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authorJoel Sherrill <joel.sherrill@OARcorp.com>2011-04-08 17:33:11 +0000
committerJoel Sherrill <joel.sherrill@OARcorp.com>2011-04-08 17:33:11 +0000
commit73f643f3f4a55310b2c8c1a9858906b2dd676e72 (patch)
treeb1df97f18dace4a5702b0bc7aafdfee8a2f25ada /gsl-1.9/doc/min.texi
parentee523abdace8337d05ec4a179fcdf5de3fe0f634 (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, 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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, 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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
+