diff options
| author | Joel Sherrill <joel.sherrill@OARcorp.com> | 2011-04-08 17:33:11 +0000 |
|---|---|---|
| committer | Joel Sherrill <joel.sherrill@OARcorp.com> | 2011-04-08 17:33:11 +0000 |
| commit | 73f643f3f4a55310b2c8c1a9858906b2dd676e72 (patch) | |
| tree | b1df97f18dace4a5702b0bc7aafdfee8a2f25ada /gsl-1.9/doc/linalg.texi | |
| parent | ee523abdace8337d05ec4a179fcdf5de3fe0f634 (diff) | |
2011-04-08 Joel Sherrill <joel.sherrill@oarcorp.com>
* AUTHORS, BUGS, COPYING, ChangeLog, INSTALL, Makefile.am, Makefile.in,
NEWS, README, SUPPORT, THANKS, TODO, acconfig.h, aclocal.m4,
autogen.sh, config.guess, config.h.in, config.sub, configure,
configure.ac, gsl-config.in, gsl-histogram.c, gsl-randist.c, gsl.m4,
gsl.pc.in, gsl.spec.in, gsl_machine.h, gsl_math.h, gsl_mode.h,
gsl_nan.h, gsl_pow_int.h, gsl_precision.h, gsl_types.h,
gsl_version.h.in, install-sh, ltmain.sh, mdate-sh, missing,
mkinstalldirs, templates_off.h, templates_on.h,
test_gsl_histogram.sh, version.c, blas/ChangeLog, blas/Makefile.am,
blas/Makefile.in, blas/TODO, blas/blas.c, blas/gsl_blas.h,
blas/gsl_blas_types.h, block/ChangeLog, block/Makefile.am,
block/Makefile.in, block/block.c, block/block_source.c, block/file.c,
block/fprintf_source.c, block/fwrite_source.c, block/gsl_block.h,
block/gsl_block_char.h, block/gsl_block_complex_double.h,
block/gsl_block_complex_float.h,
block/gsl_block_complex_long_double.h, block/gsl_block_double.h,
block/gsl_block_float.h, block/gsl_block_int.h,
block/gsl_block_long.h, block/gsl_block_long_double.h,
block/gsl_block_short.h, block/gsl_block_uchar.h,
block/gsl_block_uint.h, block/gsl_block_ulong.h,
block/gsl_block_ushort.h, block/gsl_check_range.h, block/init.c,
block/init_source.c, block/test.c, block/test_complex_io.c,
block/test_complex_source.c, block/test_io.c, block/test_source.c,
bspline/ChangeLog, bspline/Makefile.am, bspline/Makefile.in,
bspline/TODO, bspline/bspline.c, bspline/gsl_bspline.h,
bspline/test.c, cblas/ChangeLog, cblas/Makefile.am,
cblas/Makefile.in, cblas/TODO, cblas/caxpy.c, cblas/cblas.h,
cblas/ccopy.c, cblas/cdotc_sub.c, cblas/cdotu_sub.c, cblas/cgbmv.c,
cblas/cgemm.c, cblas/cgemv.c, cblas/cgerc.c, cblas/cgeru.c,
cblas/chbmv.c, cblas/chemm.c, cblas/chemv.c, cblas/cher.c,
cblas/cher2.c, cblas/cher2k.c, cblas/cherk.c, cblas/chpmv.c,
cblas/chpr.c, cblas/chpr2.c, cblas/cscal.c, cblas/csscal.c,
cblas/cswap.c, cblas/csymm.c, cblas/csyr2k.c, cblas/csyrk.c,
cblas/ctbmv.c, cblas/ctbsv.c, cblas/ctpmv.c, cblas/ctpsv.c,
cblas/ctrmm.c, cblas/ctrmv.c, cblas/ctrsm.c, cblas/ctrsv.c,
cblas/dasum.c, cblas/daxpy.c, cblas/dcopy.c, cblas/ddot.c,
cblas/dgbmv.c, cblas/dgemm.c, cblas/dgemv.c, cblas/dger.c,
cblas/dnrm2.c, cblas/drot.c, cblas/drotg.c, cblas/drotm.c,
cblas/drotmg.c, cblas/dsbmv.c, cblas/dscal.c, cblas/dsdot.c,
cblas/dspmv.c, cblas/dspr.c, cblas/dspr2.c, cblas/dswap.c,
cblas/dsymm.c, cblas/dsymv.c, cblas/dsyr.c, cblas/dsyr2.c,
cblas/dsyr2k.c, cblas/dsyrk.c, cblas/dtbmv.c, cblas/dtbsv.c,
cblas/dtpmv.c, cblas/dtpsv.c, cblas/dtrmm.c, cblas/dtrmv.c,
cblas/dtrsm.c, cblas/dtrsv.c, cblas/dzasum.c, cblas/dznrm2.c,
cblas/gsl_cblas.h, cblas/hypot.c, cblas/icamax.c, cblas/idamax.c,
cblas/isamax.c, cblas/izamax.c, cblas/sasum.c, cblas/saxpy.c,
cblas/scasum.c, cblas/scnrm2.c, cblas/scopy.c, cblas/sdot.c,
cblas/sdsdot.c, cblas/sgbmv.c, cblas/sgemm.c, cblas/sgemv.c,
cblas/sger.c, cblas/snrm2.c, cblas/source_asum_c.h,
cblas/source_asum_r.h, cblas/source_axpy_c.h, cblas/source_axpy_r.h,
cblas/source_copy_c.h, cblas/source_copy_r.h, cblas/source_dot_c.h,
cblas/source_dot_r.h, cblas/source_gbmv_c.h, cblas/source_gbmv_r.h,
cblas/source_gemm_c.h, cblas/source_gemm_r.h, cblas/source_gemv_c.h,
cblas/source_gemv_r.h, cblas/source_ger.h, cblas/source_gerc.h,
cblas/source_geru.h, cblas/source_hbmv.h, cblas/source_hemm.h,
cblas/source_hemv.h, cblas/source_her.h, cblas/source_her2.h,
cblas/source_her2k.h, cblas/source_herk.h, cblas/source_hpmv.h,
cblas/source_hpr.h, cblas/source_hpr2.h, cblas/source_iamax_c.h,
cblas/source_iamax_r.h, cblas/source_nrm2_c.h, cblas/source_nrm2_r.h,
cblas/source_rot.h, cblas/source_rotg.h, cblas/source_rotm.h,
cblas/source_rotmg.h, cblas/source_sbmv.h, cblas/source_scal_c.h,
cblas/source_scal_c_s.h, cblas/source_scal_r.h, cblas/source_spmv.h,
cblas/source_spr.h, cblas/source_spr2.h, cblas/source_swap_c.h,
