diff options
| author | Karl Berry <karl@freefriends.org> | 2018-12-10 22:58:44 +0000 |
|---|---|---|
| committer | Karl Berry <karl@freefriends.org> | 2018-12-10 22:58:44 +0000 |
| commit | 8128a1b13f192605f1711ea4f5f7da4550d0996a (patch) | |
| tree | 96bf4dc07c80082ab607d1023e59befb7f194c44 /Master | |
| parent | e167cb1f6f6ba7c4ac582ed784d8e0a9b7d81321 (diff) | |
polexpr (10dec18)
git-svn-id: svn://tug.org/texlive/trunk@49378 c570f23f-e606-0410-a88d-b1316a301751
Diffstat (limited to 'Master')
| -rw-r--r-- | Master/texmf-dist/doc/latex/polexpr/README.md | 4 | ||||
| -rw-r--r-- | Master/texmf-dist/doc/latex/polexpr/polexpr.html | 520 | ||||
| -rw-r--r-- | Master/texmf-dist/doc/latex/polexpr/polexpr.txt | 66 | ||||
| -rw-r--r-- | Master/texmf-dist/tex/latex/polexpr/polexpr.sty | 135 |
4 files changed, 417 insertions, 308 deletions
diff --git a/Master/texmf-dist/doc/latex/polexpr/README.md b/Master/texmf-dist/doc/latex/polexpr/README.md index d0275013301..fb50e104a66 100644 --- a/Master/texmf-dist/doc/latex/polexpr/README.md +++ b/Master/texmf-dist/doc/latex/polexpr/README.md @@ -75,8 +75,10 @@ Releases New feature: multiplicity of roots. - 0.7 (2018/12/08) New feature: finding all rational roots. +- 0.7.1, 0.7.2 (2018/12/09) + Emergency bugfix. -Files of 0.7 release: +Files of 0.7.2 release: - README.md, - polexpr.sty (package file), diff --git a/Master/texmf-dist/doc/latex/polexpr/polexpr.html b/Master/texmf-dist/doc/latex/polexpr/polexpr.html index 71e7ef3d019..39044a2ea54 100644 --- a/Master/texmf-dist/doc/latex/polexpr/polexpr.html +++ b/Master/texmf-dist/doc/latex/polexpr/polexpr.html @@ -362,7 +362,7 @@ ul.auto-toc { <body> <div class="document" id="package-polexpr-documentation"> <h1 class="title">Package polexpr documentation</h1> -<h2 class="subtitle" id="id1">0.7 (2018/12/08)</h2> +<h2 class="subtitle" id="id1">0.7.2 (2018/12/09)</h2> <!-- comment: -*- fill-column: 72; mode: rst; -*- --> <div class="contents topic" id="contents"> @@ -376,142 +376,143 @@ ul.auto-toc { <li><a class="reference internal" href="#a-degree-five-polynomial-with-three-rational-roots" id="id43">A degree five polynomial with three rational roots</a></li> <li><a class="reference internal" href="#a-mignotte-type-polynomial" id="id44">A Mignotte type polynomial</a></li> <li><a class="reference internal" href="#the-wilkinson-polynomial" id="id45">The Wilkinson polynomial</a></li> -<li><a class="reference internal" href="#the-degree-41-polynomial-with-2-1-9-1-8-0-0-1-1-9-2-as-roots" id="id46">The degree 41 polynomial with -2, -1.9, -1.8, ..., 0, 0.1, ..., 1.9, 2 as roots</a></li> -<li><a class="reference internal" href="#roots-of-chebyshev-polynomials" id="id47">Roots of Chebyshev polynomials</a></li> +<li><a class="reference internal" href="#the-second-wilkinson-polynomial" id="id46">The second Wilkinson polynomial</a></li> +<li><a class="reference internal" href="#the-degree-41-polynomial-with-2-1-9-1-8-0-0-1-1-9-2-as-roots" id="id47">The degree 41 polynomial with -2, -1.9, -1.8, ..., 0, 0.1, ..., 1.9, 2 as roots</a></li> +<li><a class="reference internal" href="#roots-of-chebyshev-polynomials" id="id48">Roots of Chebyshev polynomials</a></li> </ul> </li> -<li><a class="reference internal" href="#non-expandable-macros" id="id48">Non-expandable macros</a><ul> -<li><a class="reference internal" href="#poldef-polname-letter-expression-in-letter" id="id49"><tt class="docutils literal">\poldef <span class="pre">polname(letter):=</span> expression in letter;</tt></a></li> -<li><a class="reference internal" href="#poldef-letter-polname-expression-in-letter" id="id50"><tt class="docutils literal"><span class="pre">\PolDef[letter]{polname}{expression</span> in letter}</tt></a></li> -<li><a class="reference internal" href="#polgenfloatvariant-polname" id="id51"><tt class="docutils literal">\PolGenFloatVariant{polname}</tt></a></li> -<li><a class="reference internal" href="#pollet-polname-2-polname-1" id="id52"><tt class="docutils literal"><span class="pre">\PolLet{polname_2}={polname_1}</span></tt></a></li> -<li><a class="reference internal" href="#polgloballet-polname-2-polname-1" id="id53"><tt class="docutils literal"><span class="pre">\PolGlobalLet{polname_2}={polname_1}</span></tt></a></li> -<li><a class="reference internal" href="#polassign-polname-toarray-macro" id="id54"><tt class="docutils literal"><span class="pre">\PolAssign{polname}\toarray\macro</span></tt></a></li> -<li><a class="reference internal" href="#polget-polname-fromarray-macro" id="id55"><tt class="docutils literal"><span class="pre">\PolGet{polname}\fromarray\macro</span></tt></a></li> -<li><a class="reference internal" href="#polfromcsv-polname-csv" id="id56"><tt class="docutils literal"><span class="pre">\PolFromCSV{polname}{<csv>}</span></tt></a></li> -<li><a class="reference internal" href="#poltypeset-polname" id="id57"><tt class="docutils literal">\PolTypeset{polname}</tt></a><ul> -<li><a class="reference internal" href="#poltypesetcmd-raw-coeff" id="id58"><tt class="docutils literal">\PolTypesetCmd{raw_coeff}</tt></a></li> -<li><a class="reference internal" href="#poltypesetone-raw-coeff" id="id59"><tt class="docutils literal">\PolTypesetOne{raw_coeff}</tt></a></li> -<li><a class="reference internal" href="#id6" id="id60"><tt class="docutils literal">\PolTypesetMonomialCmd</tt></a></li> -<li><a class="reference internal" href="#poltypesetcmdprefix-raw-coeff" id="id61"><tt class="docutils literal">\PolTypesetCmdPrefix{raw_coeff}</tt></a></li> +<li><a class="reference internal" href="#non-expandable-macros" id="id49">Non-expandable macros</a><ul> +<li><a class="reference internal" href="#poldef-polname-letter-expression-in-letter" id="id50"><tt class="docutils literal">\poldef <span class="pre">polname(letter):=</span> expression in letter;</tt></a></li> +<li><a class="reference internal" href="#poldef-letter-polname-expression-in-letter" id="id51"><tt class="docutils literal"><span class="pre">\PolDef[letter]{polname}{expression</span> in letter}</tt></a></li> +<li><a class="reference internal" href="#polgenfloatvariant-polname" id="id52"><tt class="docutils literal">\PolGenFloatVariant{polname}</tt></a></li> +<li><a class="reference internal" href="#pollet-polname-2-polname-1" id="id53"><tt class="docutils literal"><span class="pre">\PolLet{polname_2}={polname_1}</span></tt></a></li> +<li><a class="reference internal" href="#polgloballet-polname-2-polname-1" id="id54"><tt class="docutils literal"><span class="pre">\PolGlobalLet{polname_2}={polname_1}</span></tt></a></li> +<li><a class="reference internal" href="#polassign-polname-toarray-macro" id="id55"><tt class="docutils literal"><span class="pre">\PolAssign{polname}\toarray\macro</span></tt></a></li> +<li><a class="reference internal" href="#polget-polname-fromarray-macro" id="id56"><tt class="docutils literal"><span class="pre">\PolGet{polname}\fromarray\macro</span></tt></a></li> +<li><a class="reference internal" href="#polfromcsv-polname-csv" id="id57"><tt class="docutils literal"><span class="pre">\PolFromCSV{polname}{<csv>}</span></tt></a></li> +<li><a class="reference internal" href="#poltypeset-polname" id="id58"><tt class="docutils literal">\PolTypeset{polname}</tt></a><ul> +<li><a class="reference internal" href="#poltypesetcmd-raw-coeff" id="id59"><tt class="docutils literal">\PolTypesetCmd{raw_coeff}</tt></a></li> +<li><a class="reference internal" href="#poltypesetone-raw-coeff" id="id60"><tt class="docutils literal">\PolTypesetOne{raw_coeff}</tt></a></li> +<li><a class="reference internal" href="#id6" id="id61"><tt class="docutils literal">\PolTypesetMonomialCmd</tt></a></li> +<li><a class="reference internal" href="#poltypesetcmdprefix-raw-coeff" id="id62"><tt class="docutils literal">\PolTypesetCmdPrefix{raw_coeff}</tt></a></li> </ul> </li> -<li><a class="reference internal" href="#id8" id="id62"><tt class="docutils literal"><span class="pre">\PolTypeset*{polname}</span></tt></a></li> -<li><a class="reference internal" href="#poldiff-polname-1-polname-2" id="id63"><tt class="docutils literal"><span class="pre">\PolDiff{polname_1}{polname_2}</span></tt></a></li> -<li><a class="reference internal" href="#poldiff-n-polname-1-polname-2" id="id64"><tt class="docutils literal"><span class="pre">\PolDiff[N]{polname_1}{polname_2}</span></tt></a></li> -<li><a class="reference internal" href="#polantidiff-polname-1-polname-2" id="id65"><tt class="docutils literal"><span class="pre">\PolAntiDiff{polname_1}{polname_2}</span></tt></a></li> -<li><a class="reference internal" href="#polantidiff-n-polname-1-polname-2" id="id66"><tt class="docutils literal"><span class="pre">\PolAntiDiff[N]{polname_1}{polname_2}</span></tt></a></li> -<li><a class="reference internal" href="#poldivide-polname-1-polname-2-polname-q-polname-r" id="id67"><tt class="docutils literal"><span class="pre">\PolDivide{polname_1}{polname_2}{polname_Q}{polname_R}</span></tt></a></li> -<li><a class="reference internal" href="#polquo-polname-1-polname-2-polname-q" id="id68"><tt class="docutils literal"><span class="pre">\PolQuo{polname_1}{polname_2}{polname_Q}</span></tt></a></li> -<li><a class="reference internal" href="#polrem-polname-1-polname-2-polname-r" id="id69"><tt class="docutils literal"><span class="pre">\PolRem{polname_1}{polname_2}{polname_R}</span></tt></a></li> -<li><a class="reference internal" href="#polgcd-polname-1-polname-2-polname-gcd" id="id70"><tt class="docutils literal"><span class="pre">\PolGCD{polname_1}{polname_2}{polname_GCD}</span></tt></a></li> -<li><a class="reference internal" href="#poltosturm-polname-sturmname" id="id71"><tt class="docutils literal"><span class="pre">\PolToSturm{polname}{sturmname}</span></tt></a></li> -<li><a class="reference internal" href="#id10" id="id72"><tt class="docutils literal"><span class="pre">\PolToSturm*{polname}{sturmname}</span></tt></a></li> -<li><a class="reference internal" href="#polsettosturmchainsignchangesat-macro-sturmname-fraction" id="id73"><tt class="docutils literal"><span class="pre">\PolSetToSturmChainSignChangesAt{\macro}{sturmname}{fraction}</span></tt></a></li> -<li><a class="reference internal" href="#polsettonbofzeroswithin-macro-sturmname-value-a-value-b" id="id74"><tt class="docutils literal"><span class="pre">\PolSetToNbOfZerosWithin{\macro}{sturmname}{value_a}{value_b}</span></tt></a></li> -<li><a class="reference internal" href="#polsturmisolatezeros-sturmname" id="id75"><tt class="docutils literal">\PolSturmIsolateZeros{sturmname}</tt></a></li> -<li><a class="reference internal" href="#id12" id="id76"><tt class="docutils literal"><span class="pre">\PolSturmIsolateZeros*{sturmname}</span></tt></a></li> -<li><a class="reference internal" href="#id14" id="id77"><tt class="docutils literal"><span class="pre">\PolSturmIsolateZeros**{sturmname}</span></tt></a></li> -<li><a class="reference internal" href="#polsturmisolatezerosandgetmultiplicities-sturmname" id="id78"><tt class="docutils literal">\PolSturmIsolateZerosAndGetMultiplicities{sturmname}</tt></a></li> -<li><a class="reference internal" href="#polsturmisolatezerosgetmultiplicitiesandrationalroots-sturmname" id="id79"><tt class="docutils literal">\PolSturmIsolateZerosGetMultiplicitiesAndRationalRoots{sturmname}</tt></a></li> -<li><a class="reference internal" href="#polsturmisolatezerosandfindrationalroots-sturmname" id="id80"><tt class="docutils literal">\PolSturmIsolateZerosAndFindRationalRoots{sturmname}</tt></a></li> -<li><a class="reference internal" href="#polrefineinterval-sturmname-index" id="id81"><tt class="docutils literal"><span class="pre">\PolRefineInterval*{sturmname}{index}</span></tt></a></li> -<li><a class="reference internal" href="#polrefineinterval-n-sturmname-index" id="id82"><tt class="docutils literal"><span class="pre">\PolRefineInterval[N]{sturmname}{index}</span></tt></a></li> -<li><a class="reference internal" href="#polensureintervallength-sturmname-index-e" id="id83"><tt class="docutils literal"><span class="pre">\PolEnsureIntervalLength{sturmname}{index}{E}</span></tt></a></li> -<li><a class="reference internal" href="#polensureintervallengths-sturmname-e" id="id84"><tt class="docutils literal"><span class="pre">\PolEnsureIntervalLengths{sturmname}{E}</span></tt></a></li> -<li><a class="reference internal" href="#polprintintervals-varname-sturmname" id="id85"><tt class="docutils literal"><span class="pre">\PolPrintIntervals[varname]{sturmname}</span></tt></a><ul> -<li><a class="reference internal" href="#polprintintervalsnorealroots" id="id86"><tt class="docutils literal">\PolPrintIntervalsNoRealRoots</tt></a></li> -<li><a class="reference internal" href="#polprintintervalsbeginenv" id="id87"><tt class="docutils literal">\PolPrintIntervalsBeginEnv</tt></a></li> -<li><a class="reference internal" href="#polprintintervalsendenv" id="id88"><tt class="docutils literal">\PolPrintIntervalsEndEnv</tt></a></li> -<li><a class="reference internal" href="#polprintintervalsknownroot" id="id89"><tt class="docutils literal">\PolPrintIntervalsKnownRoot</tt></a></li> -<li><a class="reference internal" href="#polprintintervalsunknownroot" id="id90"><tt class="docutils literal">\PolPrintIntervalsUnknownRoot</tt></a></li> -<li><a class="reference internal" href="#id15" id="id91"><tt class="docutils literal">\PolPrintIntervalsPrintExactZero</tt></a></li> -<li><a class="reference internal" href="#id16" id="id92"><tt class="docutils literal">\PolPrintIntervalsPrintLeftEndPoint</tt></a></li> -<li><a class="reference internal" href="#id17" id="id93"><tt class="docutils literal">\PolPrintIntervalsPrintRightEndPoint</tt></a></li> +<li><a class="reference internal" href="#id8" id="id63"><tt class="docutils literal"><span class="pre">\PolTypeset*{polname}</span></tt></a></li> +<li><a class="reference internal" href="#poldiff-polname-1-polname-2" id="id64"><tt class="docutils literal"><span class="pre">\PolDiff{polname_1}{polname_2}</span></tt></a></li> +<li><a class="reference internal" href="#poldiff-n-polname-1-polname-2" id="id65"><tt class="docutils literal"><span class="pre">\PolDiff[N]{polname_1}{polname_2}</span></tt></a></li> +<li><a class="reference internal" href="#polantidiff-polname-1-polname-2" id="id66"><tt class="docutils literal"><span class="pre">\PolAntiDiff{polname_1}{polname_2}</span></tt></a></li> +<li><a class="reference internal" href="#polantidiff-n-polname-1-polname-2" id="id67"><tt class="docutils literal"><span class="pre">\PolAntiDiff[N]{polname_1}{polname_2}</span></tt></a></li> +<li><a class="reference internal" href="#poldivide-polname-1-polname-2-polname-q-polname-r" id="id68"><tt class="docutils literal"><span class="pre">\PolDivide{polname_1}{polname_2}{polname_Q}{polname_R}</span></tt></a></li> +<li><a class="reference internal" href="#polquo-polname-1-polname-2-polname-q" id="id69"><tt class="docutils literal"><span class="pre">\PolQuo{polname_1}{polname_2}{polname_Q}</span></tt></a></li> +<li><a class="reference internal" href="#polrem-polname-1-polname-2-polname-r" id="id70"><tt class="docutils literal"><span class="pre">\PolRem{polname_1}{polname_2}{polname_R}</span></tt></a></li> +<li><a class="reference internal" href="#polgcd-polname-1-polname-2-polname-gcd" id="id71"><tt class="docutils literal"><span class="pre">\PolGCD{polname_1}{polname_2}{polname_GCD}</span></tt></a></li> +<li><a class="reference internal" href="#poltosturm-polname-sturmname" id="id72"><tt class="docutils literal"><span class="pre">\PolToSturm{polname}{sturmname}</span></tt></a></li> +<li><a class="reference internal" href="#id10" id="id73"><tt class="docutils literal"><span class="pre">\PolToSturm*{polname}{sturmname}</span></tt></a></li> +<li><a class="reference internal" href="#polsettosturmchainsignchangesat-macro-sturmname-fraction" id="id74"><tt class="docutils literal"><span class="pre">\PolSetToSturmChainSignChangesAt{\macro}{sturmname}{fraction}</span></tt></a></li> +<li><a class="reference internal" href="#polsettonbofzeroswithin-macro-sturmname-value-a-value-b" id="id75"><tt class="docutils literal"><span class="pre">\PolSetToNbOfZerosWithin{\macro}{sturmname}{value_a}{value_b}</span></tt></a></li> +<li><a class="reference internal" href="#polsturmisolatezeros-sturmname" id="id76"><tt class="docutils literal">\PolSturmIsolateZeros{sturmname}</tt></a></li> +<li><a class="reference internal" href="#id12" id="id77"><tt