding Examples
+@section Examples
+
+For any root finding algorithm we need to prepare the function to be
+solved.  For this example we will use the general quadratic equation
+described earlier.  We first need a header file (@file{demo_fn.h}) to
+define the function parameters,
+
+@example
+@verbatiminclude examples/demo_fn.h
+@end example
+
+@noindent
+We place the function definitions in a separate file (@file{demo_fn.c}),
+
+@example
+@verbatiminclude examples/demo_fn.c
+@end example
+
+@noindent
+The first program uses the function solver @code{gsl_root_fsolver_brent}
+for Brent's method and the general quadratic defined above to solve the
+following equation,
+@tex
+\beforedisplay
+$$
+x^2 - 5 = 0
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+x^2 - 5 = 0
+@end example
+
+@end ifinfo
+@noindent
+with solution @math{x = \sqrt 5 = 2.236068...}
+
+@example
+@verbatiminclude examples/roots.c
+@end example
+
+@noindent
+Here are the results of the iterations,
+
+@smallexample
+$ ./a.out 
+using brent method
+ iter [    lower,     upper]      root        err  err(est)
+    1 [1.0000000, 5.0000000] 1.0000000 -1.2360680 4.0000000
+    2 [1.0000000, 3.0000000] 3.0000000 +0.7639320 2.0000000
+    3 [2.0000000, 3.0000000] 2.0000000 -0.2360680 1.0000000
+    4 [2.2000000, 3.0000000] 2.2000000 -0.0360680 0.8000000
+    5 [2.2000000, 2.2366300] 2.2366300 +0.0005621 0.0366300
+Converged:                            
+    6 [2.2360634, 2.2366300] 2.2360634 -0.0000046 0.0005666
+@end smallexample
+
+@noindent
+If the program is modified to use the bisection solver instead of
+Brent's method, by changing @code{gsl_root_fsolver_brent} to
+@code{gsl_root_fsolver_bisection} the slower convergence of the
+Bisection method can be observed,
+
+@smallexample
+$ ./a.out 
+using bisection method
+ iter [    lower,     upper]      root        err  err(est)
+    1 [0.0000000, 2.5000000] 1.2500000 -0.9860680 2.5000000
+    2 [1.2500000, 2.5000000] 1.8750000 -0.3610680 1.2500000
+    3 [1.8750000, 2.5000000] 2.1875000 -0.0485680 0.6250000
+    4 [2.1875000, 2.5000000] 2.3437500 +0.1076820 0.3125000
+    5 [2.1875000, 2.3437500] 2.2656250 +0.0295570 0.1562500
+    6 [2.1875000, 2.2656250] 2.2265625 -0.0095055 0.0781250
+    7 [2.2265625, 2.2656250] 2.2460938 +0.0100258 0.0390625
+    8 [2.2265625, 2.2460938] 2.2363281 +0.0002601 0.0195312
+    9 [2.2265625, 2.2363281] 2.2314453 -0.0046227 0.0097656
+   10 [2.2314453, 2.2363281] 2.2338867 -0.0021813 0.0048828
+   11 [2.2338867, 2.2363281] 2.2351074 -0.0009606 0.0024414
+Converged:                            
+   12 [2.2351074, 2.2363281] 2.2357178 -0.0003502 0.0012207
+@end smallexample
+
+The next program solves the same function using a derivative solver
+instead.
+
+@example
+@verbatiminclude examples/rootnewt.c
+@end example
+
+@noindent
+Here are the results for Newton's method,
+
+@example
+$ ./a.out 
+using newton method
+iter        root        err   err(est)
+    1  3.0000000 +0.7639320 -2.0000000
+    2  2.3333333 +0.0972654 -0.6666667
+    3  2.2380952 +0.0020273 -0.0952381
+Converged:      
+    4  2.2360689 +0.0000009 -0.0020263
+@end example
+
+@noindent
+Note that the error can be estimated more accurately by taking the
+difference between the current iterate and next iterate rather than the
+previous iterate.  The other derivative solvers can be investigated by
+changing @code{gsl_root_fdfsolver_newton} to
+@code{gsl_root_fdfsolver_secant} or
+@code{gsl_root_fdfsolver_steffenson}.
+
+@node Root Finding References and Further Reading
+@section References and Further Reading
+
+For information on the Brent-Dekker algorithm see the following two
+papers,
+
+@itemize @asis
+@item
+R. P. Brent, ``An algorithm with guaranteed convergence for finding a
+zero of a function'', @cite{Computer Journal}, 14 (1971) 422--425
+
+@item
+J. C. P. Bus and T. J. Dekker, ``Two Efficient Algorithms with Guaranteed
+Convergence for Finding a Zero of a Function'', @cite{ACM Transactions of
+Mathematical Software}, Vol.@: 1 No.@: 4 (1975) 330--345
+@end itemize
+
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+%%PageTrailer
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+showpage
+
+%%Trailer
+end
+%%EOF
diff --git a/gsl-1.9/doc/siman.texi b/gsl-1.9/doc/siman.texi
new file mode 100644
index 0000000..902cad7
--- /dev/null
+++ b/gsl-1.9/doc/siman.texi
@@ -0,0 +1,371 @@
+@cindex simulated annealing
+@cindex combinatorial optimization
+@cindex optimization, combinatorial
+@cindex energy function
+@cindex cost function
+Stochastic search techniques are used when the structure of a space is
+not well understood or is not smooth, so that techniques like Newton's
+method (which requires calculating Jacobian derivative matrices) cannot
+be used. In particular, these techniques are frequently used to solve
+combinatorial optimization problems, such as the traveling salesman
+problem.
+
+The goal is to find a point in the space at which a real valued
+@dfn{energy function} (or @dfn{cost function}) is minimized.  Simulated
+annealing is a minimization technique which has given good results in
+avoiding local minima; it is based on the idea of taking a random walk
+through the space at successively lower temperatures, where the
+probability of taking a step is given by a Boltzmann distribution.
+
+The functions described in this chapter are declared in the header file
+@file{gsl_siman.h}.
+
+@menu
+* Simulated Annealing algorithm::  
+* Simulated Annealing functions::  
+* Examples with Simulated Annealing::  
+* Simulated Annealing References and Further Reading::  
+@end menu
+
+@node Simulated Annealing algorithm
+@section Simulated Annealing algorithm
+
+The simulated annealing algorithm takes random walks through the problem
+space, looking for points with low energies; in these random walks, the
+probability of taking a step is determined by the Boltzmann distribution,
+@tex
+\beforedisplay
+$$
+p = e^{-(E_{i+1} - E_i)/(kT)}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+p = e^@{-(E_@{i+1@} - E_i)/(kT)@}
+@end example
+
+@end ifinfo
+@noindent
+if 
+@c{$E_{i+1} > E_i$}
+@math{E_@{i+1@} > E_i}, and 
+@math{p = 1} when 
+@c{$E_{i+1} \le E_i$}
+@math{E_@{i+1@} <= E_i}.
+
+In other words, a step will occur if the new energy is lower.  If
+the new energy is higher, the transition can still occur, and its
+likelihood is proportional to the temperature @math{T} and inversely
+proportional to the energy difference 
+@c{$E_{i+1} - E_i$}
+@math{E_@{i+1@} - E_i}.
+
+The temperature @math{T} is initially set to a high value, and a random
+walk is carried out at that temperature.  Then the temperature is
+lowered very slightly according to a @dfn{cooling schedule}, for
+example: @c{$T \rightarrow T/\mu_T$}
+@math{T -> T/mu_T}
+where @math{\mu_T} is slightly greater than 1. 
+@cindex cooling schedule
+@cindex schedule, cooling
+
+The slight probability of taking a step that gives higher energy is what
+allows simulated annealing to frequently get out of local minima.
+
+@node Simulated Annealing functions
+@section Simulated Annealing functions
+
+@deftypefun void gsl_siman_solve (const gsl_rng * @var{r}, void * @var{x0_p}, gsl_siman_Efunc_t @var{Ef}, gsl_siman_step_t @var{take_step}, gsl_siman_metric_t @var{distance}, gsl_siman_print_t @var{print_position}, gsl_siman_copy_t @var{copyfunc}, gsl_siman_copy_construct_t @var{copy_constructor}, gsl_siman_destroy_t @var{destructor}, size_t @var{element_size}, gsl_siman_params_t @var{params})
+
+This function performs a simulated annealing search through a given
+space.  The space is specified by providing the functions @var{Ef} and
+@var{distance}.  The simulated annealing steps are generated using the
+random number generator @var{r} and the function @var{take_step}.
+
+The starting configuration of the system should be given by @var{x0_p}.
+The routine offers two modes for updating configurations, a fixed-size
+mode and a variable-size mode.  In the fixed-size mode the configuration
+is stored as a single block of memory of size @var{element_size}.
+Copies of this configuration are created, copied and destroyed
+internally using the standard library functions @code{malloc},
+@code{memcpy} and @code{free}.  The function pointers @var{copyfunc},
+@var{copy_constructor} and @var{destructor} should be null pointers in
+fixed-size mode.  In the variable-size mode the functions
+@var{copyfunc}, @var{copy_constructor} and @var{destructor} are used to
+create, copy and destroy configurations internally.  The variable
+@var{element_size} should be zero in the variable-size mode.
+
+The @var{params} structure (described below) controls the run by
+providing the temperature schedule and other tunable parameters to the
+algorithm.
+
+On exit the best result achieved during the search is placed in
+@code{*@var{x0_p}}.  If the annealing process has been successful this
+should be a good approximation to the optimal point in the space.
+
+If the function pointer @var{print_position} is not null, a debugging
+log will be printed to @code{stdout} with the following columns:
+
+@example
+number_of_iterations temperature x x-(*x0_p) Ef(x)
+@end example
+
+and the output of the function @var{print_position} itself.  If
+@var{print_position} is null then no information is printed.
+@end deftypefun
+
+@noindent
+The simulated annealing routines require several user-specified
+functions to define the configuration space and energy function.  The
+prototypes for these functions are given below.
+
+@deftp {Data Type} gsl_siman_Efunc_t
+This function type should return the energy of a configuration @var{xp}.
+
+@example
+double (*gsl_siman_Efunc_t) (void *xp)
+@end example
+@end deftp
+
+@deftp {Data Type} gsl_siman_step_t
+This function type should modify the configuration @var{xp} using a random step
+taken from the generator @var{r}, up to a maximum distance of
+@var{step_size}.
+
+@example
+void (*gsl_siman_step_t) (const gsl_rng *r, void *xp, 
+                          double step_size)
+@end example
+@end deftp
+
+@deftp {Data Type} gsl_siman_metric_t
+This function type should return the distance between two configurations
+@var{xp} and @var{yp}.
+
+@example
+double (*gsl_siman_metric_t) (void *xp, void *yp)
+@end example
+@end deftp
+
+@deftp {Data Type} gsl_siman_print_t
+This function type should print the contents of the configuration @var{xp}.
+
+@example
+void (*gsl_siman_print_t) (void *xp)
+@end example
+@end deftp
+
+@deftp {Data Type} gsl_siman_copy_t
+This function type should copy the configuration @var{source} into @var{dest}.
+
+@example
+void (*gsl_siman_copy_t) (void *source, void *dest)
+@end example
+@end deftp
+
+@deftp {Data Type} gsl_siman_copy_construct_t
+This function type should create a new copy of the configuration @var{xp}.
+
+@example
+void * (*gsl_siman_copy_construct_t) (void *xp)
+@end example
+@end deftp
+
+@deftp {Data Type} gsl_siman_destroy_t
+This function type should destroy the configuration @var{xp}, freeing its
+memory.
+
+@example
+void (*gsl_siman_destroy_t) (void *xp)
+@end example
+@end deftp
+
+@deftp {Data Type} gsl_siman_params_t
+These are the parameters that control a run of @code{gsl_siman_solve}.
+This structure contains all the information needed to control the
+search, beyond the energy function, the step function and the initial
+guess.
+
+@table @code
+@item int n_tries          
+The number of points to try for each step.
+
+@item int iters_fixed_T    
+The number of iterations at each temperature.
+
+@item double step_size     
+The maximum step size in the random walk.
+
+@item double k, t_initial, mu_t, t_min
+The parameters of the Boltzmann distribution and cooling schedule.
+@end table
+@end deftp
+
+
+@node Examples with Simulated Annealing
+@section Examples
+
+The simulated annealing package is clumsy, and it has to be because it
+is written in C, for C callers, and tries to be polymorphic at the same
+time.  But here we provide some examples which can be pasted into your
+application with little change and should make things easier.
+
+@menu
+* Trivial example::             
+* Traveling Salesman Problem::  
+@end menu
+
+@node Trivial example
+@subsection Trivial example
+
+The first example, in one dimensional cartesian space, sets up an energy
+function which is a damped sine wave; this has many local minima, but
+only one global minimum, somewhere between 1.0 and 1.5.  The initial
+guess given is 15.5, which is several local minima away from the global
+minimum.
+
+@smallexample
+@verbatiminclude examples/siman.c
+@end smallexample
+
+@need 2000
+Here are a couple of plots that are generated by running
+@code{siman_test} in the following way:
+
+@example
+$ ./siman_test | awk '!/^#/ @{print $1, $4@}' 
+ | graph -y 1.34 1.4 -W0 -X generation -Y position 
+ | plot -Tps > siman-test.eps
+$ ./siman_test | awk '!/^#/ @{print $1, $4@}' 
+ | graph -y -0.88 -0.83 -W0 -X generation -Y energy 
+ | plot -Tps > siman-energy.eps
+@end example
+
+@iftex
+@sp 1
+@center @image{siman-test,2.8in} 
+@center @image{siman-energy,2.8in}
+
+@quotation
+Example of a simulated annealing run: at higher temperatures (early in
+the plot) you see that the solution can fluctuate, but at lower
+temperatures it converges.
+@end quotation
+@end iftex
+
+@node Traveling Salesman Problem
+@subsection Traveling Salesman Problem
+@cindex TSP
+@cindex traveling salesman problem
+
+The TSP (@dfn{Traveling Salesman Problem}) is the classic combinatorial
+optimization problem.  I have provided a very simple version of it,
+based on the coordinates of twelve cities in the southwestern United
+States.  This should maybe be called the @dfn{Flying Salesman Problem},
+since I am using the great-circle distance between cities, rather than
+the driving distance.  Also: I assume the earth is a sphere, so I don't
+use geoid distances.
+
+The @code{gsl_siman_solve} routine finds a route which is 3490.62
+Kilometers long; this is confirmed by an exhaustive search of all
+possible routes with the same initial city.
+
+The full code can be found in @file{siman/siman_tsp.c}, but I include
+here some plots generated in the following way:
+
+@smallexample
+$ ./siman_tsp > tsp.output
+$ grep -v "^#" tsp.output  
+ | awk '@{print $1, $NF@}'
+ | graph -y 3300 6500 -W0 -X generation -Y distance 
+    -L "TSP - 12 southwest cities"
+ | plot -Tps > 12-cities.eps
+$ grep initial_city_coord tsp.output 
+  | awk '@{print $2, $3@}' 
+  | graph -X "longitude (- means west)" -Y "latitude" 
+     -L "TSP - initial-order" -f 0.03 -S 1 0.1 
+  | plot -Tps > initial-route.eps
+$ grep final_city_coord tsp.output 
+  | awk '@{print $2, $3@}' 
+  | graph -X "longitude (- means west)" -Y "latitude" 
+     -L "TSP - final-order" -f 0.03 -S 1 0.1 
+  | plot -Tps > final-route.eps
+@end smallexample
+
+@noindent
+This is the output showing the initial order of the cities; longitude is
+negative, since it is west and I want the plot to look like a map.
+
+@smallexample
+# initial coordinates of cities (longitude and latitude)
+###initial_city_coord: -105.95 35.68 Santa Fe
+###initial_city_coord: -112.07 33.54 Phoenix
+###initial_city_coord: -106.62 35.12 Albuquerque
+###initial_city_coord: -103.2 34.41 Clovis
+###initial_city_coord: -107.87 37.29 Durango
+###initial_city_coord: -96.77 32.79 Dallas
+###initial_city_coord: -105.92 35.77 Tesuque
+###initial_city_coord: -107.84 35.15 Grants
+###initial_city_coord: -106.28 35.89 Los Alamos
+###initial_city_coord: -106.76 32.34 Las Cruces
+###initial_city_coord: -108.58 37.35 Cortez
+###initial_city_coord: -108.74 35.52 Gallup
+###initial_city_coord: -105.95 35.68 Santa Fe
+@end smallexample
+
+The optimal route turns out to be:
+
+@smallexample
+# final coordinates of cities (longitude and latitude)
+###final_city_coord: -105.95 35.68 Santa Fe
+###final_city_coord: -106.28 35.89 Los Alamos
+###final_city_coord: -106.62 35.12 Albuquerque
+###final_city_coord: -107.84 35.15 Grants
+###final_city_coord: -107.87 37.29 Durango
+###final_city_coord: -108.58 37.35 Cortez
+###final_city_coord: -108.74 35.52 Gallup
+###final_city_coord: -112.07 33.54 Phoenix
+###final_city_coord: -106.76 32.34 Las Cruces
+###final_city_coord: -96.77 32.79 Dallas
+###final_city_coord: -103.2 34.41 Clovis
+###final_city_coord: -105.92 35.77 Tesuque
+###final_city_coord: -105.95 35.68 Santa Fe
+@end smallexample
+
+@iftex
+@sp 1
+@center @image{initial-route,2.2in} 
+@center @image{final-route,2.2in}
+
+@quotation
+Initial and final (optimal) route for the 12 southwestern cities Flying
+Salesman Problem.
+@end quotation
+@end iftex
+
+@noindent
+Here's a plot of the cost function (energy) versus generation (point in
+the calculation at which a new temperature is set) for this problem:
+
+@iftex
+@sp 1
+@center @image{12-cities,2.8in}
+
+@quotation
+Example of a simulated annealing run for the 12 southwestern cities
+Flying Salesman Problem.
+@end quotation
+@end iftex
+
+@node Simulated Annealing References and Further Reading
+@section References and Further Reading
+
+Further information is available in the following book,
+
+@itemize @asis
+@item
+@cite{Modern Heuristic Techniques for Combinatorial Problems}, Colin R. Reeves
+(ed.), McGraw-Hill, 1995 (ISBN 0-07-709239-2).
+@end itemize
diff --git a/gsl-1.9/doc/sort.texi b/gsl-1.9/doc/sort.texi
new file mode 100644
index 0000000..3e5f216
--- /dev/null
+++ b/gsl-1.9/doc/sort.texi
@@ -0,0 +1,315 @@
+@cindex sorting 
+@cindex heapsort
+This chapter describes functions for sorting data, both directly and
+indirectly (using an index).  All the functions use the @dfn{heapsort}
+algorithm.  Heapsort is an @math{O(N \log N)} algorithm which operates
+in-place and does not require any additional storage.  It also provides
+consistent performance, the running time for its worst-case (ordered
+data) being not significantly longer than the average and best cases.
+Note that the heapsort algorithm does not preserve the relative ordering
+of equal elements---it is an @dfn{unstable} sort.  However the resulting
+order of equal elements will be consistent across different platforms
+when using these functions.
+
+@menu
+* Sorting objects::             
+* Sorting vectors::             
+* Selecting the k smallest or largest elements::  
+* Computing the rank::          
+* Sorting Examples::            
+* Sorting References and Further Reading::  
+@end menu
+
+@node Sorting objects
+@section Sorting objects
+
+The following function provides a simple alternative to the standard
+library function @code{qsort}.  It is intended for systems lacking
+@code{qsort}, not as a replacement for it.  The function @code{qsort}
+should be used whenever possible, as it will be faster and can provide
+stable ordering of equal elements.  Documentation for @code{qsort} is
+available in the @cite{GNU C Library Reference Manual}.
+
+The functions described in this section are defined in the header file
+@file{gsl_heapsort.h}.
+
+@cindex comparison functions, definition
+@deftypefun void gsl_heapsort (void * @var{array}, size_t @var{count}, size_t @var{size}, gsl_comparison_fn_t @var{compare})
+
+This function sorts the @var{count} elements of the array @var{array},
+each of size @var{size}, into ascending order using the comparison
+function @var{compare}.  The type of the comparison function is defined by,
+
+@example
+int (*gsl_comparison_fn_t) (const void * a,
+                            const void * b)
+@end example
+
+@noindent
+A comparison function should return a negative integer if the first
+argument is less than the second argument, @code{0} if the two arguments
+are equal and a positive integer if the first argument is greater than
+the second argument.
+
+For example, the following function can be used to sort doubles into
+ascending numerical order.
+
+@example
+int
+compare_doubles (const double * a,
+                 const double * b)
+@{
+    if (*a > *b)
+       return 1;
+    else if (*a < *b)
+       return -1;
+    else
+       return 0;
+@}
+@end example
+
+@noindent
+The appropriate function call to perform the sort is,
+
+@example
+gsl_heapsort (array, count, sizeof(double), 
+              compare_doubles);
+@end example
+
+Note that unlike @code{qsort} the heapsort algorithm cannot be made into
+a stable sort by pointer arithmetic.  The trick of comparing pointers for
+equal elements in the comparison function does not work for the heapsort
+algorithm.  The heapsort algorithm performs an internal rearrangement of
+the data which destroys its initial ordering.
+@end deftypefun
+
+@cindex indirect sorting
+@deftypefun int gsl_heapsort_index (size_t * @var{p}, const void * @var{array}, size_t @var{count}, size_t @var{size}, gsl_comparison_fn_t @var{compare})
+
+This function indirectly sorts the @var{count} elements of the array
+@var{array}, each of size @var{size}, into ascending order using the
+comparison function @var{compare}.  The resulting permutation is stored
+in @var{p}, an array of length @var{n}.  The elements of @var{p} give the
+index of the array element which would have been stored in that position
+if the array had been sorted in place.  The first element of @var{p}
+gives the index of the least element in @var{array}, and the last
+element of @var{p} gives the index of the greatest element in
+@var{array}.  The array itself is not changed.
+@end deftypefun
+
+@node Sorting vectors
+@section Sorting vectors
+
+The following functions will sort the elements of an array or vector,
+either directly or indirectly.  They are defined for all real and integer
+types using the normal suffix rules.  For example, the @code{float}
+versions of the array functions are @code{gsl_sort_float} and
+@code{gsl_sort_float_index}.  The corresponding vector functions are
+@code{gsl_sort_vector_float} and @code{gsl_sort_vector_float_index}.  The
+prototypes are available in the header files @file{gsl_sort_float.h}
+@file{gsl_sort_vector_float.h}.  The complete set of prototypes can be
+included using the header files @file{gsl_sort.h} and
+@file{gsl_sort_vector.h}.
+
+There are no functions for sorting complex arrays or vectors, since the
+ordering of complex numbers is not uniquely defined.  To sort a complex
+vector by magnitude compute a real vector containing the magnitudes
+of the complex elements, and sort this vector indirectly.  The resulting
+index gives the appropriate ordering of the original complex vector.
+
+@cindex sorting vector elements
+@cindex vector, sorting elements of
+@deftypefun void gsl_sort (double * @var{data}, size_t @var{stride}, size_t @var{n})
+This function sorts the @var{n} elements of the array @var{data} with
+stride @var{stride} into ascending numerical order.
+@end deftypefun
+
+@deftypefun void gsl_sort_vector (gsl_vector * @var{v})
+This function sorts the elements of the vector @var{v} into ascending
+numerical order.
+@end deftypefun
+
+@cindex indirect sorting, of vector elements
+@deftypefun void gsl_sort_index (size_t * @var{p}, const double * @var{data}, size_t @var{stride}, size_t @var{n})
+This function indirectly sorts the @var{n} elements of the array
+@var{data} with stride @var{stride} into ascending order, storing the
+resulting permutation in @var{p}.  The array @var{p} must be allocated with
+a sufficient length to store the @var{n} elements of the permutation.
+The elements of @var{p} give the index of the array element which would
+have been stored in that position if the array had been sorted in place.
+The array @var{data} is not changed.
+@end deftypefun
+
+@deftypefun int gsl_sort_vector_index (gsl_permutation * @var{p}, const gsl_vector * @var{v})
+This function indirectly sorts the elements of the vector @var{v} into
+ascending order, storing the resulting permutation in @var{p}.  The
+elements of @var{p} give the index of the vector element which would
+have been stored in that position if the vector had been sorted in
+place.  The first element of @var{p} gives the index of the least element
+in @var{v}, and the last element of @var{p} gives the index of the
+greatest element in @var{v}.  The vector @var{v} is not changed.
+@end deftypefun
+
+@node Selecting the k smallest or largest elements
+@section Selecting the k smallest or largest elements
+
+The functions described in this section select the @math{k} smallest
+or largest elements of a data set of size @math{N}.  The routines use an
+@math{O(kN)} direct insertion algorithm which is suited to subsets that
+are small compared with the total size of the dataset. For example, the
+routines are useful for selecting the 10 largest values from one million
+data points, but not for selecting the largest 100,000 values.  If the
+subset is a significant part of the total dataset it may be faster
+to sort all the elements of the dataset directly with an @math{O(N \log
+N)} algorithm and obtain the smallest or largest values that way.
+
+@deftypefun int gsl_sort_smallest (double * @var{dest}, size_t @var{k}, const double * @var{src}, size_t @var{stride}, size_t @var{n})
+This function copies the @var{k} smallest elements of the array
+@var{src}, of size @var{n} and stride @var{stride}, in ascending
+numerical order into the array @var{dest}.  The size @var{k} of the subset must be
+less than or equal to @var{n}.  The data @var{src} is not modified by
+this operation.
+@end deftypefun
+
+@deftypefun int gsl_sort_largest (double * @var{dest}, size_t @var{k}, const double * @var{src}, size_t @var{stride}, size_t @var{n})
+This function copies the @var{k} largest elements of the array
+@var{src}, of size @var{n} and stride @var{stride}, in descending
+numerical order into the array @var{dest}. @var{k} must be
+less than or equal to @var{n}. The data @var{src} is not modified by
+this operation.
+@end deftypefun
+
+@deftypefun int gsl_sort_vector_smallest (double * @var{dest}, size_t @var{k}, const gsl_vector * @var{v})
+@deftypefunx int gsl_sort_vector_largest (double * @var{dest}, size_t @var{k}, const gsl_vector * @var{v})
+These functions copy the @var{k} smallest or largest elements of the
+vector @var{v} into the array @var{dest}. @var{k}
+must be less than or equal to the length of the vector @var{v}.
+@end deftypefun
+
+The following functions find the indices of the @math{k} smallest or
+largest elements of a dataset,
+
+@deftypefun int gsl_sort_smallest_index (size_t * @var{p}, size_t @var{k}, const double * @var{src}, size_t @var{stride}, size_t @var{n})
+This function stores the indices of the @var{k} smallest elements of
+the array @var{src}, of size @var{n} and stride @var{stride}, in the
+array @var{p}.  The indices are chosen so that the corresponding data is
+in ascending numerical order.  @var{k} must be
+less than or equal to @var{n}. The data @var{src} is not modified by
+this operation.
+@end deftypefun
+
+@deftypefun int gsl_sort_largest_index (size_t * @var{p}, size_t @var{k}, const double * @var{src}, size_t @var{stride}, size_t @var{n})
+This function stores the indices of the @var{k} largest elements of
+the array @var{src}, of size @var{n} and stride @var{stride}, in the
+array @var{p}.  The indices are chosen so that the corresponding data is
+in descending numerical order.  @var{k} must be
+less than or equal to @var{n}. The data @var{src} is not modified by
+this operation.
+@end deftypefun
+
+@deftypefun int gsl_sort_vector_smallest_index (size_t * @var{p}, size_t @var{k}, const gsl_vector * @var{v})
+@deftypefunx int gsl_sort_vector_largest_index (size_t * @var{p}, size_t @var{k}, const gsl_vector * @var{v})
+These functions store the indices of the @var{k} smallest or largest
+elements of the vector @var{v} in the array @var{p}. @var{k} must be less than or equal to the length of the vector
+@var{v}.
+@end deftypefun
+
+
+@node Computing the rank
+@section Computing the rank
+
+The @dfn{rank} of an element is its order in the sorted data.  The rank
+is the inverse of the index permutation, @var{p}.  It can be computed
+using the following algorithm,
+
+@example
+for (i = 0; i < p->size; i++) 
+@{
+    size_t pi = p->data[i];
+    rank->data[pi] = i;
+@}
+@end example
+
+@noindent
+This can be computed directly from the function
+@code{gsl_permutation_inverse(rank,p)}.
+
+The following function will print the rank of each element of the vector
+@var{v},
+
+@example
+void
+print_rank (gsl_vector * v)
+@{
+  size_t i;
+  size_t n = v->size;
+  gsl_permutation * perm = gsl_permutation_alloc(n);
+  gsl_permutation * rank = gsl_permutation_alloc(n);
+
+  gsl_sort_vector_index (perm, v);
+  gsl_permutation_inverse (rank, perm);
+
+  for (i = 0; i < n; i++)
+   @{
+    double vi = gsl_vector_get(v, i);
+    printf ("element = %d, value = %g, rank = %d\n",
+             i, vi, rank->data[i]);
+   @}
+
+  gsl_permutation_free (perm);
+  gsl_permutation_free (rank);
+@}
+@end example
+
+@node Sorting Examples
+@section Examples
+
+The following example shows how to use the permutation @var{p} to print
+the elements of the vector @var{v} in ascending order,
+
+@example
+gsl_sort_vector_index (p, v);
+
+for (i = 0; i < v->size; i++)
+@{
+    double vpi = gsl_vector_get (v, p->data[i]);
+    printf ("order = %d, value = %g\n", i, vpi);
+@}
+@end example
+
+@noindent
+The next example uses the function @code{gsl_sort_smallest} to select
+the 5 smallest numbers from 100000 uniform random variates stored in an
+array,
+
+@example
+@verbatiminclude examples/sortsmall.c
+@end example
+The output lists the 5 smallest values, in ascending order,
+
+@example
+$ ./a.out 
+@verbatiminclude examples/sortsmall.out
+@end example
+
+@node Sorting References and Further Reading
+@section References and Further Reading
+
+The subject of sorting is covered extensively in Knuth's
+@cite{Sorting and Searching},
+
+@itemize @asis
+@item
+Donald E. Knuth, @cite{The Art of Computer Programming: Sorting and
+Searching} (Vol 3, 3rd Ed, 1997), Addison-Wesley, ISBN 0201896850.
+@end itemize
+
+@noindent
+The Heapsort algorithm is described in the following book,
+
+@itemize @asis
+@item Robert Sedgewick, @cite{Algorithms in C}, Addison-Wesley, 
+ISBN 0201514257.
+@end itemize
+
+
diff --git a/gsl-1.9/doc/specfunc-airy.texi b/gsl-1.9/doc/specfunc-airy.texi
new file mode 100644
index 0000000..726f516
--- /dev/null
+++ b/gsl-1.9/doc/specfunc-airy.texi
@@ -0,0 +1,130 @@
+@cindex Airy functions
+@cindex Ai(x)
+@cindex Bi(x)
+
+The Airy functions @math{Ai(x)} and @math{Bi(x)} are defined by the
+integral representations,
+@tex
+\beforedisplay
+$$
+\eqalign{
+Ai(x) & = {1\over\pi} \int_0^\infty \cos(t^3/3 + xt ) \,dt, \cr
+Bi(x) & = {1\over\pi} \int_0^\infty (e^{-t^3/3} + \sin(t^3/3 + xt)) \,dt.
+}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+Ai(x) = (1/\pi) \int_0^\infty \cos((1/3) t^3 + xt) dt
+Bi(x) = (1/\pi) \int_0^\infty (e^(-(1/3) t^3) + \sin((1/3) t^3 + xt)) dt
+@end example
+
+@end ifinfo
+@noindent
+For further information see Abramowitz & Stegun, Section 10.4. The Airy
+functions are defined in the header file @file{gsl_sf_airy.h}.
+
+@menu
+* Airy Functions::              
+* Derivatives of Airy Functions::  
+* Zeros of Airy Functions::     
+* Zeros of Derivatives of Airy Functions::  
+@end menu
+
+@node Airy Functions
+@subsection Airy Functions
+
+@deftypefun double gsl_sf_airy_Ai (double @var{x}, gsl_mode_t @var{mode})
+@deftypefunx int gsl_sf_airy_Ai_e (double @var{x}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result})
+These routines compute the Airy function @math{Ai(x)} with an accuracy
+specified by @var{mode}.
+@end deftypefun
+
+@deftypefun double gsl_sf_airy_Bi (double @var{x}, gsl_mode_t @var{mode})
+@deftypefunx int gsl_sf_airy_Bi_e (double @var{x}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result})
+These routines compute the Airy function @math{Bi(x)} with an accuracy
+specified by @var{mode}.
+@end deftypefun
+
+@deftypefun double gsl_sf_airy_Ai_scaled (double @var{x}, gsl_mode_t @var{mode})
+@deftypefunx int gsl_sf_airy_Ai_scaled_e (double @var{x}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result})
+These routines compute a scaled version of the Airy function
+@math{S_A(x) Ai(x)}.  For @math{x>0} the scaling factor @math{S_A(x)} is @c{$\exp(+(2/3) x^{3/2})$}
+@math{\exp(+(2/3) x^(3/2))}, 
+and is 1
+for @math{x<0}.
+@end deftypefun
+
+@deftypefun double gsl_sf_airy_Bi_scaled (double @var{x}, gsl_mode_t @var{mode})
+@deftypefunx int gsl_sf_airy_Bi_scaled_e (double @var{x}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result})
+These routines compute a scaled version of the Airy function
+@math{S_B(x) Bi(x)}.  For @math{x>0} the scaling factor @math{S_B(x)} is @c{$\exp(-(2/3) x^{3/2})$}
+@math{exp(-(2/3) x^(3/2))}, and is 1 for @math{x<0}.
+@end deftypefun
+
+
+@node Derivatives of Airy Functions
+@subsection Derivatives of Airy Functions
+
+@deftypefun double gsl_sf_airy_Ai_deriv (double @var{x}, gsl_mode_t @var{mode})
+@deftypefunx int gsl_sf_airy_Ai_deriv_e (double @var{x}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result})
+These routines compute the Airy function derivative @math{Ai'(x)} with
+an accuracy specified by @var{mode}.
+@end deftypefun
+
+@deftypefun double gsl_sf_airy_Bi_deriv (double @var{x}, gsl_mode_t @var{mode})
+@deftypefunx int gsl_sf_airy_Bi_deriv_e (double @var{x}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result})
+These routines compute the Airy function derivative @math{Bi'(x)} with
+an accuracy specified by @var{mode}.
+@end deftypefun
+
+@deftypefun double gsl_sf_airy_Ai_deriv_scaled (double @var{x}, gsl_mode_t @var{mode})
+@deftypefunx int gsl_sf_airy_Ai_deriv_scaled_e (double @var{x}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result})
+These routines compute the scaled Airy function derivative 
+@math{S_A(x) Ai'(x)}.  
+For @math{x>0} the scaling factor @math{S_A(x)} is @c{$\exp(+(2/3) x^{3/2})$}
+@math{\exp(+(2/3) x^(3/2))}, and is 1 for @math{x<0}.
+@end deftypefun
+
+@deftypefun double gsl_sf_airy_Bi_deriv_scaled (double @var{x}, gsl_mode_t @var{mode})
+@deftypefunx int gsl_sf_airy_Bi_deriv_scaled_e (double @var{x}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result})
+These routines compute the scaled Airy function derivative 
+@math{S_B(x) Bi'(x)}.
+For @math{x>0} the scaling factor @math{S_B(x)} is @c{$\exp(-(2/3) x^{3/2})$}
+@math{exp(-(2/3) x^(3/2))}, and is 1 for @math{x<0}.
+@end deftypefun
+
+
+@node Zeros of Airy Functions
+@subsection Zeros of Airy Functions
+
+@deftypefun double gsl_sf_airy_zero_Ai (unsigned int @var{s})
+@deftypefunx int gsl_sf_airy_zero_Ai_e (unsigned int @var{s}, gsl_sf_result * @var{result})
+These routines compute the location of the @var{s}-th zero of the Airy
+function @math{Ai(x)}.
+@end deftypefun
+
+@deftypefun double gsl_sf_airy_zero_Bi (unsigned int @var{s})
+@deftypefunx int gsl_sf_airy_zero_Bi_e (unsigned int @var{s}, gsl_sf_result * @var{result})
+These routines compute the location of the @var{s}-th zero of the Airy
+function @math{Bi(x)}.
+@end deftypefun
+
+
+@node Zeros of Derivatives of Airy Functions
+@subsection Zeros of Derivatives of Airy Functions
+
+@deftypefun double gsl_sf_airy_zero_Ai_deriv (unsigned int @var{s})
+@deftypefunx int gsl_sf_airy_zero_Ai_deriv_e (unsigned int @var{s}, gsl_sf_result * @var{result})
+These routines compute the location of the @var{s}-th zero of the Airy
+function derivative @math{Ai'(x)}.
+@end deftypefun
+
+@deftypefun double gsl_sf_airy_zero_Bi_deriv (unsigned int @var{s})
+@deftypefunx int gsl_sf_airy_zero_Bi_deriv_e (unsigned int @var{s}, gsl_sf_result * @var{result})
+These routines compute the location of the @var{s}-th zero of the Airy
+function derivative @math{Bi'(x)}.
+@end deftypefun
+
diff --git a/gsl-1.9/doc/specfunc-bessel.texi b/gsl-1.9/doc/specfunc-bessel.texi
new file mode 100644
index 0000000..e3b09c9
--- /dev/null
+++ b/gsl-1.9/doc/specfunc-bessel.texi
@@ -0,0 +1,585 @@
+@cindex Bessel functions
+
+The routines described in this section compute the Cylindrical Bessel
+functions @math{J_n(x)}, @math{Y_n(x)}, Modified cylindrical Bessel
+functions @math{I_n(x)}, @math{K_n(x)}, Spherical Bessel functions
+@math{j_l(x)}, @math{y_l(x)}, and Modified Spherical Bessel functions
+@math{i_l(x)}, @math{k_l(x)}.  For more information see Abramowitz & Stegun,
+Chapters 9 and 10.  The Bessel functions are defined in the header file
+@file{gsl_sf_bessel.h}.
+
+@menu
+* Regular Cylindrical Bessel Functions::  
+* Irregular Cylindrical Bessel Functions::  
+* Regular Modified Cylindrical Bessel Functions::  
+* Irregular Modified Cylindrical Bessel Functions::  
+* Regular Spherical Bessel Functions::  
+* Irregular Spherical Bessel Functions::  
+* Regular Modified Spherical Bessel Functions::  
+* Irregular Modified Spherical Bessel Functions::  
+* Regular Bessel Function - Fractional Order::  
+* Irregular Bessel Functions - Fractional Order::  
+* Regular Modified Bessel Functions - Fractional Order::  
+* Irregular Modified Bessel Functions - Fractional Order::  
+* Zeros of Regular Bessel Functions::  
+@end menu
+
+@node Regular Cylindrical Bessel Functions
+@subsection Regular Cylindrical Bessel Functions
+@cindex Cylindrical Bessel Functions
+@cindex Regular Cylindrical Bessel Functions
+
+@deftypefun double gsl_sf_bessel_J0 (double @var{x})
+@deftypefunx int gsl_sf_bessel_J0_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the regular cylindrical Bessel function of zeroth
+order, @math{J_0(x)}.
+@comment Exceptional Return Values: none
+@end deftypefun
+
+@deftypefun double gsl_sf_bessel_J1 (double @var{x})
+@deftypefunx int gsl_sf_bessel_J1_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the regular cylindrical Bessel function of first
+order, @math{J_1(x)}.
+@comment Exceptional Return Values: GSL_EUNDRFLW
+@end deftypefun
+
+@deftypefun double gsl_sf_bessel_Jn (int @var{n}, double @var{x})
+@deftypefunx int gsl_sf_bessel_Jn_e (int @var{n}, double @var{x}, gsl_sf_result * @var{result})
+These routines compute the regular cylindrical Bessel function of 
+order @var{n}, @math{J_n(x)}.
+@comment Exceptional Return Values: GSL_EUNDRFLW
+@end deftypefun
+
+@deftypefun int gsl_sf_bessel_Jn_array (int @var{nmin}, int @var{nmax}, double @var{x}, double @var{result_array}[])
+This routine computes the values of the regular cylindrical Bessel
+functions @math{J_n(x)} for @math{n} from @var{nmin} to @var{nmax}
+inclusive, storing the results in the array @var{result_array}.  The
+values are computed using recurrence relations for efficiency, and
+therefore may differ slightly from the exact values.
+@comment Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW
+@end deftypefun
+
+
+@node Irregular Cylindrical Bessel Functions
+@subsection Irregular Cylindrical Bessel Functions
+@cindex Irregular Cylindrical Bessel Functions
+
+@deftypefun double gsl_sf_bessel_Y0 (double @var{x})
+@deftypefunx int gsl_sf_bessel_Y0_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the irregular cylindrical Bessel function of zeroth
+order, @math{Y_0(x)}, for @math{x>0}.
+@comment Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW
+@end deftypefun
+
+@deftypefun double gsl_sf_bessel_Y1 (double @var{x})
+@deftypefunx int gsl_sf_bessel_Y1_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the irregular cylindrical Bessel function of first
+order, @math{Y_1(x)}, for @math{x>0}.
+@comment Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW, GSL_EUNDRFLW
+@end deftypefun
+
+@deftypefun double gsl_sf_bessel_Yn (int @var{n},double @var{x})
+@deftypefunx int gsl_sf_bessel_Yn_e (int @var{n},double @var{x}, gsl_sf_result * @var{result})
+These routines compute the irregular cylindrical Bessel function of 
+order @var{n}, @math{Y_n(x)}, for @math{x>0}.
+@comment Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW, GSL_EUNDRFLW
+@end deftypefun
+
+@deftypefun int gsl_sf_bessel_Yn_array (int @var{nmin}, int @var{nmax}, double @var{x}, double @var{result_array}[])
+This routine computes the values of the irregular cylindrical Bessel
+functions @math{Y_n(x)} for @math{n} from @var{nmin} to @var{nmax}
+inclusive, storing the results in the array @var{result_array}.  The
+domain of the function is @math{x>0}.  The values are computed using
+recurrence relations for efficiency, and therefore may differ slightly
+from the exact values.
+@comment Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW, GSL_EUNDRFLW
+@end deftypefun
+
+
+@node Regular Modified Cylindrical Bessel Functions
+@subsection Regular Modified Cylindrical Bessel Functions
+@cindex Modified Cylindrical Bessel Functions
+@cindex Regular Modified Cylindrical Bessel Functions
+
+@deftypefun double gsl_sf_bessel_I0 (double @var{x})
+@deftypefunx int gsl_sf_bessel_I0_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the regular modified cylindrical Bessel function
+of zeroth order, @math{I_0(x)}.
+@comment Exceptional Return Values: GSL_EOVRFLW
+@end deftypefun
+
+@deftypefun double gsl_sf_bessel_I1 (double @var{x})
+@deftypefunx int gsl_sf_bessel_I1_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the regular modified cylindrical Bessel function
+of first order, @math{I_1(x)}.
+@comment Exceptional Return Values: GSL_EOVRFLW, GSL_EUNDRFLW
+@end deftypefun
+
+@deftypefun double gsl_sf_bessel_In (int @var{n}, double @var{x})
+@deftypefunx int gsl_sf_bessel_In_e (int @var{n}, double @var{x}, gsl_sf_result * @var{result})
+These routines compute the regular modified cylindrical Bessel function
+of order @var{n}, @math{I_n(x)}.
+@comment Exceptional Return Values: GSL_EOVRFLW, GSL_EUNDRFLW
+@end deftypefun
+
+@deftypefun int gsl_sf_bessel_In_array (int @var{nmin}, int @var{nmax}, double @var{x}, double @var{result_array}[])
+This routine computes the values of the regular modified cylindrical
+Bessel functions @math{I_n(x)} for @math{n} from @var{nmin} to
+@var{nmax} inclusive, storing the results in the array
+@var{result_array}.  The start of the range @var{nmin} must be positive
+or zero.  The values are computed using recurrence relations for
+efficiency, and therefore may differ slightly from the exact values.
+@comment Domain: nmin >=0, nmax >= nmin 
+@comment Conditions: n=nmin,...,nmax, nmin >=0, nmax >= nmin 
+@comment Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW, GSL_EUNDRFLW
+@end deftypefun
+
+@deftypefun double gsl_sf_bessel_I0_scaled (double @var{x})
+@deftypefunx int gsl_sf_bessel_I0_scaled_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the scaled regular modified cylindrical Bessel
+function of zeroth order @math{\exp(-|x|) I_0(x)}.
+@comment Exceptional Return Values: none
+@end deftypefun
+
+@deftypefun double gsl_sf_bessel_I1_scaled (double @var{x})
+@deftypefunx int gsl_sf_bessel_I1_scaled_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the scaled regular modified cylindrical Bessel
+function of first order @math{\exp(-|x|) I_1(x)}.
+@comment Exceptional Return Values: GSL_EUNDRFLW
+@end deftypefun
+
+@deftypefun double gsl_sf_bessel_In_scaled (int @var{n}, double @var{x})
+@deftypefunx int gsl_sf_bessel_In_scaled_e (int @var{n}, double @var{x}, gsl_sf_result * @var{result})
+These routines compute the scaled regular modified cylindrical Bessel
+function of order @var{n}, @math{\exp(-|x|) I_n(x)} 
+@comment Exceptional Return Values: GSL_EUNDRFLW
+@end deftypefun
+
+@deftypefun int gsl_sf_bessel_In_scaled_array (int @var{nmin}, int @var{nmax}, double @var{x}, double @var{result_array}[])
+This routine computes the values of the scaled regular cylindrical
+Bessel functions @math{\exp(-|x|) I_n(x)} for @math{n} from
+@var{nmin} to @var{nmax} inclusive, storing the results in the array
+@var{result_array}. The start of the range @var{nmin} must be positive
+or zero.  The values are computed using recurrence relations for
+efficiency, and therefore may differ slightly from the exact values.
+@comment Domain: nmin >=0, nmax >= nmin 
+@comment Conditions:  n=nmin,...,nmax 
+@comment Exceptional Return Values: GSL_EUNDRFLW
+@end deftypefun
+
+
+@node Irregular Modified Cylindrical Bessel Functions
+@subsection Irregular Modified Cylindrical Bessel Functions
+@cindex Irregular Modified Cylindrical Bessel Functions
+
+@deftypefun double gsl_sf_bessel_K0 (double @var{x})
+@deftypefunx int gsl_sf_bessel_K0_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the irregular modified cylindrical Bessel
+function of zeroth order, @math{K_0(x)}, for @math{x > 0}.
+@comment Domain: x > 0.0 
+@comment Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW
+@end deftypefun
+
+@deftypefun double gsl_sf_bessel_K1 (double @var{x})
+@deftypefunx int gsl_sf_bessel_K1_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the irregular modified cylindrical Bessel
+function of first order, @math{K_1(x)}, for @math{x > 0}.
+@comment Domain: x > 0.0 
+@comment Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW, GSL_EUNDRFLW
+@end deftypefun
+
+@deftypefun double gsl_sf_bessel_Kn (int @var{n}, double @var{x})
+@deftypefunx int gsl_sf_bessel_Kn_e (int @var{n}, double @var{x}, gsl_sf_result * @var{result})
+These routines compute the irregular modified cylindrical Bessel
+function of order @var{n}, @math{K_n(x)}, for @math{x > 0}.
+@comment Domain: x > 0.0 
+@comment Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW, GSL_EUNDRFLW
+@end deftypefun
+
+@deftypefun int gsl_sf_bessel_Kn_array (int @var{nmin}, int @var{nmax}, double @var{x}, double @var{result_array}[])
+This routine computes the values of the irregular modified cylindrical
+Bessel functions @math{K_n(x)} for @math{n} from @var{nmin} to
+@var{nmax} inclusive, storing the results in the array
+@var{result_array}. The start of the range @var{nmin} must be positive
+or zero. The domain of the function is @math{x>0}. The values are
+computed using recurrence relations for efficiency, and therefore
+may differ slightly from the exact values.
+@comment Conditions: n=nmin,...,nmax 
+@comment Domain: x > 0.0, nmin>=0, nmax >= nmin
+@comment Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW, GSL_EUNDRFLW
+@end deftypefun
+
+@deftypefun double gsl_sf_bessel_K0_scaled (double @var{x})
+@deftypefunx int gsl_sf_bessel_K0_scaled_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the scaled irregular modified cylindrical Bessel
+function of zeroth order @math{\exp(x) K_0(x)} for @math{x>0}.
+@comment Domain: x > 0.0 
+@comment Exceptional Return Values: GSL_EDOM
+@end deftypefun
+
+@deftypefun double gsl_sf_bessel_K1_scaled (double @var{x}) 
+@deftypefunx int gsl_sf_bessel_K1_scaled_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the scaled irregular modified cylindrical Bessel
+function of first order @math{\exp(x) K_1(x)} for @math{x>0}.
+@comment Domain: x > 0.0 
+@comment Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW
+@end deftypefun
+
+@deftypefun double gsl_sf_bessel_Kn_scaled (int @var{n}, double @var{x})
+@deftypefunx int gsl_sf_bessel_Kn_scaled_e (int @var{n}, double @var{x}, gsl_sf_result * @var{result})
+These routines compute the scaled irregular modified cylindrical Bessel
+function of order @var{n}, @math{\exp(x) K_n(x)}, for @math{x>0}.
+@comment Domain: x > 0.0 
+@comment Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW
+@end deftypefun
+
+@deftypefun int gsl_sf_bessel_Kn_scaled_array (int @var{nmin}, int @var{nmax}, double @var{x}, double @var{result_array}[])
+This routine computes the values of the scaled irregular cylindrical
+Bessel functions @math{\exp(x) K_n(x)} for @math{n} from @var{nmin} to
+@var{nmax} inclusive, storing the results in the array
+@var{result_array}. The start of the range @var{nmin} must be positive
+or zero.  The domain of the function is @math{x>0}. The values are
+computed using recurrence relations for efficiency, and therefore
+may differ slightly from the exact values.
+@comment Domain: x > 0.0, nmin >=0, nmax >= nmin 
+@comment Conditions: n=nmin,...,nmax 
+@comment Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW
+@end deftypefun
+
+
+@node Regular Spherical Bessel Functions
+@subsection Regular Spherical Bessel Functions
+@cindex Spherical Bessel Functions
+@cindex Regular Spherical Bessel Functions
+
+@deftypefun double gsl_sf_bessel_j0 (double @var{x})
+@deftypefunx int gsl_sf_bessel_j0_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the regular spherical Bessel function of zeroth
+order, @math{j_0(x) = \sin(x)/x}.
+@comment Exceptional Return Values: none
+@end deftypefun
+
+@deftypefun double gsl_sf_bessel_j1 (double @var{x})
+@deftypefunx int gsl_sf_bessel_j1_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the regular spherical Bessel function of first
+order, @math{j_1(x) = (\sin(x)/x - \cos(x))/x}.
+@comment Exceptional Return Values: GSL_EUNDRFLW
+@end deftypefun
+
+@deftypefun double gsl_sf_bessel_j2 (double @var{x})
+@deftypefunx int gsl_sf_bessel_j2_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the regular spherical Bessel function of second
+order, @math{j_2(x) = ((3/x^2 - 1)\sin(x) - 3\cos(x)/x)/x}.
+@comment Exceptional Return Values: GSL_EUNDRFLW
+@end deftypefun
+
+@deftypefun double gsl_sf_bessel_jl (int @var{l}, double @var{x})
+@deftypefunx int gsl_sf_bessel_jl_e (int @var{l}, double @var{x}, gsl_sf_result * @var{result})
+These routines compute the regular spherical Bessel function of 
+order @var{l}, @math{j_l(x)}, for @c{$l \geq 0$}
+@math{l >= 0} and @c{$x \geq 0$}
+@math{x >= 0}.
+@comment Domain: l >= 0, x >= 0.0 
+@comment Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW
+@end deftypefun
+
+@deftypefun int gsl_sf_bessel_jl_array (int @var{lmax}, double @var{x}, double @var{result_array}[])
+This routine computes the values of the regular spherical Bessel
+functions @math{j_l(x)} for @math{l} from 0 to @var{lmax}
+inclusive  for @c{$lmax \geq 0$}
+@math{lmax >= 0} and @c{$x \geq 0$}
+@math{x >= 0}, storing the results in the array @var{result_array}.
+The values are computed using recurrence relations for
+efficiency, and therefore may differ slightly from the exact values.
+@comment Domain: lmax >= 0 
+@comment Conditions: l=0,1,...,lmax 
+@comment Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW
+@end deftypefun
+
+@deftypefun int gsl_sf_bessel_jl_steed_array (int @var{lmax}, double @var{x}, double * @var{jl_x_array})
+This routine uses Steed's method to compute the values of the regular
+spherical Bessel functions @math{j_l(x)} for @math{l} from 0 to
+@var{lmax} inclusive for @c{$lmax \geq 0$}
+@math{lmax >= 0} and @c{$x \geq 0$}
+@math{x >= 0}, storing the results in the array
+@var{result_array}.
+The Steed/Barnett algorithm is described in @cite{Comp. Phys. Comm.} 21,
+297 (1981).  Steed's method is more stable than the
+recurrence used in the other functions but is also slower.
+@comment Domain: lmax >= 0 
+@comment Conditions: l=0,1,...,lmax 
+@comment Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW
+@end deftypefun
+
+
+@node Irregular Spherical Bessel Functions
+@subsection Irregular Spherical Bessel Functions
+@cindex Irregular Spherical Bessel Functions
+
+@deftypefun double gsl_sf_bessel_y0 (double @var{x})
+@deftypefunx int gsl_sf_bessel_y0_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the irregular spherical Bessel function of zeroth
+order, @math{y_0(x) = -\cos(x)/x}.
+@comment Exceptional Return Values: none
+@end deftypefun
+
+@deftypefun double gsl_sf_bessel_y1 (double @var{x})
+@deftypefunx int gsl_sf_bessel_y1_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the irregular spherical Bessel function of first
+order, @math{y_1(x) = -(\cos(x)/x + \sin(x))/x}.
+@comment Exceptional Return Values: GSL_EUNDRFLW
+@end deftypefun
+
+@deftypefun double gsl_sf_bessel_y2 (double @var{x})
+@deftypefunx int gsl_sf_bessel_y2_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the irregular spherical Bessel function of second
+order, @math{y_2(x) = (-3/x^3 + 1/x)\cos(x) - (3/x^2)\sin(x)}.
+@comment Exceptional Return Values: GSL_EUNDRFLW
+@end deftypefun
+
+@deftypefun double gsl_sf_bessel_yl (int @var{l}, double @var{x})
+@deftypefunx int gsl_sf_bessel_yl_e (int @var{l}, double @var{x}, gsl_sf_result * @var{result})
+These routines compute the irregular spherical Bessel function of 
+order @var{l}, @math{y_l(x)}, for @c{$l \geq 0$}
+@math{l >= 0}.
+@comment Exceptional Return Values: GSL_EUNDRFLW
+@end deftypefun
+
+@deftypefun int gsl_sf_bessel_yl_array (int @var{lmax}, double @var{x}, double @var{result_array}[])
+This routine computes the values of the irregular spherical Bessel
+functions @math{y_l(x)} for @math{l} from 0 to @var{lmax}
+inclusive  for @c{$lmax \geq 0$}
+@math{lmax >= 0}, storing the results in the array @var{result_array}.
+The values are computed using recurrence relations for
+efficiency, and therefore may differ slightly from the exact values.
+@comment Domain: lmax >= 0 
+@comment Conditions: l=0,1,...,lmax 
+@comment Exceptional Return Values: GSL_EUNDRFLW
+@end deftypefun
+
+
+@node Regular Modified Spherical Bessel Functions
+@subsection Regular Modified Spherical Bessel Functions
+@cindex Modified Spherical Bessel Functions
+@cindex Regular Modified Spherical Bessel Functions
+
+The regular modified spherical Bessel functions @math{i_l(x)} 
+are related to the modified Bessel functions of fractional order,
+@c{$i_l(x) = \sqrt{\pi/(2x)} I_{l+1/2}(x)$}
+@math{i_l(x) = \sqrt@{\pi/(2x)@} I_@{l+1/2@}(x)}
+
+@deftypefun double gsl_sf_bessel_i0_scaled (double @var{x})
+@deftypefunx int gsl_sf_bessel_i0_scaled_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the scaled regular modified spherical Bessel
+function of zeroth order, @math{\exp(-|x|) i_0(x)}.
+@comment Exceptional Return Values: none
+@end deftypefun
+
+@deftypefun double gsl_sf_bessel_i1_scaled (double @var{x})
+@deftypefunx int gsl_sf_bessel_i1_scaled_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the scaled regular modified spherical Bessel
+function of first order, @math{\exp(-|x|) i_1(x)}.
+@comment Exceptional Return Values: GSL_EUNDRFLW
+@end deftypefun
+
+@deftypefun double gsl_sf_bessel_i2_scaled (double @var{x})
+@deftypefunx int gsl_sf_bessel_i2_scaled_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the scaled regular modified spherical Bessel
+function of second order, @math{ \exp(-|x|) i_2(x) } 
+@comment Exceptional Return Values: GSL_EUNDRFLW
+@end deftypefun
+
+@deftypefun double gsl_sf_bessel_il_scaled (int @var{l}, double @var{x})
+@deftypefunx int gsl_sf_bessel_il_scaled_e (int @var{l}, double @var{x}, gsl_sf_result * @var{result})
+These routines compute the scaled regular modified spherical Bessel
+function of order @var{l}, @math{ \exp(-|x|) i_l(x) }
+@comment Domain: l >= 0 
+@comment Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW
+@end deftypefun
+
+@deftypefun int gsl_sf_bessel_il_scaled_array (int @var{lmax}, double @var{x}, double @var{result_array}[])
+This routine computes the values of the scaled regular modified
+cylindrical Bessel functions @math{\exp(-|x|) i_l(x)} for @math{l} from
+0 to @var{lmax} inclusive for @c{$lmax \geq 0$}
+@math{lmax >= 0}, storing the results in
+the array @var{result_array}. 
+The values are computed using recurrence relations for
+efficiency, and therefore may differ slightly from the exact values.
+@comment Domain: lmax >= 0 
+@comment Conditions: l=0,1,...,lmax 
+@comment Exceptional Return Values: GSL_EUNDRFLW
+@end deftypefun
+
+
+@node Irregular Modified Spherical Bessel Functions
+@subsection Irregular Modified Spherical Bessel Functions
+@cindex Irregular Modified Spherical Bessel Functions
+
+The irregular modified spherical Bessel functions @math{k_l(x)}
+are related to the irregular modified Bessel functions of fractional order,
+@c{$k_l(x) = \sqrt{\pi/(2x)} K_{l+1/2}(x)$}
+@math{k_l(x) = \sqrt@{\pi/(2x)@} K_@{l+1/2@}(x)}.
+
+@deftypefun double gsl_sf_bessel_k0_scaled (double @var{x})
+@deftypefunx int gsl_sf_bessel_k0_scaled_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the scaled irregular modified spherical Bessel
+function of zeroth order, @math{\exp(x) k_0(x)}, for @math{x>0}.
+@comment Domain: x > 0.0 
+@comment Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW
+@end deftypefun
+
+@deftypefun double gsl_sf_bessel_k1_scaled (double @var{x})
+@deftypefunx int gsl_sf_bessel_k1_scaled_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the scaled irregular modified spherical Bessel
+function of first order, @math{\exp(x) k_1(x)}, for @math{x>0}.
+@comment Domain: x > 0.0 
+@comment Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW, GSL_EOVRFLW
+@end deftypefun
+
+@deftypefun double gsl_sf_bessel_k2_scaled (double @var{x})
+@deftypefunx int gsl_sf_bessel_k2_scaled_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the scaled irregular modified spherical Bessel
+function of second order, @math{\exp(x) k_2(x)}, for @math{x>0}.
+@comment Domain: x > 0.0 
+@comment Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW, GSL_EOVRFLW
+@end deftypefun
+
+@deftypefun double gsl_sf_bessel_kl_scaled (int @var{l}, double @var{x})
+@deftypefunx int gsl_sf_bessel_kl_scaled_e (int @var{l}, double @var{x}, gsl_sf_result * @var{result})
+These routines compute the scaled irregular modified spherical Bessel
+function of order @var{l}, @math{\exp(x) k_l(x)}, for @math{x>0}.
+@comment Domain: x > 0.0 
+@comment Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW
+@end deftypefun
+
+@deftypefun int gsl_sf_bessel_kl_scaled_array (int @var{lmax}, double @var{x}, double @var{result_array}[])
+This routine computes the values of the scaled irregular modified
+spherical Bessel functions @math{\exp(x) k_l(x)} for @math{l} from
+0 to @var{lmax} inclusive for @c{$lmax \geq 0$}
+@math{lmax >= 0} and @math{x>0}, storing the results in
+the array @var{result_array}. 
+The values are computed using recurrence relations for
+efficiency, and therefore may differ slightly from the exact values.
+@comment Domain: lmax >= 0 
+@comment Conditions: l=0,1,...,lmax 
+@comment Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW
+@end deftypefun
+
+
+@node Regular Bessel Function - Fractional Order
+@subsection Regular Bessel Function---Fractional Order
+@cindex Fractional Order Bessel Functions
+@cindex Bessel Functions, Fractional Order
+@cindex Regular Bessel Functions, Fractional Order
+
+@deftypefun double gsl_sf_bessel_Jnu (double @var{nu}, double @var{x})
+@deftypefunx int gsl_sf_bessel_Jnu_e (double @var{nu}, double @var{x}, gsl_sf_result * @var{result})
+These routines compute the regular cylindrical Bessel function of
+fractional order @math{\nu}, @math{J_\nu(x)}.
+@comment Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW
+@end deftypefun
+
+@deftypefun int gsl_sf_bessel_sequence_Jnu_e (double @var{nu}, gsl_mode_t @var{mode}, size_t @var{size}, double @var{v}[])
+This function computes the regular cylindrical Bessel function of
+fractional order @math{\nu}, @math{J_\nu(x)}, evaluated at a series of
+@math{x} values.  The array @var{v} of length @var{size} contains the
+@math{x} values.  They are assumed to be strictly ordered and positive.
+The array is over-written with the values of @math{J_\nu(x_i)}.
+@comment Exceptional Return Values: GSL_EDOM, GSL_EINVAL
+@end deftypefun
+
+
+@node Irregular Bessel Functions - Fractional Order
+@subsection Irregular Bessel Functions---Fractional Order
+
+@deftypefun double gsl_sf_bessel_Ynu (double @var{nu}, double @var{x})
+@deftypefunx int gsl_sf_bessel_Ynu_e (double @var{nu}, double @var{x}, gsl_sf_result * @var{result})
+These routines compute the irregular cylindrical Bessel function of
+fractional order @math{\nu}, @math{Y_\nu(x)}.
+@comment Exceptional Return Values: 
+@end deftypefun
+
+
+@node Regular Modified Bessel Functions - Fractional Order
+@subsection Regular Modified Bessel Functions---Fractional Order
+@cindex Modified Bessel Functions, Fractional Order
+@cindex Regular Modified Bessel Functions, Fractional Order
+
+@deftypefun double gsl_sf_bessel_Inu (double @var{nu}, double @var{x})
+@deftypefunx int gsl_sf_bessel_Inu_e (double @var{nu}, double @var{x}, gsl_sf_result * @var{result})
+These routines compute the regular modified Bessel function of
+fractional order @math{\nu}, @math{I_\nu(x)} for @math{x>0},
+@math{\nu>0}.
+@comment Domain: x >= 0, nu >= 0 
+@comment Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW
+@end deftypefun
+
+@deftypefun double gsl_sf_bessel_Inu_scaled (double @var{nu}, double @var{x})
+@deftypefunx int gsl_sf_bessel_Inu_scaled_e (double @var{nu}, double @var{x}, gsl_sf_result * @var{result})
+These routines compute the scaled regular modified Bessel function of
+fractional order @math{\nu}, @math{\exp(-|x|)I_\nu(x)} for @math{x>0},
+@math{\nu>0}.
+@comment @math{ \exp(-|x|) I_@{\nu@}(x) } 
+@comment Domain: x >= 0, nu >= 0 
+@comment Exceptional Return Values: GSL_EDOM
+@end deftypefun
+
+
+@node Irregular Modified Bessel Functions - Fractional Order
+@subsection Irregular Modified Bessel Functions---Fractional Order
+@cindex Irregular Modified Bessel Functions, Fractional Order
+
+@deftypefun double gsl_sf_bessel_Knu (double @var{nu}, double @var{x})
+@deftypefunx int gsl_sf_bessel_Knu_e (double @var{nu}, double @var{x}, gsl_sf_result * @var{result})
+These routines compute the irregular modified Bessel function of
+fractional order @math{\nu}, @math{K_\nu(x)} for @math{x>0},
+@math{\nu>0}.
+@comment Domain: x > 0, nu >= 0 
+@comment Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW
+@end deftypefun
+
+@deftypefun double gsl_sf_bessel_lnKnu (double @var{nu}, double @var{x})
+@deftypefunx int gsl_sf_bessel_lnKnu_e (double @var{nu}, double @var{x}, gsl_sf_result * @var{result})
+These routines compute the logarithm of the irregular modified Bessel
+function of fractional order @math{\nu}, @math{\ln(K_\nu(x))} for
+@math{x>0}, @math{\nu>0}. 
+@comment Domain: x > 0, nu >= 0 
+@comment Exceptional Return Values: GSL_EDOM
+@end deftypefun
+
+@deftypefun double gsl_sf_bessel_Knu_scaled (double @var{nu}, double @var{x})
+@deftypefunx int gsl_sf_bessel_Knu_scaled_e (double @var{nu}, double @var{x}, gsl_sf_result * @var{result})
+These routines compute the scaled irregular modified Bessel function of
+fractional order @math{\nu}, @math{\exp(+|x|) K_\nu(x)} for @math{x>0},
+@math{\nu>0}.
+@comment Domain: x > 0, nu >= 0 
+@comment Exceptional Return Values: GSL_EDOM
+@end deftypefun
+
+@node Zeros of Regular Bessel Functions
+@subsection Zeros of Regular Bessel Functions
+@cindex Zeros of Regular Bessel Functions
+@cindex Regular Bessel Functions, Zeros of 
+
+@deftypefun double gsl_sf_bessel_zero_J0 (unsigned int @var{s})
+@deftypefunx int gsl_sf_bessel_zero_J0_e (unsigned int @var{s}, gsl_sf_result * @var{result})
+These routines compute the location of the @var{s}-th positive zero of
+the Bessel function @math{J_0(x)}.
+@comment Exceptional Return Values: 
+@end deftypefun
+
+@deftypefun double gsl_sf_bessel_zero_J1 (unsigned int @var{s})
+@deftypefunx int gsl_sf_bessel_zero_J1_e (unsigned int @var{s}, gsl_sf_result * @var{result})
+These routines compute the location of the @var{s}-th positive zero of
+the Bessel function @math{J_1(x)}.
+@comment Exceptional Return Values: 
+@end deftypefun
+
+@deftypefun double gsl_sf_bessel_zero_Jnu (double @var{nu}, unsigned int @var{s})
+@deftypefunx int gsl_sf_bessel_zero_Jnu_e (double @var{nu}, unsigned int @var{s}, gsl_sf_result * @var{result})
+These routines compute the location of the @var{s}-th positive zero of
+the Bessel function @math{J_\nu(x)}.  The current implementation does not
+support negative values of @var{nu}. 
+@comment Exceptional Return Values: 
+@end deftypefun
+
diff --git a/gsl-1.9/doc/specfunc-clausen.texi b/gsl-1.9/doc/specfunc-clausen.texi
new file mode 100644
index 0000000..b0987c2
--- /dev/null
+++ b/gsl-1.9/doc/specfunc-clausen.texi
@@ -0,0 +1,28 @@
+@cindex Clausen functions
+
+The Clausen function is defined by the following integral,
+@tex
+\beforedisplay
+$$
+Cl_2(x) = - \int_0^x dt \log(2 \sin(t/2))
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+Cl_2(x) = - \int_0^x dt \log(2 \sin(t/2))
+@end example
+
+@end ifinfo
+@noindent
+It is related to the dilogarithm by 
+@c{$Cl_2(\theta) = \Im(Li_2(e^{i\theta}))$}
+@math{Cl_2(\theta) = \Im Li_2(\exp(i\theta))}.  
+The Clausen functions are declared in the header file
+@file{gsl_sf_clausen.h}.
+
+@deftypefun double gsl_sf_clausen (double @var{x})
+@deftypefunx int gsl_sf_clausen_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the Clausen integral @math{Cl_2(x)}.
+@end deftypefun
diff --git a/gsl-1.9/doc/specfunc-coulomb.texi b/gsl-1.9/doc/specfunc-coulomb.texi
new file mode 100644
index 0000000..7114620
--- /dev/null
+++ b/gsl-1.9/doc/specfunc-coulomb.texi
@@ -0,0 +1,151 @@
+@cindex Coulomb wave functions
+@cindex hydrogen atom
+
+The prototypes of the Coulomb functions are declared in the header file
+@file{gsl_sf_coulomb.h}.  Both bound state and scattering solutions are
+available.
+
+@menu
+* Normalized Hydrogenic Bound States::  
+* Coulomb Wave Functions::      
+* Coulomb Wave Function Normalization Constant::  
+@end menu
+
+@node Normalized Hydrogenic Bound States
+@subsection Normalized Hydrogenic Bound States
+
+@deftypefun double gsl_sf_hydrogenicR_1 (double @var{Z}, double @var{r})
+@deftypefunx int gsl_sf_hydrogenicR_1_e (double @var{Z}, double @var{r}, gsl_sf_result * @var{result})
+These routines compute the lowest-order normalized hydrogenic bound
+state radial wavefunction @c{$R_1 := 2Z \sqrt{Z} \exp(-Z r)$}
+@math{R_1 := 2Z \sqrt@{Z@} \exp(-Z r)}.
+@end deftypefun
+
+@deftypefun double gsl_sf_hydrogenicR (int @var{n}, int @var{l}, double @var{Z}, double @var{r})
+@deftypefunx int gsl_sf_hydrogenicR_e (int @var{n}, int @var{l}, double @var{Z}, double @var{r}, gsl_sf_result * @var{result})
+These routines compute the @var{n}-th normalized hydrogenic bound state
+radial wavefunction,
+@comment
+@tex
+\beforedisplay
+$$
+R_n := {2 Z^{3/2} \over n^2}  \left({2Z r \over n}\right)^l  \sqrt{(n-l-1)! \over (n+l)!} \exp(-Z r/n) L^{2l+1}_{n-l-1}(2Z r / n).
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+R_n := 2 (Z^@{3/2@}/n^2) \sqrt@{(n-l-1)!/(n+l)!@} \exp(-Z r/n) (2Zr/n)^l
+          L^@{2l+1@}_@{n-l-1@}(2Zr/n).  
+@end example
+
+@end ifinfo
+@noindent
+where @math{L^a_b(x)} is the generalized Laguerre polynomial (@pxref{Laguerre Functions}).
+The normalization is chosen such that the wavefunction @math{\psi} is
+given by 
+@c{$\psi(n,l,r) = R_n Y_{lm}$}
+@math{\psi(n,l,r) = R_n Y_@{lm@}}.   
+@end deftypefun
+
+@node Coulomb Wave Functions
+@subsection Coulomb Wave Functions
+
+The Coulomb wave functions @math{F_L(\eta,x)}, @math{G_L(\eta,x)} are
+described in Abramowitz & Stegun, Chapter 14.  Because there can be a
+large dynamic range of values for these functions, overflows are handled
+gracefully.  If an overflow occurs, @code{GSL_EOVRFLW} is signalled and
+exponent(s) are returned through the modifiable parameters @var{exp_F},
+@var{exp_G}. The full solution can be reconstructed from the following
+relations,
+@tex
+\beforedisplay
+$$
+\eqalign{
+  F_L(\eta,x)  &=  fc[k_L] * \exp(exp_F)\cr
+  G_L(\eta,x)  &=  gc[k_L] * \exp(exp_G)\cr
+\cr
+  F_L'(\eta,x) &= fcp[k_L] * \exp(exp_F)\cr
+  G_L'(\eta,x) &= gcp[k_L] * \exp(exp_G)
+}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+F_L(eta,x)  =  fc[k_L] * exp(exp_F)
+G_L(eta,x)  =  gc[k_L] * exp(exp_G)
+
+F_L'(eta,x) = fcp[k_L] * exp(exp_F)
+G_L'(eta,x) = gcp[k_L] * exp(exp_G)
+@end example
+
+@end ifinfo
+@noindent
+
+@deftypefun int gsl_sf_coulomb_wave_FG_e (double @var{eta}, double @var{x}, double @var{L_F}, int @var{k}, gsl_sf_result * @var{F}, gsl_sf_result * @var{Fp}, gsl_sf_result * @var{G}, gsl_sf_result * @var{Gp}, double * @var{exp_F}, double * @var{exp_G})
+This function computes the Coulomb wave functions @math{F_L(\eta,x)},
+@c{$G_{L-k}(\eta,x)$} 
+@math{G_@{L-k@}(\eta,x)} and their derivatives 
+@math{F'_L(\eta,x)}, 
+@c{$G'_{L-k}(\eta,x)$}
+@math{G'_@{L-k@}(\eta,x)}
+with respect to @math{x}.  The parameters are restricted to @math{L,
+L-k > -1/2}, @math{x > 0} and integer @math{k}.  Note that @math{L}
+itself is not restricted to being an integer. The results are stored in
+the parameters @var{F}, @var{G} for the function values and @var{Fp},
+@var{Gp} for the derivative values.  If an overflow occurs,
+@code{GSL_EOVRFLW} is returned and scaling exponents are stored in
+the modifiable parameters @var{exp_F}, @var{exp_G}.
+@end deftypefun
+
+@deftypefun int gsl_sf_coulomb_wave_F_array (double @var{L_min}, int @var{kmax}, double @var{eta}, double @var{x}, double @var{fc_array}[], double * @var{F_exponent})
+This function computes the Coulomb wave function @math{F_L(\eta,x)} for
+@math{L = Lmin \dots Lmin + kmax}, storing the results in @var{fc_array}.
+In the case of overflow the exponent is stored in @var{F_exponent}.
+@end deftypefun
+
+@deftypefun int gsl_sf_coulomb_wave_FG_array (double @var{L_min}, int @var{kmax}, double @var{eta}, double @var{x}, double @var{fc_array}[], double @var{gc_array}[], double * @var{F_exponent}, double * @var{G_exponent})
+This function computes the functions @math{F_L(\eta,x)},
+@math{G_L(\eta,x)} for @math{L = Lmin \dots Lmin + kmax} storing the
+results in @var{fc_array} and @var{gc_array}.  In the case of overflow the
+exponents are stored in @var{F_exponent} and @var{G_exponent}.
+@end deftypefun
+
+@deftypefun int gsl_sf_coulomb_wave_FGp_array (double @var{L_min}, int @var{kmax}, double @var{eta}, double @var{x}, double @var{fc_array}[], double @var{fcp_array}[], double @var{gc_array}[], double @var{gcp_array}[], double * @var{F_exponent}, double * @var{G_exponent})
+This function computes the functions @math{F_L(\eta,x)},
+@math{G_L(\eta,x)} and their derivatives @math{F'_L(\eta,x)},
+@math{G'_L(\eta,x)} for @math{L = Lmin \dots Lmin + kmax} storing the
+results in @var{fc_array}, @var{gc_array}, @var{fcp_array} and @var{gcp_array}.
+In the case of overflow the exponents are stored in @var{F_exponent} 
+and @var{G_exponent}.
+@end deftypefun
+
+@deftypefun int gsl_sf_coulomb_wave_sphF_array (double @var{L_min}, int @var{kmax}, double @var{eta}, double @var{x}, double @var{fc_array}[], double @var{F_exponent}[])
+This function computes the Coulomb wave function divided by the argument
+@math{F_L(\eta, x)/x} for @math{L = Lmin \dots Lmin + kmax}, storing the
+results in @var{fc_array}.  In the case of overflow the exponent is
+stored in @var{F_exponent}. This function reduces to spherical Bessel
+functions in the limit @math{\eta \to 0}.
+@end deftypefun
+
+@node Coulomb Wave Function Normalization Constant
+@subsection Coulomb Wave Function Normalization Constant
+
+The Coulomb wave function normalization constant is defined in
+Abramowitz 14.1.7.
+
+@deftypefun int gsl_sf_coulomb_CL_e (double @var{L}, double @var{eta}, gsl_sf_result * @var{result})
+This function computes the Coulomb wave function normalization constant
+@math{C_L(\eta)} for @math{L > -1}.
