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authorNorbert Preining <norbert@preining.info>2021-03-30 03:00:56 +0000
committerNorbert Preining <norbert@preining.info>2021-03-30 03:00:56 +0000
commit93809c868bf15852ddc7b50542713131a5d8c05f (patch)
tree3c741be9d43fdee2f3ecdb17d463deb188d881a8 /macros/latex/contrib/polexpr
parent3c5dd0d2f091e5e866af1de0327171b253ec9178 (diff)
CTAN sync 202103300300
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-rw-r--r--macros/latex/contrib/polexpr/README.md102
-rw-r--r--macros/latex/contrib/polexpr/polexpr.html2911
-rw-r--r--macros/latex/contrib/polexpr/polexpr.sty3164
-rw-r--r--macros/latex/contrib/polexpr/polexpr.txt2598
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diff --git a/macros/latex/contrib/polexpr/README.md b/macros/latex/contrib/polexpr/README.md
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-Package polexpr README
-======================
-
-License
--------
-
-Copyright (C) 2018-2020 Jean-François Burnol
-
-See documentation of package [xint](http://www.ctan.org/pkg/xint) for
-contact information.
-
-This Work may be distributed and/or modified under the conditions of the
-LaTeX Project Public License version 1.3c. This version of this license
-is in
-
-> <http://www.latex-project.org/lppl/lppl-1-3c.txt>
-
-and version 1.3 or later is part of all distributions of LaTeX version
-2005/12/01 or later.
-
-This Work has the LPPL maintenance status author-maintained.
-
-The Author of this Work is Jean-François Burnol.
-
-This Work consists of the package file polexpr.sty, this README.md and
-the documentation file polexpr.txt.
-
-Abstract
---------
-
-The package provides `\poldef`. This a parser of polynomial expressions
-based upon the `\xintdeffunc` mechanism of xintexpr.
-
-The parsed expressions use the operations of algebra (inclusive of
-composition of functions) with standard operators, fractional numbers
-(possibly in scientific notation) and previously defined polynomial
-functions or other constructs as recognized by the `\xintexpr` numerical
-parser.
-
-The polynomials are then not only genuine `\xintexpr` (and
-`\xintfloatexpr`) numerical functions but additionally are known to the
-package via their coefficients. This allows dedicated macros to
-implement polynomial algorithmics.
-
-Releases
---------
-
-- 0.1 (2018/01/11)
- Initial release (files README, polexpr.sty).
-- 0.2 (2018/01/14)
- Documentation moved to polexpr.{txt,html}.
-- 0.3 (2018/01/17)
- Make polynomials known to `\xintfloatexpr` and improve
- documentation.
-- 0.3.1 (2018/01/18)
- Fix two typos in documentation.
-- 0.4 (2018/02/16)
- - Revert 0.3 automatic generation of floating point variants.
- - Move CHANGE LOG from README.md to HTML documentation.
- - A few bug fixes and breaking changes. Please refer to
- `polexpr.html`.
- - Main new feature: root localization via [Sturm
- Theorem](https://en.wikipedia.org/wiki/Sturm%27s_theorem).
-- 0.4.1 (2018/03/01)
- Synced with xint 1.3.
-- 0.4.2 (2018/03/03)
- Documentation fix.
-- 0.5 (2018/04/08)
- - new macros `\PolMakePrimitive` and `\PolIContent`.
- - main (breaking) change: `\PolToSturm` creates a chain of primitive
- integer coefficients polynomials.
-- 0.5.1 (2018/04/22)
- The `'` character can be used in polynomial names.
-- 0.6 (2018/11/20)
- New feature: multiplicity of roots.
-- 0.7 (2018/12/08), 0.7.1 (bugfix), 0.7.2 (bugfix) (2018/12/09)
- New feature: finding all rational roots.
-- 0.7.3 (2019/02/04)
- Bugfix: polynomial names ending in digits caused errors. Thanks to
- Thomas Söll for report.
-- 0.7.4 (2019/02/12)
- Bugfix: 20000000000 is too big for \numexpr, shouldn't I know that?
- Thanks to Jürgen Gilg for report.
-- 0.7.5 (2020/01/31)
- Synced with xint 1.4. Requires it.
-
-Files of 0.7.5 release:
-
-- README.md,
-- polexpr.sty (package file),
-- polexpr.txt (documentation),
-- polexpr.html (conversion via
- [DocUtils](http://docutils.sourceforge.net/docs/index.html)
- rst2html.py)
-
-Acknowledgments
----------------
-
-Thanks to Jürgen Gilg whose question about
-[xint](http://www.ctan.org/pkg/xint) usage for differentiating
-polynomials was the initial trigger leading to this package, and to
-Jürgen Gilg and Thomas Söll for testing it on some concrete problems.
diff --git a/macros/latex/contrib/polexpr/polexpr.html b/macros/latex/contrib/polexpr/polexpr.html
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-<h2 class="subtitle" id="id1">0.7.5 (2020/01/31)</h2>
-
-<!-- comment: -*- fill-column: 72; mode: rst; -*- -->
-<div class="contents topic" id="contents">
-<p class="topic-title first">Contents</p>
-<ul class="simple">
-<li><a class="reference internal" href="#basic-syntax" id="id38">Basic syntax</a></li>
-<li><a class="reference internal" href="#examples-of-localization-of-roots" id="id39">Examples of localization of roots</a><ul>
-<li><a class="reference internal" href="#a-typical-example" id="id40">A typical example</a></li>
-<li><a class="reference internal" href="#a-degree-four-polynomial-with-nearby-roots" id="id41">A degree four polynomial with nearby roots</a></li>
-<li><a class="reference internal" href="#the-degree-nine-polynomial-with-0-99-0-999-0-9999-as-triple-roots" id="id42">The degree nine polynomial with 0.99, 0.999, 0.9999 as triple roots</a></li>
-<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-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="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>
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-<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>
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-<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>
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-<li><a class="reference internal" href="#poltypesetcmd-raw-coeff" id="id59"><tt class="docutils literal">\PolTypesetCmd{raw_coeff}</tt></a></li>
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-<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>
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-<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="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="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="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="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="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="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="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="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="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">
-<h1><a class="toc-backref" href="#id38">Basic syntax</a></h1>
-<p>The syntax is:</p>
-<pre class="literal-block">
-\poldef polname(x):= expression in variable x;
-</pre>
-<p>where:</p>
-<ul class="simple">
-<li>in place of <tt class="docutils literal">x</tt> an arbitrary <em>dummy variable</em> is authorized,
-i.e. per default any of <tt class="docutils literal"><span class="pre">[a-z|A-Z]</span></tt> (more letters can be declared
-under Unicode engines.)</li>
-<li><tt class="docutils literal">polname</tt> consists of letters, digits, and the <tt class="docutils literal">_</tt> and
-<tt class="docutils literal">'</tt> characters. It must start with a letter.</li>
-</ul>
-<div class="admonition attention">
-<p class="first admonition-title">Attention!</p>
-<p class="last">The <tt class="docutils literal">'</tt> is authorized since <tt class="docutils literal">0.5.1</tt>. As a result some constructs
-recognized by the <tt class="docutils literal">\xintexpr</tt> parser, such as <tt class="docutils literal">var1 'and' var2</tt>
-will get misinterpreted and cause errors. However these constructs
-are unlikely to be frequently needed in polynomial expressions, and
-the <tt class="docutils literal">\xintexpr</tt> syntax offers alternatives, so it was deemed a
-small evil. Of course the <tt class="docutils literal">\xintexpr</tt> parser is modified only
-temporarily during execution of <tt class="docutils literal">\poldef</tt>.</p>
-</div>
-<p>One can also issue:</p>
-<pre class="literal-block">
-\PolDef{polname}{expression in variable x}
-</pre>
-<p>which admits an optional first argument to modify the variable letter
-from its default <tt class="docutils literal">x</tt>.</p>
-<dl class="docutils">
-<dt><tt class="docutils literal">\poldef <span class="pre">f(x):=</span> <span class="pre">1-x+x^2;</span></tt></dt>
-<dd>defines polynomial <tt class="docutils literal">f</tt>. Polynomial names must start with a
-letter and may contain letters, digits, underscores and the right
-tick character. The
-variable must be a single letter. The colon character is optional.
