Quasi-Random Sequences

This chapter describes functions for generating quasi-random sequences in arbitrary dimensions. A quasi-random sequence progressively covers a

System Message: WARNING/2 (d)

latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: /usr/share/texmf-dist/web2c/fmtutil.cnf mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /usr/src/.texlive2019/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /usr/src/.texlive2019/texmf-var/web2c mktexfmt [INFO]: --- remaking latex with pdftex mktexfmt: running `pdftex -ini -jobname=latex -progname=latex -translate-file=cp227.tcx *latex.ini' ... This is pdfTeX, Version 3.14159265-2.6-1.40.20 (TeX Live 2019/Mageia) (INITEX) restricted \write18 enabled. (/usr/share/texmf-dist/web2c/cp227.tcx) entering extended mode (/usr/share/texmf-dist/tex/latex/latexconfig/latex.ini (/usr/share/texmf-dist/tex/generic/config/pdftexconfig.tex) (/usr/share/texmf-dist/tex/latex/base/latex.ltx (/usr/share/texmf-dist/tex/latex/base/texsys.cfg) ./texsys.aux found \@currdir set to: ./. Assuming \openin and \input have the same search path. Defining UNIX/DOS style filename parser. catcodes, registers, parameters, LaTeX2e <2018-12-01> hacks, control, par, spacing, files, font encodings, lengths, ==================================== Local config file fonttext.cfg used ==================================== (/usr/share/texmf-dist/tex/latex/base/fonttext.cfg (/usr/share/texmf-dist/tex/latex/base/fonttext.ltx === Don't modify this file, use a .cfg file instead === (/usr/share/texmf-dist/tex/latex/base/omlenc.def) (/usr/share/texmf-dist/tex/latex/base/t1enc.def) (/usr/share/texmf-dist/tex/latex/base/ot1enc.def) (/usr/share/texmf-dist/tex/latex/base/omsenc.def) (/usr/share/texmf-dist/tex/latex/base/t1cmr.fd) (/usr/share/texmf-dist/tex/latex/base/ot1cmr.fd) (/usr/share/texmf-dist/tex/latex/base/ot1cmss.fd) (/usr/share/texmf-dist/tex/latex/base/ot1cmtt.fd))) ==================================== Local config file fontmath.cfg used ==================================== (/usr/share/texmf-dist/tex/latex/base/fontmath.cfg (/usr/share/texmf-dist/tex/latex/base/fontmath.ltx === Don't modify this file, use a .cfg file instead === (/usr/share/texmf-dist/tex/latex/base/omlcmm.fd) (/usr/share/texmf-dist/tex/latex/base/omscmsy.fd) (/usr/share/texmf-dist/tex/latex/base/omxcmex.fd) (/usr/share/texmf-dist/tex/latex/base/ucmr.fd))) ==================================== Local config file preload.cfg used ===================================== (/usr/share/texmf-dist/tex/latex/base/preload.cfg (/usr/share/texmf-dist/tex/latex/base/preload.ltx)) page nos., x-ref, environments, center, verbatim, math definitions, boxes, title, sectioning, contents, floats, footnotes, index, bibliography, output, =========================================== Local configuration file hyphen.cfg used =========================================== (/usr/share/texmf-dist/tex/generic/babel/hyphen.cfg (/usr/share/texmf-dist/tex/generic/babel/switch.def) (/usr/share/texmf-dist/tex/generic/hyphen/hyphen.tex) (/usr/share/texmf-dist/tex/generic/hyphen/dumyhyph.tex) (/usr/share/texmf-dist/tex/generic/hyphen/zerohyph.tex) (/usr/share/texmf-dist/tex/generic/dehyph-exptl/dehypht-x-2019-04-04.tex dehyph-exptl: using