+@end deftypefun
+
+@deftypefun int gsl_sf_coulomb_CL_array (double @var{Lmin}, int @var{kmax}, double @var{eta}, double @var{cl}[])
+This function computes the Coulomb wave function normalization constant
+@math{C_L(\eta)} for @math{L = Lmin \dots Lmin + kmax}, @math{Lmin > -1}.
+@end deftypefun
+
+
+
diff --git a/gsl-1.9/doc/specfunc-coupling.texi b/gsl-1.9/doc/specfunc-coupling.texi
new file mode 100644
index 0000000..3efd9fe
--- /dev/null
+++ b/gsl-1.9/doc/specfunc-coupling.texi
@@ -0,0 +1,109 @@
+@cindex coupling coefficients
+@cindex 3-j symbols
+@cindex 6-j symbols
+@cindex 9-j symbols
+@cindex Wigner coefficients
+@cindex Racah coefficients
+
+The Wigner 3-j, 6-j and 9-j symbols give the coupling coefficients for
+combined angular momentum vectors.  Since the arguments of the standard
+coupling coefficient functions are integer or half-integer, the
+arguments of the following functions are, by convention, integers equal
+to twice the actual spin value.  For information on the 3-j coefficients
+see Abramowitz & Stegun, Section 27.9.  The functions described in this
+section are declared in the header file @file{gsl_sf_coupling.h}.
+
+@menu
+* 3-j Symbols::                 
+* 6-j Symbols::                 
+* 9-j Symbols::                 
+@end menu
+
+@node 3-j Symbols
+@subsection 3-j Symbols
+
+@deftypefun double gsl_sf_coupling_3j (int @var{two_ja}, int @var{two_jb}, int @var{two_jc}, int @var{two_ma}, int @var{two_mb}, int @var{two_mc})
+@deftypefunx int gsl_sf_coupling_3j_e (int @var{two_ja}, int @var{two_jb}, int @var{two_jc}, int @var{two_ma}, int @var{two_mb}, int @var{two_mc}, gsl_sf_result * @var{result})
+These routines compute the Wigner 3-j coefficient, 
+@tex
+\beforedisplay
+$$
+\pmatrix{ja & jb & jc\cr
+         ma & mb & mc\cr}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+(ja jb jc
+ ma mb mc)
+@end example
+
+@end ifinfo
+@noindent
+where the arguments are given in half-integer units, @math{ja} =
+@var{two_ja}/2, @math{ma} = @var{two_ma}/2, etc.
+@comment Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW
+@end deftypefun
+
+
+@node 6-j Symbols
+@subsection 6-j Symbols
+
+@deftypefun double gsl_sf_coupling_6j (int @var{two_ja}, int @var{two_jb}, int @var{two_jc}, int @var{two_jd}, int @var{two_je}, int @var{two_jf})
+@deftypefunx int gsl_sf_coupling_6j_e (int @var{two_ja}, int @var{two_jb}, int @var{two_jc}, int @var{two_jd}, int @var{two_je}, int @var{two_jf}, gsl_sf_result * @var{result}) 
+These routines compute the Wigner 6-j coefficient, 
+@tex
+\beforedisplay
+$$
+\left\{\matrix{ja & jb & jc\cr
+               jd & je & jf\cr}\right\}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+@{ja jb jc
+ jd je jf@}
+@end example
+
+@end ifinfo
+@noindent
+where the arguments are given in half-integer units, @math{ja} =
+@var{two_ja}/2, @math{ma} = @var{two_ma}/2, etc.
+@comment Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW
+@end deftypefun
+
+
+@node 9-j Symbols
+@subsection 9-j Symbols
+
+@deftypefun double gsl_sf_coupling_9j (int @var{two_ja}, int @var{two_jb}, int @var{two_jc}, int @var{two_jd}, int @var{two_je}, int @var{two_jf}, int @var{two_jg}, int @var{two_jh}, int @var{two_ji})
+@deftypefunx int gsl_sf_coupling_9j_e (int @var{two_ja}, int @var{two_jb}, int @var{two_jc}, int @var{two_jd}, int @var{two_je}, int @var{two_jf}, int @var{two_jg}, int @var{two_jh}, int @var{two_ji}, gsl_sf_result * @var{result}) 
+These routines compute the Wigner 9-j coefficient, 
+@tex
+\beforedisplay
+$$
+\left\{\matrix{ja & jb & jc\cr
+               jd & je & jf\cr
+               jg & jh & ji\cr}\right\}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+@{ja jb jc
+ jd je jf
+ jg jh ji@}
+@end example
+
+@end ifinfo
+@noindent
+where the arguments are given in half-integer units, @math{ja} =
+@var{two_ja}/2, @math{ma} = @var{two_ma}/2, etc.
+@comment Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW
+@end deftypefun
+
diff --git a/gsl-1.9/doc/specfunc-dawson.texi b/gsl-1.9/doc/specfunc-dawson.texi
new file mode 100644
index 0000000..451ccf0
--- /dev/null
+++ b/gsl-1.9/doc/specfunc-dawson.texi
@@ -0,0 +1,12 @@
+@cindex Dawson function
+
+The Dawson integral is defined by @math{\exp(-x^2) \int_0^x dt
+\exp(t^2)}.  A table of Dawson's integral can be found in Abramowitz &
+Stegun, Table 7.5.  The Dawson functions are declared in the header file
+@file{gsl_sf_dawson.h}.
+
+@deftypefun double gsl_sf_dawson (double @var{x})
+@deftypefunx int gsl_sf_dawson_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the value of Dawson's integral for @var{x}.
+@comment Exceptional Return Values: GSL_EUNDRFLW
+@end deftypefun
diff --git a/gsl-1.9/doc/specfunc-debye.texi b/gsl-1.9/doc/specfunc-debye.texi
new file mode 100644
index 0000000..400dde2
--- /dev/null
+++ b/gsl-1.9/doc/specfunc-debye.texi
@@ -0,0 +1,63 @@
+@cindex Debye functions
+
+The Debye functions @math{D_n(x)} are defined by the following integral,
+@tex
+\beforedisplay
+$$
+D_n(x) = {n \over x^n} \int_0^x dt {t^n \over e^t - 1}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+D_n(x) = n/x^n \int_0^x dt (t^n/(e^t - 1))
+@end example
+
+@end ifinfo
+@noindent
+For further information see Abramowitz &
+Stegun, Section 27.1.  The Debye functions are declared in the header
+file @file{gsl_sf_debye.h}.
+
+@deftypefun double gsl_sf_debye_1 (double @var{x})
+@deftypefunx int gsl_sf_debye_1_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the first-order Debye function 
+@math{D_1(x) = (1/x) \int_0^x dt (t/(e^t - 1))}.
+@comment Exceptional Return Values: GSL_EDOM
+@end deftypefun
+
+@deftypefun double gsl_sf_debye_2 (double @var{x})
+@deftypefunx int gsl_sf_debye_2_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the second-order Debye function 
+@math{D_2(x) = (2/x^2) \int_0^x dt (t^2/(e^t - 1))}.
+@comment Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW
+@end deftypefun
+
+@deftypefun double gsl_sf_debye_3 (double @var{x})
+@deftypefunx int gsl_sf_debye_3_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the third-order Debye function 
+@math{D_3(x) = (3/x^3) \int_0^x dt (t^3/(e^t - 1))}.
+@comment Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW
+@end deftypefun
+
+@deftypefun double gsl_sf_debye_4 (double @var{x})
+@deftypefunx int gsl_sf_debye_4_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the fourth-order Debye function 
+@math{D_4(x) = (4/x^4) \int_0^x dt (t^4/(e^t - 1))}.
+@comment Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW
+@end deftypefun
+
+@deftypefun double gsl_sf_debye_5 (double @var{x})
+@deftypefunx int gsl_sf_debye_5_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the fifth-order Debye function 
+@math{D_5(x) = (5/x^5) \int_0^x dt (t^5/(e^t - 1))}.
+@comment Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW
+@end deftypefun
+
+@deftypefun double gsl_sf_debye_6 (double @var{x})
+@deftypefunx int gsl_sf_debye_6_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the sixth-order Debye function 
+@math{D_6(x) = (6/x^6) \int_0^x dt (t^6/(e^t - 1))}.
+@comment Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW
+@end deftypefun
diff --git a/gsl-1.9/doc/specfunc-dilog.texi b/gsl-1.9/doc/specfunc-dilog.texi
new file mode 100644
index 0000000..696b565
--- /dev/null
+++ b/gsl-1.9/doc/specfunc-dilog.texi
@@ -0,0 +1,33 @@
+@cindex dilogarithm
+
+The functions described in this section are declared in the header file
+@file{gsl_sf_dilog.h}.
+
+@menu
+* Real Argument::               
+* Complex Argument::            
+@end menu
+
+@node Real Argument
+@subsection Real Argument
+
+@deftypefun double gsl_sf_dilog (double @var{x})
+@deftypefunx int gsl_sf_dilog_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the dilogarithm for a real argument. In Lewin's
+notation this is @math{Li_2(x)}, the real part of the dilogarithm of a
+real @math{x}.  It is defined by the integral representation
+@math{Li_2(x) = - \Re \int_0^x ds \log(1-s) / s}.  
+Note that @math{\Im(Li_2(x)) = 0} for @c{$x \le 1$} 
+@math{x <= 1}, and @math{-\pi\log(x)} for @math{x > 1}.
+
+@end deftypefun
+
+@node Complex Argument
+@subsection Complex Argument
+
+
+@deftypefun int gsl_sf_complex_dilog_e (double @var{r}, double @var{theta}, gsl_sf_result * @var{result_re}, gsl_sf_result * @var{result_im})
+This function computes the full complex-valued dilogarithm for the
+complex argument @math{z = r \exp(i \theta)}. The real and imaginary
+parts of the result are returned in @var{result_re}, @var{result_im}.
+@end deftypefun
diff --git a/gsl-1.9/doc/specfunc-elementary.texi b/gsl-1.9/doc/specfunc-elementary.texi
new file mode 100644
index 0000000..6d6f92a
--- /dev/null
+++ b/gsl-1.9/doc/specfunc-elementary.texi
@@ -0,0 +1,23 @@
+@cindex elementary operations
+@cindex multiplication
+
+The following functions allow for the propagation of errors when
+combining quantities by multiplication.  The functions are declared in
+the header file @file{gsl_sf_elementary.h}.
+
+@comment @deftypefun double gsl_sf_multiply (double @var{x}, double @var{y})
+@deftypefun int gsl_sf_multiply_e (double @var{x}, double @var{y}, gsl_sf_result * @var{result})
+This function multiplies @var{x} and @var{y} storing the product and its
+associated error in @var{result}.
+@comment Exceptional Return Values: GSL_EOVRFLW, GSL_EUNDRFLW
+@end deftypefun
+
+
+@deftypefun int gsl_sf_multiply_err_e (double @var{x}, double @var{dx}, double @var{y}, double @var{dy}, gsl_sf_result * @var{result})
+This function multiplies @var{x} and @var{y} with associated absolute
+errors @var{dx} and @var{dy}.  The product 
+@c{$xy \pm xy \sqrt{(dx/x)^2 +(dy/y)^2}$} 
+@math{xy +/- xy \sqrt((dx/x)^2 +(dy/y)^2)} 
+is stored in @var{result}.
+@comment Exceptional Return Values: GSL_EOVRFLW, GSL_EUNDRFLW
+@end deftypefun
diff --git a/gsl-1.9/doc/specfunc-ellint.texi b/gsl-1.9/doc/specfunc-ellint.texi
new file mode 100644
index 0000000..601f9de
--- /dev/null
+++ b/gsl-1.9/doc/specfunc-ellint.texi
@@ -0,0 +1,198 @@
+@cindex elliptic integrals
+
+The functions described in this section are declared in the header
+file @file{gsl_sf_ellint.h}.  Further information about the elliptic
+integrals can be found in Abramowitz & Stegun, Chapter 17.
+
+@menu
+* Definition of Legendre Forms::  
+* Definition of Carlson Forms::  
+* Legendre Form of Complete Elliptic Integrals::  
+* Legendre Form of Incomplete Elliptic Integrals::  
+* Carlson Forms::               
+@end menu
+
+@node Definition of Legendre Forms
+@subsection Definition of Legendre Forms
+@cindex Legendre forms of elliptic integrals
+The Legendre forms of elliptic integrals @math{F(\phi,k)},
+@math{E(\phi,k)} and @math{\Pi(\phi,k,n)} are defined by,
+@tex
+\beforedisplay
+$$
+\eqalign{
+F(\phi,k)   &= \int_0^\phi dt {1 \over \sqrt{(1 - k^2 \sin^2(t))}}\cr
+E(\phi,k)   &= \int_0^\phi dt   \sqrt{(1 - k^2 \sin^2(t))}\cr
+\Pi(\phi,k,n) &= \int_0^\phi dt {1 \over (1 + n \sin^2(t)) \sqrt{1 - k^2 \sin^2(t)}}
+}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+  F(\phi,k) = \int_0^\phi dt 1/\sqrt((1 - k^2 \sin^2(t)))
+
+  E(\phi,k) = \int_0^\phi dt   \sqrt((1 - k^2 \sin^2(t)))
+
+Pi(\phi,k,n) = \int_0^\phi dt 1/((1 + n \sin^2(t))\sqrt(1 - k^2 \sin^2(t)))
+@end example
+
+@end ifinfo
+@noindent
+The complete Legendre forms are denoted by @math{K(k) = F(\pi/2, k)} and
+@math{E(k) = E(\pi/2, k)}.  
+
+The notation used here is based on Carlson, @cite{Numerische
+Mathematik} 33 (1979) 1 and differs slightly from that used by
+Abramowitz & Stegun, where the functions are given in terms of the
+parameter @math{m = k^2} and @math{n} is replaced by @math{-n}.
+
+@node Definition of Carlson Forms
+@subsection Definition of Carlson Forms
+@cindex Carlson forms of Elliptic integrals
+The Carlson symmetric forms of elliptical integrals @math{RC(x,y)},
+@math{RD(x,y,z)}, @math{RF(x,y,z)} and @math{RJ(x,y,z,p)} are defined
+by,
+@tex
+\beforedisplay
+$$
+\eqalign{
+RC(x,y)   &= 1/2 \int_0^\infty dt (t+x)^{-1/2} (t+y)^{-1}\cr
+RD(x,y,z) &= 3/2 \int_0^\infty dt (t+x)^{-1/2} (t+y)^{-1/2} (t+z)^{-3/2}\cr
+RF(x,y,z) &= 1/2 \int_0^\infty dt (t+x)^{-1/2} (t+y)^{-1/2} (t+z)^{-1/2}\cr
+RJ(x,y,z,p) &= 3/2 \int_0^\infty dt (t+x)^{-1/2} (t+y)^{-1/2} (t+z)^{-1/2} (t+p)^{-1}
+}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+    RC(x,y) = 1/2 \int_0^\infty dt (t+x)^(-1/2) (t+y)^(-1)
+
+  RD(x,y,z) = 3/2 \int_0^\infty dt (t+x)^(-1/2) (t+y)^(-1/2) (t+z)^(-3/2)
+
+  RF(x,y,z) = 1/2 \int_0^\infty dt (t+x)^(-1/2) (t+y)^(-1/2) (t+z)^(-1/2)
+
+RJ(x,y,z,p) = 3/2 \int_0^\infty dt 
+                 (t+x)^(-1/2) (t+y)^(-1/2) (t+z)^(-1/2) (t+p)^(-1)
+@end example
+@end ifinfo
+
+@node Legendre Form of Complete Elliptic Integrals
+@subsection Legendre Form of Complete Elliptic Integrals
+
+@deftypefun double gsl_sf_ellint_Kcomp (double @var{k}, gsl_mode_t @var{mode})
+@deftypefunx int gsl_sf_ellint_Kcomp_e (double @var{k}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result})
+These routines compute the complete elliptic integral @math{K(k)} to
+the accuracy specified by the mode variable @var{mode}.  
+Note that Abramowitz & Stegun define this function in terms of the
+parameter @math{m = k^2}.
+@comment Exceptional Return Values:  GSL_EDOM
+@end deftypefun
+
+@deftypefun double gsl_sf_ellint_Ecomp (double @var{k}, gsl_mode_t @var{mode})
+@deftypefunx int gsl_sf_ellint_Ecomp_e (double @var{k}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result})
+These routines compute the complete elliptic integral @math{E(k)} to the
+accuracy specified by the mode variable @var{mode}.
+Note that Abramowitz & Stegun define this function in terms of the
+parameter @math{m = k^2}.
+@comment Exceptional Return Values:  GSL_EDOM
+@end deftypefun
+
+@deftypefun double gsl_sf_ellint_Pcomp (double @var{k}, double @var{n}, gsl_mode_t @var{mode})
+@deftypefunx int gsl_sf_ellint_Pcomp_e (double @var{k}, double @var{n},  gsl_mode_t @var{mode}, gsl_sf_result * @var{result})
+These routines compute the complete elliptic integral @math{\Pi(k,n)} to the
+accuracy specified by the mode variable @var{mode}.
+Note that Abramowitz & Stegun define this function in terms of the
+parameters @math{m = k^2} and @math{\sin^2(\alpha) = k^2}, with the
+change of sign @math{n \to -n}.
+@comment Exceptional Return Values:  GSL_EDOM
+@end deftypefun
+
+@node Legendre Form of Incomplete Elliptic Integrals
+@subsection Legendre Form of Incomplete Elliptic Integrals
+
+@deftypefun double gsl_sf_ellint_F (double @var{phi}, double @var{k}, gsl_mode_t @var{mode})
+@deftypefunx int gsl_sf_ellint_F_e (double @var{phi}, double @var{k}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result})
+These routines compute the incomplete elliptic integral @math{F(\phi,k)}
+to the accuracy specified by the mode variable @var{mode}.
+Note that Abramowitz & Stegun define this function in terms of the
+parameter @math{m = k^2}.
+@comment Exceptional Return Values: GSL_EDOM
+@end deftypefun
+
+@deftypefun double gsl_sf_ellint_E (double @var{phi}, double @var{k}, gsl_mode_t @var{mode})
+@deftypefunx int gsl_sf_ellint_E_e (double @var{phi}, double @var{k}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result})
+These routines compute the incomplete elliptic integral @math{E(\phi,k)}
+to the accuracy specified by the mode variable @var{mode}.
+Note that Abramowitz & Stegun define this function in terms of the
+parameter @math{m = k^2}.
+@comment Exceptional Return Values: GSL_EDOM
+@end deftypefun
+
+@deftypefun double gsl_sf_ellint_P (double @var{phi}, double @var{k}, double @var{n}, gsl_mode_t @var{mode})
+@deftypefunx int gsl_sf_ellint_P_e (double @var{phi}, double @var{k}, double @var{n}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result})
+These routines compute the incomplete elliptic integral @math{\Pi(\phi,k,n)}
+to the accuracy specified by the mode variable @var{mode}.
+Note that Abramowitz & Stegun define this function in terms of the
+parameters @math{m = k^2} and @math{\sin^2(\alpha) = k^2}, with the
+change of sign @math{n \to -n}.
+@comment Exceptional Return Values: GSL_EDOM
+@end deftypefun
+
+@deftypefun double gsl_sf_ellint_D (double @var{phi}, double @var{k}, double @var{n}, gsl_mode_t @var{mode})
+@deftypefunx int gsl_sf_ellint_D_e (double @var{phi}, double @var{k}, double @var{n}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result})
+These functions compute the incomplete elliptic integral
+@math{D(\phi,k)} which is defined through the Carlson form @math{RD(x,y,z)}
+by the following relation, 
+@tex
+\beforedisplay
+$$
+D(\phi,k,n) = {1 \over 3} (\sin \phi)^3 RD (1-\sin^2(\phi), 1-k^2 \sin^2(\phi), 1).
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+D(\phi,k,n) = (1/3)(\sin(\phi))^3 RD (1-\sin^2(\phi), 1-k^2 \sin^2(\phi), 1).
+@end example
+@end ifinfo
+The argument @var{n} is not used and will be removed in a future release.
+
+@comment Exceptional Return Values: GSL_EDOM
+@end deftypefun
+
+
+@node Carlson Forms
+@subsection Carlson Forms
+
+@deftypefun double gsl_sf_ellint_RC (double @var{x}, double @var{y}, gsl_mode_t @var{mode})
+@deftypefunx int gsl_sf_ellint_RC_e (double @var{x}, double @var{y}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result})
+These routines compute the incomplete elliptic integral @math{RC(x,y)}
+to the accuracy specified by the mode variable @var{mode}.
+@comment Exceptional Return Values: GSL_EDOM
+@end deftypefun
+
+@deftypefun double gsl_sf_ellint_RD (double @var{x}, double @var{y}, double @var{z}, gsl_mode_t @var{mode})
+@deftypefunx int gsl_sf_ellint_RD_e (double @var{x}, double @var{y}, double @var{z}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result})
+These routines compute the incomplete elliptic integral @math{RD(x,y,z)}
+to the accuracy specified by the mode variable @var{mode}.
+@comment Exceptional Return Values: GSL_EDOM
+@end deftypefun
+
+@deftypefun double gsl_sf_ellint_RF (double @var{x}, double @var{y}, double @var{z}, gsl_mode_t @var{mode})
+@deftypefunx int gsl_sf_ellint_RF_e (double @var{x}, double @var{y}, double @var{z}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result})
+These routines compute the incomplete elliptic integral @math{RF(x,y,z)}
+to the accuracy specified by the mode variable @var{mode}.
+@comment Exceptional Return Values: GSL_EDOM
+@end deftypefun
+
+@deftypefun double gsl_sf_ellint_RJ (double @var{x}, double @var{y}, double @var{z}, double @var{p}, gsl_mode_t @var{mode})
+@deftypefunx int gsl_sf_ellint_RJ_e (double @var{x}, double @var{y}, double @var{z}, double @var{p}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result})
+These routines compute the incomplete elliptic integral @math{RJ(x,y,z,p)}
+to the accuracy specified by the mode variable @var{mode}.
+@comment Exceptional Return Values: GSL_EDOM
+@end deftypefun
diff --git a/gsl-1.9/doc/specfunc-elljac.texi b/gsl-1.9/doc/specfunc-elljac.texi
new file mode 100644
index 0000000..dead60f
--- /dev/null
+++ b/gsl-1.9/doc/specfunc-elljac.texi
@@ -0,0 +1,13 @@
+@cindex Jacobi elliptic functions
+@cindex elliptic functions (Jacobi)
+
+The Jacobian Elliptic functions are defined in Abramowitz & Stegun,
+Chapter 16.  The functions are declared in the header file
+@file{gsl_sf_elljac.h}.
+
+@deftypefun int gsl_sf_elljac_e (double @var{u}, double @var{m}, double * @var{sn}, double * @var{cn}, double * @var{dn})
+This function computes the Jacobian elliptic functions @math{sn(u|m)},
+@math{cn(u|m)}, @math{dn(u|m)} by descending Landen
+transformations.
+@comment Exceptional Return Values: GSL_EDOM
+@end deftypefun
diff --git a/gsl-1.9/doc/specfunc-erf.texi b/gsl-1.9/doc/specfunc-erf.texi
new file mode 100644
index 0000000..6b28cde
--- /dev/null
+++ b/gsl-1.9/doc/specfunc-erf.texi
@@ -0,0 +1,99 @@
+@cindex error function
+@cindex erf(x)
+@cindex erfc(x)
+
+The error function is described in Abramowitz & Stegun, Chapter 7.  The
+functions in this section are declared in the header file
+@file{gsl_sf_erf.h}.
+
+@menu
+* Error Function::              
+* Complementary Error Function::  
+* Log Complementary Error Function::  
+* Probability functions::       
+@end menu
+
+@node Error Function
+@subsection Error Function
+
+@deftypefun double gsl_sf_erf (double @var{x})
+@deftypefunx int gsl_sf_erf_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the error function @c{$\erf(x)$}
+@math{erf(x)}, where
+@c{$\erf(x) = (2/\sqrt{\pi}) \int_0^x dt \exp(-t^2)$}
+@math{erf(x) = (2/\sqrt(\pi)) \int_0^x dt \exp(-t^2)}.
+@comment Exceptional Return Values: none
+@end deftypefun
+
+@node Complementary Error Function
+@subsection Complementary Error Function
+
+@deftypefun double gsl_sf_erfc (double @var{x})
+@deftypefunx int gsl_sf_erfc_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the complementary error function 
+@c{$\erfc(x) = 1 - \erf(x) = (2/\sqrt{\pi}) \int_x^\infty \exp(-t^2)$}
+@math{erfc(x) = 1 - erf(x) = (2/\sqrt(\pi)) \int_x^\infty \exp(-t^2)}.
+@comment Exceptional Return Values: none
+@end deftypefun
+
+
+@node Log Complementary Error Function
+@subsection Log Complementary Error Function
+
+@deftypefun double gsl_sf_log_erfc (double @var{x})
+@deftypefunx int gsl_sf_log_erfc_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the logarithm of the complementary error function
+@math{\log(\erfc(x))}.
+@comment Exceptional Return Values: none
+@end deftypefun
+
+
+@node Probability functions
+@subsection Probability functions
+
+The probability functions for the Normal or Gaussian distribution are
+described in Abramowitz & Stegun, Section 26.2.
+
+@deftypefun double gsl_sf_erf_Z (double @var{x})
+@deftypefunx int gsl_sf_erf_Z_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the Gaussian probability density function 
+@c{$Z(x) = (1/\sqrt{2\pi}) \exp(-x^2/2)$} 
+@math{Z(x) = (1/\sqrt@{2\pi@}) \exp(-x^2/2)}.  
+@end deftypefun
+
+@deftypefun double gsl_sf_erf_Q (double @var{x})
+@deftypefunx int gsl_sf_erf_Q_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the upper tail of the Gaussian probability
+function 
+@c{$Q(x) = (1/\sqrt{2\pi}) \int_x^\infty dt \exp(-t^2/2)$}
+@math{Q(x) = (1/\sqrt@{2\pi@}) \int_x^\infty dt \exp(-t^2/2)}.
+@comment Exceptional Return Values: none
+@end deftypefun
+
+@cindex hazard function, normal distribution
+@cindex Mill's ratio, inverse
+The @dfn{hazard function} for the normal distribution, 
+also known as the inverse Mill's ratio, is defined as,
+@tex
+\beforedisplay
+$$
+h(x) = {Z(x)\over Q(x)} = \sqrt{2 \over \pi} {\exp(-x^2 / 2) \over \erfc(x/\sqrt 2)}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+h(x) = Z(x)/Q(x) = \sqrt@{2/\pi@} \exp(-x^2 / 2) / \erfc(x/\sqrt 2)
+@end example
+
+@end ifinfo
+@noindent
+It decreases rapidly as @math{x} approaches @math{-\infty} and asymptotes
+to @math{h(x) \sim x} as @math{x} approaches @math{+\infty}.
+
+@deftypefun double gsl_sf_hazard (double @var{x})
+@deftypefunx int gsl_sf_hazard_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the hazard function for the normal distribution.
+@comment Exceptional Return Values: GSL_EUNDRFLW
+@end deftypefun
diff --git a/gsl-1.9/doc/specfunc-exp.texi b/gsl-1.9/doc/specfunc-exp.texi
new file mode 100644
index 0000000..cbdf5ca
--- /dev/null
+++ b/gsl-1.9/doc/specfunc-exp.texi
@@ -0,0 +1,135 @@
+@cindex exponential function
+@cindex exp
+
+The functions described in this section are declared in the header file
+@file{gsl_sf_exp.h}.
+
+@menu
+* Exponential Function::        
+* Relative Exponential Functions::  
+* Exponentiation With Error Estimate::  
+@end menu
+
+@node Exponential Function
+@subsection Exponential Function
+
+@deftypefun double gsl_sf_exp (double @var{x})
+@deftypefunx int gsl_sf_exp_e (double @var{x}, gsl_sf_result * @var{result})
+These routines provide an exponential function @math{\exp(x)} using GSL
+semantics and error checking.
+@comment Exceptional Return Values: GSL_EOVRFLW, GSL_EUNDRFLW
+@end deftypefun
+
+@deftypefun int gsl_sf_exp_e10_e (double @var{x}, gsl_sf_result_e10 * @var{result})
+This function computes the exponential @math{\exp(x)} using the
+@code{gsl_sf_result_e10} type to return a result with extended range.
+This function may be useful if the value of @math{\exp(x)} would
+overflow the  numeric range of @code{double}.
+@comment Exceptional Return Values: GSL_EOVRFLW, GSL_EUNDRFLW
+@end deftypefun
+
+@deftypefun double gsl_sf_exp_mult (double @var{x}, double @var{y})
+@deftypefunx int gsl_sf_exp_mult_e (double @var{x}, double @var{y}, gsl_sf_result * @var{result})
+These routines exponentiate @var{x} and multiply by the factor @var{y}
+to return the product @math{y \exp(x)}.
+@comment Exceptional Return Values: GSL_EOVRFLW, GSL_EUNDRFLW
+@end deftypefun
+
+@deftypefun int gsl_sf_exp_mult_e10_e (const double @var{x}, const double @var{y}, gsl_sf_result_e10 * @var{result})
+This function computes the product @math{y \exp(x)} using the
+@code{gsl_sf_result_e10} type to return a result with extended numeric
+range.
+@comment Exceptional Return Values: GSL_EOVRFLW, GSL_EUNDRFLW
+@end deftypefun
+
+
+
+@node Relative Exponential Functions
+@subsection Relative Exponential Functions
+
+@deftypefun double gsl_sf_expm1 (double @var{x})
+@deftypefunx int gsl_sf_expm1_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the quantity @math{\exp(x)-1} using an algorithm
+that is accurate for small @math{x}.
+@comment Exceptional Return Values:  GSL_EOVRFLW
+@end deftypefun
+
+@deftypefun double gsl_sf_exprel (double @var{x})
+@deftypefunx int gsl_sf_exprel_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the quantity @math{(\exp(x)-1)/x} using an
+algorithm that is accurate for small @math{x}.  For small @math{x} the
+algorithm is based on the expansion @math{(\exp(x)-1)/x = 1 + x/2 +
+x^2/(2*3) + x^3/(2*3*4) + \dots}.
+@comment Exceptional Return Values:  GSL_EOVRFLW
+@end deftypefun
+
+@deftypefun double gsl_sf_exprel_2 (double @var{x})
+@deftypefunx int gsl_sf_exprel_2_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the quantity @math{2(\exp(x)-1-x)/x^2} using an
+algorithm that is accurate for small @math{x}.  For small @math{x} the
+algorithm is based on the expansion @math{2(\exp(x)-1-x)/x^2 = 
+1 + x/3 + x^2/(3*4) + x^3/(3*4*5) + \dots}.
+@comment Exceptional Return Values:  GSL_EOVRFLW
+@end deftypefun
+
+@deftypefun double gsl_sf_exprel_n (int @var{n}, double @var{x})
+@deftypefunx int gsl_sf_exprel_n_e (int @var{n}, double @var{x}, gsl_sf_result * @var{result})
+These routines compute the @math{N}-relative exponential, which is the
+@var{n}-th generalization of the functions @code{gsl_sf_exprel} and
+@code{gsl_sf_exprel2}.  The @math{N}-relative exponential is given by,
+@tex
+\beforedisplay
+$$
+\eqalign{
+\hbox{exprel}_N(x)
+            &= N!/x^N \left(\exp(x) - \sum_{k=0}^{N-1} x^k/k!\right)\cr
+            &= 1 + x/(N+1) + x^2/((N+1)(N+2)) + \dots\cr
+            &= {}_1F_1(1,1+N,x)\cr
+}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+exprel_N(x) = N!/x^N (\exp(x) - \sum_@{k=0@}^@{N-1@} x^k/k!)
+            = 1 + x/(N+1) + x^2/((N+1)(N+2)) + ...
+            = 1F1 (1,1+N,x)
+@end example
+@end ifinfo
+@comment Exceptional Return Values: 
+@end deftypefun
+
+
+
+@node Exponentiation With Error Estimate
+@subsection Exponentiation With Error Estimate
+
+
+@deftypefun int gsl_sf_exp_err_e (double @var{x}, double @var{dx}, gsl_sf_result * @var{result})
+This function exponentiates @var{x} with an associated absolute error
+@var{dx}.
+@comment Exceptional Return Values: 
+@end deftypefun
+
+@deftypefun int gsl_sf_exp_err_e10_e (double @var{x}, double @var{dx}, gsl_sf_result_e10 * @var{result})
+This function exponentiates a quantity @var{x} with an associated absolute 
+error @var{dx} using the @code{gsl_sf_result_e10} type to return a result with
+extended range.
+@comment Exceptional Return Values: 
+@end deftypefun
+
+@deftypefun int gsl_sf_exp_mult_err_e (double @var{x}, double @var{dx}, double @var{y}, double @var{dy}, gsl_sf_result * @var{result})
+This routine computes the product @math{y \exp(x)} for the quantities
+@var{x}, @var{y} with associated absolute errors @var{dx}, @var{dy}.
+@comment Exceptional Return Values: GSL_EOVRFLW, GSL_EUNDRFLW
+@end deftypefun
+
+
+@deftypefun int gsl_sf_exp_mult_err_e10_e (double @var{x}, double @var{dx}, double @var{y}, double @var{dy}, gsl_sf_result_e10 * @var{result})
+This routine computes the product @math{y \exp(x)} for the quantities
+@var{x}, @var{y} with associated absolute errors @var{dx}, @var{dy} using the
+@code{gsl_sf_result_e10} type to return a result with extended range.
+@comment Exceptional Return Values: GSL_EOVRFLW, GSL_EUNDRFLW
+@end deftypefun
+
diff --git a/gsl-1.9/doc/specfunc-expint.texi b/gsl-1.9/doc/specfunc-expint.texi
new file mode 100644
index 0000000..2d68ccb
--- /dev/null
+++ b/gsl-1.9/doc/specfunc-expint.texi
@@ -0,0 +1,155 @@
+@cindex exponential integrals
+@cindex integrals, exponential
+
+Information on the exponential integrals can be found in Abramowitz &
+Stegun, Chapter 5.  These functions are declared in the header file
+@file{gsl_sf_expint.h}.
+
+@menu
+* Exponential Integral::        
+* Ei(x)::                       
+* Hyperbolic Integrals::        
+* Ei_3(x)::                     
+* Trigonometric Integrals::     
+* Arctangent Integral::         
+@end menu
+
+@node Exponential Integral
+@subsection Exponential Integral
+@cindex E1(x), E2(x), Ei(x)
+
+@deftypefun double gsl_sf_expint_E1 (double @var{x})
+@deftypefunx int gsl_sf_expint_E1_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the exponential integral @math{E_1(x)},
+@tex
+\beforedisplay
+$$
+E_1(x) := \Re \int_1^\infty dt \exp(-xt)/t.
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+E_1(x) := \Re \int_1^\infty dt \exp(-xt)/t.
+@end example
+
+@end ifinfo
+@noindent
+@comment Domain: x != 0.0
+@comment Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW, GSL_EUNDRFLW
+@end deftypefun
+
+@deftypefun double gsl_sf_expint_E2 (double @var{x})
+@deftypefunx int gsl_sf_expint_E2_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the second-order exponential integral @math{E_2(x)},
+@tex
+\beforedisplay
+$$
+E_2(x) := \Re \int_1^\infty dt \exp(-xt)/t^2.
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+E_2(x) := \Re \int_1^\infty dt \exp(-xt)/t^2.
+@end example
+
+@end ifinfo
+@noindent
+@comment Domain: x != 0.0
+@comment Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW, GSL_EUNDRFLW
+@end deftypefun
+
+@node Ei(x)
+@subsection Ei(x)
+
+@deftypefun double gsl_sf_expint_Ei (double @var{x})
+@deftypefunx int gsl_sf_expint_Ei_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the exponential integral @math{Ei(x)},
+@tex
+\beforedisplay
+$$
+Ei(x) := - PV\left(\int_{-x}^\infty dt \exp(-t)/t\right)
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+Ei(x) := - PV(\int_@{-x@}^\infty dt \exp(-t)/t)
+@end example
+
+@end ifinfo
+@noindent
+where @math{PV} denotes the principal value of the integral.
+@comment Domain: x != 0.0
+@comment Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW, GSL_EUNDRFLW
+@end deftypefun
+
+
+@node Hyperbolic Integrals
+@subsection Hyperbolic Integrals
+@cindex hyperbolic integrals
+@cindex Shi(x)
+@cindex Chi(x)
+
+@deftypefun double gsl_sf_Shi (double @var{x})
+@deftypefunx int gsl_sf_Shi_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the integral @math{Shi(x) = \int_0^x dt \sinh(t)/t}.
+@comment Exceptional Return Values: GSL_EOVRFLW, GSL_EUNDRFLW
+@end deftypefun
+
+
+@deftypefun double gsl_sf_Chi (double @var{x})
+@deftypefunx int gsl_sf_Chi_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the integral @math{ Chi(x) := \Re[ \gamma_E + \log(x) + \int_0^x dt (\cosh[t]-1)/t] }, where @math{\gamma_E} is the Euler constant (available as the macro @code{M_EULER}).
+@comment Domain: x != 0.0
+@comment Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW, GSL_EUNDRFLW
+@end deftypefun
+
+
+@node Ei_3(x)
+@subsection Ei_3(x)
+
+@deftypefun double gsl_sf_expint_3 (double @var{x})
+@deftypefunx int gsl_sf_expint_3_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the third-order exponential integral 
+@math{Ei_3(x) = \int_0^xdt \exp(-t^3)} for @c{$x \ge 0$}
+@math{x >= 0}.
+@comment Exceptional Return Values: GSL_EDOM
+@end deftypefun
+
+@node Trigonometric Integrals
+@subsection Trigonometric Integrals
+@cindex trigonometric integrals
+@cindex Si(x)
+@cindex Ci(x)
+@deftypefun double gsl_sf_Si (const double @var{x})
+@deftypefunx int gsl_sf_Si_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the Sine integral
+@math{Si(x) = \int_0^x dt \sin(t)/t}.
+@comment Exceptional Return Values: none
+@end deftypefun
+
+ 
+@deftypefun double gsl_sf_Ci (const double @var{x})
+@deftypefunx int gsl_sf_Ci_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the Cosine integral @math{Ci(x) = -\int_x^\infty dt
+\cos(t)/t} for @math{x > 0}.  
+@comment Domain: x > 0.0
+@comment Exceptional Return Values: GSL_EDOM
+@end deftypefun
+
+
+@node Arctangent Integral
+@subsection Arctangent Integral
+@cindex arctangent integral
+@deftypefun double gsl_sf_atanint (double @var{x})
+@deftypefunx int gsl_sf_atanint_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the Arctangent integral, which is defined as @math{AtanInt(x) = \int_0^x dt \arctan(t)/t}.
+@comment Domain: 
+@comment Exceptional Return Values: 
+@end deftypefun
+
diff --git a/gsl-1.9/doc/specfunc-fermi-dirac.texi b/gsl-1.9/doc/specfunc-fermi-dirac.texi
new file mode 100644
index 0000000..f9314be
--- /dev/null
+++ b/gsl-1.9/doc/specfunc-fermi-dirac.texi
@@ -0,0 +1,122 @@
+@cindex Fermi-Dirac function
+
+The functions described in this section are declared in the header file
+@file{gsl_sf_fermi_dirac.h}.
+
+@menu
+* Complete Fermi-Dirac Integrals::  
+* Incomplete Fermi-Dirac Integrals::  
+@end menu
+
+@node Complete Fermi-Dirac Integrals
+@subsection Complete Fermi-Dirac Integrals
+@cindex complete Fermi-Dirac integrals
+@cindex Fj(x), Fermi-Dirac integral
+The complete Fermi-Dirac integral @math{F_j(x)} is given by,
+@tex
+\beforedisplay
+$$
+F_j(x)   := {1\over\Gamma(j+1)} \int_0^\infty dt {t^j  \over (\exp(t-x) + 1)}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+F_j(x)   := (1/r\Gamma(j+1)) \int_0^\infty dt (t^j / (\exp(t-x) + 1))
+@end example
+@end ifinfo
+
+@deftypefun double gsl_sf_fermi_dirac_m1 (double @var{x})
+@deftypefunx int gsl_sf_fermi_dirac_m1_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the complete Fermi-Dirac integral with an index of @math{-1}. 
+This integral is given by 
+@c{$F_{-1}(x) = e^x / (1 + e^x)$}
+@math{F_@{-1@}(x) = e^x / (1 + e^x)}.
+@comment Exceptional Return Values: GSL_EUNDRFLW
+@end deftypefun
+
+@deftypefun double gsl_sf_fermi_dirac_0 (double @var{x})
+@deftypefunx int gsl_sf_fermi_dirac_0_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the complete Fermi-Dirac integral with an index of @math{0}. 
+This integral is given by @math{F_0(x) = \ln(1 + e^x)}.
+@comment Exceptional Return Values: GSL_EUNDRFLW
+@end deftypefun
+
+@deftypefun double gsl_sf_fermi_dirac_1 (double @var{x})
+@deftypefunx int gsl_sf_fermi_dirac_1_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the complete Fermi-Dirac integral with an index of @math{1},
+@math{F_1(x) = \int_0^\infty dt (t /(\exp(t-x)+1))}.
+@comment Exceptional Return Values: GSL_EUNDRFLW, GSL_EOVRFLW
+@end deftypefun
+
+@deftypefun double gsl_sf_fermi_dirac_2 (double @var{x})
+@deftypefunx int gsl_sf_fermi_dirac_2_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the complete Fermi-Dirac integral with an index
+of @math{2},
+@math{F_2(x) = (1/2) \int_0^\infty dt (t^2 /(\exp(t-x)+1))}.
+@comment Exceptional Return Values: GSL_EUNDRFLW, GSL_EOVRFLW
+@end deftypefun
+
+@deftypefun double gsl_sf_fermi_dirac_int (int @var{j}, double @var{x})
+@deftypefunx int gsl_sf_fermi_dirac_int_e (int @var{j}, double @var{x}, gsl_sf_result * @var{result})
+These routines compute the complete Fermi-Dirac integral with an integer
+index of @math{j},
+@math{F_j(x) = (1/\Gamma(j+1)) \int_0^\infty dt (t^j /(\exp(t-x)+1))}.
+@comment Complete integral F_j(x) for integer j
+@comment Exceptional Return Values: GSL_EUNDRFLW, GSL_EOVRFLW
+@end deftypefun
+
+@deftypefun double gsl_sf_fermi_dirac_mhalf (double @var{x})
+@deftypefunx int gsl_sf_fermi_dirac_mhalf_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the complete Fermi-Dirac integral 
+@c{$F_{-1/2}(x)$}
+@math{F_@{-1/2@}(x)}.
+@comment Exceptional Return Values: GSL_EUNDRFLW, GSL_EOVRFLW
+@end deftypefun
+
+@deftypefun double gsl_sf_fermi_dirac_half (double @var{x})
+@deftypefunx int gsl_sf_fermi_dirac_half_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the complete Fermi-Dirac integral 
+@c{$F_{1/2}(x)$}
+@math{F_@{1/2@}(x)}.
+@comment Exceptional Return Values: GSL_EUNDRFLW, GSL_EOVRFLW
+@end deftypefun
+
+@deftypefun double gsl_sf_fermi_dirac_3half (double @var{x})
+@deftypefunx int gsl_sf_fermi_dirac_3half_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the complete Fermi-Dirac integral 
+@c{$F_{3/2}(x)$}
+@math{F_@{3/2@}(x)}.
+@comment Exceptional Return Values: GSL_EUNDRFLW, GSL_EOVRFLW
+@end deftypefun
+
+
+@node Incomplete Fermi-Dirac Integrals
+@subsection Incomplete Fermi-Dirac Integrals
+@cindex incomplete Fermi-Dirac integral
+@cindex Fj(x,b), incomplete Fermi-Dirac integral
+The incomplete Fermi-Dirac integral @math{F_j(x,b)} is given by,
+@tex
+\beforedisplay
+$$
+F_j(x,b)   := {1\over\Gamma(j+1)} \int_b^\infty dt {t^j  \over (\exp(t-x) + 1)}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+F_j(x,b)   := (1/\Gamma(j+1)) \int_b^\infty dt (t^j / (\Exp(t-x) + 1))
+@end example
+@end ifinfo
+
+@deftypefun double gsl_sf_fermi_dirac_inc_0 (double @var{x}, double @var{b})
+@deftypefunx int gsl_sf_fermi_dirac_inc_0_e (double @var{x}, double @var{b}, gsl_sf_result * @var{result})
+These routines compute the incomplete Fermi-Dirac integral with an index
+of zero,
+@c{$F_0(x,b) = \ln(1 + e^{b-x}) - (b-x)$}
+@math{F_0(x,b) = \ln(1 + e^@{b-x@}) - (b-x)}.
+@comment Exceptional Return Values: GSL_EUNDRFLW, GSL_EDOM
+@end deftypefun
+
diff --git a/gsl-1.9/doc/specfunc-gamma.texi b/gsl-1.9/doc/specfunc-gamma.texi
new file mode 100644
index 0000000..f15e7ef
--- /dev/null
+++ b/gsl-1.9/doc/specfunc-gamma.texi
@@ -0,0 +1,314 @@
+The  functions described in this section are declared
+in the header file @file{gsl_sf_gamma.h}.
+
+@menu
+* Gamma Functions::             
+* Factorials::                  
+* Pochhammer Symbol::           
+* Incomplete Gamma Functions::  
+* Beta Functions::              
+* Incomplete Beta Function::    
+@end menu
+
+@node Gamma Functions
+@subsection Gamma Functions
+@cindex gamma functions
+
+The Gamma function is defined by the following integral,
+@tex
+\beforedisplay
+$$
+\Gamma(x) = \int_0^{\infty} dt \, t^{x-1} \exp(-t)
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+\Gamma(x) = \int_0^\infty dt  t^@{x-1@} \exp(-t)
+@end example
+
+@end ifinfo
+@noindent
+It is related to the factorial function by @math{\Gamma(n)=(n-1)!}
+for positive integer @math{n}.  Further information on the Gamma function
+can be found in Abramowitz & Stegun, Chapter 6.  The functions
+described in this section are declared in the header file
+@file{gsl_sf_gamma.h}.
+
+@deftypefun double gsl_sf_gamma (double @var{x})
+@deftypefunx int gsl_sf_gamma_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the Gamma function @math{\Gamma(x)}, subject to @math{x}
+not being a negative integer or zero.  The function is computed using the real
+Lanczos method. The maximum value of @math{x} such that @math{\Gamma(x)} is not
+considered an overflow is given by the macro @code{GSL_SF_GAMMA_XMAX}
+and is 171.0.
+@comment exceptions: GSL_EDOM, GSL_EOVRFLW, GSL_EROUND
+@end deftypefun
+
+@deftypefun double gsl_sf_lngamma (double @var{x})
+@deftypefunx int gsl_sf_lngamma_e (double @var{x}, gsl_sf_result * @var{result})
+@cindex logarithm of Gamma function
+These routines compute the logarithm of the Gamma function,
+@math{\log(\Gamma(x))}, subject to @math{x} not being a negative
+integer or zero.  For @math{x<0} the real part of @math{\log(\Gamma(x))} is
+returned, which is equivalent to @math{\log(|\Gamma(x)|)}.  The function
+is computed using the real Lanczos method.
+@comment exceptions: GSL_EDOM, GSL_EROUND
+@end deftypefun
+
+@deftypefun int gsl_sf_lngamma_sgn_e (double @var{x}, gsl_sf_result * @var{result_lg}, double * @var{sgn})
+This routine computes the sign of the gamma function and the logarithm of
+its magnitude, subject to @math{x} not being a negative integer or zero.  The
+function is computed using the real Lanczos method.  The value of the
+gamma function can be reconstructed using the relation @math{\Gamma(x) =
+sgn * \exp(resultlg)}.
+@comment exceptions: GSL_EDOM, GSL_EROUND
+@end deftypefun
+
+@deftypefun double gsl_sf_gammastar (double @var{x})
+@deftypefunx int gsl_sf_gammastar_e (double @var{x}, gsl_sf_result * @var{result})
+@cindex Regulated Gamma function
+These routines compute the regulated Gamma Function @math{\Gamma^*(x)}
+for @math{x > 0}. The regulated gamma function is given by,
+@tex
+\beforedisplay
+$$
+\eqalign{
+\Gamma^*(x) &= \Gamma(x)/(\sqrt{2\pi} x^{(x-1/2)} \exp(-x))\cr
+            &= \left(1 + {1 \over 12x} + ...\right) \quad\hbox{for~} x\to \infty\cr
+}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+\Gamma^*(x) = \Gamma(x)/(\sqrt@{2\pi@} x^@{(x-1/2)@} \exp(-x))
+            = (1 + (1/12x) + ...)  for x \to \infty
+@end example
+@end ifinfo
+and is a useful suggestion of Temme.
+@comment exceptions: GSL_EDOM
+@end deftypefun
+
+@deftypefun double gsl_sf_gammainv (double @var{x})
+@deftypefunx int gsl_sf_gammainv_e (double @var{x}, gsl_sf_result * @var{result})
+@cindex Reciprocal Gamma function
+These routines compute the reciprocal of the gamma function,
+@math{1/\Gamma(x)} using the real Lanczos method.
+@comment exceptions: GSL_EUNDRFLW, GSL_EROUND
+@end deftypefun
+
+@deftypefun int gsl_sf_lngamma_complex_e (double @var{zr}, double @var{zi}, gsl_sf_result * @var{lnr}, gsl_sf_result * @var{arg})
+@cindex Complex Gamma function
+This routine computes @math{\log(\Gamma(z))} for complex @math{z=z_r+i
+z_i} and @math{z} not a negative integer or zero, using the complex Lanczos
+method.  The returned parameters are @math{lnr = \log|\Gamma(z)|} and
+@math{arg = \arg(\Gamma(z))} in @math{(-\pi,\pi]}.  Note that the phase
+part (@var{arg}) is not well-determined when @math{|z|} is very large,
+due to inevitable roundoff in restricting to @math{(-\pi,\pi]}.  This
+will result in a @code{GSL_ELOSS} error when it occurs.  The absolute
+value part (@var{lnr}), however, never suffers from loss of precision.
+@comment exceptions: GSL_EDOM, GSL_ELOSS
+@end deftypefun
+
+@node Factorials
+@subsection Factorials
+@cindex factorial
+
+Although factorials can be computed from the Gamma function, using
+the relation @math{n! = \Gamma(n+1)} for non-negative integer @math{n},
+it is usually more efficient to call the functions in this section,
+particularly for small values of @math{n}, whose factorial values are
+maintained in hardcoded tables.
+
+@deftypefun double gsl_sf_fact (unsigned int @var{n})
+@deftypefunx int gsl_sf_fact_e (unsigned int @var{n}, gsl_sf_result * @var{result})
+@cindex factorial
+These routines compute the factorial @math{n!}.  The factorial is
+related to the Gamma function by @math{n! = \Gamma(n+1)}.
+The maximum value of @math{n} such that @math{n!} is not
+considered an overflow is given by the macro @code{GSL_SF_FACT_NMAX}
+and is 170.
+@comment exceptions: GSL_EDOM, GSL_OVRFLW
+@end deftypefun
+
+@deftypefun double gsl_sf_doublefact (unsigned int @var{n})
+@deftypefunx int gsl_sf_doublefact_e (unsigned int @var{n}, gsl_sf_result * @var{result})
+@cindex double factorial
+These routines compute the double factorial @math{n!! = n(n-2)(n-4) \dots}. 
+The maximum value of @math{n} such that @math{n!!} is not
+considered an overflow is given by the macro @code{GSL_SF_DOUBLEFACT_NMAX}
+and is 297.
+@comment exceptions: GSL_EDOM, GSL_OVRFLW
+@end deftypefun
+
+@deftypefun double gsl_sf_lnfact (unsigned int @var{n})
+@deftypefunx int gsl_sf_lnfact_e (unsigned int @var{n}, gsl_sf_result * @var{result})
+@cindex logarithm of factorial
+These routines compute the logarithm of the factorial of @var{n},
+@math{\log(n!)}.  The algorithm is faster than computing
+@math{\ln(\Gamma(n+1))} via @code{gsl_sf_lngamma} for @math{n < 170},
+but defers for larger @var{n}.
+@comment exceptions: none
+@end deftypefun
+
+@deftypefun double gsl_sf_lndoublefact (unsigned int @var{n})
+@deftypefunx int gsl_sf_lndoublefact_e (unsigned int @var{n}, gsl_sf_result * @var{result})
+@cindex logarithm of double factorial
+These routines compute the logarithm of the double factorial of @var{n},
+@math{\log(n!!)}.
+@comment exceptions: none
+@end deftypefun
+
+@deftypefun double gsl_sf_choose (unsigned int @var{n}, unsigned int @var{m})
+@deftypefunx int gsl_sf_choose_e (unsigned int @var{n}, unsigned int @var{m}, gsl_sf_result * @var{result})
+@cindex combinatorial factor C(m,n)
+These routines compute the combinatorial factor @code{n choose m}
+@math{= n!/(m!(n-m)!)}
+@comment exceptions: GSL_EDOM, GSL_EOVRFLW
+@end deftypefun
+
+
+@deftypefun double gsl_sf_lnchoose (unsigned int @var{n}, unsigned int @var{m})
+@deftypefunx int gsl_sf_lnchoose_e (unsigned int @var{n}, unsigned int @var{m}, gsl_sf_result * @var{result})
+@cindex logarithm of combinatorial factor C(m,n)
+These routines compute the logarithm of @code{n choose m}.  This is
+equivalent to the sum @math{\log(n!) - \log(m!) - \log((n-m)!)}.
+@comment exceptions: GSL_EDOM 
+@end deftypefun
+
+@deftypefun double gsl_sf_taylorcoeff (int @var{n}, double @var{x})
+@deftypefunx int gsl_sf_taylorcoeff_e (int @var{n}, double @var{x}, gsl_sf_result * @var{result})
+@cindex Taylor coefficients, computation of
+These routines compute the Taylor coefficient @math{x^n / n!} for 
+@c{$x \ge 0$}
+@math{x >= 0}, 
+@c{$n \ge 0$}
+@math{n >= 0}.
+@comment exceptions: GSL_EDOM, GSL_EOVRFLW, GSL_EUNDRFLW
+@end deftypefun
+
+@node Pochhammer Symbol
+@subsection Pochhammer Symbol
+
+@deftypefun double gsl_sf_poch (double @var{a}, double @var{x})
+@deftypefunx int gsl_sf_poch_e (double @var{a}, double @var{x}, gsl_sf_result * @var{result})
+@cindex Pochhammer symbol
+@cindex Apell symbol, see Pochammer symbol
+These routines compute the Pochhammer symbol @math{(a)_x = \Gamma(a +
+x)/\Gamma(a)}, subject to @math{a} and @math{a+x} not being negative
+integers or zero. The Pochhammer symbol is also known as the Apell symbol and
+sometimes written as @math{(a,x)}.
+@comment exceptions:  GSL_EDOM, GSL_EOVRFLW
+@end deftypefun
+
+
+@deftypefun double gsl_sf_lnpoch (double @var{a}, double @var{x})
+@deftypefunx int gsl_sf_lnpoch_e (double @var{a}, double @var{x}, gsl_sf_result * @var{result})
+@cindex logarithm of Pochhammer symbol
+These routines compute the logarithm of the Pochhammer symbol,
+@math{\log((a)_x) = \log(\Gamma(a + x)/\Gamma(a))} for @math{a > 0},
+@math{a+x > 0}.
+@comment exceptions:  GSL_EDOM
+@end deftypefun
+
+@deftypefun int gsl_sf_lnpoch_sgn_e (double @var{a}, double @var{x}, gsl_sf_result * @var{result}, double * @var{sgn})
+These routines compute the sign of the Pochhammer symbol and the
+logarithm of its magnitude.  The computed parameters are @math{result =
+\log(|(a)_x|)} and @math{sgn = \sgn((a)_x)} where @math{(a)_x =
+\Gamma(a + x)/\Gamma(a)}, subject to @math{a}, @math{a+x} not being
+negative integers or zero.
+@comment exceptions:  GSL_EDOM
+@end deftypefun
+
+@deftypefun double gsl_sf_pochrel (double @var{a}, double @var{x})
+@deftypefunx int gsl_sf_pochrel_e (double @var{a}, double @var{x}, gsl_sf_result * @var{result})
+@cindex relative Pochhammer symbol
+These routines compute the relative Pochhammer symbol @math{((a)_x -
+1)/x} where @math{(a)_x = \Gamma(a + x)/\Gamma(a)}.
+@comment exceptions:  GSL_EDOM
+@end deftypefun
+
+
+@node Incomplete Gamma Functions
+@subsection Incomplete Gamma Functions
+
+@deftypefun double gsl_sf_gamma_inc (double @var{a}, double @var{x})
+@deftypefunx int gsl_sf_gamma_inc_e (double @var{a}, double @var{x}, gsl_sf_result * @var{result})
+@cindex non-normalized incomplete Gamma function
+@cindex unnormalized incomplete Gamma function
+These functions compute the unnormalized incomplete Gamma Function
+@c{$\Gamma(a,x) = \int_x^\infty dt\, t^{(a-1)} \exp(-t)$}
+@math{\Gamma(a,x) = \int_x^\infty dt t^@{a-1@} \exp(-t)}
+for @math{a} real and @c{$x \ge 0$}
+@math{x >= 0}.
+@comment exceptions: GSL_EDOM
+@end deftypefun
+
+@deftypefun double gsl_sf_gamma_inc_Q (double @var{a}, double @var{x})
+@deftypefunx int gsl_sf_gamma_inc_Q_e (double @var{a}, double @var{x}, gsl_sf_result * @var{result})
+@cindex incomplete Gamma function
+These routines compute the normalized incomplete Gamma Function
+@c{$Q(a,x) = 1/\Gamma(a) \int_x^\infty dt\, t^{(a-1)} \exp(-t)$}
+@math{Q(a,x) = 1/\Gamma(a) \int_x^\infty dt t^@{a-1@} \exp(-t)}
+for @math{a > 0}, @c{$x \ge 0$}
+@math{x >= 0}.
+@comment exceptions: GSL_EDOM
+@end deftypefun
+
+@deftypefun double gsl_sf_gamma_inc_P (double @var{a}, double @var{x})
+@deftypefunx int gsl_sf_gamma_inc_P_e (double @var{a}, double @var{x}, gsl_sf_result * @var{result})
+@cindex complementary incomplete Gamma function
+These routines compute the complementary normalized incomplete Gamma Function
+@c{$P(a,x) = 1 - Q(a,x) = 1/\Gamma(a) \int_0^x dt\, t^{(a-1)} \exp(-t)$}
+@math{P(a,x) = 1 - Q(a,x) = 1/\Gamma(a) \int_0^x dt t^@{a-1@} \exp(-t)}
+for @math{a > 0}, @c{$x \ge 0$}
+@math{x >= 0}. 
+
+Note that Abramowitz & Stegun call @math{P(a,x)} the incomplete gamma
+function (section 6.5).
+@comment exceptions: GSL_EDOM
+@end deftypefun
+
+@node Beta Functions
+@subsection Beta Functions
+
+@deftypefun double gsl_sf_beta (double @var{a}, double @var{b})
+@deftypefunx int gsl_sf_beta_e (double @var{a}, double @var{b}, gsl_sf_result * @var{result})
+@cindex Beta function
+These routines compute the Beta Function, @math{B(a,b) =
+\Gamma(a)\Gamma(b)/\Gamma(a+b)} subject to @math{a} and @math{b} not
+being negative integers.
+@comment exceptions: GSL_EDOM, GSL_EOVRFLW, GSL_EUNDRFLW
+@end deftypefun
+
+@deftypefun double gsl_sf_lnbeta (double @var{a}, double @var{b})
+@deftypefunx int gsl_sf_lnbeta_e (double @var{a}, double @var{b}, gsl_sf_result * @var{result})
+@cindex logarithm of Beta function
+These routines compute the logarithm of the Beta Function, @math{\log(B(a,b))}
+subject to @math{a} and @math{b} not
+being negative integers.
+@comment exceptions: GSL_EDOM
+@end deftypefun
+
+@node Incomplete Beta Function
+@subsection Incomplete Beta Function
+
+@deftypefun double gsl_sf_beta_inc (double @var{a}, double @var{b}, double @var{x})
+@deftypefunx int gsl_sf_beta_inc_e (double @var{a}, double @var{b}, double @var{x}, gsl_sf_result * @var{result})
+@cindex incomplete Beta function, normalized
+@cindex normalized incomplete Beta function
+@cindex Beta function, incomplete normalized 
+These routines compute the normalized incomplete Beta function
+@math{I_x(a,b)=B_x(a,b)/B(a,b)} where @c{$B_x(a,b) = \int_0^x t^{a-1} (1-t)^{b-1} dt$}
+@math{B_x(a,b) = \int_0^x t^@{a-1@} (1-t)^@{b-1@} dt}
+for @math{a > 0}, @math{b > 0}, and @c{$0 \le x \le 1$}
+@math{0 <= x <= 1}. 
+@end deftypefun
+
+
+
+
+
diff --git a/gsl-1.9/doc/specfunc-gegenbauer.texi b/gsl-1.9/doc/specfunc-gegenbauer.texi
new file mode 100644
index 0000000..949c3dd
--- /dev/null
+++ b/gsl-1.9/doc/specfunc-gegenbauer.texi
@@ -0,0 +1,42 @@
+@cindex Gegenbauer functions
+
+The Gegenbauer polynomials are defined in Abramowitz & Stegun, Chapter
+22, where they are known as Ultraspherical polynomials.  The functions
+described in this section are declared in the header file
+@file{gsl_sf_gegenbauer.h}.
+
+@deftypefun double gsl_sf_gegenpoly_1 (double @var{lambda}, double @var{x})
+@deftypefunx double gsl_sf_gegenpoly_2 (double @var{lambda}, double @var{x})
+@deftypefunx double gsl_sf_gegenpoly_3 (double @var{lambda}, double @var{x})
+@deftypefunx int gsl_sf_gegenpoly_1_e (double @var{lambda}, double @var{x}, gsl_sf_result * @var{result})
+@deftypefunx int gsl_sf_gegenpoly_2_e (double @var{lambda}, double @var{x}, gsl_sf_result * @var{result})
+@deftypefunx int gsl_sf_gegenpoly_3_e (double @var{lambda}, double @var{x}, gsl_sf_result * @var{result})
+These functions evaluate the Gegenbauer polynomials
+@c{$C^{(\lambda)}_n(x)$} 
+@math{C^@{(\lambda)@}_n(x)} using explicit
+representations for @math{n =1, 2, 3}.
+@comment Exceptional Return Values: none
+@end deftypefun
+
+
+@deftypefun double gsl_sf_gegenpoly_n (int @var{n}, double @var{lambda}, double @var{x})
+@deftypefunx int gsl_sf_gegenpoly_n_e (int @var{n}, double @var{lambda}, double @var{x}, gsl_sf_result * @var{result})
+These functions evaluate the Gegenbauer polynomial @c{$C^{(\lambda)}_n(x)$} 
+@math{C^@{(\lambda)@}_n(x)} for a specific value of @var{n},
+@var{lambda}, @var{x} subject to @math{\lambda > -1/2}, @c{$n \ge 0$}
+@math{n >= 0}.
+@comment Domain: lambda > -1/2, n >= 0
+@comment Exceptional Return Values: GSL_EDOM
+@end deftypefun
+
+
+@deftypefun int gsl_sf_gegenpoly_array (int @var{nmax}, double @var{lambda}, double @var{x}, double @var{result_array}[])
+This function computes an array of Gegenbauer polynomials
+@c{$C^{(\lambda)}_n(x)$} 
+@math{C^@{(\lambda)@}_n(x)} for @math{n = 0, 1, 2, \dots, nmax}, subject
+to @math{\lambda > -1/2}, @c{$nmax \ge 0$}
+@math{nmax >= 0}.
+@comment Conditions: n = 0, 1, 2, ... nmax
+@comment Domain: lambda > -1/2, nmax >= 0
+@comment Exceptional Return Values: GSL_EDOM
+@end deftypefun
diff --git a/gsl-1.9/doc/specfunc-hyperg.texi b/gsl-1.9/doc/specfunc-hyperg.texi
new file mode 100644
index 0000000..c67eb8c
--- /dev/null
+++ b/gsl-1.9/doc/specfunc-hyperg.texi
@@ -0,0 +1,108 @@
+@cindex hypergeometric functions
+@cindex confluent hypergeometric functions
+
+Hypergeometric functions are described in Abramowitz & Stegun, Chapters
+13 and 15.  These functions are declared in the header file
+@file{gsl_sf_hyperg.h}.
+
+@deftypefun double gsl_sf_hyperg_0F1 (double @var{c}, double @var{x})
+@deftypefunx int gsl_sf_hyperg_0F1_e (double @var{c}, double @var{x}, gsl_sf_result * @var{result})
+These routines compute the hypergeometric function @c{${}_0F_1(c,x)$}
+@math{0F1(c,x)}.  
+@comment It is related to Bessel functions
+@comment 0F1[c,x] =
+@comment   Gamma[c]    x^(1/2(1-c)) I_(c-1)(2 Sqrt[x])
+@comment   Gamma[c] (-x)^(1/2(1-c)) J_(c-1)(2 Sqrt[-x])
+@comment exceptions: GSL_EOVRFLW, GSL_EUNDRFLW
+@end deftypefun
+
+@deftypefun double gsl_sf_hyperg_1F1_int (int @var{m}, int @var{n}, double @var{x})
+@deftypefunx int gsl_sf_hyperg_1F1_int_e (int @var{m}, int @var{n}, double @var{x}, gsl_sf_result * @var{result})
+These routines compute the confluent hypergeometric function
+@c{${}_1F_1(m,n,x) = M(m,n,x)$}
+@math{1F1(m,n,x) = M(m,n,x)} for integer parameters @var{m}, @var{n}.
+@comment exceptions: 
+@end deftypefun
+
+@deftypefun double gsl_sf_hyperg_1F1 (double @var{a}, double @var{b}, double @var{x})
+@deftypefunx int gsl_sf_hyperg_1F1_e (double @var{a}, double @var{b}, double @var{x}, gsl_sf_result * @var{result})
+These routines compute the confluent hypergeometric function
+@c{${}_1F_1(a,b,x) = M(a,b,x)$}
+@math{1F1(a,b,x) = M(a,b,x)} for general parameters @var{a}, @var{b}.
+@comment exceptions:
+@end deftypefun
+
+@deftypefun double gsl_sf_hyperg_U_int (int @var{m}, int @var{n}, double @var{x})
+@deftypefunx int gsl_sf_hyperg_U_int_e (int @var{m}, int @var{n}, double @var{x}, gsl_sf_result * @var{result})
+These routines compute the confluent hypergeometric function
+@math{U(m,n,x)} for integer parameters @var{m}, @var{n}.
+@comment exceptions:
+@end deftypefun
+
+@deftypefun int gsl_sf_hyperg_U_int_e10_e (int @var{m}, int @var{n}, double @var{x}, gsl_sf_result_e10 * @var{result})
+This routine computes the confluent hypergeometric function
+@math{U(m,n,x)} for integer parameters @var{m}, @var{n} using the
+@code{gsl_sf_result_e10} type to return a result with extended range.
+@end deftypefun
+
+@deftypefun double gsl_sf_hyperg_U (double @var{a}, double @var{b}, double @var{x})
+@deftypefunx int gsl_sf_hyperg_U_e (double @var{a}, double @var{b}, double @var{x}, gsl_sf_result * @var{result})
+These routines compute the confluent hypergeometric function @math{U(a,b,x)}.
+@comment exceptions:
+@end deftypefun
+
+@deftypefun int gsl_sf_hyperg_U_e10_e (double @var{a}, double @var{b}, double @var{x}, gsl_sf_result_e10 * @var{result})
+This routine computes the confluent hypergeometric function
+@math{U(a,b,x)} using the @code{gsl_sf_result_e10} type to return a
+result with extended range. 
+@comment exceptions:
+@end deftypefun
+
+@deftypefun double gsl_sf_hyperg_2F1 (double @var{a}, double @var{b}, double @var{c}, double @var{x})
+@deftypefunx int gsl_sf_hyperg_2F1_e (double @var{a}, double @var{b}, double @var{c}, double @var{x}, gsl_sf_result * @var{result})
+These routines compute the Gauss hypergeometric function 
+@c{${}_2F_1(a,b,c,x)$}
+@math{2F1(a,b,c,x)} for @math{|x| < 1}.  
+
+If the arguments @math{(a,b,c,x)} are too close to a singularity then
+the function can return the error code @code{GSL_EMAXITER} when the
+series approximation converges too slowly.  This occurs in the region of
+@math{x=1}, @math{c - a - b = m} for integer m.
+@comment exceptions:
+@end deftypefun
+
+@deftypefun double gsl_sf_hyperg_2F1_conj (double @var{aR}, double @var{aI}, double @var{c}, double @var{x})
+@deftypefunx int gsl_sf_hyperg_2F1_conj_e (double @var{aR}, double @var{aI}, double @var{c}, double @var{x}, gsl_sf_result * @var{result})
+These routines compute the Gauss hypergeometric function
+@c{${}_2F_1(a_R + i a_I, aR - i aI, c, x)$}
+@math{2F1(a_R + i a_I, a_R - i a_I, c, x)} with complex parameters 
+for @math{|x| < 1}.
+exceptions:
+@end deftypefun
+
+@deftypefun double gsl_sf_hyperg_2F1_renorm (double @var{a}, double @var{b}, double @var{c}, double @var{x})
+@deftypefunx int gsl_sf_hyperg_2F1_renorm_e (double @var{a}, double @var{b}, double @var{c}, double @var{x}, gsl_sf_result * @var{result})
+These routines compute the renormalized Gauss hypergeometric function
+@c{${}_2F_1(a,b,c,x) / \Gamma(c)$}
+@math{2F1(a,b,c,x) / \Gamma(c)} for @math{|x| < 1}.
+@comment exceptions:
+@end deftypefun
+
+@deftypefun double gsl_sf_hyperg_2F1_conj_renorm (double @var{aR}, double @var{aI}, double @var{c}, double @var{x})
+@deftypefunx int gsl_sf_hyperg_2F1_conj_renorm_e (double @var{aR}, double @var{aI}, double @var{c}, double @var{x}, gsl_sf_result * @var{result})
+These routines compute the renormalized Gauss hypergeometric function
+@c{${}_2F_1(a_R + i a_I, a_R - i a_I, c, x) / \Gamma(c)$}
+@math{2F1(a_R + i a_I, a_R - i a_I, c, x) / \Gamma(c)} for @math{|x| < 1}.
+@comment exceptions:
+@end deftypefun
+
+@deftypefun double gsl_sf_hyperg_2F0 (double @var{a}, double @var{b}, double @var{x})
+@deftypefunx int gsl_sf_hyperg_2F0_e (double @var{a}, double @var{b}, double @var{x}, gsl_sf_result * @var{result})
+These routines compute the hypergeometric function @c{${}_2F_0(a,b,x)$}
+@math{2F0(a,b,x)}.  The series representation
+is a divergent hypergeometric series.  However, for @math{x < 0} we
+have 
+@c{${}_2F_0(a,b,x) = (-1/x)^a U(a,1+a-b,-1/x)$}
+@math{2F0(a,b,x) = (-1/x)^a U(a,1+a-b,-1/x)}
+@comment exceptions: GSL_EDOM
+@end deftypefun
diff --git a/gsl-1.9/doc/specfunc-laguerre.texi b/gsl-1.9/doc/specfunc-laguerre.texi
new file mode 100644
index 0000000..8672d59
--- /dev/null
+++ b/gsl-1.9/doc/specfunc-laguerre.texi
@@ -0,0 +1,40 @@
+@cindex Laguerre functions
+@cindex confluent hypergeometric function
+
+The generalized Laguerre polynomials are defined in terms of confluent
+hypergeometric functions as
+@c{$L^a_n(x) = ((a+1)_n / n!) {}_1F_1(-n,a+1,x)$}
+@math{L^a_n(x) = ((a+1)_n / n!) 1F1(-n,a+1,x)}, and are sometimes referred to as the
+associated Laguerre polynomials.  They are related to the plain
+Laguerre polynomials @math{L_n(x)} by @math{L^0_n(x) = L_n(x)} and 
+@c{$L^k_n(x) = (-1)^k (d^k/dx^k) L_{(n+k)}(x)$}
+@math{L^k_n(x) = (-1)^k (d^k/dx^k) L_(n+k)(x)}.  For
+more information see Abramowitz & Stegun, Chapter 22.
+
+The functions described in this section are
+declared in the header file @file{gsl_sf_laguerre.h}.  
+
+@deftypefun double gsl_sf_laguerre_1 (double @var{a}, double @var{x})
+@deftypefunx double gsl_sf_laguerre_2 (double @var{a}, double @var{x})
+@deftypefunx double gsl_sf_laguerre_3 (double @var{a}, double @var{x})
+@deftypefunx int gsl_sf_laguerre_1_e (double @var{a}, double @var{x}, gsl_sf_result * @var{result})
+@deftypefunx int gsl_sf_laguerre_2_e (double @var{a}, double @var{x}, gsl_sf_result * @var{result})
+@deftypefunx int gsl_sf_laguerre_3_e (double @var{a}, double @var{x}, gsl_sf_result * @var{result})
+These routines evaluate the generalized Laguerre polynomials
+@math{L^a_1(x)}, @math{L^a_2(x)}, @math{L^a_3(x)} using explicit
+representations.
+@comment Exceptional Return Values: none
+@end deftypefun
+
+
+@deftypefun double gsl_sf_laguerre_n (const int @var{n}, const double @var{a}, const double @var{x})
+@deftypefunx int gsl_sf_laguerre_n_e (int @var{n}, double @var{a}, double @var{x}, gsl_sf_result * @var{result})
+These routines evaluate the generalized Laguerre polynomials
+@math{L^a_n(x)} for @math{a > -1}, 
+@c{$n \ge 0$}
+@math{n >= 0}.
+
+@comment Domain: a > -1.0, n >= 0
+@comment Evaluate generalized Laguerre polynomials.
+@comment Exceptional Return Values: GSL_EDOM
+@end deftypefun
diff --git a/gsl-1.9/doc/specfunc-lambert.texi b/gsl-1.9/doc/specfunc-lambert.texi
new file mode 100644
index 0000000..c05d23d
--- /dev/null
+++ b/gsl-1.9/doc/specfunc-lambert.texi
@@ -0,0 +1,28 @@
+@cindex W function
+@cindex Lambert function
+
+Lambert's W functions, @math{W(x)}, are defined to be solutions
+of the equation @math{W(x) \exp(W(x)) = x}. This function has
+multiple branches for @math{x < 0}; however, it has only
+two real-valued branches. We define @math{W_0(x)} to be the
+principal branch, where @math{W > -1} for @math{x < 0}, and 
+@c{$W_{-1}(x)$}
+@math{W_@{-1@}(x)} to be the other real branch, where
+@math{W < -1} for @math{x < 0}.  The Lambert functions are
+declared in the header file @file{gsl_sf_lambert.h}.
+
+
+@deftypefun double gsl_sf_lambert_W0 (double @var{x})
+@deftypefunx int gsl_sf_lambert_W0_e (double @var{x}, gsl_sf_result * @var{result})
+These compute the principal branch of the Lambert W function, @math{W_0(x)}.
+@comment exceptions: GSL_EDOM, GSL_EMAXITER
+@end deftypefun
+
+@deftypefun double gsl_sf_lambert_Wm1 (double @var{x})
+@deftypefunx int gsl_sf_lambert_Wm1_e (double @var{x}, gsl_sf_result * @var{result})
+These compute the secondary real-valued branch of the Lambert W function, 
+@c{$W_{-1}(x)$}
+@math{W_@{-1@}(x)}.
+@comment exceptions: GSL_EDOM, GSL_EMAXITER
+@end deftypefun
+
diff --git a/gsl-1.9/doc/specfunc-legendre.texi b/gsl-1.9/doc/specfunc-legendre.texi
new file mode 100644
index 0000000..4be1005
--- /dev/null
+++ b/gsl-1.9/doc/specfunc-legendre.texi
@@ -0,0 +1,272 @@
+@cindex Legendre functions
+@cindex spherical harmonics
+@cindex conical functions
+@cindex hyperbolic space
+
+The Legendre Functions and Legendre Polynomials are described in
+Abramowitz & Stegun, Chapter 8.  These functions are declared in 
+the header file @file{gsl_sf_legendre.h}.