cblas/source_swap_r.h, cblas/source_symm_c.h, cblas/source_symm_r.h,
cblas/source_symv.h, cblas/source_syr.h, cblas/source_syr2.h,
cblas/source_syr2k_c.h, cblas/source_syr2k_r.h,
cblas/source_syrk_c.h, cblas/source_syrk_r.h, cblas/source_tbmv_c.h,
cblas/source_tbmv_r.h, cblas/source_tbsv_c.h, cblas/source_tbsv_r.h,
cblas/source_tpmv_c.h, cblas/source_tpmv_r.h, cblas/source_tpsv_c.h,
cblas/source_tpsv_r.h, cblas/source_trmm_c.h, cblas/source_trmm_r.h,
cblas/source_trmv_c.h, cblas/source_trmv_r.h, cblas/source_trsm_c.h,
cblas/source_trsm_r.h, cblas/source_trsv_c.h, cblas/source_trsv_r.h,
cblas/srot.c, cblas/srotg.c, cblas/srotm.c, cblas/srotmg.c,
cblas/ssbmv.c, cblas/sscal.c, cblas/sspmv.c, cblas/sspr.c,
cblas/sspr2.c, cblas/sswap.c, cblas/ssymm.c, cblas/ssymv.c,
cblas/ssyr.c, cblas/ssyr2.c, cblas/ssyr2k.c, cblas/ssyrk.c,
cblas/stbmv.c, cblas/stbsv.c, cblas/stpmv.c, cblas/stpsv.c,
cblas/strmm.c, cblas/strmv.c, cblas/strsm.c, cblas/strsv.c,
cblas/test.c, cblas/test_amax.c, cblas/test_asum.c,
cblas/test_axpy.c, cblas/test_copy.c, cblas/test_dot.c,
cblas/test_gbmv.c, cblas/test_gemm.c, cblas/test_gemv.c,
cblas/test_ger.c, cblas/test_hbmv.c, cblas/test_hemm.c,
cblas/test_hemv.c, cblas/test_her.c, cblas/test_her2.c,
cblas/test_her2k.c, cblas/test_herk.c, cblas/test_hpmv.c,
cblas/test_hpr.c, cblas/test_hpr2.c, cblas/test_nrm2.c,
cblas/test_rot.c, cblas/test_rotg.c, cblas/test_rotm.c,
cblas/test_rotmg.c, cblas/test_sbmv.c, cblas/test_scal.c,
cblas/test_spmv.c, cblas/test_spr.c, cblas/test_spr2.c,
cblas/test_swap.c, cblas/test_symm.c, cblas/test_symv.c,
cblas/test_syr.c, cblas/test_syr2.c, cblas/test_syr2k.c,
cblas/test_syrk.c, cblas/test_tbmv.c, cblas/test_tbsv.c,
cblas/test_tpmv.c, cblas/test_tpsv.c, cblas/test_trmm.c,
cblas/test_trmv.c, cblas/test_trsm.c, cblas/test_trsv.c,
cblas/tests.c, cblas/tests.h, cblas/xerbla.c, cblas/zaxpy.c,
cblas/zcopy.c, cblas/zdotc_sub.c, cblas/zdotu_sub.c, cblas/zdscal.c,
cblas/zgbmv.c, cblas/zgemm.c, cblas/zgemv.c, cblas/zgerc.c,
cblas/zgeru.c, cblas/zhbmv.c, cblas/zhemm.c, cblas/zhemv.c,
cblas/zher.c, cblas/zher2.c, cblas/zher2k.c, cblas/zherk.c,
cblas/zhpmv.c, cblas/zhpr.c, cblas/zhpr2.c, cblas/zscal.c,
cblas/zswap.c, cblas/zsymm.c, cblas/zsyr2k.c, cblas/zsyrk.c,
cblas/ztbmv.c, cblas/ztbsv.c, cblas/ztpmv.c, cblas/ztpsv.c,
cblas/ztrmm.c, cblas/ztrmv.c, cblas/ztrsm.c, cblas/ztrsv.c,
cdf/ChangeLog, cdf/Makefile.am, cdf/Makefile.in, cdf/beta.c,
cdf/beta_inc.c, cdf/betainv.c, cdf/binomial.c, cdf/cauchy.c,
cdf/cauchyinv.c, cdf/chisq.c, cdf/chisqinv.c, cdf/error.h,
cdf/exponential.c, cdf/exponentialinv.c, cdf/exppow.c, cdf/fdist.c,
cdf/fdistinv.c, cdf/flat.c, cdf/flatinv.c, cdf/gamma.c,
cdf/gammainv.c, cdf/gauss.c, cdf/gaussinv.c, cdf/geometric.c,
cdf/gsl_cdf.h, cdf/gumbel1.c, cdf/gumbel1inv.c, cdf/gumbel2.c,
cdf/gumbel2inv.c, cdf/hypergeometric.c, cdf/laplace.c,
cdf/laplaceinv.c, cdf/logistic.c, cdf/logisticinv.c, cdf/lognormal.c,
cdf/lognormalinv.c, cdf/nbinomial.c, cdf/pareto.c, cdf/paretoinv.c,
cdf/pascal.c, cdf/poisson.c, cdf/rat_eval.h, cdf/rayleigh.c,
cdf/rayleighinv.c, cdf/tdist.c, cdf/tdistinv.c, cdf/test.c,
cdf/test_auto.c, cdf/weibull.c, cdf/weibullinv.c, cheb/ChangeLog,
cheb/Makefile.am, cheb/Makefile.in, cheb/deriv.c, cheb/eval.c,
cheb/gsl_chebyshev.h, cheb/init.c, cheb/integ.c, cheb/test.c,
combination/ChangeLog, combination/Makefile.am,
combination/Makefile.in, combination/combination.c,
combination/file.c, combination/gsl_combination.h,
combination/init.c, combination/test.c, complex/ChangeLog,
complex/Makefile.am, complex/Makefile.in, complex/TODO,
complex/gsl_complex.h, complex/gsl_complex_math.h, complex/math.c,
complex/results.h, complex/results1.h, complex/results_real.h,
complex/test.c, const/ChangeLog, const/Makefile.am,
const/Makefile.in, const/TODO, const/gsl_const.h,
const/gsl_const_cgs.h, const/gsl_const_cgsm.h, const/gsl_const_mks.h,
const/gsl_const_mksa.h, const/gsl_const_num.h, const/test.c,
deriv/ChangeLog, deriv/Makefile.am, deriv/Makefile.in, deriv/deriv.c,
deriv/gsl_deriv.h, deriv/test.c, dht/ChangeLog, dht/Makefile.am,
dht/Makefile.in, dht/dht.c, dht/gsl_dht.h, dht/test.c,
diff/ChangeLog, diff/Makefile.am, diff/Makefile.in, diff/diff.c,
diff/gsl_diff.h, diff/test.c, doc/12-cities.eps, doc/ChangeLog,
doc/Makefile.am, doc/Makefile.in, doc/algorithm.sty,