class="docutils literal"><span class="pre">\PolSturmIsolateZeros*{sturmname}</span></tt></a></li> +<li><a class="reference internal" href="#id14" id="id78"><tt class="docutils literal"><span class="pre">\PolSturmIsolateZeros**{sturmname}</span></tt></a></li> +<li><a class="reference internal" href="#polsturmisolatezerosandgetmultiplicities-sturmname" id="id79"><tt class="docutils literal">\PolSturmIsolateZerosAndGetMultiplicities{sturmname}</tt></a></li> +<li><a class="reference internal" href="#polsturmisolatezerosgetmultiplicitiesandrationalroots-sturmname" id="id80"><tt class="docutils literal">\PolSturmIsolateZerosGetMultiplicitiesAndRationalRoots{sturmname}</tt></a></li> +<li><a class="reference internal" href="#polsturmisolatezerosandfindrationalroots-sturmname" id="id81"><tt class="docutils literal">\PolSturmIsolateZerosAndFindRationalRoots{sturmname}</tt></a></li> +<li><a class="reference internal" href="#polrefineinterval-sturmname-index" id="id82"><tt class="docutils literal"><span class="pre">\PolRefineInterval*{sturmname}{index}</span></tt></a></li> +<li><a class="reference internal" href="#polrefineinterval-n-sturmname-index" id="id83"><tt class="docutils literal"><span class="pre">\PolRefineInterval[N]{sturmname}{index}</span></tt></a></li> +<li><a class="reference internal" href="#polensureintervallength-sturmname-index-e" id="id84"><tt class="docutils literal"><span class="pre">\PolEnsureIntervalLength{sturmname}{index}{E}</span></tt></a></li> +<li><a class="reference internal" href="#polensureintervallengths-sturmname-e" id="id85"><tt class="docutils literal"><span class="pre">\PolEnsureIntervalLengths{sturmname}{E}</span></tt></a></li> +<li><a class="reference internal" href="#polprintintervals-varname-sturmname" id="id86"><tt class="docutils literal"><span class="pre">\PolPrintIntervals[varname]{sturmname}</span></tt></a><ul> +<li><a class="reference internal" href="#polprintintervalsnorealroots" id="id87"><tt class="docutils literal">\PolPrintIntervalsNoRealRoots</tt></a></li> +<li><a class="reference internal" href="#polprintintervalsbeginenv" id="id88"><tt class="docutils literal">\PolPrintIntervalsBeginEnv</tt></a></li> +<li><a class="reference internal" href="#polprintintervalsendenv" id="id89"><tt class="docutils literal">\PolPrintIntervalsEndEnv</tt></a></li> +<li><a class="reference internal" href="#polprintintervalsknownroot" id="id90"><tt class="docutils literal">\PolPrintIntervalsKnownRoot</tt></a></li> +<li><a class="reference internal" href="#polprintintervalsunknownroot" id="id91"><tt class="docutils literal">\PolPrintIntervalsUnknownRoot</tt></a></li> +<li><a class="reference internal" href="#id15" id="id92"><tt class="docutils literal">\PolPrintIntervalsPrintExactZero</tt></a></li> +<li><a class="reference internal" href="#id16" id="id93"><tt class="docutils literal">\PolPrintIntervalsPrintLeftEndPoint</tt></a></li> +<li><a class="reference internal" href="#id17" id="id94"><tt class="docutils literal">\PolPrintIntervalsPrintRightEndPoint</tt></a></li> </ul> </li> -<li><a class="reference internal" href="#id19" id="id94"><tt class="docutils literal"><span class="pre">\PolPrintIntervals*[varname]{sturmname}</span></tt></a><ul> -<li><a class="reference internal" href="#polprintintervalsprintmultiplicity" id="id95"><tt class="docutils literal">\PolPrintIntervalsPrintMultiplicity</tt></a></li> +<li><a class="reference internal" href="#id19" id="id95"><tt class="docutils literal"><span class="pre">\PolPrintIntervals*[varname]{sturmname}</span></tt></a><ul> +<li><a class="reference internal" href="#polprintintervalsprintmultiplicity" id="id96"><tt class="docutils literal">\PolPrintIntervalsPrintMultiplicity</tt></a></li> </ul> </li> -<li><a class="reference internal" href="#polmapcoeffs-macro-polname" id="id96"><tt class="docutils literal"><span class="pre">\PolMapCoeffs{\macro}{polname}</span></tt></a></li> -<li><a class="reference internal" href="#polreducecoeffs-polname" id="id97"><tt class="docutils literal">\PolReduceCoeffs{polname}</tt></a></li> -<li><a class="reference internal" href="#id21" id="id98"><tt class="docutils literal"><span class="pre">\PolReduceCoeffs*{polname}</span></tt></a></li> -<li><a class="reference internal" href="#polmakemonic-polname" id="id99"><tt class="docutils literal">\PolMakeMonic{polname}</tt></a></li> -<li><a class="reference internal" href="#polmakeprimitive-polname" id="id100"><tt class="docutils literal">\PolMakePrimitive{polname}</tt></a></li> +<li><a class="reference internal" href="#polmapcoeffs-macro-polname" id="id97"><tt class="docutils literal"><span class="pre">\PolMapCoeffs{\macro}{polname}</span></tt></a></li> +<li><a class="reference internal" href="#polreducecoeffs-polname" id="id98"><tt class="docutils literal">\PolReduceCoeffs{polname}</tt></a></li> +<li><a class="reference internal" href="#id21" id="id99"><tt class="docutils literal"><span class="pre">\PolReduceCoeffs*{polname}</span></tt></a></li> +<li><a class="reference internal" href="#polmakemonic-polname" id="id100"><tt class="docutils literal">\PolMakeMonic{polname}</tt></a></li> +<li><a class="reference internal" href="#polmakeprimitive-polname" id="id101"><tt class="docutils literal">\PolMakePrimitive{polname}</tt></a></li> </ul> </li> -<li><a class="reference internal" href="#expandable-macros" id="id101">Expandable macros</a><ul> -<li><a class="reference internal" href="#poleval-polname-atexpr-numerical-expression" id="id102"><tt class="docutils literal"><span class="pre">\PolEval{polname}\AtExpr{numerical</span> expression}</tt></a></li> -<li><a class="reference internal" href="#poleval-polname-at-fraction" id="id103"><tt class="docutils literal"><span class="pre">\PolEval{polname}\At{fraction}</span></tt></a></li> -<li><a class="reference internal" href="#polevalreduced-polname-atexpr-numerical-expression" id="id104"><tt class="docutils literal"><span class="pre">\PolEvalReduced{polname}\AtExpr{numerical</span> expression}</tt></a></li> -<li><a class="reference internal" href="#polevalreduced-polname-at-fraction" id="id105"><tt class="docutils literal"><span class="pre">\PolEvalReduced{polname}\At{fraction}</span></tt></a></li> -<li><a class="reference internal" href="#polfloateval-polname-atexpr-numerical-expression" id="id106"><tt class="docutils literal"><span class="pre">\PolFloatEval{polname}\AtExpr{numerical</span> expression}</tt></a></li> -<li><a class="reference internal" href="#polfloateval-polname-at-fraction" id="id107"><tt class="docutils literal"><span class="pre">\PolFloatEval{polname}\At{fraction}</span></tt></a></li> -<li><a class="reference internal" href="#polifcoeffisplusorminusone-a-b" id="id108"><tt class="docutils literal"><span class="pre">\PolIfCoeffIsPlusOrMinusOne{A}{B}</span></tt></a></li> -<li><a class="reference internal" href="#polleadingcoeff-polname" id="id109"><tt class="docutils literal">\PolLeadingCoeff{polname}</tt></a></li> -<li><a class="reference internal" href="#polnthcoeff-polname-number" id="id110"><tt class="docutils literal"><span class="pre">\PolNthCoeff{polname}{number}</span></tt></a></li> -<li><a class="reference internal" href="#poldegree-polname" id="id111"><tt class="docutils literal">\PolDegree{polname}</tt></a></li> -<li><a class="reference internal" href="#policontent-polname" id="id112"><tt class="docutils literal">\PolIContent{polname}</tt></a></li> -<li><a class="reference internal" href="#poltoexpr-polname" id="id113"><tt class="docutils literal">\PolToExpr{polname}</tt></a><ul> -<li><a class="reference internal" href="#poltoexproneterm-raw-coeff-number" id="id114"><tt class="docutils literal"><span class="pre">\PolToExprOneTerm{raw_coeff}{number}</span></tt></a></li> -<li><a class="reference internal" href="#poltoexpronetermstylea-raw-coeff-number" id="id115"><tt class="docutils literal"><span class="pre">\PolToExprOneTermStyleA{raw_coeff}{number}</span></tt></a></li> -<li><a class="reference internal" href="#poltoexpronetermstyleb-raw-coeff-number" id="id116"><tt class="docutils literal"><span class="pre">\PolToExprOneTermStyleB{raw_coeff}{number}</span></tt></a></li> -<li><a class="reference internal" href="#poltoexprcmd-raw-coeff" id="id117"><tt class="docutils literal">\PolToExprCmd{raw_coeff}</tt></a></li> -<li><a class="reference internal" href="#poltoexprtermprefix-raw-coeff" id="id118"><tt class="docutils literal">\PolToExprTermPrefix{raw_coeff}</tt></a></li> -<li><a class="reference internal" href="#id28" id="id119"><tt class="docutils literal">\PolToExprVar</tt></a></li> -<li><a class="reference internal" href="#id29" id="id120"><tt class="docutils literal">\PolToExprTimes</tt></a></li> +<li><a class="reference internal" href="#expandable-macros" id="id102">Expandable macros</a><ul> +<li><a class="reference internal" href="#poleval-polname-atexpr-numerical-expression" id="id103"><tt class="docutils literal"><span class="pre">\PolEval{polname}\AtExpr{numerical</span> expression}</tt></a></li> +<li><a class="reference internal" href="#poleval-polname-at-fraction" id="id104"><tt class="docutils literal"><span class="pre">\PolEval{polname}\At{fraction}</span></tt></a></li> +<li><a class="reference internal" href="#polevalreduced-polname-atexpr-numerical-expression" id="id105"><tt class="docutils literal"><span class="pre">\PolEvalReduced{polname}\AtExpr{numerical</span> expression}</tt></a></li> +<li><a class="reference internal" href="#polevalreduced-polname-at-fraction" id="id106"><tt class="docutils literal"><span class="pre">\PolEvalReduced{polname}\At{fraction}</span></tt></a></li> +<li><a class="reference internal" href="#polfloateval-polname-atexpr-numerical-expression" id="id107"><tt class="docutils literal"><span class="pre">\PolFloatEval{polname}\AtExpr{numerical</span> expression}</tt></a></li> +<li><a class="reference internal" href="#polfloateval-polname-at-fraction" id="id108"><tt class="docutils literal"><span class="pre">\PolFloatEval{polname}\At{fraction}</span></tt></a></li> +<li><a class="reference internal" href="#polifcoeffisplusorminusone-a-b" id="id109"><tt class="docutils literal"><span class="pre">\PolIfCoeffIsPlusOrMinusOne{A}{B}</span></tt></a></li> +<li><a class="reference internal" href="#polleadingcoeff-polname" id="id110"><tt class="docutils literal">\PolLeadingCoeff{polname}</tt></a></li> +<li><a class="reference internal" href="#polnthcoeff-polname-number" id="id111"><tt class="docutils literal"><span class="pre">\PolNthCoeff{polname}{number}</span></tt></a></li> +<li><a class="reference internal" href="#poldegree-polname" id="id112"><tt class="docutils literal">\PolDegree{polname}</tt></a></li> +<li><a class="reference internal" href="#policontent-polname" id="id113"><tt class="docutils literal">\PolIContent{polname}</tt></a></li> +<li><a class="reference internal" href="#poltoexpr-polname" id="id114"><tt class="docutils literal">\PolToExpr{polname}</tt></a><ul> +<li><a class="reference internal" href="#poltoexproneterm-raw-coeff-number" id="id115"><tt class="docutils literal"><span class="pre">\PolToExprOneTerm{raw_coeff}{number}</span></tt></a></li> +<li><a class="reference internal" href="#poltoexpronetermstylea-raw-coeff-number" id="id116"><tt class="docutils literal"><span class="pre">\PolToExprOneTermStyleA{raw_coeff}{number}</span></tt></a></li> +<li><a class="reference internal" href="#poltoexpronetermstyleb-raw-coeff-number" id="id117"><tt class="docutils literal"><span class="pre">\PolToExprOneTermStyleB{raw_coeff}{number}</span></tt></a></li> +<li><a class="reference internal" href="#poltoexprcmd-raw-coeff" id="id118"><tt class="docutils literal">\PolToExprCmd{raw_coeff}</tt></a></li> +<li><a class="reference internal" href="#poltoexprtermprefix-raw-coeff" id="id119"><tt class="docutils literal">\PolToExprTermPrefix{raw_coeff}</tt></a></li> +<li><a class="reference internal" href="#id28" id="id120"><tt class="docutils literal">\PolToExprVar</tt></a></li> +<li><a class="reference internal" href="#id29" id="id121"><tt class="docutils literal">\PolToExprTimes</tt></a></li> </ul> </li> -<li><a class="reference internal" href="#id31" id="id121"><tt class="docutils literal"><span class="pre">\PolToExpr*{polname}</span></tt></a></li> -<li><a class="reference internal" href="#poltofloatexpr-polname" id="id122"><tt class="docutils literal">\PolToFloatExpr{polname}</tt></a><ul> -<li><a class="reference internal" href="#poltofloatexproneterm-raw-coeff-number" id="id123"><tt class="docutils literal"><span class="pre">\PolToFloatExprOneTerm{raw_coeff}{number}</span></tt></a></li> -<li><a class="reference internal" href="#poltofloatexprcmd-raw-coeff" id="id124"><tt class="docutils literal">\PolToFloatExprCmd{raw_coeff}</tt></a></li> +<li><a class="reference internal" href="#id31" id="id122"><tt class="docutils literal"><span class="pre">\PolToExpr*{polname}</span></tt></a></li> +<li><a class="reference internal" href="#poltofloatexpr-polname" id="id123"><tt class="docutils literal">\PolToFloatExpr{polname}</tt></a><ul> +<li><a class="reference internal" href="#poltofloatexproneterm-raw-coeff-number" id="id124"><tt class="docutils literal"><span class="pre">\PolToFloatExprOneTerm{raw_coeff}{number}</span></tt></a></li> +<li><a class="reference internal" href="#poltofloatexprcmd-raw-coeff" id="id125"><tt class="docutils literal">\PolToFloatExprCmd{raw_coeff}</tt></a></li> </ul> </li> -<li><a class="reference internal" href="#id35" id="id125"><tt class="docutils literal"><span class="pre">\PolToFloatExpr*{polname}</span></tt></a></li> -<li><a class="reference internal" href="#poltolist-polname" id="id126"><tt class="docutils literal">\PolToList{polname}</tt></a></li> -<li><a class="reference internal" href="#poltocsv-polname" id="id127"><tt class="docutils literal">\PolToCSV{polname}</tt></a></li> -<li><a class="reference internal" href="#polsturmchainlength-sturmname" id="id128"><tt class="docutils literal">\PolSturmChainLength{sturmname}</tt></a></li> -<li><a class="reference internal" href="#polsturmifzeroexactlyknown-sturmname-index-a-b" id="id129"><tt class="docutils literal"><span class="pre">\PolSturmIfZeroExactlyKnown{sturmname}{index}{A}{B}</span></tt></a></li> -<li><a class="reference internal" href="#polsturmisolatedzeroleft-sturmname-index" id="id130"><tt class="docutils literal"><span class="pre">\PolSturmIsolatedZeroLeft{sturmname}{index}</span></tt></a></li> -<li><a class="reference internal" href="#polsturmisolatedzeroright-sturmname-index" id="id131"><tt class="docutils literal"><span class="pre">\PolSturmIsolatedZeroRight{sturmname}{index}</span></tt></a></li> -<li><a class="reference internal" href="#polsturmisolatedzeromultiplicity-sturmname-index" id="id132"><tt class="docutils literal"><span class="pre">\PolSturmIsolatedZeroMultiplicity{sturmname}{index}</span></tt></a></li> -<li><a class="reference internal" href="#polsturmnbofisolatedzeros-sturmname" id="id133"><tt class="docutils literal">\PolSturmNbOfIsolatedZeros{sturmname}</tt></a><ul> -<li><a class="reference internal" href="#polsturmnbofrootsof-sturmname-lessthanorequalto-value" id="id134"><tt class="docutils literal"><span class="pre">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualTo{value}</span></tt></a></li> -<li><a class="reference internal" href="#polsturmnbofrootsof-sturmname-lessthanorequaltoexpr-expression" id="id135"><tt class="docutils literal"><span class="pre">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualToExpr{expression}</span></tt></a></li> -<li><a class="reference internal" href="#polsturmnbwithmultofrootsof-sturmname-lessthanorequalto-value" id="id136"><tt class="docutils literal"><span class="pre">\PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualTo{value}</span></tt></a></li> -<li><a class="reference internal" href="#polsturmnbwithmultofrootsof-sturmname-lessthanorequaltoexpr-expression" id="id137"><tt class="docutils literal"><span class="pre">\PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualToExpr{expression}</span></tt></a></li> +<li><a class="reference internal" href="#id35" id="id126"><tt class="docutils literal"><span class="pre">\PolToFloatExpr*{polname}</span></tt></a></li> +<li><a class="reference internal" href="#poltolist-polname" id="id127"><tt class="docutils literal">\PolToList{polname}</tt></a></li> +<li><a class="reference internal" href="#poltocsv-polname" id="id128"><tt class="docutils