-The semi-colon at end of expression is mandatory.</dd>
-<dt><tt class="docutils literal"><span class="pre">\PolDef{f}{1-x+x^2}</span></tt></dt>
-<dd>does the same as <tt class="docutils literal">\poldef <span class="pre">f(x):=</span> <span class="pre">1-x+x^2;</span></tt> To use another letter
-than <tt class="docutils literal">x</tt> in the expression, one must pass it as an extra optional
-argument to <tt class="docutils literal">\PolDef</tt>. Useful if the semi-colon has been assigned
-some non-standard catcode by some package.</dd>
-<dt><tt class="docutils literal"><span class="pre">\PolLet{g}={f}</span></tt></dt>
-<dd>saves a copy of <tt class="docutils literal">f</tt> under name <tt class="docutils literal">g</tt>. Also usable without <tt class="docutils literal">=</tt>.</dd>
-<dt><tt class="docutils literal">\poldef <span class="pre">f(z):=</span> <span class="pre">f(z)^2;</span></tt></dt>
-<dd>redefines <tt class="docutils literal">f</tt> in terms of itself.</dd>
-<dt><tt class="docutils literal">\poldef <span class="pre">f(T):=</span> <span class="pre">f(f(T));</span></tt></dt>
-<dd>again redefines <tt class="docutils literal">f</tt> in terms of its (new) self.</dd>
-<dt><tt class="docutils literal">\poldef <span class="pre">k(z):=</span> <span class="pre">f(z)-g(g(z)^2)^2;</span></tt></dt>
-<dd>should now define the zero polynomial... Let's check:
-<tt class="docutils literal">\[ k(z) = <span class="pre">\PolTypeset[z]{k}</span> \]</tt></dd>
-<dt><tt class="docutils literal"><span class="pre">\PolDiff{f}{f'}</span></tt></dt>
-<dd>sets <tt class="docutils literal">f'</tt> to the derivative of <tt class="docutils literal">f</tt>. The name doesn't have to be
-<tt class="docutils literal">f'</tt> (in fact the <tt class="docutils literal">'</tt> is licit only since <tt class="docutils literal">0.5.1</tt>).</dd>
-</dl>
-<div class="admonition important">
-<p class="first admonition-title">Important</p>
-<p class="last">This is not done automatically. If some new definition needs to use
-the derivative of some available polynomial, that derivative
-polynomial must have been defined via <tt class="docutils literal">\PolDiff</tt>: something like
-<tt class="docutils literal"><span class="pre">T'(x)^2</span></tt> will not work without a prior <tt class="docutils literal"><span class="pre">\PolDiff{T}{T'}</span></tt>.</p>
-</div>
-<dl class="docutils">
-<dt><tt class="docutils literal"><span class="pre">\PolDiff{f'}{f''}</span></tt></dt>
-<dd>obtains second derivative.</dd>
-<dt><tt class="docutils literal"><span class="pre">\PolDiff[3]{f}{f'''}</span></tt></dt>
-<dd>computes the third derivative.</dd>
-</dl>
-<pre class="literal-block">
-$f(z) = \PolTypeset[z]{f} $\newline
-$f'(z) = \PolTypeset[z]{f'} $\newline
-$f''(z) = \PolTypeset[z]{f''} $\newline
-$f'''(z)= \PolTypeset[z]{f'''} $\par
-</pre>
-<div class="admonition important">
-<p class="first admonition-title">Important</p>
-<p>The package does not currently know rational functions: <tt class="docutils literal">/</tt> in
-a parsed polynomial expression does the Euclidean quotient:</p>
-<pre class="literal-block">
-(1-x^2)/(1-x)
-</pre>
-<p>does give <tt class="docutils literal">1+x</tt> but</p>
-<pre class="literal-block">
-(1/(1-x))*(1-x^2)
-</pre>
-<p>evaluates to zero. This will work as expected:</p>
-<pre class="last literal-block">
-\poldef k(x):= (x-1)(x-2)(x-3)(x-4)/(x^2-5x+4);
-</pre>
-</div>
-<div class="admonition attention" id="warningtacit">
-<p class="first admonition-title">Attention!</p>
-<p><tt class="docutils literal">1/2 x^2</tt> skips the space and is treated like <tt class="docutils literal"><span class="pre">1/(2*x^2)</span></tt> because
-of the tacit multiplication rules of xintexpr. But this means it
-gives zero! Thus one must use <tt class="docutils literal">(1/2)x^2</tt> or <tt class="docutils literal">1/2*x^2</tt> or
-<tt class="docutils literal"><span class="pre">(1/2)*x^2</span></tt> for disambiguation: <tt class="docutils literal">x - 1/2*x^2 + <span class="pre">1/3*x^3...</span></tt>. It is
-even simpler to move the denominator to the right: <tt class="docutils literal">x - x^2/2 +
-x^3/3 - ...</tt>.</p>
-<p class="last">It is worth noting that <tt class="docutils literal"><span class="pre">1/2(x-1)(x-2)</span></tt> suffers the same issue:
-<a class="reference external" href="http://www.ctan.org/pkg/xint">xint</a> tacit multiplication always &quot;ties more&quot;, hence this gets
-interpreted as <tt class="docutils literal"><span class="pre">1/(2*(x-1)*(x-2))</span></tt> which gives zero by polynomial
-division. Thus, use one of <tt class="docutils literal"><span class="pre">(1/2)(x-1)(x-2)</span></tt>, <tt class="docutils literal"><span class="pre">1/2*(x-1)(x-2)</span></tt> or
-<tt class="docutils literal"><span class="pre">(x-1)(x-2)/2</span></tt>.</p>
-</div>
-<p>After:</p>
-<pre class="literal-block">
-\poldef f_1(x):= 25(x-1)(x^2-2)(x-3)(x-4)(x-5);%
-\poldef f_2(x):= 37(x-1)(x^2-2)(x-6)(x-7)(x-8);%
-</pre>
-<p>the macro call <tt class="docutils literal"><span class="pre">\PolGCD{f_1}{f_2}{k}</span></tt> sets <tt class="docutils literal">k</tt> to the (unitary) GCD of
-<tt class="docutils literal">f_1</tt> and <tt class="docutils literal">f_2</tt> (hence to the expansion of <tt class="docutils literal"><span class="pre">(x-1)(x^2-2)</span></tt>.)</p>
-<dl class="docutils">
-<dt><tt class="docutils literal">\PolToExpr{k}</tt></dt>
-<dd>will (expandably) give in this case <tt class="docutils literal"><span class="pre">x^3-x^2-2*x+2</span></tt>. This is
-useful for console or file output (the syntax is Maple- and
-PSTricks-compatible; the letter used in output can be
-(non-expandably) changed via a redefinition of <a class="reference internal" href="#poltoexprvar">\PolToExprVar</a>.)</dd>
-<dt><tt class="docutils literal"><span class="pre">\PolToExpr*{k}</span></tt></dt>
-<dd>gives ascending powers: <tt class="docutils literal"><span class="pre">2-2*x-x^2+x^3</span></tt>.</dd>
-</dl>
-</div>
-<div class="section" id="examples-of-localization-of-roots">
-<h1><a class="toc-backref" href="#id39">Examples of localization of roots</a></h1>
-<ul>
-<li><p class="first">To make printed decimal numbers more enjoyable than via
-<tt class="docutils literal">\xintSignedFrac</tt>:</p>
-<pre class="literal-block">
-\renewcommand\PolTypesetOne[1]{\PolDecToString{\xintREZ{#1}}}%
-</pre>
-<p><tt class="docutils literal">\PolDecToString</tt> will use decimal notation to incorporate the power
-of ten part; and the <tt class="docutils literal">\xintREZ</tt> will have the effect to suppress
-trailing zeros if present in raw numerator (if those digits end up
-after decimal mark.) Notice that the above are expandable macros and
-that one can also do:</p>
-<pre class="literal-block">
-\renewcommand\PolToExprCmd[1]{\PolDecToString{\xintREZ{#1}}}%
-</pre>
-<p>to modify output of <a class="reference internal" href="#poltoexpr-polname">\PolToExpr{polname}</a>.</p>
-</li>
-<li><p class="first">For extra info in log file use <tt class="docutils literal">\xintverbosetrue</tt>.</p>
-</li>
-<li><p class="first">Only for some of these examples is the output included here.</p>
-</li>
-</ul>
-<div class="section" id="a-typical-example">
-<h2><a class="toc-backref" href="#id40">A typical example</a></h2>
-<p>In this example the polynomial is square-free.</p>
-<pre class="literal-block">
-\poldef f(x) := x^7 - x^6 - 2x + 1;
-
-\PolToSturm{f}{f}
-\PolSturmIsolateZeros{f}
-The \PolTypeset{f} polynomial has \PolSturmNbOfIsolatedZeros{f} distinct real
-roots which are located in the following intervals:
-\PolPrintIntervals{f}
-Here is the second root with ten more decimal digits:
-\PolRefineInterval[10]{f}{2}
-\[\PolSturmIsolatedZeroLeft{f}{2}&lt;Z_2&lt;\PolSturmIsolatedZeroRight{f}{2}\]
-And here is the first root with twenty digits after decimal mark:
-\PolEnsureIntervalLength{f}{1}{-20}
-\[\PolSturmIsolatedZeroLeft{f}{1}&lt;Z_1&lt;\PolSturmIsolatedZeroRight{f}{1}\]
-The first element of the Sturm chain has degree $\PolDegree{f_0}$. As
-this is the original degreee $\PolDegree{f}$ we know that $f$ is square free.
-Its derivative is up to a constant \PolTypeset{f_1} (in this example
-it is identical with it).