doc/algorithmic.sty, doc/autoconf.texi, doc/blas.texi,
doc/bspline.eps, doc/bspline.texi, doc/calc.sty, doc/cblas.texi,
doc/cheb.eps, doc/cheb.texi, doc/combination.texi, doc/complex.texi,
doc/const.texi, doc/debug.texi, doc/dht.texi, doc/diff.texi,
doc/dwt-orig.eps, doc/dwt-samp.eps, doc/dwt.texi, doc/eigen.texi,
doc/err.texi, doc/fdl.texi, doc/fft-complex-radix2-f.eps,
doc/fft-complex-radix2-t.eps, doc/fft-complex-radix2.eps,
doc/fft-real-mixedradix.eps, doc/fft.texi, doc/fftalgorithms.bib,
doc/fftalgorithms.tex, doc/final-route.eps, doc/fit-exp.eps,
doc/fit-wlinear.eps, doc/fit-wlinear2.eps, doc/fitting.texi,
doc/freemanuals.texi, doc/gpl.texi, doc/gsl-config.1,
doc/gsl-design.texi, doc/gsl-histogram.1, doc/gsl-randist.1,
doc/gsl-ref.info, doc/gsl-ref.info-1, doc/gsl-ref.info-2,
doc/gsl-ref.info-3, doc/gsl-ref.info-4, doc/gsl-ref.info-5,
doc/gsl-ref.info-6, doc/gsl-ref.texi, doc/gsl.3, doc/histogram.eps,
doc/histogram.texi, doc/histogram2d.eps, doc/ieee754.texi,
doc/initial-route.eps, doc/integration.texi, doc/interp.texi,
doc/interp2.eps, doc/interpp2.eps, doc/intro.texi, doc/landau.dat,
doc/linalg.texi, doc/math.texi, doc/mdate-sh, doc/min-interval.eps,
doc/min.texi, doc/montecarlo.texi, doc/multifit.texi,
doc/multimin.eps, doc/multimin.texi, doc/multiroots.texi,
doc/ntuple.eps, doc/ntuple.texi, doc/ode-initval.texi,
doc/permutation.texi, doc/poly.texi, doc/qrng.eps, doc/qrng.texi,
doc/rand-bernoulli.tex, doc/rand-beta.tex, doc/rand-binomial.tex,
doc/rand-bivariate-gaussian.tex, doc/rand-cauchy.tex,
doc/rand-chisq.tex, doc/rand-erlang.tex, doc/rand-exponential.tex,
doc/rand-exppow.tex, doc/rand-fdist.tex, doc/rand-flat.tex,
doc/rand-gamma.tex, doc/rand-gaussian-tail.tex,
doc/rand-gaussian.tex, doc/rand-geometric.tex, doc/rand-gumbel.tex,
doc/rand-gumbel1.tex, doc/rand-gumbel2.tex,
doc/rand-hypergeometric.tex, doc/rand-landau.tex,
doc/rand-laplace.tex, doc/rand-levy.tex, doc/rand-levyskew.tex,
doc/rand-logarithmic.tex, doc/rand-logistic.tex,
doc/rand-lognormal.tex, doc/rand-nbinomial.tex, doc/rand-pareto.tex,
doc/rand-pascal.tex, doc/rand-poisson.tex,
doc/rand-rayleigh-tail.tex, doc/rand-rayleigh.tex,
doc/rand-tdist.tex, doc/rand-weibull.tex, doc/randist.texi,
doc/random-walk.tex, doc/randplots.gnp, doc/rng.texi,
doc/roots-bisection.eps, doc/roots-false-position.eps,
doc/roots-newtons-method.eps, doc/roots-secant-method.eps,
doc/roots.texi, doc/siman-energy.eps, doc/siman-test.eps,
doc/siman.texi, doc/sort.texi, doc/specfunc-airy.texi,
doc/specfunc-bessel.texi, doc/specfunc-clausen.texi,
doc/specfunc-coulomb.texi, doc/specfunc-coupling.texi,
doc/specfunc-dawson.texi, doc/specfunc-debye.texi,
doc/specfunc-dilog.texi, doc/specfunc-elementary.texi,
doc/specfunc-ellint.texi, doc/specfunc-elljac.texi,
doc/specfunc-erf.texi, doc/specfunc-exp.texi,
doc/specfunc-expint.texi, doc/specfunc-fermi-dirac.texi,
doc/specfunc-gamma.texi, doc/specfunc-gegenbauer.texi,
doc/specfunc-hyperg.texi, doc/specfunc-laguerre.texi,
doc/specfunc-lambert.texi, doc/specfunc-legendre.texi,
doc/specfunc-log.texi, doc/specfunc-mathieu.texi,
doc/specfunc-pow-int.texi, doc/specfunc-psi.texi,
doc/specfunc-synchrotron.texi, doc/specfunc-transport.texi,
doc/specfunc-trig.texi, doc/specfunc-zeta.texi, doc/specfunc.texi,
doc/stamp-vti, doc/statistics.texi, doc/sum.texi, doc/texinfo.tex,
doc/usage.texi, doc/vdp.eps, doc/vectors.texi, doc/version-ref.texi,
doc/examples/blas.c, doc/examples/blas.out, doc/examples/block.c,
doc/examples/block.out, doc/examples/bspline.c, doc/examples/cblas.c,
doc/examples/cblas.out, doc/examples/cdf.c, doc/examples/cdf.out,
doc/examples/cheb.c, doc/examples/combination.c,
doc/examples/combination.out, doc/examples/const.c,
doc/examples/const.out, doc/examples/demo_fn.c,
doc/examples/demo_fn.h, doc/examples/diff.c, doc/examples/diff.out,
doc/examples/dwt.c, doc/examples/dwt.dat, doc/examples/ecg.dat,
doc/examples/eigen.c, doc/examples/eigen_nonsymm.c,
doc/examples/expfit.c, doc/examples/fft.c, doc/examples/fftmr.c,
doc/examples/fftreal.c, doc/examples/fitting.c,
doc/examples/fitting2.c, doc/examples/fitting3.c,
doc/examples/histogram.c, doc/examples/histogram2d.c,
doc/examples/ieee.c, doc/examples/ieeeround.c,
doc/examples/integration.c, doc/examples/integration.out,
doc/examples/interp.c, doc/examples/interpp.c, doc/examples/intro.c,