+
+@menu
+* Legendre Polynomials::        
+* Associated Legendre Polynomials and Spherical Harmonics::  
+* Conical Functions::           
+* Radial Functions for Hyperbolic Space::  
+@end menu
+
+@node Legendre Polynomials
+@subsection Legendre Polynomials
+
+@deftypefun double gsl_sf_legendre_P1 (double @var{x})
+@deftypefunx double gsl_sf_legendre_P2 (double @var{x})
+@deftypefunx double gsl_sf_legendre_P3 (double @var{x})
+@deftypefunx int gsl_sf_legendre_P1_e (double @var{x}, gsl_sf_result * @var{result})
+@deftypefunx int gsl_sf_legendre_P2_e (double @var{x}, gsl_sf_result * @var{result})
+@deftypefunx int gsl_sf_legendre_P3_e (double @var{x}, gsl_sf_result * @var{result})
+These functions evaluate the Legendre polynomials
+@c{$P_l(x)$} 
+@math{P_l(x)} using explicit
+representations for @math{l=1, 2, 3}.
+@comment Exceptional Return Values: none
+@end deftypefun
+
+@deftypefun double gsl_sf_legendre_Pl (int @var{l}, double @var{x})
+@deftypefunx int gsl_sf_legendre_Pl_e (int @var{l}, double @var{x}, gsl_sf_result * @var{result})
+These functions evaluate the Legendre polynomial @c{$P_l(x)$} 
+@math{P_l(x)} for a specific value of @var{l},
+@var{x} subject to @c{$l \ge 0$}
+@math{l >= 0}, 
+@c{$|x| \le 1$}
+@math{|x| <= 1}
+@comment Exceptional Return Values: GSL_EDOM
+@end deftypefun
+
+@deftypefun int gsl_sf_legendre_Pl_array (int @var{lmax}, double @var{x}, double @var{result_array}[])
+@deftypefunx int gsl_sf_legendre_Pl_deriv_array (int @var{lmax}, double @var{x}, double @var{result_array}[], double @var{result_deriv_array}[])
+
+These functions compute an array of Legendre polynomials
+@math{P_l(x)}, and optionally their derivatives @math{dP_l(x)/dx}, 
+for @math{l = 0, \dots, lmax}, 
+@c{$|x| \le 1$}
+@math{|x| <= 1}
+@comment Exceptional Return Values: GSL_EDOM
+@end deftypefun
+
+
+@deftypefun double gsl_sf_legendre_Q0 (double @var{x})
+@deftypefunx int gsl_sf_legendre_Q0_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the Legendre function @math{Q_0(x)} for @math{x >
+-1}, @c{$x \ne 1$}
+@math{x != 1}.
+@comment Exceptional Return Values: GSL_EDOM
+@end deftypefun
+
+
+@deftypefun double gsl_sf_legendre_Q1 (double @var{x})
+@deftypefunx int gsl_sf_legendre_Q1_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the Legendre function @math{Q_1(x)} for @math{x >
+-1}, @c{$x \ne 1$}
+@math{x != 1}.
+@comment Exceptional Return Values: GSL_EDOM
+@end deftypefun
+
+@deftypefun double gsl_sf_legendre_Ql (int @var{l}, double @var{x})
+@deftypefunx int gsl_sf_legendre_Ql_e (int @var{l}, double @var{x}, gsl_sf_result * @var{result})
+These routines compute the Legendre function @math{Q_l(x)} for @math{x >
+-1}, @c{$x \ne 1$}
+@math{x != 1} and @c{$l \ge 0$}
+@math{l >= 0}.
+@comment Exceptional Return Values: GSL_EDOM
+@end deftypefun
+
+
+@node Associated Legendre Polynomials and Spherical Harmonics
+@subsection Associated Legendre Polynomials and Spherical Harmonics
+
+The following functions compute the associated Legendre Polynomials
+@math{P_l^m(x)}.  Note that this function grows combinatorially with
+@math{l} and can overflow for @math{l} larger than about 150.  There is
+no trouble for small @math{m}, but overflow occurs when @math{m} and
+@math{l} are both large.  Rather than allow overflows, these functions
+refuse to calculate @math{P_l^m(x)} and return @code{GSL_EOVRFLW} when
+they can sense that @math{l} and @math{m} are too big.
+
+If you want to calculate a spherical harmonic, then @emph{do not} use
+these functions.  Instead use @code{gsl_sf_legendre_sphPlm} below,
+which uses a similar recursion, but with the normalized functions.
+
+@deftypefun double gsl_sf_legendre_Plm (int @var{l}, int @var{m}, double @var{x})
+@deftypefunx int gsl_sf_legendre_Plm_e (int @var{l}, int @var{m}, double @var{x}, gsl_sf_result * @var{result})
+These routines compute the associated Legendre polynomial
+@math{P_l^m(x)} for @c{$m \ge 0$}
+@math{m >= 0}, @c{$l \ge m$}
+@math{l >= m}, @c{$|x| \le 1$}
+@math{|x| <= 1}. 
+@comment Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW
+@end deftypefun
+
+@deftypefun int gsl_sf_legendre_Plm_array (int @var{lmax}, int @var{m}, double @var{x}, double @var{result_array}[])
+@deftypefunx int gsl_sf_legendre_Plm_deriv_array (int @var{lmax}, int @var{m}, double @var{x}, double @var{result_array}[], double @var{result_deriv_array}[])
+These functions compute an array of Legendre polynomials
+@math{P_l^m(x)}, and optionally their derivatives @math{dP_l^m(x)/dx},
+for @c{$m \ge 0$}
+@math{m >= 0}, @c{$l = |m|, \dots, lmax$}
+@math{l = |m|, ..., lmax}, @c{$|x| \le 1$}
+@math{|x| <= 1}.
+@comment Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW
+@end deftypefun
+
+
+@deftypefun double gsl_sf_legendre_sphPlm (int @var{l}, int @var{m}, double @var{x})
+@deftypefunx int gsl_sf_legendre_sphPlm_e (int @var{l}, int @var{m}, double @var{x}, gsl_sf_result * @var{result})
+These routines compute the normalized associated Legendre polynomial
+@c{$\sqrt{(2l+1)/(4\pi)} \sqrt{(l-m)!/(l+m)!} P_l^m(x)$}
+@math{$\sqrt@{(2l+1)/(4\pi)@} \sqrt@{(l-m)!/(l+m)!@} P_l^m(x)$} suitable
+for use in spherical harmonics.  The parameters must satisfy @c{$m \ge 0$}
+@math{m >= 0}, @c{$l \ge m$}
+@math{l >= m}, @c{$|x| \le 1$}
+@math{|x| <= 1}. Theses routines avoid the overflows
+that occur for the standard normalization of @math{P_l^m(x)}.
+@comment Exceptional Return Values: GSL_EDOM
+@end deftypefun
+
+@deftypefun int gsl_sf_legendre_sphPlm_array (int @var{lmax}, int @var{m}, double @var{x}, double @var{result_array}[])
+@deftypefunx int gsl_sf_legendre_sphPlm_deriv_array (int @var{lmax}, int @var{m}, double @var{x}, double @var{result_array}[], double @var{result_deriv_array}[])
+These functions compute an array of normalized associated Legendre functions
+@c{$\sqrt{(2l+1)/(4\pi)} \sqrt{(l-m)!/(l+m)!} P_l^m(x)$}
+@math{$\sqrt@{(2l+1)/(4\pi)@} \sqrt@{(l-m)!/(l+m)!@} P_l^m(x)$},
+and optionally their derivatives,
+for @c{$m \ge 0$}
+@math{m >= 0}, @c{$l = |m|, \dots, lmax$}
+@math{l = |m|, ..., lmax}, @c{$|x| \le 1$}
+@math{|x| <= 1.0}
+@comment Exceptional Return Values: GSL_EDOM
+@end deftypefun
+
+@deftypefun int gsl_sf_legendre_array_size (const int @var{lmax}, const int @var{m})
+This function returns the size of @var{result_array}[] needed for the array
+versions of @math{P_l^m(x)}, @math{@var{lmax} - @var{m} + 1}.
+@comment Exceptional Return Values: none
+@end deftypefun
+
+@node Conical Functions
+@subsection Conical Functions
+
+The Conical Functions @c{$P^\mu_{-(1/2)+i\lambda}(x)$}
+@math{P^\mu_@{-(1/2)+i\lambda@}(x)} and @c{$Q^\mu_{-(1/2)+i\lambda}$} 
+@math{Q^\mu_@{-(1/2)+i\lambda@}} 
+are described in Abramowitz & Stegun, Section 8.12.
+
+@deftypefun double gsl_sf_conicalP_half (double @var{lambda}, double @var{x})
+@deftypefunx int gsl_sf_conicalP_half_e (double @var{lambda}, double @var{x}, gsl_sf_result * @var{result})
+These routines compute the irregular Spherical Conical Function
+@c{$P^{1/2}_{-1/2 + i \lambda}(x)$}
+@math{P^@{1/2@}_@{-1/2 + i \lambda@}(x)} for @math{x > -1}.
+@comment Exceptional Return Values: GSL_EDOM
+@end deftypefun
+
+@deftypefun double gsl_sf_conicalP_mhalf (double @var{lambda}, double @var{x})
+@deftypefunx int gsl_sf_conicalP_mhalf_e (double @var{lambda}, double @var{x}, gsl_sf_result * @var{result})
+These routines compute the regular Spherical Conical Function
+@c{$P^{-1/2}_{-1/2 + i \lambda}(x)$}
+@math{P^@{-1/2@}_@{-1/2 + i \lambda@}(x)} for @math{x > -1}.
+@comment Exceptional Return Values: GSL_EDOM
+@end deftypefun
+
+@deftypefun double gsl_sf_conicalP_0 (double @var{lambda}, double @var{x})
+@deftypefunx int gsl_sf_conicalP_0_e (double @var{lambda}, double @var{x}, gsl_sf_result * @var{result})
+These routines compute the conical function
+@c{$P^0_{-1/2 + i \lambda}(x)$}
+@math{P^0_@{-1/2 + i \lambda@}(x)}
+for @math{x > -1}.
+@comment Exceptional Return Values: GSL_EDOM
+@end deftypefun
+
+
+@deftypefun double gsl_sf_conicalP_1 (double @var{lambda}, double @var{x})
+@deftypefunx int gsl_sf_conicalP_1_e (double @var{lambda}, double @var{x}, gsl_sf_result * @var{result})
+These routines compute the conical function 
+@c{$P^1_{-1/2 + i \lambda}(x)$}
+@math{P^1_@{-1/2 + i \lambda@}(x)} for @math{x > -1}.
+@comment Exceptional Return Values: GSL_EDOM
+@end deftypefun
+
+
+@deftypefun double gsl_sf_conicalP_sph_reg (int @var{l}, double @var{lambda}, double @var{x})
+@deftypefunx int gsl_sf_conicalP_sph_reg_e (int @var{l}, double @var{lambda}, double @var{x}, gsl_sf_result * @var{result})
+These routines compute the Regular Spherical Conical Function
+@c{$P^{-1/2-l}_{-1/2 + i \lambda}(x)$}
+@math{P^@{-1/2-l@}_@{-1/2 + i \lambda@}(x)} for @math{x > -1}, @c{$l \ge -1$}
+@math{l >= -1}.
+@comment Exceptional Return Values: GSL_EDOM
+@end deftypefun
+
+
+@deftypefun double gsl_sf_conicalP_cyl_reg (int @var{m}, double @var{lambda}, double @var{x})
+@deftypefunx int gsl_sf_conicalP_cyl_reg_e (int @var{m}, double @var{lambda}, double @var{x}, gsl_sf_result * @var{result})
+These routines compute the Regular Cylindrical Conical Function
+@c{$P^{-m}_{-1/2 + i \lambda}(x)$}
+@math{P^@{-m@}_@{-1/2 + i \lambda@}(x)} for @math{x > -1}, @c{$m \ge -1$}
+@math{m >= -1}.
+@comment Exceptional Return Values: GSL_EDOM
+@end deftypefun
+
+
+
+@node Radial Functions for Hyperbolic Space
+@subsection Radial Functions for Hyperbolic Space
+
+The following spherical functions are specializations of Legendre
+functions which give the regular eigenfunctions of the Laplacian on a
+3-dimensional hyperbolic space @math{H3d}.  Of particular interest is
+the flat limit, @math{\lambda \to \infty}, @math{\eta \to 0},
+@math{\lambda\eta} fixed.
+  
+@deftypefun double gsl_sf_legendre_H3d_0 (double @var{lambda}, double @var{eta})
+@deftypefunx int gsl_sf_legendre_H3d_0_e (double @var{lambda}, double @var{eta}, gsl_sf_result * @var{result})
+These routines compute the zeroth radial eigenfunction of the Laplacian on the
+3-dimensional hyperbolic space,
+@c{$$L^{H3d}_0(\lambda,\eta) := {\sin(\lambda\eta) \over \lambda\sinh(\eta)}$$}
+@math{L^@{H3d@}_0(\lambda,\eta) := \sin(\lambda\eta)/(\lambda\sinh(\eta))}
+for @c{$\eta \ge 0$}
+@math{\eta >= 0}.
+In the flat limit this takes the form
+@c{$L^{H3d}_0(\lambda,\eta) = j_0(\lambda\eta)$}
+@math{L^@{H3d@}_0(\lambda,\eta) = j_0(\lambda\eta)}.
+@comment Exceptional Return Values: GSL_EDOM
+@end deftypefun
+
+@deftypefun double gsl_sf_legendre_H3d_1 (double @var{lambda}, double @var{eta})
+@deftypefunx int gsl_sf_legendre_H3d_1_e (double @var{lambda}, double @var{eta}, gsl_sf_result * @var{result})
+These routines compute the first radial eigenfunction of the Laplacian on
+the 3-dimensional hyperbolic space,
+@c{$$L^{H3d}_1(\lambda,\eta) := {1\over\sqrt{\lambda^2 + 1}} {\left(\sin(\lambda \eta)\over \lambda \sinh(\eta)\right)} \left(\coth(\eta) - \lambda \cot(\lambda\eta)\right)$$}
+@math{L^@{H3d@}_1(\lambda,\eta) := 1/\sqrt@{\lambda^2 + 1@} \sin(\lambda \eta)/(\lambda \sinh(\eta)) (\coth(\eta) - \lambda \cot(\lambda\eta))}
+for @c{$\eta \ge 0$}
+@math{\eta >= 0}.
+In the flat limit this takes the form 
+@c{$L^{H3d}_1(\lambda,\eta) = j_1(\lambda\eta)$}
+@math{L^@{H3d@}_1(\lambda,\eta) = j_1(\lambda\eta)}.
+@comment Exceptional Return Values: GSL_EDOM
+@end deftypefun
+
+@deftypefun double gsl_sf_legendre_H3d (int @var{l}, double @var{lambda}, double @var{eta})
+@deftypefunx int gsl_sf_legendre_H3d_e (int @var{l}, double @var{lambda}, double @var{eta}, gsl_sf_result * @var{result})
+These routines compute the @var{l}-th radial eigenfunction of the
+Laplacian on the 3-dimensional hyperbolic space @c{$\eta \ge 0$}
+@math{\eta >= 0}, @c{$l \ge 0$}
+@math{l >= 0}. In the flat limit this takes the form
+@c{$L^{H3d}_l(\lambda,\eta) = j_l(\lambda\eta)$}
+@math{L^@{H3d@}_l(\lambda,\eta) = j_l(\lambda\eta)}.
+@comment Exceptional Return Values: GSL_EDOM
+@end deftypefun
+
+@deftypefun int gsl_sf_legendre_H3d_array (int @var{lmax}, double @var{lambda}, double @var{eta}, double @var{result_array}[])
+This function computes an array of radial eigenfunctions
+@c{$L^{H3d}_l( \lambda, \eta)$} 
+@math{L^@{H3d@}_l(\lambda, \eta)} 
+for @c{$0 \le l \le lmax$}
+@math{0 <= l <= lmax}.
+@comment Exceptional Return Values:
+@end deftypefun
+
diff --git a/gsl-1.9/doc/specfunc-log.texi b/gsl-1.9/doc/specfunc-log.texi
new file mode 100644
index 0000000..75279e8
--- /dev/null
+++ b/gsl-1.9/doc/specfunc-log.texi
@@ -0,0 +1,47 @@
+@cindex logarithm and related functions
+
+Information on the properties of the Logarithm function can be found in
+Abramowitz & Stegun, Chapter 4.  The functions described in this section
+are declared in the header file @file{gsl_sf_log.h}.
+
+@deftypefun double gsl_sf_log (double @var{x})
+@deftypefunx int gsl_sf_log_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the logarithm of @var{x}, @math{\log(x)}, for
+@math{x > 0}.
+@comment Exceptional Return Values: GSL_EDOM
+@end deftypefun
+
+
+@deftypefun double gsl_sf_log_abs (double @var{x})
+@deftypefunx int gsl_sf_log_abs_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the logarithm of the magnitude of @var{x},
+@math{\log(|x|)}, for @math{x \ne 0}.
+@comment Exceptional Return Values: GSL_EDOM
+@end deftypefun
+
+
+@deftypefun int gsl_sf_complex_log_e (double @var{zr}, double @var{zi}, gsl_sf_result * @var{lnr}, gsl_sf_result * @var{theta})
+This routine computes the complex logarithm of @math{z = z_r + i
+z_i}. The results are returned as @var{lnr}, @var{theta} such that
+@math{\exp(lnr + i \theta) = z_r + i z_i}, where @math{\theta} lies in
+the range @math{[-\pi,\pi]}.
+@comment Exceptional Return Values: GSL_EDOM
+@end deftypefun
+
+
+@deftypefun double gsl_sf_log_1plusx (double @var{x})
+@deftypefunx int gsl_sf_log_1plusx_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute @math{\log(1 + x)} for @math{x > -1} using an
+algorithm that is accurate for small @math{x}.
+@comment Domain: x > -1.0
+@comment Exceptional Return Values: GSL_EDOM
+@end deftypefun
+
+
+@deftypefun double gsl_sf_log_1plusx_mx (double @var{x})
+@deftypefunx int gsl_sf_log_1plusx_mx_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute @math{\log(1 + x) - x} for @math{x > -1} using an
+algorithm that is accurate for small @math{x}.
+@comment Domain: x > -1.0 
+@comment Exceptional Return Values: GSL_EDOM
+@end deftypefun
diff --git a/gsl-1.9/doc/specfunc-mathieu.texi b/gsl-1.9/doc/specfunc-mathieu.texi
new file mode 100644
index 0000000..dddec1d
--- /dev/null
+++ b/gsl-1.9/doc/specfunc-mathieu.texi
@@ -0,0 +1,257 @@
+@cindex Mathieu functions
+
+The routines described in this section compute the angular and radial
+Mathieu functions, and their characteristic values.  Mathieu
+functions are the solutions of the following two differential
+equations:
+@tex
+\beforedisplay
+$$
+\eqalign{
+{{d^2 y}\over{d v^2}}& + (a - 2q\cos 2v)y  = 0, \cr
+{{d^2 f}\over{d u^2}}& - (a - 2q\cosh 2u)f  = 0.
+}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+d^2y/dv^2 + (a - 2q\cos 2v)y = 0
+d^2f/du^2 - (a - 2q\cosh 2u)f = 0
+@end example
+
+@end ifinfo
+@noindent
+The angular Mathieu functions @math{ce_r(x,q)}, @math{se_r(x,q)} are
+the even and odd periodic solutions of the first equation, which is known as Mathieu's equation. These exist
+only for the discrete sequence of  characteristic values @math{a=a_r(q)}
+(even-periodic) and @math{a=b_r(q)} (odd-periodic).
+
+The radial Mathieu functions @c{$Mc^{(j)}_{r}(z,q)$}
+@math{Mc^@{(j)@}_@{r@}(z,q)}, @c{$Ms^{(j)}_@{r@}(z,q)$}
+@math{Ms^@{(j)@}_@{r@}(z,q)} are the solutions of the second equation,
+which is referred to as Mathieu's modified equation.  The
+radial Mathieu functions of the first, second, third and fourth kind
+are denoted by the parameter @math{j}, which takes the value 1, 2, 3
+or 4.
+
+@comment The angular Mathieu functions can be divided into four types as
+@comment @tex
+@comment \beforedisplay
+@comment $$
+@comment \eqalign{
+@comment x & = \sum_{m=0}^\infty A_{2m+p} \cos(2m+p)\phi, \quad p = 0, 1, \cr
+@comment x & = \sum_{m=0}^\infty B_{2m+p} \sin(2m+p)\phi, \quad p = 0, 1.
+@comment }
+@comment $$
+@comment \afterdisplay
+@comment @end tex
+@comment @ifinfo
+
+@comment @example
+@comment x = \sum_(m=0)^\infty A_(2m+p) \cos(2m+p)\phi,   p = 0, 1,
+@comment x = \sum_(m=0)^\infty B_(2m+p) \sin(2m+p)\phi,   p = 0, 1.
+@comment @end example
+
+@comment @end ifinfo
+@comment @noindent
+@comment The nomenclature used for the angular Mathieu functions is @math{ce_n}
+@comment for the first solution and @math{se_n} for the second.
+
+@comment Similar solutions exist for the radial Mathieu functions by replacing
+@comment the trigonometric functions with their corresponding hyperbolic
+@comment functions as shown below.
+@comment @tex
+@comment \beforedisplay
+@comment $$
+@comment \eqalign{
+@comment x & = \sum_{m=0}^\infty A_{2m+p} \cosh(2m+p)u, \quad p = 0, 1, \cr
+@comment x & = \sum_{m=0}^\infty B_{2m+p} \sinh(2m+p)u, \quad p = 0, 1.
+@comment }
+@comment $$
+@comment \afterdisplay
+@comment @end tex
+@comment @ifinfo
+
+@comment @example
+@comment x = \sum_(m=0)^\infty A_(2m+p) \cosh(2m+p)u,   p = 0, 1,
+@comment x = \sum_(m=0)^\infty B_(2m+p) \sinh(2m+p)u,   p = 0, 1.
+@comment @end example
+
+@comment @end ifinfo
+@comment @noindent
+@comment The nomenclature used for the radial Mathieu functions is @math{Mc_n}
+@comment for the first solution and @math{Ms_n} for the second.  The hyperbolic
+@comment series do not always converge at an acceptable rate.  Therefore most
+@comment texts on the subject suggest using the following equivalent equations
+@comment that are expanded in series of Bessel and Hankel functions.
+@comment @tex
+@comment \beforedisplay
+@comment $$
+@comment \eqalign{
+@comment Mc_{2n}^{(j)}(x,q) & = \sum_{m=0}^\infty (-1)^{r+k}
+@comment       A_{2m}^{2n}(q)\left[J_m(u_1)Z_m^{(j)}(u_2) +
+@comment                           J_m(u_1)Z_m^{(j)}(u_2)\right]/A_2^{2n} \cr
+@comment Mc_{2n+1}^{(j)}(x,q) & = \sum_{m=0}^\infty (-1)^{r+k}
+@comment       A_{2m+1}^{2n+1}(q)\left[J_m(u_1)Z_{m+1}^{(j)}(u_2) +
+@comment                               J_{m+1}(u_1)Z_m^{(j)}(u_2)\right]/A_1^{2n+1} \cr
+@comment Ms_{2n}^{(j)}(x,q) & = \sum_{m=1}^\infty (-1)^{r+k}
+@comment       B_{2m}^{2n}(q)\left[J_{m-1}(u_1)Z_{m+1}^{(j)}(u_2) +
+@comment                           J_{m+1}(u_1)Z_{m-1}^{(j)}(u_2)\right]/B_2^{2n} \cr
+@comment Ms_{2n+1}^{(j)}(x,q) & = \sum_{m=0}^\infty (-1)^{r+k}
+@comment       B_{2m+1}^{2n+1}(q)\left[J_m(u_1)Z_{m+1}^{(j)}(u_2) +
+@comment                               J_{m+1}(u_1)Z_m^{(j)}(u_2)\right]/B_1^{2n+1}
+@comment }
+@comment $$
+@comment \afterdisplay
+@comment @end tex
+@comment @ifinfo
+
+@comment @example
+@comment Mc_(2n)^(j)(x,q) = \sum_(m=0)^\infty (-1)^(r+k) A_(2m)^(2n)(q)
+@comment     [J_m(u_1)Z_m^(j)(u_2) + J_m(u_1)Z_m^(j)(u_2)]/A_2^(2n)
+@comment Mc_(2n+1)^(j)(x,q) = \sum_(m=0)^\infty (-1)^(r+k) A_(2m+1)^(2n+1)(q)
+@comment     [J_m(u_1)Z_(m+1)^(j)(u_2) + J_(m+1)(u_1)Z_m^(j)(u_2)]/A_1^(2n+1)
+@comment Ms_(2n)^(j)(x,q) = \sum_(m=1)^\infty (-1)^(r+k) B_(2m)^(2n)(q)
+@comment     [J_(m-1)(u_1)Z_(m+1)^(j)(u_2) + J_(m+1)(u_1)Z_(m-1)^(j)(u_2)]/B_2^(2n)
+@comment Ms_(2n+1)^(j)(x,q) = \sum_(m=0)^\infty (-1)^(r+k) B_(2m+1)^(2n+1)(q)
+@comment     [J_m(u_1)Z_(m+1)^(j)(u_2) + J_(m+1)(u_1)Z_m^(j)(u_2)]/B_1^(2n+1)
+@comment @end example
+
+@comment @end ifinfo
+@comment @noindent
+@comment where @c{$u_1 = \sqrt{q} \exp(-x)$} 
+@comment @math{u_1 = \sqrt@{q@} \exp(-x)} and @c{$u_2 = \sqrt@{q@} \exp(x)$}
+@comment @math{u_2 = \sqrt@{q@} \exp(x)} and
+@comment @tex
+@comment \beforedisplay
+@comment $$
+@comment \eqalign{
+@comment Z_m^{(1)}(u) & = J_m(u) \cr
+@comment Z_m^{(2)}(u) & = Y_m(u) \cr
+@comment Z_m^{(3)}(u) & = H_m^{(1)}(u) \cr
+@comment Z_m^{(4)}(u) & = H_m^{(2)}(u)
+@comment }
+@comment $$
+@comment \afterdisplay
+@comment @end tex
+@comment @ifinfo
+
+@comment @example
+@comment Z_m^(1)(u) = J_m(u)
+@comment Z_m^(2)(u) = Y_m(u)
+@comment Z_m^(3)(u) = H_m^(1)(u)
+@comment Z_m^(4)(u) = H_m^(2)(u)
+@comment @end example
+
+@comment @end ifinfo
+@comment @noindent
+@comment where @math{J_m(u)}, @math{Y_m(u)}, @math{H_m^{(1)}(u)}, and
+@comment @math{H_m^{(2)}(u)} are the regular and irregular Bessel functions and
+@comment the Hankel functions, respectively.
+
+For more information on the Mathieu functions, see Abramowitz and
+Stegun, Chapter 20.  These functions are defined in the header file
+@file{gsl_sf_mathieu.h}.
+
+@menu
+* Mathieu Function Workspace::  
+* Mathieu Function Characteristic Values::  
+* Angular Mathieu Functions::   
+* Radial Mathieu Functions::    
+@end menu
+
+@node  Mathieu Function Workspace
+@subsection  Mathieu Function Workspace
+
+The Mathieu functions can be computed for a single order or
+for multiple orders, using array-based routines.  The array-based
+routines require a preallocated workspace.
+
+@deftypefun {gsl_sf_mathieu_workspace *} gsl_sf_mathieu_alloc (size_t @var{n}, double @var{qmax})
+This function returns a workspace for the array versions of the
+Mathieu routines.  The arguments @var{n} and @var{qmax} specify the
+maximum order and @math{q}-value of Mathieu functions which can be
+computed with this workspace.  
+
+@comment This is required in order to properly
+@comment terminate the infinite eigenvalue matrix for high precision solutions.
+@comment The characteristic values for all orders @math{0 \to n} are stored in
+@comment the work structure array element @kbd{work->char_value}.
+@end deftypefun
+
+@deftypefun void gsl_sf_mathieu_free (gsl_sf_mathieu_workspace @var{*work})
+This function frees the workspace @var{work}.
+@end deftypefun
+
+@node Mathieu Function Characteristic Values
+@subsection Mathieu Function Characteristic Values
+@cindex Mathieu Function Characteristic Values
+
+@deftypefun int gsl_sf_mathieu_a (int @var{n}, double @var{q}, gsl_sf_result @var{*result})
+@deftypefunx int gsl_sf_mathieu_b (int @var{n}, double @var{q}, gsl_sf_result @var{*result})
+These routines compute the characteristic values @math{a_n(q)},
+@math{b_n(q)} of the Mathieu functions @math{ce_n(q,x)} and
+@math{se_n(q,x)}, respectively.
+@end deftypefun
+
+@deftypefun int gsl_sf_mathieu_a_array (int @var{order_min}, int @var{order_max}, double @var{q}, gsl_sf_mathieu_workspace @var{*work}, double @var{result_array}[])
+@deftypefunx int gsl_sf_mathieu_b_array (int @var{order_min}, int @var{order_max}, double @var{q}, gsl_sf_mathieu_workspace @var{*work}, double @var{result_array}[])
+These routines compute a series of Mathieu characteristic values
+@math{a_n(q)}, @math{b_n(q)} for @math{n} from @var{order_min} to
+@var{order_max} inclusive, storing the results in the array @var{result_array}.
+@end deftypefun
+
+@node Angular Mathieu Functions
+@subsection Angular Mathieu Functions
+@cindex Angular Mathieu Functions
+@cindex @math{ce(q,x)}, Mathieu function
+@cindex @math{se(q,x)}, Mathieu function
+
+@deftypefun int gsl_sf_mathieu_ce (int @var{n}, double @var{q}, double @var{x}, gsl_sf_result @var{*result})
+@deftypefunx int gsl_sf_mathieu_se (int @var{n}, double @var{q}, double @var{x}, gsl_sf_result @var{*result})
+These routines compute the angular Mathieu functions @math{ce_n(q,x)}
+and @math{se_n(q,x)}, respectively.
+@end deftypefun
+
+@deftypefun int gsl_sf_mathieu_ce_array (int @var{nmin}, int @var{nmax}, double @var{q}, double @var{x}, gsl_sf_mathieu_workspace @var{*work}, double @var{result_array}[])
+@deftypefunx int gsl_sf_mathieu_se_array (int @var{nmin}, int @var{nmax}, double @var{q}, double @var{x}, gsl_sf_mathieu_workspace @var{*work}, double @var{result_array}[])
+These routines compute a series of the angular Mathieu functions
+@math{ce_n(q,x)} and @math{se_n(q,x)} of order @math{n} from
+@var{nmin} to @var{nmax} inclusive, storing the results in the array
+@var{result_array}.
+@end deftypefun
+
+@node Radial Mathieu Functions
+@subsection Radial Mathieu Functions
+@cindex Radial Mathieu Functions
+
+@deftypefun int gsl_sf_mathieu_Mc (int @var{j}, int @var{n}, double @var{q}, double @var{x}, gsl_sf_result @var{*result})
+@deftypefunx int gsl_sf_mathieu_Ms (int @var{j}, int @var{n}, double @var{q}, double @var{x}, gsl_sf_result @var{*result})
+These routines compute the radial @var{j}-th kind Mathieu functions
+@c{$Mc_n^{(j)}(q,x)$}
+@math{Mc_n^@{(j)@}(q,x)} and 
+@c{$Ms_n^{(j)}(q,x)$}
+@math{Ms_n^@{(j)@}(q,x)} of order @var{n}.
+
+The allowed values of @var{j} are 1 and 2.
+The functions for @math{j = 3,4} can be computed as 
+@c{$M_n^{(3)} = M_n^{(1)} + iM_n^{(2)}$}
+@math{M_n^@{(3)@} = M_n^@{(1)@} + iM_n^@{(2)@}} and 
+@c{$M_n^{(4)} = M_n^{(1)} - iM_n^{(2)}$}
+@math{M_n^@{(4)@} = M_n^@{(1)@} - iM_n^@{(2)@}},
+where 
+@c{$M_n^{(j)} = Mc_n^{(j)}$}
+@math{M_n^@{(j)@} = Mc_n^@{(j)@}} or 
+@c{$Ms_n^{(j)}$}
+@math{Ms_n^@{(j)@}}.
+@end deftypefun
+
+@deftypefun int gsl_sf_mathieu_Mc_array (int @var{j}, int @var{nmin}, int @var{nmax}, double @var{q}, double @var{x}, gsl_sf_mathieu_workspace @var{*work}, double @var{result_array}[])
+@deftypefunx int gsl_sf_mathieu_Ms_array (int @var{j}, int @var{nmin}, int @var{nmax}, double @var{q}, double @var{x}, gsl_sf_mathieu_workspace @var{*work}, double @var{result_array}[])
+These routines compute a series of the radial Mathieu functions of
+kind @var{j}, with order from @var{nmin} to @var{nmax} inclusive, storing the
+results in the array @var{result_array}.
+@end deftypefun
+
diff --git a/gsl-1.9/doc/specfunc-pow-int.texi b/gsl-1.9/doc/specfunc-pow-int.texi
new file mode 100644
index 0000000..f05469a
--- /dev/null
+++ b/gsl-1.9/doc/specfunc-pow-int.texi
@@ -0,0 +1,22 @@
+@cindex power function
+@cindex integer powers
+
+The following functions are equivalent to the function @code{gsl_pow_int}
+(@pxref{Small integer powers}) with an error estimate.  These functions are
+declared in the header file @file{gsl_sf_pow_int.h}.
+
+@deftypefun double gsl_sf_pow_int (double @var{x}, int @var{n})
+@deftypefunx int gsl_sf_pow_int_e (double @var{x}, int @var{n}, gsl_sf_result * @var{result}) 
+These routines compute the power @math{x^n} for integer @var{n}.  The
+power is computed using the minimum number of multiplications. For
+example, @math{x^8} is computed as @math{((x^2)^2)^2}, requiring only 3
+multiplications.  For reasons of efficiency, these functions do not
+check for overflow or underflow conditions.
+@end deftypefun
+
+@example
+#include <gsl/gsl_sf_pow_int.h>
+/* compute 3.0**12 */
+double y = gsl_sf_pow_int(3.0, 12); 
+@end example
+
diff --git a/gsl-1.9/doc/specfunc-psi.texi b/gsl-1.9/doc/specfunc-psi.texi
new file mode 100644
index 0000000..8bbef3b
--- /dev/null
+++ b/gsl-1.9/doc/specfunc-psi.texi
@@ -0,0 +1,90 @@
+@cindex psi function
+@cindex digamma function
+@cindex polygamma functions
+
+The polygamma functions of order @math{n} are defined by
+@tex
+\beforedisplay
+$$
+\psi^{(n)}(x) = \left(d \over dx\right)^n \psi(x) = \left(d \over dx\right)^{n+1} \log(\Gamma(x))
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+\psi^@{(n)@}(x) = (d/dx)^n \psi(x) = (d/dx)^@{n+1@} \log(\Gamma(x))
+@end example
+
+@end ifinfo
+@noindent
+where @math{\psi(x) = \Gamma'(x)/\Gamma(x)} is known as the digamma function.
+These functions are declared in the header file @file{gsl_sf_psi.h}.
+
+@menu
+* Digamma Function::            
+* Trigamma Function::           
+* Polygamma Function::          
+@end menu
+
+@node Digamma Function
+@subsection Digamma Function
+
+@deftypefun double gsl_sf_psi_int (int @var{n})
+@deftypefunx int gsl_sf_psi_int_e (int @var{n}, gsl_sf_result * @var{result})
+These routines compute the digamma function @math{\psi(n)} for positive
+integer @var{n}.  The digamma function is also called the Psi function.
+@comment Domain: n integer, n > 0
+@comment Exceptional Return Values: GSL_EDOM
+@end deftypefun
+
+
+@deftypefun double gsl_sf_psi (double @var{x})
+@deftypefunx int gsl_sf_psi_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the digamma function @math{\psi(x)} for general
+@math{x}, @math{x \ne 0}.
+@comment Domain: x != 0.0, -1.0, -2.0, ...
+@comment Exceptional Return Values: GSL_EDOM, GSL_ELOSS
+@end deftypefun
+
+
+@deftypefun double gsl_sf_psi_1piy (double @var{y})
+@deftypefunx int gsl_sf_psi_1piy_e (double @var{y}, gsl_sf_result * @var{result})
+These routines compute the real part of the digamma function on the line
+@math{1+i y}, @math{\Re[\psi(1 + i y)]}.
+@comment exceptions: none
+@comment Exceptional Return Values: none
+@end deftypefun
+
+
+@node Trigamma Function
+@subsection Trigamma Function
+
+@deftypefun double gsl_sf_psi_1_int (int @var{n})
+@deftypefunx int gsl_sf_psi_1_int_e (int @var{n}, gsl_sf_result * @var{result})
+These routines compute the Trigamma function @math{\psi'(n)} for
+positive integer @math{n}.
+@comment Domain: n integer, n > 0 
+@comment Exceptional Return Values: GSL_EDOM
+@end deftypefun
+
+@deftypefun double gsl_sf_psi_1 (double @var{x})
+@deftypefunx int gsl_sf_psi_1_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the Trigamma function @math{\psi'(x)} for
+general @math{x}.
+@comment Domain: x != 0.0, -1.0, -2.0, ...
+@comment Exceptional Return Values: GSL_EDOM, GSL_ELOSS
+@end deftypefun
+
+@node Polygamma Function
+@subsection Polygamma Function
+
+@deftypefun double gsl_sf_psi_n (int @var{n}, double @var{x})
+@deftypefunx int gsl_sf_psi_n_e (int @var{n}, double @var{x}, gsl_sf_result * @var{result})
+These routines compute the polygamma function @c{$\psi^{(n)}(x)$}
+@math{\psi^@{(n)@}(x)} for
+@c{$n \ge 0$}
+@math{n >= 0}, @math{x > 0}.  
+@comment Domain: n >= 0, x > 0.0
+@comment Exceptional Return Values: GSL_EDOM
+@end deftypefun
diff --git a/gsl-1.9/doc/specfunc-synchrotron.texi b/gsl-1.9/doc/specfunc-synchrotron.texi
new file mode 100644
index 0000000..d02d4e3
--- /dev/null
+++ b/gsl-1.9/doc/specfunc-synchrotron.texi
@@ -0,0 +1,24 @@
+@cindex synchrotron functions
+
+The functions described in this section are declared in the header file
+@file{gsl_sf_synchrotron.h}.
+
+@deftypefun double gsl_sf_synchrotron_1 (double @var{x})
+@deftypefunx int gsl_sf_synchrotron_1_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the first synchrotron function 
+@c{$x \int_x^\infty dt K_{5/3}(t)$}
+@math{x \int_x^\infty dt K_@{5/3@}(t)} for @c{$x \ge 0$}
+@math{x >= 0}.
+@comment Domain: x >= 0.0
+@comment Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW
+@end deftypefun
+ 
+@deftypefun double gsl_sf_synchrotron_2 (double @var{x})
+@deftypefunx int gsl_sf_synchrotron_2_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the second synchrotron function 
+@c{$x K_{2/3}(x)$}
+@math{x K_@{2/3@}(x)} for @c{$x \ge 0$}
+@math{x >= 0}.
+@comment Domain: x >= 0.0
+@comment Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW
+@end deftypefun
diff --git a/gsl-1.9/doc/specfunc-transport.texi b/gsl-1.9/doc/specfunc-transport.texi
new file mode 100644
index 0000000..7337d5c
--- /dev/null
+++ b/gsl-1.9/doc/specfunc-transport.texi
@@ -0,0 +1,34 @@
+@cindex transport functions
+
+The transport functions @math{J(n,x)} are defined by the integral 
+representations
+@c{$J(n,x) := \int_0^x dt \, t^n e^t /(e^t - 1)^2$}
+@math{J(n,x) := \int_0^x dt t^n e^t /(e^t - 1)^2}.
+They are declared in the header file @file{gsl_sf_transport.h}.
+
+@deftypefun double gsl_sf_transport_2 (double @var{x})
+@deftypefunx int gsl_sf_transport_2_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the transport function @math{J(2,x)}.
+@comment Exceptional Return Values: GSL_EDOM
+@end deftypefun
+
+
+@deftypefun double gsl_sf_transport_3 (double @var{x})
+@deftypefunx int gsl_sf_transport_3_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the transport function @math{J(3,x)}.
+@comment Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW
+@end deftypefun
+
+
+@deftypefun double gsl_sf_transport_4 (double @var{x})
+@deftypefunx int gsl_sf_transport_4_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the transport function @math{J(4,x)}.
+@comment Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW
+@end deftypefun
+
+
+@deftypefun double gsl_sf_transport_5 (double @var{x})
+@deftypefunx int gsl_sf_transport_5_e (double @var{x}, gsl_sf_result * @var{result})
+These routines compute the transport function @math{J(5,x)}.
+@comment Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW
+@end deftypefun
diff --git a/gsl-1.9/doc/specfunc-trig.texi b/gsl-1.9/doc/specfunc-trig.texi
new file mode 100644
index 0000000..547b2a1
--- /dev/null
+++ b/gsl-1.9/doc/specfunc-trig.texi
@@ -0,0 +1,160 @@
+@cindex trigonometric functions
+
+The library includes its own trigonometric functions in order to provide
+consistency across platforms and reliable error estimates.  These
+functions are declared in the header file @file{gsl_sf_trig.h}.
+
+@menu
+* Circular Trigonometric Functions::  
+* Trigonometric Functions for Complex Arguments::  
+* Hyperbolic Trigonometric Functions::  
+* Conversion Functions::        
+* Restriction Functions::       
+* Trigonometric Functions With Error Estimates::  
+@end menu
+
+@node Circular Trigonometric Functions
+@subsection Circular Trigonometric Functions
+
+@deftypefun double gsl_sf_sin (double @var{x})
+@deftypefunx int gsl_sf_sin_e (double @var{x}, gsl_sf_result * @var{result})
+@cindex sine function, special functions
+These routines compute the sine function @math{\sin(x)}.
+@comment Exceptional Return Values:
+@end deftypefun
+
+@deftypefun double gsl_sf_cos (double @var{x})
+@deftypefunx int gsl_sf_cos_e (double @var{x}, gsl_sf_result * @var{result})
+@cindex cosine function, special functions
+These routines compute the cosine function @math{\cos(x)}.
+@comment Exceptional Return Values:
+@end deftypefun
+
+@deftypefun double gsl_sf_hypot (double @var{x}, double @var{y})
+@deftypefunx int gsl_sf_hypot_e (double @var{x}, double @var{y}, gsl_sf_result * @var{result})
+@cindex hypot function, special functions
+These routines compute the hypotenuse function @c{$\sqrt{x^2 + y^2}$}
+@math{\sqrt@{x^2 + y^2@}} avoiding overflow and underflow.
+@comment Exceptional Return Values:
+@end deftypefun
+
+@deftypefun double gsl_sf_sinc (double @var{x})
+@deftypefunx int gsl_sf_sinc_e (double @var{x}, gsl_sf_result * @var{result})
+@cindex complex sinc function, special functions
+These routines compute @math{\sinc(x) = \sin(\pi x) / (\pi x)} for any
+value of @var{x}.
+@comment Exceptional Return Values: none
+@end deftypefun
+
+@node Trigonometric Functions for Complex Arguments
+@subsection Trigonometric Functions for Complex Arguments
+
+@deftypefun int gsl_sf_complex_sin_e (double @var{zr}, double @var{zi}, gsl_sf_result * @var{szr}, gsl_sf_result * @var{szi})
+@cindex complex sine function, special functions
+This function computes the complex sine, @math{\sin(z_r + i z_i)} storing
+the real and imaginary parts in @var{szr}, @var{szi}.
+@comment Exceptional Return Values: GSL_EOVRFLW
+@end deftypefun
+
+@deftypefun int gsl_sf_complex_cos_e (double @var{zr}, double @var{zi}, gsl_sf_result * @var{czr}, gsl_sf_result * @var{czi})
+@cindex complex cosine function, special functions
+This function computes the complex cosine, @math{\cos(z_r + i z_i)} storing
+the real and imaginary parts in @var{szr}, @var{szi}.
+@comment Exceptional Return Values: GSL_EOVRFLW
+@end deftypefun
+
+@deftypefun int gsl_sf_complex_logsin_e (double @var{zr}, double @var{zi}, gsl_sf_result * @var{lszr}, gsl_sf_result * @var{lszi})
+@cindex complex log sine function, special functions
+This function computes the logarithm of the complex sine,
+@math{\log(\sin(z_r + i z_i))} storing the real and imaginary parts in
+@var{szr}, @var{szi}.
+@comment Exceptional Return Values: GSL_EDOM, GSL_ELOSS
+@end deftypefun
+
+@node Hyperbolic Trigonometric Functions
+@subsection Hyperbolic Trigonometric Functions
+
+@deftypefun double gsl_sf_lnsinh (double @var{x})
+@deftypefunx int gsl_sf_lnsinh_e (double @var{x}, gsl_sf_result * @var{result})
+@cindex logarithm of sinh function, special functions
+These routines compute @math{\log(\sinh(x))} for @math{x > 0}.
+@comment Domain: x > 0 
+@comment Exceptional Return Values: GSL_EDOM
+@end deftypefun
+
+@deftypefun double gsl_sf_lncosh (double @var{x})
+@deftypefunx int gsl_sf_lncosh_e (double @var{x}, gsl_sf_result * @var{result})
+@cindex logarithm of cosh function, special functions
+These routines compute @math{\log(\cosh(x))} for any @var{x}.
+@comment Exceptional Return Values: none
+@end deftypefun
+
+
+@node Conversion Functions
+@subsection Conversion Functions
+@cindex polar to rectangular conversion
+@cindex rectangular to polar conversion
+
+@deftypefun int gsl_sf_polar_to_rect (double @var{r}, double @var{theta}, gsl_sf_result * @var{x}, gsl_sf_result * @var{y}); 
+This function converts the polar coordinates (@var{r},@var{theta}) to
+rectilinear coordinates (@var{x},@var{y}), @math{x = r\cos(\theta)},
+@math{y = r\sin(\theta)}.
+@comment Exceptional Return Values: GSL_ELOSS
+@end deftypefun
+
+@deftypefun int gsl_sf_rect_to_polar (double @var{x}, double @var{y}, gsl_sf_result * @var{r}, gsl_sf_result * @var{theta})
+This function converts the rectilinear coordinates (@var{x},@var{y}) to
+polar coordinates (@var{r},@var{theta}), such that @math{x =
+r\cos(\theta)}, @math{y = r\sin(\theta)}.  The argument @var{theta}
+lies in the range @math{[-\pi, \pi]}.
+@comment Exceptional Return Values: GSL_EDOM
+@end deftypefun
+
+@node Restriction Functions
+@subsection Restriction Functions
+@cindex angular reduction
+@cindex reduction of angular variables
+
+@deftypefun double gsl_sf_angle_restrict_symm (double @var{theta})
+@deftypefunx int gsl_sf_angle_restrict_symm_e (double * @var{theta})
+These routines force the angle @var{theta} to lie in the range
+@math{(-\pi,\pi]}.  
+
+Note that the mathematical value of @math{\pi} is slightly greater
+than @code{M_PI}, so the machine numbers @code{M_PI} and @code{-M_PI}
+are included in the range.
+@comment Exceptional Return Values: GSL_ELOSS
+@end deftypefun
+
+@deftypefun double gsl_sf_angle_restrict_pos (double @var{theta})
+@deftypefunx int gsl_sf_angle_restrict_pos_e (double * @var{theta})
+These routines force the angle @var{theta} to lie in the range @math{[0,
+2\pi)}. 
+
+Note that the mathematical value of @math{2\pi} is slightly greater
+than @code{2*M_PI}, so the machine number @code{2*M_PI} is included in
+the range.
+
+@comment Exceptional Return Values: GSL_ELOSS
+@end deftypefun
+
+
+@node Trigonometric Functions With Error Estimates
+@subsection Trigonometric Functions With Error Estimates
+
+@deftypefun int gsl_sf_sin_err_e (double @var{x}, double @var{dx}, gsl_sf_result * @var{result})
+This routine computes the sine of an angle @var{x} with an associated 
+absolute error @var{dx},
+@c{$\sin(x \pm dx)$}
+@math{\sin(x \pm dx)}.  Note that this function is provided in the error-handling form only since
+its purpose is to compute the propagated error.
+@end deftypefun
+
+@deftypefun int gsl_sf_cos_err_e (double @var{x}, double @var{dx}, gsl_sf_result * @var{result})
+This routine computes the cosine of an angle @var{x} with an associated
+absolute error @var{dx}, 
+@c{$\cos(x \pm dx)$}
+@math{\cos(x \pm dx)}.  Note that this function is provided in the error-handling form only since
+its purpose is to compute the propagated error.
+@end deftypefun
+
diff --git a/gsl-1.9/doc/specfunc-zeta.texi b/gsl-1.9/doc/specfunc-zeta.texi
new file mode 100644
index 0000000..a773212
--- /dev/null
+++ b/gsl-1.9/doc/specfunc-zeta.texi
@@ -0,0 +1,98 @@
+@cindex Zeta functions
+
+The Riemann zeta function is defined in Abramowitz & Stegun, Section
+23.2.  The functions described in this section are declared in the
+header file @file{gsl_sf_zeta.h}.
+
+@menu
+* Riemann Zeta Function::
+* Riemann Zeta Function Minus One::
+* Hurwitz Zeta Function::
+* Eta Function::
+@end menu
+
+@node Riemann Zeta Function
+@subsection Riemann Zeta Function
+
+The Riemann zeta function is defined by the infinite sum 
+@c{$\zeta(s) = \sum_{k=1}^\infty k^{-s}$}
+@math{\zeta(s) = \sum_@{k=1@}^\infty k^@{-s@}}.  
+
+@deftypefun double gsl_sf_zeta_int (int @var{n})
+@deftypefunx int gsl_sf_zeta_int_e (int @var{n}, gsl_sf_result * @var{result})
+These routines compute the Riemann zeta function @math{\zeta(n)} 
+for integer @var{n},
+@math{n \ne 1}.
+@comment Domain: n integer, n != 1
+@comment Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW
+@end deftypefun
+
+@deftypefun double gsl_sf_zeta (double @var{s})
+@deftypefunx int gsl_sf_zeta_e (double @var{s}, gsl_sf_result * @var{result})
+These routines compute the Riemann zeta function @math{\zeta(s)}
+for arbitrary @var{s},
+@math{s \ne 1}.
+@comment Domain: s != 1.0
+@comment Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW
+@end deftypefun
+
+
+@node Riemann Zeta Function Minus One
+@subsection Riemann Zeta Function Minus One
+
+For large positive argument, the Riemann zeta function approaches one.
+In this region the fractional part is interesting, and therefore we
+need a function to evaluate it explicitly.
+
+@deftypefun double gsl_sf_zetam1_int (int @var{n})
+@deftypefunx int gsl_sf_zetam1_int_e (int @var{n}, gsl_sf_result * @var{result})
+These routines compute @math{\zeta(n) - 1} for integer @var{n},
+@math{n \ne 1}.
+@comment Domain: n integer, n != 1
+@comment Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW
+@end deftypefun
+
+@deftypefun double gsl_sf_zetam1 (double @var{s})
+@deftypefunx int gsl_sf_zetam1_e (double @var{s}, gsl_sf_result * @var{result})
+These routines compute @math{\zeta(s) - 1} for arbitrary @var{s},
+@math{s \ne 1}.
+@comment Domain: s != 1.0
+@comment Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW
+@end deftypefun
+
+
+@node Hurwitz Zeta Function
+@subsection Hurwitz Zeta Function
+
+The Hurwitz zeta function is defined by
+@c{$\zeta(s,q) = \sum_0^\infty (k+q)^{-s}$}
+@math{\zeta(s,q) = \sum_0^\infty (k+q)^@{-s@}}.
+
+@deftypefun double gsl_sf_hzeta (double @var{s}, double @var{q})
+@deftypefunx int gsl_sf_hzeta_e (double @var{s}, double @var{q}, gsl_sf_result * @var{result})
+These routines compute the Hurwitz zeta function @math{\zeta(s,q)} for
+@math{s > 1}, @math{q > 0}.
+@comment Domain: s > 1.0, q > 0.0
+@comment Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW, GSL_EOVRFLW
+@end deftypefun
+
+
+@node Eta Function
+@subsection Eta Function
+
+The eta function is defined by
+@c{$\eta(s) = (1-2^{1-s}) \zeta(s)$}
+@math{\eta(s) = (1-2^@{1-s@}) \zeta(s)}.
+
+@deftypefun double gsl_sf_eta_int (int @var{n})
+@deftypefunx int gsl_sf_eta_int_e (int @var{n}, gsl_sf_result * @var{result})
+These routines compute the eta function @math{\eta(n)} for integer @var{n}.
+@comment Exceptional Return Values: GSL_EUNDRFLW, GSL_EOVRFLW
+@end deftypefun
+
+@deftypefun double gsl_sf_eta (double @var{s})
+@deftypefunx int gsl_sf_eta_e (double @var{s}, gsl_sf_result * @var{result})
+These routines compute the eta function @math{\eta(s)} for arbitrary @var{s}.
+@comment Exceptional Return Values: GSL_EUNDRFLW, GSL_EOVRFLW
+@end deftypefun
+
diff --git a/gsl-1.9/doc/specfunc.texi b/gsl-1.9/doc/specfunc.texi
new file mode 100644
index 0000000..50c93c3
--- /dev/null
+++ b/gsl-1.9/doc/specfunc.texi
@@ -0,0 +1,352 @@
+@cindex special functions
+
+This chapter describes the GSL special function library.  The library
+includes routines for calculating the values of Airy functions, Bessel
+functions, Clausen functions, Coulomb wave functions, Coupling
+coefficients, the Dawson function, Debye functions, Dilogarithms,
+Elliptic integrals, Jacobi elliptic functions, Error functions,
+Exponential integrals, Fermi-Dirac functions, Gamma functions,
+Gegenbauer functions, Hypergeometric functions, Laguerre functions,
+Legendre functions and Spherical Harmonics, the Psi (Digamma) Function,
+Synchrotron functions, Transport functions, Trigonometric functions and
+Zeta functions.  Each routine also computes an estimate of the numerical
+error in the calculated value of the function.
+
+The functions in this chapter are declared in individual header files,
+such as @file{gsl_sf_airy.h}, @file{gsl_sf_bessel.h}, etc.  The complete
+set of header files can be included using the file @file{gsl_sf.h}.
+
+@menu
+* Special Function Usage::      
+* The gsl_sf_result struct::    
+* Special Function Modes::      
+* Airy Functions and Derivatives::  
+* Bessel Functions::            
+* Clausen Functions::           
+* Coulomb Functions::           
+* Coupling Coefficients::       
+* Dawson Function::             
+* Debye Functions::             
+* Dilogarithm::                 
+* Elementary Operations::       
+* Elliptic Integrals::          
+* Elliptic Functions (Jacobi)::  
+* Error Functions::             
+* Exponential Functions::       
+* Exponential Integrals::       
+* Fermi-Dirac Function::        
+* Gamma and Beta Functions::    
+* Gegenbauer Functions::        
+* Hypergeometric Functions::    
+* Laguerre Functions::          
+* Lambert W Functions::         
+* Legendre Functions and Spherical Harmonics::  
+* Logarithm and Related Functions::  
+* Mathieu Functions::
+* Power Function::              
+* Psi (Digamma) Function::      
+* Synchrotron Functions::       
+* Transport Functions::         
+* Trigonometric Functions::     
+* Zeta Functions::              
+* Special Functions Examples::  
+* Special Functions References and Further Reading::  
+@end menu
+
+@node Special Function Usage
+@section Usage
+
+The special functions are available in two calling conventions, a
+@dfn{natural form} which returns the numerical value of the function and
+an @dfn{error-handling form} which returns an error code.  The two types
+of function provide alternative ways of accessing the same underlying
+code.
+
+The @dfn{natural form} returns only the value of the function and can be
+used directly in mathematical expressions.  For example, the following
+function call will compute the value of the Bessel function
+@math{J_0(x)},
+
+@example
+double y = gsl_sf_bessel_J0 (x);
+@end example
+
+@noindent
+There is no way to access an error code or to estimate the error using
+this method.  To allow access to this information the alternative
+error-handling form stores the value and error in a modifiable argument,
+
+@example
+gsl_sf_result result;
+int status = gsl_sf_bessel_J0_e (x, &result);
+@end example
+
+@noindent
+The error-handling functions have the suffix @code{_e}. The returned
+status value indicates error conditions such as overflow, underflow or
+loss of precision.  If there are no errors the error-handling functions
+return @code{GSL_SUCCESS}.
+
+@node The gsl_sf_result struct
+@section The gsl_sf_result struct
+@cindex gsl_sf_result
+@cindex gsl_sf_result_e10
+
+The error handling form of the special functions always calculate an
+error estimate along with the value of the result.  Therefore,
+structures are provided for amalgamating a value and error estimate.
+These structures are declared in the header file @file{gsl_sf_result.h}.
+
+The @code{gsl_sf_result} struct contains value and error fields.
+
+@example
+typedef struct
+@{
+  double val;
+  double err;
+@} gsl_sf_result;
+@end example
+
+@noindent
+The field @var{val} contains the value and the field @var{err} contains
+an estimate of the absolute error in the value.
+
+In some cases, an overflow or underflow can be detected and handled by a
+function.  In this case, it may be possible to return a scaling exponent
+as well as an error/value pair in order to save the result from
+exceeding the dynamic range of the built-in types.  The
+@code{gsl_sf_result_e10} struct contains value and error fields as well
+as an exponent field such that the actual result is obtained as
+@code{result * 10^(e10)}.
+
+@example
+typedef struct
+@{
+  double val;
+  double err;
+  int    e10;
+@} gsl_sf_result_e10;
+@end example
+
+@node Special Function Modes
+@section Modes
+
+The goal of the library is to achieve double precision accuracy wherever
+possible.  However the cost of evaluating some special functions to
+double precision can be significant, particularly where very high order
+terms are required.  In these cases a @code{mode} argument allows the
+accuracy of the function to be reduced in order to improve performance.
+The following precision levels are available for the mode argument,
+
+@table @code
+@item GSL_PREC_DOUBLE
+Double-precision, a relative accuracy of approximately @c{$2 \times 10^{-16}$}
+@math{2 * 10^-16}.
+@item GSL_PREC_SINGLE
+Single-precision, a relative accuracy of approximately @c{$1 \times 10^{-7}$}
+@math{10^-7}.
+@item GSL_PREC_APPROX
+Approximate values, a relative accuracy of approximately @c{$5 \times 10^{-4}$}
+@math{5 * 10^-4}.
+@end table
+
+@noindent
+The approximate mode provides the fastest evaluation at the lowest
+accuracy.
+
+@node Airy Functions and Derivatives
+@section Airy Functions and Derivatives
+@include specfunc-airy.texi
+
+@node Bessel Functions
+@section Bessel Functions
+@include specfunc-bessel.texi
+
+@node Clausen Functions
+@section Clausen Functions
+@include specfunc-clausen.texi
+
+@node Coulomb Functions
+@section Coulomb Functions
+@include specfunc-coulomb.texi
+
+@node Coupling Coefficients
+@section Coupling Coefficients
+@include specfunc-coupling.texi
+
+@node Dawson Function
+@section Dawson Function
+@include specfunc-dawson.texi
+
+@node Debye Functions
+@section Debye Functions
+@include specfunc-debye.texi
+
+@node Dilogarithm
+@section Dilogarithm
+@include specfunc-dilog.texi
+
+@node Elementary Operations
+@section Elementary Operations
+@include specfunc-elementary.texi
+
+@node Elliptic Integrals
+@section Elliptic Integrals
+@include specfunc-ellint.texi
+
+@node Elliptic Functions (Jacobi)
+@section Elliptic Functions (Jacobi)
+@include specfunc-elljac.texi
+
+@node Error Functions
+@section Error Functions
+@include specfunc-erf.texi
+
+@node Exponential Functions
+@section Exponential Functions
+@include specfunc-exp.texi
+
+@node Exponential Integrals
+@section Exponential Integrals
+@include specfunc-expint.texi
+
+@node Fermi-Dirac Function
+@section Fermi-Dirac Function
+@include specfunc-fermi-dirac.texi
+
+@node Gamma and Beta Functions
+@section Gamma and Beta Functions
+@include specfunc-gamma.texi
+
+@node Gegenbauer Functions
+@section Gegenbauer Functions
+@include specfunc-gegenbauer.texi
+
+@node Hypergeometric Functions
+@section Hypergeometric Functions
+@include specfunc-hyperg.texi
+
+@node Laguerre Functions
+@section Laguerre Functions
+@include specfunc-laguerre.texi
+
+@node Lambert W Functions
+@section Lambert W Functions
+@include specfunc-lambert.texi
+
+@node Legendre Functions and Spherical Harmonics
+@section Legendre Functions and Spherical Harmonics
+@include specfunc-legendre.texi
+
+@node Logarithm and Related Functions
+@section Logarithm and Related Functions
+@include specfunc-log.texi
+
+@node Mathieu Functions
+@section Mathieu Functions
+@include specfunc-mathieu.texi
+
+@node Power Function
+@section Power Function
+@include specfunc-pow-int.texi
+
+@node Psi (Digamma) Function
+@section Psi (Digamma) Function
+@include specfunc-psi.texi
+
+@node Synchrotron Functions
+@section Synchrotron Functions
+@include specfunc-synchrotron.texi
+
+@node Transport Functions
+@section Transport Functions
+@include specfunc-transport.texi
+
+@node Trigonometric Functions
+@section Trigonometric Functions
+@include specfunc-trig.texi
+
+@node Zeta Functions
+@section Zeta Functions
+@include specfunc-zeta.texi
+
+@node Special Functions Examples
+@section Examples
+
+The following example demonstrates the use of the error handling form of
+the special functions, in this case to compute the Bessel function
+@math{J_0(5.0)},
+
+@example
+@verbatiminclude examples/specfun_e.c
+@end example
+
+@noindent
+Here are the results of running the program,
+
+@example
+$ ./a.out 
+@verbatiminclude examples/specfun_e.out
+@end example
+
+@noindent
+The next program computes the same quantity using the natural form of
+the function. In this case the error term @var{result.err} and return
+status are not accessible.
+
+@example
+@verbatiminclude examples/specfun.c
+@end example
+
+@noindent
+The results of the function are the same,
+
+@example
+$ ./a.out 
+@verbatiminclude examples/specfun.out
+@end example
+
+
+
+@node Special Functions References and Further Reading
+@section References and Further Reading
+
+The library follows the conventions of @cite{Abramowitz & Stegun} where
+possible,
+@itemize @asis
+@item
+Abramowitz & Stegun (eds.), @cite{Handbook of Mathematical Functions}
+@end itemize
+
+@noindent
+The following papers contain information on the algorithms used 
+to compute the special functions,
+@cindex MISCFUN
+@itemize @asis
+@item
+MISCFUN: A software package to compute uncommon special functions.
+@cite{ACM Trans.@: Math.@: Soft.}, vol.@: 22, 1996, 288--301
+
+@item
+G.N. Watson, A Treatise on the Theory of Bessel Functions,
+2nd Edition (Cambridge University Press, 1944).
+
+@item
+G. Nemeth, Mathematical Approximations of Special Functions,
+Nova Science Publishers, ISBN 1-56072-052-2
+
+@item
+B.C. Carlson, Special Functions of Applied Mathematics (1977)
+
+@item
+W.J. Thompson, Atlas for Computing Mathematical Functions, John Wiley & Sons,
+New York (1997).
+
+@item
+Y.Y. Luke, Algorithms for the Computation of Mathematical Functions, Academic
+Press, New York (1977).
+
+@comment @item
+@comment Fermi-Dirac functions of orders @math{-1/2}, @math{1/2}, @math{3/2}, and
+@comment @math{5/2}.  @cite{ACM Trans. Math. Soft.}, vol. 24, 1998, 1-12.
+@end itemize
diff --git a/gsl-1.9/doc/stamp-vti b/gsl-1.9/doc/stamp-vti
new file mode 100644
index 0000000..391fb64
--- /dev/null
+++ b/gsl-1.9/doc/stamp-vti
@@ -0,0 +1,4 @@
+@set UPDATED 20 February 2007
+@set UPDATED-MONTH February 2007
+@set EDITION 1.9
+@set VERSION 1.9
diff --git a/gsl-1.9/doc/statistics.texi b/gsl-1.9/doc/statistics.texi
new file mode 100644
index 0000000..c289278
--- /dev/null
+++ b/gsl-1.9/doc/statistics.texi
@@ -0,0 +1,741 @@
+@cindex statistics
+@cindex mean
+@cindex standard deviation
+@cindex variance
+@cindex estimated standard deviation
+@cindex estimated variance
+@cindex t-test
+@cindex range
+@cindex min
+@cindex max
+
+This chapter describes the statistical functions in the library.  The
+basic statistical functions include routines to compute the mean,
+variance and standard deviation.  More advanced functions allow you to
+calculate absolute deviations, skewness, and kurtosis as well as the
+median and arbitrary percentiles.  The algorithms use recurrence
+relations to compute average quantities in a stable way, without large
+intermediate values that might overflow. 
+
+The functions are available in versions for datasets in the standard
+floating-point and integer types.  The versions for double precision
+floating-point data have the prefix @code{gsl_stats} and are declared in
+the header file @file{gsl_statistics_double.h}.  The versions for integer
+data have the prefix @code{gsl_stats_int} and are declared in the header
+file @file{gsl_statistics_int.h}. 
+
+@menu
+* Mean and standard deviation and variance::  
+* Absolute deviation::          
+* Higher moments (skewness and kurtosis)::  
+* Autocorrelation::             
+* Covariance::                  
+* Weighted Samples::            
+* Maximum and Minimum values::  
+* Median and Percentiles::      
+* Example statistical programs::  
+* Statistics References and Further Reading::  
+@end menu
+
+@node Mean and standard deviation and variance
+@section Mean, Standard Deviation and Variance
+
+@deftypefun double gsl_stats_mean (const double @var{data}[], size_t @var{stride}, size_t @var{n})
+This function returns the arithmetic mean of @var{data}, a dataset of
+length @var{n} with stride @var{stride}.  The arithmetic mean, or
+@dfn{sample mean}, is denoted by @math{\Hat\mu} and defined as,
+@tex
+\beforedisplay
+$$
+{\Hat\mu} = {1 \over N} \sum x_i
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+\Hat\mu = (1/N) \sum x_i
+@end example
+
+@end ifinfo
+@noindent
+where @math{x_i} are the elements of the dataset @var{data}.  For
+samples drawn from a gaussian distribution the variance of
+@math{\Hat\mu} is @math{\sigma^2 / N}.
+@end deftypefun
+
+@deftypefun double gsl_stats_variance (const double @var{data}[], size_t @var{stride}, size_t @var{n})
+This function returns the estimated, or @dfn{sample}, variance of
+@var{data}, a dataset of length @var{n} with stride @var{stride}.  The
+estimated variance is denoted by @math{\Hat\sigma^2} and is defined by,
+@tex
+\beforedisplay
+$$
+{\Hat\sigma}^2 = {1 \over (N-1)} \sum (x_i - {\Hat\mu})^2
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+\Hat\sigma^2 = (1/(N-1)) \sum (x_i - \Hat\mu)^2
+@end example
+
+@end ifinfo
+@noindent
+where @math{x_i} are the elements of the dataset @var{data}.  Note that
+the normalization factor of @math{1/(N-1)} results from the derivation
+of @math{\Hat\sigma^2} as an unbiased estimator of the population
+variance @math{\sigma^2}.  For samples drawn from a gaussian distribution
+the variance of @math{\Hat\sigma^2} itself is @math{2 \sigma^4 / N}.
+
+This function computes the mean via a call to @code{gsl_stats_mean}.  If
+you have already computed the mean then you can pass it directly to
+@code{gsl_stats_variance_m}.
+@end deftypefun
+
+@deftypefun double gsl_stats_variance_m (const double @var{data}[], size_t @var{stride}, size_t @var{n}, double @var{mean})
+This function returns the sample variance of @var{data} relative to the
+given value of @var{mean}.  The function is computed with @math{\Hat\mu}
+replaced by the value of @var{mean} that you supply,
+@tex
+\beforedisplay
+$$
+{\Hat\sigma}^2 = {1 \over (N-1)} \sum (x_i - mean)^2
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+\Hat\sigma^2 = (1/(N-1)) \sum (x_i - mean)^2
+@end example
+@end ifinfo
+@end deftypefun
+
+@deftypefun double gsl_stats_sd (const double @var{data}[], size_t @var{stride}, size_t @var{n})
+@deftypefunx double gsl_stats_sd_m (const double @var{data}[], size_t @var{stride}, size_t @var{n}, double @var{mean})
+The standard deviation is defined as the square root of the variance.
+These functions return the square root of the corresponding variance
+functions above.
+@end deftypefun
+
+@deftypefun double gsl_stats_variance_with_fixed_mean (const double @var{data}[], size_t @var{stride}, size_t @var{n}, double @var{mean})
+This function computes an unbiased estimate of the variance of
+@var{data} when the population mean @var{mean} of the underlying
+distribution is known @emph{a priori}.  In this case the estimator for
+the variance uses the factor @math{1/N} and the sample mean
+@math{\Hat\mu} is replaced by the known population mean @math{\mu},
+@tex
+\beforedisplay
+$$
+{\Hat\sigma}^2 = {1 \over N} \sum (x_i - \mu)^2
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+\Hat\sigma^2 = (1/N) \sum (x_i - \mu)^2
+@end example
+@end ifinfo
+@end deftypefun
+
+
+@deftypefun double gsl_stats_sd_with_fixed_mean (const double @var{data}[], size_t @var{stride}, size_t @var{n}, double @var{mean})
+This function calculates the standard deviation of @var{data} for a
+fixed population mean @var{mean}.  The result is the square root of the
+corresponding variance function.
+@end deftypefun
+
+@node Absolute deviation
+@section Absolute deviation
+
+@deftypefun double gsl_stats_absdev (const double @var{data}[], size_t @var{stride}, size_t @var{n})
+This function computes the absolute deviation from the mean of
+@var{data}, a dataset of length @var{n} with stride @var{stride}.  The
+absolute deviation from the mean is defined as,
+@tex
+\beforedisplay
+$$
+absdev  = {1 \over N} \sum |x_i - {\Hat\mu}|
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+absdev  = (1/N) \sum |x_i - \Hat\mu|
+@end example
+
+@end ifinfo
+@noindent
+where @math{x_i} are the elements of the dataset @var{data}.  The
+absolute deviation from the mean provides a more robust measure of the
+width of a distribution than the variance.  This function computes the
+mean of @var{data} via a call to @code{gsl_stats_mean}.
+@end deftypefun
+
+@deftypefun double gsl_stats_absdev_m (const double @var{data}[], size_t @var{stride}, size_t @var{n}, double @var{mean})
+This function computes the absolute deviation of the dataset @var{data}
+relative to the given value of @var{mean},
+@tex
+\beforedisplay
+$$
+absdev  = {1 \over N} \sum |x_i - mean|
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+absdev  = (1/N) \sum |x_i - mean|
+@end example
+
+@end ifinfo
+@noindent
+This function is useful if you have already computed the mean of
+@var{data} (and want to avoid recomputing it), or wish to calculate the
+absolute deviation relative to another value (such as zero, or the
+median).
+@end deftypefun
+
+@node Higher moments (skewness and kurtosis)
+@section Higher moments (skewness and kurtosis)
+
+@deftypefun double gsl_stats_skew (const double @var{data}[], size_t @var{stride}, size_t @var{n})
+This function computes the skewness of @var{data}, a dataset of length
+@var{n} with stride @var{stride}.  The skewness is defined as,
+@tex
+\beforedisplay
+$$
+skew = {1 \over N} \sum 
+ {\left( x_i - {\Hat\mu} \over {\Hat\sigma} \right)}^3
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+skew = (1/N) \sum ((x_i - \Hat\mu)/\Hat\sigma)^3
+@end example
+
+@end ifinfo
+@noindent
+where @math{x_i} are the elements of the dataset @var{data}.  The skewness
+measures the asymmetry of the tails of a distribution.
+
+The function computes the mean and estimated standard deviation of
+@var{data} via calls to @code{gsl_stats_mean} and @code{gsl_stats_sd}.
+@end deftypefun
+
+@deftypefun double gsl_stats_skew_m_sd (const double @var{data}[], size_t @var{stride}, size_t @var{n}, double @var{mean}, double @var{sd})
+This function computes the skewness of the dataset @var{data} using the
+given values of the mean @var{mean} and standard deviation @var{sd},
+@tex
+\beforedisplay
+$$
+skew = {1 \over N}
+     \sum {\left( x_i - mean \over sd \right)}^3
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+skew = (1/N) \sum ((x_i - mean)/sd)^3
+@end example
+
+@end ifinfo
+@noindent
+These functions are useful if you have already computed the mean and
+standard deviation of @var{data} and want to avoid recomputing them.
+@end deftypefun
+
+@deftypefun double gsl_stats_kurtosis (const double @var{data}[], size_t @var{stride}, size_t @var{n})
+This function computes the kurtosis of @var{data}, a dataset of length
+@var{n} with stride @var{stride}.  The kurtosis is defined as,
+@tex
+\beforedisplay
+$$
+kurtosis = \left( {1 \over N} \sum 
+ {\left(x_i - {\Hat\mu} \over {\Hat\sigma} \right)}^4 
+ \right) 
+ - 3
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+kurtosis = ((1/N) \sum ((x_i - \Hat\mu)/\Hat\sigma)^4)  - 3
+@end example
+
+@end ifinfo
+@noindent
+The kurtosis measures how sharply peaked a distribution is, relative to
+its width.  The kurtosis is normalized to zero for a gaussian
+distribution.
+@end deftypefun
+
+@deftypefun double gsl_stats_kurtosis_m_sd (const double @var{data}[], size_t @var{stride}, size_t @var{n}, double @var{mean}, double @var{sd})
+This function computes the kurtosis of the dataset @var{data} using the
+given values of the mean @var{mean} and standard deviation @var{sd},
+@tex
+\beforedisplay
+$$
+kurtosis = {1 \over N}
+  \left( \sum {\left(x_i - mean \over sd \right)}^4 \right) 
+  - 3
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+kurtosis = ((1/N) \sum ((x_i - mean)/sd)^4) - 3
+@end example
+
+@end ifinfo
+@noindent
+This function is useful if you have already computed the mean and
+standard deviation of @var{data} and want to avoid recomputing them.
+@end deftypefun
+
+@node Autocorrelation
+@section Autocorrelation
+
+@deftypefun double gsl_stats_lag1_autocorrelation (const double @var{data}[], const size_t @var{stride}, const size_t @var{n})
+This function computes the lag-1 autocorrelation of the dataset @var{data}.
+@tex
+\beforedisplay
+$$
+a_1 = {\sum_{i = 1}^{n} (x_{i} - \Hat\mu) (x_{i-1} - \Hat\mu)
+\over
+\sum_{i = 1}^{n} (x_{i} - \Hat\mu) (x_{i} - \Hat\mu)}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+a_1 = @{\sum_@{i = 1@}^@{n@} (x_@{i@} - \Hat\mu) (x_@{i-1@} - \Hat\mu)
+       \over
+       \sum_@{i = 1@}^@{n@} (x_@{i@} - \Hat\mu) (x_@{i@} - \Hat\mu)@}
+@end example
+@end ifinfo
+@end deftypefun
+
+
+@deftypefun double gsl_stats_lag1_autocorrelation_m (const double @var{data}[], const size_t @var{stride}, const size_t @var{n}, const double @var{mean})
+This function computes the lag-1 autocorrelation of the dataset
+@var{data} using the given value of the mean @var{mean}.
+
+@end deftypefun
+
+@node Covariance
+@section Covariance
+@cindex covariance, of two datasets
+
+@deftypefun double gsl_stats_covariance (const double @var{data1}[], const size_t @var{stride1}, const double @var{data2}[], const size_t @var{stride2}, const size_t @var{n})
+This function computes the covariance of the datasets @var{data1} and
+@var{data2} which must both be of the same length @var{n}.
+@tex
+\beforedisplay
+$$
+covar = {1 \over (n - 1)} \sum_{i = 1}^{n} (x_{i} - \Hat x) (y_{i} - \Hat y)
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+covar = (1/(n - 1)) \sum_@{i = 1@}^@{n@} (x_i - \Hat x) (y_i - \Hat y)
+@end example
+@end ifinfo
+@end deftypefun
+
+@deftypefun double gsl_stats_covariance_m (const double @var{data1}[], const size_t @var{stride1}, const double @var{data2}[], const size_t @var{stride2}, const size_t @var{n}, const double @var{mean1}, const double @var{mean2})
+This function computes the covariance of the datasets @var{data1} and
+@var{data2} using the given values of the means, @var{mean1} and
+@var{mean2}.  This is useful if you have already computed the means of
+@var{data1} and @var{data2} and want to avoid recomputing them.