literal">\PolToCSV{polname}</tt></a></li> +<li><a class="reference internal" href="#polsturmchainlength-sturmname" id="id129"><tt class="docutils literal">\PolSturmChainLength{sturmname}</tt></a></li> +<li><a class="reference internal" href="#polsturmifzeroexactlyknown-sturmname-index-a-b" id="id130"><tt class="docutils literal"><span class="pre">\PolSturmIfZeroExactlyKnown{sturmname}{index}{A}{B}</span></tt></a></li> +<li><a class="reference internal" href="#polsturmisolatedzeroleft-sturmname-index" id="id131"><tt class="docutils literal"><span class="pre">\PolSturmIsolatedZeroLeft{sturmname}{index}</span></tt></a></li> +<li><a class="reference internal" href="#polsturmisolatedzeroright-sturmname-index" id="id132"><tt class="docutils literal"><span class="pre">\PolSturmIsolatedZeroRight{sturmname}{index}</span></tt></a></li> +<li><a class="reference internal" href="#polsturmisolatedzeromultiplicity-sturmname-index" id="id133"><tt class="docutils literal"><span class="pre">\PolSturmIsolatedZeroMultiplicity{sturmname}{index}</span></tt></a></li> +<li><a class="reference internal" href="#polsturmnbofisolatedzeros-sturmname" id="id134"><tt class="docutils literal">\PolSturmNbOfIsolatedZeros{sturmname}</tt></a><ul> +<li><a class="reference internal" href="#polsturmnbofrootsof-sturmname-lessthanorequalto-value" id="id135"><tt class="docutils literal"><span class="pre">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualTo{value}</span></tt></a></li> +<li><a class="reference internal" href="#polsturmnbofrootsof-sturmname-lessthanorequaltoexpr-expression" id="id136"><tt class="docutils literal"><span class="pre">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualToExpr{expression}</span></tt></a></li> +<li><a class="reference internal" href="#polsturmnbwithmultofrootsof-sturmname-lessthanorequalto-value" id="id137"><tt class="docutils literal"><span class="pre">\PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualTo{value}</span></tt></a></li> +<li><a class="reference internal" href="#polsturmnbwithmultofrootsof-sturmname-lessthanorequaltoexpr-expression" id="id138"><tt class="docutils literal"><span class="pre">\PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualToExpr{expression}</span></tt></a></li> </ul> </li> -<li><a class="reference internal" href="#polsturmnbofrationalroots-sturmname" id="id138"><tt class="docutils literal">\PolSturmNbOfRationalRoots{sturmname}</tt></a></li> -<li><a class="reference internal" href="#polsturmnbofrationalrootswithmultiplicities-sturmname" id="id139"><tt class="docutils literal">\PolSturmNbOfRationalRootsWithMultiplicities{sturmname}</tt></a></li> -<li><a class="reference internal" href="#polsturmrationalroot-sturmname-k" id="id140"><tt class="docutils literal"><span class="pre">\PolSturmRationalRoot{sturmname}{k}</span></tt></a></li> -<li><a class="reference internal" href="#polsturmrationalrootindex-sturmname-k" id="id141"><tt class="docutils literal"><span class="pre">\PolSturmRationalRootIndex{sturmname}{k}</span></tt></a></li> -<li><a class="reference internal" href="#polsturmrationalrootmultiplicity-sturmname-k" id="id142"><tt class="docutils literal"><span class="pre">\PolSturmRationalRootMultiplicity{sturmname}{k}</span></tt></a></li> -<li><a class="reference internal" href="#polintervalwidth-sturmname-index" id="id143"><tt class="docutils literal"><span class="pre">\PolIntervalWidth{sturmname}{index}</span></tt></a></li> -<li><a class="reference internal" href="#expandable-macros-for-use-within-execution-of-polprintintervals" id="id144">Expandable macros for use within execution of <tt class="docutils literal">\PolPrintIntervals</tt></a><ul> -<li><a class="reference internal" href="#polprintintervalsthevar" id="id145"><tt class="docutils literal">\PolPrintIntervalsTheVar</tt></a></li> -<li><a class="reference internal" href="#polprintintervalstheindex" id="id146"><tt class="docutils literal">\PolPrintIntervalsTheIndex</tt></a></li> -<li><a class="reference internal" href="#polprintintervalsthesturmname" id="id147"><tt class="docutils literal">\PolPrintIntervalsTheSturmName</tt></a></li> -<li><a class="reference internal" href="#polprintintervalstheleftendpoint" id="id148"><tt class="docutils literal">\PolPrintIntervalsTheLeftEndPoint</tt></a></li> -<li><a class="reference internal" href="#polprintintervalstherightendpoint" id="id149"><tt class="docutils literal">\PolPrintIntervalsTheRightEndPoint</tt></a></li> -<li><a class="reference internal" href="#polprintintervalsthemultiplicity" id="id150"><tt class="docutils literal">\PolPrintIntervalsTheMultiplicity</tt></a></li> +<li><a class="reference internal" href="#polsturmnbofrationalroots-sturmname" id="id139"><tt class="docutils literal">\PolSturmNbOfRationalRoots{sturmname}</tt></a></li> +<li><a class="reference internal" href="#polsturmnbofrationalrootswithmultiplicities-sturmname" id="id140"><tt class="docutils literal">\PolSturmNbOfRationalRootsWithMultiplicities{sturmname}</tt></a></li> +<li><a class="reference internal" href="#polsturmrationalroot-sturmname-k" id="id141"><tt class="docutils literal"><span class="pre">\PolSturmRationalRoot{sturmname}{k}</span></tt></a></li> +<li><a class="reference internal" href="#polsturmrationalrootindex-sturmname-k" id="id142"><tt class="docutils literal"><span class="pre">\PolSturmRationalRootIndex{sturmname}{k}</span></tt></a></li> +<li><a class="reference internal" href="#polsturmrationalrootmultiplicity-sturmname-k" id="id143"><tt class="docutils literal"><span class="pre">\PolSturmRationalRootMultiplicity{sturmname}{k}</span></tt></a></li> +<li><a class="reference internal" href="#polintervalwidth-sturmname-index" id="id144"><tt class="docutils literal"><span class="pre">\PolIntervalWidth{sturmname}{index}</span></tt></a></li> +<li><a class="reference internal" href="#expandable-macros-for-use-within-execution-of-polprintintervals" id="id145">Expandable macros for use within execution of <tt class="docutils literal">\PolPrintIntervals</tt></a><ul> +<li><a class="reference internal" href="#polprintintervalsthevar" id="id146"><tt class="docutils literal">\PolPrintIntervalsTheVar</tt></a></li> +<li><a class="reference internal" href="#polprintintervalstheindex" id="id147"><tt class="docutils literal">\PolPrintIntervalsTheIndex</tt></a></li> +<li><a class="reference internal" href="#polprintintervalsthesturmname" id="id148"><tt class="docutils literal">\PolPrintIntervalsTheSturmName</tt></a></li> +<li><a class="reference internal" href="#polprintintervalstheleftendpoint" id="id149"><tt class="docutils literal">\PolPrintIntervalsTheLeftEndPoint</tt></a></li> +<li><a class="reference internal" href="#polprintintervalstherightendpoint" id="id150"><tt class="docutils literal">\PolPrintIntervalsTheRightEndPoint</tt></a></li> +<li><a class="reference internal" href="#polprintintervalsthemultiplicity" id="id151"><tt class="docutils literal">\PolPrintIntervalsTheMultiplicity</tt></a></li> </ul> </li> -<li><a class="reference internal" href="#poldectostring-decimal-number" id="id151"><tt class="docutils literal">\PolDecToString{decimal number}</tt></a></li> +<li><a class="reference internal" href="#poldectostring-decimal-number" id="id152"><tt class="docutils literal">\PolDecToString{decimal number}</tt></a></li> </ul> </li> -<li><a class="reference internal" href="#booleans-with-default-setting-as-indicated" id="id152">Booleans (with default setting as indicated)</a><ul> -<li><a class="reference internal" href="#xintverbosefalse" id="id153"><tt class="docutils literal">\xintverbosefalse</tt></a></li> -<li><a class="reference internal" href="#poltypesetallfalse" id="id154"><tt class="docutils literal">\poltypesetallfalse</tt></a></li> -<li><a class="reference internal" href="#poltoexprallfalse" id="id155"><tt class="docutils literal">\poltoexprallfalse</tt></a></li> +<li><a class="reference internal" href="#booleans-with-default-setting-as-indicated" id="id153">Booleans (with default setting as indicated)</a><ul> +<li><a class="reference internal" href="#xintverbosefalse" id="id154"><tt class="docutils literal">\xintverbosefalse</tt></a></li> +<li><a class="reference internal" href="#poltypesetallfalse" id="id155"><tt class="docutils literal">\poltypesetallfalse</tt></a></li> +<li><a class="reference internal" href="#poltoexprallfalse" id="id156"><tt class="docutils literal">\poltoexprallfalse</tt></a></li> </ul> </li> -<li><a class="reference internal" href="#polexprsetup" id="id156"><tt class="docutils literal">\polexprsetup</tt></a></li> -<li><a class="reference internal" href="#technicalities" id="id157">Technicalities</a></li> -<li><a class="reference internal" href="#change-log" id="id158">CHANGE LOG</a></li> -<li><a class="reference internal" href="#acknowledgments" id="id159">Acknowledgments</a></li> +<li><a class="reference internal" href="#polexprsetup" id="id157"><tt class="docutils literal">\polexprsetup</tt></a></li> +<li><a class="reference internal" href="#technicalities" id="id158">Technicalities</a></li> +<li><a class="reference internal" href="#change-log" id="id159">CHANGE LOG</a></li> +<li><a class="reference internal" href="#acknowledgments" id="id160">Acknowledgments</a></li> </ul> </div> <div class="section" id="basic-syntax"> @@ -818,7 +819,7 @@ The multiplicity is 1 for the root such that 0.9997663986 < x < 0.99976639 \PolSturmIsolateZeros**{Q} \PolPrintIntervals{Q} -$Q_norr(x) = \PolTypeset{Q_norr}$ +$Q_{norr}(x) = \PolTypeset{Q_norr}$ </pre> <p>Here, all real roots are rational:</p> <pre class="literal-block"> @@ -849,7 +850,7 @@ Z_3 = 3.14159292035398230088... \PolSturmIsolateZeros{P} % finds 4 real roots This polynomial has \PolSturmNbOfIsolatedZeros{P} distinct real roots: \PolPrintIntervals{P}% -% reports -2 < Z_1 < -1, 0 < Z_2 < 0.1, 0.1 < Z_3 < 0.2, 1 < Z_4 < 2 +% reports -2 < Z_1 < -1, 0.09 < Z_2 < 0.10, 0.1 < Z_3 < 0.2, 1 < Z_4 < 2 Let us refine the second and third intervals to separate the corresponding roots: \PolRefineInterval*{P}{2}% will refine to 0.0999990 < Z_2 < 0.0999991 @@ -948,8 +949,55 @@ Z_9 = 8.9172502485... Z_10 = 20.8469081014... </pre> </div> +<div class="section" id="the-second-wilkinson-polynomial"> +<h2><a class="toc-backref" href="#id46">The second Wilkinson polynomial</a></h2> +<pre class="literal-block"> +\documentclass{article} +\usepackage{polexpr} +\begin{document} +\poldef f(x) := mul(x - 2^-i, i = 1..20); + +%\PolTypeset{f} + +\PolToSturm{f}{f} +\PolSturmIsolateZeros**{f} +\PolPrintIntervals{f} +\end{document} +</pre> +<p>This takes more time than the polynomial with 1, 2, .., 20 as roots but +less than the latter modified by the <tt class="docutils literal"><span class="pre">2**-23</span></tt> change in one +coefficient.</p> +<p>Here is the output (with release 0.7.2):</p> +<pre class="literal-block"> +Z_1 = 0.00000095367431640625 +Z_2 = 0.0000019073486328125 +Z_3 = 0.000003814697265625 +Z_4 = 0.00000762939453125 +Z_5 = 0.0000152587890625 +Z_6 = 0.000030517578125 +Z_7 = 0.00006103515625 +Z_8 = 0.0001220703125 +Z_9 = 1/4096 +Z_10 = 1/2048 +Z_11 = 1/1024 +Z_12 = 1/512 +Z_13 = 1/256 +Z_14 = 1/128 +Z_15 = 0.015625 +Z_16 = 0.03125 +Z_17 = 0.0625 +Z_18 = 0.125 +Z_19 = 0.25 +Z_20 = 0.5 +</pre> +<p>There is some incoherence in output format which has its source in the +fact that some roots are found in branches which can only find decimal +roots, whereas some are found in branches which could find general +fractions and they use <tt class="docutils literal">\xintIrr</tt> before storage of the found root. +This may evolve in future.</p> +</div> <div class="section" id="the-degree-41-polynomial-with-2-1-9-1-8-0-0-1-1-9-2-as-roots"> -<h2><a class="toc-backref" href="#id46">The degree 41 polynomial with -2, -1.9, -1.8, ..., 0, 0.1, ..., 1.9, 2 as roots</a></h2> +<h2><a class="toc-backref" href="#id47">The degree 41 polynomial with -2, -1.9, -1.8, ..., 0, 0.1, ..., 1.9, 2 as roots</a></h2> <pre class="literal-block"> \PolDef{P}{mul((x-i*1e-1), i=-20..20)}% i/10 is same but less efficient </pre> @@ -1007,7 +1055,7 @@ optional argument keeping the same.</p> </div> </div> <div class="section" id="roots-of-chebyshev-polynomials"> -<h2><a class="toc-backref" href="#id47">Roots of Chebyshev polynomials</a></h2> +<h2><a class="toc-backref" href="#id48">Roots of Chebyshev polynomials</a></h2> <pre class="literal-block"> \newcount\mycount \poldef T_0(x) := 1; @@ -1030,9 +1078,9 @@ optional argument keeping the same.</p> </div> </div> <div class="section" id="non-expandable-macros"> -<h1><a class="toc-backref" href="#id48">Non-expandable macros</a></h1> +<h1><a class="toc-backref" href="#id49">Non-expandable macros</a></h1> <div class="section" id="poldef-polname-letter-expression-in-letter"> -<span id="poldef"></span><h2><a class="toc-backref" href="#id49"><tt class="docutils literal">\poldef <span class="pre">polname(letter):=</span> expression in letter;</tt></a></h2> +<span id="poldef"></span><h2><a class="toc-backref" href="#id50"><tt class="docutils literal">\poldef <span class="pre">polname(letter):=</span> expression in letter;</tt></a></h2> <blockquote> <p>This evaluates the <em>polynomial expression</em> and stores the coefficients in a private structure accessible later via other package macros, @@ -1069,7 +1117,7 @@ manually, if needed.</p> </blockquote> </div> <div class="section" id="poldef-letter-polname-expression-in-letter"> -<span id="id2"></span><h2><a class="toc-backref" href="#id50"><tt class="docutils literal"><span class="pre">\PolDef[letter]{polname}{expression</span> in letter}</tt></a></h2> +<span id="id2"></span><h2><a class="toc-backref" href="#id51"><tt class="docutils literal"><span class="pre">\PolDef[letter]{polname}{expression</span> in letter}</tt></a></h2> <blockquote> Does the same as <a class="reference external" href="poldef;">\poldef</a> in an undelimited macro format (thus avoiding potential problems with the catcode of the @@ -1077,7 +1125,7 @@ semi-colon in presence of some packages.) In absence of the <tt class="docutils literal">[letter]</tt> optional argument, the variable is assumed to be <tt class="docutils literal">x</tt>.</blockquote> </div> <div class="section" id="polgenfloatvariant-polname"> -<span id="polgenfloatvariant"></span><h2><a class="toc-backref" href="#id51"><tt class="docutils literal">\PolGenFloatVariant{polname}</tt></a></h2> +<span id="polgenfloatvariant"></span><h2><a class="toc-backref" href="#id52"><tt class="docutils literal">\PolGenFloatVariant{polname}</tt></a></h2> <blockquote> <p>Makes the polynomial also usable in the <tt class="docutils literal">\xintfloatexpr</tt> parser. It will therein evaluates via an Horner scheme with coefficients @@ -1096,7 +1144,7 @@ context.</p> </blockquote> </div> <div class="section" id="pollet-polname-2-polname-1"> -<span id="pollet"></span><h2><a class="toc-backref" href="#id52"><tt class="docutils literal"><span class="pre">\PolLet{polname_2}={polname_1}</span></tt></a></h2> +<span id="pollet"></span><h2><a class="toc-backref" href="#id53"><tt class="docutils literal"><span class="pre">\PolLet{polname_2}={polname_1}</span></tt></a></h2> <blockquote> Makes a copy of the already defined polynomial <tt class="docutils literal">polname_1</tt> to a new one <tt class="docutils literal">polname_2</tt>. Same effect as @@ -1104,12 +1152,12 @@ new one <tt class="docutils literal">polname_2</tt>. Same effect as <tt class="docutils literal">=</tt> is optional.</blockquote> </div> <div class="section" id="polgloballet-polname-2-polname-1"> -<span id="polgloballet"></span><h2><a class="toc-backref" href="#id53"><tt class="docutils literal"><span class="pre">\PolGlobalLet{polname_2}={polname_1}</span></tt></a></h2> +<span id="polgloballet"></span><h2><a class="toc-backref" href="#id54"><tt class="docutils literal"><span class="pre">\PolGlobalLet{polname_2}={polname_1}</span></tt></a></h2> <blockquote> Acts globally.</blockquote> </div> <div class="section" id="polassign-polname-toarray-macro"> -<span id="polassign"></span><h2><a class="toc-backref" href="#id54"><tt class="docutils literal"><span class="pre">\PolAssign{polname}\toarray\macro</span></tt></a></h2> +<span id="polassign"></span><h2><a class="toc-backref" href="#id55"><tt class="docutils literal"><span class="pre">\PolAssign{polname}\toarray\macro</span></tt></a></h2> <blockquote> <p>Defines a one-argument expandable macro <tt class="docutils literal"><span class="pre">\macro{#1}</span></tt> which expands to the (raw) #1th polynomial coefficient.