doc/examples/intro.out, doc/examples/linalglu.c,
doc/examples/linalglu.out, doc/examples/matrix.c,
doc/examples/matrixw.c, doc/examples/min.c, doc/examples/min.out,
doc/examples/monte.c, doc/examples/nlfit.c, doc/examples/ntupler.c,
doc/examples/ntuplew.c, doc/examples/ode-initval.c,
doc/examples/odefixed.c, doc/examples/permseq.c,
doc/examples/permshuffle.c, doc/examples/polyroots.c,
doc/examples/polyroots.out, doc/examples/qrng.c,
doc/examples/randpoisson.2.out, doc/examples/randpoisson.c,
doc/examples/randpoisson.out, doc/examples/randwalk.c,
doc/examples/rng.c, doc/examples/rng.out, doc/examples/rngunif.2.out,
doc/examples/rngunif.c, doc/examples/rngunif.out,
doc/examples/rootnewt.c, doc/examples/roots.c, doc/examples/siman.c,
doc/examples/sortsmall.c, doc/examples/sortsmall.out,
doc/examples/specfun.c, doc/examples/specfun.out,
doc/examples/specfun_e.c, doc/examples/specfun_e.out,
doc/examples/stat.c, doc/examples/stat.out, doc/examples/statsort.c,
doc/examples/statsort.out, doc/examples/sum.c, doc/examples/sum.out,
doc/examples/vector.c, doc/examples/vectorr.c,
doc/examples/vectorview.c, doc/examples/vectorview.out,
doc/examples/vectorw.c, eigen/ChangeLog, eigen/Makefile.am,
eigen/Makefile.in, eigen/TODO, eigen/francis.c, eigen/gsl_eigen.h,
eigen/herm.c, eigen/hermv.c, eigen/jacobi.c, eigen/nonsymm.c,
eigen/nonsymmv.c, eigen/qrstep.c, eigen/schur.c, eigen/schur.h,
eigen/sort.c, eigen/symm.c, eigen/symmv.c, eigen/test.c,
err/ChangeLog, err/Makefile.am, err/Makefile.in, err/TODO,
err/error.c, err/gsl_errno.h, err/gsl_message.h, err/message.c,
err/stream.c, err/strerror.c, err/test.c, fft/ChangeLog,
fft/Makefile.am, fft/Makefile.in, fft/TODO, fft/bitreverse.c,
fft/bitreverse.h, fft/c_init.c, fft/c_main.c, fft/c_pass.h,
fft/c_pass_2.c, fft/c_pass_3.c, fft/c_pass_4.c, fft/c_pass_5.c,
fft/c_pass_6.c, fft/c_pass_7.c, fft/c_pass_n.c, fft/c_radix2.c,
fft/compare.h, fft/compare_source.c, fft/complex_internal.h,
fft/dft.c, fft/dft_source.c, fft/factorize.c, fft/factorize.h,
fft/fft.c, fft/gsl_dft_complex.h, fft/gsl_dft_complex_float.h,
fft/gsl_fft.h, fft/gsl_fft_complex.h, fft/gsl_fft_complex_float.h,
fft/gsl_fft_halfcomplex.h, fft/gsl_fft_halfcomplex_float.h,
fft/gsl_fft_real.h, fft/gsl_fft_real_float.h, fft/hc_init.c,
fft/hc_main.c, fft/hc_pass.h, fft/hc_pass_2.c, fft/hc_pass_3.c,
fft/hc_pass_4.c, fft/hc_pass_5.c, fft/hc_pass_n.c, fft/hc_radix2.c,
fft/hc_unpack.c, fft/real_init.c, fft/real_main.c, fft/real_pass.h,
fft/real_pass_2.c, fft/real_pass_3.c, fft/real_pass_4.c,
fft/real_pass_5.c, fft/real_pass_n.c, fft/real_radix2.c,
fft/real_unpack.c, fft/signals.c, fft/signals.h,
fft/signals_source.c, fft/test.c, fft/test_complex_source.c,
fft/test_real_source.c, fft/test_trap_source.c, fft/urand.c,
fit/ChangeLog, fit/Makefile.am, fit/Makefile.in, fit/gsl_fit.h,
fit/linear.c, fit/test.c, gsl/Makefile.am, gsl/Makefile.in,
histogram/ChangeLog, histogram/Makefile.am, histogram/Makefile.in,
histogram/TODO, histogram/add.c, histogram/add2d.c,
histogram/calloc_range.c, histogram/calloc_range2d.c,
histogram/copy.c, histogram/copy2d.c, histogram/file.c,
histogram/file2d.c, histogram/find.c, histogram/find2d.c,
histogram/get.c, histogram/get2d.c, histogram/gsl_histogram.h,
histogram/gsl_histogram2d.h, histogram/init.c, histogram/init2d.c,
histogram/maxval.c, histogram/maxval2d.c, histogram/oper.c,
histogram/oper2d.c, histogram/params.c, histogram/params2d.c,
histogram/pdf.c, histogram/pdf2d.c, histogram/reset.c,
histogram/reset2d.c, histogram/stat.c, histogram/stat2d.c,
histogram/test.c, histogram/test1d.c, histogram/test1d_resample.c,
histogram/test1d_trap.c, histogram/test2d.c,
histogram/test2d_resample.c, histogram/test2d_trap.c,
histogram/urand.c, ieee-utils/ChangeLog, ieee-utils/Makefile.am,
ieee-utils/Makefile.in, ieee-utils/TODO, ieee-utils/endian.c,
ieee-utils/env.c, ieee-utils/fp-aix.c, ieee-utils/fp-darwin.c,
ieee-utils/fp-darwin86.c, ieee-utils/fp-freebsd.c,
ieee-utils/fp-gnuc99.c, ieee-utils/fp-gnum68k.c,
ieee-utils/fp-gnuppc.c, ieee-utils/fp-gnusparc.c,
ieee-utils/fp-gnux86.c, ieee-utils/fp-hpux.c, ieee-utils/fp-hpux11.c,
ieee-utils/fp-irix.c, ieee-utils/fp-netbsd.c,
ieee-utils/fp-openbsd.c, ieee-utils/fp-os2emx.c,
ieee-utils/fp-solaris.c, ieee-utils/fp-sunos4.c,
ieee-utils/fp-tru64.c, ieee-utils/fp-unknown.c, ieee-utils/fp.c,