-\PolToSturm{f_1}{f_1}\PolSturmIsolateZeros{f_1}%
-The derivative has \PolSturmNbOfIsolatedZeros{f_1} distinct real
-roots:
-\PolPrintIntervals[W]{f_1}
-\PolEnsureIntervalLengths{f_1}{-10}%
-Here they are with ten digits after decimal mark:
-\PolPrintIntervals[W]{f_1}
-\PolDiff{f_1}{f''}
-\PolToSturm{f''}{f''}
-\PolSturmIsolateZeros{f''}
-The second derivative is \PolTypeset{f''}.
-It has \PolSturmNbOfIsolatedZeros{f''} distinct real
-roots:
-\PolPrintIntervals[X]{f''}
-Here is the positive one with 20 digits after decimal mark:
-\PolEnsureIntervalLength{f''}{2}{-20}%
-\[X_2 = \PolSturmIsolatedZeroLeft{f''}{2}\dots\]
-The more mathematically advanced among our dear readers will be able
-to give the exact value for $X_2$!
-</pre>
-</div>
-<div class="section" id="a-degree-four-polynomial-with-nearby-roots">
-<h2><a class="toc-backref" href="#id41">A degree four polynomial with nearby roots</a></h2>
-<p>Notice that this example is a bit outdated as <tt class="docutils literal">0.7</tt> release has
-added <tt class="docutils literal"><span class="pre">\PolSturmIsolateZeros**{sturmname}</span></tt> which would find exactly
-the roots. The steps here retain their interest when one is interested
-in finding isolating intervals for example to prepare some demonstration
-of dichotomy method.</p>
-<pre class="literal-block">
-\PolDef{Q}{(x-1.050001)(x-1.105001)(x-1.110501)(x-1.111051)}
-\PolTypeset{Q}
-\PolToSturm{Q}{Q} % it is allowed to use same prefix for Sturm chain
-\PolSturmIsolateZeros{Q}
-\PolPrintIntervals{Q}
-% reports 1.0 &lt; Z_1 &lt; 1.1, 1.10 &lt; Z_2 &lt; 1.11, 1.110 &lt; Z_3 &lt; 1.111, and 1.111 &lt; Z_4 &lt; 1.112
-% but the above bounds do not allow minimizing separation between roots
-% so we refine:
-\PolRefineInterval*{Q}{1}
-\PolRefineInterval*{Q}{2}
-\PolRefineInterval*{Q}{3}
-\PolRefineInterval*{Q}{4}
-\PolPrintIntervals{Q}
-% reports 1.05 &lt; Z_1 &lt; 1.06, 1.105 &lt; Z_2 &lt; 1.106, 1.1105 &lt; Z_3 &lt; 1.1106,
-% and 1.11105 &lt; Z_4 &lt; 1.11106.
-\PolEnsureIntervalLengths{Q}{-6}
-\PolPrintIntervals{Q}
-% of course finds here all roots exactly
-</pre>
-</div>
-<div class="section" id="the-degree-nine-polynomial-with-0-99-0-999-0-9999-as-triple-roots">
-<h2><a class="toc-backref" href="#id42">The degree nine polynomial with 0.99, 0.999, 0.9999 as triple roots</a></h2>
-<pre class="literal-block">
-% define a user command (xinttools is loaded automatically by polexpr)
-\newcommand\showmultiplicities[1]{% #1 = &quot;sturmname&quot;
-\xintFor* ##1 in {\xintSeq{1}{\PolSturmNbOfIsolatedZeros{#1}}}\do{%
- The multiplicity is \PolSturmIsolatedZeroMultiplicity{#1}{##1}
- \PolSturmIfZeroExactlyKnown{#1}{##1}%
- {at the root $x=\PolSturmIsolatedZeroLeft{#1}{##1}$}
- {for the root such that
- $\PolSturmIsolatedZeroLeft{#1}{##1}&lt;x&lt;\PolSturmIsolatedZeroRight{#1}{##1}$}
- \par
-}}%
-\PolDef{f}{(x-0.99)^3(x-0.999)^3(x-0.9999)^3}
-\renewcommand\PolTypesetOne[1]{\PolDecToString{\xintREZ{#1}}}
-\PolTypeset{f}\par
-\PolToSturm{f}{f}% it is allowed to use &quot;polname&quot; as &quot;sturmname&quot; too
-\PolSturmIsolateZerosAndGetMultiplicities{f}% use the &quot;sturmname&quot; here
-% or \PolSturmIsolateZeros*{f} which is exactly the same, but shorter..
-
-\showmultiplicities{f}
-</pre>
-<p>In this example, the output will look like this (but using math mode):</p>
-<pre class="literal-block">
-x^9 - 8.9667x^8 + 35.73400293x^7 - 83.070418400109x^6 + 124.143648875193123x^5
-- 123.683070924326075877x^4 + 82.149260397553075617891x^3
-- 35.07602992699900159127007x^2 + 8.7364078733314648368671733x
-- 0.967100824643585986488103299
-
-The multiplicity is 3 at the root x = 0.99
-The multiplicity is 3 at the root x = 0.999
-The multiplicity is 3 at the root x = 0.9999
-</pre>
-<p>On first pass, these rational roots were found (due to their relative
-magnitudes, using <tt class="docutils literal">\PolSturmIsolateZeros**</tt> was not needed here). But
-multiplicity computation works also with (decimal) roots not yet
-identified or with non-decimal or irrational roots.</p>
-<p>It is fun to modify only a tiny bit the polynomial and see if polexpr
-survives:</p>
-<pre class="literal-block">
-\PolDef{g}{f(x)+1e-27}
-\PolTypeset{g}\par
-\PolToSturm{g}{g}
-\PolSturmIsolateZeros*{g}
-
-\showmultiplicities{g}
-</pre>
-<p>This produces:</p>
-<pre class="literal-block">
-x^9 - 8.9667x^8 + 35.73400293x^7 - 83.070418400109x^6 + 124.143648875193123x^5
-- 123.683070924326075877x^4 + 82.149260397553075617891x^3
-- 35.07602992699900159127007x^2 + 8.7364078733314648368671733x
-- 0.967100824643585986488103298
-
-The multiplicity is 1 for the root such that 0.98 &lt; x &lt; 0.99
-The multiplicity is 1 for the root such that 0.9991 &lt; x &lt; 0.9992
-The multiplicity is 1 for the root such that 0.9997 &lt; x &lt; 0.9998
-</pre>
-<p>Which means that the multiplicity-3 roots each became a real and a pair of
-complex ones. Let's see them better:</p>
-<pre class="literal-block">
-\PolEnsureIntervalLengths{g}{-10}
-
-\showmultiplicities{g}
-</pre>
-<p>which produces:</p>
-<pre class="literal-block">
-The multiplicity is 1 for the root such that 0.9899888032 &lt; x &lt; 0.9899888033
-The multiplicity is 1 for the root such that 0.9991447980 &lt; x &lt; 0.9991447981
-The multiplicity is 1 for the root such that 0.9997663986 &lt; x &lt; 0.9997663987
-</pre>
-</div>
-<div class="section" id="a-degree-five-polynomial-with-three-rational-roots">
-<h2><a class="toc-backref" href="#id43">A degree five polynomial with three rational roots</a></h2>
-<pre class="literal-block">
-\poldef Q(x) := 1581755751184441 x^5
- -14907697165025339 x^4
- +48415668972339336 x^3
- -63952057791306264 x^2
- +46833913221154895 x
- -49044360626280925;
-
-\PolToSturm{Q}{Q}
-%\begin{flushleft}
- \renewcommand\PolTypesetCmdPrefix[1]{\allowbreak\xintiiifSgn{#1}{}{+}{+}}%
- $Q_0(x) = \PolTypeset{Q_0}$
-%\end{flushleft}
-\PolSturmIsolateZeros**{Q}
-\PolPrintIntervals{Q}
-
-$Q_{norr}(x) = \PolTypeset{Q_norr}$
-</pre>
-<p>Here, all real roots are rational:</p>
-<pre class="literal-block">
-Z_1 = 833719/265381
-Z_2 = 165707065/52746197
-Z_3 = 355/113
-
-Q_norr(x) = x^2 + 1
-</pre>
-<p>And let's get their decimal expansion too:</p>
-<pre class="literal-block">
-% print decimal expansion of the found roots
-\renewcommand\PolPrintIntervalsPrintExactZero
- {\xintTrunc{20}{\PolPrintIntervalsTheLeftEndPoint}\dots}
-\PolPrintIntervals{Q}
-
-Z_1 = 3.14159265358107777120...
-Z_2 = 3.14159265358979340254...
-Z_3 = 3.14159292035398230088...