</p> @@ -1132,7 +1180,7 @@ indices act the same in both.)</p> </blockquote> </div> <div class="section" id="polget-polname-fromarray-macro"> -<span id="polget"></span><h2><a class="toc-backref" href="#id55"><tt class="docutils literal"><span class="pre">\PolGet{polname}\fromarray\macro</span></tt></a></h2> +<span id="polget"></span><h2><a class="toc-backref" href="#id56"><tt class="docutils literal"><span class="pre">\PolGet{polname}\fromarray\macro</span></tt></a></h2> <blockquote> <p>Does the converse operation to <tt class="docutils literal"><span class="pre">\PolAssign{polname}\toarray\macro</span></tt>. Each individual @@ -1153,7 +1201,7 @@ via <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</ </blockquote> </div> <div class="section" id="polfromcsv-polname-csv"> -<span id="polfromcsv"></span><h2><a class="toc-backref" href="#id56"><tt class="docutils literal"><span class="pre">\PolFromCSV{polname}{<csv>}</span></tt></a></h2> +<span id="polfromcsv"></span><h2><a class="toc-backref" href="#id57"><tt class="docutils literal"><span class="pre">\PolFromCSV{polname}{<csv>}</span></tt></a></h2> <blockquote> <p>Defines a polynomial directly from the comma separated list of values (or a macro expanding to such a list) of its coefficients, the <em>first @@ -1175,7 +1223,7 @@ form via <a class="reference external" href="http://www.ctan.org/pkg/xint">xintf </blockquote> </div> <div class="section" id="poltypeset-polname"> -<span id="poltypeset"></span><h2><a class="toc-backref" href="#id57"><tt class="docutils literal">\PolTypeset{polname}</tt></a></h2> +<span id="poltypeset"></span><h2><a class="toc-backref" href="#id58"><tt class="docutils literal">\PolTypeset{polname}</tt></a></h2> <blockquote> <p>Typesets in descending powers in math mode. It uses letter <tt class="docutils literal">x</tt> but this can be changed via an optional argument:</p> @@ -1189,7 +1237,7 @@ can be re-defined for customization. Their default definitions are expandable, but this is not a requirement.</p> </blockquote> <div class="section" id="poltypesetcmd-raw-coeff"> -<span id="poltypesetcmd"></span><h3><a class="toc-backref" href="#id58"><tt class="docutils literal">\PolTypesetCmd{raw_coeff}</tt></a></h3> +<span id="poltypesetcmd"></span><h3><a class="toc-backref" href="#id59"><tt class="docutils literal">\PolTypesetCmd{raw_coeff}</tt></a></h3> <blockquote> <p>Checks if the coefficient is <tt class="docutils literal">1</tt> or <tt class="docutils literal"><span class="pre">-1</span></tt> and then skips printing the <tt class="docutils literal">1</tt>, except for the constant term. Also it sets conditional @@ -1199,7 +1247,7 @@ minus one is handled by <a class="reference internal" href="#poltypesetone-raw-c </blockquote> </div> <div class="section" id="poltypesetone-raw-coeff"> -<span id="poltypesetone"></span><h3><a class="toc-backref" href="#id59"><tt class="docutils literal">\PolTypesetOne{raw_coeff}</tt></a></h3> +<span id="poltypesetone"></span><h3><a class="toc-backref" href="#id60"><tt class="docutils literal">\PolTypesetOne{raw_coeff}</tt></a></h3> <blockquote> <p>The default is <tt class="docutils literal">\xintSignedFrac</tt> but this macro is annoying as it insists to use a power of ten, and not decimal notation.</p> @@ -1224,7 +1272,7 @@ which uses decimal notation (at least for the numerator part).</p> </blockquote> </div> <div class="section" id="id6"> -<span id="poltypesetmonomialcmd"></span><h3><a class="toc-backref" href="#id60"><tt class="docutils literal">\PolTypesetMonomialCmd</tt></a></h3> +<span id="poltypesetmonomialcmd"></span><h3><a class="toc-backref" href="#id61"><tt class="docutils literal">\PolTypesetMonomialCmd</tt></a></h3> <blockquote> This decides how a monomial (in variable <tt class="docutils literal">\PolVar</tt> and with exponent <tt class="docutils literal">\PolIndex</tt>) is to be printed. The default does nothing @@ -1234,7 +1282,7 @@ for the constant term, <tt class="docutils literal">\PolVar</tt> for the first d <tt class="docutils literal">\ifnum</tt> tests.</blockquote> </div> <div class="section" id="poltypesetcmdprefix-raw-coeff"> -<span id="poltypesetcmdprefix"></span><h3><a class="toc-backref" href="#id61"><tt class="docutils literal">\PolTypesetCmdPrefix{raw_coeff}</tt></a></h3> +<span id="poltypesetcmdprefix"></span><h3><a class="toc-backref" href="#id62"><tt class="docutils literal">\PolTypesetCmdPrefix{raw_coeff}</tt></a></h3> <blockquote> Expands to a <tt class="docutils literal">+</tt> if the <tt class="docutils literal">raw_coeff</tt> is zero or positive, and to nothing if <tt class="docutils literal">raw_coeff</tt> is negative, as in latter case the @@ -1245,13 +1293,13 @@ for the first term.</blockquote> </div> </div> <div class="section" id="id8"> -<span id="id7"></span><h2><a class="toc-backref" href="#id62"><tt class="docutils literal"><span class="pre">\PolTypeset*{polname}</span></tt></a></h2> +<span id="id7"></span><h2><a class="toc-backref" href="#id63"><tt class="docutils literal"><span class="pre">\PolTypeset*{polname}</span></tt></a></h2> <blockquote> Typesets in ascending powers. Use e.g. <tt class="docutils literal">[h]</tt> optional argument (after the <tt class="docutils literal">*</tt>) to use letter <tt class="docutils literal">h</tt> rather than <tt class="docutils literal">x</tt>.</blockquote> </div> <div class="section" id="poldiff-polname-1-polname-2"> -<span id="poldiff"></span><h2><a class="toc-backref" href="#id63"><tt class="docutils literal"><span class="pre">\PolDiff{polname_1}{polname_2}</span></tt></a></h2> +<span id="poldiff"></span><h2><a class="toc-backref" href="#id64"><tt class="docutils literal"><span class="pre">\PolDiff{polname_1}{polname_2}</span></tt></a></h2> <blockquote> <p>This sets <tt class="docutils literal">polname_2</tt> to the first derivative of <tt class="docutils literal">polname_1</tt>. It is allowed to issue <tt class="docutils literal"><span class="pre">\PolDiff{f}{f}</span></tt>, effectively replacing <tt class="docutils literal">f</tt> @@ -1261,7 +1309,7 @@ by <tt class="docutils literal">f'</tt>.</p> </blockquote> </div> <div class="section" id="poldiff-n-polname-1-polname-2"> -<span id="poldiff-n"></span><h2><a class="toc-backref" href="#id64"><tt class="docutils literal"><span class="pre">\PolDiff[N]{polname_1}{polname_2}</span></tt></a></h2> +<span id="poldiff-n"></span><h2><a class="toc-backref" href="#id65"><tt class="docutils literal"><span class="pre">\PolDiff[N]{polname_1}{polname_2}</span></tt></a></h2> <blockquote> This sets <tt class="docutils literal">polname_2</tt> to the <tt class="docutils literal">N</tt>-th derivative of <tt class="docutils literal">polname_1</tt>. Identical arguments is allowed. With <tt class="docutils literal">N=0</tt>, same effect as @@ -1269,7 +1317,7 @@ Identical arguments is allowed. With <tt class="docutils literal">N=0</tt>, same using <tt class="docutils literal">\PolAntiDiff</tt>.</blockquote> </div> <div class="section" id="polantidiff-polname-1-polname-2"> -<span id="polantidiff"></span><h2><a class="toc-backref" href="#id65"><tt class="docutils literal"><span class="pre">\PolAntiDiff{polname_1}{polname_2}</span></tt></a></h2> +<span id="polantidiff"></span><h2><a class="toc-backref" href="#id66"><tt class="docutils literal"><span class="pre">\PolAntiDiff{polname_1}{polname_2}</span></tt></a></h2> <blockquote> <p>This sets <tt class="docutils literal">polname_2</tt> to the primitive of <tt class="docutils literal">polname_1</tt> vanishing at zero.</p> @@ -1278,32 +1326,32 @@ at zero.</p> </blockquote> </div> <div class="section" id="polantidiff-n-polname-1-polname-2"> -<span id="polantidiff-n"></span><h2><a class="toc-backref" href="#id66"><tt class="docutils literal"><span class="pre">\PolAntiDiff[N]{polname_1}{polname_2}</span></tt></a></h2> +<span id="polantidiff-n"></span><h2><a class="toc-backref" href="#id67"><tt class="docutils literal"><span class="pre">\PolAntiDiff[N]{polname_1}{polname_2}</span></tt></a></h2> <blockquote> This sets <tt class="docutils literal">polname_2</tt> to the result of <tt class="docutils literal">N</tt> successive integrations on <tt class="docutils literal">polname_1</tt>. With negative <tt class="docutils literal">N</tt>, it switches to using <tt class="docutils literal">\PolDiff</tt>.</blockquote> </div> <div class="section" id="poldivide-polname-1-polname-2-polname-q-polname-r"> -<span id="poldivide"></span><h2><a class="toc-backref" href="#id67"><tt class="docutils literal"><span class="pre">\PolDivide{polname_1}{polname_2}{polname_Q}{polname_R}</span></tt></a></h2> +<span id="poldivide"></span><h2><a class="toc-backref" href="#id68"><tt class="docutils literal"><span class="pre">\PolDivide{polname_1}{polname_2}{polname_Q}{polname_R}</span></tt></a></h2> <blockquote> This sets <tt class="docutils literal">polname_Q</tt> and <tt class="docutils literal">polname_R</tt> to be the quotient and remainder in the Euclidean division of <tt class="docutils literal">polname_1</tt> by <tt class="docutils literal">polname_2</tt>.</blockquote> </div> <div class="section" id="polquo-polname-1-polname-2-polname-q"> -<span id="polquo"></span><h2><a class="toc-backref" href="#id68"><tt class="docutils literal"><span class="pre">\PolQuo{polname_1}{polname_2}{polname_Q}</span></tt></a></h2> +<span id="polquo"></span><h2><a class="toc-backref" href="#id69"><tt class="docutils literal"><span class="pre">\PolQuo{polname_1}{polname_2}{polname_Q}</span></tt></a></h2> <blockquote> This sets <tt class="docutils literal">polname_Q</tt> to be the quotient in the Euclidean division of <tt class="docutils literal">polname_1</tt> by <tt class="docutils literal">polname_2</tt>.</blockquote> </div> <div class="section" id="polrem-polname-1-polname-2-polname-r"> -<span id="polrem"></span><h2><a class="toc-backref" href="#id69"><tt class="docutils literal"><span class="pre">\PolRem{polname_1}{polname_2}{polname_R}</span></tt></a></h2> +<span id="polrem"></span><h2><a class="toc-backref" href="#id70"><tt class="docutils literal"><span class="pre">\PolRem{polname_1}{polname_2}{polname_R}</span></tt></a></h2> <blockquote> This sets <tt class="docutils literal">polname_R</tt> to be the remainder in the Euclidean division of <tt class="docutils literal">polname_1</tt> by <tt class="docutils literal">polname_2</tt>.</blockquote> </div> <div class="section" id="polgcd-polname-1-polname-2-polname-gcd"> -<span id="polgcd"></span><h2><a class="toc-backref" href="#id70"><tt class="docutils literal"><span class="pre">\PolGCD{polname_1}{polname_2}{polname_GCD}</span></tt></a></h2> +<span id="polgcd"></span><h2><a class="toc-backref" href="#id71"><tt class="docutils literal"><span class="pre">\PolGCD{polname_1}{polname_2}{polname_GCD}</span></tt></a></h2> <blockquote> This sets <tt class="docutils literal">polname_GCD</tt> to be the (monic) GCD of the two first polynomials. It is a unitary polynomial except if both <tt class="docutils literal">polname_1</tt> @@ -1329,7 +1377,7 @@ polynomial.</blockquote> no common factor among the coefficients. --> </div> <div class="section" id="poltosturm-polname-sturmname"> -<span id="poltosturm"></span><h2><a class="toc-backref" href="#id71"><tt class="docutils literal"><span class="pre">\PolToSturm{polname}{sturmname}</span></tt></a></h2> +<span id="poltosturm"></span><h2><a class="toc-backref" href="#id72"><tt class="docutils literal"><span class="pre">\PolToSturm{polname}{sturmname}</span></tt></a></h2> <blockquote> <p>With <tt class="docutils literal">polname</tt> being for example <tt class="docutils literal">P</tt>, the macro starts by computing polynomials <tt class="docutils literal">P</tt> and <tt class="docutils literal">P'</tt>, then computes the (opposite @@ -1369,15 +1417,15 @@ i.e. it is the "square-free part" of original polynomial <tt class="do <p>The polynomials <tt class="docutils literal">sturmname_k</tt> main utility is for the execution of <a class="reference internal" href="#polsturmisolatezeros-sturmname">\PolSturmIsolateZeros{sturmname}</a>. Be careful not to use these names <tt class="docutils literal">sturmname_0</tt>, <tt class="docutils literal">sturmname_1</tt>, etc... for defining other -polynomials after having done <tt class="docutils literal"><span class="pre">\PolToSturm{polname}{sturmname}</span> and -before executing <span class="pre">``\PolSturmIsolateZeros{sturmname}</span></tt> else the +polynomials after having done <tt class="docutils literal"><span class="pre">\PolToSturm{polname}{sturmname}</span></tt> and +before executing <tt class="docutils literal">\PolSturmIsolateZeros{sturmname}</tt> else the latter will behave erroneously.</p> <p><a class="reference internal" href="#polsturmchainlength-sturmname">\PolSturmChainLength{sturmname}</a> gives the index of the last element of the Sturm chain.</p> </blockquote> </div> <div class="section" id="id10"> -<span id="id9"></span><h2><a class="toc-backref" href="#id72"><tt class="docutils literal"><span class="pre">\PolToSturm*{polname}{sturmname}</span></tt></a></h2> +<span id="id9"></span><h2><a class="toc-backref" href="#id73"><tt class="docutils literal"><span class="pre">\PolToSturm*{polname}{sturmname}</span></tt></a></h2> <blockquote> <p>Does the same as <a class="reference internal" href="#poltosturm">un-starred version</a> and additionally it keeps for user usage the memory of the <em>un-normalized</em> Sturm chain @@ -1401,7 +1449,7 @@ polynomials, where each <tt class="docutils literal">f_i</tt> divides its predec </blockquote> </div> <div class="section" id="polsettosturmchainsignchangesat-macro-sturmname-fraction"> -<span id="polsettosturmchainsignchangesat"></span><h2><a class="toc-backref" href="#id73"><tt class="docutils literal"><span class="pre">\PolSetToSturmChainSignChangesAt{\macro}{sturmname}{fraction}</span></tt></a></h2> +<span id="polsettosturmchainsignchangesat"></span><h2><a class="toc-backref" href="#id74"><tt class="docutils literal"><span class="pre">\PolSetToSturmChainSignChangesAt{\macro}{sturmname}{fraction}</span></tt></a></h2> <blockquote> <p>Sets macro <tt class="docutils literal">\macro</tt> to the number of sign changes in the Sturm chain with name prefix <tt class="docutils literal">sturmname</tt>, at location <tt class="docutils literal">fraction</tt> @@ -1418,7 +1466,7 @@ use <tt class="docutils literal">[\empty]</tt> as extra optional argument.</p> </blockquote> </div> <div class="section" id="polsettonbofzeroswithin-macro-sturmname-value-a-value-b"> -<span id="polsettonbofzeroswithin"></span><h2><a class="toc-backref" href="#id74"><tt class="docutils literal"><span class="pre">\PolSetToNbOfZerosWithin{\macro}{sturmname}{value_a}{value_b}</span></tt></a></h2> +<span id="polsettonbofzeroswithin"></span><h2><a class="toc-backref" href="#id75"><tt class="docutils literal"><span class="pre">\PolSetToNbOfZerosWithin{\macro}{sturmname}{value_a}{value_b}</span></tt></a></h2> <blockquote> <p>Applies the <a class="reference external" href="https://en.wikipedia.org/wiki/Sturm%27s_theorem">Sturm Theorem</a> to set <tt class="docutils literal">\macro</tt> to the exact number of <strong>distinct</strong> roots of <tt class="docutils literal">sturmname_0</tt> in the interval <tt class="docutils literal">(value_a, @@ -1447,7 +1495,7 @@ which requires prior execution of </blockquote> </div> <div class="section" id="polsturmisolatezeros-sturmname"> -<span id="polsturmisolatezeros"></span><h2><a class="toc-backref" href="#id75"><tt class="docutils literal">\PolSturmIsolateZeros{sturmname}</tt></a></h2> +<span id="polsturmisolatezeros"></span><h2><a class="toc-backref" href="#id76"><tt class="docutils literal">\PolSturmIsolateZeros{sturmname}</tt></a></h2> <blockquote> <p>The macros locates, using <a class="reference external" href="https://en.wikipedia.org/wiki/Sturm%27s_theorem">Sturm theorem</a>, as many disjoint intervals as there are (real) roots.</p> @@ -1557,7 +1605,7 @@ are studied in numerical mathematics.