ieee-utils/gsl_ieee_utils.h, ieee-utils/make_rep.c,
ieee-utils/print.c, ieee-utils/read.c, ieee-utils/standardize.c,
ieee-utils/test.c, integration/ChangeLog, integration/Makefile.am,
integration/Makefile.in, integration/TODO, integration/append.c,
integration/err.c, integration/gsl_integration.h,
integration/initialise.c, integration/positivity.c,
integration/ptsort.c, integration/qag.c, integration/qagp.c,
integration/qags.c, integration/qawc.c, integration/qawf.c,
integration/qawo.c, integration/qaws.c, integration/qc25c.c,
integration/qc25f.c, integration/qc25s.c, integration/qcheb.c,
integration/qelg.c, integration/qk.c, integration/qk15.c,
integration/qk21.c, integration/qk31.c, integration/qk41.c,
integration/qk51.c, integration/qk61.c, integration/qmomo.c,
integration/qmomof.c, integration/qng.c, integration/qng.h,
integration/qpsrt.c, integration/qpsrt2.c, integration/reset.c,
integration/set_initial.c, integration/test.c, integration/tests.c,
integration/tests.h, integration/util.c, integration/workspace.c,
interpolation/ChangeLog, interpolation/Makefile.am,
interpolation/Makefile.in, interpolation/TODO, interpolation/accel.c,
interpolation/akima.c, interpolation/bsearch.c,
interpolation/bsearch.h, interpolation/cspline.c,
interpolation/gsl_interp.h, interpolation/gsl_spline.h,
interpolation/integ_eval.h, interpolation/interp.c,
interpolation/linear.c, interpolation/poly.c, interpolation/spline.c,
interpolation/test.c, linalg/ChangeLog, linalg/Makefile.am,
linalg/Makefile.in, linalg/TODO, linalg/apply_givens.c,
linalg/balance.c, linalg/balancemat.c, linalg/bidiag.c,
linalg/cholesky.c, linalg/exponential.c, linalg/givens.c,
linalg/gsl_linalg.h, linalg/hermtd.c, linalg/hessenberg.c,
linalg/hh.c, linalg/householder.c, linalg/householdercomplex.c,
linalg/lq.c, linalg/lu.c, linalg/luc.c, linalg/multiply.c,
linalg/ptlq.c, linalg/qr.c, linalg/qrpt.c, linalg/svd.c,
linalg/svdstep.c, linalg/symmtd.c, linalg/test.c, linalg/tridiag.c,
linalg/tridiag.h, matrix/ChangeLog, matrix/Makefile.am,
matrix/Makefile.in, matrix/TODO, matrix/copy.c, matrix/copy_source.c,
matrix/file.c, matrix/file_source.c, matrix/getset.c,
matrix/getset_source.c, matrix/gsl_matrix.h,
matrix/gsl_matrix_char.h, matrix/gsl_matrix_complex_double.h,
matrix/gsl_matrix_complex_float.h,
matrix/gsl_matrix_complex_long_double.h, matrix/gsl_matrix_double.h,
matrix/gsl_matrix_float.h, matrix/gsl_matrix_int.h,
matrix/gsl_matrix_long.h, matrix/gsl_matrix_long_double.h,
matrix/gsl_matrix_short.h, matrix/gsl_matrix_uchar.h,
matrix/gsl_matrix_uint.h, matrix/gsl_matrix_ulong.h,
matrix/gsl_matrix_ushort.h, matrix/init.c, matrix/init_source.c,
matrix/matrix.c, matrix/matrix_source.c, matrix/minmax.c,
matrix/minmax_source.c, matrix/oper.c, matrix/oper_complex_source.c,
matrix/oper_source.c, matrix/prop.c, matrix/prop_source.c,
matrix/rowcol.c, matrix/rowcol_source.c, matrix/submatrix.c,
matrix/submatrix_source.c, matrix/swap.c, matrix/swap_source.c,
matrix/test.c, matrix/test_complex_source.c, matrix/test_source.c,
matrix/test_static.c, matrix/view.c, matrix/view.h,
matrix/view_source.c, min/ChangeLog, min/Makefile.am,
min/Makefile.in, min/bracketing.c, min/brent.c, min/convergence.c,
min/fsolver.c, min/golden.c, min/gsl_min.h, min/min.h, min/test.c,
min/test.h, min/test_funcs.c, monte/ChangeLog, monte/Makefile.am,
monte/Makefile.in, monte/README, monte/TODO, monte/gsl_monte.h,
monte/gsl_monte_miser.h, monte/gsl_monte_plain.h,
monte/gsl_monte_vegas.h, monte/miser.c, monte/plain.c, monte/test.c,
monte/test_main.c, monte/vegas.c, multifit/ChangeLog,
multifit/Makefile.am, multifit/Makefile.in, multifit/TODO,
multifit/convergence.c, multifit/covar.c, multifit/fdfsolver.c,
multifit/fsolver.c, multifit/gradient.c, multifit/gsl_multifit.h,
multifit/gsl_multifit_nlin.h, multifit/lmder.c, multifit/lmiterate.c,
multifit/lmpar.c, multifit/lmset.c, multifit/lmutil.c,
multifit/multilinear.c, multifit/qrsolv.c, multifit/test.c,
multifit/test_brown.c, multifit/test_enso.c,
multifit/test_estimator.c, multifit/test_filip.c, multifit/test_fn.c,
multifit/test_hahn1.c, multifit/test_kirby2.c,
multifit/test_longley.c, multifit/test_nelson.c,
multifit/test_pontius.c, multifit/work.c, multimin/ChangeLog,
multimin/Makefile.am, multimin/Makefile.in, multimin/TODO,