-</pre>
-</div>
-<div class="section" id="a-mignotte-type-polynomial">
-<h2><a class="toc-backref" href="#id44">A Mignotte type polynomial</a></h2>
-<pre class="literal-block">
-\PolDef{P}{x^10 - (10x-1)^2}%
-\PolTypeset{P} % prints it in expanded form
-\PolToSturm{P}{P} % we can use same prefix for Sturm chain
-\PolSturmIsolateZeros{P} % finds 4 real roots
-This polynomial has \PolSturmNbOfIsolatedZeros{P} distinct real roots:
-\PolPrintIntervals{P}%
-% reports -2 &lt; Z_1 &lt; -1, 0.09 &lt; Z_2 &lt; 0.10, 0.1 &lt; Z_3 &lt; 0.2, 1 &lt; Z_4 &lt; 2
-Let us refine the second and third intervals to separate the corresponding
-roots:
-\PolRefineInterval*{P}{2}% will refine to 0.0999990 &lt; Z_2 &lt; 0.0999991
-\PolRefineInterval*{P}{3}% will refine to 0.100001 &lt; Z_3 &lt; 0.100002
-\PolPrintIntervals{P}%
-Let us now get to know all roots with 10 digits after decimal mark:
-\PolEnsureIntervalLengths{P}{-10}%
-\PolPrintIntervals{P}% now all roots are known 10 decimal digits after mark
-Finally, we display 20 digits of the second root:
-\PolEnsureIntervalLength{P}{2}{-20}% makes Z_2 known with 20 digits after mark
-\[\PolSturmIsolatedZeroLeft{P}{2}&lt;Z_2&lt;\PolSturmIsolatedZeroRight{P}{2}\]
-</pre>
-<p>The last line produces:</p>
-<pre class="literal-block">
-0.09999900004999650028 &lt; Z_2 &lt; 0.09999900004999650029
-</pre>
-</div>
-<div class="section" id="the-wilkinson-polynomial">
-<h2><a class="toc-backref" href="#id45">The Wilkinson polynomial</a></h2>
-<p>See <a class="reference external" href="https://en.wikipedia.org/wiki/Wilkinson%27s_polynomial">Wilkinson polynomial</a>.</p>
-<pre class="literal-block">
-\documentclass{article}
-\usepackage{polexpr}
-\begin{document}
-%\xintverbosetrue % for the curious...
-
-\poldef f(x) := mul((x - i), i = 1..20);
-
-\renewcommand\PolTypesetCmdPrefix[1]{\allowbreak\xintiiifSgn{#1}{}{+}{+}}%
-\renewcommand\PolTypesetOne[1]{\xintDecToString{#1}}%
-
-\noindent\PolTypeset{f}
-
-\PolToSturm{f}{f}
-\PolSturmIsolateZeros{f}
-\PolPrintIntervals{f}
-
-\clearpage
-
-\poldef g(x) := f(x) - 2**{-23} x**19;
-
-% be patient!
-\PolToSturm{g}{g}
-\noindent\PolTypeset{g_0}% integer coefficient primitive polynomial
-
-\PolSturmIsolateZeros{g}
-\PolEnsureIntervalLengths{g}{-10}
-
-\renewcommand\PolPrintIntervalsPrintMultiplicity{}
-\PolPrintIntervals*{g}
-
-\end{document}
-</pre>
-<p>The first polynomial:</p>
-<pre class="literal-block">
-f(x) = x**20
-- 210 x**19
-+ 20615 x**18
-- 1256850 x**17
-+ 53327946 x**16
-- 1672280820 x**15
-+ 40171771630 x**14
-- 756111184500 x**13
-+ 11310276995381 x**12
-- 135585182899530 x**11
-+ 1307535010540395 x**10
-- 10142299865511450 x**9
-+ 63030812099294896 x**8
-- 311333643161390640 x**7
-+ 1206647803780373360 x**6
-- 3599979517947607200 x**5
-+ 8037811822645051776 x**4
-- 12870931245150988800 x**3
-+ 13803759753640704000 x**2
-- 8752948036761600000 x
-+ 2432902008176640000
-</pre>
-<p>is handled fast enough (a few seconds), but the modified one <tt class="docutils literal">f(x) -
-<span class="pre">2**-23</span> <span class="pre">x**19</span></tt> takes about 20x longer (the Sturm chain polynomials
-have integer coefficients with up to 321 digits, whereas (surprisingly
-perhaps) those of the Sturm chain polynomials derived from <tt class="docutils literal">f</tt> never
-have more than 21 digits ...).</p>
-<p>Once the Sturm chain is computed and the zeros isolated, obtaining their
-decimal digits is relatively faster. Here is for the ten real roots of
-<tt class="docutils literal">f(x) - <span class="pre">2**-23</span> <span class="pre">x**19</span></tt> as computed by the code above:</p>
-<pre class="literal-block">
-Z_1 = 0.9999999999...
-Z_2 = 2.0000000000...
-Z_3 = 2.9999999999...
-Z_4 = 4.0000000002...
-Z_5 = 4.9999999275...
-Z_6 = 6.0000069439...
-Z_7 = 6.9996972339...
-Z_8 = 8.0072676034...
-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="#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>
-<p>In the defining expression we could have used <tt class="docutils literal">i/10</tt> but this gives
-less efficient internal form for the coefficients (the <tt class="docutils literal">10</tt>'s end up
-in denominators). Using <tt class="docutils literal">\PolToExpr{P}</tt> after having done</p>
-<pre class="literal-block">
-\renewcommand\PolToExprCmd[1]{\PolDecToString{\xintREZ{#1}}}
-</pre>
-<p>we get this expanded form:</p>
-<pre class="literal-block">
-x^41
--28.7*x^39
-+375.7117*x^37
--2975.11006*x^35
-+15935.28150578*x^33
--61167.527674162*x^31
-+173944.259366417394*x^29
--373686.963560544648*x^27
-+613012.0665016658846445*x^25
--771182.31133138163125495*x^23
-+743263.86672885754888959569*x^21
--545609.076599482896371978698*x^19
-+301748.325708943677229642930528*x^17
--123655.8987669450434698869844544*x^15
-+36666.1782054884005855608205864192*x^13
--7607.85821367459445649518380016128*x^11
-+1053.15135918687298508885950223794176*x^9
--90.6380005918141132650786081964032*x^7
-+4.33701563847327366842552218288128*x^5
--0.0944770968420804735498178265088*x^3
-+0.00059190121813899276854174416896*x
-</pre>
-<p>which shows coefficients with up to 36 significant digits...</p>
-<p>Stress test: not a hard challenge to <tt class="docutils literal">xint + polexpr</tt>, but be a bit patient!</p>
-<pre class="literal-block">
-\PolDef{P}{mul((x-i*1e-1), i=-20..20)}%
-\PolToSturm{P}{S} % dutifully computes S_0, ..., S_{41}
-% the [1] optional argument limits the search to interval (-10,10)
-\PolSturmIsolateZeros[1]{S} % finds *exactly* (but a bit slowly) all 41 roots!
-\PolPrintIntervals{S} % nice, isn't it?
-</pre>
-<div class="admonition note">
-<p class="first admonition-title">Note</p>
-<p>Release <tt class="docutils literal">0.5</tt> has <em>experimental</em> addition of optional argument
-<tt class="docutils literal">E</tt> to <tt class="docutils literal">\PolSturmIsolateZeros</tt>. It instructs to search roots only
-in interval <tt class="docutils literal"><span class="pre">(-10^E,</span> 10^E)</tt>. Important: the extremities are
-<em>assumed to not be roots</em>. In this example, the <tt class="docutils literal">[1]</tt> in
-<tt class="docutils literal"><span class="pre">\PolSturmIsolateZeros[1]{S}</span></tt> gives some speed gain; without it, it
-turns out in this case that <tt class="docutils literal">polexpr</tt> would have started with
-<tt class="docutils literal"><span class="pre">(-10^6,</span> 10^6)</tt> interval.</p>
-<p class="last">Please note that this will probably get replaced in future by the
-specification of a general interval. Do not rely on meaning of this
-optional argument keeping the same.</p>
-</div>
-</div>
-<div class="section" id="roots-of-chebyshev-polynomials">
-<h2><a class="toc-backref" href="#id48">Roots of Chebyshev polynomials</a></h2>
-<pre class="literal-block">
-\newcount\mycount
-\poldef T_0(x) := 1;
-\poldef T_1(x) := x;
-\mycount 2
-\xintloop
- \poldef T_\the\mycount(x) :=
- 2x*T_\the\numexpr\mycount-1(x)
- - T_\the\numexpr\mycount-2(x);
-\ifnum\mycount&lt;15
-\advance\mycount 1
-\repeat
-
-\[T_{15} = \PolTypeset[X]{T_15}\]
-\PolToSturm{T_15}{T_15}
-\PolSturmIsolateZeros{T_15}
-\PolEnsureIntervalLengths{T_15}{-10}
-\PolPrintIntervals{T_15}
-</pre>
-</div>
-</div>
-<div class="section" id="non-expandable-macros">
-<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="#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,
-under the user-chosen <tt class="docutils literal">polname</tt>. Of course the <em>expression</em> can
-use other previously defined polynomials. Names must start with a
-letter and are constituted of letters, digits, underscores and
-(since <tt class="docutils literal">0.5.1</tt>) the right tick <tt class="docutils literal">'</tt>.