an 8-bit TeX engine. (/usr/share/texmf-dist/tex/generic/dehyph-exptl/dehypht-x-2019-04-04.pat German Hyphenation Patterns (Traditional Orthography) `dehypht-x' 2019-04-04 (W L))) (/usr/share/texmf-dist/tex/generic/dehyph-exptl/dehyphn-x-2019-04-04.tex dehyph-exptl: using an 8-bit TeX engine. (/usr/share/texmf-dist/tex/generic/dehyph-exptl/dehyphn-x-2019-04-04.pat German Hyphenation Patterns (Reformed Orthography, 2006) `dehyphn-x' 2019-04-04 (WL))) (/usr/share/texmf-dist/tex/generic/hyph-utf8/loadhyph/loadhyph-af.tex EC Afrikaans hyphenation patterns (/usr/share/texmf-dist/tex/generic/hyph-utf8/conversions/conv-utf8-ec.tex) (/usr/share/texmf-dist/tex/generic/hyph-utf8/patterns/tex/hyph-af.tex)) (/usr/share/texmf-dist/tex/generic/hyph-utf8/loadhyph/loadhyph-grc.tex Hyphenation patterns for Ancient Greek (/usr/share/texmf-dist/tex/generic/hyphen/grahyph5.tex Hyphenation patterns for Ancient Greek)) (/usr/share/texmf-dist/tex/generic/hyphen/ibyhyph.tex Greek hyphenation patterns for Ibycus encoding, v3.0) (/usr/share/texmf-dist/tex/generic/hyphen/zerohyph.tex) (/usr/share/texmf-dist/tex/generic/hyph-utf8/loadhyph/loadhyph-hy.tex No Armenian hyphenation patterns - only for Unicode engines) (/usr/share/texmf-dist/tex/generic/hyph-utf8/loadhyph/loadhyph-eu.tex EC Basque hyphenation patterns (/usr/share/texmf-dist/tex/generic/hyph-utf8/conversions/conv-utf8-ec.tex) (/usr/share/texmf-dist/tex/generic/hyph-utf8/patterns/tex/hyph-eu.tex)) (/usr/share/texmf-dist/tex/generic/hyph-utf8/loadhyph/loadhyph-be.tex T2A Belarusian hyphenation patterns (/usr/share/texmf-dist/tex/generic/hyph-utf8/conversions/conv-utf8-t2a.tex) (/usr/share/texmf-dist/tex/generic/hyph-utf8/patterns/tex/hyph-be.tex)) (/usr/share/texmf-dist/tex/generic/hyph-utf8/loadhyph/loadhyph-bg.tex T2A Bulgarian hyphenation patterns (/usr/share/texmf-dist/tex/generic/hyph-utf8/conversions/conv-utf8-t2a.tex) (/usr/share/texmf-dist/tex/generic/hyph-utf8/patterns/tex/hyph-bg.tex Bulgarian hyphenation patterns (options: --safe-morphology --standalone-tex, ve rsion 21 October 2017))) (/usr/share/texmf-dist/tex/generic/hyph-utf8/loadhyph/loadhyph-ca.tex EC Catalan hyphenation patterns (/usr/share/texmf-dist/tex/generic/hyph-utf8/conversions/conv-utf8-ec.tex) (/usr/share/texmf-dist/tex/generic/hyph-utf8/patterns/tex/hyph-ca.tex)) (/usr/share/texmf-dist/tex/generic/hyph-utf8/loadhyph/loadhyph-cu.tex No Church Slavonic hyphenation patterns - only for Unicode engines) (/usr/share/texmf-dist/tex/generic/hyph-utf8/loadhyph/loadhyph-cop.tex Coptic hyphenation patterns (/usr/share/texmf-dist/tex/generic/hyph-utf8/patterns/tex-8bit/copthyph.tex)) (/usr/share/texmf-dist/tex/generic/hyph-utf8/loadhyph/loadhyph-hr.tex EC Croatian hyphenation patterns (/usr/share/texmf-dist/tex/generic/hyph-utf8/conversions/conv-utf8-ec.tex) (/usr/share/texmf-dist/tex/generic/hyph-utf8/patterns/tex/hyph-hr.tex)) (/usr/share/texmf-dist/tex/generic/hyph-utf8/loadhyph/loadhyph-cs.tex EC Czech hyphenation patterns (/usr/share/texmf-dist/tex/generic/hyph-utf8/conversions/conv-utf8-ec.tex) (/usr/share/texmf-dist/tex/generic/hyph-utf8/patterns/tex/hyph-cs.tex)) (/usr/share/texmf-dist/tex/generic/hyph-utf8/loadhyph/loadhyph-da.tex EC Danish hyphenation patterns (/usr/share/texmf-dist/tex/generic/hyph-utf8/conversions/conv-utf8-ec.tex) (/usr/share/texmf-dist/tex/generic/hyph-utf8/patterns/tex/hyph-da.tex)) (/usr/share/texmf-dist/tex/generic/hyph-utf8/loadhyph/loadhyph-nl.tex EC Dutch hyphenation patterns (/usr/share/texmf-dist/tex/generic/hyph-utf8/conversions/conv-utf8-ec.tex) (/usr/share/texmf-dist/tex/generic/hyph-utf8/patterns/tex/hyph-nl.tex)) (/usr/share/texmf-dist/tex/generic/hyph-utf8/loadhyph/loadhyph-en-gb.tex ASCII Hyphenation patterns for British English (/usr/share/texmf-dist/tex/generic/hyph-utf8/patterns/tex/hyph-en-gb.tex)) (/usr/share/texmf-dist/tex/generic/hyph-utf8/loadhyph/loadhyph-en-us.tex ASCII Hyphenation patterns for American English (/usr/share/texmf-dist/tex/generic/hyph-utf8/patterns/tex/hyph-en-us.tex)) (/usr/share/texmf-dist/tex/generic/hyph-utf8/loadhyph/loadhyph-eo.tex IL3 Esperanto hyphenation patterns (/usr/share/texmf-dist/tex/generic/hyph-utf8/conversions/conv-utf8-il3.tex) (/usr/share/texmf-dist/tex/generic/hyph-utf8/patterns/tex/hyph-eo.tex)) (/usr/share/texmf-dist/tex/generic/hyph-utf8/loadhyph/loadhyph-et.tex EC Estonian hyphenation patterns (/usr/share/texmf-dist/tex/generic/hyph-utf8/conversions/conv-utf8-ec.tex) (/usr/share/texmf-dist/tex/generic/hyph-utf8/patterns/tex/hyph-et.tex)) (/usr/share/texmf-dist/tex/generic/hyph-utf8/loadhyph/loadhyph-mul-ethi.tex No Pan-Ethiopic hyphenation patterns - only for Unicode engines) (/usr/share/texmf-dist/tex/generic/hyphen/zerohyph.tex) (/usr/share/texmf-dist/tex/generic/hyph-utf8/loadhyph/loadhyph-fi.tex EC Finnish hyphenation patterns (/usr/share/texmf-dist/tex/generic/hyph-utf8/conversions/conv-utf8-ec.tex) (/usr/share/texmf-dist/tex/generic/hyph-utf8/patterns/tex/hyph-fi.tex)) (/usr/share/texmf-dist/tex/generic/hyph-utf8/loadhyph/loadhyph-fr.tex EC French hyphenation patterns (/usr/share/texmf-dist/tex/generic/hyph-utf8/conversions/conv-utf8-ec.tex) (/usr/share/texmf-dist/tex/generic/hyph-utf8/patterns/tex/hyph-fr.tex)) (/usr/share/texmf-dist/tex/generic/hyph-utf8/loadhyph/loadhyph-fur.tex EC Friulan hyphenation patterns (/usr/share/texmf-dist/tex/generic/hyph-utf8/conversions/conv-utf8-ec.tex) (/usr/share/texmf-dist/tex/generic/hyph-utf8/patterns/tex/hyph-fur.tex)) (/usr/share/texmf-dist/tex/generic/hyph-utf8/loadhyph/loadhyph-gl.tex EC Galician hyphenation patterns (/usr/share/texmf-dist/tex/generic/hyph-utf8/conversions/conv-utf8-ec.tex) (/usr/share/texmf-dist/tex/generic/hyph-utf8/patterns/tex/hyph-gl.tex)) (/usr/share/texmf-dist/tex/generic/hyph-utf8/loadhyph/loadhyph-ka.tex T8M Georgian hyphenation patterns (/usr/share/texmf-dist/tex/generic/hyph-utf8/conversions/conv-utf8-t8m.tex) (/usr/share/texmf-dist/tex/generic/hyph-utf8/patterns/tex/hyph-ka.tex)) (/usr/share/texmf-dist/tex/generic/hyph-utf8/loadhyph/loadhyph-de-1901.tex EC German hyphenation patterns (traditional orthography) ! 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-dimensional space with a set of points that are uniformly distributed. Quasi-random sequences are also known as low-discrepancy sequences. The quasi-random sequence generators use an interface that is similar to the interface for random number generators, except that seeding is not required—each generator produces a single sequence.