multimin/conjugate_fr.c, multimin/conjugate_pr.c,
multimin/convergence.c, multimin/diff.c,
multimin/directional_minimize.c, multimin/fdfminimizer.c,
multimin/fminimizer.c, multimin/gsl_multimin.h,
multimin/linear_minimize.c, multimin/linear_wrapper.c,
multimin/simplex.c, multimin/steepest_descent.c, multimin/test.c,
multimin/test_funcs.c, multimin/test_funcs.h, multimin/vector_bfgs.c,
multimin/vector_bfgs2.c, multiroots/ChangeLog,
multiroots/Makefile.am, multiroots/Makefile.in, multiroots/broyden.c,
multiroots/convergence.c, multiroots/dnewton.c, multiroots/dogleg.c,
multiroots/enorm.c, multiroots/fdfsolver.c, multiroots/fdjac.c,
multiroots/fsolver.c, multiroots/gnewton.c,
multiroots/gsl_multiroots.h, multiroots/hybrid.c,
multiroots/hybridj.c, multiroots/newton.c, multiroots/test.c,
multiroots/test_funcs.c, multiroots/test_funcs.h, ntuple/ChangeLog,
ntuple/Makefile.am, ntuple/Makefile.in, ntuple/gsl_ntuple.h,
ntuple/ntuple.c, ntuple/test.c, ode-initval/ChangeLog,
ode-initval/Makefile.am, ode-initval/Makefile.in, ode-initval/TODO,
ode-initval/bsimp.c, ode-initval/control.c, ode-initval/cscal.c,
ode-initval/cstd.c, ode-initval/evolve.c, ode-initval/gear1.c,
ode-initval/gear2.c, ode-initval/gsl_odeiv.h,
ode-initval/odeiv_util.h, ode-initval/rk2.c, ode-initval/rk2imp.c,
ode-initval/rk2simp.c, ode-initval/rk4.c, ode-initval/rk4imp.c,
ode-initval/rk8pd.c, ode-initval/rkck.c, ode-initval/rkf45.c,
ode-initval/step.c, ode-initval/test.c, permutation/ChangeLog,
permutation/Makefile.am, permutation/Makefile.in,
permutation/canonical.c, permutation/file.c,
permutation/gsl_permutation.h, permutation/gsl_permute.h,
permutation/gsl_permute_char.h,
permutation/gsl_permute_complex_double.h,
permutation/gsl_permute_complex_float.h,
permutation/gsl_permute_complex_long_double.h,
permutation/gsl_permute_double.h, permutation/gsl_permute_float.h,
permutation/gsl_permute_int.h, permutation/gsl_permute_long.h,
permutation/gsl_permute_long_double.h,
permutation/gsl_permute_short.h, permutation/gsl_permute_uchar.h,
permutation/gsl_permute_uint.h, permutation/gsl_permute_ulong.h,
permutation/gsl_permute_ushort.h, permutation/gsl_permute_vector.h,
permutation/gsl_permute_vector_char.h,
permutation/gsl_permute_vector_complex_double.h,
permutation/gsl_permute_vector_complex_float.h,
permutation/gsl_permute_vector_complex_long_double.h,
permutation/gsl_permute_vector_double.h,
permutation/gsl_permute_vector_float.h,
permutation/gsl_permute_vector_int.h,
permutation/gsl_permute_vector_long.h,
permutation/gsl_permute_vector_long_double.h,
permutation/gsl_permute_vector_short.h,
permutation/gsl_permute_vector_uchar.h,
permutation/gsl_permute_vector_uint.h,
permutation/gsl_permute_vector_ulong.h,
permutation/gsl_permute_vector_ushort.h, permutation/init.c,
permutation/permutation.c, permutation/permute.c,
permutation/permute_source.c, permutation/test.c, poly/ChangeLog,
poly/Makefile.am, poly/Makefile.in, poly/TODO, poly/balance.c,
poly/companion.c, poly/dd.c, poly/eval.c, poly/gsl_poly.h, poly/qr.c,
poly/solve_cubic.c, poly/solve_quadratic.c, poly/test.c,
poly/zsolve.c, poly/zsolve_cubic.c, poly/zsolve_init.c,
poly/zsolve_quadratic.c, qrng/ChangeLog, qrng/Makefile.am,
qrng/Makefile.in, qrng/TODO, qrng/gsl_qrng.h, qrng/niederreiter-2.c,
qrng/qrng.c, qrng/sobol.c, qrng/test.c, randist/ChangeLog,
randist/Makefile.am, randist/Makefile.in, randist/TODO,
randist/bernoulli.c, randist/beta.c, randist/bigauss.c,
randist/binomial.c, randist/binomial_tpe.c, randist/cauchy.c,
randist/chisq.c, randist/dirichlet.c, randist/discrete.c,
randist/erlang.c, randist/exponential.c, randist/exppow.c,
randist/fdist.c, randist/flat.c, randist/gamma.c, randist/gauss.c,
randist/gausstail.c, randist/gausszig.c, randist/geometric.c,
randist/gsl_randist.h, randist/gumbel.c, randist/hyperg.c,
randist/landau.c, randist/laplace.c, randist/levy.c,
randist/logarithmic.c, randist/logistic.c, randist/lognormal.c,
randist/multinomial.c, randist/nbinomial.c, randist/pareto.c,
randist/pascal.c, randist/poisson.c, randist/rayleigh.c,
randist/shuffle.c, randist/sphere.c, randist/tdist.c, randist/test.c,
randist/weibull.c, rng/ChangeLog, rng/Makefile.am, rng/Makefile.in,
rng/TODO, rng/borosh13.c, rng/cmrg.c, rng/coveyou.c, rng/default.c,
rng/file.c, rng/fishman18.c, rng/fishman20.c, rng/fishman2x.c,