+@end deftypefun
+
+
+@node Weighted Samples
+@section Weighted Samples
+
+The functions described in this section allow the computation of
+statistics for weighted samples.  The functions accept an array of
+samples, @math{x_i}, with associated weights, @math{w_i}.  Each sample
+@math{x_i} is considered as having been drawn from a Gaussian
+distribution with variance @math{\sigma_i^2}.  The sample weight
+@math{w_i} is defined as the reciprocal of this variance, @math{w_i =
+1/\sigma_i^2}.  Setting a weight to zero corresponds to removing a
+sample from a dataset.
+
+@deftypefun double gsl_stats_wmean (const double @var{w}[], size_t @var{wstride}, const double @var{data}[], size_t @var{stride}, size_t @var{n})
+This function returns the weighted mean of the dataset @var{data} with
+stride @var{stride} and length @var{n}, using the set of weights @var{w}
+with stride @var{wstride} and length @var{n}.  The weighted mean is defined as,
+@tex
+\beforedisplay
+$$
+{\Hat\mu} = {{\sum w_i x_i} \over {\sum w_i}}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+\Hat\mu = (\sum w_i x_i) / (\sum w_i)
+@end example
+@end ifinfo
+@end deftypefun
+
+
+@deftypefun double gsl_stats_wvariance (const double @var{w}[], size_t @var{wstride}, const double @var{data}[], size_t @var{stride}, size_t @var{n})
+This function returns the estimated variance of the dataset @var{data}
+with stride @var{stride} and length @var{n}, using the set of weights
+@var{w} with stride @var{wstride} and length @var{n}.  The estimated
+variance of a weighted dataset is defined as,
+@tex
+\beforedisplay
+$$
+\Hat\sigma^2 = {{\sum w_i} \over {(\sum w_i)^2 - \sum (w_i^2)}} 
+                \sum w_i (x_i - \Hat\mu)^2
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+\Hat\sigma^2 = ((\sum w_i)/((\sum w_i)^2 - \sum (w_i^2))) 
+                \sum w_i (x_i - \Hat\mu)^2
+@end example
+
+@end ifinfo
+@noindent
+Note that this expression reduces to an unweighted variance with the
+familiar @math{1/(N-1)} factor when there are @math{N} equal non-zero
+weights.
+@end deftypefun
+
+@deftypefun double gsl_stats_wvariance_m (const double @var{w}[], size_t @var{wstride}, const double @var{data}[], size_t @var{stride}, size_t @var{n}, double @var{wmean})
+This function returns the estimated variance of the weighted dataset
+@var{data} using the given weighted mean @var{wmean}.
+@end deftypefun
+
+@deftypefun double gsl_stats_wsd (const double @var{w}[], size_t @var{wstride}, const double @var{data}[], size_t @var{stride}, size_t @var{n})
+The standard deviation is defined as the square root of the variance.
+This function returns the square root of the corresponding variance
+function @code{gsl_stats_wvariance} above.
+@end deftypefun
+
+@deftypefun double gsl_stats_wsd_m (const double @var{w}[], size_t @var{wstride}, const double @var{data}[], size_t @var{stride}, size_t @var{n}, double @var{wmean})
+This function returns the square root of the corresponding variance
+function @code{gsl_stats_wvariance_m} above.
+@end deftypefun
+
+@deftypefun double gsl_stats_wvariance_with_fixed_mean (const double @var{w}[], size_t @var{wstride}, const double @var{data}[], size_t @var{stride}, size_t @var{n}, const double @var{mean})
+This function computes an unbiased estimate of the variance of weighted
+dataset @var{data} when the population mean @var{mean} of the underlying
+distribution is known @emph{a priori}.  In this case the estimator for
+the variance replaces the sample mean @math{\Hat\mu} by the known
+population mean @math{\mu},
+@tex
+\beforedisplay
+$$
+\Hat\sigma^2 = {{\sum w_i (x_i - \mu)^2} \over {\sum w_i}}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+\Hat\sigma^2 = (\sum w_i (x_i - \mu)^2) / (\sum w_i)
+@end example
+@end ifinfo
+@end deftypefun
+
+@deftypefun double gsl_stats_wsd_with_fixed_mean (const double @var{w}[], size_t @var{wstride}, const double @var{data}[], size_t @var{stride}, size_t @var{n}, const double @var{mean})
+The standard deviation is defined as the square root of the variance.
+This function returns the square root of the corresponding variance
+function above.
+@end deftypefun
+
+@deftypefun double gsl_stats_wabsdev (const double @var{w}[], size_t @var{wstride}, const double @var{data}[], size_t @var{stride}, size_t @var{n})
+This function computes the weighted absolute deviation from the weighted
+mean of @var{data}.  The absolute deviation from the mean is defined as,
+@tex
+\beforedisplay
+$$
+absdev = {{\sum w_i |x_i - \Hat\mu|} \over {\sum w_i}}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+absdev = (\sum w_i |x_i - \Hat\mu|) / (\sum w_i)
+@end example
+@end ifinfo
+@end deftypefun
+
+@deftypefun double gsl_stats_wabsdev_m (const double @var{w}[], size_t @var{wstride}, const double @var{data}[], size_t @var{stride}, size_t @var{n}, double @var{wmean})
+This function computes the absolute deviation of the weighted dataset
+@var{data} about the given weighted mean @var{wmean}.
+@end deftypefun
+
+@deftypefun double gsl_stats_wskew (const double @var{w}[], size_t @var{wstride}, const double @var{data}[], size_t @var{stride}, size_t @var{n})
+This function computes the weighted skewness of the dataset @var{data}.
+@tex
+\beforedisplay
+$$
+skew = {{\sum w_i ((x_i - xbar)/\sigma)^3} \over {\sum w_i}}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+skew = (\sum w_i ((x_i - xbar)/\sigma)^3) / (\sum w_i)
+@end example
+@end ifinfo
+@end deftypefun
+
+@deftypefun double gsl_stats_wskew_m_sd (const double @var{w}[], size_t @var{wstride}, const double @var{data}[], size_t @var{stride}, size_t @var{n}, double @var{wmean}, double @var{wsd})
+This function computes the weighted skewness of the dataset @var{data}
+using the given values of the weighted mean and weighted standard
+deviation, @var{wmean} and @var{wsd}.
+@end deftypefun
+
+@deftypefun double gsl_stats_wkurtosis (const double @var{w}[], size_t @var{wstride}, const double @var{data}[], size_t @var{stride}, size_t @var{n})
+This function computes the weighted kurtosis of the dataset @var{data}.
+
+@tex
+\beforedisplay
+$$
+kurtosis = {{\sum w_i ((x_i - xbar)/sigma)^4} \over {\sum w_i}} - 3
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+kurtosis = ((\sum w_i ((x_i - xbar)/sigma)^4) / (\sum w_i)) - 3
+@end example
+@end ifinfo
+@end deftypefun
+
+@deftypefun double gsl_stats_wkurtosis_m_sd (const double @var{w}[], size_t @var{wstride}, const double @var{data}[], size_t @var{stride}, size_t @var{n}, double @var{wmean}, double @var{wsd})
+This function computes the weighted kurtosis of the dataset @var{data}
+using the given values of the weighted mean and weighted standard
+deviation, @var{wmean} and @var{wsd}.
+@end deftypefun
+
+@node Maximum and Minimum values
+@section Maximum and Minimum values
+
+The following functions find the maximum and minimum values of a
+dataset (or their indices).  If the data contains @code{NaN}s then a
+@code{NaN} will be returned, since the maximum or minimum value is
+undefined.  For functions which return an index, the location of the
+first @code{NaN} in the array is returned.
+
+@deftypefun double gsl_stats_max (const double @var{data}[], size_t @var{stride}, size_t @var{n})
+This function returns the maximum value in @var{data}, a dataset of
+length @var{n} with stride @var{stride}.  The maximum value is defined
+as the value of the element @math{x_i} which satisfies @c{$x_i \ge x_j$}
+@math{x_i >= x_j} for all @math{j}.
+
+If you want instead to find the element with the largest absolute
+magnitude you will need to apply @code{fabs} or @code{abs} to your data
+before calling this function.
+@end deftypefun
+
+@deftypefun double gsl_stats_min (const double @var{data}[], size_t @var{stride}, size_t @var{n})
+This function returns the minimum value in @var{data}, a dataset of
+length @var{n} with stride @var{stride}.  The minimum value is defined
+as the value of the element @math{x_i} which satisfies @c{$x_i \le x_j$}
+@math{x_i <= x_j} for all @math{j}.
+
+If you want instead to find the element with the smallest absolute
+magnitude you will need to apply @code{fabs} or @code{abs} to your data
+before calling this function.
+@end deftypefun
+
+@deftypefun void gsl_stats_minmax (double * @var{min}, double * @var{max}, const double @var{data}[], size_t @var{stride}, size_t @var{n})
+This function finds both the minimum and maximum values @var{min},
+@var{max} in @var{data} in a single pass.
+@end deftypefun
+
+@deftypefun size_t gsl_stats_max_index (const double @var{data}[], size_t @var{stride}, size_t @var{n})
+This function returns the index of the maximum value in @var{data}, a
+dataset of length @var{n} with stride @var{stride}.  The maximum value is
+defined as the value of the element @math{x_i} which satisfies 
+@c{$x_i \ge x_j$}
+@math{x_i >= x_j} for all @math{j}.  When there are several equal maximum
+elements then the first one is chosen.
+@end deftypefun
+
+@deftypefun size_t gsl_stats_min_index (const double @var{data}[], size_t @var{stride}, size_t @var{n})
+This function returns the index of the minimum value in @var{data}, a
+dataset of length @var{n} with stride @var{stride}.  The minimum value
+is defined as the value of the element @math{x_i} which satisfies
+@c{$x_i \ge x_j$}
+@math{x_i >= x_j} for all @math{j}.  When there are several equal
+minimum elements then the first one is chosen.
+@end deftypefun
+
+@deftypefun void gsl_stats_minmax_index (size_t * @var{min_index}, size_t * @var{max_index}, const double @var{data}[], size_t @var{stride}, size_t @var{n})
+This function returns the indexes @var{min_index}, @var{max_index} of
+the minimum and maximum values in @var{data} in a single pass.
+@end deftypefun
+
+@node Median and Percentiles
+@section Median and Percentiles
+
+The median and percentile functions described in this section operate on
+sorted data.  For convenience we use @dfn{quantiles}, measured on a scale
+of 0 to 1, instead of percentiles (which use a scale of 0 to 100).
+
+@deftypefun double gsl_stats_median_from_sorted_data (const double @var{sorted_data}[], size_t @var{stride}, size_t @var{n})
+This function returns the median value of @var{sorted_data}, a dataset
+of length @var{n} with stride @var{stride}.  The elements of the array
+must be in ascending numerical order.  There are no checks to see
+whether the data are sorted, so the function @code{gsl_sort} should
+always be used first.
+
+When the dataset has an odd number of elements the median is the value
+of element @math{(n-1)/2}.  When the dataset has an even number of
+elements the median is the mean of the two nearest middle values,
+elements @math{(n-1)/2} and @math{n/2}.  Since the algorithm for
+computing the median involves interpolation this function always returns
+a floating-point number, even for integer data types.
+@end deftypefun
+
+@deftypefun double gsl_stats_quantile_from_sorted_data (const double @var{sorted_data}[], size_t @var{stride}, size_t @var{n}, double @var{f})
+This function returns a quantile value of @var{sorted_data}, a
+double-precision array of length @var{n} with stride @var{stride}.  The
+elements of the array must be in ascending numerical order.  The
+quantile is determined by the @var{f}, a fraction between 0 and 1.  For
+example, to compute the value of the 75th percentile @var{f} should have
+the value 0.75.
+
+There are no checks to see whether the data are sorted, so the function
+@code{gsl_sort} should always be used first.
+
+The quantile is found by interpolation, using the formula
+@tex
+\beforedisplay
+$$
+\hbox{quantile} = (1 - \delta) x_i + \delta x_{i+1}
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+quantile = (1 - \delta) x_i + \delta x_@{i+1@}
+@end example
+
+@end ifinfo
+@noindent
+where @math{i} is @code{floor}(@math{(n - 1)f}) and @math{\delta} is
+@math{(n-1)f - i}.
+
+Thus the minimum value of the array (@code{data[0*stride]}) is given by
+@var{f} equal to zero, the maximum value (@code{data[(n-1)*stride]}) is
+given by @var{f} equal to one and the median value is given by @var{f}
+equal to 0.5.  Since the algorithm for computing quantiles involves
+interpolation this function always returns a floating-point number, even
+for integer data types.
+@end deftypefun
+
+
+@comment @node Statistical tests
+@comment @section Statistical tests
+
+@comment FIXME, do more work on the statistical tests
+
+@comment -@deftypefun double gsl_stats_ttest (const double @var{data1}[], double @var{data2}[], size_t @var{n1}, size_t @var{n2})
+@comment -@deftypefunx Statistics double gsl_stats_int_ttest (const double @var{data1}[], double @var{data2}[], size_t @var{n1}, size_t @var{n2})
+
+@comment The function @code{gsl_stats_ttest} computes the t-test statistic for
+@comment the two arrays @var{data1}[] and @var{data2}[], of lengths @var{n1} and
+@comment -@var{n2} respectively.
+
+@comment The t-test statistic measures the difference between the means of two
+@comment datasets.
+
+@node Example statistical programs
+@section Examples
+Here is a basic example of how to use the statistical functions:
+
+@example
+@verbatiminclude examples/stat.c
+@end example
+
+The program should produce the following output,
+
+@example
+@verbatiminclude examples/stat.out
+@end example
+
+
+Here is an example using sorted data,
+
+@example
+@verbatiminclude examples/statsort.c
+@end example
+
+This program should produce the following output,
+
+@example
+@verbatiminclude examples/statsort.out
+@end example
+
+@node Statistics References and Further Reading
+@section References and Further Reading
+
+The standard reference for almost any topic in statistics is the
+multi-volume @cite{Advanced Theory of Statistics} by Kendall and Stuart.
+
+@itemize @asis
+@item
+Maurice Kendall, Alan Stuart, and J. Keith Ord.
+@cite{The Advanced Theory of Statistics} (multiple volumes)
+reprinted as @cite{Kendall's Advanced Theory of Statistics}.
+Wiley, ISBN 047023380X.
+@end itemize
+
+@noindent
+Many statistical concepts can be more easily understood by a Bayesian
+approach.  The following book by Gelman, Carlin, Stern and Rubin gives a
+comprehensive coverage of the subject.
+
+@itemize @asis
+@item
+Andrew Gelman, John B. Carlin, Hal S. Stern, Donald B. Rubin.
+@cite{Bayesian Data Analysis}.
+Chapman & Hall, ISBN 0412039915.
+@end itemize
+
+@noindent
+For physicists the Particle Data Group provides useful reviews of
+Probability and Statistics in the ``Mathematical Tools'' section of its
+Annual Review of Particle Physics. 
+
+@itemize @asis
+@item
+@cite{Review of Particle Properties}
+R.M. Barnett et al., Physical Review D54, 1 (1996)
+@end itemize
+
+@noindent
+The Review of Particle Physics is available online at
+the website @uref{http://pdg.lbl.gov/}.
+
+
diff --git a/gsl-1.9/doc/sum.texi b/gsl-1.9/doc/sum.texi
new file mode 100644
index 0000000..9b11675
--- /dev/null
+++ b/gsl-1.9/doc/sum.texi
@@ -0,0 +1,186 @@
+@cindex acceleration of series
+@cindex summation, acceleration
+@cindex series, acceleration
+@cindex u-transform for series
+@cindex Levin u-transform
+@cindex convergence, accelerating a series
+
+The functions described in this chapter accelerate the convergence of a
+series using the Levin @math{u}-transform.  This method takes a small number of
+terms from the start of a series and uses a systematic approximation to
+compute an extrapolated value and an estimate of its error.  The
+@math{u}-transform works for both convergent and divergent series, including
+asymptotic series.
+
+These functions are declared in the header file @file{gsl_sum.h}.
+
+@menu
+* Acceleration functions::      
+* Acceleration functions without error estimation::  
+* Example of accelerating a series::  
+* Series Acceleration References::  
+@end menu
+
+@node Acceleration functions
+@section Acceleration functions
+
+The following functions compute the full Levin @math{u}-transform of a series
+with its error estimate.  The error estimate is computed by propagating
+rounding errors from each term through to the final extrapolation. 
+
+These functions are intended for summing analytic series where each term
+is known to high accuracy, and the rounding errors are assumed to
+originate from finite precision. They are taken to be relative errors of
+order @code{GSL_DBL_EPSILON} for each term.
+
+The calculation of the error in the extrapolated value is an
+@math{O(N^2)} process, which is expensive in time and memory.  A faster
+but less reliable method which estimates the error from the convergence
+of the extrapolated value is described in the next section.  For the
+method described here a full table of intermediate values and
+derivatives through to @math{O(N)} must be computed and stored, but this
+does give a reliable error estimate.
+
+@deftypefun {gsl_sum_levin_u_workspace *} gsl_sum_levin_u_alloc (size_t @var{n})
+This function allocates a workspace for a Levin @math{u}-transform of @var{n}
+terms.  The size of the workspace is @math{O(2n^2 + 3n)}.
+@end deftypefun
+
+@deftypefun void gsl_sum_levin_u_free (gsl_sum_levin_u_workspace * @var{w})
+This function frees the memory associated with the workspace @var{w}.
+@end deftypefun
+
+@deftypefun int gsl_sum_levin_u_accel (const double * @var{array}, size_t @var{array_size}, gsl_sum_levin_u_workspace * @var{w}, double * @var{sum_accel}, double * @var{abserr})
+This function takes the terms of a series in @var{array} of size
+@var{array_size} and computes the extrapolated limit of the series using
+a Levin @math{u}-transform.  Additional working space must be provided in
+@var{w}.  The extrapolated sum is stored in @var{sum_accel}, with an
+estimate of the absolute error stored in @var{abserr}.  The actual
+term-by-term sum is returned in @code{w->sum_plain}. The algorithm
+calculates the truncation error (the difference between two successive
+extrapolations) and round-off error (propagated from the individual
+terms) to choose an optimal number of terms for the extrapolation.  
+All the terms of the series passed in through @var{array} should be non-zero.
+@end deftypefun
+
+
+@node Acceleration functions without error estimation
+@section Acceleration functions without error estimation
+
+The functions described in this section compute the Levin @math{u}-transform of
+series and attempt to estimate the error from the ``truncation error'' in
+the extrapolation, the difference between the final two approximations.
+Using this method avoids the need to compute an intermediate table of
+derivatives because the error is estimated from the behavior of the
+extrapolated value itself. Consequently this algorithm is an @math{O(N)}
+process and only requires @math{O(N)} terms of storage.  If the series
+converges sufficiently fast then this procedure can be acceptable.  It
+is appropriate to use this method when there is a need to compute many
+extrapolations of series with similar convergence properties at high-speed.
+For example, when numerically integrating a function defined by a
+parameterized series where the parameter varies only slightly. A
+reliable error estimate should be computed first using the full
+algorithm described above in order to verify the consistency of the
+results.
+
+@deftypefun {gsl_sum_levin_utrunc_workspace *} gsl_sum_levin_utrunc_alloc (size_t @var{n})
+This function allocates a workspace for a Levin @math{u}-transform of @var{n}
+terms, without error estimation.  The size of the workspace is
+@math{O(3n)}.
+@end deftypefun
+
+@deftypefun void gsl_sum_levin_utrunc_free (gsl_sum_levin_utrunc_workspace * @var{w})
+This function frees the memory associated with the workspace @var{w}.
+@end deftypefun
+
+@deftypefun int gsl_sum_levin_utrunc_accel (const double * @var{array}, size_t @var{array_size}, gsl_sum_levin_utrunc_workspace * @var{w}, double * @var{sum_accel}, double * @var{abserr_trunc})
+This function takes the terms of a series in @var{array} of size
+@var{array_size} and computes the extrapolated limit of the series using
+a Levin @math{u}-transform.  Additional working space must be provided in
+@var{w}.  The extrapolated sum is stored in @var{sum_accel}.  The actual
+term-by-term sum is returned in @code{w->sum_plain}. The algorithm
+terminates when the difference between two successive extrapolations
+reaches a minimum or is sufficiently small. The difference between these
+two values is used as estimate of the error and is stored in
+@var{abserr_trunc}.  To improve the reliability of the algorithm the
+extrapolated values are replaced by moving averages when calculating the
+truncation error, smoothing out any fluctuations.
+@end deftypefun
+
+
+@node Example of accelerating a series
+@section Examples
+
+The following code calculates an estimate of @math{\zeta(2) = \pi^2 / 6}
+using the series,
+@tex
+\beforedisplay
+$$
+\zeta(2) = 1 + 1/2^2 + 1/3^2 + 1/4^2 + \dots
+$$
+\afterdisplay
+@end tex
+@ifinfo
+
+@example
+\zeta(2) = 1 + 1/2^2 + 1/3^2 + 1/4^2 + ...
+@end example
+
+@end ifinfo
+@noindent
+After @var{N} terms the error in the sum is @math{O(1/N)}, making direct
+summation of the series converge slowly.
+
+@example
+@verbatiminclude examples/sum.c
+@end example
+
+@noindent
+The output below shows that the Levin @math{u}-transform is able to obtain an 
+estimate of the sum to 1 part in 
+@c{$10^{10}$}
+@math{10^10} using the first eleven terms of the series.  The
+error estimate returned by the function is also accurate, giving
+the correct number of significant digits. 
+
+@example
+$ ./a.out 
+@verbatiminclude examples/sum.out
+@end example
+
+@noindent
+Note that a direct summation of this series would require 
+@c{$10^{10}$}
+@math{10^10} terms to achieve the same precision as the accelerated 
+sum does in 13 terms.
+
+@node Series Acceleration References
+@section References and Further Reading
+
+The algorithms used by these functions are described in the following papers,
+
+@itemize @asis
+@item
+T. Fessler, W.F. Ford, D.A. Smith,
+@sc{hurry}: An acceleration algorithm for scalar sequences and series
+@cite{ACM Transactions on Mathematical Software}, 9(3):346--354, 1983.
+and Algorithm 602 9(3):355--357, 1983.
+@end itemize
+
+@noindent
+The theory of the @math{u}-transform was presented by Levin,
+
+@itemize @asis
+@item
+D. Levin,
+Development of Non-Linear Transformations for Improving Convergence of
+Sequences, @cite{Intern.@: J.@: Computer Math.} B3:371--388, 1973.
+@end itemize
+
+@noindent
+A review paper on the Levin Transform is available online,
+@itemize @asis
+@item
+Herbert H. H. Homeier, Scalar Levin-Type Sequence Transformations,
+@uref{http://arxiv.org/abs/math/0005209}.
+@end itemize
diff --git a/gsl-1.9/doc/texinfo.tex b/gsl-1.9/doc/texinfo.tex
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+% texinfo.tex -- TeX macros to handle Texinfo files.
+%
+% Load plain if necessary, i.e., if running under initex.
+\expandafter\ifx\csname fmtname\endcsname\relax\input plain\fi
+%
+\def\texinfoversion{2004-11-25.16}
+%
+% Copyright (C) 1985, 1986, 1988, 1990, 1991, 1992, 1993, 1994, 1995,
+% 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004 Free Software
+% Foundation, Inc.
+%
+% This texinfo.tex file is free software; you can redistribute it and/or
+% modify it under the terms of the GNU General Public License as
+% published by the Free Software Foundation; either version 2, or (at
+% your option) any later version.
+%
+% This texinfo.tex file is distributed in the hope that it will be
+% useful, but WITHOUT ANY WARRANTY; without even the implied warranty
+% of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
+% General Public License for more details.
+%
+% You should have received a copy of the GNU General Public License
+% along with this texinfo.tex file; see the file COPYING.  If not, write
+% to the Free Software Foundation, Inc., 59 Temple Place - Suite 330,
+% Boston, MA 02111-1307, USA.
+%
+% As a special exception, when this file is read by TeX when processing
+% a Texinfo source document, you may use the result without
+% restriction.  (This has been our intent since Texinfo was invented.)
+%
+% Please try the latest version of texinfo.tex before submitting bug
+% reports; you can get the latest version from:
+%   http://www.gnu.org/software/texinfo/ (the Texinfo home page), or
+%   ftp://tug.org/tex/texinfo.tex
+%     (and all CTAN mirrors, see http://www.ctan.org).
+% The texinfo.tex in any given distribution could well be out
+% of date, so if that's what you're using, please check.
+%
+% Send bug reports to bug-texinfo@gnu.org.  Please include including a
+% complete document in each bug report with which we can reproduce the
+% problem.  Patches are, of course, greatly appreciated.
+%
+% To process a Texinfo manual with TeX, it's most reliable to use the
+% texi2dvi shell script that comes with the distribution.  For a simple
+% manual foo.texi, however, you can get away with this:
+%   tex foo.texi
+%   texindex foo.??
+%   tex foo.texi
+%   tex foo.texi
+%   dvips foo.dvi -o  # or whatever; this makes foo.ps.
+% The extra TeX runs get the cross-reference information correct.
+% Sometimes one run after texindex suffices, and sometimes you need more
+% than two; texi2dvi does it as many times as necessary.
+%
+% It is possible to adapt texinfo.tex for other languages, to some
+% extent.  You can get the existing language-specific files from the
+% full Texinfo distribution.
+%
+% The GNU Texinfo home page is http://www.gnu.org/software/texinfo.
+
+
+\message{Loading texinfo [version \texinfoversion]:}
+
+% If in a .fmt file, print the version number
+% and turn on active characters that we couldn't do earlier because
+% they might have appeared in the input file name.
+\everyjob{\message{[Texinfo version \texinfoversion]}%
+  \catcode`+=\active \catcode`\_=\active}
+
+\message{Basics,}
+\chardef\other=12
+
+% We never want plain's \outer definition of \+ in Texinfo.
+% For @tex, we can use \tabalign.
+\let\+ = \relax
+
+% Save some plain tex macros whose names we will redefine.
+\let\ptexb=\b
+\let\ptexbullet=\bullet
+\let\ptexc=\c
+\let\ptexcomma=\,
+\let\ptexdot=\.
+\let\ptexdots=\dots
+\let\ptexend=\end
+\let\ptexequiv=\equiv
+\let\ptexexclam=\!
+\let\ptexfootnote=\footnote
+\let\ptexgtr=>
+\let\ptexhat=^
+\let\ptexi=\i
+\let\ptexindent=\indent
+\let\ptexinsert=\insert
+\let\ptexlbrace=\{
+\let\ptexless=<
+\let\ptexnewwrite\newwrite
+\let\ptexnoindent=\noindent
+\let\ptexplus=+
+\let\ptexrbrace=\}
+\let\ptexslash=\/
+\let\ptexstar=\*
+\let\ptext=\t
+
+% If this character appears in an error message or help string, it
+% starts a new line in the output.
+\newlinechar = `^^J
+
+% Use TeX 3.0's \inputlineno to get the line number, for better error
+% messages, but if we're using an old version of TeX, don't do anything.
+%
+\ifx\inputlineno\thisisundefined
+  \let\linenumber = \empty % Pre-3.0.
+\else
+  \def\linenumber{l.\the\inputlineno:\space}
+\fi
+
+% Set up fixed words for English if not already set.
+\ifx\putwordAppendix\undefined  \gdef\putwordAppendix{Appendix}\fi
+\ifx\putwordChapter\undefined   \gdef\putwordChapter{Chapter}\fi
+\ifx\putwordfile\undefined      \gdef\putwordfile{file}\fi
+\ifx\putwordin\undefined        \gdef\putwordin{in}\fi
+\ifx\putwordIndexIsEmpty\undefined     \gdef\putwordIndexIsEmpty{(Index is empty)}\fi
+\ifx\putwordIndexNonexistent\undefined \gdef\putwordIndexNonexistent{(Index is nonexistent)}\fi
+\ifx\putwordInfo\undefined      \gdef\putwordInfo{Info}\fi
+\ifx\putwordInstanceVariableof\undefined \gdef\putwordInstanceVariableof{Instance Variable of}\fi
+\ifx\putwordMethodon\undefined  \gdef\putwordMethodon{Method on}\fi
+\ifx\putwordNoTitle\undefined   \gdef\putwordNoTitle{No Title}\fi
+\ifx\putwordof\undefined        \gdef\putwordof{of}\fi
+\ifx\putwordon\undefined        \gdef\putwordon{on}\fi
+\ifx\putwordpage\undefined      \gdef\putwordpage{page}\fi
+\ifx\putwordsection\undefined   \gdef\putwordsection{section}\fi
+\ifx\putwordSection\undefined   \gdef\putwordSection{Section}\fi
+\ifx\putwordsee\undefined       \gdef\putwordsee{see}\fi
+\ifx\putwordSee\undefined       \gdef\putwordSee{See}\fi
+\ifx\putwordShortTOC\undefined  \gdef\putwordShortTOC{Short Contents}\fi
+\ifx\putwordTOC\undefined       \gdef\putwordTOC{Table of Contents}\fi
+%
+\ifx\putwordMJan\undefined \gdef\putwordMJan{January}\fi
+\ifx\putwordMFeb\undefined \gdef\putwordMFeb{February}\fi
+\ifx\putwordMMar\undefined \gdef\putwordMMar{March}\fi
+\ifx\putwordMApr\undefined \gdef\putwordMApr{April}\fi
+\ifx\putwordMMay\undefined \gdef\putwordMMay{May}\fi
+\ifx\putwordMJun\undefined \gdef\putwordMJun{June}\fi
+\ifx\putwordMJul\undefined \gdef\putwordMJul{July}\fi
+\ifx\putwordMAug\undefined \gdef\putwordMAug{August}\fi
+\ifx\putwordMSep\undefined \gdef\putwordMSep{September}\fi
+\ifx\putwordMOct\undefined \gdef\putwordMOct{October}\fi
+\ifx\putwordMNov\undefined \gdef\putwordMNov{November}\fi
+\ifx\putwordMDec\undefined \gdef\putwordMDec{December}\fi
+%
+\ifx\putwordDefmac\undefined    \gdef\putwordDefmac{Macro}\fi
+\ifx\putwordDefspec\undefined   \gdef\putwordDefspec{Special Form}\fi
+\ifx\putwordDefvar\undefined    \gdef\putwordDefvar{Variable}\fi
+\ifx\putwordDefopt\undefined    \gdef\putwordDefopt{User Option}\fi
+\ifx\putwordDeffunc\undefined   \gdef\putwordDeffunc{Function}\fi
+
+% In some macros, we cannot use the `\? notation---the left quote is
+% in some cases the escape char.
+\chardef\colonChar = `\:
+\chardef\commaChar = `\,
+\chardef\dotChar   = `\.
+\chardef\exclamChar= `\!
+\chardef\questChar = `\?
+\chardef\semiChar  = `\;
+\chardef\underChar = `\_
+
+\chardef\spaceChar = `\ %
+\chardef\spacecat = 10
+\def\spaceisspace{\catcode\spaceChar=\spacecat}
+
+% Ignore a token.
+%
+\def\gobble#1{}
+
+% The following is used inside several \edef's.
+\def\makecsname#1{\expandafter\noexpand\csname#1\endcsname}
+
+% Hyphenation fixes.
+\hyphenation{
+  Flor-i-da Ghost-script Ghost-view Mac-OS Post-Script
+  ap-pen-dix bit-map bit-maps
+  data-base data-bases eshell fall-ing half-way long-est man-u-script
+  man-u-scripts mini-buf-fer mini-buf-fers over-view par-a-digm
+  par-a-digms rath-er rec-tan-gu-lar ro-bot-ics se-vere-ly set-up spa-ces
+  spell-ing spell-ings
+  stand-alone strong-est time-stamp time-stamps which-ever white-space
+  wide-spread wrap-around
+}
+
+% Margin to add to right of even pages, to left of odd pages.
+\newdimen\bindingoffset
+\newdimen\normaloffset
+\newdimen\pagewidth \newdimen\pageheight
+
+% For a final copy, take out the rectangles
+% that mark overfull boxes (in case you have decided
+% that the text looks ok even though it passes the margin).
+%
+\def\finalout{\overfullrule=0pt}
+
+% @| inserts a changebar to the left of the current line.  It should
+% surround any changed text.  This approach does *not* work if the
+% change spans more than two lines of output.  To handle that, we would
+% have adopt a much more difficult approach (putting marks into the main
+% vertical list for the beginning and end of each change).
+%
+\def\|{%
+  % \vadjust can only be used in horizontal mode.
+  \leavevmode
+  %
+  % Append this vertical mode material after the current line in the output.
+  \vadjust{%
+    % We want to insert a rule with the height and depth of the current
+    % leading; that is exactly what \strutbox is supposed to record.
+    \vskip-\baselineskip
+    %
+    % \vadjust-items are inserted at the left edge of the type.  So
+    % the \llap here moves out into the left-hand margin.
+    \llap{%
+      %
+      % For a thicker or thinner bar, change the `1pt'.
+      \vrule height\baselineskip width1pt
+      %
+      % This is the space between the bar and the text.
+      \hskip 12pt
+    }%
+  }%
+}
+
+% Sometimes it is convenient to have everything in the transcript file
+% and nothing on the terminal.  We don't just call \tracingall here,
+% since that produces some useless output on the terminal.  We also make
+% some effort to order the tracing commands to reduce output in the log
+% file; cf. trace.sty in LaTeX.
+%
+\def\gloggingall{\begingroup \globaldefs = 1 \loggingall \endgroup}%
+\def\loggingall{%
+  \tracingstats2
+  \tracingpages1
+  \tracinglostchars2  % 2 gives us more in etex
+  \tracingparagraphs1
+  \tracingoutput1
+  \tracingmacros2
+  \tracingrestores1
+  \showboxbreadth\maxdimen \showboxdepth\maxdimen
+  \ifx\eTeXversion\undefined\else % etex gives us more logging
+    \tracingscantokens1
+    \tracingifs1
+    \tracinggroups1
+    \tracingnesting2
+    \tracingassigns1
+  \fi
+  \tracingcommands3  % 3 gives us more in etex
+  \errorcontextlines16
+}%
+
+% add check for \lastpenalty to plain's definitions.  If the last thing
+% we did was a \nobreak, we don't want to insert more space.
+%
+\def\smallbreak{\ifnum\lastpenalty<10000\par\ifdim\lastskip<\smallskipamount
+  \removelastskip\penalty-50\smallskip\fi\fi}
+\def\medbreak{\ifnum\lastpenalty<10000\par\ifdim\lastskip<\medskipamount
+  \removelastskip\penalty-100\medskip\fi\fi}
+\def\bigbreak{\ifnum\lastpenalty<10000\par\ifdim\lastskip<\bigskipamount
+  \removelastskip\penalty-200\bigskip\fi\fi}
+
+% For @cropmarks command.
+% Do @cropmarks to get crop marks.
+%
+\newif\ifcropmarks
+\let\cropmarks = \cropmarkstrue
+%
+% Dimensions to add cropmarks at corners.
+% Added by P. A. MacKay, 12 Nov. 1986
+%
+\newdimen\outerhsize \newdimen\outervsize % set by the paper size routines
+\newdimen\cornerlong  \cornerlong=1pc
+\newdimen\cornerthick \cornerthick=.3pt
+\newdimen\topandbottommargin \topandbottommargin=.75in
+
+% Main output routine.
+\chardef\PAGE = 255
+\output = {\onepageout{\pagecontents\PAGE}}
+
+\newbox\headlinebox
+\newbox\footlinebox
+
+% \onepageout takes a vbox as an argument.  Note that \pagecontents
+% does insertions, but you have to call it yourself.
+\def\onepageout#1{%
+  \ifcropmarks \hoffset=0pt \else \hoffset=\normaloffset \fi
+  %
+  \ifodd\pageno  \advance\hoffset by \bindingoffset
+  \else \advance\hoffset by -\bindingoffset\fi
+  %
+  % Do this outside of the \shipout so @code etc. will be expanded in
+  % the headline as they should be, not taken literally (outputting ''code).
+  \setbox\headlinebox = \vbox{\let\hsize=\pagewidth \makeheadline}%
+  \setbox\footlinebox = \vbox{\let\hsize=\pagewidth \makefootline}%
+  %
+  {%
+    % Have to do this stuff outside the \shipout because we want it to
+    % take effect in \write's, yet the group defined by the \vbox ends
+    % before the \shipout runs.
+    %
+    \escapechar = `\\     % use backslash in output files.
+    \indexdummies         % don't expand commands in the output.
+    \normalturnoffactive  % \ in index entries must not stay \, e.g., if
+                   % the page break happens to be in the middle of an example.
+    \shipout\vbox{%
+      % Do this early so pdf references go to the beginning of the page.
+      \ifpdfmakepagedest \pdfdest name{\the\pageno} xyz\fi
+      %
+      \ifcropmarks \vbox to \outervsize\bgroup
+        \hsize = \outerhsize
+        \vskip-\topandbottommargin
+        \vtop to0pt{%
+          \line{\ewtop\hfil\ewtop}%
+          \nointerlineskip
+          \line{%
+            \vbox{\moveleft\cornerthick\nstop}%
+            \hfill
+            \vbox{\moveright\cornerthick\nstop}%
+          }%
+          \vss}%
+        \vskip\topandbottommargin
+        \line\bgroup
+          \hfil % center the page within the outer (page) hsize.
+          \ifodd\pageno\hskip\bindingoffset\fi
+          \vbox\bgroup
+      \fi
+      %
+      \unvbox\headlinebox
+      \pagebody{#1}%
+      \ifdim\ht\footlinebox > 0pt
+        % Only leave this space if the footline is nonempty.
+        % (We lessened \vsize for it in \oddfootingxxx.)
+        % The \baselineskip=24pt in plain's \makefootline has no effect.
+        \vskip 2\baselineskip
+        \unvbox\footlinebox
+      \fi
+      %
+      \ifcropmarks
+          \egroup % end of \vbox\bgroup
+        \hfil\egroup % end of (centering) \line\bgroup
+        \vskip\topandbottommargin plus1fill minus1fill
+        \boxmaxdepth = \cornerthick
+        \vbox to0pt{\vss
+          \line{%
+            \vbox{\moveleft\cornerthick\nsbot}%
+            \hfill
+            \vbox{\moveright\cornerthick\nsbot}%
+          }%
+          \nointerlineskip
+          \line{\ewbot\hfil\ewbot}%
+        }%
+      \egroup % \vbox from first cropmarks clause
+      \fi
+    }% end of \shipout\vbox
+  }% end of group with \normalturnoffactive
+  \advancepageno
+  \ifnum\outputpenalty>-20000 \else\dosupereject\fi
+}
+
+\newinsert\margin \dimen\margin=\maxdimen
+
+\def\pagebody#1{\vbox to\pageheight{\boxmaxdepth=\maxdepth #1}}
+{\catcode`\@ =11
+\gdef\pagecontents#1{\ifvoid\topins\else\unvbox\topins\fi
+% marginal hacks, juha@viisa.uucp (Juha Takala)
+\ifvoid\margin\else % marginal info is present
+  \rlap{\kern\hsize\vbox to\z@{\kern1pt\box\margin \vss}}\fi
+\dimen@=\dp#1 \unvbox#1
+\ifvoid\footins\else\vskip\skip\footins\footnoterule \unvbox\footins\fi
+\ifr@ggedbottom \kern-\dimen@ \vfil \fi}
+}
+
+% Here are the rules for the cropmarks.  Note that they are
+% offset so that the space between them is truly \outerhsize or \outervsize
+% (P. A. MacKay, 12 November, 1986)
+%
+\def\ewtop{\vrule height\cornerthick depth0pt width\cornerlong}
+\def\nstop{\vbox
+  {\hrule height\cornerthick depth\cornerlong width\cornerthick}}
+\def\ewbot{\vrule height0pt depth\cornerthick width\cornerlong}
+\def\nsbot{\vbox
+  {\hrule height\cornerlong depth\cornerthick width\cornerthick}}
+
+% Parse an argument, then pass it to #1.  The argument is the rest of
+% the input line (except we remove a trailing comment).  #1 should be a
+% macro which expects an ordinary undelimited TeX argument.
+%
+\def\parsearg{\parseargusing{}}
+\def\parseargusing#1#2{%
+  \def\next{#2}%
+  \begingroup
+    \obeylines
+    \spaceisspace
+    #1%
+    \parseargline\empty% Insert the \empty token, see \finishparsearg below.
+}
+
+{\obeylines %
+  \gdef\parseargline#1^^M{%
+    \endgroup % End of the group started in \parsearg.
+    \argremovecomment #1\comment\ArgTerm%
+  }%
+}
+
+% First remove any @comment, then any @c comment.
+\def\argremovecomment#1\comment#2\ArgTerm{\argremovec #1\c\ArgTerm}
+\def\argremovec#1\c#2\ArgTerm{\argcheckspaces#1\^^M\ArgTerm}
+
+% Each occurence of `\^^M' or `<space>\^^M' is replaced by a single space.
+%
+% \argremovec might leave us with trailing space, e.g.,
+%    @end itemize  @c foo
+% This space token undergoes the same procedure and is eventually removed
+% by \finishparsearg.
+%
+\def\argcheckspaces#1\^^M{\argcheckspacesX#1\^^M \^^M}
+\def\argcheckspacesX#1 \^^M{\argcheckspacesY#1\^^M}
+\def\argcheckspacesY#1\^^M#2\^^M#3\ArgTerm{%
+  \def\temp{#3}%
+  \ifx\temp\empty
+    % We cannot use \next here, as it holds the macro to run;
+    % thus we reuse \temp.
+    \let\temp\finishparsearg
+  \else
+    \let\temp\argcheckspaces
+  \fi
+  % Put the space token in:
+  \temp#1 #3\ArgTerm
+}
+
+% If a _delimited_ argument is enclosed in braces, they get stripped; so
+% to get _exactly_ the rest of the line, we had to prevent such situation.
+% We prepended an \empty token at the very beginning and we expand it now,
+% just before passing the control to \next.
+% (Similarily, we have to think about #3 of \argcheckspacesY above: it is
+% either the null string, or it ends with \^^M---thus there is no danger
+% that a pair of braces would be stripped.
+%
+% But first, we have to remove the trailing space token.
+%
+\def\finishparsearg#1 \ArgTerm{\expandafter\next\expandafter{#1}}
+
+% \parseargdef\foo{...}
+%	is roughly equivalent to
+% \def\foo{\parsearg\Xfoo}
+% \def\Xfoo#1{...}
+%
+% Actually, I use \csname\string\foo\endcsname, ie. \\foo, as it is my
+% favourite TeX trick.  --kasal, 16nov03
+
+\def\parseargdef#1{%
+  \expandafter \doparseargdef \csname\string#1\endcsname #1%
+}
+\def\doparseargdef#1#2{%
+  \def#2{\parsearg#1}%
+  \def#1##1%
+}
+
+% Several utility definitions with active space:
+{
+  \obeyspaces
+  \gdef\obeyedspace{ }
+
+  % Make each space character in the input produce a normal interword
+  % space in the output.  Don't allow a line break at this space, as this
+  % is used only in environments like @example, where each line of input
+  % should produce a line of output anyway.
+  %
+  \gdef\sepspaces{\obeyspaces\let =\tie}
+
+  % If an index command is used in an @example environment, any spaces
+  % therein should become regular spaces in the raw index file, not the
+  % expansion of \tie (\leavevmode \penalty \@M \ ).
+  \gdef\unsepspaces{\let =\space}
+}
+
+
+\def\flushcr{\ifx\par\lisppar \def\next##1{}\else \let\next=\relax \fi \next}
+
+% Define the framework for environments in texinfo.tex.  It's used like this:
+%
+%   \envdef\foo{...}
+%   \def\Efoo{...}
+%
+% It's the responsibility of \envdef to insert \begingroup before the
+% actual body; @end closes the group after calling \Efoo.  \envdef also
+% defines \thisenv, so the current environment is known; @end checks
+% whether the environment name matches.  The \checkenv macro can also be
+% used to check whether the current environment is the one expected.
+%
+% Non-false conditionals (@iftex, @ifset) don't fit into this, so they
+% are not treated as enviroments; they don't open a group.  (The
+% implementation of @end takes care not to call \endgroup in this
+% special case.)
+
+
+% At runtime, environments start with this:
+\def\startenvironment#1{\begingroup\def\thisenv{#1}}
+% initialize
+\let\thisenv\empty
+
+% ... but they get defined via ``\envdef\foo{...}'':
+\long\def\envdef#1#2{\def#1{\startenvironment#1#2}}
+\def\envparseargdef#1#2{\parseargdef#1{\startenvironment#1#2}}
+
+% Check whether we're in the right environment:
+\def\checkenv#1{%
+  \def\temp{#1}%
+  \ifx\thisenv\temp
+  \else
+    \badenverr
+  \fi
+}
+
+% Evironment mismatch, #1 expected:
+\def\badenverr{%
+  \errhelp = \EMsimple
+  \errmessage{This command can appear only \inenvironment\temp,
+    not \inenvironment\thisenv}%
+}
+\def\inenvironment#1{%
+  \ifx#1\empty
+    out of any environment%
+  \else
+    in environment \expandafter\string#1%
+  \fi
+}
+
+% @end foo executes the definition of \Efoo.
+% But first, it executes a specialized version of \checkenv
+%
+\parseargdef\end{%
+  \if 1\csname iscond.#1\endcsname
+  \else
+    % The general wording of \badenverr may not be ideal, but... --kasal, 06nov03
+    \expandafter\checkenv\csname#1\endcsname
+    \csname E#1\endcsname
+    \endgroup
+  \fi
+}
+
+\newhelp\EMsimple{Press RETURN to continue.}
+
+
+%% Simple single-character @ commands
+
+% @@ prints an @
+% Kludge this until the fonts are right (grr).
+\def\@{{\tt\char64}}
+
+% This is turned off because it was never documented
+% and you can use @w{...} around a quote to suppress ligatures.
+%% Define @` and @' to be the same as ` and '
+%% but suppressing ligatures.
+%\def\`{{`}}
+%\def\'{{'}}
+
+% Used to generate quoted braces.
+\def\mylbrace {{\tt\char123}}
+\def\myrbrace {{\tt\char125}}
+\let\{=\mylbrace
+\let\}=\myrbrace
+\begingroup
+  % Definitions to produce \{ and \} commands for indices,
+  % and @{ and @} for the aux file.
+  \catcode`\{ = \other \catcode`\} = \other
+  \catcode`\[ = 1 \catcode`\] = 2
+  \catcode`\! = 0 \catcode`\\ = \other
+  !gdef!lbracecmd[\{]%
+  !gdef!rbracecmd[\}]%
+  !gdef!lbraceatcmd[@{]%
+  !gdef!rbraceatcmd[@}]%
+!endgroup
+
+% @comma{} to avoid , parsing problems.
+\let\comma = ,
+
+% Accents: @, @dotaccent @ringaccent @ubaraccent @udotaccent
+% Others are defined by plain TeX: @` @' @" @^ @~ @= @u @v @H.
+\let\, = \c
+\let\dotaccent = \.
+\def\ringaccent#1{{\accent23 #1}}
+\let\tieaccent = \t
+\let\ubaraccent = \b
+\let\udotaccent = \d
+
+% Other special characters: @questiondown @exclamdown @ordf @ordm
+% Plain TeX defines: @AA @AE @O @OE @L (plus lowercase versions) @ss.
+\def\questiondown{?`}
+\def\exclamdown{!`}
+\def\ordf{\leavevmode\raise1ex\hbox{\selectfonts\lllsize \underbar{a}}}
+\def\ordm{\leavevmode\raise1ex\hbox{\selectfonts\lllsize \underbar{o}}}
+
+% Dotless i and dotless j, used for accents.
+\def\imacro{i}
+\def\jmacro{j}
+\def\dotless#1{%
+  \def\temp{#1}%
+  \ifx\temp\imacro \ptexi
+  \else\ifx\temp\jmacro \j
+  \else \errmessage{@dotless can be used only with i or j}%
+  \fi\fi
+}
+
+% The \TeX{} logo, as in plain, but resetting the spacing so that a
+% period following counts as ending a sentence.  (Idea found in latex.)
+%
+\edef\TeX{\TeX \spacefactor=1000 }
+
+% @LaTeX{} logo.  Not quite the same results as the definition in
+% latex.ltx, since we use a different font for the raised A; it's most
+% convenient for us to use an explicitly smaller font, rather than using
+% the \scriptstyle font (since we don't reset \scriptstyle and
+% \scriptscriptstyle).
+%
+\def\LaTeX{%
+  L\kern-.36em
+  {\setbox0=\hbox{T}%
+   \vbox to \ht0{\hbox{\selectfonts\lllsize A}\vss}}%
+  \kern-.15em
+  \TeX
+}
+
+% Be sure we're in horizontal mode when doing a tie, since we make space
+% equivalent to this in @example-like environments. Otherwise, a space
+% at the beginning of a line will start with \penalty -- and
+% since \penalty is valid in vertical mode, we'd end up putting the
+% penalty on the vertical list instead of in the new paragraph.
+{\catcode`@ = 11
+ % Avoid using \@M directly, because that causes trouble
+ % if the definition is written into an index file.
+ \global\let\tiepenalty = \@M
+ \gdef\tie{\leavevmode\penalty\tiepenalty\ }
+}
+
+% @: forces normal size whitespace following.
+\def\:{\spacefactor=1000 }
+
+% @* forces a line break.
+\def\*{\hfil\break\hbox{}\ignorespaces}
+
+% @/ allows a line break.
+\let\/=\allowbreak
+
+% @. is an end-of-sentence period.
+\def\.{.\spacefactor=3000 }
+
+% @! is an end-of-sentence bang.
+\def\!{!\spacefactor=3000 }
+
+% @? is an end-of-sentence query.
+\def\?{?\spacefactor=3000 }
+
+% @w prevents a word break.  Without the \leavevmode, @w at the
+% beginning of a paragraph, when TeX is still in vertical mode, would
+% produce a whole line of output instead of starting the paragraph.
+\def\w#1{\leavevmode\hbox{#1}}
+
+% @group ... @end group forces ... to be all on one page, by enclosing
+% it in a TeX vbox.  We use \vtop instead of \vbox to construct the box
+% to keep its height that of a normal line.  According to the rules for
+% \topskip (p.114 of the TeXbook), the glue inserted is
+% max (\topskip - \ht (first item), 0).  If that height is large,
+% therefore, no glue is inserted, and the space between the headline and
+% the text is small, which looks bad.
+%
+% Another complication is that the group might be very large.  This can
+% cause the glue on the previous page to be unduly stretched, because it
+% does not have much material.  In this case, it's better to add an
+% explicit \vfill so that the extra space is at the bottom.  The
+% threshold for doing this is if the group is more than \vfilllimit
+% percent of a page (\vfilllimit can be changed inside of @tex).
+%
+\newbox\groupbox
+\def\vfilllimit{0.7}
+%
+\envdef\group{%
+  \ifnum\catcode`\^^M=\active \else
+    \errhelp = \groupinvalidhelp
+    \errmessage{@group invalid in context where filling is enabled}%
+  \fi
+  \startsavinginserts
+  %
+  \setbox\groupbox = \vtop\bgroup
+    % Do @comment since we are called inside an environment such as
+    % @example, where each end-of-line in the input causes an
+    % end-of-line in the output.  We don't want the end-of-line after
+    % the `@group' to put extra space in the output.  Since @group
+    % should appear on a line by itself (according to the Texinfo
+    % manual), we don't worry about eating any user text.
+    \comment
+}
+%
+% The \vtop produces a box with normal height and large depth; thus, TeX puts
+% \baselineskip glue before it, and (when the next line of text is done)
+% \lineskip glue after it.  Thus, space below is not quite equal to space
+% above.  But it's pretty close.
+\def\Egroup{%
+    % To get correct interline space between the last line of the group
+    % and the first line afterwards, we have to propagate \prevdepth.
+    \endgraf % Not \par, as it may have been set to \lisppar.
+    \global\dimen1 = \prevdepth
+  \egroup           % End the \vtop.
+  % \dimen0 is the vertical size of the group's box.
+  \dimen0 = \ht\groupbox  \advance\dimen0 by \dp\groupbox
+  % \dimen2 is how much space is left on the page (more or less).
+  \dimen2 = \pageheight   \advance\dimen2 by -\pagetotal
+  % if the group doesn't fit on the current page, and it's a big big
+  % group, force a page break.
+  \ifdim \dimen0 > \dimen2
+    \ifdim \pagetotal < \vfilllimit\pageheight
+      \page
+    \fi
+  \fi
+  \box\groupbox
+  \prevdepth = \dimen1
+  \checkinserts
+}
+%
+% TeX puts in an \escapechar (i.e., `@') at the beginning of the help
+% message, so this ends up printing `@group can only ...'.
+%
+\newhelp\groupinvalidhelp{%
+group can only be used in environments such as @example,^^J%
+where each line of input produces a line of output.}
+
+% @need space-in-mils
+% forces a page break if there is not space-in-mils remaining.
+
+\newdimen\mil  \mil=0.001in
+
+% Old definition--didn't work.
+%\parseargdef\need{\par %
+%% This method tries to make TeX break the page naturally
+%% if the depth of the box does not fit.
+%{\baselineskip=0pt%
+%\vtop to #1\mil{\vfil}\kern -#1\mil\nobreak
+%\prevdepth=-1000pt
+%}}
+
+\parseargdef\need{%
+  % Ensure vertical mode, so we don't make a big box in the middle of a
+  % paragraph.
+  \par
+  %
+  % If the @need value is less than one line space, it's useless.
+  \dimen0 = #1\mil
+  \dimen2 = \ht\strutbox
+  \advance\dimen2 by \dp\strutbox
+  \ifdim\dimen0 > \dimen2
+    %
+    % Do a \strut just to make the height of this box be normal, so the
+    % normal leading is inserted relative to the preceding line.
+    % And a page break here is fine.
+    \vtop to #1\mil{\strut\vfil}%
+    %
+    % TeX does not even consider page breaks if a penalty added to the
+    % main vertical list is 10000 or more.  But in order to see if the
+    % empty box we just added fits on the page, we must make it consider
+    % page breaks.  On the other hand, we don't want to actually break the
+    % page after the empty box.  So we use a penalty of 9999.
+    %
+    % There is an extremely small chance that TeX will actually break the
+    % page at this \penalty, if there are no other feasible breakpoints in
+    % sight.  (If the user is using lots of big @group commands, which
+    % almost-but-not-quite fill up a page, TeX will have a hard time doing
+    % good page breaking, for example.)  However, I could not construct an
+    % example where a page broke at this \penalty; if it happens in a real
+    % document, then we can reconsider our strategy.
+    \penalty9999
+    %
+    % Back up by the size of the box, whether we did a page break or not.
+    \kern -#1\mil
+    %
+    % Do not allow a page break right after this kern.
+    \nobreak
+  \fi
+}
+
+% @br   forces paragraph break (and is undocumented).
+
+\let\br = \par
+
+% @page forces the start of a new page.
+%
+\def\page{\par\vfill\supereject}
+
+% @exdent text....
+% outputs text on separate line in roman font, starting at standard page margin
+
+% This records the amount of indent in the innermost environment.
+% That's how much \exdent should take out.
+\newskip\exdentamount
+
+% This defn is used inside fill environments such as @defun.
+\parseargdef\exdent{\hfil\break\hbox{\kern -\exdentamount{\rm#1}}\hfil\break}
+
+% This defn is used inside nofill environments such as @example.
+\parseargdef\nofillexdent{{\advance \leftskip by -\exdentamount
+  \leftline{\hskip\leftskip{\rm#1}}}}
+
+% @inmargin{WHICH}{TEXT} puts TEXT in the WHICH margin next to the current
+% paragraph.  For more general purposes, use the \margin insertion
+% class.  WHICH is `l' or `r'.
+%
+\newskip\inmarginspacing \inmarginspacing=1cm
+\def\strutdepth{\dp\strutbox}
+%
+\def\doinmargin#1#2{\strut\vadjust{%
+  \nobreak
+  \kern-\strutdepth
+  \vtop to \strutdepth{%
+    \baselineskip=\strutdepth
+    \vss
+    % if you have multiple lines of stuff to put here, you'll need to
+    % make the vbox yourself of the appropriate size.
+    \ifx#1l%
+      \llap{\ignorespaces #2\hskip\inmarginspacing}%
+    \else
+      \rlap{\hskip\hsize \hskip\inmarginspacing \ignorespaces #2}%
+    \fi
+    \null
+  }%
+}}
+\def\inleftmargin{\doinmargin l}
+\def\inrightmargin{\doinmargin r}
+%
+% @inmargin{TEXT [, RIGHT-TEXT]}
+% (if RIGHT-TEXT is given, use TEXT for left page, RIGHT-TEXT for right;
+% else use TEXT for both).
+%
+\def\inmargin#1{\parseinmargin #1,,\finish}
+\def\parseinmargin#1,#2,#3\finish{% not perfect, but better than nothing.
+  \setbox0 = \hbox{\ignorespaces #2}%
+  \ifdim\wd0 > 0pt
+    \def\lefttext{#1}%  have both texts
+    \def\righttext{#2}%
+  \else
+    \def\lefttext{#1}%  have only one text
+    \def\righttext{#1}%
+  \fi
+  %
+  \ifodd\pageno
+    \def\temp{\inrightmargin\righttext}% odd page -> outside is right margin
+  \else
+    \def\temp{\inleftmargin\lefttext}%
+  \fi
+  \temp
+}
+
+% @include file    insert text of that file as input.
+%
+\def\include{\parseargusing\filenamecatcodes\includezzz}
+\def\includezzz#1{%
+  \pushthisfilestack
+  \def\thisfile{#1}%
+  {%
+    \makevalueexpandable
+    \def\temp{\input #1 }%
+    \expandafter
+  }\temp
+  \popthisfilestack
+}
+\def\filenamecatcodes{%
+  \catcode`\\=\other
+  \catcode`~=\other
+  \catcode`^=\other
+  \catcode`_=\other
+  \catcode`|=\other
+  \catcode`<=\other
+  \catcode`>=\other
+  \catcode`+=\other
+  \catcode`-=\other
+}
+
+\def\pushthisfilestack{%
+  \expandafter\pushthisfilestackX\popthisfilestack\StackTerm
+}
+\def\pushthisfilestackX{%
+  \expandafter\pushthisfilestackY\thisfile\StackTerm
+}
+\def\pushthisfilestackY #1\StackTerm #2\StackTerm {%
+  \gdef\popthisfilestack{\gdef\thisfile{#1}\gdef\popthisfilestack{#2}}%
+}
+
+\def\popthisfilestack{\errthisfilestackempty}
+\def\errthisfilestackempty{\errmessage{Internal error:
+  the stack of filenames is empty.}}
+
+\def\thisfile{}
+
+% @center line
+% outputs that line, centered.
+%
+\parseargdef\center{%
+  \ifhmode
+    \let\next\centerH
+  \else
+    \let\next\centerV
+  \fi
+  \next{\hfil \ignorespaces#1\unskip \hfil}%
+}
+\def\centerH#1{%
+  {%
+    \hfil\break
+    \advance\hsize by -\leftskip
+    \advance\hsize by -\rightskip
+    \line{#1}%
+    \break
+  }%
+}
+\def\centerV#1{\line{\kern\leftskip #1\kern\rightskip}}
+
+% @sp n   outputs n lines of vertical space
+
+\parseargdef\sp{\vskip #1\baselineskip}
+
+% @comment ...line which is ignored...
+% @c is the same as @comment
+% @ignore ... @end ignore  is another way to write a comment
+
+\def\comment{\begingroup \catcode`\^^M=\other%
+\catcode`\@=\other \catcode`\{=\other \catcode`\}=\other%
+\commentxxx}
+{\catcode`\^^M=\other \gdef\commentxxx#1^^M{\endgroup}}
+
+\let\c=\comment
+
+% @paragraphindent NCHARS
+% We'll use ems for NCHARS, close enough.
+% NCHARS can also be the word `asis' or `none'.
+% We cannot feasibly implement @paragraphindent asis, though.
+%
+\def\asisword{asis} % no translation, these are keywords
+\def\noneword{none}
+%
+\parseargdef\paragraphindent{%
+  \def\temp{#1}%
+  \ifx\temp\asisword
+  \else
+    \ifx\temp\noneword
+      \defaultparindent = 0pt
+    \else
+      \defaultparindent = #1em
+    \fi
+  \fi
+  \parindent = \defaultparindent
+}
+
+% @exampleindent NCHARS
+% We'll use ems for NCHARS like @paragraphindent.
+% It seems @exampleindent asis isn't necessary, but
+% I preserve it to make it similar to @paragraphindent.
+\parseargdef\exampleindent{%
+  \def\temp{#1}%
+  \ifx\temp\asisword
+  \else
+    \ifx\temp\noneword
+      \lispnarrowing = 0pt
+    \else
+      \lispnarrowing = #1em
+    \fi
+  \fi
+}
+
+% @firstparagraphindent WORD
+% If WORD is `none', then suppress indentation of the first paragraph
+% after a section heading.  If WORD is `insert', then do indent at such
+% paragraphs.
+%
+% The paragraph indentation is suppressed or not by calling
+% \suppressfirstparagraphindent, which the sectioning commands do.
+% We switch the definition of this back and forth according to WORD.
+% By default, we suppress indentation.
+%
+\def\suppressfirstparagraphindent{\dosuppressfirstparagraphindent}
+\def\insertword{insert}
+%
+\parseargdef\firstparagraphindent{%
+  \def\temp{#1}%
+  \ifx\temp\noneword
+    \let\suppressfirstparagraphindent = \dosuppressfirstparagraphindent
+  \else\ifx\temp\insertword
+    \let\suppressfirstparagraphindent = \relax
+  \else
+    \errhelp = \EMsimple
+    \errmessage{Unknown @firstparagraphindent option `\temp'}%
+  \fi\fi
+}
+
+% Here is how we actually suppress indentation.  Redefine \everypar to
+% \kern backwards by \parindent, and then reset itself to empty.
+%
+% We also make \indent itself not actually do anything until the next
+% paragraph.
+%
+\gdef\dosuppressfirstparagraphindent{%
+  \gdef\indent{%
+    \restorefirstparagraphindent
+    \indent
+  }%
+  \gdef\noindent{%
+    \restorefirstparagraphindent
+    \noindent
+  }%
+  \global\everypar = {%
+    \kern -\parindent
+    \restorefirstparagraphindent
+  }%
+}
+
+\gdef\restorefirstparagraphindent{%
+  \global \let \indent = \ptexindent
+  \global \let \noindent = \ptexnoindent
+  \global \everypar = {}%
+}
+
+
+% @asis just yields its argument.  Used with @table, for example.
+%
+\def\asis#1{#1}
+
+% @math outputs its argument in math mode.
+%
+% One complication: _ usually means subscripts, but it could also mean
+% an actual _ character, as in @math{@var{some_variable} + 1}.  So make
+% _ active, and distinguish by seeing if the current family is \slfam,
+% which is what @var uses.
+{
+  \catcode\underChar = \active
+  \gdef\mathunderscore{%
+    \catcode\underChar=\active
+    \def_{\ifnum\fam=\slfam \_\else\sb\fi}%
+  }
+}
+% Another complication: we want \\ (and @\) to output a \ character.
+% FYI, plain.tex uses \\ as a temporary control sequence (why?), but
+% this is not advertised and we don't care.  Texinfo does not
+% otherwise define @\.
+%
+% The \mathchar is class=0=ordinary, family=7=ttfam, position=5C=\.
+\def\mathbackslash{\ifnum\fam=\ttfam \mathchar"075C \else\backslash \fi}
+%
+\def\math{%
+  \tex
+  \mathunderscore
+  \let\\ = \mathbackslash
+  \mathactive
+  $\finishmath
+}
+\def\finishmath#1{#1$\endgroup}  % Close the group opened by \tex.
+
+% Some active characters (such as <) are spaced differently in math.
+% We have to reset their definitions in case the @math was an argument
+% to a command which sets the catcodes (such as @item or @section).
+%
+{
+  \catcode`^ = \active
+  \catcode`< = \active
+  \catcode`> = \active
+  \catcode`+ = \active
+  \gdef\mathactive{%
+    \let^ = \ptexhat
+    \let< = \ptexless
+    \let> = \ptexgtr
+    \let+ = \ptexplus
+  }
+}
+
+% @bullet and @minus need the same treatment as @math, just above.
+\def\bullet{$\ptexbullet$}
+\def\minus{$-$}
+
+% @dots{} outputs an ellipsis using the current font.
+% We do .5em per period so that it has the same spacing in a typewriter
+% font as three actual period characters.
+%
+\def\dots{%
+  \leavevmode
+  \hbox to 1.5em{%
+    \hskip 0pt plus 0.25fil
+    .\hfil.\hfil.%
+    \hskip 0pt plus 0.5fil
+  }%
+}
+
+% @enddots{} is an end-of-sentence ellipsis.
+%
+\def\enddots{%
+  \dots
+  \spacefactor=3000
+}
+
+% @comma{} is so commas can be inserted into text without messing up
+% Texinfo's parsing.
+%
+\let\comma = ,
+
+% @refill is a no-op.
+\let\refill=\relax
+
+% If working on a large document in chapters, it is convenient to
+% be able to disable indexing, cross-referencing, and contents, for test runs.
+% This is done with @novalidate (before @setfilename).
+%
+\newif\iflinks \linkstrue % by default we want the aux files.
+\let\novalidate = \linksfalse
+
+% @setfilename is done at the beginning of every texinfo file.
+% So open here the files we need to have open while reading the input.
+% This makes it possible to make a .fmt file for texinfo.
+\def\setfilename{%
+   \fixbackslash  % Turn off hack to swallow `\input texinfo'.
+   \iflinks
+     \tryauxfile
+     % Open the new aux file.  TeX will close it automatically at exit.
+     \immediate\openout\auxfile=\jobname.aux
+   \fi % \openindices needs to do some work in any case.
+   \openindices
+   \let\setfilename=\comment % Ignore extra @setfilename cmds.
+   %
+   % If texinfo.cnf is present on the system, read it.
+   % Useful for site-wide @afourpaper, etc.
+   \openin 1 texinfo.cnf
+   \ifeof 1 \else \input texinfo.cnf \fi
+   \closein 1
+   %
+   \comment % Ignore the actual filename.
+}
+
+% Called from \setfilename.
+%
+\def\openindices{%
+  \newindex{cp}%
+  \newcodeindex{fn}%
+  \newcodeindex{vr}%
+  \newcodeindex{tp}%
+  \newcodeindex{ky}%
+  \newcodeindex{pg}%
+}
+
+% @bye.
+\outer\def\bye{\pagealignmacro\tracingstats=1\ptexend}
+
+
+\message{pdf,}
+% adobe `portable' document format
+\newcount\tempnum
+\newcount\lnkcount
+\newtoks\filename
+\newcount\filenamelength
+\newcount\pgn
+\newtoks\toksA
+\newtoks\toksB
+\newtoks\toksC
+\newtoks\toksD
+\newbox\boxA
+\newcount\countA
+\newif\ifpdf
+\newif\ifpdfmakepagedest
+
+% when pdftex is run in dvi mode, \pdfoutput is defined (so \pdfoutput=1
+% can be set).  So we test for \relax and 0 as well as \undefined,
+% borrowed from ifpdf.sty.
+\ifx\pdfoutput\undefined
+\else
+  \ifx\pdfoutput\relax
+  \else
+    \ifcase\pdfoutput
+    \else
+      \pdftrue
+    \fi
+  \fi
+\fi
+%
+\ifpdf
+  \input pdfcolor
+  \pdfcatalog{/PageMode /UseOutlines}%
+  \def\dopdfimage#1#2#3{%
+    \def\imagewidth{#2}%
+    \def\imageheight{#3}%
+    % without \immediate, pdftex seg faults when the same image is
+    % included twice.  (Version 3.14159-pre-1.0-unofficial-20010704.)
+    \ifnum\pdftexversion < 14
+      \immediate\pdfimage
+    \else
+      \immediate\pdfximage
+    \fi
+      \ifx\empty\imagewidth\else width \imagewidth \fi
+      \ifx\empty\imageheight\else height \imageheight \fi
+      \ifnum\pdftexversion<13
+         #1.pdf%
+       \else
+         {#1.pdf}%
+       \fi
+    \ifnum\pdftexversion < 14 \else
+      \pdfrefximage \pdflastximage
+    \fi}
+  \def\pdfmkdest#1{{%
+    % We have to set dummies so commands such as @code in a section title
+    % aren't expanded.
+    \atdummies
+    \normalturnoffactive
+    \pdfdest name{#1} xyz%
+  }}
+  \def\pdfmkpgn#1{#1}
+  \let\linkcolor = \Blue  % was Cyan, but that seems light?
+  \def\endlink{\Black\pdfendlink}
+  % Adding outlines to PDF; macros for calculating structure of outlines
+  % come from Petr Olsak
+  \def\expnumber#1{\expandafter\ifx\csname#1\endcsname\relax 0%
+    \else \csname#1\endcsname \fi}
+  \def\advancenumber#1{\tempnum=\expnumber{#1}\relax
+    \advance\tempnum by 1
+    \expandafter\xdef\csname#1\endcsname{\the\tempnum}}
+  %
+  % #1 is the section text.  #2 is the pdf expression for the number
+  % of subentries (or empty, for subsubsections).  #3 is the node
+  % text, which might be empty if this toc entry had no
+  % corresponding node.  #4 is the page number.
+  %
+  \def\dopdfoutline#1#2#3#4{%
+    % Generate a link to the node text if that exists; else, use the
+    % page number.  We could generate a destination for the section
+    % text in the case where a section has no node, but it doesn't
+    % seem worthwhile, since most documents are normally structured.
+    \def\pdfoutlinedest{#3}%
+    \ifx\pdfoutlinedest\empty \def\pdfoutlinedest{#4}\fi
+    %
+    \pdfoutline goto name{\pdfmkpgn{\pdfoutlinedest}}#2{#1}%
+  }
+  %
+  \def\pdfmakeoutlines{%
+    \begingroup
+      % Thanh's hack / proper braces in bookmarks
+      \edef\mylbrace{\iftrue \string{\else}\fi}\let\{=\mylbrace
+      \edef\myrbrace{\iffalse{\else\string}\fi}\let\}=\myrbrace
+      %
+      % Read toc silently, to get counts of subentries for \pdfoutline.
+      \def\numchapentry##1##2##3##4{%
+	\def\thischapnum{##2}%
+	\def\thissecnum{0}%
+	\def\thissubsecnum{0}%
+      }%
+      \def\numsecentry##1##2##3##4{%
+	\advancenumber{chap\thischapnum}%
+	\def\thissecnum{##2}%
+	\def\thissubsecnum{0}%
+      }%
+      \def\numsubsecentry##1##2##3##4{%
+	\advancenumber{sec\thissecnum}%
+	\def\thissubsecnum{##2}%
+      }%
+      \def\numsubsubsecentry##1##2##3##4{%
+	\advancenumber{subsec\thissubsecnum}%
+      }%
+      \def\thischapnum{0}%
+      \def\thissecnum{0}%
+      \def\thissubsecnum{0}%
+      %
+      % use \def rather than \let here because we redefine \chapentry et
+      % al. a second time, below.
+      \def\appentry{\numchapentry}%
+      \def\appsecentry{\numsecentry}%
+      \def\appsubsecentry{\numsubsecentry}%
+      \def\appsubsubsecentry{\numsubsubsecentry}%
+      \def\unnchapentry{\numchapentry}%
+      \def\unnsecentry{\numsecentry}%
+      \def\unnsubsecentry{\numsubsecentry}%
+      \def\unnsubsubsecentry{\numsubsubsecentry}%
+      \input \jobname.toc
+      %
+      % Read toc second time, this time actually producing the outlines.
+      % The `-' means take the \expnumber as the absolute number of
+      % subentries, which we calculated on our first read of the .toc above.
+      %
+      % We use the node names as the destinations.
+      \def\numchapentry##1##2##3##4{%
+        \dopdfoutline{##1}{count-\expnumber{chap##2}}{##3}{##4}}%
+      \def\numsecentry##1##2##3##4{%
+        \dopdfoutline{##1}{count-\expnumber{sec##2}}{##3}{##4}}%
+      \def\numsubsecentry##1##2##3##4{%
+        \dopdfoutline{##1}{count-\expnumber{subsec##2}}{##3}{##4}}%
+      \def\numsubsubsecentry##1##2##3##4{% count is always zero
+        \dopdfoutline{##1}{}{##3}{##4}}%
+      %
+      % PDF outlines are displayed using system fonts, instead of
+      % document fonts.  Therefore we cannot use special characters,
+      % since the encoding is unknown.  For example, the eogonek from
+      % Latin 2 (0xea) gets translated to a | character.  Info from
+      % Staszek Wawrykiewicz, 19 Jan 2004 04:09:24 +0100.
+      %
+      % xx to do this right, we have to translate 8-bit characters to
+      % their "best" equivalent, based on the @documentencoding.  Right
+      % now, I guess we'll just let the pdf reader have its way.
+      \indexnofonts
+      \turnoffactive
+      \input \jobname.toc
+    \endgroup
+  }
+  %
+  \def\makelinks #1,{%
+    \def\params{#1}\def\E{END}%
+    \ifx\params\E
+      \let\nextmakelinks=\relax
+    \else
+      \let\nextmakelinks=\makelinks
+      \ifnum\lnkcount>0,\fi
+      \picknum{#1}%
+      \startlink attr{/Border [0 0 0]}
+        goto name{\pdfmkpgn{\the\pgn}}%
+      \linkcolor #1%
+      \advance\lnkcount by 1%
+      \endlink
+    \fi
+    \nextmakelinks
+  }
+  \def\picknum#1{\expandafter\pn#1}
+  \def\pn#1{%
+    \def\p{#1}%
+    \ifx\p\lbrace
+      \let\nextpn=\ppn
+    \else
+      \let\nextpn=\ppnn
+      \def\first{#1}
+    \fi
+    \nextpn
+  }
+  \def\ppn#1{\pgn=#1\gobble}
+  \def\ppnn{\pgn=\first}
+  \def\pdfmklnk#1{\lnkcount=0\makelinks #1,END,}
+  \def\skipspaces#1{\def\PP{#1}\def\D{|}%
+    \ifx\PP\D\let\nextsp\relax
+    \else\let\nextsp\skipspaces
+      \ifx\p\space\else\addtokens{\filename}{\PP}%
+        \advance\filenamelength by 1
+      \fi
+    \fi
+    \nextsp}
+  \def\getfilename#1{\filenamelength=0\expandafter\skipspaces#1|\relax}
+  \ifnum\pdftexversion < 14
+    \let \startlink \pdfannotlink
+  \else
+    \let \startlink \pdfstartlink
+  \fi
+  \def\pdfurl#1{%
+    \begingroup
+      \normalturnoffactive\def\@{@}%
+      \makevalueexpandable
+      \leavevmode\Red
+      \startlink attr{/Border [0 0 0]}%
+        user{/Subtype /Link /A << /S /URI /URI (#1) >>}%
+    \endgroup}
+  \def\pdfgettoks#1.{\setbox\boxA=\hbox{\toksA={#1.}\toksB={}\maketoks}}
+  \def\addtokens#1#2{\edef\addtoks{\noexpand#1={\the#1#2}}\addtoks}
+  \def\adn#1{\addtokens{\toksC}{#1}\global\countA=1\let\next=\maketoks}
+  \def\poptoks#1#2|ENDTOKS|{\let\first=#1\toksD={#1}\toksA={#2}}
+  \def\maketoks{%
+    \expandafter\poptoks\the\toksA|ENDTOKS|\relax
+    \ifx\first0\adn0
+    \else\ifx\first1\adn1 \else\ifx\first2\adn2 \else\ifx\first3\adn3
+    \else\ifx\first4\adn4 \else\ifx\first5\adn5 \else\ifx\first6\adn6
+    \else\ifx\first7\adn7 \else\ifx\first8\adn8 \else\ifx\first9\adn9
+    \else
+      \ifnum0=\countA\else\makelink\fi
+      \ifx\first.\let\next=\done\else
+        \let\next=\maketoks
+        \addtokens{\toksB}{\the\toksD}
+        \ifx\first,\addtokens{\toksB}{\space}\fi
+      \fi
+    \fi\fi\fi\fi\fi\fi\fi\fi\fi\fi
+    \next}
+  \def\makelink{\addtokens{\toksB}%
+    {\noexpand\pdflink{\the\toksC}}\toksC={}\global\countA=0}
+  \def\pdflink#1{%
+    \startlink attr{/Border [0 0 0]} goto name{\pdfmkpgn{#1}}
+    \linkcolor #1\endlink}
+  \def\done{\edef\st{\global\noexpand\toksA={\the\toksB}}\st}
+\else
+  \let\pdfmkdest = \gobble
+  \let\pdfurl = \gobble
+  \let\endlink = \relax
+  \let\linkcolor = \relax
+  \let\pdfmakeoutlines = \relax
+\fi  % \ifx\pdfoutput
+
+
+\message{fonts,}
+
+% Change the current font style to #1, remembering it in \curfontstyle.