The functions described in this section are declared in the header file gsl_qrng.h.

Quasi-random number generator initialization

type gsl_qrng

This is a workspace for computing quasi-random sequences.

gsl_qrng *gsl_qrng_alloc(const gsl_qrng_type *T, unsigned int d)

This function returns a pointer to a newly-created instance of a quasi-random sequence generator of type T and dimension d. If there is insufficient memory to create the generator then the function returns a null pointer and the error handler is invoked with an error code of GSL_ENOMEM.

void gsl_qrng_free(gsl_qrng *q)

This function frees all the memory associated with the generator q.

void gsl_qrng_init(gsl_qrng *q)

This function reinitializes the generator q to its starting point. Note that quasi-random sequences do not use a seed and always produce the same set of values.

Sampling from a quasi-random number generator

int gsl_qrng_get(const gsl_qrng *q, double x[])

This function stores the next point from the sequence generator q in the array x. The space available for x must match the dimension of the generator. The point x will lie in the range

System Message: WARNING/2 (0 < x_i < 1)

latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: /usr/share/texmf-dist/web2c/fmtutil.cnf mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /usr/src/.texlive2019/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /usr/src/.texlive2019/texmf-var/web2c mktexfmt [INFO]: --- remaking latex with pdftex mktexfmt: running `pdftex -ini -jobname=latex -progname=latex -translate-file=cp227.tcx *latex.ini' ... This is pdfTeX, Version 3.14159265-2.6-1.40.20 (TeX Live 2019/Mageia) (INITEX) restricted \write18 enabled. (/usr/share/texmf-dist/web2c/cp227.tcx) entering extended mode (/usr/share/texmf-dist/tex/latex/latexconfig/latex.ini (/usr/share/texmf-dist/tex/generic/config/pdftexconfig.tex) (/usr/share/texmf-dist/tex/latex/base/latex.ltx (/usr/share/texmf-dist/tex/latex/base/texsys.cfg) ./texsys.aux found \@currdir set to: ./. Assuming \openin and \input have the same search path. Defining UNIX/DOS style filename parser. catcodes, registers, parameters, LaTeX2e <2018-12-01> hacks, control, par, spacing, files, font encodings, lengths, ==================================== Local config file fonttext.cfg used ==================================== (/usr/share/texmf-dist/tex/latex/base/fonttext.cfg (/usr/share/texmf-dist/tex/latex/base/fonttext.ltx === Don't modify this file, use a .cfg file instead === (/usr/share/texmf-dist/tex/latex/base/omlenc.def) (/usr/share/texmf-dist/tex/latex/base/t1enc.def) (/usr/share/texmf-dist/tex/latex/base/ot1enc.def) (/usr/share/texmf-dist/tex/latex/base/omsenc.def) (/usr/share/texmf-dist/tex/latex/base/t1cmr.fd) (/usr/share/texmf-dist/tex/latex/base/ot1cmr.fd) (/usr/share/texmf-dist/tex/latex/base/ot1cmss.fd) (/usr/share/texmf-dist/tex/latex/base/ot1cmtt.fd))) ==================================== Local config file fontmath.cfg used ==================================== (/usr/share/texmf-dist/tex/latex/base/fontmath.cfg (/usr/share/texmf-dist/tex/latex/base/fontmath.ltx === Don't modify this file, use a .cfg file instead === (/usr/share/texmf-dist/tex/latex/base/omlcmm.fd) (/usr/share/texmf-dist/tex/latex/base/omscmsy.fd) (/usr/share/texmf-dist/tex/latex/base/omxcmex.fd) (/usr/share/texmf-dist/tex/latex/base/ucmr.fd))) ==================================== Local config file preload.cfg used ===================================== (/usr/share/texmf-dist/tex/latex/base/preload.cfg (/usr/share/texmf-dist/tex/latex/base/preload.ltx)) page nos., x-ref, environments, center, verbatim, math definitions, boxes, title, sectioning, contents, floats, footnotes, index, bibliography, output, =========================================== Local configuration file hyphen.cfg used =========================================== (/usr/share/texmf-dist/tex/generic/babel/hyphen.cfg (/usr/share/texmf-dist/tex/generic/babel/switch.def) (/usr/share/texmf-dist/tex/generic/hyphen/hyphen.tex) (/usr/share/texmf-dist/tex/generic/hyphen/dumyhyph.tex) (/usr/share/texmf-dist/tex/generic/hyphen/zerohyph.tex) (/usr/share/texmf-dist/tex/generic/dehyph-exptl/dehypht-x-2019-04-04.tex dehyph-exptl: using an 8-bit TeX engine. (/usr/share/texmf-dist/tex/generic/dehyph-exptl/dehypht-x-2019-04-04.pat German Hyphenation Patterns (Traditional Orthography) `dehypht-x' 2019-04-04 (W L))) (/usr/share/texmf-dist/tex/generic/dehyph-exptl/dehyphn-x-2019-04-04.tex dehyph-exptl: using an 8-bit TeX engine. (/usr/share/texmf-dist/tex/generic/dehyph-exptl/dehyphn-x-2019-04-04.pat German Hyphenation Patterns (Reformed Orthography, 2006) `dehyphn-x' 2019-04-04 (WL))) (/usr/share/texmf-dist/tex/generic/hyph-utf8/loadhyph/loadhyph-af.tex EC Afrikaans hyphenation patterns (/usr/share/texmf-dist/tex/generic/hyph-utf8/conversions/conv-utf8-ec.tex) (/usr/share/texmf-dist/tex/generic/hyph-utf8/patterns/tex/hyph-af.tex)) (/usr/share/texmf-dist/tex/generic/hyph-utf8/loadhyph/loadhyph-grc.tex Hyphenation patterns for Ancient Greek (/usr/share/texmf-dist/tex/generic/hyphen/grahyph5.tex Hyphenation patterns for Ancient Greek)) (/usr/share/texmf-dist/tex/generic/hyphen/ibyhyph.tex Greek hyphenation patterns for Ibycus encoding, v3.0) (/usr/share/texmf-dist/tex/generic/hyphen/zerohyph.tex) (/usr/share/texmf-dist/tex/generic/hyph-utf8/loadhyph/loadhyph-hy.tex No Armenian hyphenation patterns - only for Unicode engines) (/usr/share/texmf-dist/tex/generic/hyph-utf8/loadhyph/loadhyph-eu.tex EC Basque hyphenation patterns (/usr/share/texmf-dist/tex/generic/hyph-utf8/conversions/conv-utf8-ec.tex) (/usr/share/texmf-dist/tex/generic/hyph-utf8/patterns/tex/hyph-eu.tex)) (/usr/share/texmf-dist/tex/generic/hyph-utf8/loadhyph/loadhyph-be.tex T2A Belarusian hyphenation patterns (/usr/share/texmf-dist/tex/generic/hyph-utf8/conversions/conv-utf8-t2a.tex) (/usr/share/texmf-dist/tex/generic/hyph-utf8/patterns/tex/hyph-be.tex)) (/usr/share/texmf-dist/tex/generic/hyph-utf8/loadhyph/loadhyph-bg.tex T2A Bulgarian hyphenation patterns (/usr/share/texmf-dist/tex/generic/hyph-utf8/conversions/conv-utf8-t2a.tex) (/usr/share/texmf-dist/tex/generic/hyph-utf8/patterns/tex/hyph-bg.tex Bulgarian hyphenation patterns (options: --safe-morphology --standalone-tex, ve rsion 21 October 2017))) (/usr/share/texmf-dist/tex/generic/hyph-utf8/loadhyph/loadhyph-ca.tex EC Catalan hyphenation patterns (/usr/share/texmf-dist/tex/generic/hyph-utf8/conversions/conv-utf8-ec.tex) (/usr/share/texmf-dist/tex/generic/hyph-utf8/patterns/tex/hyph-ca.tex)) (/usr/share/texmf-dist/tex/generic/hyph-utf8/loadhyph/loadhyph-cu.tex No Church Slavonic hyphenation patterns - only for Unicode engines) (/usr/share/texmf-dist/tex/generic/hyph-utf8/loadhyph/loadhyph-cop.tex Coptic hyphenation patterns (/usr/share/texmf-dist/tex/generic/hyph-utf8/patterns/tex-8bit/copthyph.tex)) (/usr/share/texmf-dist/tex/generic/hyph-utf8/loadhyph/loadhyph-hr.tex EC Croatian hyphenation patterns (/usr/share/texmf-dist/tex/generic/hyph-utf8/conversions/conv-utf8-ec.tex) (/usr/share/texmf-dist/tex/generic/hyph-utf8/patterns/tex/hyph-hr.tex)) (/usr/share/texmf-dist/tex/generic/hyph-utf8/loadhyph/loadhyph-cs.tex EC Czech hyphenation patterns (/usr/share/texmf-dist/tex/generic/hyph-utf8/conversions/conv-utf8-ec.tex) (/usr/share/texmf-dist/tex/generic/hyph-utf8/patterns/tex/hyph-cs.tex)) (/usr/share/texmf-dist/tex/generic/hyph-utf8/loadhyph/loadhyph-da.tex EC Danish hyphenation patterns (/usr/share/texmf-dist/tex/generic/hyph-utf8/conversions/conv-utf8-ec.tex) (/usr/share/texmf-dist/tex/generic/hyph-utf8/patterns/tex/hyph-da.tex)) (/usr/share/texmf-dist/tex/generic/hyph-utf8/loadhyph/loadhyph-nl.tex EC Dutch hyphenation patterns (/usr/share/texmf-dist/tex/generic/hyph-utf8/conversions/conv-utf8-ec.tex) (/usr/share/texmf-dist/tex/generic/hyph-utf8/patterns/tex/hyph-nl.tex)) (/usr/share/texmf-dist/tex/generic/hyph-utf8/loadhyph/loadhyph-en-gb.tex ASCII Hyphenation patterns for British English (/usr/share/texmf-dist/tex/generic/hyph-utf8/patterns/tex/hyph-en-gb.tex)) (/usr/share/texmf-dist/tex/generic/hyph-utf8/loadhyph/loadhyph-en-us.tex ASCII Hyphenation patterns for American English (/usr/share/texmf-dist/tex/generic/hyph-utf8/patterns/tex/hyph-en-us.tex)) (/usr/share/texmf-dist/tex/generic/hyph-utf8/loadhyph/loadhyph-eo.tex IL3 Esperanto hyphenation patterns (/usr/share/texmf-dist/tex/generic/hyph-utf8/conversions/conv-utf8-il3.tex) (/usr/share/texmf-dist/tex/generic/hyph-utf8/patterns/tex/hyph-eo.tex)) (/usr/share/texmf-dist/tex/generic/hyph-utf8/loadhyph/loadhyph-et.tex EC Estonian hyphenation patterns (/usr/share/texmf-dist/tex/generic/hyph-utf8/conversions/conv-utf8-ec.tex) (/usr/share/texmf-dist/tex/generic/hyph-utf8/patterns/tex/hyph-et.tex)) (/usr/share/texmf-dist/tex/generic/hyph-utf8/loadhyph/loadhyph-mul-ethi.tex No Pan-Ethiopic hyphenation patterns - only for Unicode engines) (/usr/share/texmf-dist/tex/generic/hyphen/zerohyph.tex) (/usr/share/texmf-dist/tex/generic/hyph-utf8/loadhyph/loadhyph-fi.tex EC Finnish hyphenation patterns (/usr/share/texmf-dist/tex/generic/hyph-utf8/conversions/conv-utf8-ec.tex) (/usr/share/texmf-dist/tex/generic/hyph-utf8/patterns/tex/hyph-fi.tex)) (/usr/share/texmf-dist/tex/generic/hyph-utf8/loadhyph/loadhyph-fr.tex EC French hyphenation patterns (/usr/share/texmf-dist/tex/generic/hyph-utf8/conversions/conv-utf8-ec.tex) (/usr/share/texmf-dist/tex/generic/hyph-utf8/patterns/tex/hyph-fr.tex)) (/usr/share/texmf-dist/tex/generic/hyph-utf8/loadhyph/loadhyph-fur.tex EC Friulan hyphenation patterns (/usr/share/texmf-dist/tex/generic/hyph-utf8/conversions/conv-utf8-ec.tex) (/usr/share/texmf-dist/tex/generic/hyph-utf8/patterns/tex/hyph-fur.tex)) (/usr/share/texmf-dist/tex/generic/hyph-utf8/loadhyph/loadhyph-gl.tex EC Galician hyphenation patterns (/usr/share/texmf-dist/tex/generic/hyph-utf8/conversions/conv-utf8-ec.tex) (/usr/share/texmf-dist/tex/generic/hyph-utf8/patterns/tex/hyph-gl.tex)) (/usr/share/texmf-dist/tex/generic/hyph-utf8/loadhyph/loadhyph-ka.tex T8M Georgian hyphenation patterns (/usr/share/texmf-dist/tex/generic/hyph-utf8/conversions/conv-utf8-t8m.tex) (/usr/share/texmf-dist/tex/generic/hyph-utf8/patterns/tex/hyph-ka.tex)) (/usr/share/texmf-dist/tex/generic/hyph-utf8/loadhyph/loadhyph-de-1901.tex EC German hyphenation patterns (traditional orthography) ! 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. An inline version of this function is used when HAVE_INLINE is defined.