rng/gfsr4.c, rng/gsl_rng.h, rng/knuthran.c, rng/knuthran2.c,
rng/knuthran2002.c, rng/lecuyer21.c, rng/minstd.c, rng/mrg.c,
rng/mt.c, rng/r250.c, rng/ran0.c, rng/ran1.c, rng/ran2.c, rng/ran3.c,
rng/rand.c, rng/rand48.c, rng/random.c, rng/randu.c, rng/ranf.c,
rng/ranlux.c, rng/ranlxd.c, rng/ranlxs.c, rng/ranmar.c, rng/rng.c,
rng/schrage.c, rng/slatec.c, rng/taus.c, rng/taus113.c, rng/test.c,
rng/transputer.c, rng/tt.c, rng/types.c, rng/uni.c, rng/uni32.c,
rng/vax.c, rng/waterman14.c, rng/zuf.c, roots/ChangeLog,
roots/Makefile.am, roots/Makefile.in, roots/TODO, roots/bisection.c,
roots/brent.c, roots/convergence.c, roots/falsepos.c,
roots/fdfsolver.c, roots/fsolver.c, roots/gsl_roots.h,
roots/newton.c, roots/roots.h, roots/secant.c, roots/steffenson.c,
roots/test.c, roots/test.h, roots/test_funcs.c, siman/ChangeLog,
siman/Makefile.am, siman/Makefile.in, siman/TODO, siman/gsl_siman.h,
siman/siman.c, siman/siman_test_driver.sh, siman/siman_tsp.c,
siman/test.c, sort/ChangeLog, sort/Makefile.am, sort/Makefile.in,
sort/TODO, sort/gsl_heapsort.h, sort/gsl_sort.h,
sort/gsl_sort_char.h, sort/gsl_sort_double.h, sort/gsl_sort_float.h,
sort/gsl_sort_int.h, sort/gsl_sort_long.h,
sort/gsl_sort_long_double.h, sort/gsl_sort_short.h,
sort/gsl_sort_uchar.h, sort/gsl_sort_uint.h, sort/gsl_sort_ulong.h,
sort/gsl_sort_ushort.h, sort/gsl_sort_vector.h,
sort/gsl_sort_vector_char.h, sort/gsl_sort_vector_double.h,
sort/gsl_sort_vector_float.h, sort/gsl_sort_vector_int.h,
sort/gsl_sort_vector_long.h, sort/gsl_sort_vector_long_double.h,
sort/gsl_sort_vector_short.h, sort/gsl_sort_vector_uchar.h,
sort/gsl_sort_vector_uint.h, sort/gsl_sort_vector_ulong.h,
sort/gsl_sort_vector_ushort.h, sort/sort.c, sort/sortind.c,
sort/sortvec.c, sort/sortvec_source.c, sort/sortvecind.c,
sort/sortvecind_source.c, sort/subset.c, sort/subset_source.c,
sort/subsetind.c, sort/subsetind_source.c, sort/test.c,
sort/test_heapsort.c, sort/test_source.c, specfunc/ChangeLog,
specfunc/Makefile.am, specfunc/Makefile.in, specfunc/TODO,
specfunc/airy.c, specfunc/airy_der.c, specfunc/airy_zero.c,
specfunc/atanint.c, specfunc/bessel.c, specfunc/bessel.h,
specfunc/bessel_I0.c, specfunc/bessel_I1.c, specfunc/bessel_In.c,
specfunc/bessel_Inu.c, specfunc/bessel_J0.c, specfunc/bessel_J1.c,
specfunc/bessel_Jn.c, specfunc/bessel_Jnu.c, specfunc/bessel_K0.c,
specfunc/bessel_K1.c, specfunc/bessel_Kn.c, specfunc/bessel_Knu.c,
specfunc/bessel_Y0.c, specfunc/bessel_Y1.c, specfunc/bessel_Yn.c,
specfunc/bessel_Ynu.c, specfunc/bessel_amp_phase.c,
specfunc/bessel_amp_phase.h, specfunc/bessel_i.c,
specfunc/bessel_j.c, specfunc/bessel_k.c, specfunc/bessel_olver.c,
specfunc/bessel_olver.h, specfunc/bessel_sequence.c,
specfunc/bessel_temme.c, specfunc/bessel_temme.h,
specfunc/bessel_y.c, specfunc/bessel_zero.c, specfunc/beta.c,
specfunc/beta_inc.c, specfunc/cheb_eval.c, specfunc/cheb_eval_mode.c,
specfunc/chebyshev.h, specfunc/check.h, specfunc/clausen.c,
specfunc/coulomb.c, specfunc/coulomb_bound.c, specfunc/coupling.c,
specfunc/dawson.c, specfunc/debye.c, specfunc/dilog.c,
specfunc/elementary.c, specfunc/ellint.c, specfunc/elljac.c,
specfunc/erfc.c, specfunc/error.h, specfunc/eval.h, specfunc/exp.c,
specfunc/expint.c, specfunc/expint3.c, specfunc/fermi_dirac.c,
specfunc/gamma.c, specfunc/gamma_inc.c, specfunc/gegenbauer.c,
specfunc/gsl_sf.h, specfunc/gsl_sf_airy.h, specfunc/gsl_sf_bessel.h,
specfunc/gsl_sf_clausen.h, specfunc/gsl_sf_coulomb.h,
specfunc/gsl_sf_coupling.h, specfunc/gsl_sf_dawson.h,
specfunc/gsl_sf_debye.h, specfunc/gsl_sf_dilog.h,
specfunc/gsl_sf_elementary.h, specfunc/gsl_sf_ellint.h,
specfunc/gsl_sf_elljac.h, specfunc/gsl_sf_erf.h,
specfunc/gsl_sf_exp.h, specfunc/gsl_sf_expint.h,
specfunc/gsl_sf_fermi_dirac.h, specfunc/gsl_sf_gamma.h,
specfunc/gsl_sf_gegenbauer.h, specfunc/gsl_sf_hyperg.h,
specfunc/gsl_sf_laguerre.h, specfunc/gsl_sf_lambert.h,
specfunc/gsl_sf_legendre.h, specfunc/gsl_sf_log.h,
specfunc/gsl_sf_mathieu.h, specfunc/gsl_sf_pow_int.h,
specfunc/gsl_sf_psi.h, specfunc/gsl_sf_result.h,
specfunc/gsl_sf_synchrotron.h, specfunc/gsl_sf_transport.h,
specfunc/gsl_sf_trig.h, specfunc/gsl_sf_zeta.h,
specfunc/gsl_specfunc.h, specfunc/hyperg.c, specfunc/hyperg.h,
specfunc/hyperg_0F1.c, specfunc/hyperg_1F1.c, specfunc/hyperg_2F0.c,