-The whole <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> syntax is authorized:</p>
-<pre class="literal-block">
-\poldef sin(z) := add((-1)^i z^(2i+1)/(2i+1)!, i = 0..10);
-</pre>
-<p>With fractional coefficients, beware the <a class="reference internal" href="#warningtacit">tacit multiplication issue</a>.</p>
-<p>As a side effect the function <tt class="docutils literal">polname()</tt> is recognized as a
-genuine <tt class="docutils literal"><span class="pre">\xintexpr...\relax</span></tt> function for (exact) numerical
-evaluation (or within an <tt class="docutils literal">\xintdefvar</tt> assignment.) It computes
-values not according to the original expression but via the Horner
-scheme corresponding to the polynomial coefficients.</p>
-<div class="admonition attention">
-<p class="first admonition-title">Attention!</p>
-<p>Release <tt class="docutils literal">0.3</tt> also did the necessary set-up to let the
-polynomial be known to the <tt class="docutils literal">\xintfloatexpr</tt> (or
-<tt class="docutils literal">\xintdeffloatvar</tt>) parser.</p>
-<p>Since <tt class="docutils literal">0.4</tt> this isn't done automatically. Even more, a
-previously existing floating point variant of the same name will
-be let undefined again, to avoid hard to debug mismatches between
-exact and floating point polynomials. This also applies when the
-polynomial is produced not via <tt class="docutils literal">\poldef</tt> or <tt class="docutils literal">\PolDef</tt> but as
-a product of the other package macros.</p>
-<p class="last">See <a class="reference internal" href="#polgenfloatvariant-polname">\PolGenFloatVariant{polname}</a>.</p>
-</div>
-<p>The original expression is lost after parsing, and in particular
-the package provides no way to typeset it. This has to be done
-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="#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
-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="#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
-already pre-rounded to the float precision.</p>
-<p>See also <a class="reference internal" href="#poltofloatexpr-polname">\PolToFloatExpr{polname}</a>.</p>
-<div class="admonition attention">
-<p class="first admonition-title">Attention!</p>
-<p>Release <tt class="docutils literal">0.3</tt> did this automatically on <tt class="docutils literal">\PolDef</tt> and
-<tt class="docutils literal">\poldef</tt> but this was removed at <tt class="docutils literal">0.4</tt> for optimization.</p>
-<p class="last">Any operation, for example generating the derivative polynomial,
-or dividing two polynomials or using the <tt class="docutils literal">\PolLet</tt>, <strong>must</strong> be
-followed by explicit usage of <tt class="docutils literal">\PolGenFloatVariant{polname}</tt> if
-the new polynomial is to be used in <tt class="docutils literal">\xintfloatexpr</tt> or alike
-context.</p>
-</div>
-</blockquote>
-</div>
-<div class="section" id="pollet-polname-2-polname-1">
-<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
-<tt class="docutils literal"><span class="pre">\PolDef{polname_2}{polname_1(x)}</span></tt> but with less overhead. The
-<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="#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="#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>
-<ul class="simple">
-<li>Attention, coefficients here are indexed starting at 1.</li>
-<li>With #1=-1, -2, ..., <tt class="docutils literal"><span class="pre">\macro{#1}</span></tt> returns leading coefficients.</li>
-<li>With #1=0, returns the number of coefficients, i.e. <tt class="docutils literal">1 + deg f</tt>
-for non-zero polynomials.</li>
-<li>Out-of-range #1's return <tt class="docutils literal">0/1[0]</tt>.</li>
-</ul>
-<p>See also <a class="reference internal" href="#polnthcoeff-polname-number">\PolNthCoeff{polname}{number}</a>. The main difference is that
-with <tt class="docutils literal">\PolAssign</tt>, <tt class="docutils literal">\macro</tt> is made a prefix to <tt class="docutils literal">1 + deg f</tt>
-already defined (hidden to user) macros holding individually the
-coefficients but <a class="reference internal" href="#polnthcoeff-polname-number">\PolNthCoeff{polname}{number}</a> does each time the job
-to expandably recover the <tt class="docutils literal">Nth</tt> coefficient, and due to
-expandability can not store it in a macro for future usage (of course,
-it can be an argument in an <tt class="docutils literal">\edef</tt>.) The other difference
-is the shift by one in indexing, mentioned above (negative
-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="#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
-<tt class="docutils literal">\macro{number}</tt> gets expanded in an <tt class="docutils literal">\edef</tt> and then normalized
-via <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a>'s macro <tt class="docutils literal">\xintRaw</tt>.</p>
-<p>The leading zeros are removed from the polynomial.</p>
-<p>(contrived) Example:</p>
-<pre class="literal-block">
-\xintAssignArray{1}{-2}{5}{-3}\to\foo
-\PolGet{f}\fromarray\foo
-</pre>
-<p>This will define <tt class="docutils literal">f</tt> as would have <tt class="docutils literal">\poldef <span class="pre">f(x):=1-2x+5x^2-3x^3;</span></tt>.</p>
-<div class="admonition note">
-<p class="first admonition-title">Note</p>
-<p class="last">Prior to <tt class="docutils literal">0.5</tt>, coefficients were not normalized via
-<tt class="docutils literal">\xintRaw</tt> for internal storage.</p>
-</div>
-</blockquote>
-</div>
-<div class="section" id="polfromcsv-polname-csv">
-<span id="polfromcsv"></span><h2><a class="toc-backref" href="#id57"><tt class="docutils literal"><span class="pre">\PolFromCSV{polname}{&lt;csv&gt;}</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
-item</em> gives the constant term, the <em>last item</em> gives the leading
-coefficient, except if zero, then it is dropped (iteratively). List
-items are each expanded in an <tt class="docutils literal">\edef</tt> and then put into normalized
-form via <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a>'s macro <tt class="docutils literal">\xintRaw</tt>.</p>
-<p>As leading zero coefficients are removed:</p>
-<pre class="literal-block">
-\PolFromCSV{f}{0, 0, 0, 0, 0, 0, 0, 0, 0, 0}
-</pre>
-<p>defines the zero polynomial, which holds only one coefficient.</p>
-<p>See also expandable macro <a class="reference internal" href="#poltocsv-polname">\PolToCSV</a>.</p>
-<div class="admonition note">
-<p class="first admonition-title">Note</p>
-<p class="last">Prior to <tt class="docutils literal">0.5</tt>, coefficients were not normalized via
-<tt class="docutils literal">\xintRaw</tt> for internal storage.</p>
-</div>
-</blockquote>
-</div>
-<div class="section" id="poltypeset-polname">
-<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>
-<pre class="literal-block">
-\PolTypeset[z]{polname}
-</pre>
-<p>By default zero coefficients are skipped (issue <tt class="docutils literal">\poltypesetalltrue</tt>
-to get all of them in output).</p>
-<p>These commands (whose meanings will be found in the package code)
-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="#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
-<a class="reference internal" href="#polifcoeffisplusorminusone-a-b">\PolIfCoeffIsPlusOrMinusOne{A}{B}</a>.</p>
-<p>The actual printing of the coefficients, when not equal to plus or
-minus one is handled by <a class="reference internal" href="#poltypesetone-raw-coeff">\PolTypesetOne{raw_coeff}</a>.</p>
-</blockquote>
-</div>
-<div class="section" id="poltypesetone-raw-coeff">
-<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>
-<p>One can do things such as for example: <a class="footnote-reference" href="#id5" id="id4">[1]</a></p>
-<pre class="literal-block">
-\renewcommand\PolTypesetOne[1]{\num{\xintPFloat[5]{#1}}}
-\renewcommand\PolTypesetOne[1]{\num{\xintRound{4}{#1}}}
-</pre>
-<p>where e.g. we used the <tt class="docutils literal">\num</tt> macro of <tt class="docutils literal">siunitx</tt> as it
-understands floating point notation.</p>
-<table class="docutils footnote" frame="void" id="id5" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id4">[1]</a></td><td>the difference in the syntaxes of <tt class="docutils literal">\xintPFloat</tt> and
-<tt class="docutils literal">\xintRound</tt> is explained from the fact that
-<tt class="docutils literal">\xintPFloat</tt> by default uses the prevailing precision
-hence the extra argument like here <tt class="docutils literal">5</tt> is an optional one.</td></tr>
-</tbody>
-</table>
-<p>One can also give a try to using <a class="reference internal" href="#poldectostring-decimal-number">\PolDecToString{decimal number}</a>
-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="#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
-for the constant term, <tt class="docutils literal">\PolVar</tt> for the first degree and
-<tt class="docutils literal"><span class="pre">\PolVar^{\PolIndex}</span></tt> for higher degrees monomials. Beware that
-<tt class="docutils literal">\PolIndex</tt> expands to digit tokens and needs termination in
-<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="#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
-<tt class="docutils literal">\xintSignedFrac</tt> used by <a class="reference internal" href="#poltypesetcmd-raw-coeff">\PolTypesetCmd{raw_coeff}</a> will put
-the <tt class="docutils literal">-</tt> sign in front of the fraction (if it is a fraction) and
-this will thus serve as separator in the typeset formula. Not used
-for the first term.</blockquote>
-</div>
-</div>
-<div class="section" id="id8">
-<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="#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>
-by <tt class="docutils literal">f'</tt>.</p>
-<p>Coefficients of the result <tt class="docutils literal">polname_2</tt> are irreducible fractions
-(see <a class="reference internal" href="#technicalities">Technicalities</a> for the whole story.)</p>
-</blockquote>
-</div>
-<div class="section" id="poldiff-n-polname-1-polname-2">
-<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
-<tt class="docutils literal"><span class="pre">\PolLet{polname_2}={polname_1}</span></tt>. With negative <tt class="docutils literal">N</tt>, switches to
-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="#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>
-<p>Coefficients of the result <tt class="docutils literal">polname_2</tt> are irreducible fractions
-(see <a class="reference internal" href="#technicalities">Technicalities</a> for the whole story.)</p>
-</blockquote>
-</div>
-<div class="section" id="polantidiff-n-polname-1-polname-2">
-<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="#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="#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="#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="#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>
-and <tt class="docutils literal">polname_2</tt> vanish, then <tt class="docutils literal">polname_GCD</tt> is the zero
-polynomial.</blockquote>
-<!-- ``\PolIGCD{polname_1}{polname_2}{polname_iGCD}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- **NOT YET**
-
- This **assumes** that the two polynomials have integer coefficients.