</p> </blockquote> </div> <div class="section" id="id12"> -<span id="id11"></span><h2><a class="toc-backref" href="#id76"><tt class="docutils literal"><span class="pre">\PolSturmIsolateZeros*{sturmname}</span></tt></a></h2> +<span id="id11"></span><h2><a class="toc-backref" href="#id77"><tt class="docutils literal"><span class="pre">\PolSturmIsolateZeros*{sturmname}</span></tt></a></h2> <blockquote> <p>The macro does the same as <a class="reference internal" href="#polsturmisolatezeros-sturmname">\PolSturmIsolateZeros{sturmname}</a> and then in addition it does the extra work to determine all @@ -1583,7 +1631,7 @@ roots</a> for an example.</p> </blockquote> </div> <div class="section" id="id14"> -<span id="id13"></span><h2><a class="toc-backref" href="#id77"><tt class="docutils literal"><span class="pre">\PolSturmIsolateZeros**{sturmname}</span></tt></a></h2> +<span id="id13"></span><h2><a class="toc-backref" href="#id78"><tt class="docutils literal"><span class="pre">\PolSturmIsolateZeros**{sturmname}</span></tt></a></h2> <blockquote> <p>The macro does the same as <a class="reference internal" href="#id12">\PolSturmIsolateZeros*{sturmname}</a> and in addition it does the extra work to determine all the <em>rational</em> @@ -1608,17 +1656,17 @@ roots</a> for an example.</p> </blockquote> </div> <div class="section" id="polsturmisolatezerosandgetmultiplicities-sturmname"> -<span id="polsturmisolatezerosandgetmultiplicities"></span><h2><a class="toc-backref" href="#id78"><tt class="docutils literal">\PolSturmIsolateZerosAndGetMultiplicities{sturmname}</tt></a></h2> +<span id="polsturmisolatezerosandgetmultiplicities"></span><h2><a class="toc-backref" href="#id79"><tt class="docutils literal">\PolSturmIsolateZerosAndGetMultiplicities{sturmname}</tt></a></h2> <blockquote> This is another name for <a class="reference internal" href="#id12">\PolSturmIsolateZeros*{sturmname}</a>.</blockquote> </div> <div class="section" id="polsturmisolatezerosgetmultiplicitiesandrationalroots-sturmname"> -<span id="polsturmisolatezerosgetmultiplicitiesandrationalroots"></span><h2><a class="toc-backref" href="#id79"><tt class="docutils literal">\PolSturmIsolateZerosGetMultiplicitiesAndRationalRoots{sturmname}</tt></a></h2> +<span id="polsturmisolatezerosgetmultiplicitiesandrationalroots"></span><h2><a class="toc-backref" href="#id80"><tt class="docutils literal">\PolSturmIsolateZerosGetMultiplicitiesAndRationalRoots{sturmname}</tt></a></h2> <blockquote> This is another name for <a class="reference internal" href="#id14">\PolSturmIsolateZeros**{sturmname}</a>.</blockquote> </div> <div class="section" id="polsturmisolatezerosandfindrationalroots-sturmname"> -<h2><a class="toc-backref" href="#id80"><tt class="docutils literal">\PolSturmIsolateZerosAndFindRationalRoots{sturmname}</tt></a></h2> +<h2><a class="toc-backref" href="#id81"><tt class="docutils literal">\PolSturmIsolateZerosAndFindRationalRoots{sturmname}</tt></a></h2> <blockquote> <p>This works exactly like <a class="reference internal" href="#id14">\PolSturmIsolateZeros**{sturmname}</a> (inclusive of declaring the polynomials <tt class="docutils literal">sturmname_sqf_norr</tt> and @@ -1648,7 +1696,7 @@ rational root anyhow).</p> </blockquote> </div> <div class="section" id="polrefineinterval-sturmname-index"> -<span id="polrefineinterval"></span><h2><a class="toc-backref" href="#id81"><tt class="docutils literal"><span class="pre">\PolRefineInterval*{sturmname}{index}</span></tt></a></h2> +<span id="polrefineinterval"></span><h2><a class="toc-backref" href="#id82"><tt class="docutils literal"><span class="pre">\PolRefineInterval*{sturmname}{index}</span></tt></a></h2> <blockquote> The <tt class="docutils literal">index</tt>-th interval (starting indexing at one) is further subdivided as many times as is necessary in order for the newer @@ -1657,21 +1705,21 @@ the original interval. This means that the <tt class="docutils literal">k</tt>th strictly separated from the other roots.</blockquote> </div> <div class="section" id="polrefineinterval-n-sturmname-index"> -<span id="polrefineinterval-n"></span><h2><a class="toc-backref" href="#id82"><tt class="docutils literal"><span class="pre">\PolRefineInterval[N]{sturmname}{index}</span></tt></a></h2> +<span id="polrefineinterval-n"></span><h2><a class="toc-backref" href="#id83"><tt class="docutils literal"><span class="pre">\PolRefineInterval[N]{sturmname}{index}</span></tt></a></h2> <blockquote> The <tt class="docutils literal">index</tt>-th interval (starting count at one) is further subdivided once, reducing its length by a factor of 10. This is done <tt class="docutils literal">N</tt> times if the optional argument <tt class="docutils literal">[N]</tt> is present.</blockquote> </div> <div class="section" id="polensureintervallength-sturmname-index-e"> -<span id="polensureintervallength"></span><h2><a class="toc-backref" href="#id83"><tt class="docutils literal"><span class="pre">\PolEnsureIntervalLength{sturmname}{index}{E}</span></tt></a></h2> +<span id="polensureintervallength"></span><h2><a class="toc-backref" href="#id84"><tt class="docutils literal"><span class="pre">\PolEnsureIntervalLength{sturmname}{index}{E}</span></tt></a></h2> <blockquote> The <tt class="docutils literal">index</tt>-th interval is subdivided until its length becomes at most <tt class="docutils literal">10^E</tt>. This means (for <tt class="docutils literal">E<0</tt>) that the first <tt class="docutils literal"><span class="pre">-E</span></tt> digits after decimal mark of the <tt class="docutils literal">k</tt>th root will then be known exactly.</blockquote> </div> <div class="section" id="polensureintervallengths-sturmname-e"> -<span id="polensureintervallengths"></span><h2><a class="toc-backref" href="#id84"><tt class="docutils literal"><span class="pre">\PolEnsureIntervalLengths{sturmname}{E}</span></tt></a></h2> +<span id="polensureintervallengths"></span><h2><a class="toc-backref" href="#id85"><tt class="docutils literal"><span class="pre">\PolEnsureIntervalLengths{sturmname}{E}</span></tt></a></h2> <blockquote> <p>The intervals as obtained from <tt class="docutils literal">\PolSturmIsolateZeros</tt> are (if necessary) subdivided further by (base 10) dichotomy in order for @@ -1682,7 +1730,7 @@ than <tt class="docutils literal">10^E</tt> in output only if it did not change </blockquote> </div> <div class="section" id="polprintintervals-varname-sturmname"> -<span id="polprintintervals"></span><h2><a class="toc-backref" href="#id85"><tt class="docutils literal"><span class="pre">\PolPrintIntervals[varname]{sturmname}</span></tt></a></h2> +<span id="polprintintervals"></span><h2><a class="toc-backref" href="#id86"><tt class="docutils literal"><span class="pre">\PolPrintIntervals[varname]{sturmname}</span></tt></a></h2> <blockquote> <p>This is a convenience macro which prints the bounds for the roots <tt class="docutils literal">Z_1</tt>, <tt class="docutils literal">Z_2</tt>, ... (the optional argument <tt class="docutils literal">varname</tt> allows to @@ -1701,7 +1749,7 @@ modified.</p> <p>See next macros which govern its output.</p> </blockquote> <div class="section" id="polprintintervalsnorealroots"> -<h3><a class="toc-backref" href="#id86"><tt class="docutils literal">\PolPrintIntervalsNoRealRoots</tt></a></h3> +<h3><a class="toc-backref" href="#id87"><tt class="docutils literal">\PolPrintIntervalsNoRealRoots</tt></a></h3> <blockquote> <p>Executed in place of an <tt class="docutils literal">array</tt> environment, when there are no real roots. Default definition:</p> @@ -1711,7 +1759,7 @@ real roots. Default definition:</p> </blockquote> </div> <div class="section" id="polprintintervalsbeginenv"> -<h3><a class="toc-backref" href="#id87"><tt class="docutils literal">\PolPrintIntervalsBeginEnv</tt></a></h3> +<h3><a class="toc-backref" href="#id88"><tt class="docutils literal">\PolPrintIntervalsBeginEnv</tt></a></h3> <blockquote> <p>Default definition:</p> <pre class="literal-block"> @@ -1720,7 +1768,7 @@ real roots. Default definition:</p> </blockquote> </div> <div class="section" id="polprintintervalsendenv"> -<h3><a class="toc-backref" href="#id88"><tt class="docutils literal">\PolPrintIntervalsEndEnv</tt></a></h3> +<h3><a class="toc-backref" href="#id89"><tt class="docutils literal">\PolPrintIntervalsEndEnv</tt></a></h3> <blockquote> <p>Default definition:</p> <pre class="literal-block"> @@ -1729,7 +1777,7 @@ real roots. Default definition:</p> </blockquote> </div> <div class="section" id="polprintintervalsknownroot"> -<h3><a class="toc-backref" href="#id89"><tt class="docutils literal">\PolPrintIntervalsKnownRoot</tt></a></h3> +<h3><a class="toc-backref" href="#id90"><tt class="docutils literal">\PolPrintIntervalsKnownRoot</tt></a></h3> <blockquote> <p>Default definition:</p> <pre class="literal-block"> @@ -1741,7 +1789,7 @@ real roots. Default definition:</p> </blockquote> </div> <div class="section" id="polprintintervalsunknownroot"> -<h3><a class="toc-backref" href="#id90"><tt class="docutils literal">\PolPrintIntervalsUnknownRoot</tt></a></h3> +<h3><a class="toc-backref" href="#id91"><tt class="docutils literal">\PolPrintIntervalsUnknownRoot</tt></a></h3> <blockquote> <p>Default definition:</p> <pre class="literal-block"> @@ -1754,7 +1802,7 @@ real roots. Default definition:</p> </blockquote> </div> <div class="section" id="id15"> -<span id="polprintintervalsprintexactzero"></span><h3><a class="toc-backref" href="#id91"><tt class="docutils literal">\PolPrintIntervalsPrintExactZero</tt></a></h3> +<span id="polprintintervalsprintexactzero"></span><h3><a class="toc-backref" href="#id92"><tt class="docutils literal">\PolPrintIntervalsPrintExactZero</tt></a></h3> <blockquote> <p>Default definition:</p> <pre class="literal-block"> @@ -1763,7 +1811,7 @@ real roots. Default definition:</p> </blockquote> </div> <div class="section" id="id16"> -<span id="polprintintervalsprintleftendpoint"></span><h3><a class="toc-backref" href="#id92"><tt class="docutils literal">\PolPrintIntervalsPrintLeftEndPoint</tt></a></h3> +<span id="polprintintervalsprintleftendpoint"></span><h3><a class="toc-backref" href="#id93"><tt class="docutils literal">\PolPrintIntervalsPrintLeftEndPoint</tt></a></h3> <blockquote> <p>Default definition:</p> <pre class="literal-block"> @@ -1772,7 +1820,7 @@ real roots. Default definition:</p> </blockquote> </div> <div class="section" id="id17"> -<span id="polprintintervalsprintrightendpoint"></span><h3><a class="toc-backref" href="#id93"><tt class="docutils literal">\PolPrintIntervalsPrintRightEndPoint</tt></a></h3> +<span id="polprintintervalsprintrightendpoint"></span><h3><a class="toc-backref" href="#id94"><tt class="docutils literal">\PolPrintIntervalsPrintRightEndPoint</tt></a></h3> <blockquote> <p>Default definition is:</p> <pre class="literal-block"> @@ -1782,7 +1830,7 @@ real roots. Default definition:</p> </div> </div> <div class="section" id="id19"> -<span id="id18"></span><h2><a class="toc-backref" href="#id94"><tt class="docutils literal"><span class="pre">\PolPrintIntervals*[varname]{sturmname}</span></tt></a></h2> +<span id="id18"></span><h2><a class="toc-backref" href="#id95"><tt class="docutils literal"><span class="pre">\PolPrintIntervals*[varname]{sturmname}</span></tt></a></h2> <blockquote> <p>This starred variant produces an alternative output (which displays the root multiplicity), and is provided as an @@ -1801,7 +1849,7 @@ definition:</p> <p>Multiplicities are printed using this auxiliary macro:</p> </blockquote> <div class="section" id="polprintintervalsprintmultiplicity"> -<h3><a class="toc-backref" href="#id95"><tt class="docutils literal">\PolPrintIntervalsPrintMultiplicity</tt></a></h3> +<h3><a class="toc-backref" href="#id96"><tt class="docutils literal">\PolPrintIntervalsPrintMultiplicity</tt></a></h3> <blockquote> <p>whose default definition is:</p> <pre class="literal-block"> @@ -1811,7 +1859,7 @@ definition:</p> </div> </div> <div class="section" id="polmapcoeffs-macro-polname"> -<span id="polmapcoeffs"></span><h2><a class="toc-backref" href="#id96"><tt class="docutils literal"><span class="pre">\PolMapCoeffs{\macro}{polname}</span></tt></a></h2> +<span id="polmapcoeffs"></span><h2><a class="toc-backref" href="#id97"><tt class="docutils literal"><span class="pre">\PolMapCoeffs{\macro}{polname}</span></tt></a></h2> <blockquote> <p>It modifies ('in-place': original coefficients get lost) each coefficient of the defined polynomial via the <em>expandable</em> macro @@ -1831,7 +1879,7 @@ will have to be expressed in terms of macros from <a class="reference external" </blockquote> </div> <div class="section" id="polreducecoeffs-polname"> -<span id="polreducecoeffs"></span><h2><a class="toc-backref" href="#id97"><tt class="docutils literal">\PolReduceCoeffs{polname}</tt></a></h2> +<span id="polreducecoeffs"></span><h2><a class="toc-backref" href="#id98"><tt class="docutils literal">\PolReduceCoeffs{polname}</tt></a></h2> <blockquote> About the same as <tt class="docutils literal"><span class="pre">\PolMapCoeffs{\xintIrr}{polname}</span></tt> (but maintaining a <tt class="docutils literal">[0]</tt> postfix for speedier <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> parsing when @@ -1839,7 +1887,7 @@ polynomial function is used for computations.) This is a one-argument macro, working 'in-place'.</blockquote> </div> <div class="section" id="id21"> -<span id="id20"></span><h2><a class="toc-backref" href="#id98"><tt class="docutils literal"><span class="pre">\PolReduceCoeffs*{polname}</span></tt></a></h2> +<span id="id20"></span><h2><a class="toc-backref" href="#id99"><tt class="docutils literal"><span class="pre">\PolReduceCoeffs*{polname}</span></tt></a></h2> <blockquote> <p>This starred variant leaves un-touched the decimal exponent in the internal representation of the fractional coefficients, i.e. if a @@ -1858,7 +1906,7 @@ expansion speed of the <a class="reference external" href="http://www.ctan.org/p </blockquote> </div> <div class="section" id="polmakemonic-polname"> -<span id="polmakemonic"></span><h2><a class="toc-backref" href="#id99"><tt class="docutils literal">\PolMakeMonic{polname}</tt></a></h2> +<span id="polmakemonic"></span><h2><a class="toc-backref" href="#id100"><tt class="docutils literal">\PolMakeMonic{polname}</tt></a></h2> <blockquote> Divides by the leading coefficient. It is recommended to execute <a class="reference internal" href="#id21">\PolReduceCoeffs*{polname}</a> immediately afterwards. This is not @@ -1867,7 +1915,7 @@ coefficients and we want to keep the leading one as common denominator.</blockquote> </div> <div class="section" id="polmakeprimitive-polname"> -<span id="polmakeprimitive"></span><h2><a class="toc-backref" href="#id100"><tt class="docutils literal">\PolMakePrimitive{polname}</tt></a></h2> +<span id="polmakeprimitive"></span><h2><a class="toc-backref" href="#id101"><tt class="docutils literal">\PolMakePrimitive{polname}</tt></a></h2> <blockquote> Divides by the integer content see (<a class="reference internal" href="#policontent">\PolIContent</a>). This thus produces a polynomial with integer coefficients having no common factor. The sign of the leading @@ -1875,36 +1923,36 @@ coefficient is not modified.</blockquote> </div> </div> <div class="section" id="expandable-macros"> -<h1><a class="toc-backref" href="#id101">Expandable macros</a></h1> +<h1><a class="toc-backref" href="#id102">Expandable macros</a></h1> <p>All these macros expand completely in two steps except <tt class="docutils literal">\PolToExpr</tt> and <tt class="docutils literal">\PolToFloatExpr</tt> (and their auxiliaries) which need a <tt class="docutils literal">\write</tt>, <tt class="docutils literal">\edef</tt> or a <tt class="docutils literal"><span class="pre">\csname...\endcsname</span></tt> context.</p> <div class="section" id="poleval-polname-atexpr-numerical-expression"> -<span id="polevalatexpr"></span><h2><a class="toc-backref" href="#id102"><tt class="docutils literal"><span class="pre">\PolEval{polname}\AtExpr{numerical</span> expression}</tt></a></h2> +<span id="polevalatexpr"></span><h2><a class="toc-backref" href="#id103"><tt class="docutils literal"><span class="pre">\PolEval{polname}\AtExpr{numerical</span> expression}</tt></a></h2> <blockquote> It boils down to <tt class="docutils literal">\xinttheexpr polname(numerical <span class="pre">expression)\relax</span></tt>.