+% For now, we do not accumulate font styles: @b{@i{foo}} prints foo in
+% italics, not bold italics.
+%
+\def\setfontstyle#1{%
+  \def\curfontstyle{#1}% not as a control sequence, because we are \edef'd.
+  \csname ten#1\endcsname  % change the current font
+}
+
+% Select #1 fonts with the current style.
+%
+\def\selectfonts#1{\csname #1fonts\endcsname \csname\curfontstyle\endcsname}
+
+\def\rm{\fam=0 \setfontstyle{rm}}
+\def\it{\fam=\itfam \setfontstyle{it}}
+\def\sl{\fam=\slfam \setfontstyle{sl}}
+\def\bf{\fam=\bffam \setfontstyle{bf}}\def\bfstylename{bf}
+\def\tt{\fam=\ttfam \setfontstyle{tt}}
+
+% Texinfo sort of supports the sans serif font style, which plain TeX does not.
+% So we set up a \sf.
+\newfam\sffam
+\def\sf{\fam=\sffam \setfontstyle{sf}}
+\let\li = \sf % Sometimes we call it \li, not \sf.
+
+% We don't need math for this font style.
+\def\ttsl{\setfontstyle{ttsl}}
+
+% Default leading.
+\newdimen\textleading  \textleading = 13.2pt
+
+% Set the baselineskip to #1, and the lineskip and strut size
+% correspondingly.  There is no deep meaning behind these magic numbers
+% used as factors; they just match (closely enough) what Knuth defined.
+%
+\def\lineskipfactor{.08333}
+\def\strutheightpercent{.70833}
+\def\strutdepthpercent {.29167}
+%
+\def\setleading#1{%
+  \normalbaselineskip = #1\relax
+  \normallineskip = \lineskipfactor\normalbaselineskip
+  \normalbaselines
+  \setbox\strutbox =\hbox{%
+    \vrule width0pt height\strutheightpercent\baselineskip
+                    depth \strutdepthpercent \baselineskip
+  }%
+}
+
+% Set the font macro #1 to the font named #2, adding on the
+% specified font prefix (normally `cm').
+% #3 is the font's design size, #4 is a scale factor
+\def\setfont#1#2#3#4{\font#1=\fontprefix#2#3 scaled #4}
+
+% Use cm as the default font prefix.
+% To specify the font prefix, you must define \fontprefix
+% before you read in texinfo.tex.
+\ifx\fontprefix\undefined
+\def\fontprefix{cm}
+\fi
+% Support font families that don't use the same naming scheme as CM.
+\def\rmshape{r}
+\def\rmbshape{bx}               %where the normal face is bold
+\def\bfshape{b}
+\def\bxshape{bx}
+\def\ttshape{tt}
+\def\ttbshape{tt}
+\def\ttslshape{sltt}
+\def\itshape{ti}
+\def\itbshape{bxti}
+\def\slshape{sl}
+\def\slbshape{bxsl}
+\def\sfshape{ss}
+\def\sfbshape{ss}
+\def\scshape{csc}
+\def\scbshape{csc}
+
+% Text fonts (11.2pt, magstep1).
+\def\textnominalsize{11pt}
+\edef\mainmagstep{\magstephalf}
+\setfont\textrm\rmshape{10}{\mainmagstep}
+\setfont\texttt\ttshape{10}{\mainmagstep}
+\setfont\textbf\bfshape{10}{\mainmagstep}
+\setfont\textit\itshape{10}{\mainmagstep}
+\setfont\textsl\slshape{10}{\mainmagstep}
+\setfont\textsf\sfshape{10}{\mainmagstep}
+\setfont\textsc\scshape{10}{\mainmagstep}
+\setfont\textttsl\ttslshape{10}{\mainmagstep}
+\font\texti=cmmi10 scaled \mainmagstep
+\font\textsy=cmsy10 scaled \mainmagstep
+
+% A few fonts for @defun names and args.
+\setfont\defbf\bfshape{10}{\magstep1}
+\setfont\deftt\ttshape{10}{\magstep1}
+\setfont\defttsl\ttslshape{10}{\magstep1}
+\def\df{\let\tentt=\deftt \let\tenbf = \defbf \let\tenttsl=\defttsl \bf}
+
+% Fonts for indices, footnotes, small examples (9pt).
+\def\smallnominalsize{9pt}
+\setfont\smallrm\rmshape{9}{1000}
+\setfont\smalltt\ttshape{9}{1000}
+\setfont\smallbf\bfshape{10}{900}
+\setfont\smallit\itshape{9}{1000}
+\setfont\smallsl\slshape{9}{1000}
+\setfont\smallsf\sfshape{9}{1000}
+\setfont\smallsc\scshape{10}{900}
+\setfont\smallttsl\ttslshape{10}{900}
+\font\smalli=cmmi9
+\font\smallsy=cmsy9
+
+% Fonts for small examples (8pt).
+\def\smallernominalsize{8pt}
+\setfont\smallerrm\rmshape{8}{1000}
+\setfont\smallertt\ttshape{8}{1000}
+\setfont\smallerbf\bfshape{10}{800}
+\setfont\smallerit\itshape{8}{1000}
+\setfont\smallersl\slshape{8}{1000}
+\setfont\smallersf\sfshape{8}{1000}
+\setfont\smallersc\scshape{10}{800}
+\setfont\smallerttsl\ttslshape{10}{800}
+\font\smalleri=cmmi8
+\font\smallersy=cmsy8
+
+% Fonts for title page (20.4pt):
+\def\titlenominalsize{20pt}
+\setfont\titlerm\rmbshape{12}{\magstep3}
+\setfont\titleit\itbshape{10}{\magstep4}
+\setfont\titlesl\slbshape{10}{\magstep4}
+\setfont\titlett\ttbshape{12}{\magstep3}
+\setfont\titlettsl\ttslshape{10}{\magstep4}
+\setfont\titlesf\sfbshape{17}{\magstep1}
+\let\titlebf=\titlerm
+\setfont\titlesc\scbshape{10}{\magstep4}
+\font\titlei=cmmi12 scaled \magstep3
+\font\titlesy=cmsy10 scaled \magstep4
+\def\authorrm{\secrm}
+\def\authortt{\sectt}
+
+% Chapter (and unnumbered) fonts (17.28pt).
+\def\chapnominalsize{17pt}
+\setfont\chaprm\rmbshape{12}{\magstep2}
+\setfont\chapit\itbshape{10}{\magstep3}
+\setfont\chapsl\slbshape{10}{\magstep3}
+\setfont\chaptt\ttbshape{12}{\magstep2}
+\setfont\chapttsl\ttslshape{10}{\magstep3}
+\setfont\chapsf\sfbshape{17}{1000}
+\let\chapbf=\chaprm
+\setfont\chapsc\scbshape{10}{\magstep3}
+\font\chapi=cmmi12 scaled \magstep2
+\font\chapsy=cmsy10 scaled \magstep3
+
+% Section fonts (14.4pt).
+\def\secnominalsize{14pt}
+\setfont\secrm\rmbshape{12}{\magstep1}
+\setfont\secit\itbshape{10}{\magstep2}
+\setfont\secsl\slbshape{10}{\magstep2}
+\setfont\sectt\ttbshape{12}{\magstep1}
+\setfont\secttsl\ttslshape{10}{\magstep2}
+\setfont\secsf\sfbshape{12}{\magstep1}
+\let\secbf\secrm
+\setfont\secsc\scbshape{10}{\magstep2}
+\font\seci=cmmi12 scaled \magstep1
+\font\secsy=cmsy10 scaled \magstep2
+
+% Subsection fonts (13.15pt).
+\def\ssecnominalsize{13pt}
+\setfont\ssecrm\rmbshape{12}{\magstephalf}
+\setfont\ssecit\itbshape{10}{1315}
+\setfont\ssecsl\slbshape{10}{1315}
+\setfont\ssectt\ttbshape{12}{\magstephalf}
+\setfont\ssecttsl\ttslshape{10}{1315}
+\setfont\ssecsf\sfbshape{12}{\magstephalf}
+\let\ssecbf\ssecrm
+\setfont\ssecsc\scbshape{10}{1315}
+\font\sseci=cmmi12 scaled \magstephalf
+\font\ssecsy=cmsy10 scaled 1315
+
+% Reduced fonts for @acro in text (10pt).
+\def\reducednominalsize{10pt}
+\setfont\reducedrm\rmshape{10}{1000}
+\setfont\reducedtt\ttshape{10}{1000}
+\setfont\reducedbf\bfshape{10}{1000}
+\setfont\reducedit\itshape{10}{1000}
+\setfont\reducedsl\slshape{10}{1000}
+\setfont\reducedsf\sfshape{10}{1000}
+\setfont\reducedsc\scshape{10}{1000}
+\setfont\reducedttsl\ttslshape{10}{1000}
+\font\reducedi=cmmi10
+\font\reducedsy=cmsy10
+
+% In order for the font changes to affect most math symbols and letters,
+% we have to define the \textfont of the standard families.  Since
+% texinfo doesn't allow for producing subscripts and superscripts except
+% in the main text, we don't bother to reset \scriptfont and
+% \scriptscriptfont (which would also require loading a lot more fonts).
+%
+\def\resetmathfonts{%
+  \textfont0=\tenrm \textfont1=\teni \textfont2=\tensy
+  \textfont\itfam=\tenit \textfont\slfam=\tensl \textfont\bffam=\tenbf
+  \textfont\ttfam=\tentt \textfont\sffam=\tensf
+}
+
+% The font-changing commands redefine the meanings of \tenSTYLE, instead
+% of just \STYLE.  We do this because \STYLE needs to also set the
+% current \fam for math mode.  Our \STYLE (e.g., \rm) commands hardwire
+% \tenSTYLE to set the current font.
+%
+% Each font-changing command also sets the names \lsize (one size lower)
+% and \lllsize (three sizes lower).  These relative commands are used in
+% the LaTeX logo and acronyms.
+%
+% This all needs generalizing, badly.
+%
+\def\textfonts{%
+  \let\tenrm=\textrm \let\tenit=\textit \let\tensl=\textsl
+  \let\tenbf=\textbf \let\tentt=\texttt \let\smallcaps=\textsc
+  \let\tensf=\textsf \let\teni=\texti \let\tensy=\textsy
+  \let\tenttsl=\textttsl
+  \def\curfontsize{text}%
+  \def\lsize{reduced}\def\lllsize{smaller}%
+  \resetmathfonts \setleading{\textleading}}
+\def\titlefonts{%
+  \let\tenrm=\titlerm \let\tenit=\titleit \let\tensl=\titlesl
+  \let\tenbf=\titlebf \let\tentt=\titlett \let\smallcaps=\titlesc
+  \let\tensf=\titlesf \let\teni=\titlei \let\tensy=\titlesy
+  \let\tenttsl=\titlettsl
+  \def\curfontsize{title}%
+  \def\lsize{chap}\def\lllsize{subsec}%
+  \resetmathfonts \setleading{25pt}}
+\def\titlefont#1{{\titlefonts\rm #1}}
+\def\chapfonts{%
+  \let\tenrm=\chaprm \let\tenit=\chapit \let\tensl=\chapsl
+  \let\tenbf=\chapbf \let\tentt=\chaptt \let\smallcaps=\chapsc
+  \let\tensf=\chapsf \let\teni=\chapi \let\tensy=\chapsy
+  \let\tenttsl=\chapttsl
+  \def\curfontsize{chap}%
+  \def\lsize{sec}\def\lllsize{text}%
+  \resetmathfonts \setleading{19pt}}
+\def\secfonts{%
+  \let\tenrm=\secrm \let\tenit=\secit \let\tensl=\secsl
+  \let\tenbf=\secbf \let\tentt=\sectt \let\smallcaps=\secsc
+  \let\tensf=\secsf \let\teni=\seci \let\tensy=\secsy
+  \let\tenttsl=\secttsl
+  \def\curfontsize{sec}%
+  \def\lsize{subsec}\def\lllsize{reduced}%
+  \resetmathfonts \setleading{16pt}}
+\def\subsecfonts{%
+  \let\tenrm=\ssecrm \let\tenit=\ssecit \let\tensl=\ssecsl
+  \let\tenbf=\ssecbf \let\tentt=\ssectt \let\smallcaps=\ssecsc
+  \let\tensf=\ssecsf \let\teni=\sseci \let\tensy=\ssecsy
+  \let\tenttsl=\ssecttsl
+  \def\curfontsize{ssec}%
+  \def\lsize{text}\def\lllsize{small}%
+  \resetmathfonts \setleading{15pt}}
+\let\subsubsecfonts = \subsecfonts
+\def\reducedfonts{%
+  \let\tenrm=\reducedrm \let\tenit=\reducedit \let\tensl=\reducedsl
+  \let\tenbf=\reducedbf \let\tentt=\reducedtt \let\reducedcaps=\reducedsc
+  \let\tensf=\reducedsf \let\teni=\reducedi \let\tensy=\reducedsy
+  \let\tenttsl=\reducedttsl
+  \def\curfontsize{reduced}%
+  \def\lsize{small}\def\lllsize{smaller}%
+  \resetmathfonts \setleading{10.5pt}}
+\def\smallfonts{%
+  \let\tenrm=\smallrm \let\tenit=\smallit \let\tensl=\smallsl
+  \let\tenbf=\smallbf \let\tentt=\smalltt \let\smallcaps=\smallsc
+  \let\tensf=\smallsf \let\teni=\smalli \let\tensy=\smallsy
+  \let\tenttsl=\smallttsl
+  \def\curfontsize{small}%
+  \def\lsize{smaller}\def\lllsize{smaller}%
+  \resetmathfonts \setleading{10.5pt}}
+\def\smallerfonts{%
+  \let\tenrm=\smallerrm \let\tenit=\smallerit \let\tensl=\smallersl
+  \let\tenbf=\smallerbf \let\tentt=\smallertt \let\smallcaps=\smallersc
+  \let\tensf=\smallersf \let\teni=\smalleri \let\tensy=\smallersy
+  \let\tenttsl=\smallerttsl
+  \def\curfontsize{smaller}%
+  \def\lsize{smaller}\def\lllsize{smaller}%
+  \resetmathfonts \setleading{9.5pt}}
+
+% Set the fonts to use with the @small... environments.
+\let\smallexamplefonts = \smallfonts
+
+% About \smallexamplefonts.  If we use \smallfonts (9pt), @smallexample
+% can fit this many characters:
+%   8.5x11=86   smallbook=72  a4=90  a5=69
+% If we use \scriptfonts (8pt), then we can fit this many characters:
+%   8.5x11=90+  smallbook=80  a4=90+  a5=77
+% For me, subjectively, the few extra characters that fit aren't worth
+% the additional smallness of 8pt.  So I'm making the default 9pt.
+%
+% By the way, for comparison, here's what fits with @example (10pt):
+%   8.5x11=71  smallbook=60  a4=75  a5=58
+%
+% I wish the USA used A4 paper.
+% --karl, 24jan03.
+
+
+% Set up the default fonts, so we can use them for creating boxes.
+%
+\textfonts \rm
+
+% Define these so they can be easily changed for other fonts.
+\def\angleleft{$\langle$}
+\def\angleright{$\rangle$}
+
+% Count depth in font-changes, for error checks
+\newcount\fontdepth \fontdepth=0
+
+% Fonts for short table of contents.
+\setfont\shortcontrm\rmshape{12}{1000}
+\setfont\shortcontbf\bfshape{10}{\magstep1}  % no cmb12
+\setfont\shortcontsl\slshape{12}{1000}
+\setfont\shortconttt\ttshape{12}{1000}
+
+%% Add scribe-like font environments, plus @l for inline lisp (usually sans
+%% serif) and @ii for TeX italic
+
+% \smartitalic{ARG} outputs arg in italics, followed by an italic correction
+% unless the following character is such as not to need one.
+\def\smartitalicx{\ifx\next,\else\ifx\next-\else\ifx\next.\else
+                    \ptexslash\fi\fi\fi}
+\def\smartslanted#1{{\ifusingtt\ttsl\sl #1}\futurelet\next\smartitalicx}
+\def\smartitalic#1{{\ifusingtt\ttsl\it #1}\futurelet\next\smartitalicx}
+
+% like \smartslanted except unconditionally uses \ttsl.
+% @var is set to this for defun arguments.
+\def\ttslanted#1{{\ttsl #1}\futurelet\next\smartitalicx}
+
+% like \smartslanted except unconditionally use \sl.  We never want
+% ttsl for book titles, do we?
+\def\cite#1{{\sl #1}\futurelet\next\smartitalicx}
+
+\let\i=\smartitalic
+\let\slanted=\smartslanted
+\let\var=\smartslanted
+\let\dfn=\smartslanted
+\let\emph=\smartitalic
+
+% @b, explicit bold.
+\def\b#1{{\bf #1}}
+\let\strong=\b
+
+% @sansserif, explicit sans.
+\def\sansserif#1{{\sf #1}}
+
+% We can't just use \exhyphenpenalty, because that only has effect at
+% the end of a paragraph.  Restore normal hyphenation at the end of the
+% group within which \nohyphenation is presumably called.
+%
+\def\nohyphenation{\hyphenchar\font = -1  \aftergroup\restorehyphenation}
+\def\restorehyphenation{\hyphenchar\font = `- }
+
+% Set sfcode to normal for the chars that usually have another value.
+% Can't use plain's \frenchspacing because it uses the `\x notation, and
+% sometimes \x has an active definition that messes things up.
+%
+\catcode`@=11
+  \def\frenchspacing{%
+    \sfcode\dotChar  =\@m \sfcode\questChar=\@m \sfcode\exclamChar=\@m
+    \sfcode\colonChar=\@m \sfcode\semiChar =\@m \sfcode\commaChar =\@m
+  }
+\catcode`@=\other
+
+\def\t#1{%
+  {\tt \rawbackslash \frenchspacing #1}%
+  \null
+}
+\def\samp#1{`\tclose{#1}'\null}
+\setfont\keyrm\rmshape{8}{1000}
+\font\keysy=cmsy9
+\def\key#1{{\keyrm\textfont2=\keysy \leavevmode\hbox{%
+  \raise0.4pt\hbox{\angleleft}\kern-.08em\vtop{%
+    \vbox{\hrule\kern-0.4pt
+     \hbox{\raise0.4pt\hbox{\vphantom{\angleleft}}#1}}%
+    \kern-0.4pt\hrule}%
+  \kern-.06em\raise0.4pt\hbox{\angleright}}}}
+% The old definition, with no lozenge:
+%\def\key #1{{\ttsl \nohyphenation \uppercase{#1}}\null}
+\def\ctrl #1{{\tt \rawbackslash \hat}#1}
+
+% @file, @option are the same as @samp.
+\let\file=\samp
+\let\option=\samp
+
+% @code is a modification of @t,
+% which makes spaces the same size as normal in the surrounding text.
+\def\tclose#1{%
+  {%
+    % Change normal interword space to be same as for the current font.
+    \spaceskip = \fontdimen2\font
+    %
+    % Switch to typewriter.
+    \tt
+    %
+    % But `\ ' produces the large typewriter interword space.
+    \def\ {{\spaceskip = 0pt{} }}%
+    %
+    % Turn off hyphenation.
+    \nohyphenation
+    %
+    \rawbackslash
+    \frenchspacing
+    #1%
+  }%
+  \null
+}
+
+% We *must* turn on hyphenation at `-' and `_' in @code.
+% Otherwise, it is too hard to avoid overfull hboxes
+% in the Emacs manual, the Library manual, etc.
+
+% Unfortunately, TeX uses one parameter (\hyphenchar) to control
+% both hyphenation at - and hyphenation within words.
+% We must therefore turn them both off (\tclose does that)
+% and arrange explicitly to hyphenate at a dash.
+%  -- rms.
+{
+  \catcode`\-=\active
+  \catcode`\_=\active
+  %
+  \global\def\code{\begingroup
+    \catcode`\-=\active \let-\codedash
+    \catcode`\_=\active \let_\codeunder
+    \codex
+  }
+}
+
+\def\realdash{-}
+\def\codedash{-\discretionary{}{}{}}
+\def\codeunder{%
+  % this is all so @math{@code{var_name}+1} can work.  In math mode, _
+  % is "active" (mathcode"8000) and \normalunderscore (or \char95, etc.)
+  % will therefore expand the active definition of _, which is us
+  % (inside @code that is), therefore an endless loop.
+  \ifusingtt{\ifmmode
+               \mathchar"075F % class 0=ordinary, family 7=ttfam, pos 0x5F=_.
+             \else\normalunderscore \fi
+             \discretionary{}{}{}}%
+            {\_}%
+}
+\def\codex #1{\tclose{#1}\endgroup}
+
+% @kbd is like @code, except that if the argument is just one @key command,
+% then @kbd has no effect.
+
+% @kbdinputstyle -- arg is `distinct' (@kbd uses slanted tty font always),
+%   `example' (@kbd uses ttsl only inside of @example and friends),
+%   or `code' (@kbd uses normal tty font always).
+\parseargdef\kbdinputstyle{%
+  \def\arg{#1}%
+  \ifx\arg\worddistinct
+    \gdef\kbdexamplefont{\ttsl}\gdef\kbdfont{\ttsl}%
+  \else\ifx\arg\wordexample
+    \gdef\kbdexamplefont{\ttsl}\gdef\kbdfont{\tt}%
+  \else\ifx\arg\wordcode
+    \gdef\kbdexamplefont{\tt}\gdef\kbdfont{\tt}%
+  \else
+    \errhelp = \EMsimple
+    \errmessage{Unknown @kbdinputstyle option `\arg'}%
+  \fi\fi\fi
+}
+\def\worddistinct{distinct}
+\def\wordexample{example}
+\def\wordcode{code}
+
+% Default is `distinct.'
+\kbdinputstyle distinct
+
+\def\xkey{\key}
+\def\kbdfoo#1#2#3\par{\def\one{#1}\def\three{#3}\def\threex{??}%
+\ifx\one\xkey\ifx\threex\three \key{#2}%
+\else{\tclose{\kbdfont\look}}\fi
+\else{\tclose{\kbdfont\look}}\fi}
+
+% For @indicateurl, @env, @command quotes seem unnecessary, so use \code.
+\let\indicateurl=\code
+\let\env=\code
+\let\command=\code
+
+% @uref (abbreviation for `urlref') takes an optional (comma-separated)
+% second argument specifying the text to display and an optional third
+% arg as text to display instead of (rather than in addition to) the url
+% itself.  First (mandatory) arg is the url.  Perhaps eventually put in
+% a hypertex \special here.
+%
+\def\uref#1{\douref #1,,,\finish}
+\def\douref#1,#2,#3,#4\finish{\begingroup
+  \unsepspaces
+  \pdfurl{#1}%
+  \setbox0 = \hbox{\ignorespaces #3}%
+  \ifdim\wd0 > 0pt
+    \unhbox0 % third arg given, show only that
+  \else
+    \setbox0 = \hbox{\ignorespaces #2}%
+    \ifdim\wd0 > 0pt
+      \ifpdf
+        \unhbox0             % PDF: 2nd arg given, show only it
+      \else
+        \unhbox0\ (\code{#1})% DVI: 2nd arg given, show both it and url
+      \fi
+    \else
+      \code{#1}% only url given, so show it
+    \fi
+  \fi
+  \endlink
+\endgroup}
+
+% @url synonym for @uref, since that's how everyone uses it.
+%
+\let\url=\uref
+
+% rms does not like angle brackets --karl, 17may97.
+% So now @email is just like @uref, unless we are pdf.
+%
+%\def\email#1{\angleleft{\tt #1}\angleright}
+\ifpdf
+  \def\email#1{\doemail#1,,\finish}
+  \def\doemail#1,#2,#3\finish{\begingroup
+    \unsepspaces
+    \pdfurl{mailto:#1}%
+    \setbox0 = \hbox{\ignorespaces #2}%
+    \ifdim\wd0>0pt\unhbox0\else\code{#1}\fi
+    \endlink
+  \endgroup}
+\else
+  \let\email=\uref
+\fi
+
+% Check if we are currently using a typewriter font.  Since all the
+% Computer Modern typewriter fonts have zero interword stretch (and
+% shrink), and it is reasonable to expect all typewriter fonts to have
+% this property, we can check that font parameter.
+%
+\def\ifmonospace{\ifdim\fontdimen3\font=0pt }
+
+% Typeset a dimension, e.g., `in' or `pt'.  The only reason for the
+% argument is to make the input look right: @dmn{pt} instead of @dmn{}pt.
+%
+\def\dmn#1{\thinspace #1}
+
+\def\kbd#1{\def\look{#1}\expandafter\kbdfoo\look??\par}
+
+% @l was never documented to mean ``switch to the Lisp font'',
+% and it is not used as such in any manual I can find.  We need it for
+% Polish suppressed-l.  --karl, 22sep96.
+%\def\l#1{{\li #1}\null}
+
+% Explicit font changes: @r, @sc, undocumented @ii.
+\def\r#1{{\rm #1}}              % roman font
+\def\sc#1{{\smallcaps#1}}       % smallcaps font
+\def\ii#1{{\it #1}}             % italic font
+
+% @acronym for "FBI", "NATO", and the like.
+% We print this one point size smaller, since it's intended for
+% all-uppercase.
+% 
+\def\acronym#1{\doacronym #1,,\finish}
+\def\doacronym#1,#2,#3\finish{%
+  {\selectfonts\lsize #1}%
+  \def\temp{#2}%
+  \ifx\temp\empty \else
+    \space ({\unsepspaces \ignorespaces \temp \unskip})%
+  \fi
+}
+
+% @abbr for "Comput. J." and the like.
+% No font change, but don't do end-of-sentence spacing.
+% 
+\def\abbr#1{\doabbr #1,,\finish}
+\def\doabbr#1,#2,#3\finish{%
+  {\frenchspacing #1}%
+  \def\temp{#2}%
+  \ifx\temp\empty \else
+    \space ({\unsepspaces \ignorespaces \temp \unskip})%
+  \fi
+}
+
+% @pounds{} is a sterling sign, which Knuth put in the CM italic font.
+%
+\def\pounds{{\it\$}}
+
+% @euro{} comes from a separate font, depending on the current style.
+% We use the free feym* fonts from the eurosym package by Henrik
+% Theiling, which support regular, slanted, bold and bold slanted (and
+% "outlined" (blackboard board, sort of) versions, which we don't need).
+% It is available from http://www.ctan.org/tex-archive/fonts/eurosym.
+% 
+% Although only regular is the truly official Euro symbol, we ignore
+% that.  The Euro is designed to be slightly taller than the regular
+% font height.
+% 
+% feymr - regular
+% feymo - slanted
+% feybr - bold
+% feybo - bold slanted
+% 
+% There is no good (free) typewriter version, to my knowledge.
+% A feymr10 euro is ~7.3pt wide, while a normal cmtt10 char is ~5.25pt wide.
+% Hmm.
+% 
+% Also doesn't work in math.  Do we need to do math with euro symbols?
+% Hope not.
+% 
+% 
+\def\euro{{\eurofont e}}
+\def\eurofont{%
+  % We set the font at each command, rather than predefining it in
+  % \textfonts and the other font-switching commands, so that
+  % installations which never need the symbold don't have to have the
+  % font installed.
+  % 
+  % There is only one designed size (nominal 10pt), so we always scale
+  % that to the current nominal size.
+  % 
+  % By the way, simply using "at 1em" works for cmr10 and the like, but
+  % does not work for cmbx10 and other extended/shrunken fonts.
+  % 
+  \def\eurosize{\csname\curfontsize nominalsize\endcsname}%
+  %
+  \ifx\curfontstyle\bfstylename 
+    % bold:
+    \font\thiseurofont = \ifusingit{feybo10}{feybr10} at \eurosize
+  \else 
+    % regular:
+    \font\thiseurofont = \ifusingit{feymo10}{feymr10} at \eurosize
+  \fi
+  \thiseurofont
+}
+
+% @registeredsymbol - R in a circle.  The font for the R should really
+% be smaller yet, but lllsize is the best we can do for now.
+% Adapted from the plain.tex definition of \copyright.
+%
+\def\registeredsymbol{%
+  $^{{\ooalign{\hfil\raise.07ex\hbox{\selectfonts\lllsize R}%
+               \hfil\crcr\Orb}}%
+    }$%
+}
+
+% Laurent Siebenmann reports \Orb undefined with:
+%  Textures 1.7.7 (preloaded format=plain 93.10.14)  (68K)  16 APR 2004 02:38
+% so we'll define it if necessary.
+% 
+\ifx\Orb\undefined
+\def\Orb{\mathhexbox20D}
+\fi
+
+
+\message{page headings,}
+
+\newskip\titlepagetopglue \titlepagetopglue = 1.5in
+\newskip\titlepagebottomglue \titlepagebottomglue = 2pc
+
+% First the title page.  Must do @settitle before @titlepage.
+\newif\ifseenauthor
+\newif\iffinishedtitlepage
+
+% Do an implicit @contents or @shortcontents after @end titlepage if the
+% user says @setcontentsaftertitlepage or @setshortcontentsaftertitlepage.
+%
+\newif\ifsetcontentsaftertitlepage
+ \let\setcontentsaftertitlepage = \setcontentsaftertitlepagetrue
+\newif\ifsetshortcontentsaftertitlepage
+ \let\setshortcontentsaftertitlepage = \setshortcontentsaftertitlepagetrue
+
+\parseargdef\shorttitlepage{\begingroup\hbox{}\vskip 1.5in \chaprm \centerline{#1}%
+        \endgroup\page\hbox{}\page}
+
+\envdef\titlepage{%
+  % Open one extra group, as we want to close it in the middle of \Etitlepage.
+  \begingroup
+    \parindent=0pt \textfonts
+    % Leave some space at the very top of the page.
+    \vglue\titlepagetopglue
+    % No rule at page bottom unless we print one at the top with @title.
+    \finishedtitlepagetrue
+    %
+    % Most title ``pages'' are actually two pages long, with space
+    % at the top of the second.  We don't want the ragged left on the second.
+    \let\oldpage = \page
+    \def\page{%
+      \iffinishedtitlepage\else
+	 \finishtitlepage
+      \fi
+      \let\page = \oldpage
+      \page
+      \null
+    }%
+}
+
+\def\Etitlepage{%
+    \iffinishedtitlepage\else
+	\finishtitlepage
+    \fi
+    % It is important to do the page break before ending the group,
+    % because the headline and footline are only empty inside the group.
+    % If we use the new definition of \page, we always get a blank page
+    % after the title page, which we certainly don't want.
+    \oldpage
+  \endgroup
+  %
+  % Need this before the \...aftertitlepage checks so that if they are
+  % in effect the toc pages will come out with page numbers.
+  \HEADINGSon
+  %
+  % If they want short, they certainly want long too.
+  \ifsetshortcontentsaftertitlepage
+    \shortcontents
+    \contents
+    \global\let\shortcontents = \relax
+    \global\let\contents = \relax
+  \fi
+  %
+  \ifsetcontentsaftertitlepage
+    \contents
+    \global\let\contents = \relax
+    \global\let\shortcontents = \relax
+  \fi
+}
+
+\def\finishtitlepage{%
+  \vskip4pt \hrule height 2pt width \hsize
+  \vskip\titlepagebottomglue
+  \finishedtitlepagetrue
+}
+
+%%% Macros to be used within @titlepage:
+
+\let\subtitlerm=\tenrm
+\def\subtitlefont{\subtitlerm \normalbaselineskip = 13pt \normalbaselines}
+
+\def\authorfont{\authorrm \normalbaselineskip = 16pt \normalbaselines
+		\let\tt=\authortt}
+
+\parseargdef\title{%
+  \checkenv\titlepage
+  \leftline{\titlefonts\rm #1}
+  % print a rule at the page bottom also.
+  \finishedtitlepagefalse
+  \vskip4pt \hrule height 4pt width \hsize \vskip4pt
+}
+
+\parseargdef\subtitle{%
+  \checkenv\titlepage
+  {\subtitlefont \rightline{#1}}%
+}
+
+% @author should come last, but may come many times.
+% It can also be used inside @quotation.
+%
+\parseargdef\author{%
+  \def\temp{\quotation}%
+  \ifx\thisenv\temp
+    \def\quotationauthor{#1}% printed in \Equotation.
+  \else
+    \checkenv\titlepage
+    \ifseenauthor\else \vskip 0pt plus 1filll \seenauthortrue \fi
+    {\authorfont \leftline{#1}}%
+  \fi
+}
+
+
+%%% Set up page headings and footings.
+
+\let\thispage=\folio
+
+\newtoks\evenheadline    % headline on even pages
+\newtoks\oddheadline     % headline on odd pages
+\newtoks\evenfootline    % footline on even pages
+\newtoks\oddfootline     % footline on odd pages
+
+% Now make TeX use those variables
+\headline={{\textfonts\rm \ifodd\pageno \the\oddheadline
+                            \else \the\evenheadline \fi}}
+\footline={{\textfonts\rm \ifodd\pageno \the\oddfootline
+                            \else \the\evenfootline \fi}\HEADINGShook}
+\let\HEADINGShook=\relax
+
+% Commands to set those variables.
+% For example, this is what  @headings on  does
+% @evenheading @thistitle|@thispage|@thischapter
+% @oddheading @thischapter|@thispage|@thistitle
+% @evenfooting @thisfile||
+% @oddfooting ||@thisfile
+
+
+\def\evenheading{\parsearg\evenheadingxxx}
+\def\evenheadingxxx #1{\evenheadingyyy #1\|\|\|\|\finish}
+\def\evenheadingyyy #1\|#2\|#3\|#4\finish{%
+\global\evenheadline={\rlap{\centerline{#2}}\line{#1\hfil#3}}}
+
+\def\oddheading{\parsearg\oddheadingxxx}
+\def\oddheadingxxx #1{\oddheadingyyy #1\|\|\|\|\finish}
+\def\oddheadingyyy #1\|#2\|#3\|#4\finish{%
+\global\oddheadline={\rlap{\centerline{#2}}\line{#1\hfil#3}}}
+
+\parseargdef\everyheading{\oddheadingxxx{#1}\evenheadingxxx{#1}}%
+
+\def\evenfooting{\parsearg\evenfootingxxx}
+\def\evenfootingxxx #1{\evenfootingyyy #1\|\|\|\|\finish}
+\def\evenfootingyyy #1\|#2\|#3\|#4\finish{%
+\global\evenfootline={\rlap{\centerline{#2}}\line{#1\hfil#3}}}
+
+\def\oddfooting{\parsearg\oddfootingxxx}
+\def\oddfootingxxx #1{\oddfootingyyy #1\|\|\|\|\finish}
+\def\oddfootingyyy #1\|#2\|#3\|#4\finish{%
+  \global\oddfootline = {\rlap{\centerline{#2}}\line{#1\hfil#3}}%
+  %
+  % Leave some space for the footline.  Hopefully ok to assume
+  % @evenfooting will not be used by itself.
+  \global\advance\pageheight by -\baselineskip
+  \global\advance\vsize by -\baselineskip
+}
+
+\parseargdef\everyfooting{\oddfootingxxx{#1}\evenfootingxxx{#1}}
+
+
+% @headings double      turns headings on for double-sided printing.
+% @headings single      turns headings on for single-sided printing.
+% @headings off         turns them off.
+% @headings on          same as @headings double, retained for compatibility.
+% @headings after       turns on double-sided headings after this page.
+% @headings doubleafter turns on double-sided headings after this page.
+% @headings singleafter turns on single-sided headings after this page.
+% By default, they are off at the start of a document,
+% and turned `on' after @end titlepage.
+
+\def\headings #1 {\csname HEADINGS#1\endcsname}
+
+\def\HEADINGSoff{%
+\global\evenheadline={\hfil} \global\evenfootline={\hfil}
+\global\oddheadline={\hfil} \global\oddfootline={\hfil}}
+\HEADINGSoff
+% When we turn headings on, set the page number to 1.
+% For double-sided printing, put current file name in lower left corner,
+% chapter name on inside top of right hand pages, document
+% title on inside top of left hand pages, and page numbers on outside top
+% edge of all pages.
+\def\HEADINGSdouble{%
+\global\pageno=1
+\global\evenfootline={\hfil}
+\global\oddfootline={\hfil}
+\global\evenheadline={\line{\folio\hfil\thistitle}}
+\global\oddheadline={\line{\thischapter\hfil\folio}}
+\global\let\contentsalignmacro = \chapoddpage
+}
+\let\contentsalignmacro = \chappager
+
+% For single-sided printing, chapter title goes across top left of page,
+% page number on top right.
+\def\HEADINGSsingle{%
+\global\pageno=1
+\global\evenfootline={\hfil}
+\global\oddfootline={\hfil}
+\global\evenheadline={\line{\thischapter\hfil\folio}}
+\global\oddheadline={\line{\thischapter\hfil\folio}}
+\global\let\contentsalignmacro = \chappager
+}
+\def\HEADINGSon{\HEADINGSdouble}
+
+\def\HEADINGSafter{\let\HEADINGShook=\HEADINGSdoublex}
+\let\HEADINGSdoubleafter=\HEADINGSafter
+\def\HEADINGSdoublex{%
+\global\evenfootline={\hfil}
+\global\oddfootline={\hfil}
+\global\evenheadline={\line{\folio\hfil\thistitle}}
+\global\oddheadline={\line{\thischapter\hfil\folio}}
+\global\let\contentsalignmacro = \chapoddpage
+}
+
+\def\HEADINGSsingleafter{\let\HEADINGShook=\HEADINGSsinglex}
+\def\HEADINGSsinglex{%
+\global\evenfootline={\hfil}
+\global\oddfootline={\hfil}
+\global\evenheadline={\line{\thischapter\hfil\folio}}
+\global\oddheadline={\line{\thischapter\hfil\folio}}
+\global\let\contentsalignmacro = \chappager
+}
+
+% Subroutines used in generating headings
+% This produces Day Month Year style of output.
+% Only define if not already defined, in case a txi-??.tex file has set
+% up a different format (e.g., txi-cs.tex does this).
+\ifx\today\undefined
+\def\today{%
+  \number\day\space
+  \ifcase\month
+  \or\putwordMJan\or\putwordMFeb\or\putwordMMar\or\putwordMApr
+  \or\putwordMMay\or\putwordMJun\or\putwordMJul\or\putwordMAug
+  \or\putwordMSep\or\putwordMOct\or\putwordMNov\or\putwordMDec
+  \fi
+  \space\number\year}
+\fi
+
+% @settitle line...  specifies the title of the document, for headings.
+% It generates no output of its own.
+\def\thistitle{\putwordNoTitle}
+\def\settitle{\parsearg{\gdef\thistitle}}
+
+
+\message{tables,}
+% Tables -- @table, @ftable, @vtable, @item(x).
+
+% default indentation of table text
+\newdimen\tableindent \tableindent=.8in
+% default indentation of @itemize and @enumerate text
+\newdimen\itemindent  \itemindent=.3in
+% margin between end of table item and start of table text.
+\newdimen\itemmargin  \itemmargin=.1in
+
+% used internally for \itemindent minus \itemmargin
+\newdimen\itemmax
+
+% Note @table, @ftable, and @vtable define @item, @itemx, etc., with
+% these defs.
+% They also define \itemindex
+% to index the item name in whatever manner is desired (perhaps none).
+
+\newif\ifitemxneedsnegativevskip
+
+\def\itemxpar{\par\ifitemxneedsnegativevskip\nobreak\vskip-\parskip\nobreak\fi}
+
+\def\internalBitem{\smallbreak \parsearg\itemzzz}
+\def\internalBitemx{\itemxpar \parsearg\itemzzz}
+
+\def\itemzzz #1{\begingroup %
+  \advance\hsize by -\rightskip
+  \advance\hsize by -\tableindent
+  \setbox0=\hbox{\itemindicate{#1}}%
+  \itemindex{#1}%
+  \nobreak % This prevents a break before @itemx.
+  %
+  % If the item text does not fit in the space we have, put it on a line
+  % by itself, and do not allow a page break either before or after that
+  % line.  We do not start a paragraph here because then if the next
+  % command is, e.g., @kindex, the whatsit would get put into the
+  % horizontal list on a line by itself, resulting in extra blank space.
+  \ifdim \wd0>\itemmax
+    %
+    % Make this a paragraph so we get the \parskip glue and wrapping,
+    % but leave it ragged-right.
+    \begingroup
+      \advance\leftskip by-\tableindent
+      \advance\hsize by\tableindent
+      \advance\rightskip by0pt plus1fil
+      \leavevmode\unhbox0\par
+    \endgroup
+    %
+    % We're going to be starting a paragraph, but we don't want the
+    % \parskip glue -- logically it's part of the @item we just started.
+    \nobreak \vskip-\parskip
+    %
+    % Stop a page break at the \parskip glue coming up.  However, if
+    % what follows is an environment such as @example, there will be no
+    % \parskip glue; then the negative vskip we just inserted would
+    % cause the example and the item to crash together.  So we use this
+    % bizarre value of 10001 as a signal to \aboveenvbreak to insert
+    % \parskip glue after all.  Section titles are handled this way also.
+    % 
+    \penalty 10001
+    \endgroup
+    \itemxneedsnegativevskipfalse
+  \else
+    % The item text fits into the space.  Start a paragraph, so that the
+    % following text (if any) will end up on the same line.
+    \noindent
+    % Do this with kerns and \unhbox so that if there is a footnote in
+    % the item text, it can migrate to the main vertical list and
+    % eventually be printed.
+    \nobreak\kern-\tableindent
+    \dimen0 = \itemmax  \advance\dimen0 by \itemmargin \advance\dimen0 by -\wd0
+    \unhbox0
+    \nobreak\kern\dimen0
+    \endgroup
+    \itemxneedsnegativevskiptrue
+  \fi
+}
+
+\def\item{\errmessage{@item while not in a list environment}}
+\def\itemx{\errmessage{@itemx while not in a list environment}}
+
+% @table, @ftable, @vtable.
+\envdef\table{%
+  \let\itemindex\gobble
+  \tablecheck{table}%
+}
+\envdef\ftable{%
+  \def\itemindex ##1{\doind {fn}{\code{##1}}}%
+  \tablecheck{ftable}%
+}
+\envdef\vtable{%
+  \def\itemindex ##1{\doind {vr}{\code{##1}}}%
+  \tablecheck{vtable}%
+}
+\def\tablecheck#1{%
+  \ifnum \the\catcode`\^^M=\active
+    \endgroup
+    \errmessage{This command won't work in this context; perhaps the problem is
+      that we are \inenvironment\thisenv}%
+    \def\next{\doignore{#1}}%
+  \else
+    \let\next\tablex
+  \fi
+  \next
+}
+\def\tablex#1{%
+  \def\itemindicate{#1}%
+  \parsearg\tabley
+}
+\def\tabley#1{%
+  {%
+    \makevalueexpandable
+    \edef\temp{\noexpand\tablez #1\space\space\space}%
+    \expandafter
+  }\temp \endtablez
+}
+\def\tablez #1 #2 #3 #4\endtablez{%
+  \aboveenvbreak
+  \ifnum 0#1>0 \advance \leftskip by #1\mil \fi
+  \ifnum 0#2>0 \tableindent=#2\mil \fi
+  \ifnum 0#3>0 \advance \rightskip by #3\mil \fi
+  \itemmax=\tableindent
+  \advance \itemmax by -\itemmargin
+  \advance \leftskip by \tableindent
+  \exdentamount=\tableindent
+  \parindent = 0pt
+  \parskip = \smallskipamount
+  \ifdim \parskip=0pt \parskip=2pt \fi
+  \let\item = \internalBitem
+  \let\itemx = \internalBitemx
+}
+\def\Etable{\endgraf\afterenvbreak}
+\let\Eftable\Etable
+\let\Evtable\Etable
+\let\Eitemize\Etable
+\let\Eenumerate\Etable
+
+% This is the counter used by @enumerate, which is really @itemize
+
+\newcount \itemno
+
+\envdef\itemize{\parsearg\doitemize}
+
+\def\doitemize#1{%
+  \aboveenvbreak
+  \itemmax=\itemindent
+  \advance\itemmax by -\itemmargin
+  \advance\leftskip by \itemindent
+  \exdentamount=\itemindent
+  \parindent=0pt
+  \parskip=\smallskipamount
+  \ifdim\parskip=0pt \parskip=2pt \fi
+  \def\itemcontents{#1}%
+  % @itemize with no arg is equivalent to @itemize @bullet.
+  \ifx\itemcontents\empty\def\itemcontents{\bullet}\fi
+  \let\item=\itemizeitem
+}
+
+% Definition of @item while inside @itemize and @enumerate.
+%
+\def\itemizeitem{%
+  \advance\itemno by 1  % for enumerations
+  {\let\par=\endgraf \smallbreak}% reasonable place to break
+  {%
+   % If the document has an @itemize directly after a section title, a
+   % \nobreak will be last on the list, and \sectionheading will have
+   % done a \vskip-\parskip.  In that case, we don't want to zero
+   % parskip, or the item text will crash with the heading.  On the
+   % other hand, when there is normal text preceding the item (as there
+   % usually is), we do want to zero parskip, or there would be too much
+   % space.  In that case, we won't have a \nobreak before.  At least
+   % that's the theory.
+   \ifnum\lastpenalty<10000 \parskip=0in \fi
+   \noindent
+   \hbox to 0pt{\hss \itemcontents \kern\itemmargin}%
+   \vadjust{\penalty 1200}}% not good to break after first line of item.
+  \flushcr
+}
+
+% \splitoff TOKENS\endmark defines \first to be the first token in
+% TOKENS, and \rest to be the remainder.
+%
+\def\splitoff#1#2\endmark{\def\first{#1}\def\rest{#2}}%
+
+% Allow an optional argument of an uppercase letter, lowercase letter,
+% or number, to specify the first label in the enumerated list.  No
+% argument is the same as `1'.
+%
+\envparseargdef\enumerate{\enumeratey #1  \endenumeratey}
+\def\enumeratey #1 #2\endenumeratey{%
+  % If we were given no argument, pretend we were given `1'.
+  \def\thearg{#1}%
+  \ifx\thearg\empty \def\thearg{1}\fi
+  %
+  % Detect if the argument is a single token.  If so, it might be a
+  % letter.  Otherwise, the only valid thing it can be is a number.
+  % (We will always have one token, because of the test we just made.
+  % This is a good thing, since \splitoff doesn't work given nothing at
+  % all -- the first parameter is undelimited.)
+  \expandafter\splitoff\thearg\endmark
+  \ifx\rest\empty
+    % Only one token in the argument.  It could still be anything.
+    % A ``lowercase letter'' is one whose \lccode is nonzero.
+    % An ``uppercase letter'' is one whose \lccode is both nonzero, and
+    %   not equal to itself.
+    % Otherwise, we assume it's a number.
+    %
+    % We need the \relax at the end of the \ifnum lines to stop TeX from
+    % continuing to look for a <number>.
+    %
+    \ifnum\lccode\expandafter`\thearg=0\relax
+      \numericenumerate % a number (we hope)
+    \else
+      % It's a letter.
+      \ifnum\lccode\expandafter`\thearg=\expandafter`\thearg\relax
+        \lowercaseenumerate % lowercase letter
+      \else
+        \uppercaseenumerate % uppercase letter
+      \fi
+    \fi
+  \else
+    % Multiple tokens in the argument.  We hope it's a number.
+    \numericenumerate
+  \fi
+}
+
+% An @enumerate whose labels are integers.  The starting integer is
+% given in \thearg.
+%
+\def\numericenumerate{%
+  \itemno = \thearg
+  \startenumeration{\the\itemno}%
+}
+
+% The starting (lowercase) letter is in \thearg.
+\def\lowercaseenumerate{%
+  \itemno = \expandafter`\thearg
+  \startenumeration{%
+    % Be sure we're not beyond the end of the alphabet.
+    \ifnum\itemno=0
+      \errmessage{No more lowercase letters in @enumerate; get a bigger
+                  alphabet}%
+    \fi
+    \char\lccode\itemno
+  }%
+}
+
+% The starting (uppercase) letter is in \thearg.
+\def\uppercaseenumerate{%
+  \itemno = \expandafter`\thearg
+  \startenumeration{%
+    % Be sure we're not beyond the end of the alphabet.
+    \ifnum\itemno=0
+      \errmessage{No more uppercase letters in @enumerate; get a bigger
+                  alphabet}
+    \fi
+    \char\uccode\itemno
+  }%
+}
+
+% Call \doitemize, adding a period to the first argument and supplying the
+% common last two arguments.  Also subtract one from the initial value in
+% \itemno, since @item increments \itemno.
+%
+\def\startenumeration#1{%
+  \advance\itemno by -1
+  \doitemize{#1.}\flushcr
+}
+
+% @alphaenumerate and @capsenumerate are abbreviations for giving an arg
+% to @enumerate.
+%
+\def\alphaenumerate{\enumerate{a}}
+\def\capsenumerate{\enumerate{A}}
+\def\Ealphaenumerate{\Eenumerate}
+\def\Ecapsenumerate{\Eenumerate}
+
+
+% @multitable macros
+% Amy Hendrickson, 8/18/94, 3/6/96
+%
+% @multitable ... @end multitable will make as many columns as desired.
+% Contents of each column will wrap at width given in preamble.  Width
+% can be specified either with sample text given in a template line,
+% or in percent of \hsize, the current width of text on page.
+
+% Table can continue over pages but will only break between lines.
+
+% To make preamble:
+%
+% Either define widths of columns in terms of percent of \hsize:
+%   @multitable @columnfractions .25 .3 .45
+%   @item ...
+%
+%   Numbers following @columnfractions are the percent of the total
+%   current hsize to be used for each column. You may use as many
+%   columns as desired.
+
+
+% Or use a template:
+%   @multitable {Column 1 template} {Column 2 template} {Column 3 template}
+%   @item ...
+%   using the widest term desired in each column.
+
+% Each new table line starts with @item, each subsequent new column
+% starts with @tab. Empty columns may be produced by supplying @tab's
+% with nothing between them for as many times as empty columns are needed,
+% ie, @tab@tab@tab will produce two empty columns.
+
+% @item, @tab do not need to be on their own lines, but it will not hurt
+% if they are.
+
+% Sample multitable:
+
+%   @multitable {Column 1 template} {Column 2 template} {Column 3 template}
+%   @item first col stuff @tab second col stuff @tab third col
+%   @item
+%   first col stuff
+%   @tab
+%   second col stuff
+%   @tab
+%   third col
+%   @item first col stuff @tab second col stuff
+%   @tab Many paragraphs of text may be used in any column.
+%
+%         They will wrap at the width determined by the template.
+%   @item@tab@tab This will be in third column.
+%   @end multitable
+
+% Default dimensions may be reset by user.
+% @multitableparskip is vertical space between paragraphs in table.
+% @multitableparindent is paragraph indent in table.
+% @multitablecolmargin is horizontal space to be left between columns.
+% @multitablelinespace is space to leave between table items, baseline
+%                                                            to baseline.
+%   0pt means it depends on current normal line spacing.
+%
+\newskip\multitableparskip
+\newskip\multitableparindent
+\newdimen\multitablecolspace
+\newskip\multitablelinespace
+\multitableparskip=0pt
+\multitableparindent=6pt
+\multitablecolspace=12pt
+\multitablelinespace=0pt
+
+% Macros used to set up halign preamble:
+%
+\let\endsetuptable\relax
+\def\xendsetuptable{\endsetuptable}
+\let\columnfractions\relax
+\def\xcolumnfractions{\columnfractions}
+\newif\ifsetpercent
+
+% #1 is the @columnfraction, usually a decimal number like .5, but might
+% be just 1.  We just use it, whatever it is.
+%
+\def\pickupwholefraction#1 {%
+  \global\advance\colcount by 1
+  \expandafter\xdef\csname col\the\colcount\endcsname{#1\hsize}%
+  \setuptable
+}
+
+\newcount\colcount
+\def\setuptable#1{%
+  \def\firstarg{#1}%
+  \ifx\firstarg\xendsetuptable
+    \let\go = \relax
+  \else
+    \ifx\firstarg\xcolumnfractions
+      \global\setpercenttrue
+    \else
+      \ifsetpercent
+         \let\go\pickupwholefraction
+      \else
+         \global\advance\colcount by 1
+         \setbox0=\hbox{#1\unskip\space}% Add a normal word space as a
+                   % separator; typically that is always in the input, anyway.
+         \expandafter\xdef\csname col\the\colcount\endcsname{\the\wd0}%
+      \fi
+    \fi
+    \ifx\go\pickupwholefraction
+      % Put the argument back for the \pickupwholefraction call, so
+      % we'll always have a period there to be parsed.
+      \def\go{\pickupwholefraction#1}%
+    \else
+      \let\go = \setuptable
+    \fi%
+  \fi
+  \go
+}
+
+% multitable-only commands.
+%
+% @headitem starts a heading row, which we typeset in bold.
+% Assignments have to be global since we are inside the implicit group
+% of an alignment entry.  Note that \everycr resets \everytab.
+\def\headitem{\checkenv\multitable \crcr \global\everytab={\bf}\the\everytab}%
+%
+% A \tab used to include \hskip1sp.  But then the space in a template
+% line is not enough.  That is bad.  So let's go back to just `&' until
+% we encounter the problem it was intended to solve again.
+%					--karl, nathan@acm.org, 20apr99.
+\def\tab{\checkenv\multitable &\the\everytab}%
+
+% @multitable ... @end multitable definitions:
+%
+\newtoks\everytab  % insert after every tab.
+%
+\envdef\multitable{%
+  \vskip\parskip
+  \startsavinginserts
+  %
+  % @item within a multitable starts a normal row.
+  % We use \def instead of \let so that if one of the multitable entries
+  % contains an @itemize, we don't choke on the \item (seen as \crcr aka
+  % \endtemplate) expanding \doitemize.
+  \def\item{\crcr}%
+  %
+  \tolerance=9500
+  \hbadness=9500
+  \setmultitablespacing
+  \parskip=\multitableparskip
+  \parindent=\multitableparindent
+  \overfullrule=0pt
+  \global\colcount=0
+  %
+  \everycr = {%
+    \noalign{%
+      \global\everytab={}%
+      \global\colcount=0 % Reset the column counter.
+      % Check for saved footnotes, etc.
+      \checkinserts
+      % Keeps underfull box messages off when table breaks over pages.
+      %\filbreak
+	% Maybe so, but it also creates really weird page breaks when the
+	% table breaks over pages. Wouldn't \vfil be better?  Wait until the
+	% problem manifests itself, so it can be fixed for real --karl.
+    }%
+  }%
+  %
+  \parsearg\domultitable
+}
+\def\domultitable#1{%
+  % To parse everything between @multitable and @item:
+  \setuptable#1 \endsetuptable
+  %
+  % This preamble sets up a generic column definition, which will
+  % be used as many times as user calls for columns.
+  % \vtop will set a single line and will also let text wrap and
+  % continue for many paragraphs if desired.
+  \halign\bgroup &%
+    \global\advance\colcount by 1
+    \multistrut
+    \vtop{%
+      % Use the current \colcount to find the correct column width:
+      \hsize=\expandafter\csname col\the\colcount\endcsname
+      %
+      % In order to keep entries from bumping into each other
+      % we will add a \leftskip of \multitablecolspace to all columns after
+      % the first one.
+      %
+      % If a template has been used, we will add \multitablecolspace
+      % to the width of each template entry.
+      %
+      % If the user has set preamble in terms of percent of \hsize we will
+      % use that dimension as the width of the column, and the \leftskip
+      % will keep entries from bumping into each other.  Table will start at
+      % left margin and final column will justify at right margin.
+      %
+      % Make sure we don't inherit \rightskip from the outer environment.
+      \rightskip=0pt
+      \ifnum\colcount=1
+	% The first column will be indented with the surrounding text.
+	\advance\hsize by\leftskip
+      \else
+	\ifsetpercent \else
+	  % If user has not set preamble in terms of percent of \hsize
+	  % we will advance \hsize by \multitablecolspace.
+	  \advance\hsize by \multitablecolspace
+	\fi
+       % In either case we will make \leftskip=\multitablecolspace:
+      \leftskip=\multitablecolspace
+      \fi
+      % Ignoring space at the beginning and end avoids an occasional spurious
+      % blank line, when TeX decides to break the line at the space before the
+      % box from the multistrut, so the strut ends up on a line by itself.
+      % For example:
+      % @multitable @columnfractions .11 .89
+      % @item @code{#}
+      % @tab Legal holiday which is valid in major parts of the whole country.
+      % Is automatically provided with highlighting sequences respectively
+      % marking characters.
+      \noindent\ignorespaces##\unskip\multistrut
+    }\cr
+}
+\def\Emultitable{%
+  \crcr
+  \egroup % end the \halign
+  \global\setpercentfalse
+}
+
+\def\setmultitablespacing{%
+  \def\multistrut{\strut}% just use the standard line spacing
+  %
+  % Compute \multitablelinespace (if not defined by user) for use in
+  % \multitableparskip calculation.  We used define \multistrut based on
+  % this, but (ironically) that caused the spacing to be off.
+  % See bug-texinfo report from Werner Lemberg, 31 Oct 2004 12:52:20 +0100.
+\ifdim\multitablelinespace=0pt
+\setbox0=\vbox{X}\global\multitablelinespace=\the\baselineskip
+\global\advance\multitablelinespace by-\ht0
+\fi
+%% Test to see if parskip is larger than space between lines of
+%% table. If not, do nothing.
+%%        If so, set to same dimension as multitablelinespace.
+\ifdim\multitableparskip>\multitablelinespace
+\global\multitableparskip=\multitablelinespace
+\global\advance\multitableparskip-7pt %% to keep parskip somewhat smaller
+                                      %% than skip between lines in the table.
+\fi%
+\ifdim\multitableparskip=0pt
+\global\multitableparskip=\multitablelinespace
+\global\advance\multitableparskip-7pt %% to keep parskip somewhat smaller
+                                      %% than skip between lines in the table.
+\fi}
+
+
+\message{conditionals,}
+
+% @iftex, @ifnotdocbook, @ifnothtml, @ifnotinfo, @ifnotplaintext,
+% @ifnotxml always succeed.  They currently do nothing; we don't
+% attempt to check whether the conditionals are properly nested.  But we
+% have to remember that they are conditionals, so that @end doesn't
+% attempt to close an environment group.
+%
+\def\makecond#1{%
+  \expandafter\let\csname #1\endcsname = \relax
+  \expandafter\let\csname iscond.#1\endcsname = 1
+}
+\makecond{iftex}
+\makecond{ifnotdocbook}
+\makecond{ifnothtml}
+\makecond{ifnotinfo}
+\makecond{ifnotplaintext}
+\makecond{ifnotxml}
+
+% Ignore @ignore, @ifhtml, @ifinfo, and the like.
+%
+\def\direntry{\doignore{direntry}}
+\def\documentdescription{\doignore{documentdescription}}
+\def\docbook{\doignore{docbook}}
+\def\html{\doignore{html}}
+\def\ifdocbook{\doignore{ifdocbook}}
+\def\ifhtml{\doignore{ifhtml}}
+\def\ifinfo{\doignore{ifinfo}}
+\def\ifnottex{\doignore{ifnottex}}
+\def\ifplaintext{\doignore{ifplaintext}}
+\def\ifxml{\doignore{ifxml}}
+\def\ignore{\doignore{ignore}}
+\def\menu{\doignore{menu}}
+\def\xml{\doignore{xml}}
+
+% Ignore text until a line `@end #1', keeping track of nested conditionals.
+%
+% A count to remember the depth of nesting.
+\newcount\doignorecount
+
+\def\doignore#1{\begingroup
+  % Scan in ``verbatim'' mode:
+  \catcode`\@ = \other
+  \catcode`\{ = \other
+  \catcode`\} = \other
+  %
+  % Make sure that spaces turn into tokens that match what \doignoretext wants.
+  \spaceisspace
+  %
+  % Count number of #1's that we've seen.
+  \doignorecount = 0
+  %
+  % Swallow text until we reach the matching `@end #1'.
+  \dodoignore{#1}%
+}
+
+{ \catcode`_=11 % We want to use \_STOP_ which cannot appear in texinfo source.
+  \obeylines %
+  %
+  \gdef\dodoignore#1{%
+    % #1 contains the command name as a string, e.g., `ifinfo'.
+    %
+    % Define a command to find the next `@end #1', which must be on a line
+    % by itself.
+    \long\def\doignoretext##1^^M@end #1{\doignoretextyyy##1^^M@#1\_STOP_}%
+    % And this command to find another #1 command, at the beginning of a
+    % line.  (Otherwise, we would consider a line `@c @ifset', for
+    % example, to count as an @ifset for nesting.)
+    \long\def\doignoretextyyy##1^^M@#1##2\_STOP_{\doignoreyyy{##2}\_STOP_}%
+    %
+    % And now expand that command.
+    \obeylines %
+    \doignoretext ^^M%
+  }%
+}
+
+\def\doignoreyyy#1{%
+  \def\temp{#1}%
+  \ifx\temp\empty			% Nothing found.
+    \let\next\doignoretextzzz
+  \else					% Found a nested condition, ...
+    \advance\doignorecount by 1
+    \let\next\doignoretextyyy		% ..., look for another.
+    % If we're here, #1 ends with ^^M\ifinfo (for example).
+  \fi
+  \next #1% the token \_STOP_ is present just after this macro.
+}
+
+% We have to swallow the remaining "\_STOP_".
+%
+\def\doignoretextzzz#1{%
+  \ifnum\doignorecount = 0	% We have just found the outermost @end.
+    \let\next\enddoignore
+  \else				% Still inside a nested condition.
+    \advance\doignorecount by -1
+    \let\next\doignoretext      % Look for the next @end.
+  \fi
+  \next
+}
+
+% Finish off ignored text.
+\def\enddoignore{\endgroup\ignorespaces}
+
+
+% @set VAR sets the variable VAR to an empty value.
+% @set VAR REST-OF-LINE sets VAR to the value REST-OF-LINE.
+%
+% Since we want to separate VAR from REST-OF-LINE (which might be
+% empty), we can't just use \parsearg; we have to insert a space of our
+% own to delimit the rest of the line, and then take it out again if we
+% didn't need it.
+% We rely on the fact that \parsearg sets \catcode`\ =10.
+%
+\parseargdef\set{\setyyy#1 \endsetyyy}
+\def\setyyy#1 #2\endsetyyy{%
+  {%
+    \makevalueexpandable
+    \def\temp{#2}%
+    \edef\next{\gdef\makecsname{SET#1}}%
+    \ifx\temp\empty
+      \next{}%
+    \else
+      \setzzz#2\endsetzzz
+    \fi
+  }%
+}
+% Remove the trailing space \setxxx inserted.
+\def\setzzz#1 \endsetzzz{\next{#1}}
+
+% @clear VAR clears (i.e., unsets) the variable VAR.
+%
+\parseargdef\clear{%
+  {%
+    \makevalueexpandable
+    \global\expandafter\let\csname SET#1\endcsname=\relax
+  }%
+}
+
+% @value{foo} gets the text saved in variable foo.
+\def\value{\begingroup\makevalueexpandable\valuexxx}
+\def\valuexxx#1{\expandablevalue{#1}\endgroup}
+{
+  \catcode`\- = \active \catcode`\_ = \active
+  %
+  \gdef\makevalueexpandable{%
+    \let\value = \expandablevalue
+    % We don't want these characters active, ...
+    \catcode`\-=\other \catcode`\_=\other
+    % ..., but we might end up with active ones in the argument if
+    % we're called from @code, as @code{@value{foo-bar_}}, though.
+    % So \let them to their normal equivalents.
+    \let-\realdash \let_\normalunderscore
+  }
+}
+
+% We have this subroutine so that we can handle at least some @value's
+% properly in indexes (we call \makevalueexpandable in \indexdummies).
+% The command has to be fully expandable (if the variable is set), since
+% the result winds up in the index file.  This means that if the
+% variable's value contains other Texinfo commands, it's almost certain
+% it will fail (although perhaps we could fix that with sufficient work
+% to do a one-level expansion on the result, instead of complete).
+%
+\def\expandablevalue#1{%
+  \expandafter\ifx\csname SET#1\endcsname\relax
+    {[No value for ``#1'']}%
+    \message{Variable `#1', used in @value, is not set.}%
+  \else
+    \csname SET#1\endcsname
+  \fi
+}
+
+% @ifset VAR ... @end ifset reads the `...' iff VAR has been defined
+% with @set.
+%
+% To get special treatment of `@end ifset,' call \makeond and the redefine.
+%
+\makecond{ifset}
+\def\ifset{\parsearg{\doifset{\let\next=\ifsetfail}}}
+\def\doifset#1#2{%
+  {%
+    \makevalueexpandable
+    \let\next=\empty
+    \expandafter\ifx\csname SET#2\endcsname\relax
+      #1% If not set, redefine \next.
+    \fi
+    \expandafter
+  }\next
+}
+\def\ifsetfail{\doignore{ifset}}
+
+% @ifclear VAR ... @end ifclear reads the `...' iff VAR has never been
+% defined with @set, or has been undefined with @clear.
+%
+% The `\else' inside the `\doifset' parameter is a trick to reuse the
+% above code: if the variable is not set, do nothing, if it is set,
+% then redefine \next to \ifclearfail.
+%
+\makecond{ifclear}
+\def\ifclear{\parsearg{\doifset{\else \let\next=\ifclearfail}}}
+\def\ifclearfail{\doignore{ifclear}}
+
+% @dircategory CATEGORY  -- specify a category of the dir file
+% which this file should belong to.  Ignore this in TeX.
+\let\dircategory=\comment
+
+% @defininfoenclose.
+\let\definfoenclose=\comment
+
+
+\message{indexing,}
+% Index generation facilities
+
+% Define \newwrite to be identical to plain tex's \newwrite
+% except not \outer, so it can be used within macros and \if's.
+\edef\newwrite{\makecsname{ptexnewwrite}}
+
+% \newindex {foo} defines an index named foo.
+% It automatically defines \fooindex such that
+% \fooindex ...rest of line... puts an entry in the index foo.
+% It also defines \fooindfile to be the number of the output channel for
+% the file that accumulates this index.  The file's extension is foo.
+% The name of an index should be no more than 2 characters long
+% for the sake of vms.
+%
+\def\newindex#1{%
+  \iflinks
+    \expandafter\newwrite \csname#1indfile\endcsname
+    \openout \csname#1indfile\endcsname \jobname.#1 % Open the file
+  \fi
+  \expandafter\xdef\csname#1index\endcsname{%     % Define @#1index
+    \noexpand\doindex{#1}}
+}
+
+% @defindex foo  ==  \newindex{foo}
+%
+\def\defindex{\parsearg\newindex}
+
+% Define @defcodeindex, like @defindex except put all entries in @code.
+%
+\def\defcodeindex{\parsearg\newcodeindex}
+%
+\def\newcodeindex#1{%
+  \iflinks
+    \expandafter\newwrite \csname#1indfile\endcsname
+    \openout \csname#1indfile\endcsname \jobname.#1
+  \fi
+  \expandafter\xdef\csname#1index\endcsname{%
+    \noexpand\docodeindex{#1}}%
+}
+
+
+% @synindex foo bar    makes index foo feed into index bar.
+% Do this instead of @defindex foo if you don't want it as a separate index.
+%
+% @syncodeindex foo bar   similar, but put all entries made for index foo
+% inside @code.
+%
+\def\synindex#1 #2 {\dosynindex\doindex{#1}{#2}}
+\def\syncodeindex#1 #2 {\dosynindex\docodeindex{#1}{#2}}
+
+% #1 is \doindex or \docodeindex, #2 the index getting redefined (foo),
+% #3 the target index (bar).
+\def\dosynindex#1#2#3{%
+  % Only do \closeout if we haven't already done it, else we'll end up
+  % closing the target index.
+  \expandafter \ifx\csname donesynindex#2\endcsname \undefined
+    % The \closeout helps reduce unnecessary open files; the limit on the
+    % Acorn RISC OS is a mere 16 files.
+    \expandafter\closeout\csname#2indfile\endcsname
+    \expandafter\let\csname\donesynindex#2\endcsname = 1
+  \fi
+  % redefine \fooindfile:
+  \expandafter\let\expandafter\temp\expandafter=\csname#3indfile\endcsname
+  \expandafter\let\csname#2indfile\endcsname=\temp
+  % redefine \fooindex:
+  \expandafter\xdef\csname#2index\endcsname{\noexpand#1{#3}}%
+}
+
+% Define \doindex, the driver for all \fooindex macros.
+% Argument #1 is generated by the calling \fooindex macro,
+%  and it is "foo", the name of the index.
+
+% \doindex just uses \parsearg; it calls \doind for the actual work.
+% This is because \doind is more useful to call from other macros.
+
+% There is also \dosubind {index}{topic}{subtopic}
+% which makes an entry in a two-level index such as the operation index.
+
+\def\doindex#1{\edef\indexname{#1}\parsearg\singleindexer}
+\def\singleindexer #1{\doind{\indexname}{#1}}
+
+% like the previous two, but they put @code around the argument.
+\def\docodeindex#1{\edef\indexname{#1}\parsearg\singlecodeindexer}
+\def\singlecodeindexer #1{\doind{\indexname}{\code{#1}}}
+
+% Take care of Texinfo commands that can appear in an index entry.
+% Since there are some commands we want to expand, and others we don't,
+% we have to laboriously prevent expansion for those that we don't.
+%
+\def\indexdummies{%
+  \def\@{@}% change to @@ when we switch to @ as escape char in index files.
+  \def\ {\realbackslash\space }%
+  % Need these in case \tex is in effect and \{ is a \delimiter again.
+  % But can't use \lbracecmd and \rbracecmd because texindex assumes
+  % braces and backslashes are used only as delimiters.
+  \let\{ = \mylbrace
+  \let\} = \myrbrace
+  %
+  % \definedummyword defines \#1 as \realbackslash #1\space, thus
+  % effectively preventing its expansion.  This is used only for control
+  % words, not control letters, because the \space would be incorrect
+  % for control characters, but is needed to separate the control word
+  % from whatever follows.
+  %
+  % For control letters, we have \definedummyletter, which omits the
+  % space.
+  %
+  % These can be used both for control words that take an argument and
+  % those that do not.  If it is followed by {arg} in the input, then
+  % that will dutifully get written to the index (or wherever).
+  %
+  \def\definedummyword##1{%
+    \expandafter\def\csname ##1\endcsname{\realbackslash ##1\space}%
+  }%
+  \def\definedummyletter##1{%
+    \expandafter\def\csname ##1\endcsname{\realbackslash ##1}%
+  }%
+  \let\definedummyaccent\definedummyletter
+  %
+  % Do the redefinitions.
+  \commondummies
+}
+
+% For the aux file, @ is the escape character.  So we want to redefine
+% everything using @ instead of \realbackslash.  When everything uses
+% @, this will be simpler.
+%
+\def\atdummies{%
+  \def\@{@@}%
+  \def\ {@ }%
+  \let\{ = \lbraceatcmd
+  \let\} = \rbraceatcmd
+  %
+  % (See comments in \indexdummies.)
+  \def\definedummyword##1{%
+    \expandafter\def\csname ##1\endcsname{@##1\space}%
+  }%
+  \def\definedummyletter##1{%
+    \expandafter\def\csname ##1\endcsname{@##1}%
+  }%
+  \let\definedummyaccent\definedummyletter
+  %
+  % Do the redefinitions.
+  \commondummies
+}
+
+% Called from \indexdummies and \atdummies.  \definedummyword and
+% \definedummyletter must be defined first.
+%
+\def\commondummies{%
+  %
+  \normalturnoffactive
+  %
+  \commondummiesnofonts
+  %
+  \definedummyletter{_}%
+  %
+  % Non-English letters.
+  \definedummyword{AA}%
+  \definedummyword{AE}%
+  \definedummyword{L}%
+  \definedummyword{OE}%
+  \definedummyword{O}%
+  \definedummyword{aa}%
+  \definedummyword{ae}%
+  \definedummyword{l}%
+  \definedummyword{oe}%
+  \definedummyword{o}%
+  \definedummyword{ss}%
+  \definedummyword{exclamdown}%
+  \definedummyword{questiondown}%
+  \definedummyword{ordf}%
+  \definedummyword{ordm}%
+  %
+  % Although these internal commands shouldn't show up, sometimes they do.
+  \definedummyword{bf}%
+  \definedummyword{gtr}%
+  \definedummyword{hat}%
+  \definedummyword{less}%
+  \definedummyword{sf}%
+  \definedummyword{sl}%
+  \definedummyword{tclose}%
+  \definedummyword{tt}%
+  %
+  \definedummyword{LaTeX}%
+  \definedummyword{TeX}%
+  %
+  % Assorted special characters.
+  \definedummyword{bullet}%
+  \definedummyword{comma}%
+  \definedummyword{copyright}%
+  \definedummyword{registeredsymbol}%
+  \definedummyword{dots}%
+  \definedummyword{enddots}%
+  \definedummyword{equiv}%
+  \definedummyword{error}%
+  \definedummyword{euro}%
+  \definedummyword{expansion}%
+  \definedummyword{minus}%
+  \definedummyword{pounds}%
+  \definedummyword{point}%
+  \definedummyword{print}%
+  \definedummyword{result}%
+  %
+  % Handle some cases of @value -- where it does not contain any
+  % (non-fully-expandable) commands.
+  \makevalueexpandable
+  %
+  % Normal spaces, not active ones.
+  \unsepspaces
+  %
+  % No macro expansion.
+  \turnoffmacros
+}
+
+% \commondummiesnofonts: common to \commondummies and \indexnofonts.
+%
+% Better have this without active chars.
+{
+  \catcode`\~=\other
+  \gdef\commondummiesnofonts{%
+    % Control letters and accents.
+    \definedummyletter{!}%
+    \definedummyaccent{"}%
+    \definedummyaccent{'}%
+    \definedummyletter{*}%
+    \definedummyaccent{,}%
+    \definedummyletter{.}%
+    \definedummyletter{/}%
+    \definedummyletter{:}%
+    \definedummyaccent{=}%
+    \definedummyletter{?}%
+    \definedummyaccent{^}%
+    \definedummyaccent{`}%
+    \definedummyaccent{~}%
+    \definedummyword{u}%
+    \definedummyword{v}%
+    \definedummyword{H}%
+    \definedummyword{dotaccent}%
+    \definedummyword{ringaccent}%
+    \definedummyword{tieaccent}%
+    \definedummyword{ubaraccent}%
+    \definedummyword{udotaccent}%
+    \definedummyword{dotless}%
+    %
+    % Texinfo font commands.
+    \definedummyword{b}%
+    \definedummyword{i}%
+    \definedummyword{r}%
+    \definedummyword{sc}%
+    \definedummyword{t}%
+    %
+    % Commands that take arguments.
+    \definedummyword{acronym}%
+    \definedummyword{cite}%
+    \definedummyword{code}%
+    \definedummyword{command}%
+    \definedummyword{dfn}%
+    \definedummyword{emph}%
+    \definedummyword{env}%
+    \definedummyword{file}%
+    \definedummyword{kbd}%
+    \definedummyword{key}%
+    \definedummyword{math}%
+    \definedummyword{option}%
+    \definedummyword{samp}%
+    \definedummyword{strong}%
+    \definedummyword{tie}%
+    \definedummyword{uref}%
+    \definedummyword{url}%
+    \definedummyword{var}%
+    \definedummyword{verb}%
+    \definedummyword{w}%
+  }
+}
+
+% \indexnofonts is used when outputting the strings to sort the index
+% by, and when constructing control sequence names.  It eliminates all
+% control sequences and just writes whatever the best ASCII sort string
+% would be for a given command (usually its argument).
+%
+\def\indexnofonts{%
+  % Accent commands should become @asis.
+  \def\definedummyaccent##1{%
+    \expandafter\let\csname ##1\endcsname\asis
+  }%
+  % We can just ignore other control letters.
+  \def\definedummyletter##1{%
+    \expandafter\def\csname ##1\endcsname{}%
+  }%
+  % Hopefully, all control words can become @asis.
+  \let\definedummyword\definedummyaccent
+  %
+  \commondummiesnofonts
+  %
+  % Don't no-op \tt, since it isn't a user-level command
+  % and is used in the definitions of the active chars like <, >, |, etc.
+  % Likewise with the other plain tex font commands.
+  %\let\tt=\asis
+  %
+  \def\ { }%
+  \def\@{@}%
+  % how to handle braces?
+  \def\_{\normalunderscore}%
+  %
+  % Non-English letters.