specfunc/hyperg_2F1.c, specfunc/hyperg_U.c, specfunc/laguerre.c,
specfunc/lambert.c, specfunc/legendre.h, specfunc/legendre_H3d.c,
specfunc/legendre_Qn.c, specfunc/legendre_con.c,
specfunc/legendre_poly.c, specfunc/log.c, specfunc/mathieu_angfunc.c,
specfunc/mathieu_charv.c, specfunc/mathieu_coeff.c,
specfunc/mathieu_radfunc.c, specfunc/mathieu_workspace.c,
specfunc/poch.c, specfunc/pow_int.c, specfunc/psi.c,
specfunc/recurse.h, specfunc/result.c, specfunc/shint.c,
specfunc/sinint.c, specfunc/synchrotron.c, specfunc/test_airy.c,
specfunc/test_bessel.c, specfunc/test_coulomb.c,
specfunc/test_dilog.c, specfunc/test_gamma.c, specfunc/test_hyperg.c,
specfunc/test_legendre.c, specfunc/test_mathieu.c,
specfunc/test_sf.c, specfunc/test_sf.h, specfunc/transport.c,
specfunc/trig.c, specfunc/zeta.c, statistics/ChangeLog,
statistics/Makefile.am, statistics/Makefile.in, statistics/TODO,
statistics/absdev.c, statistics/absdev_source.c,
statistics/covariance.c, statistics/covariance_source.c,
statistics/gsl_statistics.h, statistics/gsl_statistics_char.h,
statistics/gsl_statistics_double.h,
statistics/gsl_statistics_float.h, statistics/gsl_statistics_int.h,
statistics/gsl_statistics_long.h,
statistics/gsl_statistics_long_double.h,
statistics/gsl_statistics_short.h, statistics/gsl_statistics_uchar.h,
statistics/gsl_statistics_uint.h, statistics/gsl_statistics_ulong.h,
statistics/gsl_statistics_ushort.h, statistics/kurtosis.c,
statistics/kurtosis_source.c, statistics/lag1.c,
statistics/lag1_source.c, statistics/mean.c,
statistics/mean_source.c, statistics/median.c,
statistics/median_source.c, statistics/minmax.c,
statistics/minmax_source.c, statistics/p_variance.c,
statistics/p_variance_source.c, statistics/quantiles.c,
statistics/quantiles_source.c, statistics/skew.c,
statistics/skew_source.c, statistics/test.c,
statistics/test_float_source.c, statistics/test_int_source.c,
statistics/test_nist.c, statistics/ttest.c,
statistics/ttest_source.c, statistics/variance.c,
statistics/variance_source.c, statistics/wabsdev.c,
statistics/wabsdev_source.c, statistics/wkurtosis.c,
statistics/wkurtosis_source.c, statistics/wmean.c,
statistics/wmean_source.c, statistics/wskew.c,
statistics/wskew_source.c, statistics/wvariance.c,
statistics/wvariance_source.c, sum/ChangeLog, sum/Makefile.am,
sum/Makefile.in, sum/gsl_sum.h, sum/levin_u.c, sum/levin_utrunc.c,
sum/test.c, sum/work_u.c, sum/work_utrunc.c, sys/ChangeLog,
sys/Makefile.am, sys/Makefile.in, sys/coerce.c, sys/expm1.c,
sys/fcmp.c, sys/fdiv.c, sys/gsl_sys.h, sys/hypot.c, sys/infnan.c,
sys/invhyp.c, sys/ldfrexp.c, sys/log1p.c, sys/minmax.c,
sys/pow_int.c, sys/prec.c, sys/test.c, test/ChangeLog,
test/Makefile.am, test/Makefile.in, test/gsl_test.h, test/results.c,
utils/Makefile.am, utils/Makefile.in, utils/README, utils/memcpy.c,
utils/memmove.c, utils/placeholder.c, utils/strdup.c, utils/strtol.c,
utils/strtoul.c, utils/system.h, vector/ChangeLog,
vector/Makefile.am, vector/Makefile.in, vector/TODO, vector/copy.c,
vector/copy_source.c, vector/file.c, vector/file_source.c,
vector/gsl_vector.h, vector/gsl_vector_char.h,
vector/gsl_vector_complex.h, vector/gsl_vector_complex_double.h,
vector/gsl_vector_complex_float.h,
vector/gsl_vector_complex_long_double.h, vector/gsl_vector_double.h,
vector/gsl_vector_float.h, vector/gsl_vector_int.h,
vector/gsl_vector_long.h, vector/gsl_vector_long_double.h,
vector/gsl_vector_short.h, vector/gsl_vector_uchar.h,
vector/gsl_vector_uint.h, vector/gsl_vector_ulong.h,
vector/gsl_vector_ushort.h, vector/init.c, vector/init_source.c,
vector/minmax.c, vector/minmax_source.c, vector/oper.c,
vector/oper_source.c, vector/prop.c, vector/prop_source.c,
vector/reim.c, vector/reim_source.c, vector/subvector.c,
vector/subvector_source.c, vector/swap.c, vector/swap_source.c,
vector/test.c, vector/test_complex_source.c, vector/test_source.c,
vector/test_static.c, vector/vector.c, vector/vector_source.c,
vector/view.c, vector/view.h, vector/view_source.c,
wavelet/ChangeLog, wavelet/Makefile.am, wavelet/Makefile.in,
wavelet/TODO, wavelet/bspline.c, wavelet/daubechies.c, wavelet/dwt.c,
wavelet/gsl_wavelet.h, wavelet/gsl_wavelet2d.h, wavelet/haar.c,
wavelet/test.c, wavelet/wavelet.c: New files.
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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 + + + |