</blockquote> </div> <div class="section" id="poleval-polname-at-fraction"> -<span id="polevalat"></span><h2><a class="toc-backref" href="#id103"><tt class="docutils literal"><span class="pre">\PolEval{polname}\At{fraction}</span></tt></a></h2> +<span id="polevalat"></span><h2><a class="toc-backref" href="#id104"><tt class="docutils literal"><span class="pre">\PolEval{polname}\At{fraction}</span></tt></a></h2> <blockquote> Evaluates the polynomial at value <tt class="docutils literal">fraction</tt> which must be in (or expand to) a format acceptable to the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> macros.</blockquote> </div> <div class="section" id="polevalreduced-polname-atexpr-numerical-expression"> -<span id="polevalreducedatexpr"></span><h2><a class="toc-backref" href="#id104"><tt class="docutils literal"><span class="pre">\PolEvalReduced{polname}\AtExpr{numerical</span> expression}</tt></a></h2> +<span id="polevalreducedatexpr"></span><h2><a class="toc-backref" href="#id105"><tt class="docutils literal"><span class="pre">\PolEvalReduced{polname}\AtExpr{numerical</span> expression}</tt></a></h2> <blockquote> Boils down to <tt class="docutils literal">\xinttheexpr reduce(polname(numerical <span class="pre">expression))\relax</span></tt>.</blockquote> </div> <div class="section" id="polevalreduced-polname-at-fraction"> -<span id="polevalreducedat"></span><h2><a class="toc-backref" href="#id105"><tt class="docutils literal"><span class="pre">\PolEvalReduced{polname}\At{fraction}</span></tt></a></h2> +<span id="polevalreducedat"></span><h2><a class="toc-backref" href="#id106"><tt class="docutils literal"><span class="pre">\PolEvalReduced{polname}\At{fraction}</span></tt></a></h2> <blockquote> Evaluates the polynomial at value <tt class="docutils literal">fraction</tt> which must be in (or expand to) a format acceptable to the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> macros, and produce an irreducible fraction.</blockquote> </div> <div class="section" id="polfloateval-polname-atexpr-numerical-expression"> -<span id="polfloatevalatexpr"></span><h2><a class="toc-backref" href="#id106"><tt class="docutils literal"><span class="pre">\PolFloatEval{polname}\AtExpr{numerical</span> expression}</tt></a></h2> +<span id="polfloatevalatexpr"></span><h2><a class="toc-backref" href="#id107"><tt class="docutils literal"><span class="pre">\PolFloatEval{polname}\AtExpr{numerical</span> expression}</tt></a></h2> <blockquote> <p>Boils down to <tt class="docutils literal">\xintthefloatexpr polname(numerical <span class="pre">expression)\relax</span></tt>.</p> <p>This is done via a Horner Scheme (see <a class="reference internal" href="#poldef">\poldef</a> and @@ -1936,14 +1984,14 @@ nested expressions.</td></tr> </blockquote> </div> <div class="section" id="polfloateval-polname-at-fraction"> -<span id="polfloatevalat"></span><h2><a class="toc-backref" href="#id107"><tt class="docutils literal"><span class="pre">\PolFloatEval{polname}\At{fraction}</span></tt></a></h2> +<span id="polfloatevalat"></span><h2><a class="toc-backref" href="#id108"><tt class="docutils literal"><span class="pre">\PolFloatEval{polname}\At{fraction}</span></tt></a></h2> <blockquote> Evaluates the polynomial at value <tt class="docutils literal">fraction</tt> which must be in (or expand to) a format acceptable to the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> macros, and produces a floating point number.</blockquote> </div> <div class="section" id="polifcoeffisplusorminusone-a-b"> -<span id="polifcoeffisplusorminusone"></span><h2><a class="toc-backref" href="#id108"><tt class="docutils literal"><span class="pre">\PolIfCoeffIsPlusOrMinusOne{A}{B}</span></tt></a></h2> +<span id="polifcoeffisplusorminusone"></span><h2><a class="toc-backref" href="#id109"><tt class="docutils literal"><span class="pre">\PolIfCoeffIsPlusOrMinusOne{A}{B}</span></tt></a></h2> <blockquote> <p>This macro is a priori undefined.</p> <p>It is defined via the default <a class="reference internal" href="#poltypesetcmd-raw-coeff">\PolTypesetCmd{raw_coeff}</a> to be @@ -1955,25 +2003,25 @@ plus or minus one, and <tt class="docutils literal">B</tt> if not.</p> </blockquote> </div> <div class="section" id="polleadingcoeff-polname"> -<span id="polleadingcoeff"></span><h2><a class="toc-backref" href="#id109"><tt class="docutils literal">\PolLeadingCoeff{polname}</tt></a></h2> +<span id="polleadingcoeff"></span><h2><a class="toc-backref" href="#id110"><tt class="docutils literal">\PolLeadingCoeff{polname}</tt></a></h2> <blockquote> Expands to the leading coefficient.</blockquote> </div> <div class="section" id="polnthcoeff-polname-number"> -<span id="polnthcoeff"></span><h2><a class="toc-backref" href="#id110"><tt class="docutils literal"><span class="pre">\PolNthCoeff{polname}{number}</span></tt></a></h2> +<span id="polnthcoeff"></span><h2><a class="toc-backref" href="#id111"><tt class="docutils literal"><span class="pre">\PolNthCoeff{polname}{number}</span></tt></a></h2> <blockquote> It expands to the raw <tt class="docutils literal">N</tt>-th coefficient (<tt class="docutils literal">0/1[0]</tt> if the index number is out of range). With <tt class="docutils literal"><span class="pre">N=-1</span></tt>, <tt class="docutils literal"><span class="pre">-2</span></tt>, ... expands to the leading coefficients.</blockquote> </div> <div class="section" id="poldegree-polname"> -<span id="poldegree"></span><h2><a class="toc-backref" href="#id111"><tt class="docutils literal">\PolDegree{polname}</tt></a></h2> +<span id="poldegree"></span><h2><a class="toc-backref" href="#id112"><tt class="docutils literal">\PolDegree{polname}</tt></a></h2> <blockquote> It expands to the degree. This is <tt class="docutils literal"><span class="pre">-1</span></tt> if zero polynomial but this may change in future. Should it then expand to <tt class="docutils literal"><span class="pre">-\infty</span></tt> ?</blockquote> </div> <div class="section" id="policontent-polname"> -<span id="policontent"></span><h2><a class="toc-backref" href="#id112"><tt class="docutils literal">\PolIContent{polname}</tt></a></h2> +<span id="policontent"></span><h2><a class="toc-backref" href="#id113"><tt class="docutils literal">\PolIContent{polname}</tt></a></h2> <blockquote> <p>It expands to the contents of the polynomial, i.e. to the positive fraction such that dividing by this fraction produces a polynomial @@ -1982,7 +2030,7 @@ with integer coefficients having no common prime divisor.</p> </blockquote> </div> <div class="section" id="poltoexpr-polname"> -<span id="poltoexpr"></span><h2><a class="toc-backref" href="#id113"><tt class="docutils literal">\PolToExpr{polname}</tt></a></h2> +<span id="poltoexpr"></span><h2><a class="toc-backref" href="#id114"><tt class="docutils literal">\PolToExpr{polname}</tt></a></h2> <blockquote> <p>Expands <a class="footnote-reference" href="#id27" id="id26">[4]</a> to <tt class="docutils literal"><span class="pre">coeff_N*x^N+...</span></tt> (descending powers.)</p> <table class="docutils footnote" frame="void" id="id27" rules="none"> @@ -2007,7 +2055,7 @@ of <tt class="docutils literal">\PolToExpr{f}</tt>, but a simple <tt class="docu the identical result.</p> </blockquote> <div class="section" id="poltoexproneterm-raw-coeff-number"> -<span id="poltoexproneterm"></span><h3><a class="toc-backref" href="#id114"><tt class="docutils literal"><span class="pre">\PolToExprOneTerm{raw_coeff}{number}</span></tt></a></h3> +<span id="poltoexproneterm"></span><h3><a class="toc-backref" href="#id115"><tt class="docutils literal"><span class="pre">\PolToExprOneTerm{raw_coeff}{number}</span></tt></a></h3> <blockquote> <p>This two argument expandable command takes care of the monomial and its coefficient. The default definition is done in order for @@ -2020,13 +2068,13 @@ or a minus one. See <a class="reference internal" href="#poltoexprtimes">\PolToE </blockquote> </div> <div class="section" id="poltoexpronetermstylea-raw-coeff-number"> -<span id="poltoexpronetermstylea"></span><h3><a class="toc-backref" href="#id115"><tt class="docutils literal"><span class="pre">\PolToExprOneTermStyleA{raw_coeff}{number}</span></tt></a></h3> +<span id="poltoexpronetermstylea"></span><h3><a class="toc-backref" href="#id116"><tt class="docutils literal"><span class="pre">\PolToExprOneTermStyleA{raw_coeff}{number}</span></tt></a></h3> <blockquote> Holds the default package meaning of <a class="reference internal" href="#poltoexproneterm-raw-coeff-number">\PolToExprOneTerm{raw_coeff}{number}</a>.</blockquote> </div> <div class="section" id="poltoexpronetermstyleb-raw-coeff-number"> -<span id="poltoexpronetermstyleb"></span><h3><a class="toc-backref" href="#id116"><tt class="docutils literal"><span class="pre">\PolToExprOneTermStyleB{raw_coeff}{number}</span></tt></a></h3> +<span id="poltoexpronetermstyleb"></span><h3><a class="toc-backref" href="#id117"><tt class="docutils literal"><span class="pre">\PolToExprOneTermStyleB{raw_coeff}{number}</span></tt></a></h3> <blockquote> <p>For output in this style:</p> <pre class="literal-block"> @@ -2040,7 +2088,7 @@ To revert to package default, issue </blockquote> </div> <div class="section" id="poltoexprcmd-raw-coeff"> -<span id="poltoexprcmd"></span><h3><a class="toc-backref" href="#id117"><tt class="docutils literal">\PolToExprCmd{raw_coeff}</tt></a></h3> +<span id="poltoexprcmd"></span><h3><a class="toc-backref" href="#id118"><tt class="docutils literal">\PolToExprCmd{raw_coeff}</tt></a></h3> <blockquote> It is the one-argument macro used by the package definition of <tt class="docutils literal">\PolToExprOneTerm</tt> for the coefficients themselves (when not @@ -2050,21 +2098,21 @@ to <tt class="docutils literal"><span class="pre">\xintIrr{#1}</span></tt> or to output forcefully reduced coefficients.</blockquote> </div> <div class="section" id="poltoexprtermprefix-raw-coeff"> -<span id="poltoexprtermprefix"></span><h3><a class="toc-backref" href="#id118"><tt class="docutils literal">\PolToExprTermPrefix{raw_coeff}</tt></a></h3> +<span id="poltoexprtermprefix"></span><h3><a class="toc-backref" href="#id119"><tt class="docutils literal">\PolToExprTermPrefix{raw_coeff}</tt></a></h3> <blockquote> Defined identically as <a class="reference internal" href="#poltypesetcmdprefix-raw-coeff">\PolTypesetCmdPrefix{raw_coeff}</a>. It prefixes with a plus sign for non-negative coefficients, because they don't carry one by themselves.</blockquote> </div> <div class="section" id="id28"> -<span id="poltoexprvar"></span><h3><a class="toc-backref" href="#id119"><tt class="docutils literal">\PolToExprVar</tt></a></h3> +<span id="poltoexprvar"></span><h3><a class="toc-backref" href="#id120"><tt class="docutils literal">\PolToExprVar</tt></a></h3> <blockquote> This expands to the variable to use in output (it does not have to be a single letter, may be an expandable macro.) Initial definition is <tt class="docutils literal">x</tt>.</blockquote> </div> <div class="section" id="id29"> -<span id="poltoexprtimes"></span><h3><a class="toc-backref" href="#id120"><tt class="docutils literal">\PolToExprTimes</tt></a></h3> +<span id="poltoexprtimes"></span><h3><a class="toc-backref" href="#id121"><tt class="docutils literal">\PolToExprTimes</tt></a></h3> <blockquote> This expands to the symbol used for multiplication of an <tt class="docutils literal"><span class="pre">x^{number}</span></tt> by the corresponding coefficient. The default is @@ -2074,13 +2122,13 @@ algebra software).</blockquote> </div> </div> <div class="section" id="id31"> -<span id="id30"></span><h2><a class="toc-backref" href="#id121"><tt class="docutils literal"><span class="pre">\PolToExpr*{polname}</span></tt></a></h2> +<span id="id30"></span><h2><a class="toc-backref" href="#id122"><tt class="docutils literal"><span class="pre">\PolToExpr*{polname}</span></tt></a></h2> <blockquote> Expands to <tt class="docutils literal"><span class="pre">coeff_0+coeff_1*x+coeff_2*x^2+...</span></tt> (ascending powers). Customizable like <a class="reference internal" href="#poltoexpr-polname">\PolToExpr{polname}</a> via the same macros.</blockquote> </div> <div class="section" id="poltofloatexpr-polname"> -<span id="poltofloatexpr"></span><h2><a class="toc-backref" href="#id122"><tt class="docutils literal">\PolToFloatExpr{polname}</tt></a></h2> +<span id="poltofloatexpr"></span><h2><a class="toc-backref" href="#id123"><tt class="docutils literal">\PolToFloatExpr{polname}</tt></a></h2> <blockquote> <p>Similar to <a class="reference internal" href="#poltoexpr-polname">\PolToExpr{polname}</a> but uses <a class="reference external" href="\PolToFloatExprCmd{raw_coeff}">\PolToFloatExprCmd</a> which by default rounds and converts the coefficients to floating @@ -2099,13 +2147,13 @@ those output by <tt class="docutils literal">\PolToFloatExpr{polname}</tt>.</p> </div> </blockquote> <div class="section" id="poltofloatexproneterm-raw-coeff-number"> -<span id="poltofloatexproneterm"></span><h3><a class="toc-backref" href="#id123"><tt class="docutils literal"><span class="pre">\PolToFloatExprOneTerm{raw_coeff}{number}</span></tt></a></h3> +<span id="poltofloatexproneterm"></span><h3><a class="toc-backref" href="#id124"><tt class="docutils literal"><span class="pre">\PolToFloatExprOneTerm{raw_coeff}{number}</span></tt></a></h3> <blockquote> Similar to <a class="reference external" href="\PolToExprOneTerm{raw_coeff}{number}">\PolToExprOneTerm</a>. But does not treat especially coefficients equal to plus or minus one.</blockquote> </div> <div class="section" id="poltofloatexprcmd-raw-coeff"> -<span id="id33"></span><h3><a class="toc-backref" href="#id124"><tt class="docutils literal">\PolToFloatExprCmd{raw_coeff}</tt></a></h3> +<span id="id33"></span><h3><a class="toc-backref" href="#id125"><tt class="docutils literal">\PolToFloatExprCmd{raw_coeff}</tt></a></h3> <blockquote> <p>It is the one-argument macro used by <tt class="docutils literal">\PolToFloatExprOneTerm</tt>. Its package definition is <tt class="docutils literal"><span class="pre">\xintFloat{#1}</span></tt>.</p> @@ -2128,12 +2176,12 @@ in <tt class="docutils literal">xintfrac</tt> raw format.</p> </div> </div> <div class="section" id="id35"> -<span id="id34"></span><h2><a class="toc-backref" href="#id125"><tt class="docutils literal"><span class="pre">\PolToFloatExpr*{polname}</span></tt></a></h2> +<span id="id34"></span><h2><a class="toc-backref" href="#id126"><tt class="docutils literal"><span class="pre">\PolToFloatExpr*{polname}</span></tt></a></h2> <blockquote> Typesets in ascending powers.</blockquote> </div> <div class="section" id="poltolist-polname"> -<span id="poltolist"></span><h2><a class="toc-backref" href="#id126"><tt class="docutils literal">\PolToList{polname}</tt></a></h2> +<span id="poltolist"></span><h2><a class="toc-backref" href="#id127"><tt class="docutils literal">\PolToList{polname}</tt></a></h2> <blockquote> Expands to <tt class="docutils literal"><span class="pre">{coeff_0}{coeff_1}...{coeff_N}</span></tt> with <tt class="docutils literal">N</tt> = degree, and <tt class="docutils literal">coeff_N</tt> the leading coefficient @@ -2141,14 +2189,14 @@ Expands to <tt class="docutils literal"><span class="pre">{coeff_0}{coeff_1}...{ empty output.)</blockquote> </div> <div class="section" id="poltocsv-polname"> -<span id="poltocsv"></span><h2><a class="toc-backref" href="#id127"><tt class="docutils literal">\PolToCSV{polname}</tt></a></h2> +<span id="poltocsv"></span><h2><a class="toc-backref" href="#id128"><tt class="docutils literal">\PolToCSV{polname}</tt></a></h2> <blockquote> Expands to <tt class="docutils literal">coeff_0, coeff_1, coeff_2, <span class="pre">.....,</span> coeff_N</tt>, starting with constant term and ending with leading coefficient. Converse to <a class="reference internal" href="#polfromcsv-polname-csv">\PolFromCSV</a>.</blockquote> </div> <div class="section" id="polsturmchainlength-sturmname"> -<span id="polsturmchainlength"></span><h2><a class="toc-backref" href="#id128"><tt class="docutils literal">\PolSturmChainLength{sturmname}</tt></a></h2> +<span id="polsturmchainlength"></span><h2><a class="toc-backref" href="#id129"><tt class="docutils literal">\PolSturmChainLength{sturmname}</tt></a></h2> <blockquote> <p>Returns the integer <tt class="docutils literal">N</tt> such that <tt class="docutils literal">sturmname_N</tt> is the last one in the Sturm chain <tt class="docutils literal">sturmname_0</tt>, <tt class="docutils literal">sturmname_1</tt>, ...