- It then computes the greatest common divisor in the integer
- polynomial ring, normalized to have a positive leading coefficient
- (if the inputs are not both zero).
-
-``\PolIContent{polname}``
-~~~~~~~~~~~~~~~~~~~~~~~~~
-
- **NOT YET**
-
- This computes a positive rational number such that dividing the
- polynomial with it returns an integer coefficients polynomial with
- no common factor among the coefficients. -->
-</div>
-<div class="section" id="poltosturm-polname-sturmname">
-<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
-of the) remainder in euclidean division, iteratively.</p>
-<p>The last non-zero remainder <tt class="docutils literal">P_N_</tt> (where <tt class="docutils literal">N</tt> is obtainable as
-<a class="reference internal" href="#polsturmchainlength-sturmname">\PolSturmChainLength{sturmname}</a>) is up to a factor
-the GCD of <tt class="docutils literal">P</tt> and <tt class="docutils literal">P'</tt> hence it is a constant if and only if
-<tt class="docutils literal">P</tt> is square-free.</p>
-<div class="admonition note">
-<p class="first admonition-title">Note</p>
-<ul class="last simple">
-<li>Since <tt class="docutils literal">0.5</tt> all these polynomials are divided by their rational
-content, so they have integer coefficients with no common factor,
-and the last one if a constant is either <tt class="docutils literal">1</tt> or <tt class="docutils literal"><span class="pre">-1</span></tt>.</li>
-<li>After this normalization to primitive polynomials, they are
-stored internally as <tt class="docutils literal">sturmname_k_</tt>, <tt class="docutils literal">k=0,1, ...</tt>.</li>
-<li>These polynomials are used internally only. To keep them as
-genuine declared polynomials also after the macro call, use the
-starred variant <a class="reference internal" href="#id9">PolToSturm*</a>.</li>
-</ul>
-</div>
-<div class="admonition note">
-<p class="first admonition-title">Note</p>
-<p class="last">It is perfectly allowed to use the polynomial name as Sturm chain name:
-<tt class="docutils literal"><span class="pre">\PolToSturm{f}(f}</span></tt>.</p>
-</div>
-<p>The macro then declares <tt class="docutils literal">sturmname_0</tt>, <tt class="docutils literal">sturmname_1</tt>, ..., which are
-the (non-declared) <tt class="docutils literal">sturmname_k_</tt> divided by the last one. Division is
-not done if this last one is the constant <tt class="docutils literal">1</tt> or <tt class="docutils literal"><span class="pre">-1</span></tt>, i.e. if the
-original polynomial was square-free. These polynomials are primitive
-polynomials too, i.e. with integer coefficients having no common factor.</p>
-<p>Thus <tt class="docutils literal">sturmname_0</tt> has exactly the same real and complex roots as
-polynomial <tt class="docutils literal">polname</tt>, but with each root now of multiplicity one:
-i.e. it is the &quot;square-free part&quot; of original polynomial <tt class="docutils literal">polname</tt>.</p>
-<p>Notice that <tt class="docutils literal">sturmname_1</tt> isn't necessarily the derivative of
-<tt class="docutils literal">sturmname_0</tt> due to the various normalizations.</p>
-<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></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="#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
-polynomials <tt class="docutils literal">sturmname_k_</tt>, <tt class="docutils literal">k=0,1, <span class="pre">...,</span> N</tt>, with
-<tt class="docutils literal">N</tt> being <a class="reference internal" href="#polsturmchainlength-sturmname">\PolSturmChainLength{sturmname}</a>.</p>
-<div class="admonition note">
-<p class="first admonition-title">Note</p>
-<p class="last">This behaviour was modified at <tt class="docutils literal">0.6</tt>, anyhow the macro was
-broken at <tt class="docutils literal">0.5</tt>.</p>
-</div>
-<div class="admonition hint">
-<p class="first admonition-title">Hint</p>
-<p class="last">The square-free part of <tt class="docutils literal">polname</tt> is <tt class="docutils literal">sturmname_0</tt>, and their
-quotient is the polynomial with name
-<tt class="docutils literal">sturname_\PolSturmChainLength{sturmname}_</tt>. It thus easy to
-set-up a loop iteratively computing the latter until the last one
-is a constant, thus obtaining the decomposition of an <tt class="docutils literal">f</tt> as
-a product <tt class="docutils literal">c f_1 f_2 f_3 ...</tt> of a constant and square-free (primitive)
-polynomials, where each <tt class="docutils literal">f_i</tt> divides its predecessor.</p>
-</div>
-</blockquote>
-</div>
-<div class="section" id="polsettosturmchainsignchangesat-macro-sturmname-fraction">
-<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>
-(which must be in format as acceptable by the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> macros.)</p>
-<div class="admonition note">
-<p class="first admonition-title">Note</p>
-<p>The author was lazy and did not provide rather an expandable
-variant, where one would do <tt class="docutils literal"><span class="pre">\edef\macro{\PolNbOf...}</span></tt>.</p>
-<p>This will presumably get added in a future release.</p>
-<p class="last">After some hesitation it was decided the macro would by default
-act globally. To make the scope of its macro definition local,
-use <tt class="docutils literal">[\empty]</tt> as extra optional argument.</p>
-</div>
-</blockquote>
-</div>
-<div class="section" id="polsettonbofzeroswithin-macro-sturmname-value-a-value-b">
-<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,
-value_b]</tt> (the macro first re-orders the value for <tt class="docutils literal">value_a &lt;=
-value_b</tt> to hold).</p>
-<div class="admonition note">
-<p class="first admonition-title">Note</p>
-<p>The author was lazy and did not provide rather an expandable
-variant, where one would do <tt class="docutils literal"><span class="pre">\edef\macro{\PolNbOf...}</span></tt>.</p>
-<p>This will presumably get added in future.</p>
-<p class="last">After some hesitation it was decided the macro would by default
-act globally. To make the scope of its macro definition local,
-use <tt class="docutils literal">[\empty]</tt> as extra optional argument.</p>
-</div>
-<p>See also the expandable
-<a class="reference internal" href="#polsturmnbofrootsof-sturmname-lessthanorequalto-value">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualTo{value}</a>, from
-which it is immediate (with <tt class="docutils literal">\numexpr</tt>) to create an expandable
-variant of this macro. However the difference is that this macro
-requires only <a class="reference internal" href="#poltosturm">\PolToSturm</a> to have been executed,
-whereas the expandable variant requires prior execution of
-<a class="reference internal" href="#polsturmisolatezeros">\PolSturmIsolateZeros</a>.</p>
-<p>See also the expandable
-<a class="reference internal" href="#polsturmnbwithmultofrootsof-sturmname-lessthanorequalto-value">\PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualTo{value}</a>
-which requires prior execution of
-<a class="reference internal" href="#id11">\PolSturmIsolateZeros*</a>.</p>
-</blockquote>
-</div>
-<div class="section" id="polsturmisolatezeros-sturmname">
-<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>
-<div class="admonition important">
-<p class="first admonition-title">Important</p>
-<p>The Sturm chain must have been produced by an earlier
-<a class="reference internal" href="#poltosturm-polname-sturmname">\PolToSturm{polname}{sturmname}</a>.</p>
-<p>Why does this macro ask for argument the name of Sturm chain,
-rather than the name of a polynomial? well this is mainly for
-legacy reason, and because it is accompanied by other macros for
-which it is simpler to assume the argument will be the name of an
-already computed Sturm chain.</p>
-<p>Notice that <tt class="docutils literal"><span class="pre">\PolToSturm{f}{f}</span></tt> is perfectly legal (the
-<tt class="docutils literal">sturmname</tt> can be same as the <tt class="docutils literal">polname</tt>): it defines
-polynomials <tt class="docutils literal">f_0</tt>, <tt class="docutils literal">f_1</tt>, ... having <tt class="docutils literal">f</tt> has name prefix.</p>
-<p class="last">Such a prior call
-to <tt class="docutils literal">\PolToSturm</tt> must have been made at any rate for
-<tt class="docutils literal">\PolSturmIsolateZeros</tt> to be usable.</p>
-</div>
-<p>After its execution they are two types of such intervals (stored in
-memory and accessible via macros or <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> variables, see below):</p>
-<ul class="simple">
-<li>singleton <tt class="docutils literal">{a}</tt>: then <tt class="docutils literal">a</tt> is a root, (necessarily a decimal
-number, but not all such decimal numbers are exactly identified yet).</li>
-<li>open intervals <tt class="docutils literal">(a,b)</tt>: then there is exactly one root <tt class="docutils literal">z</tt>
-such that <tt class="docutils literal">a &lt; z &lt; b</tt>, and the end points are guaranteed to not
-be roots.</li>
-</ul>
-<p>The interval boundaries are decimal numbers, originating
-in iterated decimal subdivision from initial intervals
-<tt class="docutils literal"><span class="pre">(-10^E,</span> 0)</tt> and <tt class="docutils literal">(0, 10^E)</tt> with <tt class="docutils literal">E</tt> chosen initially large
-enough so that all roots are enclosed; if zero is a root it is always
-identified as such. The non-singleton intervals are of the
-type <tt class="docutils literal">(a/10^f, <span class="pre">(a+1)/10^f)</span></tt> with <tt class="docutils literal">a</tt> an integer, which is
-neither <tt class="docutils literal">0</tt> nor <tt class="docutils literal"><span class="pre">-1</span></tt>. Hence either <tt class="docutils literal">a</tt> and <tt class="docutils literal">a+1</tt> are both positive
-or they are both negative.</p>
-<p>One does not <em>a priori</em> know what will be the lengths of these
-intervals (except that they are always powers of ten), they
-vary depending on how many digits two successive roots have in
-common in their respective decimal expansions.</p>
-<div class="admonition important">
-<p class="first admonition-title">Important</p>
-<p>If some two consecutive intervals share an end-point, no
-information is yet gained about the separation between the two
-roots which could at this stage be arbitrarily small.</p>
-<p class="last">See <a class="reference internal" href="#polrefineinterval-sturmname-index">\PolRefineInterval*{sturmname}{index}</a> which addresses
-this issue.</p>
-</div>
-<!-- This procedure is covariant
-with the independent variable ``x`` becoming ``-x``.