+  \def\AA{AA}%
+  \def\AE{AE}%
+  \def\L{L}%
+  \def\OE{OE}%
+  \def\O{O}%
+  \def\aa{aa}%
+  \def\ae{ae}%
+  \def\l{l}%
+  \def\oe{oe}%
+  \def\o{o}%
+  \def\ss{ss}%
+  \def\exclamdown{!}%
+  \def\questiondown{?}%
+  \def\ordf{a}%
+  \def\ordm{o}%
+  %
+  \def\LaTeX{LaTeX}%
+  \def\TeX{TeX}%
+  %
+  % Assorted special characters.
+  % (The following {} will end up in the sort string, but that's ok.)
+  \def\bullet{bullet}%
+  \def\comma{,}%
+  \def\copyright{copyright}%
+  \def\registeredsymbol{R}%
+  \def\dots{...}%
+  \def\enddots{...}%
+  \def\equiv{==}%
+  \def\error{error}%
+  \def\euro{euro}%
+  \def\expansion{==>}%
+  \def\minus{-}%
+  \def\pounds{pounds}%
+  \def\point{.}%
+  \def\print{-|}%
+  \def\result{=>}%
+  %
+  % Don't write macro names.
+  \emptyusermacros
+}
+
+\let\indexbackslash=0  %overridden during \printindex.
+\let\SETmarginindex=\relax % put index entries in margin (undocumented)?
+
+% Most index entries go through here, but \dosubind is the general case.
+% #1 is the index name, #2 is the entry text.
+\def\doind#1#2{\dosubind{#1}{#2}{}}
+
+% Workhorse for all \fooindexes.
+% #1 is name of index, #2 is stuff to put there, #3 is subentry --
+% empty if called from \doind, as we usually are (the main exception
+% is with most defuns, which call us directly).
+%
+\def\dosubind#1#2#3{%
+  \iflinks
+  {%
+    % Store the main index entry text (including the third arg).
+    \toks0 = {#2}%
+    % If third arg is present, precede it with a space.
+    \def\thirdarg{#3}%
+    \ifx\thirdarg\empty \else
+      \toks0 = \expandafter{\the\toks0 \space #3}%
+    \fi
+    %
+    \edef\writeto{\csname#1indfile\endcsname}%
+    %
+    \ifvmode
+      \dosubindsanitize
+    \else
+      \dosubindwrite
+    \fi
+  }%
+  \fi
+}
+
+% Write the entry in \toks0 to the index file:
+%
+\def\dosubindwrite{%
+  % Put the index entry in the margin if desired.
+  \ifx\SETmarginindex\relax\else
+    \insert\margin{\hbox{\vrule height8pt depth3pt width0pt \the\toks0}}%
+  \fi
+  %
+  % Remember, we are within a group.
+  \indexdummies % Must do this here, since \bf, etc expand at this stage
+  \escapechar=`\\
+  \def\backslashcurfont{\indexbackslash}% \indexbackslash isn't defined now
+      % so it will be output as is; and it will print as backslash.
+  %
+  % Process the index entry with all font commands turned off, to
+  % get the string to sort by.
+  {\indexnofonts
+   \edef\temp{\the\toks0}% need full expansion
+   \xdef\indexsorttmp{\temp}%
+  }%
+  %
+  % Set up the complete index entry, with both the sort key and
+  % the original text, including any font commands.  We write
+  % three arguments to \entry to the .?? file (four in the
+  % subentry case), texindex reduces to two when writing the .??s
+  % sorted result.
+  \edef\temp{%
+    \write\writeto{%
+      \string\entry{\indexsorttmp}{\noexpand\folio}{\the\toks0}}%
+  }%
+  \temp
+}
+
+% Take care of unwanted page breaks:
+%
+% If a skip is the last thing on the list now, preserve it
+% by backing up by \lastskip, doing the \write, then inserting
+% the skip again.  Otherwise, the whatsit generated by the
+% \write will make \lastskip zero.  The result is that sequences
+% like this:
+% @end defun
+% @tindex whatever
+% @defun ...
+% will have extra space inserted, because the \medbreak in the
+% start of the @defun won't see the skip inserted by the @end of
+% the previous defun.
+%
+% But don't do any of this if we're not in vertical mode.  We
+% don't want to do a \vskip and prematurely end a paragraph.
+%
+% Avoid page breaks due to these extra skips, too.
+%
+% But wait, there is a catch there:
+% We'll have to check whether \lastskip is zero skip.  \ifdim is not
+% sufficient for this purpose, as it ignores stretch and shrink parts
+% of the skip.  The only way seems to be to check the textual
+% representation of the skip.
+%
+% The following is almost like \def\zeroskipmacro{0.0pt} except that
+% the ``p'' and ``t'' characters have catcode \other, not 11 (letter).
+%
+\edef\zeroskipmacro{\expandafter\the\csname z@skip\endcsname}
+%
+% ..., ready, GO:
+%
+\def\dosubindsanitize{%
+  % \lastskip and \lastpenalty cannot both be nonzero simultaneously.
+  \skip0 = \lastskip
+  \edef\lastskipmacro{\the\lastskip}%
+  \count255 = \lastpenalty
+  %
+  % If \lastskip is nonzero, that means the last item was a
+  % skip.  And since a skip is discardable, that means this
+  % -\skip0 glue we're inserting is preceded by a
+  % non-discardable item, therefore it is not a potential
+  % breakpoint, therefore no \nobreak needed.
+  \ifx\lastskipmacro\zeroskipmacro
+  \else
+    \vskip-\skip0
+  \fi
+  %
+  \dosubindwrite
+  %
+  \ifx\lastskipmacro\zeroskipmacro
+    % If \lastskip was zero, perhaps the last item was a penalty, and
+    % perhaps it was >=10000, e.g., a \nobreak.  In that case, we want
+    % to re-insert the same penalty (values >10000 are used for various
+    % signals); since we just inserted a non-discardable item, any
+    % following glue (such as a \parskip) would be a breakpoint.  For example:
+    % 
+    %   @deffn deffn-whatever
+    %   @vindex index-whatever
+    %   Description.
+    % would allow a break between the index-whatever whatsit
+    % and the "Description." paragraph.
+    \ifnum\count255>9999 \penalty\count255 \fi
+  \else
+    % On the other hand, if we had a nonzero \lastskip,
+    % this make-up glue would be preceded by a non-discardable item
+    % (the whatsit from the \write), so we must insert a \nobreak.
+    \nobreak\vskip\skip0
+  \fi
+}
+
+% The index entry written in the file actually looks like
+%  \entry {sortstring}{page}{topic}
+% or
+%  \entry {sortstring}{page}{topic}{subtopic}
+% The texindex program reads in these files and writes files
+% containing these kinds of lines:
+%  \initial {c}
+%     before the first topic whose initial is c
+%  \entry {topic}{pagelist}
+%     for a topic that is used without subtopics
+%  \primary {topic}
+%     for the beginning of a topic that is used with subtopics
+%  \secondary {subtopic}{pagelist}
+%     for each subtopic.
+
+% Define the user-accessible indexing commands
+% @findex, @vindex, @kindex, @cindex.
+
+\def\findex {\fnindex}
+\def\kindex {\kyindex}
+\def\cindex {\cpindex}
+\def\vindex {\vrindex}
+\def\tindex {\tpindex}
+\def\pindex {\pgindex}
+
+\def\cindexsub {\begingroup\obeylines\cindexsub}
+{\obeylines %
+\gdef\cindexsub "#1" #2^^M{\endgroup %
+\dosubind{cp}{#2}{#1}}}
+
+% Define the macros used in formatting output of the sorted index material.
+
+% @printindex causes a particular index (the ??s file) to get printed.
+% It does not print any chapter heading (usually an @unnumbered).
+%
+\parseargdef\printindex{\begingroup
+  \dobreak \chapheadingskip{10000}%
+  %
+  \smallfonts \rm
+  \tolerance = 9500
+  \everypar = {}% don't want the \kern\-parindent from indentation suppression.
+  %
+  % See if the index file exists and is nonempty.
+  % Change catcode of @ here so that if the index file contains
+  % \initial {@}
+  % as its first line, TeX doesn't complain about mismatched braces
+  % (because it thinks @} is a control sequence).
+  \catcode`\@ = 11
+  \openin 1 \jobname.#1s
+  \ifeof 1
+    % \enddoublecolumns gets confused if there is no text in the index,
+    % and it loses the chapter title and the aux file entries for the
+    % index.  The easiest way to prevent this problem is to make sure
+    % there is some text.
+    \putwordIndexNonexistent
+  \else
+    %
+    % If the index file exists but is empty, then \openin leaves \ifeof
+    % false.  We have to make TeX try to read something from the file, so
+    % it can discover if there is anything in it.
+    \read 1 to \temp
+    \ifeof 1
+      \putwordIndexIsEmpty
+    \else
+      % Index files are almost Texinfo source, but we use \ as the escape
+      % character.  It would be better to use @, but that's too big a change
+      % to make right now.
+      \def\indexbackslash{\backslashcurfont}%
+      \catcode`\\ = 0
+      \escapechar = `\\
+      \begindoublecolumns
+      \input \jobname.#1s
+      \enddoublecolumns
+    \fi
+  \fi
+  \closein 1
+\endgroup}
+
+% These macros are used by the sorted index file itself.
+% Change them to control the appearance of the index.
+
+\def\initial#1{{%
+  % Some minor font changes for the special characters.
+  \let\tentt=\sectt \let\tt=\sectt \let\sf=\sectt
+  %
+  % Remove any glue we may have, we'll be inserting our own.
+  \removelastskip
+  %
+  % We like breaks before the index initials, so insert a bonus.
+  \nobreak
+  \vskip 0pt plus 3\baselineskip
+  \penalty 0
+  \vskip 0pt plus -3\baselineskip
+  %
+  % Typeset the initial.  Making this add up to a whole number of
+  % baselineskips increases the chance of the dots lining up from column
+  % to column.  It still won't often be perfect, because of the stretch
+  % we need before each entry, but it's better.
+  %
+  % No shrink because it confuses \balancecolumns.
+  \vskip 1.67\baselineskip plus .5\baselineskip
+  \leftline{\secbf #1}%
+  % Do our best not to break after the initial.
+  \nobreak
+  \vskip .33\baselineskip plus .1\baselineskip
+}}
+
+% \entry typesets a paragraph consisting of the text (#1), dot leaders, and
+% then page number (#2) flushed to the right margin.  It is used for index
+% and table of contents entries.  The paragraph is indented by \leftskip.
+%
+% A straightforward implementation would start like this:
+%	\def\entry#1#2{...
+% But this frozes the catcodes in the argument, and can cause problems to
+% @code, which sets - active.  This problem was fixed by a kludge---
+% ``-'' was active throughout whole index, but this isn't really right.
+%
+% The right solution is to prevent \entry from swallowing the whole text.
+%                                 --kasal, 21nov03
+\def\entry{%
+  \begingroup
+    %
+    % Start a new paragraph if necessary, so our assignments below can't
+    % affect previous text.
+    \par
+    %
+    % Do not fill out the last line with white space.
+    \parfillskip = 0in
+    %
+    % No extra space above this paragraph.
+    \parskip = 0in
+    %
+    % Do not prefer a separate line ending with a hyphen to fewer lines.
+    \finalhyphendemerits = 0
+    %
+    % \hangindent is only relevant when the entry text and page number
+    % don't both fit on one line.  In that case, bob suggests starting the
+    % dots pretty far over on the line.  Unfortunately, a large
+    % indentation looks wrong when the entry text itself is broken across
+    % lines.  So we use a small indentation and put up with long leaders.
+    %
+    % \hangafter is reset to 1 (which is the value we want) at the start
+    % of each paragraph, so we need not do anything with that.
+    \hangindent = 2em
+    %
+    % When the entry text needs to be broken, just fill out the first line
+    % with blank space.
+    \rightskip = 0pt plus1fil
+    %
+    % A bit of stretch before each entry for the benefit of balancing
+    % columns.
+    \vskip 0pt plus1pt
+    %
+    % Swallow the left brace of the text (first parameter):
+    \afterassignment\doentry
+    \let\temp =
+}
+\def\doentry{%
+    \bgroup % Instead of the swallowed brace.
+      \noindent
+      \aftergroup\finishentry
+      % And now comes the text of the entry.
+}
+\def\finishentry#1{%
+    % #1 is the page number.
+    %
+    % The following is kludged to not output a line of dots in the index if
+    % there are no page numbers.  The next person who breaks this will be
+    % cursed by a Unix daemon.
+    \def\tempa{{\rm }}%
+    \def\tempb{#1}%
+    \edef\tempc{\tempa}%
+    \edef\tempd{\tempb}%
+    \ifx\tempc\tempd
+      \ %
+    \else
+      %
+      % If we must, put the page number on a line of its own, and fill out
+      % this line with blank space.  (The \hfil is overwhelmed with the
+      % fill leaders glue in \indexdotfill if the page number does fit.)
+      \hfil\penalty50
+      \null\nobreak\indexdotfill % Have leaders before the page number.
+      %
+      % The `\ ' here is removed by the implicit \unskip that TeX does as
+      % part of (the primitive) \par.  Without it, a spurious underfull
+      % \hbox ensues.
+      \ifpdf
+	\pdfgettoks#1.%
+	\ \the\toksA
+      \else
+	\ #1%
+      \fi
+    \fi
+    \par
+  \endgroup
+}
+
+% Like \dotfill except takes at least 1 em.
+\def\indexdotfill{\cleaders
+  \hbox{$\mathsurround=0pt \mkern1.5mu ${\it .}$ \mkern1.5mu$}\hskip 1em plus 1fill}
+
+\def\primary #1{\line{#1\hfil}}
+
+\newskip\secondaryindent \secondaryindent=0.5cm
+\def\secondary#1#2{{%
+  \parfillskip=0in
+  \parskip=0in
+  \hangindent=1in
+  \hangafter=1
+  \noindent\hskip\secondaryindent\hbox{#1}\indexdotfill
+  \ifpdf
+    \pdfgettoks#2.\ \the\toksA % The page number ends the paragraph.
+  \else
+    #2
+  \fi
+  \par
+}}
+
+% Define two-column mode, which we use to typeset indexes.
+% Adapted from the TeXbook, page 416, which is to say,
+% the manmac.tex format used to print the TeXbook itself.
+\catcode`\@=11
+
+\newbox\partialpage
+\newdimen\doublecolumnhsize
+
+\def\begindoublecolumns{\begingroup % ended by \enddoublecolumns
+  % Grab any single-column material above us.
+  \output = {%
+    %
+    % Here is a possibility not foreseen in manmac: if we accumulate a
+    % whole lot of material, we might end up calling this \output
+    % routine twice in a row (see the doublecol-lose test, which is
+    % essentially a couple of indexes with @setchapternewpage off).  In
+    % that case we just ship out what is in \partialpage with the normal
+    % output routine.  Generally, \partialpage will be empty when this
+    % runs and this will be a no-op.  See the indexspread.tex test case.
+    \ifvoid\partialpage \else
+      \onepageout{\pagecontents\partialpage}%
+    \fi
+    %
+    \global\setbox\partialpage = \vbox{%
+      % Unvbox the main output page.
+      \unvbox\PAGE
+      \kern-\topskip \kern\baselineskip
+    }%
+  }%
+  \eject % run that output routine to set \partialpage
+  %
+  % Use the double-column output routine for subsequent pages.
+  \output = {\doublecolumnout}%
+  %
+  % Change the page size parameters.  We could do this once outside this
+  % routine, in each of @smallbook, @afourpaper, and the default 8.5x11
+  % format, but then we repeat the same computation.  Repeating a couple
+  % of assignments once per index is clearly meaningless for the
+  % execution time, so we may as well do it in one place.
+  %
+  % First we halve the line length, less a little for the gutter between
+  % the columns.  We compute the gutter based on the line length, so it
+  % changes automatically with the paper format.  The magic constant
+  % below is chosen so that the gutter has the same value (well, +-<1pt)
+  % as it did when we hard-coded it.
+  %
+  % We put the result in a separate register, \doublecolumhsize, so we
+  % can restore it in \pagesofar, after \hsize itself has (potentially)
+  % been clobbered.
+  %
+  \doublecolumnhsize = \hsize
+    \advance\doublecolumnhsize by -.04154\hsize
+    \divide\doublecolumnhsize by 2
+  \hsize = \doublecolumnhsize
+  %
+  % Double the \vsize as well.  (We don't need a separate register here,
+  % since nobody clobbers \vsize.)
+  \vsize = 2\vsize
+}
+
+% The double-column output routine for all double-column pages except
+% the last.
+%
+\def\doublecolumnout{%
+  \splittopskip=\topskip \splitmaxdepth=\maxdepth
+  % Get the available space for the double columns -- the normal
+  % (undoubled) page height minus any material left over from the
+  % previous page.
+  \dimen@ = \vsize
+  \divide\dimen@ by 2
+  \advance\dimen@ by -\ht\partialpage
+  %
+  % box0 will be the left-hand column, box2 the right.
+  \setbox0=\vsplit255 to\dimen@ \setbox2=\vsplit255 to\dimen@
+  \onepageout\pagesofar
+  \unvbox255
+  \penalty\outputpenalty
+}
+%
+% Re-output the contents of the output page -- any previous material,
+% followed by the two boxes we just split, in box0 and box2.
+\def\pagesofar{%
+  \unvbox\partialpage
+  %
+  \hsize = \doublecolumnhsize
+  \wd0=\hsize \wd2=\hsize
+  \hbox to\pagewidth{\box0\hfil\box2}%
+}
+%
+% All done with double columns.
+\def\enddoublecolumns{%
+  \output = {%
+    % Split the last of the double-column material.  Leave it on the
+    % current page, no automatic page break.
+    \balancecolumns
+    %
+    % If we end up splitting too much material for the current page,
+    % though, there will be another page break right after this \output
+    % invocation ends.  Having called \balancecolumns once, we do not
+    % want to call it again.  Therefore, reset \output to its normal
+    % definition right away.  (We hope \balancecolumns will never be
+    % called on to balance too much material, but if it is, this makes
+    % the output somewhat more palatable.)
+    \global\output = {\onepageout{\pagecontents\PAGE}}%
+  }%
+  \eject
+  \endgroup % started in \begindoublecolumns
+  %
+  % \pagegoal was set to the doubled \vsize above, since we restarted
+  % the current page.  We're now back to normal single-column
+  % typesetting, so reset \pagegoal to the normal \vsize (after the
+  % \endgroup where \vsize got restored).
+  \pagegoal = \vsize
+}
+%
+% Called at the end of the double column material.
+\def\balancecolumns{%
+  \setbox0 = \vbox{\unvbox255}% like \box255 but more efficient, see p.120.
+  \dimen@ = \ht0
+  \advance\dimen@ by \topskip
+  \advance\dimen@ by-\baselineskip
+  \divide\dimen@ by 2 % target to split to
+  %debug\message{final 2-column material height=\the\ht0, target=\the\dimen@.}%
+  \splittopskip = \topskip
+  % Loop until we get a decent breakpoint.
+  {%
+    \vbadness = 10000
+    \loop
+      \global\setbox3 = \copy0
+      \global\setbox1 = \vsplit3 to \dimen@
+    \ifdim\ht3>\dimen@
+      \global\advance\dimen@ by 1pt
+    \repeat
+  }%
+  %debug\message{split to \the\dimen@, column heights: \the\ht1, \the\ht3.}%
+  \setbox0=\vbox to\dimen@{\unvbox1}%
+  \setbox2=\vbox to\dimen@{\unvbox3}%
+  %
+  \pagesofar
+}
+\catcode`\@ = \other
+
+
+\message{sectioning,}
+% Chapters, sections, etc.
+
+% \unnumberedno is an oxymoron, of course.  But we count the unnumbered
+% sections so that we can refer to them unambiguously in the pdf
+% outlines by their "section number".  We avoid collisions with chapter
+% numbers by starting them at 10000.  (If a document ever has 10000
+% chapters, we're in trouble anyway, I'm sure.)
+\newcount\unnumberedno \unnumberedno = 10000
+\newcount\chapno
+\newcount\secno        \secno=0
+\newcount\subsecno     \subsecno=0
+\newcount\subsubsecno  \subsubsecno=0
+
+% This counter is funny since it counts through charcodes of letters A, B, ...
+\newcount\appendixno  \appendixno = `\@
+%
+% \def\appendixletter{\char\the\appendixno}
+% We do the following ugly conditional instead of the above simple
+% construct for the sake of pdftex, which needs the actual
+% letter in the expansion, not just typeset.
+%
+\def\appendixletter{%
+  \ifnum\appendixno=`A A%
+  \else\ifnum\appendixno=`B B%
+  \else\ifnum\appendixno=`C C%
+  \else\ifnum\appendixno=`D D%
+  \else\ifnum\appendixno=`E E%
+  \else\ifnum\appendixno=`F F%
+  \else\ifnum\appendixno=`G G%
+  \else\ifnum\appendixno=`H H%
+  \else\ifnum\appendixno=`I I%
+  \else\ifnum\appendixno=`J J%
+  \else\ifnum\appendixno=`K K%
+  \else\ifnum\appendixno=`L L%
+  \else\ifnum\appendixno=`M M%
+  \else\ifnum\appendixno=`N N%
+  \else\ifnum\appendixno=`O O%
+  \else\ifnum\appendixno=`P P%
+  \else\ifnum\appendixno=`Q Q%
+  \else\ifnum\appendixno=`R R%
+  \else\ifnum\appendixno=`S S%
+  \else\ifnum\appendixno=`T T%
+  \else\ifnum\appendixno=`U U%
+  \else\ifnum\appendixno=`V V%
+  \else\ifnum\appendixno=`W W%
+  \else\ifnum\appendixno=`X X%
+  \else\ifnum\appendixno=`Y Y%
+  \else\ifnum\appendixno=`Z Z%
+  % The \the is necessary, despite appearances, because \appendixletter is
+  % expanded while writing the .toc file.  \char\appendixno is not
+  % expandable, thus it is written literally, thus all appendixes come out
+  % with the same letter (or @) in the toc without it.
+  \else\char\the\appendixno
+  \fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi
+  \fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi}
+
+% Each @chapter defines this as the name of the chapter.
+% page headings and footings can use it.  @section does likewise.
+% However, they are not reliable, because we don't use marks.
+\def\thischapter{}
+\def\thissection{}
+
+\newcount\absseclevel % used to calculate proper heading level
+\newcount\secbase\secbase=0 % @raisesections/@lowersections modify this count
+
+% @raisesections: treat @section as chapter, @subsection as section, etc.
+\def\raisesections{\global\advance\secbase by -1}
+\let\up=\raisesections % original BFox name
+
+% @lowersections: treat @chapter as section, @section as subsection, etc.
+\def\lowersections{\global\advance\secbase by 1}
+\let\down=\lowersections % original BFox name
+
+% we only have subsub.
+\chardef\maxseclevel = 3
+%
+% A numbered section within an unnumbered changes to unnumbered too.
+% To achive this, remember the "biggest" unnum. sec. we are currently in:
+\chardef\unmlevel = \maxseclevel
+%
+% Trace whether the current chapter is an appendix or not:
+% \chapheadtype is "N" or "A", unnumbered chapters are ignored.
+\def\chapheadtype{N}
+
+% Choose a heading macro
+% #1 is heading type
+% #2 is heading level
+% #3 is text for heading
+\def\genhead#1#2#3{%
+  % Compute the abs. sec. level:
+  \absseclevel=#2
+  \advance\absseclevel by \secbase
+  % Make sure \absseclevel doesn't fall outside the range:
+  \ifnum \absseclevel < 0
+    \absseclevel = 0
+  \else
+    \ifnum \absseclevel > 3
+      \absseclevel = 3
+    \fi
+  \fi
+  % The heading type:
+  \def\headtype{#1}%
+  \if \headtype U%
+    \ifnum \absseclevel < \unmlevel
+      \chardef\unmlevel = \absseclevel
+    \fi
+  \else
+    % Check for appendix sections:
+    \ifnum \absseclevel = 0
+      \edef\chapheadtype{\headtype}%
+    \else
+      \if \headtype A\if \chapheadtype N%
+	\errmessage{@appendix... within a non-appendix chapter}%
+      \fi\fi
+    \fi
+    % Check for numbered within unnumbered:
+    \ifnum \absseclevel > \unmlevel
+      \def\headtype{U}%
+    \else
+      \chardef\unmlevel = 3
+    \fi
+  \fi
+  % Now print the heading:
+  \if \headtype U%
+    \ifcase\absseclevel
+	\unnumberedzzz{#3}%
+    \or \unnumberedseczzz{#3}%
+    \or \unnumberedsubseczzz{#3}%
+    \or \unnumberedsubsubseczzz{#3}%
+    \fi
+  \else
+    \if \headtype A%
+      \ifcase\absseclevel
+	  \appendixzzz{#3}%
+      \or \appendixsectionzzz{#3}%
+      \or \appendixsubseczzz{#3}%
+      \or \appendixsubsubseczzz{#3}%
+      \fi
+    \else
+      \ifcase\absseclevel
+	  \chapterzzz{#3}%
+      \or \seczzz{#3}%
+      \or \numberedsubseczzz{#3}%
+      \or \numberedsubsubseczzz{#3}%
+      \fi
+    \fi
+  \fi
+  \suppressfirstparagraphindent
+}
+
+% an interface:
+\def\numhead{\genhead N}
+\def\apphead{\genhead A}
+\def\unnmhead{\genhead U}
+
+% @chapter, @appendix, @unnumbered.  Increment top-level counter, reset
+% all lower-level sectioning counters to zero.
+%
+% Also set \chaplevelprefix, which we prepend to @float sequence numbers
+% (e.g., figures), q.v.  By default (before any chapter), that is empty.
+\let\chaplevelprefix = \empty
+%
+\outer\parseargdef\chapter{\numhead0{#1}} % normally numhead0 calls chapterzzz
+\def\chapterzzz#1{%
+  % section resetting is \global in case the chapter is in a group, such
+  % as an @include file.
+  \global\secno=0 \global\subsecno=0 \global\subsubsecno=0
+    \global\advance\chapno by 1
+  %
+  % Used for \float.
+  \gdef\chaplevelprefix{\the\chapno.}%
+  \resetallfloatnos
+  %
+  \message{\putwordChapter\space \the\chapno}%
+  %
+  % Write the actual heading.
+  \chapmacro{#1}{Ynumbered}{\the\chapno}%
+  %
+  % So @section and the like are numbered underneath this chapter.
+  \global\let\section = \numberedsec
+  \global\let\subsection = \numberedsubsec
+  \global\let\subsubsection = \numberedsubsubsec
+}
+
+\outer\parseargdef\appendix{\apphead0{#1}} % normally apphead0 calls appendixzzz
+\def\appendixzzz#1{%
+  \global\secno=0 \global\subsecno=0 \global\subsubsecno=0
+    \global\advance\appendixno by 1
+  \gdef\chaplevelprefix{\appendixletter.}%
+  \resetallfloatnos
+  %
+  \def\appendixnum{\putwordAppendix\space \appendixletter}%
+  \message{\appendixnum}%
+  %
+  \chapmacro{#1}{Yappendix}{\appendixletter}%
+  %
+  \global\let\section = \appendixsec
+  \global\let\subsection = \appendixsubsec
+  \global\let\subsubsection = \appendixsubsubsec
+}
+
+\outer\parseargdef\unnumbered{\unnmhead0{#1}} % normally unnmhead0 calls unnumberedzzz
+\def\unnumberedzzz#1{%
+  \global\secno=0 \global\subsecno=0 \global\subsubsecno=0
+    \global\advance\unnumberedno by 1
+  %
+  % Since an unnumbered has no number, no prefix for figures.
+  \global\let\chaplevelprefix = \empty
+  \resetallfloatnos
+  %
+  % This used to be simply \message{#1}, but TeX fully expands the
+  % argument to \message.  Therefore, if #1 contained @-commands, TeX
+  % expanded them.  For example, in `@unnumbered The @cite{Book}', TeX
+  % expanded @cite (which turns out to cause errors because \cite is meant
+  % to be executed, not expanded).
+  %
+  % Anyway, we don't want the fully-expanded definition of @cite to appear
+  % as a result of the \message, we just want `@cite' itself.  We use
+  % \the<toks register> to achieve this: TeX expands \the<toks> only once,
+  % simply yielding the contents of <toks register>.  (We also do this for
+  % the toc entries.)
+  \toks0 = {#1}%
+  \message{(\the\toks0)}%
+  %
+  \chapmacro{#1}{Ynothing}{\the\unnumberedno}%
+  %
+  \global\let\section = \unnumberedsec
+  \global\let\subsection = \unnumberedsubsec
+  \global\let\subsubsection = \unnumberedsubsubsec
+}
+
+% @centerchap is like @unnumbered, but the heading is centered.
+\outer\parseargdef\centerchap{%
+  % Well, we could do the following in a group, but that would break
+  % an assumption that \chapmacro is called at the outermost level.
+  % Thus we are safer this way:		--kasal, 24feb04
+  \let\centerparametersmaybe = \centerparameters
+  \unnmhead0{#1}%
+  \let\centerparametersmaybe = \relax
+}
+
+% @top is like @unnumbered.
+\let\top\unnumbered
+
+% Sections.
+\outer\parseargdef\numberedsec{\numhead1{#1}} % normally calls seczzz
+\def\seczzz#1{%
+  \global\subsecno=0 \global\subsubsecno=0  \global\advance\secno by 1
+  \sectionheading{#1}{sec}{Ynumbered}{\the\chapno.\the\secno}%
+}
+
+\outer\parseargdef\appendixsection{\apphead1{#1}} % normally calls appendixsectionzzz
+\def\appendixsectionzzz#1{%
+  \global\subsecno=0 \global\subsubsecno=0  \global\advance\secno by 1
+  \sectionheading{#1}{sec}{Yappendix}{\appendixletter.\the\secno}%
+}
+\let\appendixsec\appendixsection
+
+\outer\parseargdef\unnumberedsec{\unnmhead1{#1}} % normally calls unnumberedseczzz
+\def\unnumberedseczzz#1{%
+  \global\subsecno=0 \global\subsubsecno=0  \global\advance\secno by 1
+  \sectionheading{#1}{sec}{Ynothing}{\the\unnumberedno.\the\secno}%
+}
+
+% Subsections.
+\outer\parseargdef\numberedsubsec{\numhead2{#1}} % normally calls numberedsubseczzz
+\def\numberedsubseczzz#1{%
+  \global\subsubsecno=0  \global\advance\subsecno by 1
+  \sectionheading{#1}{subsec}{Ynumbered}{\the\chapno.\the\secno.\the\subsecno}%
+}
+
+\outer\parseargdef\appendixsubsec{\apphead2{#1}} % normally calls appendixsubseczzz
+\def\appendixsubseczzz#1{%
+  \global\subsubsecno=0  \global\advance\subsecno by 1
+  \sectionheading{#1}{subsec}{Yappendix}%
+                 {\appendixletter.\the\secno.\the\subsecno}%
+}
+
+\outer\parseargdef\unnumberedsubsec{\unnmhead2{#1}} %normally calls unnumberedsubseczzz
+\def\unnumberedsubseczzz#1{%
+  \global\subsubsecno=0  \global\advance\subsecno by 1
+  \sectionheading{#1}{subsec}{Ynothing}%
+                 {\the\unnumberedno.\the\secno.\the\subsecno}%
+}
+
+% Subsubsections.
+\outer\parseargdef\numberedsubsubsec{\numhead3{#1}} % normally numberedsubsubseczzz
+\def\numberedsubsubseczzz#1{%
+  \global\advance\subsubsecno by 1
+  \sectionheading{#1}{subsubsec}{Ynumbered}%
+                 {\the\chapno.\the\secno.\the\subsecno.\the\subsubsecno}%
+}
+
+\outer\parseargdef\appendixsubsubsec{\apphead3{#1}} % normally appendixsubsubseczzz
+\def\appendixsubsubseczzz#1{%
+  \global\advance\subsubsecno by 1
+  \sectionheading{#1}{subsubsec}{Yappendix}%
+                 {\appendixletter.\the\secno.\the\subsecno.\the\subsubsecno}%
+}
+
+\outer\parseargdef\unnumberedsubsubsec{\unnmhead3{#1}} %normally unnumberedsubsubseczzz
+\def\unnumberedsubsubseczzz#1{%
+  \global\advance\subsubsecno by 1
+  \sectionheading{#1}{subsubsec}{Ynothing}%
+                 {\the\unnumberedno.\the\secno.\the\subsecno.\the\subsubsecno}%
+}
+
+% These macros control what the section commands do, according
+% to what kind of chapter we are in (ordinary, appendix, or unnumbered).
+% Define them by default for a numbered chapter.
+\let\section = \numberedsec
+\let\subsection = \numberedsubsec
+\let\subsubsection = \numberedsubsubsec
+
+% Define @majorheading, @heading and @subheading
+
+% NOTE on use of \vbox for chapter headings, section headings, and such:
+%       1) We use \vbox rather than the earlier \line to permit
+%          overlong headings to fold.
+%       2) \hyphenpenalty is set to 10000 because hyphenation in a
+%          heading is obnoxious; this forbids it.
+%       3) Likewise, headings look best if no \parindent is used, and
+%          if justification is not attempted.  Hence \raggedright.
+
+
+\def\majorheading{%
+  {\advance\chapheadingskip by 10pt \chapbreak }%
+  \parsearg\chapheadingzzz
+}
+
+\def\chapheading{\chapbreak \parsearg\chapheadingzzz}
+\def\chapheadingzzz#1{%
+  {\chapfonts \vbox{\hyphenpenalty=10000\tolerance=5000
+                    \parindent=0pt\raggedright
+                    \rm #1\hfill}}%
+  \bigskip \par\penalty 200\relax
+  \suppressfirstparagraphindent
+}
+
+% @heading, @subheading, @subsubheading.
+\parseargdef\heading{\sectionheading{#1}{sec}{Yomitfromtoc}{}
+  \suppressfirstparagraphindent}
+\parseargdef\subheading{\sectionheading{#1}{subsec}{Yomitfromtoc}{}
+  \suppressfirstparagraphindent}
+\parseargdef\subsubheading{\sectionheading{#1}{subsubsec}{Yomitfromtoc}{}
+  \suppressfirstparagraphindent}
+
+% These macros generate a chapter, section, etc. heading only
+% (including whitespace, linebreaking, etc. around it),
+% given all the information in convenient, parsed form.
+
+%%% Args are the skip and penalty (usually negative)
+\def\dobreak#1#2{\par\ifdim\lastskip<#1\removelastskip\penalty#2\vskip#1\fi}
+
+%%% Define plain chapter starts, and page on/off switching for it
+% Parameter controlling skip before chapter headings (if needed)
+
+\newskip\chapheadingskip
+
+\def\chapbreak{\dobreak \chapheadingskip {-4000}}
+\def\chappager{\par\vfill\supereject}
+\def\chapoddpage{\chappager \ifodd\pageno \else \hbox to 0pt{} \chappager\fi}
+
+\def\setchapternewpage #1 {\csname CHAPPAG#1\endcsname}
+
+\def\CHAPPAGoff{%
+\global\let\contentsalignmacro = \chappager
+\global\let\pchapsepmacro=\chapbreak
+\global\let\pagealignmacro=\chappager}
+
+\def\CHAPPAGon{%
+\global\let\contentsalignmacro = \chappager
+\global\let\pchapsepmacro=\chappager
+\global\let\pagealignmacro=\chappager
+\global\def\HEADINGSon{\HEADINGSsingle}}
+
+\def\CHAPPAGodd{%
+\global\let\contentsalignmacro = \chapoddpage
+\global\let\pchapsepmacro=\chapoddpage
+\global\let\pagealignmacro=\chapoddpage
+\global\def\HEADINGSon{\HEADINGSdouble}}
+
+\CHAPPAGon
+
+% Chapter opening.
+%
+% #1 is the text, #2 is the section type (Ynumbered, Ynothing,
+% Yappendix, Yomitfromtoc), #3 the chapter number.
+%
+% To test against our argument.
+\def\Ynothingkeyword{Ynothing}
+\def\Yomitfromtockeyword{Yomitfromtoc}
+\def\Yappendixkeyword{Yappendix}
+%
+\def\chapmacro#1#2#3{%
+  \pchapsepmacro
+  {%
+    \chapfonts \rm
+    %
+    % Have to define \thissection before calling \donoderef, because the
+    % xref code eventually uses it.  On the other hand, it has to be called
+    % after \pchapsepmacro, or the headline will change too soon.
+    \gdef\thissection{#1}%
+    \gdef\thischaptername{#1}%
+    %
+    % Only insert the separating space if we have a chapter/appendix
+    % number, and don't print the unnumbered ``number''.
+    \def\temptype{#2}%
+    \ifx\temptype\Ynothingkeyword
+      \setbox0 = \hbox{}%
+      \def\toctype{unnchap}%
+      \def\thischapter{#1}%
+    \else\ifx\temptype\Yomitfromtockeyword
+      \setbox0 = \hbox{}% contents like unnumbered, but no toc entry
+      \def\toctype{omit}%
+      \xdef\thischapter{}%
+    \else\ifx\temptype\Yappendixkeyword
+      \setbox0 = \hbox{\putwordAppendix{} #3\enspace}%
+      \def\toctype{app}%
+      % We don't substitute the actual chapter name into \thischapter
+      % because we don't want its macros evaluated now.  And we don't
+      % use \thissection because that changes with each section.
+      %
+      \xdef\thischapter{\putwordAppendix{} \appendixletter:
+                        \noexpand\thischaptername}%
+    \else
+      \setbox0 = \hbox{#3\enspace}%
+      \def\toctype{numchap}%
+      \xdef\thischapter{\putwordChapter{} \the\chapno:
+                        \noexpand\thischaptername}%
+    \fi\fi\fi
+    %
+    % Write the toc entry for this chapter.  Must come before the
+    % \donoderef, because we include the current node name in the toc
+    % entry, and \donoderef resets it to empty.
+    \writetocentry{\toctype}{#1}{#3}%
+    %
+    % For pdftex, we have to write out the node definition (aka, make
+    % the pdfdest) after any page break, but before the actual text has
+    % been typeset.  If the destination for the pdf outline is after the
+    % text, then jumping from the outline may wind up with the text not
+    % being visible, for instance under high magnification.
+    \donoderef{#2}%
+    %
+    % Typeset the actual heading.
+    \vbox{\hyphenpenalty=10000 \tolerance=5000 \parindent=0pt \raggedright
+          \hangindent=\wd0 \centerparametersmaybe
+          \unhbox0 #1\par}%
+  }%
+  \nobreak\bigskip % no page break after a chapter title
+  \nobreak
+}
+
+% @centerchap -- centered and unnumbered.
+\let\centerparametersmaybe = \relax
+\def\centerparameters{%
+  \advance\rightskip by 3\rightskip
+  \leftskip = \rightskip
+  \parfillskip = 0pt
+}
+
+
+% I don't think this chapter style is supported any more, so I'm not
+% updating it with the new noderef stuff.  We'll see.  --karl, 11aug03.
+%
+\def\setchapterstyle #1 {\csname CHAPF#1\endcsname}
+%
+\def\unnchfopen #1{%
+\chapoddpage {\chapfonts \vbox{\hyphenpenalty=10000\tolerance=5000
+                       \parindent=0pt\raggedright
+                       \rm #1\hfill}}\bigskip \par\nobreak
+}
+\def\chfopen #1#2{\chapoddpage {\chapfonts
+\vbox to 3in{\vfil \hbox to\hsize{\hfil #2} \hbox to\hsize{\hfil #1} \vfil}}%
+\par\penalty 5000 %
+}
+\def\centerchfopen #1{%
+\chapoddpage {\chapfonts \vbox{\hyphenpenalty=10000\tolerance=5000
+                       \parindent=0pt
+                       \hfill {\rm #1}\hfill}}\bigskip \par\nobreak
+}
+\def\CHAPFopen{%
+  \global\let\chapmacro=\chfopen
+  \global\let\centerchapmacro=\centerchfopen}
+
+
+% Section titles.  These macros combine the section number parts and
+% call the generic \sectionheading to do the printing.
+%
+\newskip\secheadingskip
+\def\secheadingbreak{\dobreak \secheadingskip{-1000}}
+
+% Subsection titles.
+\newskip\subsecheadingskip
+\def\subsecheadingbreak{\dobreak \subsecheadingskip{-500}}
+
+% Subsubsection titles.
+\def\subsubsecheadingskip{\subsecheadingskip}
+\def\subsubsecheadingbreak{\subsecheadingbreak}
+
+
+% Print any size, any type, section title.
+%
+% #1 is the text, #2 is the section level (sec/subsec/subsubsec), #3 is
+% the section type for xrefs (Ynumbered, Ynothing, Yappendix), #4 is the
+% section number.
+%
+\def\sectionheading#1#2#3#4{%
+  {%
+    % Switch to the right set of fonts.
+    \csname #2fonts\endcsname \rm
+    %
+    % Insert space above the heading.
+    \csname #2headingbreak\endcsname
+    %
+    % Only insert the space after the number if we have a section number.
+    \def\sectionlevel{#2}%
+    \def\temptype{#3}%
+    %
+    \ifx\temptype\Ynothingkeyword
+      \setbox0 = \hbox{}%
+      \def\toctype{unn}%
+      \gdef\thissection{#1}%
+    \else\ifx\temptype\Yomitfromtockeyword
+      % for @headings -- no section number, don't include in toc,
+      % and don't redefine \thissection.
+      \setbox0 = \hbox{}%
+      \def\toctype{omit}%
+      \let\sectionlevel=\empty
+    \else\ifx\temptype\Yappendixkeyword
+      \setbox0 = \hbox{#4\enspace}%
+      \def\toctype{app}%
+      \gdef\thissection{#1}%
+    \else
+      \setbox0 = \hbox{#4\enspace}%
+      \def\toctype{num}%
+      \gdef\thissection{#1}%
+    \fi\fi\fi
+    %
+    % Write the toc entry (before \donoderef).  See comments in \chfplain.
+    \writetocentry{\toctype\sectionlevel}{#1}{#4}%
+    %
+    % Write the node reference (= pdf destination for pdftex).
+    % Again, see comments in \chfplain.
+    \donoderef{#3}%
+    %
+    % Output the actual section heading.
+    \vbox{\hyphenpenalty=10000 \tolerance=5000 \parindent=0pt \raggedright
+          \hangindent=\wd0  % zero if no section number
+          \unhbox0 #1}%
+  }%
+  % Add extra space after the heading -- half of whatever came above it.
+  % Don't allow stretch, though.
+  \kern .5 \csname #2headingskip\endcsname
+  %
+  % Do not let the kern be a potential breakpoint, as it would be if it
+  % was followed by glue.
+  \nobreak
+  %
+  % We'll almost certainly start a paragraph next, so don't let that
+  % glue accumulate.  (Not a breakpoint because it's preceded by a
+  % discardable item.)
+  \vskip-\parskip
+  % 
+  % This is purely so the last item on the list is a known \penalty >
+  % 10000.  This is so \startdefun can avoid allowing breakpoints after
+  % section headings.  Otherwise, it would insert a valid breakpoint between:
+  % 
+  %   @section sec-whatever
+  %   @deffn def-whatever
+  \penalty 10001
+}
+
+
+\message{toc,}
+% Table of contents.
+\newwrite\tocfile
+
+% Write an entry to the toc file, opening it if necessary.
+% Called from @chapter, etc.
+%
+% Example usage: \writetocentry{sec}{Section Name}{\the\chapno.\the\secno}
+% We append the current node name (if any) and page number as additional
+% arguments for the \{chap,sec,...}entry macros which will eventually
+% read this.  The node name is used in the pdf outlines as the
+% destination to jump to.
+%
+% We open the .toc file for writing here instead of at @setfilename (or
+% any other fixed time) so that @contents can be anywhere in the document.
+% But if #1 is `omit', then we don't do anything.  This is used for the
+% table of contents chapter openings themselves.
+%
+\newif\iftocfileopened
+\def\omitkeyword{omit}%
+%
+\def\writetocentry#1#2#3{%
+  \edef\writetoctype{#1}%
+  \ifx\writetoctype\omitkeyword \else
+    \iftocfileopened\else
+      \immediate\openout\tocfile = \jobname.toc
+      \global\tocfileopenedtrue
+    \fi
+    %
+    \iflinks
+      \toks0 = {#2}%
+      \toks2 = \expandafter{\lastnode}%
+      \edef\temp{\write\tocfile{\realbackslash #1entry{\the\toks0}{#3}%
+                               {\the\toks2}{\noexpand\folio}}}%
+      \temp
+    \fi
+  \fi
+  %
+  % Tell \shipout to create a pdf destination on each page, if we're
+  % writing pdf.  These are used in the table of contents.  We can't
+  % just write one on every page because the title pages are numbered
+  % 1 and 2 (the page numbers aren't printed), and so are the first
+  % two pages of the document.  Thus, we'd have two destinations named
+  % `1', and two named `2'.
+  \ifpdf \global\pdfmakepagedesttrue \fi
+}
+
+\newskip\contentsrightmargin \contentsrightmargin=1in
+\newcount\savepageno
+\newcount\lastnegativepageno \lastnegativepageno = -1
+
+% Prepare to read what we've written to \tocfile.
+%
+\def\startcontents#1{%
+  % If @setchapternewpage on, and @headings double, the contents should
+  % start on an odd page, unlike chapters.  Thus, we maintain
+  % \contentsalignmacro in parallel with \pagealignmacro.
+  % From: Torbjorn Granlund <tege@matematik.su.se>
+  \contentsalignmacro
+  \immediate\closeout\tocfile
+  %
+  % Don't need to put `Contents' or `Short Contents' in the headline.
+  % It is abundantly clear what they are.
+  \def\thischapter{}%
+  \chapmacro{#1}{Yomitfromtoc}{}%
+  %
+  \savepageno = \pageno
+  \begingroup                  % Set up to handle contents files properly.
+    \catcode`\\=0  \catcode`\{=1  \catcode`\}=2  \catcode`\@=11
+    % We can't do this, because then an actual ^ in a section
+    % title fails, e.g., @chapter ^ -- exponentiation.  --karl, 9jul97.
+    %\catcode`\^=7 % to see ^^e4 as \"a etc. juha@piuha.ydi.vtt.fi
+    \raggedbottom             % Worry more about breakpoints than the bottom.
+    \advance\hsize by -\contentsrightmargin % Don't use the full line length.
+    %
+    % Roman numerals for page numbers.
+    \ifnum \pageno>0 \global\pageno = \lastnegativepageno \fi
+}
+
+
+% Normal (long) toc.
+\def\contents{%
+  \startcontents{\putwordTOC}%
+    \openin 1 \jobname.toc
+    \ifeof 1 \else
+      \input \jobname.toc
+    \fi
+    \vfill \eject
+    \contentsalignmacro % in case @setchapternewpage odd is in effect
+    \ifeof 1 \else
+      \pdfmakeoutlines
+    \fi
+    \closein 1
+  \endgroup
+  \lastnegativepageno = \pageno
+  \global\pageno = \savepageno
+}
+
+% And just the chapters.
+\def\summarycontents{%
+  \startcontents{\putwordShortTOC}%
+    %
+    \let\numchapentry = \shortchapentry
+    \let\appentry = \shortchapentry
+    \let\unnchapentry = \shortunnchapentry
+    % We want a true roman here for the page numbers.
+    \secfonts
+    \let\rm=\shortcontrm \let\bf=\shortcontbf
+    \let\sl=\shortcontsl \let\tt=\shortconttt
+    \rm
+    \hyphenpenalty = 10000
+    \advance\baselineskip by 1pt % Open it up a little.
+    \def\numsecentry##1##2##3##4{}
+    \let\appsecentry = \numsecentry
+    \let\unnsecentry = \numsecentry
+    \let\numsubsecentry = \numsecentry
+    \let\appsubsecentry = \numsecentry
+    \let\unnsubsecentry = \numsecentry
+    \let\numsubsubsecentry = \numsecentry
+    \let\appsubsubsecentry = \numsecentry
+    \let\unnsubsubsecentry = \numsecentry
+    \openin 1 \jobname.toc
+    \ifeof 1 \else
+      \input \jobname.toc
+    \fi
+    \closein 1
+    \vfill \eject
+    \contentsalignmacro % in case @setchapternewpage odd is in effect
+  \endgroup
+  \lastnegativepageno = \pageno
+  \global\pageno = \savepageno
+}
+\let\shortcontents = \summarycontents
+
+% Typeset the label for a chapter or appendix for the short contents.
+% The arg is, e.g., `A' for an appendix, or `3' for a chapter.
+%
+\def\shortchaplabel#1{%
+  % This space should be enough, since a single number is .5em, and the
+  % widest letter (M) is 1em, at least in the Computer Modern fonts.
+  % But use \hss just in case.
+  % (This space doesn't include the extra space that gets added after
+  % the label; that gets put in by \shortchapentry above.)
+  %
+  % We'd like to right-justify chapter numbers, but that looks strange
+  % with appendix letters.  And right-justifying numbers and
+  % left-justifying letters looks strange when there is less than 10
+  % chapters.  Have to read the whole toc once to know how many chapters
+  % there are before deciding ...
+  \hbox to 1em{#1\hss}%
+}
+
+% These macros generate individual entries in the table of contents.
+% The first argument is the chapter or section name.
+% The last argument is the page number.
+% The arguments in between are the chapter number, section number, ...
+
+% Chapters, in the main contents.
+\def\numchapentry#1#2#3#4{\dochapentry{#2\labelspace#1}{#4}}
+%
+% Chapters, in the short toc.
+% See comments in \dochapentry re vbox and related settings.
+\def\shortchapentry#1#2#3#4{%
+  \tocentry{\shortchaplabel{#2}\labelspace #1}{\doshortpageno\bgroup#4\egroup}%
+}
+
+% Appendices, in the main contents.
+% Need the word Appendix, and a fixed-size box.
+%
+\def\appendixbox#1{%
+  % We use M since it's probably the widest letter.
+  \setbox0 = \hbox{\putwordAppendix{} M}%
+  \hbox to \wd0{\putwordAppendix{} #1\hss}}
+%
+\def\appentry#1#2#3#4{\dochapentry{\appendixbox{#2}\labelspace#1}{#4}}
+
+% Unnumbered chapters.
+\def\unnchapentry#1#2#3#4{\dochapentry{#1}{#4}}
+\def\shortunnchapentry#1#2#3#4{\tocentry{#1}{\doshortpageno\bgroup#4\egroup}}
+
+% Sections.
+\def\numsecentry#1#2#3#4{\dosecentry{#2\labelspace#1}{#4}}
+\let\appsecentry=\numsecentry
+\def\unnsecentry#1#2#3#4{\dosecentry{#1}{#4}}
+
+% Subsections.
+\def\numsubsecentry#1#2#3#4{\dosubsecentry{#2\labelspace#1}{#4}}
+\let\appsubsecentry=\numsubsecentry
+\def\unnsubsecentry#1#2#3#4{\dosubsecentry{#1}{#4}}
+
+% And subsubsections.
+\def\numsubsubsecentry#1#2#3#4{\dosubsubsecentry{#2\labelspace#1}{#4}}
+\let\appsubsubsecentry=\numsubsubsecentry
+\def\unnsubsubsecentry#1#2#3#4{\dosubsubsecentry{#1}{#4}}
+
+% This parameter controls the indentation of the various levels.
+% Same as \defaultparindent.
+\newdimen\tocindent \tocindent = 15pt
+
+% Now for the actual typesetting. In all these, #1 is the text and #2 is the
+% page number.
+%
+% If the toc has to be broken over pages, we want it to be at chapters
+% if at all possible; hence the \penalty.
+\def\dochapentry#1#2{%
+   \penalty-300 \vskip1\baselineskip plus.33\baselineskip minus.25\baselineskip
+   \begingroup
+     \chapentryfonts
+     \tocentry{#1}{\dopageno\bgroup#2\egroup}%
+   \endgroup
+   \nobreak\vskip .25\baselineskip plus.1\baselineskip
+}
+
+\def\dosecentry#1#2{\begingroup
+  \secentryfonts \leftskip=\tocindent
+  \tocentry{#1}{\dopageno\bgroup#2\egroup}%
+\endgroup}
+
+\def\dosubsecentry#1#2{\begingroup
+  \subsecentryfonts \leftskip=2\tocindent
+  \tocentry{#1}{\dopageno\bgroup#2\egroup}%
+\endgroup}
+
+\def\dosubsubsecentry#1#2{\begingroup
+  \subsubsecentryfonts \leftskip=3\tocindent
+  \tocentry{#1}{\dopageno\bgroup#2\egroup}%
+\endgroup}
+
+% We use the same \entry macro as for the index entries.
+\let\tocentry = \entry
+
+% Space between chapter (or whatever) number and the title.
+\def\labelspace{\hskip1em \relax}
+
+\def\dopageno#1{{\rm #1}}
+\def\doshortpageno#1{{\rm #1}}
+
+\def\chapentryfonts{\secfonts \rm}
+\def\secentryfonts{\textfonts}
+\def\subsecentryfonts{\textfonts}
+\def\subsubsecentryfonts{\textfonts}
+
+
+\message{environments,}
+% @foo ... @end foo.
+
+% @point{}, @result{}, @expansion{}, @print{}, @equiv{}.
+%
+% Since these characters are used in examples, it should be an even number of
+% \tt widths. Each \tt character is 1en, so two makes it 1em.
+%
+\def\point{$\star$}
+\def\result{\leavevmode\raise.15ex\hbox to 1em{\hfil$\Rightarrow$\hfil}}
+\def\expansion{\leavevmode\raise.1ex\hbox to 1em{\hfil$\mapsto$\hfil}}
+\def\print{\leavevmode\lower.1ex\hbox to 1em{\hfil$\dashv$\hfil}}
+\def\equiv{\leavevmode\lower.1ex\hbox to 1em{\hfil$\ptexequiv$\hfil}}
+
+% The @error{} command.
+% Adapted from the TeXbook's \boxit.
+%
+\newbox\errorbox
+%
+{\tentt \global\dimen0 = 3em}% Width of the box.
+\dimen2 = .55pt % Thickness of rules
+% The text. (`r' is open on the right, `e' somewhat less so on the left.)
+\setbox0 = \hbox{\kern-.75pt \tensf error\kern-1.5pt}
+%
+\setbox\errorbox=\hbox to \dimen0{\hfil
+   \hsize = \dimen0 \advance\hsize by -5.8pt % Space to left+right.
+   \advance\hsize by -2\dimen2 % Rules.
+   \vbox{%
+      \hrule height\dimen2
+      \hbox{\vrule width\dimen2 \kern3pt          % Space to left of text.
+         \vtop{\kern2.4pt \box0 \kern2.4pt}% Space above/below.
+         \kern3pt\vrule width\dimen2}% Space to right.
+      \hrule height\dimen2}
+    \hfil}
+%
+\def\error{\leavevmode\lower.7ex\copy\errorbox}
+
+% @tex ... @end tex    escapes into raw Tex temporarily.
+% One exception: @ is still an escape character, so that @end tex works.
+% But \@ or @@ will get a plain tex @ character.
+
+\envdef\tex{%
+  \catcode `\\=0 \catcode `\{=1 \catcode `\}=2
+  \catcode `\$=3 \catcode `\&=4 \catcode `\#=6
+  \catcode `\^=7 \catcode `\_=8 \catcode `\~=\active \let~=\tie
+  \catcode `\%=14
+  \catcode `\+=\other
+  \catcode `\"=\other
+  \catcode `\|=\other
+  \catcode `\<=\other
+  \catcode `\>=\other
+  \escapechar=`\\
+  %
+  \let\b=\ptexb
+  \let\bullet=\ptexbullet
+  \let\c=\ptexc
+  \let\,=\ptexcomma
+  \let\.=\ptexdot
+  \let\dots=\ptexdots
+  \let\equiv=\ptexequiv
+  \let\!=\ptexexclam
+  \let\i=\ptexi
+  \let\indent=\ptexindent
+  \let\noindent=\ptexnoindent
+  \let\{=\ptexlbrace
+  \let\+=\tabalign
+  \let\}=\ptexrbrace
+  \let\/=\ptexslash
+  \let\*=\ptexstar
+  \let\t=\ptext
+  %
+  \def\endldots{\mathinner{\ldots\ldots\ldots\ldots}}%
+  \def\enddots{\relax\ifmmode\endldots\else$\mathsurround=0pt \endldots\,$\fi}%
+  \def\@{@}%
+}
+% There is no need to define \Etex.
+
+% Define @lisp ... @end lisp.
+% @lisp environment forms a group so it can rebind things,
+% including the definition of @end lisp (which normally is erroneous).
+
+% Amount to narrow the margins by for @lisp.
+\newskip\lispnarrowing \lispnarrowing=0.4in
+
+% This is the definition that ^^M gets inside @lisp, @example, and other
+% such environments.  \null is better than a space, since it doesn't
+% have any width.
+\def\lisppar{\null\endgraf}
+
+% This space is always present above and below environments.
+\newskip\envskipamount \envskipamount = 0pt
+
+% Make spacing and below environment symmetrical.  We use \parskip here
+% to help in doing that, since in @example-like environments \parskip
+% is reset to zero; thus the \afterenvbreak inserts no space -- but the
+% start of the next paragraph will insert \parskip.
+%
+\def\aboveenvbreak{{%
+  % =10000 instead of <10000 because of a special case in \itemzzz and
+  % \sectionheading, q.v.
+  \ifnum \lastpenalty=10000 \else
+    \advance\envskipamount by \parskip
+    \endgraf
+    \ifdim\lastskip<\envskipamount
+      \removelastskip
+      % it's not a good place to break if the last penalty was \nobreak
+      % or better ...
+      \ifnum\lastpenalty<10000 \penalty-50 \fi
+      \vskip\envskipamount
+    \fi
+  \fi
+}}
+
+\let\afterenvbreak = \aboveenvbreak
+
+% \nonarrowing is a flag.  If "set", @lisp etc don't narrow margins.
+\let\nonarrowing=\relax
+
+% @cartouche ... @end cartouche: draw rectangle w/rounded corners around
+% environment contents.
+\font\circle=lcircle10
+\newdimen\circthick
+\newdimen\cartouter\newdimen\cartinner
+\newskip\normbskip\newskip\normpskip\newskip\normlskip
+\circthick=\fontdimen8\circle
+%
+\def\ctl{{\circle\char'013\hskip -6pt}}% 6pt from pl file: 1/2charwidth
+\def\ctr{{\hskip 6pt\circle\char'010}}
+\def\cbl{{\circle\char'012\hskip -6pt}}
+\def\cbr{{\hskip 6pt\circle\char'011}}
+\def\carttop{\hbox to \cartouter{\hskip\lskip
+        \ctl\leaders\hrule height\circthick\hfil\ctr
+        \hskip\rskip}}
+\def\cartbot{\hbox to \cartouter{\hskip\lskip
+        \cbl\leaders\hrule height\circthick\hfil\cbr
+        \hskip\rskip}}
+%
+\newskip\lskip\newskip\rskip
+
+\envdef\cartouche{%
+  \ifhmode\par\fi  % can't be in the midst of a paragraph.
+  \startsavinginserts
+  \lskip=\leftskip \rskip=\rightskip
+  \leftskip=0pt\rightskip=0pt % we want these *outside*.
+  \cartinner=\hsize \advance\cartinner by-\lskip
+  \advance\cartinner by-\rskip
+  \cartouter=\hsize
+  \advance\cartouter by 18.4pt	% allow for 3pt kerns on either
+				% side, and for 6pt waste from
+				% each corner char, and rule thickness
+  \normbskip=\baselineskip \normpskip=\parskip \normlskip=\lineskip
+  % Flag to tell @lisp, etc., not to narrow margin.
+  \let\nonarrowing=\comment
+  \vbox\bgroup
+      \baselineskip=0pt\parskip=0pt\lineskip=0pt
+      \carttop
+      \hbox\bgroup
+	  \hskip\lskip
+	  \vrule\kern3pt
+	  \vbox\bgroup
+	      \kern3pt
+	      \hsize=\cartinner
+	      \baselineskip=\normbskip
+	      \lineskip=\normlskip
+	      \parskip=\normpskip
+	      \vskip -\parskip
+	      \comment % For explanation, see the end of \def\group.
+}
+\def\Ecartouche{%
+              \ifhmode\par\fi
+	      \kern3pt
+	  \egroup
+	  \kern3pt\vrule
+	  \hskip\rskip
+      \egroup
+      \cartbot
+  \egroup
+  \checkinserts
+}
+
+
+% This macro is called at the beginning of all the @example variants,
+% inside a group.
+\def\nonfillstart{%
+  \aboveenvbreak
+  \hfuzz = 12pt % Don't be fussy
+  \sepspaces % Make spaces be word-separators rather than space tokens.
+  \let\par = \lisppar % don't ignore blank lines
+  \obeylines % each line of input is a line of output
+  \parskip = 0pt
+  \parindent = 0pt
+  \emergencystretch = 0pt % don't try to avoid overfull boxes
+  % @cartouche defines \nonarrowing to inhibit narrowing
+  % at next level down.
+  \ifx\nonarrowing\relax
+    \advance \leftskip by \lispnarrowing
+    \exdentamount=\lispnarrowing
+  \fi
+  \let\exdent=\nofillexdent
+}
+
+% If you want all examples etc. small: @set dispenvsize small.
+% If you want even small examples the full size: @set dispenvsize nosmall.
+% This affects the following displayed environments:
+%    @example, @display, @format, @lisp
+%
+\def\smallword{small}
+\def\nosmallword{nosmall}
+\let\SETdispenvsize\relax
+\def\setnormaldispenv{%
+  \ifx\SETdispenvsize\smallword
+    \smallexamplefonts \rm
+  \fi
+}
+\def\setsmalldispenv{%
+  \ifx\SETdispenvsize\nosmallword
+  \else
+    \smallexamplefonts \rm
+  \fi
+}
+
+% We often define two environments, @foo and @smallfoo.
+% Let's do it by one command:
+\def\makedispenv #1#2{
+  \expandafter\envdef\csname#1\endcsname {\setnormaldispenv #2}
+  \expandafter\envdef\csname small#1\endcsname {\setsmalldispenv #2}
+  \expandafter\let\csname E#1\endcsname \afterenvbreak
+  \expandafter\let\csname Esmall#1\endcsname \afterenvbreak
+}
+
+% Define two synonyms:
+\def\maketwodispenvs #1#2#3{
+  \makedispenv{#1}{#3}
+  \makedispenv{#2}{#3}
+}
+
+% @lisp: indented, narrowed, typewriter font; @example: same as @lisp.
+%
+% @smallexample and @smalllisp: use smaller fonts.
+% Originally contributed by Pavel@xerox.
+%
+\maketwodispenvs {lisp}{example}{%
+  \nonfillstart
+  \tt
+  \let\kbdfont = \kbdexamplefont % Allow @kbd to do something special.
+  \gobble       % eat return
+}
+
+% @display/@smalldisplay: same as @lisp except keep current font.
+%
+\makedispenv {display}{%
+  \nonfillstart
+  \gobble
+}
+
+% @format/@smallformat: same as @display except don't narrow margins.
+%
+\makedispenv{format}{%
+  \let\nonarrowing = t%
+  \nonfillstart
+  \gobble
+}
+
+% @flushleft: same as @format, but doesn't obey \SETdispenvsize.
+\envdef\flushleft{%
+  \let\nonarrowing = t%
+  \nonfillstart
+  \gobble
+}
+\let\Eflushleft = \afterenvbreak
+
+% @flushright.
+%
+\envdef\flushright{%
+  \let\nonarrowing = t%
+  \nonfillstart
+  \advance\leftskip by 0pt plus 1fill
+  \gobble
+}
+\let\Eflushright = \afterenvbreak
+
+
+% @quotation does normal linebreaking (hence we can't use \nonfillstart)
+% and narrows the margins.  We keep \parskip nonzero in general, since
+% we're doing normal filling.  So, when using \aboveenvbreak and
+% \afterenvbreak, temporarily make \parskip 0.
+%
+\envdef\quotation{%
+  {\parskip=0pt \aboveenvbreak}% because \aboveenvbreak inserts \parskip
+  \parindent=0pt
+  %
+  % @cartouche defines \nonarrowing to inhibit narrowing at next level down.
+  \ifx\nonarrowing\relax
+    \advance\leftskip by \lispnarrowing
+    \advance\rightskip by \lispnarrowing
+    \exdentamount = \lispnarrowing
+    \let\nonarrowing = \relax
+  \fi
+  \parsearg\quotationlabel
+}
+
+% We have retained a nonzero parskip for the environment, since we're
+% doing normal filling.
+%
+\def\Equotation{%
+  \par
+  \ifx\quotationauthor\undefined\else
+    % indent a bit.
+    \leftline{\kern 2\leftskip \sl ---\quotationauthor}%
+  \fi
+  {\parskip=0pt \afterenvbreak}%
+}
+
+% If we're given an argument, typeset it in bold with a colon after.
+\def\quotationlabel#1{%
+  \def\temp{#1}%
+  \ifx\temp\empty \else
+    {\bf #1: }%
+  \fi
+}
+
+
+% LaTeX-like @verbatim...@end verbatim and @verb{<char>...<char>}
+% If we want to allow any <char> as delimiter,
+% we need the curly braces so that makeinfo sees the @verb command, eg:
+% `@verbx...x' would look like the '@verbx' command.  --janneke@gnu.org
+%
+% [Knuth]: Donald Ervin Knuth, 1996.  The TeXbook.
+%
+% [Knuth] p.344; only we need to do the other characters Texinfo sets
+% active too.  Otherwise, they get lost as the first character on a
+% verbatim line.
+\def\dospecials{%
+  \do\ \do\\\do\{\do\}\do\$\do\&%
+  \do\#\do\^\do\^^K\do\_\do\^^A\do\%\do\~%
+  \do\<\do\>\do\|\do\@\do+\do\"%
+}
+%
+% [Knuth] p. 380
+\def\uncatcodespecials{%
+  \def\do##1{\catcode`##1=\other}\dospecials}
+%
+% [Knuth] pp. 380,381,391
+% Disable Spanish ligatures ?` and !` of \tt font
+\begingroup
+  \catcode`\`=\active\gdef`{\relax\lq}
+\endgroup
+%
+% Setup for the @verb command.
+%
+% Eight spaces for a tab
+\begingroup
+  \catcode`\^^I=\active
+  \gdef\tabeightspaces{\catcode`\^^I=\active\def^^I{\ \ \ \ \ \ \ \ }}
+\endgroup
+%
+\def\setupverb{%
+  \tt  % easiest (and conventionally used) font for verbatim
+  \def\par{\leavevmode\endgraf}%
+  \catcode`\`=\active
+  \tabeightspaces
+  % Respect line breaks,
+  % print special symbols as themselves, and
+  % make each space count
+  % must do in this order:
+  \obeylines \uncatcodespecials \sepspaces
+}
+
+% Setup for the @verbatim environment
+%
+% Real tab expansion
+\newdimen\tabw \setbox0=\hbox{\tt\space} \tabw=8\wd0 % tab amount
+%
+\def\starttabbox{\setbox0=\hbox\bgroup}
+\begingroup
+  \catcode`\^^I=\active
+  \gdef\tabexpand{%
+    \catcode`\^^I=\active
+    \def^^I{\leavevmode\egroup
+      \dimen0=\wd0 % the width so far, or since the previous tab
+      \divide\dimen0 by\tabw
+      \multiply\dimen0 by\tabw % compute previous multiple of \tabw
+      \advance\dimen0 by\tabw  % advance to next multiple of \tabw
+      \wd0=\dimen0 \box0 \starttabbox
+    }%
+  }
+\endgroup
+\def\setupverbatim{%
+  \nonfillstart
+  \advance\leftskip by -\defbodyindent
+  % Easiest (and conventionally used) font for verbatim
+  \tt
+  \def\par{\leavevmode\egroup\box0\endgraf}%
+  \catcode`\`=\active
+  \tabexpand
+  % Respect line breaks,
+  % print special symbols as themselves, and
+  % make each space count
+  % must do in this order:
+  \obeylines \uncatcodespecials \sepspaces
+  \everypar{\starttabbox}%
+}
+
+% Do the @verb magic: verbatim text is quoted by unique
+% delimiter characters.  Before first delimiter expect a
+% right brace, after last delimiter expect closing brace:
+%
+%    \def\doverb'{'<char>#1<char>'}'{#1}
+%
+% [Knuth] p. 382; only eat outer {}
+\begingroup
+  \catcode`[=1\catcode`]=2\catcode`\{=\other\catcode`\}=\other
+  \gdef\doverb{#1[\def\next##1#1}[##1\endgroup]\next]
+\endgroup
+%
+\def\verb{\begingroup\setupverb\doverb}
+%
+%
+% Do the @verbatim magic: define the macro \doverbatim so that
+% the (first) argument ends when '@end verbatim' is reached, ie:
+%
+%     \def\doverbatim#1@end verbatim{#1}
+%
+% For Texinfo it's a lot easier than for LaTeX,
+% because texinfo's \verbatim doesn't stop at '\end{verbatim}':
+% we need not redefine '\', '{' and '}'.
+%
+% Inspired by LaTeX's verbatim command set [latex.ltx]
+%
+\begingroup
+  \catcode`\ =\active
+  \obeylines %
+  % ignore everything up to the first ^^M, that's the newline at the end
+  % of the @verbatim input line itself.  Otherwise we get an extra blank
+  % line in the output.
+  \xdef\doverbatim#1^^M#2@end verbatim{#2\noexpand\end\gobble verbatim}%
+  % We really want {...\end verbatim} in the body of the macro, but
+  % without the active space; thus we have to use \xdef and \gobble.
+\endgroup
+%
+\envdef\verbatim{%
+    \setupverbatim\doverbatim
+}
+\let\Everbatim = \afterenvbreak
+
+
+% @verbatiminclude FILE - insert text of file in verbatim environment.
+%
+\def\verbatiminclude{\parseargusing\filenamecatcodes\doverbatiminclude}
+%
+\def\doverbatiminclude#1{%
+  {%
+    \makevalueexpandable
+    \setupverbatim
+    \input #1
+    \afterenvbreak
+  }%
+}
+
+% @copying ... @end copying.
+% Save the text away for @insertcopying later.
+%
+% We save the uninterpreted tokens, rather than creating a box.
+% Saving the text in a box would be much easier, but then all the
+% typesetting commands (@smallbook, font changes, etc.) have to be done
+% beforehand -- and a) we want @copying to be done first in the source
+% file; b) letting users define the frontmatter in as flexible order as
+% possible is very desirable.
+%
+\def\copying{\checkenv{}\begingroup\scanargctxt\docopying}
+\def\docopying#1@end copying{\endgroup\def\copyingtext{#1}}
+%
+\def\insertcopying{%
+  \begingroup
+    \parindent = 0pt  % paragraph indentation looks wrong on title page
+    \scanexp\copyingtext
+  \endgroup
+}
+
+\message{defuns,}
+% @defun etc.
+
+\newskip\defbodyindent \defbodyindent=.4in
+\newskip\defargsindent \defargsindent=50pt
+\newskip\deflastargmargin \deflastargmargin=18pt
+
+% Start the processing of @deffn:
+\def\startdefun{%
+  \ifnum\lastpenalty<10000
+    \medbreak
+  \else
+    % If there are two @def commands in a row, we'll have a \nobreak,
+    % which is there to keep the function description together with its
+    % header.  But if there's nothing but headers, we need to allow a
+    % break somewhere.  Check specifically for penalty 10002, inserted
+    % by \defargscommonending, instead of 10000, since the sectioning
+    % commands also insert a nobreak penalty, and we don't want to allow
+    % a break between a section heading and a defun.
+    % 
+    \ifnum\lastpenalty=10002 \penalty2000 \fi
+    %
+    % Similarly, after a section heading, do not allow a break.
+    % But do insert the glue.
+    \medskip  % preceded by discardable penalty, so not a breakpoint
+  \fi
+  %
+  \parindent=0in
+  \advance\leftskip by \defbodyindent
+  \exdentamount=\defbodyindent
+}
+
+\def\dodefunx#1{%
+  % First, check whether we are in the right environment:
+  \checkenv#1%
+  %
+  % As above, allow line break if we have multiple x headers in a row.
+  % It's not a great place, though.
+  \ifnum\lastpenalty=10002 \penalty3000 \fi
+  %
+  % And now, it's time to reuse the body of the original defun:
+  \expandafter\gobbledefun#1%
+}
+\def\gobbledefun#1\startdefun{}
+
+% \printdefunline \deffnheader{text}
+%
+\def\printdefunline#1#2{%
+  \begingroup
+    % call \deffnheader:
+    #1#2 \endheader
+    % common ending:
+    \interlinepenalty = 10000
+    \advance\rightskip by 0pt plus 1fil
+    \endgraf
+    \nobreak\vskip -\parskip
+    \penalty 10002  % signal to \startdefun and \dodefunx
+    % Some of the @defun-type tags do not enable magic parentheses,
+    % rendering the following check redundant.  But we don't optimize.
+    \checkparencounts
+  \endgroup
+}
+
+\def\Edefun{\endgraf\medbreak}
+
+% \makedefun{deffn} creates \deffn, \deffnx and \Edeffn;
+% the only thing remainnig is to define \deffnheader.
+%
+\def\makedefun#1{%
+  \expandafter\let\csname E#1\endcsname = \Edefun
+  \edef\temp{\noexpand\domakedefun
+    \makecsname{#1}\makecsname{#1x}\makecsname{#1header}}%
+  \temp
+}
+
+% \domakedefun \deffn \deffnx \deffnheader
+%
+% Define \deffn and \deffnx, without parameters.
+% \deffnheader has to be defined explicitly.
+%
+\def\domakedefun#1#2#3{%
+  \envdef#1{%
+    \startdefun
+    \parseargusing\activeparens{\printdefunline#3}%
+  }%
+  \def#2{\dodefunx#1}%
+  \def#3%
+}
+
+%%% Untyped functions:
+
+% @deffn category name args
+\makedefun{deffn}{\deffngeneral{}}
+
+% @deffn category class name args
+\makedefun{defop}#1 {\defopon{#1\ \putwordon}}
+
+% \defopon {category on}class name args
+\def\defopon#1#2 {\deffngeneral{\putwordon\ \code{#2}}{#1\ \code{#2}} }
+
+% \deffngeneral {subind}category name args
+%
+\def\deffngeneral#1#2 #3 #4\endheader{%
+  % Remember that \dosubind{fn}{foo}{} is equivalent to \doind{fn}{foo}.
+  \dosubind{fn}{\code{#3}}{#1}%
+  \defname{#2}{}{#3}\magicamp\defunargs{#4\unskip}%
+}
+
+%%% Typed functions:
+
+% @deftypefn category type name args
+\makedefun{deftypefn}{\deftypefngeneral{}}
+
+% @deftypeop category class type name args
+\makedefun{deftypeop}#1 {\deftypeopon{#1\ \putwordon}}
+
+% \deftypeopon {category on}class type name args
+\def\deftypeopon#1#2 {\deftypefngeneral{\putwordon\ \code{#2}}{#1\ \code{#2}} }
+
+% \deftypefngeneral {subind}category type name args
+%
+\def\deftypefngeneral#1#2 #3 #4 #5\endheader{%
+  \dosubind{fn}{\code{#4}}{#1}%
+  \defname{#2}{#3}{#4}\defunargs{#5\unskip}%
+}
+
+%%% Typed variables:
+
+% @deftypevr category type var args
+\makedefun{deftypevr}{\deftypecvgeneral{}}
+
+% @deftypecv category class type var args
+\makedefun{deftypecv}#1 {\deftypecvof{#1\ \putwordof}}
+
+% \deftypecvof {category of}class type var args
+\def\deftypecvof#1#2 {\deftypecvgeneral{\putwordof\ \code{#2}}{#1\ \code{#2}} }
+
+% \deftypecvgeneral {subind}category type var args
+%
+\def\deftypecvgeneral#1#2 #3 #4 #5\endheader{%
+  \dosubind{vr}{\code{#4}}{#1}%
+  \defname{#2}{#3}{#4}\defunargs{#5\unskip}%
+}
+
+%%% Untyped variables:
+
+% @defvr category var args
+\makedefun{defvr}#1 {\deftypevrheader{#1} {} }
+
+% @defcv category class var args
+\makedefun{defcv}#1 {\defcvof{#1\ \putwordof}}
+
+% \defcvof {category of}class var args
+\def\defcvof#1#2 {\deftypecvof{#1}#2 {} }
+
+%%% Type:
+% @deftp category name args
+\makedefun{deftp}#1 #2 #3\endheader{%
+  \doind{tp}{\code{#2}}%
+  \defname{#1}{}{#2}\defunargs{#3\unskip}%
+}
+
+% Remaining @defun-like shortcuts:
+\makedefun{defun}{\deffnheader{\putwordDeffunc} }
+\makedefun{defmac}{\deffnheader{\putwordDefmac} }
+\makedefun{defspec}{\deffnheader{\putwordDefspec} }
+\makedefun{deftypefun}{\deftypefnheader{\putwordDeffunc} }
+\makedefun{defvar}{\defvrheader{\putwordDefvar} }
+\makedefun{defopt}{\defvrheader{\putwordDefopt} }
+\makedefun{deftypevar}{\deftypevrheader{\putwordDefvar} }
+\makedefun{defmethod}{\defopon\putwordMethodon}
+\makedefun{deftypemethod}{\deftypeopon\putwordMethodon}
+\makedefun{defivar}{\defcvof\putwordInstanceVariableof}
+\makedefun{deftypeivar}{\deftypecvof\putwordInstanceVariableof}
+
+% \defname, which formats the name of the @def (not the args).