</p> @@ -2156,7 +2204,7 @@ in the Sturm chain <tt class="docutils literal">sturmname_0</tt>, <tt class="doc </blockquote> </div> <div class="section" id="polsturmifzeroexactlyknown-sturmname-index-a-b"> -<span id="polsturmifzeroexactlyknown"></span><h2><a class="toc-backref" href="#id129"><tt class="docutils literal"><span class="pre">\PolSturmIfZeroExactlyKnown{sturmname}{index}{A}{B}</span></tt></a></h2> +<span id="polsturmifzeroexactlyknown"></span><h2><a class="toc-backref" href="#id130"><tt class="docutils literal"><span class="pre">\PolSturmIfZeroExactlyKnown{sturmname}{index}{A}{B}</span></tt></a></h2> <blockquote> <p>Executes <tt class="docutils literal">A</tt> if the <tt class="docutils literal">index</tt>-th interval reduces to a singleton, i.e. the root is known exactly, else <tt class="docutils literal">B</tt>.</p> @@ -2168,7 +2216,7 @@ to <tt class="docutils literal"><span class="pre">\the\numexpr...\relax</span></ </blockquote> </div> <div class="section" id="polsturmisolatedzeroleft-sturmname-index"> -<span id="polsturmisolatedzeroleft"></span><h2><a class="toc-backref" href="#id130"><tt class="docutils literal"><span class="pre">\PolSturmIsolatedZeroLeft{sturmname}{index}</span></tt></a></h2> +<span id="polsturmisolatedzeroleft"></span><h2><a class="toc-backref" href="#id131"><tt class="docutils literal"><span class="pre">\PolSturmIsolatedZeroLeft{sturmname}{index}</span></tt></a></h2> <blockquote> <p>Expands to the left end-point for the <tt class="docutils literal">index</tt>-th interval, as computed by some earlier <a class="reference internal" href="#polsturmisolatezeros-sturmname">\PolSturmIsolateZeros{sturmname}</a>.</p> @@ -2181,7 +2229,7 @@ computed by some earlier <a class="reference internal" href="#polsturmisolatezer </blockquote> </div> <div class="section" id="polsturmisolatedzeroright-sturmname-index"> -<span id="polsturmisolatedzeroright"></span><h2><a class="toc-backref" href="#id131"><tt class="docutils literal"><span class="pre">\PolSturmIsolatedZeroRight{sturmname}{index}</span></tt></a></h2> +<span id="polsturmisolatedzeroright"></span><h2><a class="toc-backref" href="#id132"><tt class="docutils literal"><span class="pre">\PolSturmIsolatedZeroRight{sturmname}{index}</span></tt></a></h2> <blockquote> <p>Expands to the right end-point for the <tt class="docutils literal">index</tt>-th interval as computed by some earlier <a class="reference internal" href="#polsturmisolatezeros-sturmname">\PolSturmIsolateZeros{sturmname}</a> and @@ -2190,7 +2238,7 @@ possibly refined afterwards.</p> </blockquote> </div> <div class="section" id="polsturmisolatedzeromultiplicity-sturmname-index"> -<span id="polsturmisolatedzeromultiplicity"></span><h2><a class="toc-backref" href="#id132"><tt class="docutils literal"><span class="pre">\PolSturmIsolatedZeroMultiplicity{sturmname}{index}</span></tt></a></h2> +<span id="polsturmisolatedzeromultiplicity"></span><h2><a class="toc-backref" href="#id133"><tt class="docutils literal"><span class="pre">\PolSturmIsolatedZeroMultiplicity{sturmname}{index}</span></tt></a></h2> <blockquote> <p>Expands to the multiplicity of the unique root contained in the <tt class="docutils literal">index</tt>-th interval.</p> @@ -2203,7 +2251,7 @@ roots</a> for an example of use.</p> </blockquote> </div> <div class="section" id="polsturmnbofisolatedzeros-sturmname"> -<span id="polsturmnbofisolatedzeros"></span><h2><a class="toc-backref" href="#id133"><tt class="docutils literal">\PolSturmNbOfIsolatedZeros{sturmname}</tt></a></h2> +<span id="polsturmnbofisolatedzeros"></span><h2><a class="toc-backref" href="#id134"><tt class="docutils literal">\PolSturmNbOfIsolatedZeros{sturmname}</tt></a></h2> <blockquote> Expands to the number of real roots of the polynomial <tt class="docutils literal"><sturmname>_0</tt>, i.e. the number of distinct real roots of the @@ -2218,7 +2266,7 @@ enough. They can be re-coded at user level on the basis of the other documented package macros anyway.</p> </div> <div class="section" id="polsturmnbofrootsof-sturmname-lessthanorequalto-value"> -<h3><a class="toc-backref" href="#id134"><tt class="docutils literal"><span class="pre">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualTo{value}</span></tt></a></h3> +<h3><a class="toc-backref" href="#id135"><tt class="docutils literal"><span class="pre">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualTo{value}</span></tt></a></h3> <blockquote> <p>Expands to the number of distinct roots (of the polynomial used to create the Sturm chain) less than or equal to the <tt class="docutils literal">value</tt> (i.e. a @@ -2234,7 +2282,7 @@ of the above constraint.</p> </blockquote> </div> <div class="section" id="polsturmnbofrootsof-sturmname-lessthanorequaltoexpr-expression"> -<h3><a class="toc-backref" href="#id135"><tt class="docutils literal"><span class="pre">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualToExpr{expression}</span></tt></a></h3> +<h3><a class="toc-backref" href="#id136"><tt class="docutils literal"><span class="pre">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualToExpr{expression}</span></tt></a></h3> <blockquote> <p>Expands to the number of distinct roots (of the polynomial used to create the Sturm chain) which are less than or equal to the @@ -2247,7 +2295,7 @@ beforehand.</p> </blockquote> </div> <div class="section" id="polsturmnbwithmultofrootsof-sturmname-lessthanorequalto-value"> -<h3><a class="toc-backref" href="#id136"><tt class="docutils literal"><span class="pre">\PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualTo{value}</span></tt></a></h3> +<h3><a class="toc-backref" href="#id137"><tt class="docutils literal"><span class="pre">\PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualTo{value}</span></tt></a></h3> <blockquote> <p>Expands to the number counted with multiplicities of the roots (of the polynomial used to create the Sturm chain) which are less than @@ -2260,7 +2308,7 @@ variant) must have been executed beforehand.</p> </blockquote> </div> <div class="section" id="polsturmnbwithmultofrootsof-sturmname-lessthanorequaltoexpr-expression"> -<h3><a class="toc-backref" href="#id137"><tt class="docutils literal"><span class="pre">\PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualToExpr{expression}</span></tt></a></h3> +<h3><a class="toc-backref" href="#id138"><tt class="docutils literal"><span class="pre">\PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualToExpr{expression}</span></tt></a></h3> <blockquote> <p>Expands to the total number of roots (counted with multiplicities) which are less than or equal to the given <tt class="docutils literal">expression</tt>.</p> @@ -2273,7 +2321,7 @@ variant) must have been executed beforehand.</p> </div> </div> <div class="section" id="polsturmnbofrationalroots-sturmname"> -<h2><a class="toc-backref" href="#id138"><tt class="docutils literal">\PolSturmNbOfRationalRoots{sturmname}</tt></a></h2> +<h2><a class="toc-backref" href="#id139"><tt class="docutils literal">\PolSturmNbOfRationalRoots{sturmname}</tt></a></h2> <blockquote> <p>Expands to the number of rational roots (without multiplicities).</p> <div class="admonition attention"> @@ -2284,7 +2332,7 @@ beforehand.</p> </blockquote> </div> <div class="section" id="polsturmnbofrationalrootswithmultiplicities-sturmname"> -<h2><a class="toc-backref" href="#id139"><tt class="docutils literal">\PolSturmNbOfRationalRootsWithMultiplicities{sturmname}</tt></a></h2> +<h2><a class="toc-backref" href="#id140"><tt class="docutils literal">\PolSturmNbOfRationalRootsWithMultiplicities{sturmname}</tt></a></h2> <blockquote> <p>Expands to the number of rational roots (counted with multiplicities).</p> <div class="admonition attention"> @@ -2295,7 +2343,7 @@ beforehand.</p> </blockquote> </div> <div class="section" id="polsturmrationalroot-sturmname-k"> -<h2><a class="toc-backref" href="#id140"><tt class="docutils literal"><span class="pre">\PolSturmRationalRoot{sturmname}{k}</span></tt></a></h2> +<h2><a class="toc-backref" href="#id141"><tt class="docutils literal"><span class="pre">\PolSturmRationalRoot{sturmname}{k}</span></tt></a></h2> <blockquote> <p>Expands to the <tt class="docutils literal">k</tt>th rational root (they are ordered and indexed starting at 1 for the most negative).</p> @@ -2307,7 +2355,7 @@ beforehand.</p> </blockquote> </div> <div class="section" id="polsturmrationalrootindex-sturmname-k"> -<h2><a class="toc-backref" href="#id141"><tt class="docutils literal"><span class="pre">\PolSturmRationalRootIndex{sturmname}{k}</span></tt></a></h2> +<h2><a class="toc-backref" href="#id142"><tt class="docutils literal"><span class="pre">\PolSturmRationalRootIndex{sturmname}{k}</span></tt></a></h2> <blockquote> <p>Expands to <tt class="docutils literal">index</tt> of the <tt class="docutils literal">k</tt>th rational root as part of the ordered real roots (without multiplicities). I.e., above macro @@ -2324,7 +2372,7 @@ beforehand.</p> </blockquote> </div> <div class="section" id="polsturmrationalrootmultiplicity-sturmname-k"> -<h2><a class="toc-backref" href="#id142"><tt class="docutils literal"><span class="pre">\PolSturmRationalRootMultiplicity{sturmname}{k}</span></tt></a></h2> +<h2><a class="toc-backref" href="#id143"><tt class="docutils literal"><span class="pre">\PolSturmRationalRootMultiplicity{sturmname}{k}</span></tt></a></h2> <blockquote> <p>Expands to the multiplicity of the <tt class="docutils literal">k</tt>th rational root.</p> <div class="admonition attention"> @@ -2335,13 +2383,13 @@ beforehand.</p> </blockquote> </div> <div class="section" id="polintervalwidth-sturmname-index"> -<span id="polintervalwidth"></span><h2><a class="toc-backref" href="#id143"><tt class="docutils literal"><span class="pre">\PolIntervalWidth{sturmname}{index}</span></tt></a></h2> +<span id="polintervalwidth"></span><h2><a class="toc-backref" href="#id144"><tt class="docutils literal"><span class="pre">\PolIntervalWidth{sturmname}{index}</span></tt></a></h2> <blockquote> The <tt class="docutils literal">10^E</tt> width of the current <tt class="docutils literal">index</tt>-th root localization interval. Output is in <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> raw <tt class="docutils literal">1/1[E]</tt> format (if not zero).</blockquote> </div> <div class="section" id="expandable-macros-for-use-within-execution-of-polprintintervals"> -<h2><a class="toc-backref" href="#id144">Expandable macros for use within execution of <tt class="docutils literal">\PolPrintIntervals</tt></a></h2> +<h2><a class="toc-backref" href="#id145">Expandable macros for use within execution of <tt class="docutils literal">\PolPrintIntervals</tt></a></h2> <p>These macros are for usage within custom user redefinitions of <a class="reference internal" href="#polprintintervalsknownroot">\PolPrintIntervalsKnownRoot</a>, <a class="reference internal" href="#polprintintervalsunknownroot">\PolPrintIntervalsUnknownRoot</a>, or in redefinitions of <a class="reference internal" href="#polprintintervalsprintexactzero">PolPrintIntervalsPrintExactZero</a> (used in the @@ -2357,26 +2405,26 @@ latter).</p> <tt class="docutils literal"><span class="pre">\PolIfEndPointIsZero{A}{B}</span></tt>.</p> </div> <div class="section" id="polprintintervalsthevar"> -<h3><a class="toc-backref" href="#id145"><tt class="docutils literal">\PolPrintIntervalsTheVar</tt></a></h3> +<h3><a class="toc-backref" href="#id146"><tt class="docutils literal">\PolPrintIntervalsTheVar</tt></a></h3> <blockquote> Expands to the name (default <tt class="docutils literal">Z</tt>) used for representing the roots, which was passed as optional argument <tt class="docutils literal">varname</tt> to <a class="reference internal" href="#polprintintervals-varname-sturmname">\PolPrintIntervals[varname]{sturmname}</a>.</blockquote> </div> <div class="section" id="polprintintervalstheindex"> -<h3><a class="toc-backref" href="#id146"><tt class="docutils literal">\PolPrintIntervalsTheIndex</tt></a></h3> +<h3><a class="toc-backref" href="#id147"><tt class="docutils literal">\PolPrintIntervalsTheIndex</tt></a></h3> <blockquote> Expands to the index of the considered interval (indexing starting at 1 for the leftmost interval).</blockquote> </div> <div class="section" id="polprintintervalsthesturmname"> -<h3><a class="toc-backref" href="#id147"><tt class="docutils literal">\PolPrintIntervalsTheSturmName</tt></a></h3> +<h3><a class="toc-backref" href="#id148"><tt class="docutils literal">\PolPrintIntervalsTheSturmName</tt></a></h3> <blockquote> Expands to the argument which was passed as <tt class="docutils literal">sturmname</tt> to <a class="reference internal" href="#polprintintervals-varname-sturmname">\PolPrintIntervals[varname]{sturmname}</a>.</blockquote> </div> <div class="section" id="polprintintervalstheleftendpoint"> -<h3><a class="toc-backref" href="#id148"><tt class="docutils literal">\PolPrintIntervalsTheLeftEndPoint</tt></a></h3> +<h3><a class="toc-backref" href="#id149"><tt class="docutils literal">\PolPrintIntervalsTheLeftEndPoint</tt></a></h3> <blockquote> The left end point of the interval, as would be produced by <a class="reference internal" href="#polsturmisolatedzeroleft">\PolSturmIsolatedZeroLeft</a> if it was @@ -2385,14 +2433,14 @@ by <a class="reference internal" href="#polprintintervalsthesturmname">\PolPrint <a class="reference internal" href="#polprintintervalstheindex">\PolPrintIntervalsTheIndex</a>.</blockquote> </div> <div class="section" id="polprintintervalstherightendpoint"> -<h3><a class="toc-backref" href="#id149"><tt class="docutils literal">\PolPrintIntervalsTheRightEndPoint</tt></a></h3> +<h3><a class="toc-backref" href="#id150"><tt class="docutils literal">\PolPrintIntervalsTheRightEndPoint</tt></a></h3> <blockquote> The right end point of the interval, as would be produced by <a class="reference internal" href="#polsturmisolatedzeroright">\PolSturmIsolatedZeroRight</a> for this Sturm chain name and index.</blockquote> </div> <div class="section" id="polprintintervalsthemultiplicity"> -<h3><a class="toc-backref" href="#id150"><tt class="docutils literal">\PolPrintIntervalsTheMultiplicity</tt></a></h3> +<h3><a class="toc-backref" href="#id151"><tt class="docutils literal">\PolPrintIntervalsTheMultiplicity</tt></a></h3> <blockquote> The multiplicity of the unique root within the interval of index <a class="reference internal" href="#polprintintervalstheindex">\PolPrintIntervalsTheIndex</a>. Makes sense only if the starred (or @@ -2400,7 +2448,7 @@ double-starred) variant of <a class="reference internal" href="#polsturmisolatez </div> </div> <div class="section" id="poldectostring-decimal-number"> -<span id="poldectostring"></span><h2><a class="toc-backref" href="#id151"><tt class="docutils literal">\PolDecToString{decimal number}</tt></a></h2> +<span id="poldectostring"></span><h2><a class="toc-backref" href="#id152"><tt class="docutils literal">\PolDecToString{decimal number}</tt></a></h2> <blockquote> <p>This is a utility macro to print decimal numbers. It has been backported to <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> (release <tt class="docutils literal">1.3</tt> of <tt class="docutils literal">2018/03/01</tt>) under @@ -2417,9 +2465,9 @@ zeroes, one can use <tt class="docutils literal"><span class="pre">\PolDecToStri </div> </div> <div class="section" id="booleans-with-default-setting-as-indicated"> -<h1><a class="toc-backref" href="#id152">Booleans (with default setting as indicated)</a></h1> +<h1><a class="toc-backref" href="#id153">Booleans (with default setting as indicated)</a></h1> <div class="section" id="xintverbosefalse"> -<h2><a class="toc-backref" href="#id153"><tt class="docutils literal">\xintverbosefalse</tt></a></h2> +<h2><a class="toc-backref" href="#id154"><tt class="docutils literal">\xintverbosefalse</tt></a></h2> <blockquote> <p>This is actually an <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> configuration. Setting it to <tt class="docutils literal">true</tt> triggers the writing of information to the log when new @@ -2432,20 +2480,20 @@ unstable and undocumented internal structures.</p> </blockquote> </div> <div class="section" id="poltypesetallfalse"> -<h2><a class="toc-backref" href="#id154"><tt class="docutils literal">\poltypesetallfalse</tt></a></h2> +<h2><a class="toc-backref" href="#id155"><tt class="docutils literal">\poltypesetallfalse</tt></a></h2> <blockquote> If <tt class="docutils literal">true</tt>, <a class="reference internal" href="#poltypeset-polname">\PolTypeset{polname}</a> will also typeset the vanishing coefficients.