-Hmm, pas sûr et trop fatigué -->
-<p>The interval boundaries (and exactly found roots) are made available
-for future computations in <tt class="docutils literal">\xintexpr</tt>-essions or polynomial
-definitions as variables <tt class="docutils literal">&lt;sturmname&gt;L_1</tt>,
-<tt class="docutils literal">&lt;sturmname&gt;L_2</tt>, etc..., for the left end-points and
-<tt class="docutils literal">&lt;sturmname&gt;R_1</tt>, <tt class="docutils literal">&lt;sturmname&gt;R_2</tt>, ..., for the right
-end-points.</p>
-<p>Thus for example, if <tt class="docutils literal">sturmname</tt> is <tt class="docutils literal">f</tt>, one can use the
-<a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> variables <tt class="docutils literal">fL_1</tt>, <tt class="docutils literal">fL_2</tt>, ... to refer in expressions
-to the left end-points (or to the exact root, if left and right end
-points coincide). Additionally, <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> variable <tt class="docutils literal">fZ_1_isknown</tt>
-will have value <tt class="docutils literal">1</tt> if the root in the first interval is known,
-and <tt class="docutils literal">0</tt> otherwise. And similarly for the other intervals.</p>
-<p>Also, macros <a class="reference internal" href="#polsturmisolatedzeroleft-sturmname-index">\PolSturmIsolatedZeroLeft{sturmname}{index}</a> and
-<a class="reference internal" href="#polsturmisolatedzeroright-sturmname-index">\PolSturmIsolatedZeroRight{sturmname}{index}</a> are provided which
-expand to these same values, written in decimal notation (i.e.
-pre-processed by <a class="reference internal" href="#poldectostring">\PolDecToString</a>.) And there
-is also <a class="reference internal" href="#polsturmifzeroexactlyknown-sturmname-index-a-b">\PolSturmIfZeroExactlyKnown{sturmname}{index}{A}{B}</a>.</p>
-<div class="admonition important">
-<p class="first admonition-title">Important</p>
-<p class="last">Trailing zeroes in the stored decimal numbers accessible via the
-macros are significant: they are also present in the decimal
-expansion of the exact root.</p>
-</div>
-<p>These variables and macros are automatically updated when one next
-uses macros such as <a class="reference internal" href="#polrefineinterval-sturmname-index">\PolRefineInterval*{sturmname}{index}</a>.</p>
-<p>The start of decimal expansion of a positive <tt class="docutils literal">k</tt>-th root is given
-by <a class="reference internal" href="#polsturmisolatedzeroleft">\PolSturmIsolatedZeroLeft{sturmname}{k}</a>, and for a negative root it is given
-by <a class="reference internal" href="#polsturmisolatedzeroright">PolSturmIsolatedZeroRight{sturmname}{k}</a>. These two decimal
-numbers are either both zero or both of the same sign.</p>
-<p>The number of distinct roots is obtainable expandably as
-<a class="reference internal" href="#polsturmnbofisolatedzeros-sturmname">\PolSturmNbOfIsolatedZeros{sturmname}</a>.</p>
-<p>Furthermore
-<a class="reference internal" href="#polsturmnbofrootsof-sturmname-lessthanorequalto-value">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualTo{value}</a> and
-<a class="reference internal" href="#polsturmnbofrootsof-sturmname-lessthanorequaltoexpr-expression">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualToExpr{expression}</a>.
-will expandably compute respectively the number of real roots at
-most equal to <tt class="docutils literal">value</tt> or <tt class="docutils literal">expression</tt>, and the same but with
-multiplicities.</p>
-<div class="admonition note">
-<p class="first admonition-title">Note</p>
-<p class="last">In the current implementation the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> variables
-and <a class="reference external" href="http://www.ctan.org/pkg/xint">xinttools</a> arrays are globally defined. On the
-other hand the Sturm sequence polynomials obey the current scope.</p>
-</div>
-<div class="admonition note">
-<p class="first admonition-title">Note</p>
-<p>As all computations are done <em>exactly</em> there can be no errors...
-apart those due to bad coding by author. The results are exact
-bounds for the mathematically exact real roots.</p>
-<p class="last">Future releases will perhaps also provide macros based on Newton
-or Regula Falsi methods. Exact computations with such methods
-lead however quickly to very big fractions, and this forces usage
-of some rounding scheme for the abscissas if computation times
-are to remain reasonable. This raises issues of its own, which
-are studied in numerical mathematics.</p>
-</div>
-</blockquote>
-</div>
-<div class="section" id="id12">
-<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
-multiplicities (of the real roots):
-after executing this macro,
-<a class="reference internal" href="#polsturmisolatedzeromultiplicity-sturmname-index">\PolSturmIsolatedZeroMultiplicity{sturmname}{index}</a> will expand
-to the multiplicity of the root located in the <tt class="docutils literal">index</tt>-th
-interval (intervals are enumerated from left to right, with index
-starting at <tt class="docutils literal">1</tt>).</p>
-<p>Furthermore, if for example the <tt class="docutils literal">sturmname</tt> is <tt class="docutils literal">f</tt>, <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a>
-variables <tt class="docutils literal">fM_1</tt>, <tt class="docutils literal">fM_2</tt>... hold the multiplicities thus
-computed.</p>
-<div class="admonition note">
-<p class="first admonition-title">Note</p>
-<p class="last">It is <strong>not</strong> necessary to have executed the <a class="reference internal" href="#id9">PolToSturm*</a> starred
-variant, as the non-starred variant keeps internally the memory of the
-original GCD (and even of the full non-normalized original Sturm
-chain), even though it does not make the declarations as <em>user-level</em>
-genuine polynomials.</p>
-</div>
-<p>See <a class="reference internal" href="#the-degree-nine-polynomial-with-0-99-0-999-0-9999-as-triple-roots">The degree nine polynomial with 0.99, 0.999, 0.9999 as triple
-roots</a> for an example.</p>
-</blockquote>
-</div>
-<div class="section" id="id14">
-<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>
-roots.</p>
-<div class="admonition note">
-<p class="first admonition-title">Note</p>
-<p class="last">After execution of this macro, a root is &quot;known&quot; if and only if
-it is rational.</p>
-</div>
-<p>Furthermore, primitive polynomial <tt class="docutils literal">sturmname_sqf_norr</tt> is created
-to match the (square-free) <tt class="docutils literal">sturmname_0</tt> from which all rational
-roots have been removed (see <a class="reference internal" href="#polexprsetup">\polexprsetup</a> for customizing this
-name). The number of distinct rational roots is thus the difference
-between the degrees of these two polynomials (see also
-<a class="reference internal" href="#polsturmnbofrationalroots-sturmname">\PolSturmNbOfRationalRoots{sturmname}</a>).</p>
-<p>And <tt class="docutils literal">sturmname_norr</tt> is <tt class="docutils literal">sturmname_0_</tt> from which all rational
-roots have been removed (see <a class="reference internal" href="#polexprsetup">\polexprsetup</a>), i.e. it contains
-the irrational roots of the original polynomial, with the same
-multiplicities.</p>
-<p>See <a class="reference internal" href="#a-degree-five-polynomial-with-three-rational-roots">A degree five polynomial with three rational
-roots</a> for an example.</p>
-</blockquote>
-</div>
-<div class="section" id="polsturmisolatezerosandgetmultiplicities-sturmname">
-<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="#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="#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
-<tt class="docutils literal">sturmname_norr</tt> with no rational roots) except that it does <em>not</em>
-compute the multiplicities of the <em>non-rational</em> roots.</p>
-<div class="admonition note">
-<p class="first admonition-title">Note</p>
-<p class="last">There is no macro to find the rational roots but not compute