Auxiliary quasi-random number generator functions

const char *gsl_qrng_name(const gsl_qrng *q)

This function returns a pointer to the name of the generator.

size_t gsl_qrng_size(const gsl_qrng *q)
void *gsl_qrng_state(const gsl_qrng *q)

These functions return a pointer to the state of generator r and its size. You can use this information to access the state directly. For example, the following code will write the state of a generator to a stream:

void * state = gsl_qrng_state (q);
size_t n = gsl_qrng_size (q);
fwrite (state, n, 1, stream);

Saving and restoring quasi-random number generator state

int gsl_qrng_memcpy(gsl_qrng *dest, const gsl_qrng *src)

This function copies the quasi-random sequence generator src into the pre-existing generator dest, making dest into an exact copy of src. The two generators must be of the same type.

gsl_qrng *gsl_qrng_clone(const gsl_qrng *q)

This function returns a pointer to a newly created generator which is an exact copy of the generator q.

Quasi-random number generator algorithms

The following quasi-random sequence algorithms are available,

type gsl_qrng_type
gsl_qrng_type *gsl_qrng_niederreiter_2

This generator uses the algorithm described in Bratley, Fox, Niederreiter, ACM Trans. Model. Comp. Sim. 2, 195 (1992). It is valid up to 12 dimensions.

gsl_qrng_type *gsl_qrng_sobol

This generator uses the Sobol sequence described in Antonov, Saleev, USSR Comput. Maths. Math. Phys. 19, 252 (1980). It is valid up to 40 dimensions.

gsl_qrng_type *gsl_qrng_halton
gsl_qrng_type *gsl_qrng_reversehalton

These generators use the Halton and reverse Halton sequences described in J.H. Halton, Numerische Mathematik, 2, 84-90 (1960) and B. Vandewoestyne and R. Cools Computational and Applied Mathematics, 189, 1&2, 341-361 (2006). They are valid up to 1229 dimensions.

Examples

The following program prints the first 1024 points of the 2-dimensional Sobol sequence.

#include <stdio.h>
#include <gsl/gsl_qrng.h>

int
main (void)
{
  int i;
  gsl_qrng * q = gsl_qrng_alloc (gsl_qrng_sobol, 2);

  for (i = 0; i < 1024; i++)
    {
      double v[2];
      gsl_qrng_get (q, v);
      printf ("%.5f %.5f\n", v[0], v[1]);
    }

  gsl_qrng_free (q);
  return 0;
}

Here is the output from the program:

$ ./a.out
0.50000 0.50000
0.75000 0.25000
0.25000 0.75000
0.37500 0.37500
0.87500 0.87500
0.62500 0.12500
0.12500 0.62500
....

It can be seen that successive points progressively fill-in the spaces between previous points.

Fig. 3 shows the distribution in the x-y plane of the first 1024 points from the Sobol sequence,

_images/qrng.png

Fig. 3 Distribution of the first 1024 points from the quasi-random Sobol sequence

References

The implementations of the quasi-random sequence routines are based on the algorithms described in the following paper,

  • P. Bratley and B.L. Fox and H. Niederreiter, “Algorithm 738: Programs to Generate Niederreiter’s Low-discrepancy Sequences”, ACM Transactions on Mathematical Software, Vol.: 20, No.: 4, December, 1994, p.: 494–495.