</blockquote> </div> <div class="section" id="poltoexprallfalse"> -<h2><a class="toc-backref" href="#id155"><tt class="docutils literal">\poltoexprallfalse</tt></a></h2> +<h2><a class="toc-backref" href="#id156"><tt class="docutils literal">\poltoexprallfalse</tt></a></h2> <blockquote> If <tt class="docutils literal">true</tt>, <a class="reference internal" href="#poltoexpr-polname">\PolToExpr{polname}</a> and <a class="reference internal" href="#poltofloatexpr-polname">\PolToFloatExpr{polname}</a> will also include the vanishing coefficients in their outputs.</blockquote> </div> </div> <div class="section" id="polexprsetup"> -<h1><a class="toc-backref" href="#id156"><tt class="docutils literal">\polexprsetup</tt></a></h1> +<h1><a class="toc-backref" href="#id157"><tt class="docutils literal">\polexprsetup</tt></a></h1> <blockquote> <p>Serves to customize the package. Currently only two keys are recognized:</p> @@ -2466,7 +2514,7 @@ sqfnorr=_sqf_norr}</tt> as default.</p> </blockquote> </div> <div class="section" id="technicalities"> -<h1><a class="toc-backref" href="#id157">Technicalities</a></h1> +<h1><a class="toc-backref" href="#id158">Technicalities</a></h1> <ul> <li><p class="first">The catcode of the semi-colon is reset temporarily by <a class="reference internal" href="#poldef">\poldef</a> macro in case some other package (for example the French babel module) may have made it active. This will fail though if the @@ -2523,7 +2571,7 @@ documented and unstable. Don't use them.</p> </ul> </div> <div class="section" id="change-log"> -<h1><a class="toc-backref" href="#id158">CHANGE LOG</a></h1> +<h1><a class="toc-backref" href="#id159">CHANGE LOG</a></h1> <ul> <li><p class="first">v0.1 (2018/01/11): initial release. Features:</p> <ul class="simple"> @@ -2768,7 +2816,11 @@ acts like the <a class="reference internal" href="#polsturmisolatezeros">non-sta </li> </ul> </li> -<li><p class="first">v0.7 (2018/12/08)</p> +<li><p class="first">v0.7 (2018/12/08) (not released)</p> +</li> +<li><p class="first">v0.7.1 (2018/12/09) (bugfix)</p> +</li> +<li><p class="first">v0.7.2 (2018/12/09)</p> <ul class="simple"> <li>breaking changes:<ul> <li>although <a class="reference internal" href="#polprintintervals-varname-sturmname">\PolPrintIntervals[varname]{sturmname}</a> default output @@ -2821,7 +2873,7 @@ existed (thanks to Thomas Söll for report).</li> </ul> </div> <div class="section" id="acknowledgments"> -<h1><a class="toc-backref" href="#id159">Acknowledgments</a></h1> +<h1><a class="toc-backref" href="#id160">Acknowledgments</a></h1> <p>Thanks to Jürgen Gilg whose question about <a class="reference external" href="http://www.ctan.org/pkg/xint">xint</a> usage for differentiating polynomials was the initial trigger leading to this package, and to Jürgen Gilg and Thomas Söll for testing it on some diff --git a/Master/texmf-dist/doc/latex/polexpr/polexpr.txt b/Master/texmf-dist/doc/latex/polexpr/polexpr.txt index 51008763dcf..d77361edc4a 100644 --- a/Master/texmf-dist/doc/latex/polexpr/polexpr.txt +++ b/Master/texmf-dist/doc/latex/polexpr/polexpr.txt @@ -4,8 +4,8 @@ Package polexpr documentation =============================== -0.7 (2018/12/08) -================ +0.7.2 (2018/12/09) +================== .. contents:: @@ -334,7 +334,7 @@ A degree five polynomial with three rational roots \PolSturmIsolateZeros**{Q} \PolPrintIntervals{Q} - $Q_norr(x) = \PolTypeset{Q_norr}$ + $Q_{norr}(x) = \PolTypeset{Q_norr}$ Here, all real roots are rational:: @@ -367,7 +367,7 @@ A Mignotte type polynomial \PolSturmIsolateZeros{P} % finds 4 real roots This polynomial has \PolSturmNbOfIsolatedZeros{P} distinct real roots: \PolPrintIntervals{P}% - % reports -2 < Z_1 < -1, 0 < Z_2 < 0.1, 0.1 < Z_3 < 0.2, 1 < Z_4 < 2 + % reports -2 < Z_1 < -1, 0.09 < Z_2 < 0.10, 0.1 < Z_3 < 0.2, 1 < Z_4 < 2 Let us refine the second and third intervals to separate the corresponding roots: \PolRefineInterval*{P}{2}% will refine to 0.0999990 < Z_2 < 0.0999991 @@ -470,6 +470,56 @@ decimal digits is relatively faster. Here is for the ten real roots of Z_9 = 8.9172502485... Z_10 = 20.8469081014... +The second Wilkinson polynomial +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +:: + + \documentclass{article} + \usepackage{polexpr} + \begin{document} + \poldef f(x) := mul(x - 2^-i, i = 1..20); + + %\PolTypeset{f} + + \PolToSturm{f}{f} + \PolSturmIsolateZeros**{f} + \PolPrintIntervals{f} + \end{document} + +This takes more time than the polynomial with 1, 2, .., 20 as roots but +less than the latter modified by the ``2**-23`` change in one +coefficient. + +Here is the output (with release 0.7.2):: + + Z_1 = 0.00000095367431640625 + Z_2 = 0.0000019073486328125 + Z_3 = 0.000003814697265625 + Z_4 = 0.00000762939453125 + Z_5 = 0.0000152587890625 + Z_6 = 0.000030517578125 + Z_7 = 0.00006103515625 + Z_8 = 0.0001220703125 + Z_9 = 1/4096 + Z_10 = 1/2048 + Z_11 = 1/1024 + Z_12 = 1/512 + Z_13 = 1/256 + Z_14 = 1/128 + Z_15 = 0.015625 + Z_16 = 0.03125 + Z_17 = 0.0625 + Z_18 = 0.125 + Z_19 = 0.25 + Z_20 = 0.5 + +There is some incoherence in output format which has its source in the +fact that some roots are found in branches which can only find decimal +roots, whereas some are found in branches which could find general +fractions and they use ``\xintIrr`` before storage of the found root. +This may evolve in future. + The degree 41 polynomial with -2, -1.9, -1.8, ..., 0, 0.1, ..., 1.9, 2 as roots ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ @@ -961,7 +1011,7 @@ Non-expandable macros The polynomials ``sturmname_k`` main utility is for the execution of `\\PolSturmIsolateZeros{sturmname}`_. Be careful not to use these names ``sturmname_0``, ``sturmname_1``, etc... for defining other - polynomials after having done ``\PolToSturm{polname}{sturmname} and + polynomials after having done ``\PolToSturm{polname}{sturmname}`` and before executing ``\PolSturmIsolateZeros{sturmname}`` else the latter will behave erroneously. @@ -2448,7 +2498,11 @@ CHANGE LOG - `\\PolSturmNbWithMultOfRootsOf{sturmname}\\LessThanOrEqualTo{value}`_ - `\\PolSturmNbWithMultOfRootsOf{sturmname}\\LessThanOrEqualToExpr{expression}`_ -- v0.7 (2018/12/08) +- v0.7 (2018/12/08) (not released) + +- v0.7.1 (2018/12/09) (bugfix) + +- v0.7.2 (2018/12/09) * breaking changes: diff --git a/Master/texmf-dist/tex/latex/polexpr/polexpr.sty b/Master/texmf-dist/tex/latex/polexpr/polexpr.sty index 601446cf671..9bc286115d8 100644 --- a/Master/texmf-dist/tex/latex/polexpr/polexpr.sty +++ b/Master/texmf-dist/tex/latex/polexpr/polexpr.sty @@ -1,7 +1,7 @@ % author: Jean-François Burnol % License: LPPL 1.3c (author-maintained) \ProvidesPackage{polexpr}% - [2018/12/08 v0.7 Polynomial expressions with rational coefficients (JFB)]% + [2018/12/09 v0.7.2 Polynomial expressions with rational coefficients (JFB)]% \RequirePackage{xintexpr}[2018/06/17]% xint 1.3c for \ifxintglobaldefs boolean \edef\POL@restorecatcodes {\catcode`\noexpand\_ \the\catcode`\_ % @@ -914,12 +914,13 @@ {\let\POL@findrat@E\POL@findrat@Dexp} % aussi ok pour 1[0] {\edef\POL@findrat@E{\the\numexpr\xintLen{\POL@findrat@Dint}% +\POL@findrat@Dexp}}% +% ATTENTION QUE LA CONVENTION DE SIGNE POUR \POL@findrat@E EST OPPOSÉE À CELLE +% POUR LE CODE PLUS ANCIEN FAISANT "REFINE" \POL@initarray\POL@IfMultIsKnown\xint_secondoftwo \let\POL@findrat@nbofirrroots\POL@isolz@NbOfRoots % find all rational roots, and their multiplicities, % factor them out in passing from original (Sturm root) polynomial \ifnum\POL@findrat@E<7 - \PolEnsureIntervalLength{\POL@sturmname}{1}{-\POL@findrat@E}% \def\POL@findrat@index{1}% \POL@findrat@loop@secondpass@direct \else @@ -928,7 +929,6 @@ \POL@findrat@loop@firstpass % and now we do the final pass finding them all \def\POL@findrat@index{1}% - \PolEnsureIntervalLength{\POL@sturmname}{1}{-\POL@findrat@E}% \POL@findrat@loop@secondpass \fi % declare the new polynomials @@ -947,7 +947,7 @@ {}% \ifnum\xintiloopindex<\POL@isolz@NbOfRoots\space \repeat }% - \begingroup\globaldefs1 + \begingroup\globaldefs\@ne % attention de ne surtout pas faire un \expandafter ici, car en cas d'un % seul item, \xintAssignArray l'unbraces... \xintAssignArray\POL@temp\to#1% @@ -956,36 +956,20 @@ \def\POL@findrat@loop@firstpass{% \PolSturmIfZeroExactlyKnown{\POL@sturmname}{\POL@findrat@index}% \POL@findrat@loop@decimal% get its multiplicity - \POL@findrat@loop@a + \POL@findrat@loop@aa % refine interval and check \edef\POL@findrat@index{\the\numexpr\POL@findrat@index+\@ne}% \ifnum\POL@findrat@index>\POL@isolz@NbOfRoots \else \expandafter\POL@findrat@loop@firstpass \fi }% -\def\POL@findrat@loop@secondpass{% +\def\POL@findrat@loop@aa{% + % we do a first pass to identify roots with denominators < 1000 + \PolEnsureIntervalLength{\POL@sturmname}{\POL@findrat@index}{-6}% + % attention that perhaps now the root is known! \PolSturmIfZeroExactlyKnown{\POL@sturmname}{\POL@findrat@index}% - {}% nothing more to be done, already stored - \POL@findrat@loop@b - \edef\POL@findrat@index{\the\numexpr\POL@findrat@index+\@ne}% - \ifnum\POL@findrat@index>\POL@isolz@NbOfRoots - \else - \PolEnsureIntervalLength - {\POL@sturmname}{\POL@findrat@index}{-\POL@findrat@E}% dynamic - \expandafter\POL@findrat@loop@secondpass - \fi -}% -\def\POL@findrat@loop@secondpass@direct{% - \PolSturmIfZeroExactlyKnown{\POL@sturmname}{\POL@findrat@index}% - \POL@findrat@loop@decimal - \POL@findrat@loop@b - \edef\POL@findrat@index{\the\numexpr\POL@findrat@index+\@ne}% - \ifnum\POL@findrat@index>\POL@isolz@NbOfRoots - \else - \PolEnsureIntervalLength - {\POL@sturmname}{\POL@findrat@index}{-\POL@findrat@E}% dynamic - \expandafter\POL@findrat@loop@secondpass@direct - \fi + \POL@findrat@loop@decimal + \POL@findrat@loop@a }% \def\POL@findrat@loop@decimal{% we have an already found decimal root % we do not go via @storeit, as it is already stored @@ -1002,6 +986,7 @@ \edef\POLuserpol@_findrat@oneterm{1.\noexpand\empty {\xintiiOpp\POL@findrat@xN/1[0]}{\POL@findrat@xD/1[0]}}% \POL@divide{\POL@sturmname\POL@sqfnorr}{_findrat@oneterm}% the one without mult. + %\expandafter\POL@split\POL@R;\POL@degR\POL@polR \POL@findrat@loop@updatequotients \POL@findrat@loop@getmultiplicity }% @@ -1018,10 +1003,7 @@ \def\POL@findrat@getE #1/1[#2]{#2}% /1 as it should be there. % so an error will arise if not but cf \POL@refine@getE where I did not put it \def\POL@findrat@loop@a{% - % we do a first pass to identify roots with denominators < 1000 - \PolEnsureIntervalLength{\POL@sturmname}{\POL@findrat@index}{-6}% % attention that the width may have been already smaller than 10^{-6} - % also attention that one of the bound may be zero \POL@get@IsoLeft@rawin \POL@get@IsoRight@rawin \edef\POL@findrat@localW @@ -1046,28 +1028,20 @@ \fi }% \def\POL@findrat@gcdloop{% - % we must be careful with sign (attention one of the bounds may be zero) - \let\POL@findrat@IfNeg\xint_secondoftwo + % we must be careful with sign + % but we are certain no extremity is a root + \let\POL@findrat@ifnegative\xint_secondoftwo \xintiiifSgn\POL@IsoLeft@Int \POL@findrat@gcdloop@n - \POL@findrat@gcdloop@zero + \POL@error@thisisimpossible \POL@findrat@gcdloop@p }% \def\POL@findrat@gcdloop@n{% - \let\POL@findrat@IfNeg\xint_firstoftwo + \let\POL@findrat@ifnegative\xint_firstoftwo \let\POL@temp\POL@IsoRight@Int \edef\POL@IsoRight@Int{\xintiiOpp{\POL@IsoLeft@Int}}% \edef\POL@IsoLeft@Int{\xintiiOpp{\POL@temp}}% - \xintiiifSgn\POL@IsoLeft@Int - \POL@error % impossible branch - \POL@findrat@gcdloop@zero - \POL@findrat@gcdloop@p -}% -\def\POL@findrat@gcdloop@zero{% - % the continued fraction would be the one of 1/2, so only 1/2 to test... - \edef\POL@findrat@x - {1/2\romannumeral\xintreplicate{\POL@findrat@localW}{0}[0]}% - \POL@findrat@gcdloop@testit + \POL@findrat@gcdloop@p }% \def\POL@findrat@gcdloop@p{% \edef\POL@findrat@gcdloop@Ap{\xintDec{\xintDouble\POL@IsoRight@Int}}% @@ -1140,7 +1114,7 @@ }% \def\POL@findrat@gcdloop@testit{% % zero should never occur here - \POL@findrat@IfNeg{\edef\POL@findrat@x{-\POL@findrat@x}}{}% + \POL@findrat@ifnegative{\edef\POL@findrat@x{-\POL@findrat@x}}{}% \POL@xintfrac@getNDE\POL@findrat@x\POL@findrat@xN\POL@findrat@xD\POL@_ \edef\POLuserpol@_findrat@oneterm{1.\noexpand\empty {\xintiiOpp{\POL@findrat@xN}/1[0]}{\POL@findrat@xD/1[0]}}% @@ -1151,10 +1125,36 @@ \POL@findrat@loop@updatequotients \POL@findrat@loop@getmultiplicity % will continue updating the mult. one \def\POL@findrat@gcdloop@done{1}% - \else \fi }% % This is second phase +\def\POL@findrat@loop@secondpass{% + \PolSturmIfZeroExactlyKnown{\POL@sturmname}{\POL@findrat@index}% + {}% nothing more to be done, already stored + \POL@findrat@loop@bb % refine interval and check + \edef\POL@findrat@index{\the\numexpr\POL@findrat@index+\@ne}% + \ifnum\POL@findrat@index>\POL@isolz@NbOfRoots + \else + \expandafter\POL@findrat@loop@secondpass + \fi +}% +\def\POL@findrat@loop@secondpass@direct{% + \PolSturmIfZeroExactlyKnown{\POL@sturmname}{\POL@findrat@index}% + \POL@findrat@loop@decimal + \POL@findrat@loop@bb + \edef\POL@findrat@index{\the\numexpr\POL@findrat@index+\@ne}% + \ifnum\POL@findrat@index>\POL@isolz@NbOfRoots + \else + \expandafter\POL@findrat@loop@secondpass@direct + \fi +}% +\def\POL@findrat@loop@bb{% + \PolEnsureIntervalLength{\POL@sturmname}{\POL@findrat@index}{-\POL@findrat@E}% + % ATTENTION THAT PERHAPS NOW THE ROOT IS KNOWN! + \PolSturmIfZeroExactlyKnown{\POL@sturmname}{\POL@findrat@index}% + \POL@findrat@loop@decimal + \POL@findrat@loop@b +}% \def\POL@findrat@loop@b{% \edef\POL@findrat@Lscaled{\xintMul{\POL@findrat@D}% {\POL@xintexprGetVar{\POL@sturmname L_\POL@findrat@index}}}% @@ -1206,11 +1206,13 @@ \let\csname POL_ZeroIsKnown\POL@sturmname\POL@findrat@index\endcsname \xint_stop_atfirstoftwo \begingroup\xintglobaldefstrue - \xintdefvar - \POL@sturmname L_\POL@findrat@index,% - \POL@sturmname R_\POL@findrat@index,% - \POL@sturmname Z_\POL@findrat@index _isknown - := qfrac(\POL@findrat@x),qfrac(\POL@findrat@x),1;% + % skip some overhead of \xintdefvar... + \XINT_expr_defvar_one{\POL@sturmname L_\POL@findrat@index}% + {\csname .=\POL@findrat@x\endcsname}% + \XINT_expr_defvar_one{\POL@sturmname R_\POL@findrat@index}% + {\csname .=\POL@findrat@x\endcsname}% + \XINT_expr_defvar_one{\POL@sturmname Z_\POL@findrat@index _isknown}% + {\csname .=1\endcsname}% \endgroup }% \def\POL@findrat@loop@updatequotients{% @@ -1320,8 +1322,9 @@ % added at 0.7 \let\x\POL@isolz@NbOfRoots \xintloop - \xintdefvar \POL@sturmname M_\x - := \csname POL_ZeroMult\POL@sturmname\x\endcsname ;% + % skip some overhead of \xintdefvar... + \XINT_expr_defvar_one{\POL@sturmname M_\x}% + {\csname .=\csname POL_ZeroMult\POL@sturmname\x\endcsname\endcsname}% \edef\x{\the\numexpr\x-\@ne}% \ifnum\x>\z@ \repeat @@ -1694,11 +1697,11 @@ \repeat }% \def\POL@isolz@check{% \POL@IsoRightSign must be ready for use here -% \ifxintverbose -% \xintMessage{polexpr}{Info}% -% {\the\numexpr\POL@IsoLeftSV-\POL@IsoRightSV\relax\space roots -% in (\POL@IsoLeft@raw,\POL@IsoRight@raw] (E = \POL@isolz@E)}% -% \fi +\ifxintverbose + \xintMessage{polexpr}{Info}% + {\the\numexpr\POL@IsoLeftSV-\POL@IsoRightSV\relax\space roots + in (\POL@IsoLeft@raw,\POL@IsoRight@raw] (E = \POL@isolz@E)}% +\fi \ifcase\numexpr\POL@IsoLeftSV-\POL@IsoRightSV\relax % no root in ]left, right] \global\POL@isolz@nextwillneedrefinefalse @@ -1896,18 +1899,16 @@ \xint_stop_atfirstoftwo \fi \begingroup\xintglobaldefstrue - \xintdefvar - \POL@sturmname L_\POL@isolz@IntervalIndex,% - \POL@sturmname R_\POL@isolz@IntervalIndex,% + % skip some overhead of \xintdefvar... + \XINT_expr_defvar_one{\POL@sturmname L_\POL@isolz@IntervalIndex}% + {\csname .=\POL@IsoLeft@rawout\endcsname}% + \XINT_expr_defvar_one{\POL@sturmname R_\POL@isolz@IntervalIndex}% + {\csname .=\POL@IsoRight@rawout\endcsname}% % added at 0.7 - \POL@sturmname Z_\POL@isolz@IntervalIndex _isknown - := qfrac(\POL@IsoLeft@rawout),% - qfrac(\POL@IsoRight@rawout),% - \ifnum\POL@IsoRightSign=\z@ 1\else 0\fi;% + \XINT_expr_defvar_one{\POL@sturmname Z_\POL@isolz@IntervalIndex _isknown}% + {\csname .=\ifnum\POL@IsoRightSign=\z@ 1\else 0\fi\endcsname}% \endgroup }% - - %% \PolRefineInterval \def\POL@xintexprGetVar#1{\expandafter\expandafter\expandafter \XINT_expr_unlock\csname XINT_expr_var_#1\endcsname}% |