+% #1 is the category, such as "Function".
+% #2 is the return type, if any.
+% #3 is the function name.
+%
+% We are followed by (but not passed) the arguments, if any.
+%
+\def\defname#1#2#3{%
+  % Get the values of \leftskip and \rightskip as they were outside the @def...
+  \advance\leftskip by -\defbodyindent
+  %
+  % How we'll format the type name.  Putting it in brackets helps
+  % distinguish it from the body text that may end up on the next line
+  % just below it.
+  \def\temp{#1}%
+  \setbox0=\hbox{\kern\deflastargmargin \ifx\temp\empty\else [\rm\temp]\fi}
+  %
+  % Figure out line sizes for the paragraph shape.
+  % The first line needs space for \box0; but if \rightskip is nonzero,
+  % we need only space for the part of \box0 which exceeds it:
+  \dimen0=\hsize  \advance\dimen0 by -\wd0  \advance\dimen0 by \rightskip
+  % The continuations:
+  \dimen2=\hsize  \advance\dimen2 by -\defargsindent
+  % (plain.tex says that \dimen1 should be used only as global.)
+  \parshape 2 0in \dimen0 \defargsindent \dimen2
+  %
+  % Put the type name to the right margin.
+  \noindent
+  \hbox to 0pt{%
+    \hfil\box0 \kern-\hsize
+    % \hsize has to be shortened this way:
+    \kern\leftskip
+    % Intentionally do not respect \rightskip, since we need the space.
+  }%
+  %
+  % Allow all lines to be underfull without complaint:
+  \tolerance=10000 \hbadness=10000
+  \exdentamount=\defbodyindent
+  {%
+    % defun fonts. We use typewriter by default (used to be bold) because:
+    % . we're printing identifiers, they should be in tt in principle.
+    % . in languages with many accents, such as Czech or French, it's
+    %   common to leave accents off identifiers.  The result looks ok in
+    %   tt, but exceedingly strange in rm.
+    % . we don't want -- and --- to be treated as ligatures.
+    % . this still does not fix the ?` and !` ligatures, but so far no
+    %   one has made identifiers using them :).
+    \df \tt
+    \def\temp{#2}% return value type
+    \ifx\temp\empty\else \tclose{\temp} \fi
+    #3% output function name
+  }%
+  {\rm\enskip}% hskip 0.5 em of \tenrm
+  %
+  \boldbrax
+  % arguments will be output next, if any.
+}
+
+% Print arguments in slanted roman (not ttsl), inconsistently with using
+% tt for the name.  This is because literal text is sometimes needed in
+% the argument list (groff manual), and ttsl and tt are not very
+% distinguishable.  Prevent hyphenation at `-' chars.
+%
+\def\defunargs#1{%
+  % use sl by default (not ttsl),
+  % tt for the names.
+  \df \sl \hyphenchar\font=0
+  %
+  % On the other hand, if an argument has two dashes (for instance), we
+  % want a way to get ttsl.  Let's try @var for that.
+  \let\var=\ttslanted
+  #1%
+  \sl\hyphenchar\font=45
+}
+
+% We want ()&[] to print specially on the defun line.
+%
+\def\activeparens{%
+  \catcode`\(=\active \catcode`\)=\active
+  \catcode`\[=\active \catcode`\]=\active
+  \catcode`\&=\active
+}
+
+% Make control sequences which act like normal parenthesis chars.
+\let\lparen = ( \let\rparen = )
+
+% Be sure that we always have a definition for `(', etc.  For example,
+% if the fn name has parens in it, \boldbrax will not be in effect yet,
+% so TeX would otherwise complain about undefined control sequence.
+{
+  \activeparens
+  \global\let(=\lparen \global\let)=\rparen
+  \global\let[=\lbrack \global\let]=\rbrack
+  \global\let& = \&
+
+  \gdef\boldbrax{\let(=\opnr\let)=\clnr\let[=\lbrb\let]=\rbrb}
+  \gdef\magicamp{\let&=\amprm}
+}
+
+\newcount\parencount
+
+% If we encounter &foo, then turn on ()-hacking afterwards
+\newif\ifampseen
+\def\amprm#1 {\ampseentrue{\bf\&#1 }}
+
+\def\parenfont{%
+  \ifampseen
+    % At the first level, print parens in roman,
+    % otherwise use the default font.
+    \ifnum \parencount=1 \rm \fi
+  \else
+    % The \sf parens (in \boldbrax) actually are a little bolder than
+    % the contained text.  This is especially needed for [ and ] .
+    \sf
+  \fi
+}
+\def\infirstlevel#1{%
+  \ifampseen
+    \ifnum\parencount=1
+      #1%
+    \fi
+  \fi
+}
+\def\bfafterword#1 {#1 \bf}
+
+\def\opnr{%
+  \global\advance\parencount by 1
+  {\parenfont(}%
+  \infirstlevel \bfafterword
+}
+\def\clnr{%
+  {\parenfont)}%
+  \infirstlevel \sl
+  \global\advance\parencount by -1
+}
+
+\newcount\brackcount
+\def\lbrb{%
+  \global\advance\brackcount by 1
+  {\bf[}%
+}
+\def\rbrb{%
+  {\bf]}%
+  \global\advance\brackcount by -1
+}
+
+\def\checkparencounts{%
+  \ifnum\parencount=0 \else \badparencount \fi
+  \ifnum\brackcount=0 \else \badbrackcount \fi
+}
+\def\badparencount{%
+  \errmessage{Unbalanced parentheses in @def}%
+  \global\parencount=0
+}
+\def\badbrackcount{%
+  \errmessage{Unbalanced square braces in @def}%
+  \global\brackcount=0
+}
+
+
+\message{macros,}
+% @macro.
+
+% To do this right we need a feature of e-TeX, \scantokens,
+% which we arrange to emulate with a temporary file in ordinary TeX.
+\ifx\eTeXversion\undefined
+  \newwrite\macscribble
+  \def\scantokens#1{%
+    \toks0={#1}%
+    \immediate\openout\macscribble=\jobname.tmp
+    \immediate\write\macscribble{\the\toks0}%
+    \immediate\closeout\macscribble
+    \input \jobname.tmp
+  }
+\fi
+
+\def\scanmacro#1{%
+  \begingroup
+    \newlinechar`\^^M
+    \let\xeatspaces\eatspaces
+    % Undo catcode changes of \startcontents and \doprintindex
+    % When called from @insertcopying or (short)caption, we need active
+    % backslash to get it printed correctly.  Previously, we had
+    % \catcode`\\=\other instead.  We'll see whether a problem appears
+    % with macro expansion.				--kasal, 19aug04
+    \catcode`\@=0 \catcode`\\=\active \escapechar=`\@
+    % ... and \example
+    \spaceisspace
+    %
+    % Append \endinput to make sure that TeX does not see the ending newline.
+    %
+    % I've verified that it is necessary both for e-TeX and for ordinary TeX
+    %							--kasal, 29nov03
+    \scantokens{#1\endinput}%
+  \endgroup
+}
+
+\def\scanexp#1{%
+  \edef\temp{\noexpand\scanmacro{#1}}%
+  \temp
+}
+
+\newcount\paramno   % Count of parameters
+\newtoks\macname    % Macro name
+\newif\ifrecursive  % Is it recursive?
+\def\macrolist{}    % List of all defined macros in the form
+                    % \do\macro1\do\macro2...
+
+% Utility routines.
+% This does \let #1 = #2, with \csnames; that is,
+%   \let \csname#1\endcsname = \csname#2\endcsname
+% (except of course we have to play expansion games).
+% 
+\def\cslet#1#2{%
+  \expandafter\let
+  \csname#1\expandafter\endcsname
+  \csname#2\endcsname
+}
+
+% Trim leading and trailing spaces off a string.
+% Concepts from aro-bend problem 15 (see CTAN).
+{\catcode`\@=11
+\gdef\eatspaces #1{\expandafter\trim@\expandafter{#1 }}
+\gdef\trim@ #1{\trim@@ @#1 @ #1 @ @@}
+\gdef\trim@@ #1@ #2@ #3@@{\trim@@@\empty #2 @}
+\def\unbrace#1{#1}
+\unbrace{\gdef\trim@@@ #1 } #2@{#1}
+}
+
+% Trim a single trailing ^^M off a string.
+{\catcode`\^^M=\other \catcode`\Q=3%
+\gdef\eatcr #1{\eatcra #1Q^^MQ}%
+\gdef\eatcra#1^^MQ{\eatcrb#1Q}%
+\gdef\eatcrb#1Q#2Q{#1}%
+}
+
+% Macro bodies are absorbed as an argument in a context where
+% all characters are catcode 10, 11 or 12, except \ which is active
+% (as in normal texinfo). It is necessary to change the definition of \.
+
+% It's necessary to have hard CRs when the macro is executed. This is
+% done by  making ^^M (\endlinechar) catcode 12 when reading the macro
+% body, and then making it the \newlinechar in \scanmacro.
+
+\def\scanctxt{%
+  \catcode`\"=\other
+  \catcode`\+=\other
+  \catcode`\<=\other
+  \catcode`\>=\other
+  \catcode`\@=\other
+  \catcode`\^=\other
+  \catcode`\_=\other
+  \catcode`\|=\other
+  \catcode`\~=\other
+}
+
+\def\scanargctxt{%
+  \scanctxt
+  \catcode`\\=\other
+  \catcode`\^^M=\other
+}
+
+\def\macrobodyctxt{%
+  \scanctxt
+  \catcode`\{=\other
+  \catcode`\}=\other
+  \catcode`\^^M=\other
+  \usembodybackslash
+}
+
+\def\macroargctxt{%
+  \scanctxt
+  \catcode`\\=\other
+}
+
+% \mbodybackslash is the definition of \ in @macro bodies.
+% It maps \foo\ => \csname macarg.foo\endcsname => #N
+% where N is the macro parameter number.
+% We define \csname macarg.\endcsname to be \realbackslash, so
+% \\ in macro replacement text gets you a backslash.
+
+{\catcode`@=0 @catcode`@\=@active
+ @gdef@usembodybackslash{@let\=@mbodybackslash}
+ @gdef@mbodybackslash#1\{@csname macarg.#1@endcsname}
+}
+\expandafter\def\csname macarg.\endcsname{\realbackslash}
+
+\def\macro{\recursivefalse\parsearg\macroxxx}
+\def\rmacro{\recursivetrue\parsearg\macroxxx}
+
+\def\macroxxx#1{%
+  \getargs{#1}%           now \macname is the macname and \argl the arglist
+  \ifx\argl\empty       % no arguments
+     \paramno=0%
+  \else
+     \expandafter\parsemargdef \argl;%
+  \fi
+  \if1\csname ismacro.\the\macname\endcsname
+     \message{Warning: redefining \the\macname}%
+  \else
+     \expandafter\ifx\csname \the\macname\endcsname \relax
+     \else \errmessage{Macro name \the\macname\space already defined}\fi
+     \global\cslet{macsave.\the\macname}{\the\macname}%
+     \global\expandafter\let\csname ismacro.\the\macname\endcsname=1%
+     % Add the macroname to \macrolist
+     \toks0 = \expandafter{\macrolist\do}%
+     \xdef\macrolist{\the\toks0
+       \expandafter\noexpand\csname\the\macname\endcsname}%
+  \fi
+  \begingroup \macrobodyctxt
+  \ifrecursive \expandafter\parsermacbody
+  \else \expandafter\parsemacbody
+  \fi}
+
+\parseargdef\unmacro{%
+  \if1\csname ismacro.#1\endcsname
+    \global\cslet{#1}{macsave.#1}%
+    \global\expandafter\let \csname ismacro.#1\endcsname=0%
+    % Remove the macro name from \macrolist:
+    \begingroup
+      \expandafter\let\csname#1\endcsname \relax
+      \let\do\unmacrodo
+      \xdef\macrolist{\macrolist}%
+    \endgroup
+  \else
+    \errmessage{Macro #1 not defined}%
+  \fi
+}
+
+% Called by \do from \dounmacro on each macro.  The idea is to omit any
+% macro definitions that have been changed to \relax.
+%
+\def\unmacrodo#1{%
+  \ifx#1\relax
+    % remove this
+  \else
+    \noexpand\do \noexpand #1%
+  \fi
+}
+
+% This makes use of the obscure feature that if the last token of a
+% <parameter list> is #, then the preceding argument is delimited by
+% an opening brace, and that opening brace is not consumed.
+\def\getargs#1{\getargsxxx#1{}}
+\def\getargsxxx#1#{\getmacname #1 \relax\getmacargs}
+\def\getmacname #1 #2\relax{\macname={#1}}
+\def\getmacargs#1{\def\argl{#1}}
+
+% Parse the optional {params} list.  Set up \paramno and \paramlist
+% so \defmacro knows what to do.  Define \macarg.blah for each blah
+% in the params list, to be ##N where N is the position in that list.
+% That gets used by \mbodybackslash (above).
+
+% We need to get `macro parameter char #' into several definitions.
+% The technique used is stolen from LaTeX:  let \hash be something
+% unexpandable, insert that wherever you need a #, and then redefine
+% it to # just before using the token list produced.
+%
+% The same technique is used to protect \eatspaces till just before
+% the macro is used.
+
+\def\parsemargdef#1;{\paramno=0\def\paramlist{}%
+        \let\hash\relax\let\xeatspaces\relax\parsemargdefxxx#1,;,}
+\def\parsemargdefxxx#1,{%
+  \if#1;\let\next=\relax
+  \else \let\next=\parsemargdefxxx
+    \advance\paramno by 1%
+    \expandafter\edef\csname macarg.\eatspaces{#1}\endcsname
+        {\xeatspaces{\hash\the\paramno}}%
+    \edef\paramlist{\paramlist\hash\the\paramno,}%
+  \fi\next}
+
+% These two commands read recursive and nonrecursive macro bodies.
+% (They're different since rec and nonrec macros end differently.)
+
+\long\def\parsemacbody#1@end macro%
+{\xdef\temp{\eatcr{#1}}\endgroup\defmacro}%
+\long\def\parsermacbody#1@end rmacro%
+{\xdef\temp{\eatcr{#1}}\endgroup\defmacro}%
+
+% This defines the macro itself. There are six cases: recursive and
+% nonrecursive macros of zero, one, and many arguments.
+% Much magic with \expandafter here.
+% \xdef is used so that macro definitions will survive the file
+% they're defined in; @include reads the file inside a group.
+\def\defmacro{%
+  \let\hash=##% convert placeholders to macro parameter chars
+  \ifrecursive
+    \ifcase\paramno
+    % 0
+      \expandafter\xdef\csname\the\macname\endcsname{%
+        \noexpand\scanmacro{\temp}}%
+    \or % 1
+      \expandafter\xdef\csname\the\macname\endcsname{%
+         \bgroup\noexpand\macroargctxt
+         \noexpand\braceorline
+         \expandafter\noexpand\csname\the\macname xxx\endcsname}%
+      \expandafter\xdef\csname\the\macname xxx\endcsname##1{%
+         \egroup\noexpand\scanmacro{\temp}}%
+    \else % many
+      \expandafter\xdef\csname\the\macname\endcsname{%
+         \bgroup\noexpand\macroargctxt
+         \noexpand\csname\the\macname xx\endcsname}%
+      \expandafter\xdef\csname\the\macname xx\endcsname##1{%
+          \expandafter\noexpand\csname\the\macname xxx\endcsname ##1,}%
+      \expandafter\expandafter
+      \expandafter\xdef
+      \expandafter\expandafter
+        \csname\the\macname xxx\endcsname
+          \paramlist{\egroup\noexpand\scanmacro{\temp}}%
+    \fi
+  \else
+    \ifcase\paramno
+    % 0
+      \expandafter\xdef\csname\the\macname\endcsname{%
+        \noexpand\norecurse{\the\macname}%
+        \noexpand\scanmacro{\temp}\egroup}%
+    \or % 1
+      \expandafter\xdef\csname\the\macname\endcsname{%
+         \bgroup\noexpand\macroargctxt
+         \noexpand\braceorline
+         \expandafter\noexpand\csname\the\macname xxx\endcsname}%
+      \expandafter\xdef\csname\the\macname xxx\endcsname##1{%
+        \egroup
+        \noexpand\norecurse{\the\macname}%
+        \noexpand\scanmacro{\temp}\egroup}%
+    \else % many
+      \expandafter\xdef\csname\the\macname\endcsname{%
+         \bgroup\noexpand\macroargctxt
+         \expandafter\noexpand\csname\the\macname xx\endcsname}%
+      \expandafter\xdef\csname\the\macname xx\endcsname##1{%
+          \expandafter\noexpand\csname\the\macname xxx\endcsname ##1,}%
+      \expandafter\expandafter
+      \expandafter\xdef
+      \expandafter\expandafter
+      \csname\the\macname xxx\endcsname
+      \paramlist{%
+          \egroup
+          \noexpand\norecurse{\the\macname}%
+          \noexpand\scanmacro{\temp}\egroup}%
+    \fi
+  \fi}
+
+\def\norecurse#1{\bgroup\cslet{#1}{macsave.#1}}
+
+% \braceorline decides whether the next nonwhitespace character is a
+% {.  If so it reads up to the closing }, if not, it reads the whole
+% line.  Whatever was read is then fed to the next control sequence
+% as an argument (by \parsebrace or \parsearg)
+\def\braceorline#1{\let\next=#1\futurelet\nchar\braceorlinexxx}
+\def\braceorlinexxx{%
+  \ifx\nchar\bgroup\else
+    \expandafter\parsearg
+  \fi \next}
+
+% We want to disable all macros during \shipout so that they are not
+% expanded by \write.
+\def\turnoffmacros{\begingroup \def\do##1{\let\noexpand##1=\relax}%
+  \edef\next{\macrolist}\expandafter\endgroup\next}
+
+% For \indexnofonts, we need to get rid of all macros, leaving only the
+% arguments (if present).  Of course this is not nearly correct, but it
+% is the best we can do for now.  makeinfo does not expand macros in the
+% argument to @deffn, which ends up writing an index entry, and texindex
+% isn't prepared for an index sort entry that starts with \.
+% 
+% Since macro invocations are followed by braces, we can just redefine them
+% to take a single TeX argument.  The case of a macro invocation that
+% goes to end-of-line is not handled.
+% 
+\def\emptyusermacros{\begingroup
+  \def\do##1{\let\noexpand##1=\noexpand\asis}%
+  \edef\next{\macrolist}\expandafter\endgroup\next}
+
+
+% @alias.
+% We need some trickery to remove the optional spaces around the equal
+% sign.  Just make them active and then expand them all to nothing.
+\def\alias{\parseargusing\obeyspaces\aliasxxx}
+\def\aliasxxx #1{\aliasyyy#1\relax}
+\def\aliasyyy #1=#2\relax{%
+  {%
+    \expandafter\let\obeyedspace=\empty
+    \xdef\next{\global\let\makecsname{#1}=\makecsname{#2}}%
+  }%
+  \next
+}
+
+
+\message{cross references,}
+
+\newwrite\auxfile
+
+\newif\ifhavexrefs    % True if xref values are known.
+\newif\ifwarnedxrefs  % True if we warned once that they aren't known.
+
+% @inforef is relatively simple.
+\def\inforef #1{\inforefzzz #1,,,,**}
+\def\inforefzzz #1,#2,#3,#4**{\putwordSee{} \putwordInfo{} \putwordfile{} \file{\ignorespaces #3{}},
+  node \samp{\ignorespaces#1{}}}
+
+% @node's only job in TeX is to define \lastnode, which is used in
+% cross-references.  The @node line might or might not have commas, and
+% might or might not have spaces before the first comma, like:
+% @node foo , bar , ...
+% We don't want such trailing spaces in the node name.
+%
+\parseargdef\node{\checkenv{}\donode #1 ,\finishnodeparse}
+%
+% also remove a trailing comma, in case of something like this:
+% @node Help-Cross,  ,  , Cross-refs
+\def\donode#1 ,#2\finishnodeparse{\dodonode #1,\finishnodeparse}
+\def\dodonode#1,#2\finishnodeparse{\gdef\lastnode{#1}}
+
+\let\nwnode=\node
+\let\lastnode=\empty
+
+% Write a cross-reference definition for the current node.  #1 is the
+% type (Ynumbered, Yappendix, Ynothing).
+%
+\def\donoderef#1{%
+  \ifx\lastnode\empty\else
+    \setref{\lastnode}{#1}%
+    \global\let\lastnode=\empty
+  \fi
+}
+
+% @anchor{NAME} -- define xref target at arbitrary point.
+%
+\newcount\savesfregister
+%
+\def\savesf{\relax \ifhmode \savesfregister=\spacefactor \fi}
+\def\restoresf{\relax \ifhmode \spacefactor=\savesfregister \fi}
+\def\anchor#1{\savesf \setref{#1}{Ynothing}\restoresf \ignorespaces}
+
+% \setref{NAME}{SNT} defines a cross-reference point NAME (a node or an
+% anchor), which consists of three parts:
+% 1) NAME-title - the current sectioning name taken from \thissection,
+%                 or the anchor name.
+% 2) NAME-snt   - section number and type, passed as the SNT arg, or
+%                 empty for anchors.
+% 3) NAME-pg    - the page number.
+%
+% This is called from \donoderef, \anchor, and \dofloat.  In the case of
+% floats, there is an additional part, which is not written here:
+% 4) NAME-lof   - the text as it should appear in a @listoffloats.
+%
+\def\setref#1#2{%
+  \pdfmkdest{#1}%
+  \iflinks
+    {%
+      \atdummies  % preserve commands, but don't expand them
+      \turnoffactive
+      \otherbackslash
+      \edef\writexrdef##1##2{%
+	\write\auxfile{@xrdef{#1-% #1 of \setref, expanded by the \edef
+	  ##1}{##2}}% these are parameters of \writexrdef
+      }%
+      \toks0 = \expandafter{\thissection}%
+      \immediate \writexrdef{title}{\the\toks0 }%
+      \immediate \writexrdef{snt}{\csname #2\endcsname}% \Ynumbered etc.
+      \writexrdef{pg}{\folio}% will be written later, during \shipout
+    }%
+  \fi
+}
+
+% @xref, @pxref, and @ref generate cross-references.  For \xrefX, #1 is
+% the node name, #2 the name of the Info cross-reference, #3 the printed
+% node name, #4 the name of the Info file, #5 the name of the printed
+% manual.  All but the node name can be omitted.
+%
+\def\pxref#1{\putwordsee{} \xrefX[#1,,,,,,,]}
+\def\xref#1{\putwordSee{} \xrefX[#1,,,,,,,]}
+\def\ref#1{\xrefX[#1,,,,,,,]}
+\def\xrefX[#1,#2,#3,#4,#5,#6]{\begingroup
+  \unsepspaces
+  \def\printedmanual{\ignorespaces #5}%
+  \def\printedrefname{\ignorespaces #3}%
+  \setbox1=\hbox{\printedmanual\unskip}%
+  \setbox0=\hbox{\printedrefname\unskip}%
+  \ifdim \wd0 = 0pt
+    % No printed node name was explicitly given.
+    \expandafter\ifx\csname SETxref-automatic-section-title\endcsname\relax
+      % Use the node name inside the square brackets.
+      \def\printedrefname{\ignorespaces #1}%
+    \else
+      % Use the actual chapter/section title appear inside
+      % the square brackets.  Use the real section title if we have it.
+      \ifdim \wd1 > 0pt
+        % It is in another manual, so we don't have it.
+        \def\printedrefname{\ignorespaces #1}%
+      \else
+        \ifhavexrefs
+          % We know the real title if we have the xref values.
+          \def\printedrefname{\refx{#1-title}{}}%
+        \else
+          % Otherwise just copy the Info node name.
+          \def\printedrefname{\ignorespaces #1}%
+        \fi%
+      \fi
+    \fi
+  \fi
+  %
+  % Make link in pdf output.
+  \ifpdf
+    \leavevmode
+    \getfilename{#4}%
+    {\turnoffactive \otherbackslash
+     \ifnum\filenamelength>0
+       \startlink attr{/Border [0 0 0]}%
+         goto file{\the\filename.pdf} name{#1}%
+     \else
+       \startlink attr{/Border [0 0 0]}%
+         goto name{\pdfmkpgn{#1}}%
+     \fi
+    }%
+    \linkcolor
+  \fi
+  %
+  % Float references are printed completely differently: "Figure 1.2"
+  % instead of "[somenode], p.3".  We distinguish them by the
+  % LABEL-title being set to a magic string.
+  {%
+    % Have to otherify everything special to allow the \csname to
+    % include an _ in the xref name, etc.
+    \indexnofonts
+    \turnoffactive
+    \otherbackslash
+    \expandafter\global\expandafter\let\expandafter\Xthisreftitle
+      \csname XR#1-title\endcsname
+  }%
+  \iffloat\Xthisreftitle
+    % If the user specified the print name (third arg) to the ref,
+    % print it instead of our usual "Figure 1.2".
+    \ifdim\wd0 = 0pt
+      \refx{#1-snt}%
+    \else
+      \printedrefname
+    \fi
+    %
+    % if the user also gave the printed manual name (fifth arg), append
+    % "in MANUALNAME".
+    \ifdim \wd1 > 0pt
+      \space \putwordin{} \cite{\printedmanual}%
+    \fi
+  \else
+    % node/anchor (non-float) references.
+    %
+    % If we use \unhbox0 and \unhbox1 to print the node names, TeX does not
+    % insert empty discretionaries after hyphens, which means that it will
+    % not find a line break at a hyphen in a node names.  Since some manuals
+    % are best written with fairly long node names, containing hyphens, this
+    % is a loss.  Therefore, we give the text of the node name again, so it
+    % is as if TeX is seeing it for the first time.
+    \ifdim \wd1 > 0pt
+      \putwordsection{} ``\printedrefname'' \putwordin{} \cite{\printedmanual}%
+    \else
+      % _ (for example) has to be the character _ for the purposes of the
+      % control sequence corresponding to the node, but it has to expand
+      % into the usual \leavevmode...\vrule stuff for purposes of
+      % printing. So we \turnoffactive for the \refx-snt, back on for the
+      % printing, back off for the \refx-pg.
+      {\turnoffactive \otherbackslash
+       % Only output a following space if the -snt ref is nonempty; for
+       % @unnumbered and @anchor, it won't be.
+       \setbox2 = \hbox{\ignorespaces \refx{#1-snt}{}}%
+       \ifdim \wd2 > 0pt \refx{#1-snt}\space\fi
+      }%
+      % output the `[mynode]' via a macro so it can be overridden.
+      \xrefprintnodename\printedrefname
+      %
+      % But we always want a comma and a space:
+      ,\space
+      %
+      % output the `page 3'.
+      \turnoffactive \otherbackslash \putwordpage\tie\refx{#1-pg}{}%
+    \fi
+  \fi
+  \endlink
+\endgroup}
+
+% This macro is called from \xrefX for the `[nodename]' part of xref
+% output.  It's a separate macro only so it can be changed more easily,
+% since square brackets don't work well in some documents.  Particularly
+% one that Bob is working on :).
+%
+\def\xrefprintnodename#1{[#1]}
+
+% Things referred to by \setref.
+%
+\def\Ynothing{}
+\def\Yomitfromtoc{}
+\def\Ynumbered{%
+  \ifnum\secno=0
+    \putwordChapter@tie \the\chapno
+  \else \ifnum\subsecno=0
+    \putwordSection@tie \the\chapno.\the\secno
+  \else \ifnum\subsubsecno=0
+    \putwordSection@tie \the\chapno.\the\secno.\the\subsecno
+  \else
+    \putwordSection@tie \the\chapno.\the\secno.\the\subsecno.\the\subsubsecno
+  \fi\fi\fi
+}
+\def\Yappendix{%
+  \ifnum\secno=0
+     \putwordAppendix@tie @char\the\appendixno{}%
+  \else \ifnum\subsecno=0
+     \putwordSection@tie @char\the\appendixno.\the\secno
+  \else \ifnum\subsubsecno=0
+    \putwordSection@tie @char\the\appendixno.\the\secno.\the\subsecno
+  \else
+    \putwordSection@tie
+      @char\the\appendixno.\the\secno.\the\subsecno.\the\subsubsecno
+  \fi\fi\fi
+}
+
+% Define \refx{NAME}{SUFFIX} to reference a cross-reference string named NAME.
+% If its value is nonempty, SUFFIX is output afterward.
+%
+\def\refx#1#2{%
+  {%
+    \indexnofonts
+    \otherbackslash
+    \expandafter\global\expandafter\let\expandafter\thisrefX
+      \csname XR#1\endcsname
+  }%
+  \ifx\thisrefX\relax
+    % If not defined, say something at least.
+    \angleleft un\-de\-fined\angleright
+    \iflinks
+      \ifhavexrefs
+        \message{\linenumber Undefined cross reference `#1'.}%
+      \else
+        \ifwarnedxrefs\else
+          \global\warnedxrefstrue
+          \message{Cross reference values unknown; you must run TeX again.}%
+        \fi
+      \fi
+    \fi
+  \else
+    % It's defined, so just use it.
+    \thisrefX
+  \fi
+  #2% Output the suffix in any case.
+}
+
+% This is the macro invoked by entries in the aux file.  Usually it's
+% just a \def (we prepend XR to the control sequence name to avoid
+% collisions).  But if this is a float type, we have more work to do.
+%
+\def\xrdef#1#2{%
+  \expandafter\gdef\csname XR#1\endcsname{#2}% remember this xref value.
+  %
+  % Was that xref control sequence that we just defined for a float?
+  \expandafter\iffloat\csname XR#1\endcsname
+    % it was a float, and we have the (safe) float type in \iffloattype.
+    \expandafter\let\expandafter\floatlist
+      \csname floatlist\iffloattype\endcsname
+    %
+    % Is this the first time we've seen this float type?
+    \expandafter\ifx\floatlist\relax
+      \toks0 = {\do}% yes, so just \do
+    \else
+      % had it before, so preserve previous elements in list.
+      \toks0 = \expandafter{\floatlist\do}%
+    \fi
+    %
+    % Remember this xref in the control sequence \floatlistFLOATTYPE,
+    % for later use in \listoffloats.
+    \expandafter\xdef\csname floatlist\iffloattype\endcsname{\the\toks0{#1}}%
+  \fi
+}
+
+% Read the last existing aux file, if any.  No error if none exists.
+%
+\def\tryauxfile{%
+  \openin 1 \jobname.aux
+  \ifeof 1 \else
+    \readauxfile
+    \global\havexrefstrue
+  \fi
+  \closein 1
+}
+
+\def\readauxfile{\begingroup
+  \catcode`\^^@=\other
+  \catcode`\^^A=\other
+  \catcode`\^^B=\other
+  \catcode`\^^C=\other
+  \catcode`\^^D=\other
+  \catcode`\^^E=\other
+  \catcode`\^^F=\other
+  \catcode`\^^G=\other
+  \catcode`\^^H=\other
+  \catcode`\^^K=\other
+  \catcode`\^^L=\other
+  \catcode`\^^N=\other
+  \catcode`\^^P=\other
+  \catcode`\^^Q=\other
+  \catcode`\^^R=\other
+  \catcode`\^^S=\other
+  \catcode`\^^T=\other
+  \catcode`\^^U=\other
+  \catcode`\^^V=\other
+  \catcode`\^^W=\other
+  \catcode`\^^X=\other
+  \catcode`\^^Z=\other
+  \catcode`\^^[=\other
+  \catcode`\^^\=\other
+  \catcode`\^^]=\other
+  \catcode`\^^^=\other
+  \catcode`\^^_=\other
+  % It was suggested to set the catcode of ^ to 7, which would allow ^^e4 etc.
+  % in xref tags, i.e., node names.  But since ^^e4 notation isn't
+  % supported in the main text, it doesn't seem desirable.  Furthermore,
+  % that is not enough: for node names that actually contain a ^
+  % character, we would end up writing a line like this: 'xrdef {'hat
+  % b-title}{'hat b} and \xrdef does a \csname...\endcsname on the first
+  % argument, and \hat is not an expandable control sequence.  It could
+  % all be worked out, but why?  Either we support ^^ or we don't.
+  %
+  % The other change necessary for this was to define \auxhat:
+  % \def\auxhat{\def^{'hat }}% extra space so ok if followed by letter
+  % and then to call \auxhat in \setq.
+  %
+  \catcode`\^=\other
+  %
+  % Special characters.  Should be turned off anyway, but...
+  \catcode`\~=\other
+  \catcode`\[=\other
+  \catcode`\]=\other
+  \catcode`\"=\other
+  \catcode`\_=\other
+  \catcode`\|=\other
+  \catcode`\<=\other
+  \catcode`\>=\other
+  \catcode`\$=\other
+  \catcode`\#=\other
+  \catcode`\&=\other
+  \catcode`\%=\other
+  \catcode`+=\other % avoid \+ for paranoia even though we've turned it off
+  %
+  % This is to support \ in node names and titles, since the \
+  % characters end up in a \csname.  It's easier than
+  % leaving it active and making its active definition an actual \
+  % character.  What I don't understand is why it works in the *value*
+  % of the xrdef.  Seems like it should be a catcode12 \, and that
+  % should not typeset properly.  But it works, so I'm moving on for
+  % now.  --karl, 15jan04.
+  \catcode`\\=\other
+  %
+  % Make the characters 128-255 be printing characters.
+  {%
+    \count 1=128
+    \def\loop{%
+      \catcode\count 1=\other
+      \advance\count 1 by 1
+      \ifnum \count 1<256 \loop \fi
+    }%
+  }%
+  %
+  % @ is our escape character in .aux files, and we need braces.
+  \catcode`\{=1
+  \catcode`\}=2
+  \catcode`\@=0
+  %
+  \input \jobname.aux
+\endgroup}
+
+
+\message{insertions,}
+% including footnotes.
+
+\newcount \footnoteno
+
+% The trailing space in the following definition for supereject is
+% vital for proper filling; pages come out unaligned when you do a
+% pagealignmacro call if that space before the closing brace is
+% removed. (Generally, numeric constants should always be followed by a
+% space to prevent strange expansion errors.)
+\def\supereject{\par\penalty -20000\footnoteno =0 }
+
+% @footnotestyle is meaningful for info output only.
+\let\footnotestyle=\comment
+
+{\catcode `\@=11
+%
+% Auto-number footnotes.  Otherwise like plain.
+\gdef\footnote{%
+  \let\indent=\ptexindent
+  \let\noindent=\ptexnoindent
+  \global\advance\footnoteno by \@ne
+  \edef\thisfootno{$^{\the\footnoteno}$}%
+  %
+  % In case the footnote comes at the end of a sentence, preserve the
+  % extra spacing after we do the footnote number.
+  \let\@sf\empty
+  \ifhmode\edef\@sf{\spacefactor\the\spacefactor}\ptexslash\fi
+  %
+  % Remove inadvertent blank space before typesetting the footnote number.
+  \unskip
+  \thisfootno\@sf
+  \dofootnote
+}%
+
+% Don't bother with the trickery in plain.tex to not require the
+% footnote text as a parameter.  Our footnotes don't need to be so general.
+%
+% Oh yes, they do; otherwise, @ifset (and anything else that uses
+% \parseargline) fails inside footnotes because the tokens are fixed when
+% the footnote is read.  --karl, 16nov96.
+%
+\gdef\dofootnote{%
+  \insert\footins\bgroup
+  % We want to typeset this text as a normal paragraph, even if the
+  % footnote reference occurs in (for example) a display environment.
+  % So reset some parameters.
+  \hsize=\pagewidth
+  \interlinepenalty\interfootnotelinepenalty
+  \splittopskip\ht\strutbox % top baseline for broken footnotes
+  \splitmaxdepth\dp\strutbox
+  \floatingpenalty\@MM
+  \leftskip\z@skip
+  \rightskip\z@skip
+  \spaceskip\z@skip
+  \xspaceskip\z@skip
+  \parindent\defaultparindent
+  %
+  \smallfonts \rm
+  %
+  % Because we use hanging indentation in footnotes, a @noindent appears
+  % to exdent this text, so make it be a no-op.  makeinfo does not use
+  % hanging indentation so @noindent can still be needed within footnote
+  % text after an @example or the like (not that this is good style).
+  \let\noindent = \relax
+  %
+  % Hang the footnote text off the number.  Use \everypar in case the
+  % footnote extends for more than one paragraph.
+  \everypar = {\hang}%
+  \textindent{\thisfootno}%
+  %
+  % Don't crash into the line above the footnote text.  Since this
+  % expands into a box, it must come within the paragraph, lest it
+  % provide a place where TeX can split the footnote.
+  \footstrut
+  \futurelet\next\fo@t
+}
+}%end \catcode `\@=11
+
+% In case a @footnote appears in a vbox, save the footnote text and create
+% the real \insert just after the vbox finished.  Otherwise, the insertion
+% would be lost.
+% Similarily, if a @footnote appears inside an alignment, save the footnote
+% text to a box and make the \insert when a row of the table is finished.
+% And the same can be done for other insert classes.  --kasal, 16nov03.
+
+% Replace the \insert primitive by a cheating macro.
+% Deeper inside, just make sure that the saved insertions are not spilled
+% out prematurely.
+%
+\def\startsavinginserts{%
+  \ifx \insert\ptexinsert
+    \let\insert\saveinsert
+  \else
+    \let\checkinserts\relax
+  \fi
+}
+
+% This \insert replacement works for both \insert\footins{foo} and
+% \insert\footins\bgroup foo\egroup, but it doesn't work for \insert27{foo}.
+%
+\def\saveinsert#1{%
+  \edef\next{\noexpand\savetobox \makeSAVEname#1}%
+  \afterassignment\next
+  % swallow the left brace
+  \let\temp =
+}
+\def\makeSAVEname#1{\makecsname{SAVE\expandafter\gobble\string#1}}
+\def\savetobox#1{\global\setbox#1 = \vbox\bgroup \unvbox#1}
+
+\def\checksaveins#1{\ifvoid#1\else \placesaveins#1\fi}
+
+\def\placesaveins#1{%
+  \ptexinsert \csname\expandafter\gobblesave\string#1\endcsname
+    {\box#1}%
+}
+
+% eat @SAVE -- beware, all of them have catcode \other:
+{
+  \def\dospecials{\do S\do A\do V\do E} \uncatcodespecials  %  ;-)
+  \gdef\gobblesave @SAVE{}
+}
+
+% initialization:
+\def\newsaveins #1{%
+  \edef\next{\noexpand\newsaveinsX \makeSAVEname#1}%
+  \next
+}
+\def\newsaveinsX #1{%
+  \csname newbox\endcsname #1%
+  \expandafter\def\expandafter\checkinserts\expandafter{\checkinserts
+    \checksaveins #1}%
+}
+
+% initialize:
+\let\checkinserts\empty
+\newsaveins\footins
+\newsaveins\margin
+
+
+% @image.  We use the macros from epsf.tex to support this.
+% If epsf.tex is not installed and @image is used, we complain.
+%
+% Check for and read epsf.tex up front.  If we read it only at @image
+% time, we might be inside a group, and then its definitions would get
+% undone and the next image would fail.
+\openin 1 = epsf.tex
+\ifeof 1 \else
+  % Do not bother showing banner with epsf.tex v2.7k (available in
+  % doc/epsf.tex and on ctan).
+  \def\epsfannounce{\toks0 = }%
+  \input epsf.tex
+\fi
+\closein 1
+%
+% We will only complain once about lack of epsf.tex.
+\newif\ifwarnednoepsf
+\newhelp\noepsfhelp{epsf.tex must be installed for images to
+  work.  It is also included in the Texinfo distribution, or you can get
+  it from ftp://tug.org/tex/epsf.tex.}
+%
+\def\image#1{%
+  \ifx\epsfbox\undefined
+    \ifwarnednoepsf \else
+      \errhelp = \noepsfhelp
+      \errmessage{epsf.tex not found, images will be ignored}%
+      \global\warnednoepsftrue
+    \fi
+  \else
+    \imagexxx #1,,,,,\finish
+  \fi
+}
+%
+% Arguments to @image:
+% #1 is (mandatory) image filename; we tack on .eps extension.
+% #2 is (optional) width, #3 is (optional) height.
+% #4 is (ignored optional) html alt text.
+% #5 is (ignored optional) extension.
+% #6 is just the usual extra ignored arg for parsing this stuff.
+\newif\ifimagevmode
+\def\imagexxx#1,#2,#3,#4,#5,#6\finish{\begingroup
+  \catcode`\^^M = 5     % in case we're inside an example
+  \normalturnoffactive  % allow _ et al. in names
+  % If the image is by itself, center it.
+  \ifvmode
+    \imagevmodetrue
+    \nobreak\bigskip
+    % Usually we'll have text after the image which will insert
+    % \parskip glue, so insert it here too to equalize the space
+    % above and below.
+    \nobreak\vskip\parskip
+    \nobreak
+    \line\bgroup\hss
+  \fi
+  %
+  % Output the image.
+  \ifpdf
+    \dopdfimage{#1}{#2}{#3}%
+  \else
+    % \epsfbox itself resets \epsf?size at each figure.
+    \setbox0 = \hbox{\ignorespaces #2}\ifdim\wd0 > 0pt \epsfxsize=#2\relax \fi
+    \setbox0 = \hbox{\ignorespaces #3}\ifdim\wd0 > 0pt \epsfysize=#3\relax \fi
+    \epsfbox{#1.eps}%
+  \fi
+  %
+  \ifimagevmode \hss \egroup \bigbreak \fi  % space after the image
+\endgroup}
+
+
+% @float FLOATTYPE,LABEL,LOC ... @end float for displayed figures, tables,
+% etc.  We don't actually implement floating yet, we always include the
+% float "here".  But it seemed the best name for the future.
+%
+\envparseargdef\float{\eatcommaspace\eatcommaspace\dofloat#1, , ,\finish}
+
+% There may be a space before second and/or third parameter; delete it.
+\def\eatcommaspace#1, {#1,}
+
+% #1 is the optional FLOATTYPE, the text label for this float, typically
+% "Figure", "Table", "Example", etc.  Can't contain commas.  If omitted,
+% this float will not be numbered and cannot be referred to.
+%
+% #2 is the optional xref label.  Also must be present for the float to
+% be referable.
+%
+% #3 is the optional positioning argument; for now, it is ignored.  It
+% will somehow specify the positions allowed to float to (here, top, bottom).
+%
+% We keep a separate counter for each FLOATTYPE, which we reset at each
+% chapter-level command.
+\let\resetallfloatnos=\empty
+%
+\def\dofloat#1,#2,#3,#4\finish{%
+  \let\thiscaption=\empty
+  \let\thisshortcaption=\empty
+  %
+  % don't lose footnotes inside @float.
+  %
+  % BEWARE: when the floats start float, we have to issue warning whenever an
+  % insert appears inside a float which could possibly float. --kasal, 26may04
+  %
+  \startsavinginserts
+  %
+  % We can't be used inside a paragraph.
+  \par
+  %
+  \vtop\bgroup
+    \def\floattype{#1}%
+    \def\floatlabel{#2}%
+    \def\floatloc{#3}% we do nothing with this yet.
+    %
+    \ifx\floattype\empty
+      \let\safefloattype=\empty
+    \else
+      {%
+        % the floattype might have accents or other special characters,
+        % but we need to use it in a control sequence name.
+        \indexnofonts
+        \turnoffactive
+        \xdef\safefloattype{\floattype}%
+      }%
+    \fi
+    %
+    % If label is given but no type, we handle that as the empty type.
+    \ifx\floatlabel\empty \else
+      % We want each FLOATTYPE to be numbered separately (Figure 1,
+      % Table 1, Figure 2, ...).  (And if no label, no number.)
+      %
+      \expandafter\getfloatno\csname\safefloattype floatno\endcsname
+      \global\advance\floatno by 1
+      %
+      {%
+        % This magic value for \thissection is output by \setref as the
+        % XREFLABEL-title value.  \xrefX uses it to distinguish float
+        % labels (which have a completely different output format) from
+        % node and anchor labels.  And \xrdef uses it to construct the
+        % lists of floats.
+        %
+        \edef\thissection{\floatmagic=\safefloattype}%
+        \setref{\floatlabel}{Yfloat}%
+      }%
+    \fi
+    %
+    % start with \parskip glue, I guess.
+    \vskip\parskip
+    %
+    % Don't suppress indentation if a float happens to start a section.
+    \restorefirstparagraphindent
+}
+
+% we have these possibilities:
+% @float Foo,lbl & @caption{Cap}: Foo 1.1: Cap
+% @float Foo,lbl & no caption:    Foo 1.1
+% @float Foo & @caption{Cap}:     Foo: Cap
+% @float Foo & no caption:        Foo
+% @float ,lbl & Caption{Cap}:     1.1: Cap
+% @float ,lbl & no caption:       1.1
+% @float & @caption{Cap}:         Cap
+% @float & no caption:
+%
+\def\Efloat{%
+    \let\floatident = \empty
+    %
+    % In all cases, if we have a float type, it comes first.
+    \ifx\floattype\empty \else \def\floatident{\floattype}\fi
+    %
+    % If we have an xref label, the number comes next.
+    \ifx\floatlabel\empty \else
+      \ifx\floattype\empty \else % if also had float type, need tie first.
+        \appendtomacro\floatident{\tie}%
+      \fi
+      % the number.
+      \appendtomacro\floatident{\chaplevelprefix\the\floatno}%
+    \fi
+    %
+    % Start the printed caption with what we've constructed in
+    % \floatident, but keep it separate; we need \floatident again.
+    \let\captionline = \floatident
+    %
+    \ifx\thiscaption\empty \else
+      \ifx\floatident\empty \else
+	\appendtomacro\captionline{: }% had ident, so need a colon between
+      \fi
+      %
+      % caption text.
+      \appendtomacro\captionline{\scanexp\thiscaption}%
+    \fi
+    %
+    % If we have anything to print, print it, with space before.
+    % Eventually this needs to become an \insert.
+    \ifx\captionline\empty \else
+      \vskip.5\parskip
+      \captionline
+      %
+      % Space below caption.
+      \vskip\parskip
+    \fi
+    %
+    % If have an xref label, write the list of floats info.  Do this
+    % after the caption, to avoid chance of it being a breakpoint.
+    \ifx\floatlabel\empty \else
+      % Write the text that goes in the lof to the aux file as
+      % \floatlabel-lof.  Besides \floatident, we include the short
+      % caption if specified, else the full caption if specified, else nothing.
+      {%
+        \atdummies \turnoffactive \otherbackslash
+        % since we read the caption text in the macro world, where ^^M
+        % is turned into a normal character, we have to scan it back, so
+        % we don't write the literal three characters "^^M" into the aux file.
+	\scanexp{%
+	  \xdef\noexpand\gtemp{%
+	    \ifx\thisshortcaption\empty
+	      \thiscaption
+	    \else
+	      \thisshortcaption
+	    \fi
+	  }%
+	}%
+        \immediate\write\auxfile{@xrdef{\floatlabel-lof}{\floatident
+	  \ifx\gtemp\empty \else : \gtemp \fi}}%
+      }%
+    \fi
+  \egroup  % end of \vtop
+  %
+  % place the captured inserts
+  %
+  % BEWARE: when the floats start float, we have to issue warning whenever an
+  % insert appears inside a float which could possibly float. --kasal, 26may04
+  %
+  \checkinserts
+}
+
+% Append the tokens #2 to the definition of macro #1, not expanding either.
+%
+\def\appendtomacro#1#2{%
+  \expandafter\def\expandafter#1\expandafter{#1#2}%
+}
+
+% @caption, @shortcaption
+%
+\def\caption{\docaption\thiscaption}
+\def\shortcaption{\docaption\thisshortcaption}
+\def\docaption{\checkenv\float \bgroup\scanargctxt\defcaption}
+\def\defcaption#1#2{\egroup \def#1{#2}}
+
+% The parameter is the control sequence identifying the counter we are
+% going to use.  Create it if it doesn't exist and assign it to \floatno.
+\def\getfloatno#1{%
+  \ifx#1\relax
+      % Haven't seen this figure type before.
+      \csname newcount\endcsname #1%
+      %
+      % Remember to reset this floatno at the next chap.
+      \expandafter\gdef\expandafter\resetallfloatnos
+        \expandafter{\resetallfloatnos #1=0 }%
+  \fi
+  \let\floatno#1%
+}
+
+% \setref calls this to get the XREFLABEL-snt value.  We want an @xref
+% to the FLOATLABEL to expand to "Figure 3.1".  We call \setref when we
+% first read the @float command.
+%
+\def\Yfloat{\floattype@tie \chaplevelprefix\the\floatno}%
+
+% Magic string used for the XREFLABEL-title value, so \xrefX can
+% distinguish floats from other xref types.
+\def\floatmagic{!!float!!}
+
+% #1 is the control sequence we are passed; we expand into a conditional
+% which is true if #1 represents a float ref.  That is, the magic
+% \thissection value which we \setref above.
+%
+\def\iffloat#1{\expandafter\doiffloat#1==\finish}
+%
+% #1 is (maybe) the \floatmagic string.  If so, #2 will be the
+% (safe) float type for this float.  We set \iffloattype to #2.
+%
+\def\doiffloat#1=#2=#3\finish{%
+  \def\temp{#1}%
+  \def\iffloattype{#2}%
+  \ifx\temp\floatmagic
+}
+
+% @listoffloats FLOATTYPE - print a list of floats like a table of contents.
+%
+\parseargdef\listoffloats{%
+  \def\floattype{#1}% floattype
+  {%
+    % the floattype might have accents or other special characters,
+    % but we need to use it in a control sequence name.
+    \indexnofonts
+    \turnoffactive
+    \xdef\safefloattype{\floattype}%
+  }%
+  %
+  % \xrdef saves the floats as a \do-list in \floatlistSAFEFLOATTYPE.
+  \expandafter\ifx\csname floatlist\safefloattype\endcsname \relax
+    \ifhavexrefs
+      % if the user said @listoffloats foo but never @float foo.
+      \message{\linenumber No `\safefloattype' floats to list.}%
+    \fi
+  \else
+    \begingroup
+      \leftskip=\tocindent  % indent these entries like a toc
+      \let\do=\listoffloatsdo
+      \csname floatlist\safefloattype\endcsname
+    \endgroup
+  \fi
+}
+
+% This is called on each entry in a list of floats.  We're passed the
+% xref label, in the form LABEL-title, which is how we save it in the
+% aux file.  We strip off the -title and look up \XRLABEL-lof, which
+% has the text we're supposed to typeset here.
+%
+% Figures without xref labels will not be included in the list (since
+% they won't appear in the aux file).
+%
+\def\listoffloatsdo#1{\listoffloatsdoentry#1\finish}
+\def\listoffloatsdoentry#1-title\finish{{%
+  % Can't fully expand XR#1-lof because it can contain anything.  Just
+  % pass the control sequence.  On the other hand, XR#1-pg is just the
+  % page number, and we want to fully expand that so we can get a link
+  % in pdf output.
+  \toksA = \expandafter{\csname XR#1-lof\endcsname}%
+  %
+  % use the same \entry macro we use to generate the TOC and index.
+  \edef\writeentry{\noexpand\entry{\the\toksA}{\csname XR#1-pg\endcsname}}%
+  \writeentry
+}}
+
+\message{localization,}
+% and i18n.
+
+% @documentlanguage is usually given very early, just after
+% @setfilename.  If done too late, it may not override everything
+% properly.  Single argument is the language abbreviation.
+% It would be nice if we could set up a hyphenation file here.
+%
+\parseargdef\documentlanguage{%
+  \tex % read txi-??.tex file in plain TeX.
+    % Read the file if it exists.
+    \openin 1 txi-#1.tex
+    \ifeof 1
+      \errhelp = \nolanghelp
+      \errmessage{Cannot read language file txi-#1.tex}%
+    \else
+      \input txi-#1.tex
+    \fi
+    \closein 1
+  \endgroup
+}
+\newhelp\nolanghelp{The given language definition file cannot be found or
+is empty.  Maybe you need to install it?  In the current directory
+should work if nowhere else does.}
+
+
+% @documentencoding should change something in TeX eventually, most
+% likely, but for now just recognize it.
+\let\documentencoding = \comment
+
+
+% Page size parameters.
+%
+\newdimen\defaultparindent \defaultparindent = 15pt
+
+\chapheadingskip = 15pt plus 4pt minus 2pt
+\secheadingskip = 12pt plus 3pt minus 2pt
+\subsecheadingskip = 9pt plus 2pt minus 2pt
+
+% Prevent underfull vbox error messages.
+\vbadness = 10000
+
+% Don't be so finicky about underfull hboxes, either.
+\hbadness = 2000
+
+% Following George Bush, just get rid of widows and orphans.
+\widowpenalty=10000
+\clubpenalty=10000
+
+% Use TeX 3.0's \emergencystretch to help line breaking, but if we're
+% using an old version of TeX, don't do anything.  We want the amount of
+% stretch added to depend on the line length, hence the dependence on
+% \hsize.  We call this whenever the paper size is set.
+%
+\def\setemergencystretch{%
+  \ifx\emergencystretch\thisisundefined
+    % Allow us to assign to \emergencystretch anyway.
+    \def\emergencystretch{\dimen0}%
+  \else
+    \emergencystretch = .15\hsize
+  \fi
+}
+
+% Parameters in order: 1) textheight; 2) textwidth; 3) voffset;
+% 4) hoffset; 5) binding offset; 6) topskip; 7) physical page height; 8)
+% physical page width.
+%
+% We also call \setleading{\textleading}, so the caller should define
+% \textleading.  The caller should also set \parskip.
+%
+\def\internalpagesizes#1#2#3#4#5#6#7#8{%
+  \voffset = #3\relax
+  \topskip = #6\relax
+  \splittopskip = \topskip
+  %
+  \vsize = #1\relax
+  \advance\vsize by \topskip
+  \outervsize = \vsize
+  \advance\outervsize by 2\topandbottommargin
+  \pageheight = \vsize
+  %
+  \hsize = #2\relax
+  \outerhsize = \hsize
+  \advance\outerhsize by 0.5in
+  \pagewidth = \hsize
+  %
+  \normaloffset = #4\relax
+  \bindingoffset = #5\relax
+  %
+  \ifpdf
+    \pdfpageheight #7\relax
+    \pdfpagewidth #8\relax
+  \fi
+  %
+  \setleading{\textleading}
+  %
+  \parindent = \defaultparindent
+  \setemergencystretch
+}
+
+% @letterpaper (the default).
+\def\letterpaper{{\globaldefs = 1
+  \parskip = 3pt plus 2pt minus 1pt
+  \textleading = 13.2pt
+  %
+  % If page is nothing but text, make it come out even.
+  \internalpagesizes{46\baselineskip}{6in}%
+                    {\voffset}{.25in}%
+                    {\bindingoffset}{36pt}%
+                    {11in}{8.5in}%
+}}
+
+% Use @smallbook to reset parameters for 7x9.5 (or so) format.
+\def\smallbook{{\globaldefs = 1
+  \parskip = 2pt plus 1pt
+  \textleading = 12pt
+  %
+  \internalpagesizes{7.5in}{5in}%
+                    {\voffset}{.25in}%
+                    {\bindingoffset}{16pt}%
+                    {9.25in}{7in}%
+  %
+  \lispnarrowing = 0.3in
+  \tolerance = 700
+  \hfuzz = 1pt
+  \contentsrightmargin = 0pt
+  \defbodyindent = .5cm
+}}
+
+% Use @afourpaper to print on European A4 paper.
+\def\afourpaper{{\globaldefs = 1
+  \parskip = 3pt plus 2pt minus 1pt
+  \textleading = 13.2pt
+  %
+  % Double-side printing via postscript on Laserjet 4050
+  % prints double-sided nicely when \bindingoffset=10mm and \hoffset=-6mm.
+  % To change the settings for a different printer or situation, adjust
+  % \normaloffset until the front-side and back-side texts align.  Then
+  % do the same for \bindingoffset.  You can set these for testing in
+  % your texinfo source file like this:
+  % @tex
+  % \global\normaloffset = -6mm
+  % \global\bindingoffset = 10mm
+  % @end tex
+  \internalpagesizes{51\baselineskip}{160mm}
+                    {\voffset}{\hoffset}%
+                    {\bindingoffset}{44pt}%
+                    {297mm}{210mm}%
+  %
+  \tolerance = 700
+  \hfuzz = 1pt
+  \contentsrightmargin = 0pt
+  \defbodyindent = 5mm
+}}
+
+% Use @afivepaper to print on European A5 paper.
+% From romildo@urano.iceb.ufop.br, 2 July 2000.
+% He also recommends making @example and @lisp be small.
+\def\afivepaper{{\globaldefs = 1
+  \parskip = 2pt plus 1pt minus 0.1pt
+  \textleading = 12.5pt
+  %
+  \internalpagesizes{160mm}{120mm}%
+                    {\voffset}{\hoffset}%
+                    {\bindingoffset}{8pt}%
+                    {210mm}{148mm}%
+  %
+  \lispnarrowing = 0.2in
+  \tolerance = 800
+  \hfuzz = 1.2pt
+  \contentsrightmargin = 0pt
+  \defbodyindent = 2mm
+  \tableindent = 12mm
+}}
+
+% A specific text layout, 24x15cm overall, intended for A4 paper.
+\def\afourlatex{{\globaldefs = 1
+  \afourpaper
+  \internalpagesizes{237mm}{150mm}%
+                    {\voffset}{4.6mm}%
+                    {\bindingoffset}{7mm}%
+                    {297mm}{210mm}%
+  %
+  % Must explicitly reset to 0 because we call \afourpaper.
+  \globaldefs = 0
+}}
+
+% Use @afourwide to print on A4 paper in landscape format.
+\def\afourwide{{\globaldefs = 1
+  \afourpaper
+  \internalpagesizes{241mm}{165mm}%
+                    {\voffset}{-2.95mm}%
+                    {\bindingoffset}{7mm}%
+                    {297mm}{210mm}%
+  \globaldefs = 0
+}}
+
+% @pagesizes TEXTHEIGHT[,TEXTWIDTH]
+% Perhaps we should allow setting the margins, \topskip, \parskip,
+% and/or leading, also. Or perhaps we should compute them somehow.
+%
+\parseargdef\pagesizes{\pagesizesyyy #1,,\finish}
+\def\pagesizesyyy#1,#2,#3\finish{{%
+  \setbox0 = \hbox{\ignorespaces #2}\ifdim\wd0 > 0pt \hsize=#2\relax \fi
+  \globaldefs = 1
+  %
+  \parskip = 3pt plus 2pt minus 1pt
+  \setleading{\textleading}%
+  %
+  \dimen0 = #1
+  \advance\dimen0 by \voffset
+  %
+  \dimen2 = \hsize
+  \advance\dimen2 by \normaloffset
+  %
+  \internalpagesizes{#1}{\hsize}%
+                    {\voffset}{\normaloffset}%
+                    {\bindingoffset}{44pt}%
+                    {\dimen0}{\dimen2}%
+}}
+
+% Set default to letter.
+%
+\letterpaper
+
+
+\message{and turning on texinfo input format.}
+
+% Define macros to output various characters with catcode for normal text.
+\catcode`\"=\other
+\catcode`\~=\other
+\catcode`\^=\other
+\catcode`\_=\other
+\catcode`\|=\other
+\catcode`\<=\other
+\catcode`\>=\other
+\catcode`\+=\other
+\catcode`\$=\other
+\def\normaldoublequote{"}
+\def\normaltilde{~}
+\def\normalcaret{^}
+\def\normalunderscore{_}
+\def\normalverticalbar{|}
+\def\normalless{<}
+\def\normalgreater{>}
+\def\normalplus{+}
+\def\normaldollar{$}%$ font-lock fix
+
+% This macro is used to make a character print one way in \tt
+% (where it can probably be output as-is), and another way in other fonts,
+% where something hairier probably needs to be done.
+%
+% #1 is what to print if we are indeed using \tt; #2 is what to print
+% otherwise.  Since all the Computer Modern typewriter fonts have zero
+% interword stretch (and shrink), and it is reasonable to expect all
+% typewriter fonts to have this, we can check that font parameter.
+%
+\def\ifusingtt#1#2{\ifdim \fontdimen3\font=0pt #1\else #2\fi}
+
+% Same as above, but check for italic font.  Actually this also catches
+% non-italic slanted fonts since it is impossible to distinguish them from
+% italic fonts.  But since this is only used by $ and it uses \sl anyway
+% this is not a problem.
+\def\ifusingit#1#2{\ifdim \fontdimen1\font>0pt #1\else #2\fi}
+
+% Turn off all special characters except @
+% (and those which the user can use as if they were ordinary).
+% Most of these we simply print from the \tt font, but for some, we can
+% use math or other variants that look better in normal text.
+
+\catcode`\"=\active
+\def\activedoublequote{{\tt\char34}}
+\let"=\activedoublequote
+\catcode`\~=\active
+\def~{{\tt\char126}}
+\chardef\hat=`\^
+\catcode`\^=\active
+\def^{{\tt \hat}}
+
+\catcode`\_=\active
+\def_{\ifusingtt\normalunderscore\_}
+% Subroutine for the previous macro.
+\def\_{\leavevmode \kern.07em \vbox{\hrule width.3em height.1ex}\kern .07em }
+
+\catcode`\|=\active
+\def|{{\tt\char124}}
+\chardef \less=`\<
+\catcode`\<=\active
+\def<{{\tt \less}}
+\chardef \gtr=`\>
+\catcode`\>=\active
+\def>{{\tt \gtr}}
+\catcode`\+=\active
+\def+{{\tt \char 43}}
+\catcode`\$=\active
+\def${\ifusingit{{\sl\$}}\normaldollar}%$ font-lock fix
+
+% If a .fmt file is being used, characters that might appear in a file
+% name cannot be active until we have parsed the command line.
+% So turn them off again, and have \everyjob (or @setfilename) turn them on.
+% \otherifyactive is called near the end of this file.
+\def\otherifyactive{\catcode`+=\other \catcode`\_=\other}
+
+\catcode`\@=0
+
+% \backslashcurfont outputs one backslash character in current font,
+% as in \char`\\.
+\global\chardef\backslashcurfont=`\\
+\global\let\rawbackslashxx=\backslashcurfont  % let existing .??s files work
+
+% \rawbackslash defines an active \ to do \backslashcurfont.
+% \otherbackslash defines an active \ to be a literal `\' character with
+% catcode other.
+{\catcode`\\=\active
+ @gdef@rawbackslash{@let\=@backslashcurfont}
+ @gdef@otherbackslash{@let\=@realbackslash}
+}
+
+% \realbackslash is an actual character `\' with catcode other.
+{\catcode`\\=\other @gdef@realbackslash{\}}
+
+% \normalbackslash outputs one backslash in fixed width font.
+\def\normalbackslash{{\tt\backslashcurfont}}
+
+\catcode`\\=\active
+
+% Used sometimes to turn off (effectively) the active characters
+% even after parsing them.
+@def@turnoffactive{%
+  @let"=@normaldoublequote
+  @let\=@realbackslash
+  @let~=@normaltilde
+  @let^=@normalcaret
+  @let_=@normalunderscore
+  @let|=@normalverticalbar
+  @let<=@normalless
+  @let>=@normalgreater
+  @let+=@normalplus
+  @let$=@normaldollar %$ font-lock fix
+  @unsepspaces
+}
+
+% Same as @turnoffactive except outputs \ as {\tt\char`\\} instead of
+% the literal character `\'.  (Thus, \ is not expandable when this is in
+% effect.)
+%
+@def@normalturnoffactive{@turnoffactive @let\=@normalbackslash}
+
+% Make _ and + \other characters, temporarily.
+% This is canceled by @fixbackslash.
+@otherifyactive
+
+% If a .fmt file is being used, we don't want the `\input texinfo' to show up.
+% That is what \eatinput is for; after that, the `\' should revert to printing
+% a backslash.
+%
+@gdef@eatinput input texinfo{@fixbackslash}
+@global@let\ = @eatinput
+
+% On the other hand, perhaps the file did not have a `\input texinfo'. Then
+% the first `\{ in the file would cause an error. This macro tries to fix
+% that, assuming it is called before the first `\' could plausibly occur.
+% Also back turn on active characters that might appear in the input
+% file name, in case not using a pre-dumped format.
+%
+@gdef@fixbackslash{%
+  @ifx\@eatinput @let\ = @normalbackslash @fi
+  @catcode`+=@active
+  @catcode`@_=@active
+}
+
+% Say @foo, not \foo, in error messages.
+@escapechar = `@@
+
+% These look ok in all fonts, so just make them not special.
+@catcode`@& = @other
+@catcode`@# = @other
+@catcode`@% = @other
+
+
+@c Local variables:
+@c eval: (add-hook 'write-file-hooks 'time-stamp)
+@c page-delimiter: "^\\\\message"
+@c time-stamp-start: "def\\\\texinfoversion{"
+@c time-stamp-format: "%:y-%02m-%02d.%02H"
+@c time-stamp-end: "}"
+@c End:
+
+@c vim:sw=2:
+
+@ignore
+   arch-tag: e1b36e32-c96e-4135-a41a-0b2efa2ea115
+@end ignore
diff --git a/gsl-1.9/doc/usage.texi b/gsl-1.9/doc/usage.texi
new file mode 100644
index 0000000..adcd494
--- /dev/null
+++ b/gsl-1.9/doc/usage.texi
@@ -0,0 +1,489 @@
+@cindex standards conformance, ANSI C
+@cindex ANSI C, use of
+@cindex C extensions, compatible use of
+@cindex compatibility
+This chapter describes how to compile programs that use GSL, and
+introduces its conventions.  
+
+@menu
+* An Example Program::          
+* Compiling and Linking::       
+* Shared Libraries::            
+* ANSI C Compliance::           
+* Inline functions::            
+* Long double::                 
+* Portability functions::       
+* Alternative optimized functions::  
+* Support for different numeric types::  
+* Compatibility with C++::      
+* Aliasing of arrays::          
+* Thread-safety::               
+* Deprecated Functions::        
+* Code Reuse::                  
+@end menu
+
+@node   An Example Program
+@section An Example Program
+
+The following short program demonstrates the use of the library by
+computing the value of the Bessel function @math{J_0(x)} for @math{x=5},
+
+@example
+@verbatiminclude examples/intro.c
+@end example
+
+@noindent
+The output is shown below, and should be correct to double-precision
+accuracy,
+
+@example
+@verbatiminclude examples/intro.out
+@end example
+
+@noindent
+The steps needed to compile this program are described in the following
+sections.
+
+@node Compiling and Linking
+@section Compiling and Linking
+@cindex compiling programs, include paths
+@cindex including GSL header files
+@cindex header files, including
+The library header files are installed in their own @file{gsl}
+directory.  You should write any preprocessor include statements with a
+@file{gsl/} directory prefix thus,
+
+@example
+#include <gsl/gsl_math.h>
+@end example
+
+@noindent
+If the directory is not installed on the standard search path of your
+compiler you will also need to provide its location to the preprocessor
+as a command line flag.  The default location of the @file{gsl}
+directory is @file{/usr/local/include/gsl}.  A typical compilation
+command for a source file @file{example.c} with the GNU C compiler
+@code{gcc} is,
+
+@example
+$ gcc -Wall -I/usr/local/include -c example.c
+@end example
+
+@noindent
+This results in an object file @file{example.o}.   The default
+include path for @code{gcc} searches @file{/usr/local/include} automatically so
+the @code{-I} option can actually be omitted when GSL is installed 
+in its default location.
+
+@menu
+* Linking programs with the library::  
+* Linking with an alternative BLAS library::  
+@end menu
+
+@node Linking programs with the library
+@subsection Linking programs with the library
+@cindex compiling programs, library paths
+@cindex linking with GSL libraries
+@cindex libraries, linking with
+The library is installed as a single file, @file{libgsl.a}.  A shared
+version of the library @file{libgsl.so} is also installed on systems
+that support shared libraries.  The default location of these files is
+@file{/usr/local/lib}.  If this directory is not on the standard search
+path of your linker you will also need to provide its location as a
+command line flag.
+
+To link against the library you need to specify
+both the main library and a supporting @sc{cblas} library, which
+provides standard basic linear algebra subroutines.  A suitable
+@sc{cblas} implementation is provided in the library
+@file{libgslcblas.a} if your system does not provide one.  The following
+example shows how to link an application with the library,
+
+@example
+$ gcc -L/usr/local/lib example.o -lgsl -lgslcblas -lm
+@end example
+
+@noindent
+The default library path for @code{gcc} searches @file{/usr/local/lib}
+automatically so the @code{-L} option can be omitted when GSL is
+installed in its default location.
+
+@node Linking with an alternative BLAS library
+@subsection Linking with an alternative BLAS library
+
+The following command line shows how you would link the same application
+with an alternative @sc{cblas} library called @file{libcblas},
+
+@example
+$ gcc example.o -lgsl -lcblas -lm
+@end example
+
+@noindent
+For the best performance an optimized platform-specific @sc{cblas}
+library should be used for @code{-lcblas}.  The library must conform to
+the @sc{cblas} standard.  The @sc{atlas} package provides a portable
+high-performance @sc{blas} library with a @sc{cblas} interface.  It is
+free software and should be installed for any work requiring fast vector
+and matrix operations.  The following command line will link with the
+@sc{atlas} library and its @sc{cblas} interface,
+
+@example
+$ gcc example.o -lgsl -lcblas -latlas -lm
+@end example
+
+@noindent
+For more information see @ref{BLAS Support}.
+
+@comment  The program @code{gsl-config} provides information on the local version
+@comment  of the library.  For example, the following command shows that the
+@comment  library has been installed under the directory @file{/usr/local},
+
+@comment  @example
+@comment  $ gsl-config --prefix
+@comment  /usr/local
+@comment  @end example
+@comment  @noindent
+@comment  Further information is available using the command @code{gsl-config --help}.
+
+@node Shared Libraries
+@section Shared Libraries
+@cindex shared libraries
+@cindex libraries, shared
+@cindex LD_LIBRARY_PATH
+To run a program linked with the shared version of the library the
+operating system must be able to locate the corresponding @file{.so}
+file at runtime.  If the library cannot be found, the following error
+will occur:
+
+@example
+$ ./a.out 
+./a.out: error while loading shared libraries: 
+libgsl.so.0: cannot open shared object file: No such 
+file or directory
+@end example
+
+@noindent
+To avoid this error, define the shell variable @code{LD_LIBRARY_PATH} to
+include the directory where the library is installed.
+
+For example, in the Bourne shell (@code{/bin/sh} or @code{/bin/bash}),
+the library search path can be set with the following commands:
+
+@example
+$ LD_LIBRARY_PATH=/usr/local/lib:$LD_LIBRARY_PATH 
+$ export LD_LIBRARY_PATH
+$ ./example
+@end example
+
+@noindent
+In the C-shell (@code{/bin/csh} or @code{/bin/tcsh}) the equivalent
+command is,
+
+@example
+% setenv LD_LIBRARY_PATH /usr/local/lib:$LD_LIBRARY_PATH
+@end example
+
+@noindent
+The standard prompt for the C-shell in the example above is the percent
+character @samp{%}, and should not be typed as part of the command.
+
+To save retyping these commands each session they should be placed in an
+individual or system-wide login file.
+
+To compile a statically linked version of the program, use the
+@code{-static} flag in @code{gcc},
+
+@example
+$ gcc -static example.o -lgsl -lgslcblas -lm
+@end example
+
+@node ANSI C Compliance
+@section ANSI C Compliance
+
+The library is written in ANSI C and is intended to conform to the ANSI
+C standard (C89).  It should be portable to any system with a working
+ANSI C compiler.
+
+The library does not rely on any non-ANSI extensions in the interface it
+exports to the user.  Programs you write using GSL can be ANSI
+compliant.  Extensions which can be used in a way compatible with pure
+ANSI C are supported, however, via conditional compilation.  This allows
+the library to take advantage of compiler extensions on those platforms
+which support them.
+
+When an ANSI C feature is known to be broken on a particular system the
+library will exclude any related functions at compile-time.  This should
+make it impossible to link a program that would use these functions and
+give incorrect results.
+
+To avoid namespace conflicts all exported function names and variables
+have the prefix @code{gsl_}, while exported macros have the prefix
+@code{GSL_}.
+
+@node Inline functions
+@section Inline functions
+
+@cindex inline functions
+@cindex HAVE_INLINE
+The @code{inline} keyword is not part of the original ANSI C standard
+(C89) and the library does not export any inline function definitions by
+default. However, the library provides optional inline versions of
+performance-critical functions by conditional compilation.  The inline
+versions of these functions can be included by defining the macro
+@code{HAVE_INLINE} when compiling an application,
+
+@example
+$ gcc -Wall -c -DHAVE_INLINE example.c
+@end example
+
+@noindent
+If you use @code{autoconf} this macro can be defined automatically.  If
+you do not define the macro @code{HAVE_INLINE} then the slower
+non-inlined versions of the functions will be used instead.
+
+Note that the actual usage of the inline keyword is @code{extern
+inline}, which eliminates unnecessary function definitions in @sc{gcc}.
+If the form @code{extern inline} causes problems with other compilers a
+stricter autoconf test can be used, see @ref{Autoconf Macros}.
+
+@node Long double
+@section Long double
+@cindex long double
+The extended numerical type @code{long double} is part of the ANSI C
+standard and should be available in every modern compiler.  However, the
+precision of @code{long double} is platform dependent, and this should
+be considered when using it.  The IEEE standard only specifies the
+minimum precision of extended precision numbers, while the precision of
+@code{double} is the same on all platforms.
+
+In some system libraries the @code{stdio.h} formatted input/output
+functions @code{printf} and @code{scanf} are not implemented correctly
+for @code{long double}.  Undefined or incorrect results are avoided by
+testing these functions during the @code{configure} stage of library
+compilation and eliminating certain GSL functions which depend on them
+if necessary.  The corresponding line in the @code{configure} output
+looks like this,
+
+@example
+checking whether printf works with long double... no
+@end example
+
+@noindent
+Consequently when @code{long double} formatted input/output does not
+work on a given system it should be impossible to link a program which
+uses GSL functions dependent on this.
+
+If it is necessary to work on a system which does not support formatted
+@code{long double} input/output then the options are to use binary
+formats or to convert @code{long double} results into @code{double} for
+reading and writing.
+
+@node Portability functions
+@section Portability functions
+
+To help in writing portable applications GSL provides some
+implementations of functions that are found in other libraries, such as
+the BSD math library.  You can write your application to use the native
+versions of these functions, and substitute the GSL versions via a
+preprocessor macro if they are unavailable on another platform. 
+
+For example, after determining whether the BSD function @code{hypot} is
+available you can include the following macro definitions in a file
+@file{config.h} with your application,
+
+@example
+/* Substitute gsl_hypot for missing system hypot */
+
+#ifndef HAVE_HYPOT
+#define hypot gsl_hypot
+#endif
+@end example
+
+@noindent
+The application source files can then use the include command
+@code{#include <config.h>} to replace each occurrence of @code{hypot} by
+@code{gsl_hypot} when @code{hypot} is not available.  This substitution
+can be made automatically if you use @code{autoconf}, see @ref{Autoconf
+Macros}.
+
+In most circumstances the best strategy is to use the native versions of
+these functions when available, and fall back to GSL versions otherwise,
+since this allows your application to take advantage of any
+platform-specific optimizations in the system library.  This is the
+strategy used within GSL itself.
+
+@node Alternative optimized functions
+@section Alternative optimized functions
+
+@cindex alternative optimized functions
+@cindex optimized functions, alternatives
+The main implementation of some functions in the library will not be
+optimal on all architectures.  For example, there are several ways to
+compute a Gaussian random variate and their relative speeds are
+platform-dependent.  In cases like this the library provides alternative
+implementations of these functions with the same interface.  If you
+write your application using calls to the standard implementation you
+can select an alternative version later via a preprocessor definition.
+It is also possible to introduce your own optimized functions this way
+while retaining portability.  The following lines demonstrate the use of
+a platform-dependent choice of methods for sampling from the Gaussian
+distribution,
+
+@example
+#ifdef SPARC
+#define gsl_ran_gaussian gsl_ran_gaussian_ratio_method
+#endif
+#ifdef INTEL
+#define gsl_ran_gaussian my_gaussian
+#endif
+@end example
+
+@noindent
+These lines would be placed in the configuration header file
+@file{config.h} of the application, which should then be included by all
+the source files.  Note that the alternative implementations will not
+produce bit-for-bit identical results, and in the case of random number
+distributions will produce an entirely different stream of random
+variates.
+
+@node Support for different numeric types
+@section Support for different numeric types
+
+Many functions in the library are defined for different numeric types.
+This feature is implemented by varying the name of the function with a
+type-related modifier---a primitive form of C++ templates.  The
+modifier is inserted into the function name after the initial module
+prefix.  The following table shows the function names defined for all
+the numeric types of an imaginary module @code{gsl_foo} with function
+@code{fn},
+
+@example
+gsl_foo_fn               double        
+gsl_foo_long_double_fn   long double   
+gsl_foo_float_fn         float         
+gsl_foo_long_fn          long          
+gsl_foo_ulong_fn         unsigned long 
+gsl_foo_int_fn           int           
+gsl_foo_uint_fn          unsigned int  
+gsl_foo_short_fn         short         
+gsl_foo_ushort_fn        unsigned short
+gsl_foo_char_fn          char          
+gsl_foo_uchar_fn         unsigned char 
+@end example
+
+@noindent
+The normal numeric precision @code{double} is considered the default and
+does not require a suffix.  For example, the function
+@code{gsl_stats_mean} computes the mean of double precision numbers,
+while the function @code{gsl_stats_int_mean} computes the mean of
+integers.
+
+A corresponding scheme is used for library defined types, such as
+@code{gsl_vector} and @code{gsl_matrix}.  In this case the modifier is
+appended to the type name.  For example, if a module defines a new
+type-dependent struct or typedef @code{gsl_foo} it is modified for other
+types in the following way,
+
+@example
+gsl_foo                  double        
+gsl_foo_long_double      long double   
+gsl_foo_float            float         
+gsl_foo_long             long          
+gsl_foo_ulong            unsigned long 
+gsl_foo_int              int           
+gsl_foo_uint             unsigned int  
+gsl_foo_short            short         
+gsl_foo_ushort           unsigned short
+gsl_foo_char             char          
+gsl_foo_uchar            unsigned char 
+@end example
+
+@noindent
+When a module contains type-dependent definitions the library provides
+individual header files for each type.  The filenames are modified as
+shown in the below.  For convenience the default header includes the
+definitions for all the types.  To include only the double precision
+header file, or any other specific type, use its individual filename.
+
+@example
+#include <gsl/gsl_foo.h>               All types
+#include <gsl/gsl_foo_double.h>        double        
+#include <gsl/gsl_foo_long_double.h>   long double   
+#include <gsl/gsl_foo_float.h>         float         
+#include <gsl/gsl_foo_long.h>          long          
+#include <gsl/gsl_foo_ulong.h>         unsigned long 
+#include <gsl/gsl_foo_int.h>           int           
+#include <gsl/gsl_foo_uint.h>          unsigned int  
+#include <gsl/gsl_foo_short.h>         short         
+#include <gsl/gsl_foo_ushort.h>        unsigned short
+#include <gsl/gsl_foo_char.h>          char          
+#include <gsl/gsl_foo_uchar.h>         unsigned char 
+@end example
+
+@node Compatibility with C++
+@section Compatibility with C++
+@cindex C++, compatibility
+The library header files automatically define functions to have
+@code{extern "C"} linkage when included in C++ programs.  This allows
+the functions to be called directly from C++.
+
+To use C++ exception handling within user-defined functions passed to
+the library as parameters, the library must be built with the
+additional @code{CFLAGS} compilation option @option{-fexceptions}.
+
+@node Aliasing of arrays
+@section Aliasing of arrays
+@cindex aliasing of arrays
+The library assumes that arrays, vectors and matrices passed as
+modifiable arguments are not aliased and do not overlap with each other.
+This removes the need for the library to handle overlapping memory
+regions as a special case, and allows additional optimizations to be
+used.  If overlapping memory regions are passed as modifiable arguments
+then the results of such functions will be undefined.  If the arguments
+will not be modified (for example, if a function prototype declares them
+as @code{const} arguments) then overlapping or aliased memory regions
+can be safely used.
+
+@node Thread-safety
+@section Thread-safety
+
+The library can be used in multi-threaded programs.  All the functions
+are thread-safe, in the sense that they do not use static variables.
+Memory is always associated with objects and not with functions.  For
+functions which use @dfn{workspace} objects as temporary storage the
+workspaces should be allocated on a per-thread basis.  For functions
+which use @dfn{table} objects as read-only memory the tables can be used
+by multiple threads simultaneously.  Table arguments are always declared
+@code{const} in function prototypes, to indicate that they may be
+safely accessed by different threads.
+
+There are a small number of static global variables which are used to
+control the overall behavior of the library (e.g. whether to use
+range-checking, the function to call on fatal error, etc).  These
+variables are set directly by the user, so they should be initialized
+once at program startup and not modified by different threads.
+
+@node Deprecated Functions
+@section Deprecated Functions
+@cindex deprecated functions
+
+From time to time, it may be necessary for the definitions of some
+functions to be altered or removed from the library.  In these
+circumstances the functions will first be declared @dfn{deprecated} and
+then removed from subsequent versions of the library.  Functions that
+are deprecated can be disabled in the current release by setting the
+preprocessor definition @code{GSL_DISABLE_DEPRECATED}.  This allows
+existing code to be tested for forwards compatibility.
+
+@node Code Reuse
+@section Code Reuse
+@cindex code reuse in applications
+@cindex source code, reuse in applications
+Where possible the routines in the library have been written to avoid
+dependencies between modules and files.  This should make it possible to
+extract individual functions for use in your own applications, without
+needing to have the whole library installed.  You may need to define
+certain macros such as @code{GSL_ERROR} and remove some @code{#include}
+statements in order to compile the files as standalone units. Reuse of
+the library code in this way is encouraged, subject to the terms of the
+GNU General Public License.
diff --git a/gsl-1.9/doc/vdp.eps b/gsl-1.9/doc/vdp.eps
new file mode 100644
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--- /dev/null
+++ b/gsl-1.9/doc/vdp.eps
@@ -0,0 +1,1082 @@
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+stroke
+grestore
+end
+showpage
+%%Trailer
+%%DocumentFonts: Helvetica
diff --git a/gsl-1.9/doc/vectors.texi b/gsl-1.9/doc/vectors.texi
new file mode 100644
index 0000000..e17a1af
--- /dev/null
+++ b/gsl-1.9/doc/vectors.texi
@@ -0,0 +1,1616 @@
+@cindex blocks
+@cindex vectors
+@cindex matrices
+
+The functions described in this chapter provide a simple vector and
+matrix interface to ordinary C arrays. The memory management of these
+arrays is implemented using a single underlying type, known as a
+block. By writing your functions in terms of vectors and matrices you
+can pass a single structure containing both data and dimensions as an
+argument without needing additional function parameters.  The structures
+are compatible with the vector and matrix formats used by @sc{blas}
+routines.
+
+@menu
+* Data types::                  
+* Blocks::                      
+* Vectors::                     
+* Matrices::                    
+* Vector and Matrix References and Further Reading::  
+@end menu
+
+@node Data types
+@section Data types
+
+All the functions are available for each of the standard data-types.
+The versions for @code{double} have the prefix @code{gsl_block},
+@code{gsl_vector} and @code{gsl_matrix}.  Similarly the versions for
+single-precision @code{float} arrays have the prefix
+@code{gsl_block_float}, @code{gsl_vector_float} and
+@code{gsl_matrix_float}.  The full list of available types is given
+below,
+
+@example
+gsl_block                       double         
+gsl_block_float                 float         
+gsl_block_long_double           long double   
+gsl_block_int                   int           
+gsl_block_uint                  unsigned int  
+gsl_block_long                  long          
+gsl_block_ulong                 unsigned long 
+gsl_block_short                 short         
+gsl_block_ushort                unsigned short
+gsl_block_char                  char          
+gsl_block_uchar                 unsigned char 
+gsl_block_complex               complex double        
+gsl_block_complex_float         complex float         
+gsl_block_complex_long_double   complex long double   
+@end example
+
+@noindent
+Corresponding types exist for the @code{gsl_vector} and
+@code{gsl_matrix} functions.
+
+
+
+@node Blocks
+@section Blocks
+
+For consistency all memory is allocated through a @code{gsl_block}
+structure.  The structure contains two components, the size of an area of
+memory and a pointer to the memory.  The @code{gsl_block} structure looks
+like this,
+
+@example
+typedef struct
+@{
+  size_t size;
+  double * data;
+@} gsl_block;
+@end example
+@comment
+
+@noindent
+Vectors and matrices are made by @dfn{slicing} an underlying block. A
+slice is a set of elements formed from an initial offset and a
+combination of indices and step-sizes. In the case of a matrix the
+step-size for the column index represents the row-length.  The step-size
+for a vector is known as the @dfn{stride}.
+
+The functions for allocating and deallocating blocks are defined in
+@file{gsl_block.h}
+
+@menu
+* Block allocation::            
+* Reading and writing blocks::  
+* Example programs for blocks::  
+@end menu
+
+@node Block allocation
+@subsection Block allocation
+
+The functions for allocating memory to a block follow the style of
+@code{malloc} and @code{free}.  In addition they also perform their own
+error checking.  If there is insufficient memory available to allocate a
+block then the functions call the GSL error handler (with an error
+number of @code{GSL_ENOMEM}) in addition to returning a null
+pointer.  Thus if you use the library error handler to abort your program
+then it isn't necessary to check every @code{alloc}.  
+
+@deftypefun {gsl_block *} gsl_block_alloc (size_t @var{n})
+This function allocates memory for a block of @var{n} double-precision
+elements, returning a pointer to the block struct.  The block is not
+initialized and so the values of its elements are undefined.  Use the
+function @code{gsl_block_calloc} if you want to ensure that all the
+elements are initialized to zero.
+
+A null pointer is returned if insufficient memory is available to create
+the block.
+@end deftypefun
+
+@deftypefun {gsl_block *} gsl_block_calloc (size_t @var{n})
+This function allocates memory for a block and initializes all the
+elements of the block to zero.
+@end deftypefun
+
+@deftypefun void gsl_block_free (gsl_block * @var{b})
+This function frees the memory used by a block @var{b} previously
+allocated with @code{gsl_block_alloc} or @code{gsl_block_calloc}.  The
+block @var{b} must be a valid block object (a null pointer is not
+allowed).
+@end deftypefun
+
+@node Reading and writing blocks
+@subsection Reading and writing blocks
+
+The library provides functions for reading and writing blocks to a file
+as binary data or formatted text.
+
+@deftypefun int gsl_block_fwrite (FILE * @var{stream}, const gsl_block * @var{b})
+This function writes the elements of the block @var{b} to the stream
+@var{stream} in binary format.  The return value is 0 for success and
+@code{GSL_EFAILED} if there was a problem writing to the file.  Since the
+data is written in the native binary format it may not be portable
+between different architectures.
+@end deftypefun
+
+@deftypefun int gsl_block_fread (FILE * @var{stream}, gsl_block * @var{b})
+This function reads into the block @var{b} from the open stream
+@var{stream} in binary format.  The block @var{b} must be preallocated
+with the correct length since the function uses the size of @var{b} to
+determine how many bytes to read.  The return value is 0 for success and
+@code{GSL_EFAILED} if there was a problem reading from the file.  The
+data is assumed to have been written in the native binary format on the
+same architecture.
+@end deftypefun
+
+@deftypefun int gsl_block_fprintf (FILE * @var{stream}, const gsl_block * @var{b}, const char * @var{format})
+This function writes the elements of the block @var{b} line-by-line to
+the stream @var{stream} using the format specifier @var{format}, which
+should be one of the @code{%g}, @code{%e} or @code{%f} formats for
+floating point numbers and @code{%d} for integers.  The function returns
+0 for success and @code{GSL_EFAILED} if there was a problem writing to
+the file.
+@end deftypefun
+
+@deftypefun int gsl_block_fscanf (FILE * @var{stream}, gsl_block * @var{b})
+This function reads formatted data from the stream @var{stream} into the
+block @var{b}.  The block @var{b} must be preallocated with the correct
+length since the function uses the size of @var{b} to determine how many
+numbers to read.  The function returns 0 for success and
+@code{GSL_EFAILED} if there was a problem reading from the file.
+@end deftypefun
+@comment
+
+@node Example programs for blocks
+@subsection Example programs for blocks
+
+The following program shows how to allocate a block,
+
+@example
+@verbatiminclude examples/block.c
+@end example
+@comment
+
+@noindent
+Here is the output from the program,
+
+@example
+@verbatiminclude examples/block.out
+@end example
+@comment
+
+@node Vectors
+@section Vectors
+@cindex vectors
+@cindex stride, of vector index
+
+Vectors are defined by a @code{gsl_vector} structure which describes a
+slice of a block.  Different vectors can be created which point to the
+same block.  A vector slice is a set of equally-spaced elements of an
+area of memory.
+
+The @code{gsl_vector} structure contains five components, the
+@dfn{size}, the @dfn{stride}, a pointer to the memory where the elements
+are stored, @var{data}, a pointer to the block owned by the vector,
+@var{block}, if any, and an ownership flag, @var{owner}.  The structure
+is very simple and looks like this,
+
+@example
+typedef struct
+@{
+  size_t size;
+  size_t stride;
+  double * data;
+  gsl_block * block;
+  int owner;
+@} gsl_vector;
+@end example
+@comment
+
+@noindent
+The @var{size} is simply the number of vector elements.  The range of
+valid indices runs from 0 to @code{size-1}.  The @var{stride} is the
+step-size from one element to the next in physical memory, measured in
+units of the appropriate datatype.  The pointer @var{data} gives the
+location of the first element of the vector in memory.  The pointer
+@var{block} stores the location of the memory block in which the vector
+elements are located (if any).  If the vector owns this block then the
+@var{owner} field is set to one and the block will be deallocated when the
+vector is freed.  If the vector points to a block owned by another
+object then the @var{owner} field is zero and any underlying block will not be
+deallocated with the vector.
+
+The functions for allocating and accessing vectors are defined in
+@file{gsl_vector.h}
+
+@menu
+* Vector allocation::           
+* Accessing vector elements::   
+* Initializing vector elements::  
+* Reading and writing vectors::  
+* Vector views::                
+* Copying vectors::             
+* Exchanging elements::         
+* Vector operations::           
+* Finding maximum and minimum elements of vectors::  
+* Vector properties::           
+* Example programs for vectors::  
+@end menu
+
+@node Vector allocation
+@subsection Vector allocation
+
+The functions for allocating memory to a vector follow the style of
+@code{malloc} and @code{free}.  In addition they also perform their own
+error checking.  If there is insufficient memory available to allocate a
+vector then the functions call the GSL error handler (with an error
+number of @code{GSL_ENOMEM}) in addition to returning a null
+pointer.  Thus if you use the library error handler to abort your program
+then it isn't necessary to check every @code{alloc}.
+
+@deftypefun {gsl_vector *} gsl_vector_alloc (size_t @var{n})
+This function creates a vector of length @var{n}, returning a pointer to
+a newly initialized vector struct. A new block is allocated for the
+elements of the vector, and stored in the @var{block} component of the
+vector struct.  The block is ``owned'' by the vector, and will be
+deallocated when the vector is deallocated.
+@end deftypefun
+
+@deftypefun {gsl_vector *} gsl_vector_calloc (size_t @var{n})
+This function allocates memory for a vector of length @var{n} and
+initializes all the elements of the vector to zero.
+@end deftypefun
+
+@deftypefun void gsl_vector_free (gsl_vector * @var{v})
+This function frees a previously allocated vector @var{v}.  If the
+vector was created using @code{gsl_vector_alloc} then the block
+underlying the vector will also be deallocated.  If the vector has
+been created from another object then the memory is still owned by
+that object and will not be deallocated.  The vector @var{v} must be a
+valid vector object (a null pointer is not allowed).
+@end deftypefun
+
+@node Accessing vector elements
+@subsection Accessing vector elements
+@cindex vectors, range-checking
+@cindex range-checking for vectors
+@cindex bounds checking, extension to GCC
+@cindex gcc extensions, range-checking
+@cindex Fortran range checking, equivalent in gcc
+
+Unlike @sc{fortran} compilers, C compilers do not usually provide
+support for range checking of vectors and matrices.  Range checking is
+available in the GNU C Compiler bounds-checking extension, but it is not
+part of the default installation of GCC.  The functions
+@code{gsl_vector_get} and @code{gsl_vector_set} can perform portable
+range checking for you and report an error if you attempt to access
+elements outside the allowed range.
+
+The functions for accessing the elements of a vector or matrix are
+defined in @file{gsl_vector.h} and declared @code{extern inline} to
+eliminate function-call overhead.  You must compile your program with
+the macro @code{HAVE_INLINE} defined to use these functions.  
+
+If necessary you can turn off range checking completely without
+modifying any source files by recompiling your program with the
+preprocessor definition @code{GSL_RANGE_CHECK_OFF}.  Provided your
+compiler supports inline functions the effect of turning off range
+checking is to replace calls to @code{gsl_vector_get(v,i)} by
+@code{v->data[i*v->stride]} and calls to @code{gsl_vector_set(v,i,x)} by
+@code{v->data[i*v->stride]=x}.  Thus there should be no performance
+penalty for using the range checking functions when range checking is
+turned off.
+
+@deftypefun double gsl_vector_get (const gsl_vector * @var{v}, size_t @var{i})
+This function returns the @var{i}-th element of a vector @var{v}.  If
+@var{i} lies outside the allowed range of 0 to @math{@var{n}-1} then the error
+handler is invoked and 0 is returned.
+@end deftypefun
+
+@deftypefun void gsl_vector_set (gsl_vector * @var{v}, size_t @var{i}, double @var{x})
+This function sets the value of the @var{i}-th element of a vector
+@var{v} to @var{x}.  If @var{i} lies outside the allowed range of 0 to
+@math{@var{n}-1} then the error handler is invoked.
+@end deftypefun
+
+@deftypefun {double *} gsl_vector_ptr (gsl_vector * @var{v}, size_t @var{i})
+@deftypefunx {const double *} gsl_vector_const_ptr (const gsl_vector * @var{v}, size_t @var{i})
+These functions return a pointer to the @var{i}-th element of a vector
+@var{v}.  If @var{i} lies outside the allowed range of 0 to @math{@var{n}-1}
+then the error handler is invoked and a null pointer is returned.
+@end deftypefun
+
+@node Initializing vector elements
+@subsection Initializing vector elements
+@cindex vectors, initializing
+@cindex initializing vectors
+
+@deftypefun void gsl_vector_set_all (gsl_vector * @var{v}, double @var{x})
+This function sets all the elements of the vector @var{v} to the value
+@var{x}.
+@end deftypefun
+
+@deftypefun void gsl_vector_set_zero (gsl_vector * @var{v})
+This function sets all the elements of the vector @var{v} to zero.
+@end deftypefun
+
+@deftypefun int gsl_vector_set_basis (gsl_vector * @var{v}, size_t @var{i})
+This function makes a basis vector by setting all the elements of the
+vector @var{v} to zero except for the @var{i}-th element which is set to
+one.
+@end deftypefun
+
+@node Reading and writing vectors
+@subsection Reading and writing vectors
+
+The library provides functions for reading and writing vectors to a file
+as binary data or formatted text.
+
+@deftypefun int gsl_vector_fwrite (FILE * @var{stream}, const gsl_vector * @var{v})
+This function writes the elements of the vector @var{v} to the stream
+@var{stream} in binary format.  The return value is 0 for success and
+@code{GSL_EFAILED} if there was a problem writing to the file.  Since the
+data is written in the native binary format it may not be portable
+between different architectures.
+@end deftypefun
+
+@deftypefun int gsl_vector_fread (FILE * @var{stream}, gsl_vector * @var{v})
+This function reads into the vector @var{v} from the open stream
+@var{stream} in binary format.  The vector @var{v} must be preallocated
+with the correct length since the function uses the size of @var{v} to
+determine how many bytes to read.  The return value is 0 for success and
+@code{GSL_EFAILED} if there was a problem reading from the file.  The
+data is assumed to have been written in the native binary format on the
+same architecture.
+@end deftypefun
+
+@deftypefun int gsl_vector_fprintf (FILE * @var{stream}, const gsl_vector * @var{v}, const char * @var{format})
+This function writes the elements of the vector @var{v} line-by-line to
+the stream @var{stream} using the format specifier @var{format}, which
+should be one of the @code{%g}, @code{%e} or @code{%f} formats for
+floating point numbers and @code{%d} for integers.  The function returns
+0 for success and @code{GSL_EFAILED} if there was a problem writing to
+the file.
+@end deftypefun
+
+@deftypefun int gsl_vector_fscanf (FILE * @var{stream}, gsl_vector * @var{v})
+This function reads formatted data from the stream @var{stream} into the
+vector @var{v}.  The vector @var{v} must be preallocated with the correct
+length since the function uses the size of @var{v} to determine how many
+numbers to read.  The function returns 0 for success and
+@code{GSL_EFAILED} if there was a problem reading from the file.
+@end deftypefun
+
+@node Vector views
+@subsection Vector views
+
+In addition to creating vectors from slices of blocks it is also
+possible to slice vectors and create vector views.  For example, a
+subvector of another vector can be described with a view, or two views
+can be made which provide access to the even and odd elements of a
+vector.
+
+A vector view is a temporary object, stored on the stack, which can be
+used to operate on a subset of vector elements.  Vector views can be
+defined for both constant and non-constant vectors, using separate types
+that preserve constness.  A vector view has the type
+@code{gsl_vector_view} and a constant vector view has the type
+@code{gsl_vector_const_view}.  In both cases the elements of the view
+can be accessed as a @code{gsl_vector} using the @code{vector} component
+of the view object.  A pointer to a vector of type @code{gsl_vector *}
+or @code{const gsl_vector *} can be obtained by taking the address of
+this component with the @code{&} operator.  
+
+When using this pointer it is important to ensure that the view itself
+remains in scope---the simplest way to do so is by always writing the
+pointer as @code{&}@var{view}@code{.vector}, and never storing this value
+in another variable.  
+
+@deftypefun gsl_vector_view gsl_vector_subvector (gsl_vector * @var{v}, size_t @var{offset}, size_t @var{n})
+@deftypefunx {gsl_vector_const_view} gsl_vector_const_subvector (const gsl_vector * @var{v}, size_t @var{offset}, size_t @var{n})
+These functions return a vector view of a subvector of another vector
+@var{v}.  The start of the new vector is offset by @var{offset} elements
+from the start of the original vector.  The new vector has @var{n}
+elements.  Mathematically, the @var{i}-th element of the new vector
+@var{v'} is given by,
+
+@example
+v'(i) = v->data[(offset + i)*v->stride]
+@end example
+
+@noindent
+where the index @var{i} runs from 0 to @code{n-1}.
+
+The @code{data} pointer of the returned vector struct is set to null if
+the combined parameters (@var{offset},@var{n}) overrun the end of the
+original vector.
+
+The new vector is only a view of the block underlying the original
+vector, @var{v}.  The block containing the elements of @var{v} is not
+owned by the new vector.  When the view goes out of scope the original
+vector @var{v} and its block will continue to exist.  The original
+memory can only be deallocated by freeing the original vector.  Of
+course, the original vector should not be deallocated while the view is
+still in use.
+
+The function @code{gsl_vector_const_subvector} is equivalent to
+@code{gsl_vector_subvector} but can be used for vectors which are
+declared @code{const}.
+@end deftypefun
+
+
+@deftypefun gsl_vector_view gsl_vector_subvector_with_stride (gsl_vector * @var{v}, size_t @var{offset}, size_t @var{stride}, size_t @var{n})
+@deftypefunx {gsl_vector_const_view} gsl_vector_const_subvector_with_stride (const gsl_vector * @var{v}, size_t @var{offset}, size_t @var{stride}, size_t @var{n})
+These functions return a vector view of a subvector of another vector
+@var{v} with an additional stride argument. The subvector is formed in
+the same way as for @code{gsl_vector_subvector} but the new vector has
+@var{n} elements with a step-size of @var{stride} from one element to
+the next in the original vector.  Mathematically, the @var{i}-th element
+of the new vector @var{v'} is given by,
+
+@example
+v'(i) = v->data[(offset + i*stride)*v->stride]
+@end example
+
+@noindent
+where the index @var{i} runs from 0 to @code{n-1}.
+
+Note that subvector views give direct access to the underlying elements
+of the original vector. For example, the following code will zero the
+even elements of the vector @code{v} of length @code{n}, while leaving the
+odd elements untouched,
+
+@example
+gsl_vector_view v_even 
+  = gsl_vector_subvector_with_stride (v, 0, 2, n/2);
+gsl_vector_set_zero (&v_even.vector);
+@end example
+
+@noindent
+A vector view can be passed to any subroutine which takes a vector
+argument just as a directly allocated vector would be, using
+@code{&}@var{view}@code{.vector}.  For example, the following code
+computes the norm of the odd elements of @code{v} using the @sc{blas}
+routine @sc{dnrm2},
+
+@example
+gsl_vector_view v_odd 
+  = gsl_vector_subvector_with_stride (v, 1, 2, n/2);
+double r = gsl_blas_dnrm2 (&v_odd.vector);
+@end example
+
+The function @code{gsl_vector_const_subvector_with_stride} is equivalent
+to @code{gsl_vector_subvector_with_stride} but can be used for vectors
+which are declared @code{const}.
+@end deftypefun
+
+@deftypefun gsl_vector_view gsl_vector_complex_real (gsl_vector_complex * @var{v})
+@deftypefunx {gsl_vector_const_view} gsl_vector_complex_const_real (const gsl_vector_complex * @var{v})
+These functions return a vector view of the real parts of the complex
+vector @var{v}.
+
+The function @code{gsl_vector_complex_const_real} is equivalent to
+@code{gsl_vector_complex_real} but can be used for vectors which are
+declared @code{const}.
+@end deftypefun
+
+@deftypefun gsl_vector_view gsl_vector_complex_imag (gsl_vector_complex * @var{v})
+@deftypefunx {gsl_vector_const_view} gsl_vector_complex_const_imag (const gsl_vector_complex * @var{v})
+These functions return a vector view of the imaginary parts of the
+complex vector @var{v}.
+
+The function @code{gsl_vector_complex_const_imag} is equivalent to
+@code{gsl_vector_complex_imag} but can be used for vectors which are
+declared @code{const}.
+@end deftypefun
+
+@deftypefun gsl_vector_view gsl_vector_view_array (double * @var{base}, size_t @var{n}) 
+@deftypefunx {gsl_vector_const_view} gsl_vector_const_view_array (const double * @var{base}, size_t @var{n})
+These functions return a vector view of an array.  The start of the new
+vector is given by @var{base} and has @var{n} elements.  Mathematically,
+the @var{i}-th element of the new vector @var{v'} is given by,
+
+@example
+v'(i) = base[i]
+@end example
+
+@noindent
+where the index @var{i} runs from 0 to @code{n-1}.
+
+The array containing the elements of @var{v} is not owned by the new
+vector view.  When the view goes out of scope the original array will
+continue to exist.  The original memory can only be deallocated by
+freeing the original pointer @var{base}.  Of course, the original array
+should not be deallocated while the view is still in use.
+
+The function @code{gsl_vector_const_view_array} is equivalent to
+@code{gsl_vector_view_array} but can be used for arrays which are
+declared @code{const}.
+@end deftypefun
+
+@deftypefun gsl_vector_view gsl_vector_view_array_with_stride (double * @var{base}, size_t @var{stride}, size_t @var{n})
+@deftypefunx gsl_vector_const_view gsl_vector_const_view_array_with_stride (const double * @var{base}, size_t @var{stride}, size_t @var{n})
+These functions return a vector view of an array @var{base} with an
+additional stride argument. The subvector is formed in the same way as
+for @code{gsl_vector_view_array} but the new vector has @var{n} elements
+with a step-size of @var{stride} from one element to the next in the
+original array.  Mathematically, the @var{i}-th element of the new
+vector @var{v'} is given by,
+
+@example
+v'(i) = base[i*stride]
+@end example
+
+@noindent
+where the index @var{i} runs from 0 to @code{n-1}.
+
+Note that the view gives direct access to the underlying elements of the
+original array.  A vector view can be passed to any subroutine which
+takes a vector argument just as a directly allocated vector would be,
+using @code{&}@var{view}@code{.vector}.
+
+The function @code{gsl_vector_const_view_array_with_stride} is
+equivalent to @code{gsl_vector_view_array_with_stride} but can be used
+for arrays which are declared @code{const}.
+@end deftypefun
+
+
+@comment @node Modifying subvector views
+@comment @subsection Modifying subvector views
+@comment 
+@comment @deftypefun int gsl_vector_view_from_vector (gsl_vector * @var{v}, gsl_vector * @var{base}, size_t @var{offset}, size_t @var{n}, size_t @var{stride})
+@comment This function modifies and existing vector view @var{v} to form a new
+@comment view of a vector @var{base}, starting from element @var{offset}.  The
+@comment vector has @var{n} elements separated by stride @var{stride}.  Any
+@comment existing view in @var{v} will be lost as a result of this function.
+@comment @end deftypefun
+@comment 
+@comment @deftypefun int gsl_vector_view_from_array (gsl_vector * @var{v}, double * @var{base}, size_t @var{offset}, size_t @var{n}, size_t @var{stride})
+@comment This function modifies and existing vector view @var{v} to form a new
+@comment view of an array @var{base}, starting from element @var{offset}.  The
+@comment vector has @var{n} elements separated by stride @var{stride}.  Any
+@comment existing view in @var{v} will be lost as a result of this function.
+@comment @end deftypefun
+
+@comment @deftypefun {gsl_vector *} gsl_vector_alloc_from_block (gsl_block * @var{b}, size_t @var{offset}, size_t @var{n}, size_t @var{stride})
+@comment This function creates a vector as a slice of an existing block @var{b},
+@comment returning a pointer to a newly initialized vector struct.  The start of
+@comment the vector is offset by @var{offset} elements from the start of the
+@comment block.  The vector has @var{n} elements, with a step-size of @var{stride}
+@comment from one element to the next.  Mathematically, the @var{i}-th element of
+@comment the vector is given by,
+@comment 
+@comment @example
+@comment v(i) = b->data[offset + i*stride]
+@comment @end example
+@comment @noindent
+@comment where the index @var{i} runs from 0 to @code{n-1}.
+@comment 
+@comment A null pointer is returned if the combined parameters
+@comment (@var{offset},@var{n},@var{stride}) overrun the end of the block or if
+@comment insufficient memory is available to store the vector.
+@comment 
+@comment The vector is only a view of the block @var{b}, and the block is not
+@comment owned by the vector.  When the vector is deallocated the block @var{b}
+@comment will continue to exist.  This memory can only be deallocated by freeing
+@comment the block itself.  Of course, this block should not be deallocated while
+@comment the vector is still in use.
+@comment @end deftypefun
+@comment 
+@comment @deftypefun {gsl_vector *} gsl_vector_alloc_from_vector (gsl_vector * @var{v}, size_t @var{offset}, size_t @var{n}, size_t @var{stride})
+@comment This function creates a vector as a slice of another vector @var{v},
+@comment returning a pointer to a newly initialized vector struct.  The start of
+@comment the new vector is offset by @var{offset} elements from the start of the
+@comment original vector.  The new vector has @var{n} elements, with a step-size
+@comment of @var{stride} from one element to the next in the original vector.
+@comment Mathematically, the @var{i}-th element of the new vector @var{v'} is
+@comment given by,
+@comment 
+@comment @example
+@comment v'(i) = v->data[(offset + i*stride)*v->stride]
+@comment @end example
+@comment @noindent
+@comment where the index @var{i} runs from 0 to @code{n-1}.
+@comment 
+@comment A null pointer is returned if the combined parameters
+@comment (@var{offset},@var{n},@var{stride}) overrun the end of the original
+@comment vector or if insufficient memory is available store the new vector.
+@comment 
+@comment The new vector is only a view of the block underlying the original
+@comment vector, @var{v}.  The block is not owned by the new vector.  When the new
+@comment vector is deallocated the original vector @var{v} and its block will
+@comment continue to exist.  The original memory can only be deallocated by
+@comment freeing the original vector.  Of course, the original vector should not
+@comment be deallocated while the new vector is still in use.
+@comment @end deftypefun
+
+@node Copying vectors
+@subsection Copying vectors
+
+Common operations on vectors such as addition and multiplication are
+available in the @sc{blas} part of the library (@pxref{BLAS
+Support}).  However, it is useful to have a small number of utility
+functions which do not require the full @sc{blas} code.  The following
+functions fall into this category.
+
+@deftypefun int gsl_vector_memcpy (gsl_vector * @var{dest}, const gsl_vector * @var{src})
+This function copies the elements of the vector @var{src} into the
+vector @var{dest}.  The two vectors must have the same length.
+@end deftypefun
+
+@deftypefun int gsl_vector_swap (gsl_vector * @var{v}, gsl_vector * @var{w})
+This function exchanges the elements of the vectors @var{v} and @var{w}
+by copying.  The two vectors must have the same length.
+@end deftypefun
+
+
+@node Exchanging elements
+@subsection Exchanging elements
+
+The following function can be used to exchange, or permute, the elements
+of a vector.
+
+@deftypefun int gsl_vector_swap_elements (gsl_vector * @var{v}, size_t @var{i}, size_t @var{j})
+This function exchanges the @var{i}-th and @var{j}-th elements of the
+vector @var{v} in-place.
+@end deftypefun
+
+@deftypefun int gsl_vector_reverse (gsl_vector * @var{v})
+This function reverses the order of the elements of the vector @var{v}.
+@end deftypefun
+
+
+@node Vector operations
+@subsection Vector operations
+
+The following operations are only defined for real vectors.
+
+@deftypefun int gsl_vector_add (gsl_vector * @var{a}, const gsl_vector * @var{b})
+This function adds the elements of vector @var{b} to the elements of
+vector @var{a}, @math{a'_i = a_i + b_i}. The two vectors must have the
+same length.
+@end deftypefun
+
+@deftypefun int gsl_vector_sub (gsl_vector * @var{a}, const gsl_vector * @var{b})
+This function subtracts the elements of vector @var{b} from the elements of
+vector @var{a}, @math{a'_i = a_i - b_i}. The two vectors must have the
+same length.
+@end deftypefun
+
+@deftypefun int gsl_vector_mul (gsl_vector * @var{a}, const gsl_vector * @var{b})
+This function multiplies the elements of vector @var{a} by the elements of
+vector @var{b}, @math{a'_i = a_i * b_i}. The two vectors must have the
+same length.
+@end deftypefun
+
+@deftypefun int gsl_vector_div (gsl_vector * @var{a}, const gsl_vector * @var{b})
+This function divides the elements of vector @var{a} by the elements of
+vector @var{b}, @math{a'_i = a_i / b_i}. The two vectors must have the
+same length.
+@end deftypefun
+
+@deftypefun int gsl_vector_scale (gsl_vector * @var{a}, const double @var{x})
+This function multiplies the elements of vector @var{a} by the constant
+factor @var{x}, @math{a'_i = x a_i}.
+@end deftypefun
+
+@deftypefun int gsl_vector_add_constant (gsl_vector * @var{a}, const double @var{x})
+This function adds the constant value @var{x} to the elements of the
+vector @var{a}, @math{a'_i = a_i + x}.
+@end deftypefun
+
+@node Finding maximum and minimum elements of vectors
+@subsection Finding maximum and minimum elements of vectors
+
+@deftypefun double gsl_vector_max (const gsl_vector * @var{v})
+This function returns the maximum value in the vector @var{v}.
+@end deftypefun
+
+@deftypefun double gsl_vector_min (const gsl_vector * @var{v})
+This function returns the minimum value in the vector @var{v}.
+@end deftypefun
+
+@deftypefun void gsl_vector_minmax (const gsl_vector * @var{v}, double * @var{min_out}, double * @var{max_out})
+This function returns the minimum and maximum values in the vector
+@var{v}, storing them in @var{min_out} and @var{max_out}.
+@end deftypefun
+
+@deftypefun size_t gsl_vector_max_index (const gsl_vector * @var{v})
+This function returns the index of the maximum value in the vector @var{v}.
+When there are several equal maximum elements then the lowest index is
+returned.
+@end deftypefun
+
+@deftypefun size_t gsl_vector_min_index (const gsl_vector * @var{v})
+This function returns the index of the minimum value in the vector @var{v}.
+When there are several equal minimum elements then the lowest index is
+returned.
+@end deftypefun
+
+@deftypefun void gsl_vector_minmax_index (const gsl_vector * @var{v}, size_t * @var{imin}, size_t * @var{imax})
+This function returns the indices of the minimum and maximum values in
+the vector @var{v}, storing them in @var{imin} and @var{imax}. When
+there are several equal minimum or maximum elements then the lowest
+indices are returned.
+@end deftypefun
+
+@node Vector properties
+@subsection Vector properties
+
+@deftypefun int gsl_vector_isnull (const gsl_vector * @var{v})
+@deftypefunx int gsl_vector_ispos (const gsl_vector * @var{v})
+@deftypefunx int gsl_vector_isneg (const gsl_vector * @var{v})
+These functions return 1 if all the elements of the vector @var{v}
+are zero, strictly positive, or strictly negative respectively, and 0
+otherwise.  To test for a non-negative vector, use the expression
+@code{!gsl_vector_isneg(v)}.
+@end deftypefun
+
+@node Example programs for vectors
+@subsection Example programs for vectors
+
+This program shows how to allocate, initialize and read from a vector
+using the functions @code{gsl_vector_alloc}, @code{gsl_vector_set} and
+@code{gsl_vector_get}.
+
+@example
+@verbatiminclude examples/vector.c
+@end example
+@comment
+
+@noindent
+Here is the output from the program.  The final loop attempts to read
+outside the range of the vector @code{v}, and the error is trapped by
+the range-checking code in @code{gsl_vector_get}.
+
+@example
+$ ./a.out
+v_0 = 1.23
+v_1 = 2.23
+v_2 = 3.23
+gsl: vector_source.c:12: ERROR: index out of range
+Default GSL error handler invoked.
+Aborted (core dumped)
+@end example
+@comment
+
+@noindent
+The next program shows how to write a vector to a file.
+
+@example
+@verbatiminclude examples/vectorw.c
+@end example
+@comment
+
+@noindent
+After running this program the file @file{test.dat} should contain the
+elements of @code{v}, written using the format specifier
+@code{%.5g}.  The vector could then be read back in using the function
+@code{gsl_vector_fscanf (f, v)} as follows:
+
+@example
+@verbatiminclude examples/vectorr.c
+@end example
+
+
+@node Matrices
+@section Matrices
+@cindex matrices
+@cindex physical dimension, matrices
+@cindex trailing dimension, matrices
+@cindex leading dimension, matrices
+
+Matrices are defined by a @code{gsl_matrix} structure which describes a
+generalized slice of a block.  Like a vector it represents a set of
+elements in an area of memory, but uses two indices instead of one.
+
+The @code{gsl_matrix} structure contains six components, the two
+dimensions of the matrix, a physical dimension, a pointer to the memory
+where the elements of the matrix are stored, @var{data}, a pointer to
+the block owned by the matrix @var{block}, if any, and an ownership
+flag, @var{owner}.  The physical dimension determines the memory layout
+and can differ from the matrix dimension to allow the use of
+submatrices.  The @code{gsl_matrix} structure is very simple and looks
+like this,
+
+@example
+typedef struct
+@{
+  size_t size1;
+  size_t size2;
+  size_t tda;
+  double * data;
+  gsl_block * block;
+  int owner;
+@} gsl_matrix;
+@end example
+@comment
+
+@noindent
+Matrices are stored in row-major order, meaning that each row of
+elements forms a contiguous block in memory.  This is the standard
+``C-language ordering'' of two-dimensional arrays. Note that @sc{fortran}
+stores arrays in column-major order. The number of rows is @var{size1}.
+The range of valid row indices runs from 0 to @code{size1-1}.  Similarly
+@var{size2} is the number of columns.  The range of valid column indices
+runs from 0 to @code{size2-1}.  The physical row dimension @var{tda}, or
+@dfn{trailing dimension}, specifies the size of a row of the matrix as
+laid out in memory.
+
+For example, in the following matrix @var{size1} is 3, @var{size2} is 4,
+and @var{tda} is 8.  The physical memory layout of the matrix begins in
+the top left hand-corner and proceeds from left to right along each row
+in turn.
+
+@example
+@group
+00 01 02 03 XX XX XX XX
+10 11 12 13 XX XX XX XX
+20 21 22 23 XX XX XX XX
+@end group
+@end example
+
+@noindent
+Each unused memory location is represented by ``@code{XX}''.  The
+pointer @var{data} gives the location of the first element of the matrix
+in memory.  The pointer @var{block} stores the location of the memory
+block in which the elements of the matrix are located (if any).  If the
+matrix owns this block then the @var{owner} field is set to one and the
+block will be deallocated when the matrix is freed.  If the matrix is
+only a slice of a block owned by another object then the @var{owner} field is
+zero and any underlying block will not be freed.
+
+The functions for allocating and accessing matrices are defined in
+@file{gsl_matrix.h}
+
+@menu
+* Matrix allocation::           
+* Accessing matrix elements::   
+* Initializing matrix elements::  
+* Reading and writing matrices::  
+* Matrix views::                
+* Creating row and column views::  
+* Copying matrices::            
+* Copying rows and columns::    
+* Exchanging rows and columns::  
+* Matrix operations::           
+* Finding maximum and minimum elements of matrices::  
+* Matrix properties::           
+* Example programs for matrices::  
+@end menu
+
+@node Matrix allocation
+@subsection Matrix allocation
+
+The functions for allocating memory to a matrix follow the style of
+@code{malloc} and @code{free}.  They also perform their own error
+checking.  If there is insufficient memory available to allocate a vector
+then the functions call the GSL error handler (with an error number of
+@code{GSL_ENOMEM}) in addition to returning a null pointer.  Thus if you
+use the library error handler to abort your program then it isn't
+necessary to check every @code{alloc}.
+
+@deftypefun {gsl_matrix *} gsl_matrix_alloc (size_t @var{n1}, size_t @var{n2})
+This function creates a matrix of size @var{n1} rows by @var{n2}
+columns, returning a pointer to a newly initialized matrix struct. A new
+block is allocated for the elements of the matrix, and stored in the
+@var{block} component of the matrix struct.  The block is ``owned'' by the
+matrix, and will be deallocated when the matrix is deallocated.
+@end deftypefun
+
+@deftypefun {gsl_matrix *} gsl_matrix_calloc (size_t @var{n1}, size_t @var{n2})
+This function allocates memory for a matrix of size @var{n1} rows by
+@var{n2} columns and initializes all the elements of the matrix to zero.
+@end deftypefun
+
+@deftypefun void gsl_matrix_free (gsl_matrix * @var{m})
+This function frees a previously allocated matrix @var{m}.  If the
+matrix was created using @code{gsl_matrix_alloc} then the block
+underlying the matrix will also be deallocated.  If the matrix has
+been created from another object then the memory is still owned by
+that object and will not be deallocated.  The matrix @var{m} must be a
+valid matrix object (a null pointer is not allowed).
+@end deftypefun
+
+@node Accessing matrix elements
+@subsection Accessing matrix elements
+@cindex matrices, range-checking
+@cindex range-checking for matrices
+
+The functions for accessing the elements of a matrix use the same range
+checking system as vectors.  You can turn off range checking by recompiling
+your program with the preprocessor definition
+@code{GSL_RANGE_CHECK_OFF}.
+
+The elements of the matrix are stored in ``C-order'', where the second
+index moves continuously through memory.  More precisely, the element
+accessed by the function @code{gsl_matrix_get(m,i,j)} and
+@code{gsl_matrix_set(m,i,j,x)} is 
+
+@example
+m->data[i * m->tda + j]
+@end example
+@comment 
+
+@noindent
+where @var{tda} is the physical row-length of the matrix.
+
+@deftypefun double gsl_matrix_get (const gsl_matrix * @var{m}, size_t @var{i}, size_t @var{j})
+This function returns the @math{(i,j)}-th element of a matrix
+@var{m}.  If @var{i} or @var{j} lie outside the allowed range of 0 to
+@math{@var{n1}-1} and 0 to @math{@var{n2}-1} then the error handler is invoked and 0
+is returned.
+@end deftypefun
+
+@deftypefun void gsl_matrix_set (gsl_matrix * @var{m}, size_t @var{i}, size_t @var{j}, double @var{x})
+This function sets the value of the @math{(i,j)}-th element of a
+matrix @var{m} to @var{x}.  If @var{i} or @var{j} lies outside the
+allowed range of 0 to @math{@var{n1}-1} and 0 to @math{@var{n2}-1} then the error
+handler is invoked.
+@end deftypefun
+
+@deftypefun {double *} gsl_matrix_ptr (gsl_matrix * @var{m}, size_t @var{i}, size_t @var{j})
+@deftypefunx {const double *} gsl_matrix_const_ptr (const gsl_matrix * @var{m}, size_t @var{i}, size_t @var{j})
+These functions return a pointer to the @math{(i,j)}-th element of a
+matrix @var{m}.  If @var{i} or @var{j} lie outside the allowed range of
+0 to @math{@var{n1}-1} and 0 to @math{@var{n2}-1} then the error handler is invoked
+and a null pointer is returned.
+@end deftypefun
+
+@node Initializing matrix elements
+@subsection Initializing matrix elements
+@cindex matrices, initializing
+@cindex initializing matrices
+@cindex identity matrix
+@cindex matrix, identity
+@cindex zero matrix
+@cindex matrix, zero
+@cindex constant matrix
+@cindex matrix, constant
+
+@deftypefun void gsl_matrix_set_all (gsl_matrix * @var{m}, double @var{x})
+This function sets all the elements of the matrix @var{m} to the value
+@var{x}.
+@end deftypefun
+
+@deftypefun void gsl_matrix_set_zero (gsl_matrix * @var{m})
+This function sets all the elements of the matrix @var{m} to zero.
+@end deftypefun
+
+@deftypefun void gsl_matrix_set_identity (gsl_matrix * @var{m})
+This function sets the elements of the matrix @var{m} to the
+corresponding elements of the identity matrix, @math{m(i,j) =
+\delta(i,j)}, i.e. a unit diagonal with all off-diagonal elements zero.
+This applies to both square and rectangular matrices.
+@end deftypefun
+
+@node Reading and writing matrices
+@subsection Reading and writing matrices
+
+The library provides functions for reading and writing matrices to a file
+as binary data or formatted text.
+
+@deftypefun int gsl_matrix_fwrite (FILE * @var{stream}, const gsl_matrix * @var{m})
+This function writes the elements of the matrix @var{m} to the stream
+@var{stream} in binary format.  The return value is 0 for success and
+@code{GSL_EFAILED} if there was a problem writing to the file.  Since the
+data is written in the native binary format it may not be portable
+between different architectures.
+@end deftypefun
+
+@deftypefun int gsl_matrix_fread (FILE * @var{stream}, gsl_matrix * @var{m})
+This function reads into the matrix @var{m} from the open stream
+@var{stream} in binary format.  The matrix @var{m} must be preallocated
+with the correct dimensions since the function uses the size of @var{m} to
+determine how many bytes to read.  The return value is 0 for success and
+@code{GSL_EFAILED} if there was a problem reading from the file.  The
+data is assumed to have been written in the native binary format on the
+same architecture.
+@end deftypefun
+
+@deftypefun int gsl_matrix_fprintf (FILE * @var{stream}, const gsl_matrix * @var{m}, const char * @var{format})
+This function writes the elements of the matrix @var{m} line-by-line to
+the stream @var{stream} using the format specifier @var{format}, which
+should be one of the @code{%g}, @code{%e} or @code{%f} formats for
+floating point numbers and @code{%d} for integers.  The function returns
+0 for success and @code{GSL_EFAILED} if there was a problem writing to
+the file.
+@end deftypefun
+
+@deftypefun int gsl_matrix_fscanf (FILE * @var{stream}, gsl_matrix * @var{m})
+This function reads formatted data from the stream @var{stream} into the
+matrix @var{m}.  The matrix @var{m} must be preallocated with the correct
+dimensions since the function uses the size of @var{m} to determine how many
+numbers to read.  The function returns 0 for success and
+@code{GSL_EFAILED} if there was a problem reading from the file.
+@end deftypefun
+
+@node Matrix views
+@subsection Matrix views
+
+A matrix view is a temporary object, stored on the stack, which can be
+used to operate on a subset of matrix elements.  Matrix views can be
+defined for both constant and non-constant matrices using separate types
+that preserve constness.  A matrix view has the type
+@code{gsl_matrix_view} and a constant matrix view has the type
+@code{gsl_matrix_const_view}.  In both cases the elements of the view
+can by accessed using the @code{matrix} component of the view object.  A
+pointer @code{gsl_matrix *} or @code{const gsl_matrix *} can be obtained
+by taking the address of the @code{matrix} component with the @code{&}
+operator.  In addition to matrix views it is also possible to create
+vector views of a matrix, such as row or column views.
+
+@deftypefun gsl_matrix_view gsl_matrix_submatrix (gsl_matrix * @var{m}, size_t @var{k1}, size_t @var{k2}, size_t @var{n1}, size_t @var{n2})
+@deftypefunx gsl_matrix_const_view gsl_matrix_const_submatrix (const gsl_matrix * @var{m}, size_t @var{k1}, size_t @var{k2}, size_t @var{n1}, size_t @var{n2})
+These functions return a matrix view of a submatrix of the matrix
+@var{m}.  The upper-left element of the submatrix is the element
+(@var{k1},@var{k2}) of the original matrix.  The submatrix has @var{n1}
+rows and @var{n2} columns.  The physical number of columns in memory
+given by @var{tda} is unchanged.  Mathematically, the
+@math{(i,j)}-th element of the new matrix is given by,
+
+@example
+m'(i,j) = m->data[(k1*m->tda + k2) + i*m->tda + j]
+@end example
+
+@noindent
+where the index @var{i} runs from 0 to @code{n1-1} and the index @var{j}
+runs from 0 to @code{n2-1}.
+
+The @code{data} pointer of the returned matrix struct is set to null if
+the combined parameters (@var{i},@var{j},@var{n1},@var{n2},@var{tda})
+overrun the ends of the original matrix.
+
+The new matrix view is only a view of the block underlying the existing
+matrix, @var{m}.  The block containing the elements of @var{m} is not
+owned by the new matrix view.  When the view goes out of scope the
+original matrix @var{m} and its block will continue to exist.  The
+original memory can only be deallocated by freeing the original matrix.
+Of course, the original matrix should not be deallocated while the view
+is still in use.
+
+The function @code{gsl_matrix_const_submatrix} is equivalent to
+@code{gsl_matrix_submatrix} but can be used for matrices which are
+declared @code{const}.
+@end deftypefun
+
+
+@deftypefun gsl_matrix_view gsl_matrix_view_array (double * @var{base}, size_t @var{n1}, size_t @var{n2})
+@deftypefunx gsl_matrix_const_view gsl_matrix_const_view_array (const double * @var{base}, size_t @var{n1}, size_t @var{n2})
+These functions return a matrix view of the array @var{base}.  The
+matrix has @var{n1} rows and @var{n2} columns.  The physical number of
+columns in memory is also given by @var{n2}.  Mathematically, the
+@math{(i,j)}-th element of the new matrix is given by,
+
+@example
+m'(i,j) = base[i*n2 + j]
+@end example
+
+@noindent
+where the index @var{i} runs from 0 to @code{n1-1} and the index @var{j}
+runs from 0 to @code{n2-1}.
+
+The new matrix is only a view of the array @var{base}.  When the view
+goes out of scope the original array @var{base} will continue to exist.
+The original memory can only be deallocated by freeing the original
+array.  Of course, the original array should not be deallocated while
+the view is still in use.
+
+The function @code{gsl_matrix_const_view_array} is equivalent to
+@code{gsl_matrix_view_array} but can be used for matrices which are
+declared @code{const}.
+@end deftypefun
+
+
+@deftypefun gsl_matrix_view gsl_matrix_view_array_with_tda (double * @var{base}, size_t @var{n1}, size_t @var{n2}, size_t @var{tda})
+@deftypefunx gsl_matrix_const_view gsl_matrix_const_view_array_with_tda (const double * @var{base}, size_t @var{n1}, size_t @var{n2}, size_t @var{tda})
+These functions return a matrix view of the array @var{base} with a
+physical number of columns @var{tda} which may differ from the corresponding
+dimension of the matrix.  The matrix has @var{n1} rows and @var{n2}
+columns, and the physical number of columns in memory is given by
+@var{tda}.  Mathematically, the @math{(i,j)}-th element of the new
+matrix is given by,
+
+@example
+m'(i,j) = base[i*tda + j]
+@end example
+
+@noindent
+where the index @var{i} runs from 0 to @code{n1-1} and the index @var{j}
+runs from 0 to @code{n2-1}.
+
+The new matrix is only a view of the array @var{base}.  When the view
+goes out of scope the original array @var{base} will continue to exist.
+The original memory can only be deallocated by freeing the original
+array.  Of course, the original array should not be deallocated while
+the view is still in use.
+
+The function @code{gsl_matrix_const_view_array_with_tda} is equivalent
+to @code{gsl_matrix_view_array_with_tda} but can be used for matrices
+which are declared @code{const}.
+@end deftypefun
+
+@deftypefun gsl_matrix_view gsl_matrix_view_vector (gsl_vector * @var{v}, size_t @var{n1}, size_t @var{n2})
+@deftypefunx gsl_matrix_const_view gsl_matrix_const_view_vector (const gsl_vector * @var{v}, size_t @var{n1}, size_t @var{n2})
+These functions return a matrix view of the vector @var{v}.  The matrix
+has @var{n1} rows and @var{n2} columns. The vector must have unit
+stride. The physical number of columns in memory is also given by
+@var{n2}.  Mathematically, the @math{(i,j)}-th element of the new
+matrix is given by,
+
+@example
+m'(i,j) = v->data[i*n2 + j]
+@end example
+
+@noindent
+where the index @var{i} runs from 0 to @code{n1-1} and the index @var{j}
+runs from 0 to @code{n2-1}.
+
+The new matrix is only a view of the vector @var{v}.  When the view
+goes out of scope the original vector @var{v} will continue to exist.
+The original memory can only be deallocated by freeing the original
+vector.  Of course, the original vector should not be deallocated while
+the view is still in use.
+
+The function @code{gsl_matrix_const_view_vector} is equivalent to
+@code{gsl_matrix_view_vector} but can be used for matrices which are
+declared @code{const}.
+@end deftypefun
+
+
+@deftypefun gsl_matrix_view gsl_matrix_view_vector_with_tda (gsl_vector * @var{v}, size_t @var{n1}, size_t @var{n2}, size_t @var{tda})
+@deftypefunx gsl_matrix_const_view gsl_matrix_const_view_vector_with_tda (const gsl_vector * @var{v}, size_t @var{n1}, size_t @var{n2}, size_t @var{tda})
+These functions return a matrix view of the vector @var{v} with a
+physical number of columns @var{tda} which may differ from the
+corresponding matrix dimension.  The vector must have unit stride. The
+matrix has @var{n1} rows and @var{n2} columns, and the physical number
+of columns in memory is given by @var{tda}.  Mathematically, the
+@math{(i,j)}-th element of the new matrix is given by,
+
+@example
+m'(i,j) = v->data[i*tda + j]
+@end example
+
+@noindent
+where the index @var{i} runs from 0 to @code{n1-1} and the index @var{j}
+runs from 0 to @code{n2-1}.
+
+The new matrix is only a view of the vector @var{v}.  When the view
+goes out of scope the original vector @var{v} will continue to exist.
+The original memory can only be deallocated by freeing the original
+vector.  Of course, the original vector should not be deallocated while
+the view is still in use.
+
+The function @code{gsl_matrix_const_view_vector_with_tda} is equivalent
+to @code{gsl_matrix_view_vector_with_tda} but can be used for matrices
+which are declared @code{const}.
+@end deftypefun
+
+
+@comment @node Modifying matrix views
+@comment @subsection Modifying matrix views
+@comment 
+@comment @deftypefun int gsl_matrix_view_from_matrix (gsl_matrix * @var{m}, gsl_matrix * @var{mm}, const size_t @var{k1}, const size_t @var{k2}, const size_t @var{n1}, const size_t @var{n2})
+@comment This function modifies and existing matrix view @var{m} to form a new
+@comment view of a matrix @var{mm}, starting from element (@var{k1},@var{k2}).
+@comment The matrix view has @var{n1} rows and @var{n2} columns.  Any existing
+@comment view in @var{m} will be lost as a result of this function.
+@comment @end deftypefun
+@comment 
+@comment @deftypefun int gsl_matrix_view_from_vector (gsl_matrix * @var{m}, gsl_vector * @var{v}, const size_t @var{offset}, const size_t @var{n1}, const size_t @var{n2})
+@comment This function modifies and existing matrix view @var{m} to form a new
+@comment view of a vector @var{v}, starting from element @var{offset}.  The
+@comment vector has @var{n1} rows and @var{n2} columns.  Any
+@comment existing view in @var{m} will be lost as a result of this function.
+@comment @end deftypefun
+@comment 
+@comment @deftypefun int gsl_matrix_view_from_array (gsl_matrix * @var{m}, double * @var{base}, const size_t @var{offset}, const size_t @var{n1}, const size_t @var{n2})
+@comment This function modifies and existing matrix view @var{m} to form a new
+@comment view of an array @var{base}, starting from element @var{offset}.  The
+@comment matrix has @var{n1} rows and @var{n2} columns.  Any
+@comment existing view in @var{m} will be lost as a result of this function.
+@comment @end deftypefun
+@comment 
+@comment @deftypefun {gsl_matrix *} gsl_matrix_alloc_from_block (gsl_block * @var{b}, size_t @var{offset}, size_t @var{n1}, size_t @var{n2}, size_t @var{tda})
+@comment This function creates a matrix as a slice of the block @var{b},
+@comment returning a pointer to a newly initialized matrix struct.  The start of
+@comment the matrix is offset by @var{offset} elements from the start of the
+@comment block.  The matrix has @var{n1} rows and @var{n2} columns, with the
+@comment physical number of columns in memory given by @var{tda}.
+@comment Mathematically, the (@var{i},@var{j})-th element of the matrix is given by,
+@comment 
+@comment @example
+@comment m(i,j) = b->data[offset + i*tda + j]
+@comment @end example
+@comment @noindent
+@comment where the index @var{i} runs from 0 to @code{n1-1} and the index @var{j}
+@comment runs from 0 to @code{n2-1}.
+@comment 
+@comment A null pointer is returned if the combined parameters
+@comment (@var{offset},@var{n1},@var{n2},@var{tda}) overrun the end of the block
+@comment or if insufficient memory is available to store the matrix.
+@comment 
+@comment The matrix is only a view of the block @var{b}, and the block is not
+@comment owned by the matrix.  When the matrix is deallocated the block @var{b}
+@comment will continue to exist.  This memory can only be deallocated by freeing
+@comment the block itself.  Of course, this block should not be deallocated while
+@comment the matrix is still in use.
+@comment @end deftypefun
+@comment 
+@comment @deftypefun {gsl_matrix *} gsl_matrix_alloc_from_matrix (gsl_matrix * @var{m}, size_t @var{k1}, size_t @var{k2}, size_t @var{n1}, size_t @var{n2})
+@comment 
+@comment This function creates a matrix as a submatrix of the matrix @var{m},
+@comment returning a pointer to a newly initialized matrix struct.  The upper-left
+@comment element of the submatrix is the element (@var{k1},@var{k2}) of the
+@comment original matrix.  The submatrix has @var{n1} rows and @var{n2} columns.
+@comment The physical number of columns in memory given by @var{tda} is
+@comment unchanged.  Mathematically, the (@var{i},@var{j})-th element of the
+@comment new matrix is given by,
+@comment 
+@comment @example
+@comment m'(i,j) = m->data[(k1*m->tda + k2) + i*m->tda + j]
+@comment @end example
+@comment @noindent
+@comment where the index @var{i} runs from 0 to @code{n1-1} and the index @var{j}
+@comment runs from 0 to @code{n2-1}.
+@comment 
+@comment A null pointer is returned if the combined parameters
+@comment (@var{k1},@var{k2},@var{n1},@var{n2},@var{tda}) overrun the end of the
+@comment original matrix or if insufficient memory is available to store the matrix.
+@comment 
+@comment The new matrix is only a view of the block underlying the existing
+@comment matrix, @var{m}.  The block is not owned by the new matrix.  When the new
+@comment matrix is deallocated the original matrix @var{m} and its block will
+@comment continue to exist.  The original memory can only be deallocated by
+@comment freeing the original matrix.  Of course, the original matrix should not
+@comment be deallocated while the new matrix is still in use.
+@comment @end deftypefun
+
+@node Creating row and column views
+@subsection Creating row and column views
+
+In general there are two ways to access an object, by reference or by
+copying.  The functions described in this section create vector views
+which allow access to a row or column of a matrix by reference.
+Modifying elements of the view is equivalent to modifying the matrix,
+since both the vector view and the matrix point to the same memory
+block.
+
+@deftypefun gsl_vector_view gsl_matrix_row (gsl_matrix * @var{m}, size_t @var{i})
+@deftypefunx {gsl_vector_const_view} gsl_matrix_const_row (const gsl_matrix * @var{m}, size_t @var{i})
+These functions return a vector view of the @var{i}-th row of the matrix
+@var{m}.  The @code{data} pointer of the new vector is set to null if
+@var{i} is out of range.
+
+The function @code{gsl_vector_const_row} is equivalent to
+@code{gsl_matrix_row} but can be used for matrices which are declared
+@code{const}.
+@end deftypefun
+
+@deftypefun gsl_vector_view gsl_matrix_column (gsl_matrix * @var{m}, size_t @var{j})
+@deftypefunx {gsl_vector_const_view} gsl_matrix_const_column (const gsl_matrix * @var{m}, size_t @var{j})
+These functions return a vector view of the @var{j}-th column of the
+matrix @var{m}.  The @code{data} pointer of the new vector is set to
+null if @var{j} is out of range.
+
+The function @code{gsl_vector_const_column} is equivalent to
+@code{gsl_matrix_column} but can be used for matrices which are declared
+@code{const}.
+@end deftypefun
+
+@cindex matrix diagonal
+@cindex diagonal, of a matrix
+@deftypefun gsl_vector_view gsl_matrix_diagonal (gsl_matrix * @var{m})
+@deftypefunx {gsl_vector_const_view} gsl_matrix_const_diagonal (const gsl_matrix * @var{m})
+These functions returns a vector view of the diagonal of the matrix
+@var{m}. The matrix @var{m} is not required to be square. For a
+rectangular matrix the length of the diagonal is the same as the smaller
+dimension of the matrix.
+
+The function @code{gsl_matrix_const_diagonal} is equivalent to
+@code{gsl_matrix_diagonal} but can be used for matrices which are
+declared @code{const}.
+@end deftypefun
+
+@cindex matrix subdiagonal
+@cindex subdiagonal, of a matrix
+@deftypefun gsl_vector_view gsl_matrix_subdiagonal (gsl_matrix * @var{m}, size_t @var{k}) 
+@deftypefunx {gsl_vector_const_view} gsl_matrix_const_subdiagonal (const gsl_matrix * @var{m}, size_t @var{k})
+These functions return a vector view of the @var{k}-th subdiagonal of
+the matrix @var{m}. The matrix @var{m} is not required to be square.
+The diagonal of the matrix corresponds to @math{k = 0}.
+
+The function @code{gsl_matrix_const_subdiagonal} is equivalent to
+@code{gsl_matrix_subdiagonal} but can be used for matrices which are
+declared @code{const}.
+@end deftypefun
+
+@cindex matrix superdiagonal
+@cindex superdiagonal, matrix
+@deftypefun gsl_vector_view gsl_matrix_superdiagonal (gsl_matrix * @var{m}, size_t @var{k}) 
+@deftypefunx {gsl_vector_const_view} gsl_matrix_const_superdiagonal (const gsl_matrix * @var{m}, size_t @var{k})
+These functions return a vector view of the @var{k}-th superdiagonal of
+the matrix @var{m}. The matrix @var{m} is not required to be square. The
+diagonal of the matrix corresponds to @math{k = 0}.
+
+The function @code{gsl_matrix_const_superdiagonal} is equivalent to
+@code{gsl_matrix_superdiagonal} but can be used for matrices which are
+declared @code{const}.
+@end deftypefun
+
+@comment @deftypefun {gsl_vector *} gsl_vector_alloc_row_from_matrix (gsl_matrix * @var{m}, size_t @var{i})
+@comment This function allocates a new @code{gsl_vector} struct which points to
+@comment the @var{i}-th row of the matrix @var{m}.
+@comment @end deftypefun
+@comment 
+@comment @deftypefun {gsl_vector *} gsl_vector_alloc_col_from_matrix (gsl_matrix * @var{m}, size_t @var{j})
+@comment This function allocates a new @code{gsl_vector} struct which points to
+@comment the @var{j}-th column of the matrix @var{m}.
+@comment @end deftypefun
+
+@node Copying matrices
+@subsection Copying matrices
+
+@deftypefun int gsl_matrix_memcpy (gsl_matrix * @var{dest}, const gsl_matrix * @var{src})
+This function copies the elements of the matrix @var{src} into the
+matrix @var{dest}.  The two matrices must have the same size.
+@end deftypefun
+
+@deftypefun int gsl_matrix_swap (gsl_matrix * @var{m1}, gsl_matrix * @var{m2})
+This function exchanges the elements of the matrices @var{m1} and
+@var{m2} by copying.  The two matrices must have the same size.
+@end deftypefun
+
+@node Copying rows and columns
+@subsection Copying rows and columns
+
+The functions described in this section copy a row or column of a matrix
+into a vector.  This allows the elements of the vector and the matrix to
+be modified independently.  Note that if the matrix and the vector point
+to overlapping regions of memory then the result will be undefined.  The
+same effect can be achieved with more generality using
+@code{gsl_vector_memcpy} with vector views of rows and columns.
+
+@deftypefun int gsl_matrix_get_row (gsl_vector * @var{v}, const gsl_matrix * @var{m}, size_t @var{i})
+This function copies the elements of the @var{i}-th row of the matrix
+@var{m} into the vector @var{v}.  The length of the vector must be the
+same as the length of the row.
+@end deftypefun
+
+@deftypefun int gsl_matrix_get_col (gsl_vector * @var{v}, const gsl_matrix * @var{m}, size_t @var{j})
+This function copies the elements of the @var{j}-th column of the matrix
+@var{m} into the vector @var{v}.  The length of the vector must be the
+same as the length of the column.
+@end deftypefun
+
+@deftypefun int gsl_matrix_set_row (gsl_matrix * @var{m}, size_t @var{i}, const gsl_vector * @var{v})
+This function copies the elements of the vector @var{v} into the
+@var{i}-th row of the matrix @var{m}.  The length of the vector must be
+the same as the length of the row.
+@end deftypefun
+
+@deftypefun int gsl_matrix_set_col (gsl_matrix * @var{m}, size_t @var{j}, const gsl_vector * @var{v})
+This function copies the elements of the vector @var{v} into the
+@var{j}-th column of the matrix @var{m}.  The length of the vector must be
+the same as the length of the column.
+@end deftypefun
+
+@node Exchanging rows and columns
+@subsection Exchanging rows and columns
+
+The following functions can be used to exchange the rows and columns of
+a matrix.
+
+@deftypefun int gsl_matrix_swap_rows (gsl_matrix * @var{m}, size_t @var{i}, size_t @var{j})
+This function exchanges the @var{i}-th and @var{j}-th rows of the matrix
+@var{m} in-place.
+@end deftypefun
+
+@deftypefun int gsl_matrix_swap_columns (gsl_matrix * @var{m}, size_t @var{i}, size_t @var{j})
+This function exchanges the @var{i}-th and @var{j}-th columns of the
+matrix @var{m} in-place.
+@end deftypefun
+
+@deftypefun int gsl_matrix_swap_rowcol (gsl_matrix * @var{m}, size_t @var{i}, size_t @var{j})
+This function exchanges the @var{i}-th row and @var{j}-th column of the
+matrix @var{m} in-place.  The matrix must be square for this operation to
+be possible.
+@end deftypefun
+
+@deftypefun int gsl_matrix_transpose_memcpy (gsl_matrix * @var{dest}, const gsl_matrix * @var{src})
+This function makes the matrix @var{dest} the transpose of the matrix
+@var{src} by copying the elements of @var{src} into @var{dest}.  This
+function works for all matrices provided that the dimensions of the matrix
+@var{dest} match the transposed dimensions of the matrix @var{src}.
+@end deftypefun
+
+@deftypefun int gsl_matrix_transpose (gsl_matrix * @var{m})
+This function replaces the matrix @var{m} by its transpose by copying
+the elements of the matrix in-place.  The matrix must be square for this
+operation to be possible.
+@end deftypefun
+
+@node Matrix operations
+@subsection Matrix operations
+
+The following operations are defined for real and complex matrices.
+
+@deftypefun int gsl_matrix_add (gsl_matrix * @var{a}, const gsl_matrix * @var{b})
+This function adds the elements of matrix @var{b} to the elements of
+matrix @var{a}, @math{a'(i,j) = a(i,j) + b(i,j)}. The two matrices must have the
+same dimensions.
+@end deftypefun
+
+@deftypefun int gsl_matrix_sub (gsl_matrix * @var{a}, const gsl_matrix * @var{b})
+This function subtracts the elements of matrix @var{b} from the elements of
+matrix @var{a}, @math{a'(i,j) = a(i,j) - b(i,j)}. The two matrices must have the
+same dimensions.
+@end deftypefun
+
+@deftypefun int gsl_matrix_mul_elements (gsl_matrix * @var{a}, const gsl_matrix * @var{b})
+This function multiplies the elements of matrix @var{a} by the elements of
+matrix @var{b}, @math{a'(i,j) = a(i,j) * b(i,j)}. The two matrices must have the
+same dimensions.
+@end deftypefun
+
+@deftypefun int gsl_matrix_div_elements (gsl_matrix * @var{a}, const gsl_matrix * @var{b})
+This function divides the elements of matrix @var{a} by the elements of
+matrix @var{b}, @math{a'(i,j) = a(i,j) / b(i,j)}. The two matrices must have the
+same dimensions.
+@end deftypefun
+
+@deftypefun int gsl_matrix_scale (gsl_matrix * @var{a}, const double @var{x})
+This function multiplies the elements of matrix @var{a} by the constant
+factor @var{x}, @math{a'(i,j) = x a(i,j)}.
+@end deftypefun
+
+@deftypefun int gsl_matrix_add_constant (gsl_matrix * @var{a}, const double @var{x})
+This function adds the constant value @var{x} to the elements of the
+matrix @var{a}, @math{a'(i,j) = a(i,j) + x}.
+@end deftypefun
+
+@node  Finding maximum and minimum elements of matrices
+@subsection Finding maximum and minimum elements of matrices
+
+The following operations are only defined for real matrices.
+
+@deftypefun double gsl_matrix_max (const gsl_matrix * @var{m})
+This function returns the maximum value in the matrix @var{m}.
+@end deftypefun
+
+@deftypefun double gsl_matrix_min (const gsl_matrix * @var{m})
+This function returns the minimum value in the matrix @var{m}.
+@end deftypefun
+
+@deftypefun void gsl_matrix_minmax (const gsl_matrix * @var{m}, double * @var{min_out}, double * @var{max_out})
+This function returns the minimum and maximum values in the matrix
+@var{m}, storing them in @var{min_out} and @var{max_out}.
+@end deftypefun
+
+@deftypefun void gsl_matrix_max_index (const gsl_matrix * @var{m}, size_t * @var{imax}, size_t * @var{jmax})
+This function returns the indices of the maximum value in the matrix
+@var{m}, storing them in @var{imax} and @var{jmax}.  When there are
+several equal maximum elements then the first element found is returned,
+searching in row-major order.
+@end deftypefun
+
+@deftypefun void gsl_matrix_min_index (const gsl_matrix * @var{m}, size_t * @var{imin}, size_t * @var{jmin})
+This function returns the indices of the minimum value in the matrix
+@var{m}, storing them in @var{imin} and @var{jmin}.  When there are
+several equal minimum elements then the first element found is returned,
+searching in row-major order.
+@end deftypefun
+
+@deftypefun void gsl_matrix_minmax_index (const gsl_matrix * @var{m}, size_t * @var{imin}, size_t * @var{jmin}, size_t * @var{imax}, size_t * @var{jmax})
+This function returns the indices of the minimum and maximum values in
+the matrix @var{m}, storing them in (@var{imin},@var{jmin}) and
+(@var{imax},@var{jmax}). When there are several equal minimum or maximum
+elements then the first elements found are returned, searching in
+row-major order.
+@end deftypefun
+
+@node Matrix properties
+@subsection Matrix properties
+
+@deftypefun int gsl_matrix_isnull (const gsl_matrix * @var{m})
+@deftypefunx int gsl_matrix_ispos (const gsl_matrix * @var{m})
+@deftypefunx int gsl_matrix_isneg (const gsl_matrix * @var{m})
+These functions return 1 if all the elements of the matrix @var{m} are
+zero, strictly positive, or strictly negative respectively, and 0
+otherwise. To test for a non-negative matrix, use the expression
+@code{!gsl_matrix_isneg(m)}.  To test whether a matrix is
+positive-definite, use the Cholesky decomposition (@pxref{Cholesky Decomposition}).
+@end deftypefun
+
+@node Example programs for matrices
+@subsection Example programs for matrices
+
+The program below shows how to allocate, initialize and read from a matrix
+using the functions @code{gsl_matrix_alloc}, @code{gsl_matrix_set} and
+@code{gsl_matrix_get}.
+
+@example
+@verbatiminclude examples/matrix.c
+@end example
+@comment
+
+@noindent
+Here is the output from the program.  The final loop attempts to read
+outside the range of the matrix @code{m}, and the error is trapped by
+the range-checking code in @code{gsl_matrix_get}.
+
+@example
+$ ./a.out
+m(0,0) = 0.23
+m(0,1) = 1.23
+m(0,2) = 2.23
+m(1,0) = 100.23
+m(1,1) = 101.23
+m(1,2) = 102.23
+...
+m(9,2) = 902.23
+gsl: matrix_source.c:13: ERROR: first index out of range
+Default GSL error handler invoked.
+Aborted (core dumped)
+@end example
+@comment
+
+@noindent
+The next program shows how to write a matrix to a file.
+
+@example
+@verbatiminclude examples/matrixw.c
+@end example
+@comment
+
+@noindent
+After running this program the file @file{test.dat} should contain the
+elements of @code{m}, written in binary format.  The matrix which is read
+back in using the function @code{gsl_matrix_fread} should be exactly
+equal to the original matrix.
+
+The following program demonstrates the use of vector views.  The program
+computes the column norms of a matrix.
+
+@example
+@verbatiminclude examples/vectorview.c
+@end example
+
+@noindent
+Here is the output of the program, 
+
+@example
+$ ./a.out
+@verbatiminclude examples/vectorview.out
+@end example
+
+@noindent
+The results can be confirmed using @sc{gnu octave},
+
+@example
+$ octave
+GNU Octave, version 2.0.16.92
+octave> m = sin(0:9)' * ones(1,10) 
+               + ones(10,1) * cos(0:9); 
+octave> sqrt(sum(m.^2))
+ans =
+  4.3146  3.1205  2.1932  3.2611  2.5342  2.5728
+  4.2047  3.6520  2.0852  3.0731
+@end example
+
+
+@node Vector and Matrix References and Further Reading
+@section References and Further Reading
+
+The block, vector and matrix objects in GSL follow the @code{valarray}
+model of C++.  A description of this model can be found in the following
+reference,
+
+@itemize @asis
+@item
+B. Stroustrup,
+@cite{The C++ Programming Language} (3rd Ed), 
+Section 22.4 Vector Arithmetic.
+Addison-Wesley 1997, ISBN 0-201-88954-4.
+@end itemize
diff --git a/gsl-1.9/doc/version-ref.texi b/gsl-1.9/doc/version-ref.texi
new file mode 100644
index 0000000..391fb64
--- /dev/null
+++ b/gsl-1.9/doc/version-ref.texi
@@ -0,0 +1,4 @@
+@set UPDATED 20 February 2007
+@set UPDATED-MONTH February 2007
+@set EDITION 1.9
+@set VERSION 1.9