-their multiplicities at the same time.</p>
-</div>
-<div class="admonition attention">
-<p class="first admonition-title">Attention!</p>
-<p>This macro does <em>not</em> define <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> variables
-<tt class="docutils literal">sturmnameM_1</tt>, <tt class="docutils literal">sturmnameM_2</tt>, ... holding the
-multiplicities and it leaves the multiplicity array (whose accessor
-is <a class="reference internal" href="#polsturmisolatedzeromultiplicity-sturmname-index">\PolSturmIsolatedZeroMultiplicity{sturmname}{index}</a>) into
-a broken state, as all non-rational roots will supposedly have
-multiplicity one. This means that the output of
-<a class="reference internal" href="#id18">\PolPrintIntervals*</a> for example will be
-erroneous for the intervals with irrational roots.</p>
-<p class="last">I decided to document it because finding multiplicities of the
-non rational roots is somewhat costly, and one may be interested
-only into finding the rational roots (of course random
-polynomials with integer coefficients will not have <em>any</em>
-rational root anyhow).</p>
-</div>
-</blockquote>
-</div>
-<div class="section" id="polrefineinterval-sturmname-index">
-<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
-interval to have both its end-points distinct from the end-points of
-the original interval. This means that the <tt class="docutils literal">k</tt>th root is then
-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="#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="#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&lt;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="#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
-each of them to have length at most <tt class="docutils literal">10^E</tt> (length will be shorter
-than <tt class="docutils literal">10^E</tt> in output only if it did not change or became zero.)</p>
-<p>This means that decimal expansions of all roots will be known with
-<tt class="docutils literal"><span class="pre">-E</span></tt> digits (for <tt class="docutils literal">E&lt;0</tt>) after decimal mark.</p>
-</blockquote>
-</div>
-<div class="section" id="polprintintervals-varname-sturmname">
-<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
-specify a replacement for the default <tt class="docutils literal">Z</tt>). This will be done (by
-default) in a
-math mode <tt class="docutils literal">array</tt>, one interval per row, and pattern <tt class="docutils literal">rcccl</tt>,
-where the second and fourth column hold the <tt class="docutils literal">&lt;</tt> sign, except when
-the interval reduces to a singleton, which means the root is known
-exactly.</p>
-<div class="admonition attention">
-<p class="first admonition-title">Attention!</p>
-<p class="last">This macro was refactored at 0.7, its default output remained
-identical but the ways to customize it got completely
-modified.</p>
-</div>
-<p>See next macros which govern its output.</p>
-</blockquote>
-<div class="section" id="polprintintervalsnorealroots">
-<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>
-<pre class="literal-block">
-\newcommand\PolPrintIntervalsNoRealRoots{}
-</pre>
-</blockquote>
-</div>
-<div class="section" id="polprintintervalsbeginenv">
-<h3><a class="toc-backref" href="#id88"><tt class="docutils literal">\PolPrintIntervalsBeginEnv</tt></a></h3>
-<blockquote>
-<p>Default definition:</p>
-<pre class="literal-block">
-\newcommand\PolPrintIntervalsBeginEnv{\[\begin{array}{rcccl}}
-</pre>
-</blockquote>
-</div>
-<div class="section" id="polprintintervalsendenv">
-<h3><a class="toc-backref" href="#id89"><tt class="docutils literal">\PolPrintIntervalsEndEnv</tt></a></h3>
-<blockquote>
-<p>Default definition:</p>
-<pre class="literal-block">
-\newcommand\PolPrintIntervalsEndEnv{\end{array}\]}
-</pre>
-</blockquote>
-</div>
-<div class="section" id="polprintintervalsknownroot">
-<h3><a class="toc-backref" href="#id90"><tt class="docutils literal">\PolPrintIntervalsKnownRoot</tt></a></h3>
-<blockquote>
-<p>Default definition:</p>
-<pre class="literal-block">
-\newcommand\PolPrintIntervalsKnownRoot{%
- &amp;&amp;\PolPrintIntervalsTheVar_{\PolPrintIntervalsTheIndex}%
- &amp;=&amp;\PolPrintIntervalsPrintExactZero
-}
-</pre>
-</blockquote>
-</div>
-<div class="section" id="polprintintervalsunknownroot">
-<h3><a class="toc-backref" href="#id91"><tt class="docutils literal">\PolPrintIntervalsUnknownRoot</tt></a></h3>
-<blockquote>
-<p>Default definition:</p>
-<pre class="literal-block">
-\newcommand\PolPrintIntervalsUnknownRoot{%
- \PolPrintIntervalsPrintLeftEndPoint&amp;&lt;&amp;%
- \PolPrintIntervalsTheVar_{\PolPrintIntervalsTheIndex}&amp;&lt;&amp;%
- \PolPrintIntervalsPrintRightEndPoint
-}
-</pre>
-</blockquote>
-</div>
-<div class="section" id="id15">
-<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">
-\newcommand\PolPrintIntervalsPrintExactZero{\PolPrintIntervalsTheLeftEndPoint}
-</pre>
-</blockquote>
-</div>
-<div class="section" id="id16">
-<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">
-\newcommand\PolPrintIntervalsPrintLeftEndPoint{\PolPrintIntervalsTheLeftEndPoint}
-</pre>
-</blockquote>
-</div>
-<div class="section" id="id17">
-<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">
-\newcommand\PolPrintIntervalsPrintRightEndPoint{\PolPrintIntervalsTheRightEndPoint}
-</pre>
-</blockquote>
-</div>
-</div>
-<div class="section" id="id19">
-<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
-example of customization.</p>
-<p>As replacement for <a class="reference internal" href="#polprintintervalsknownroot">\PolPrintIntervalsKnownRoot</a>,
-<a class="reference internal" href="#polprintintervalsprintexactzero">\PolPrintIntervalsPrintExactZero</a>,
-<a class="reference internal" href="#polprintintervalsunknownroot">\PolPrintIntervalsUnknownRoot</a> it uses its own
-<tt class="docutils literal"><span class="pre">\POL&#64;&#64;PrintIntervals...</span></tt> macros. We only reproduce here one
-definition:</p>
-<pre class="literal-block">
-\newcommand\POL&#64;&#64;PrintIntervalsPrintExactZero{%
- \displaystyle
- \xintSignedFrac{\PolPrintIntervalsTheLeftEndPoint}%
-}%
-</pre>
-<p>Multiplicities are printed using this auxiliary macro:</p>
-</blockquote>
-<div class="section" id="polprintintervalsprintmultiplicity">
-<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">
-\newcommand\PolPrintIntervalsPrintMultiplicity{(\mbox{mult. }\PolPrintIntervalsTheMultiplicity)}
-</pre>
-</blockquote>
-</div>
-</div>
-<div class="section" id="polmapcoeffs-macro-polname">
-<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
-<tt class="docutils literal">\macro</tt>. The degree is adjusted as necessary if some leading
-coefficients vanish after the operation. In replacement text of
-<tt class="docutils literal">\macro</tt>, <tt class="docutils literal">\index</tt> expands to the coefficient index (which is
-defined to be zero for the constant term).</p>
-<p>Notice that <tt class="docutils literal">\macro</tt> will have to handle inputs of the shape
-<tt class="docutils literal">A/B[N]</tt> (<a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> internal notation). This means that it probably
-will have to be expressed in terms of macros from <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> package.</p>
-<p>Example:</p>
-<pre class="literal-block">
-\def\foo#1{\xintMul{#1}{\the\numexpr\index*\index\relax}}
-</pre>
-<p>(or with <tt class="docutils literal"><span class="pre">\xintSqr{\index}</span></tt>) to replace <tt class="docutils literal">n</tt>-th coefficient
-<tt class="docutils literal">f_n</tt> by <tt class="docutils literal">f_n*n^2</tt>.</p>
-</blockquote>
-</div>
-<div class="section" id="polreducecoeffs-polname">
-<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
-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="#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
-coefficient is internally <tt class="docutils literal">A/B[N]</tt>, then <tt class="docutils literal">A/B</tt> is reduced to
-smallest terms, but the <tt class="docutils literal">10^N</tt> part is kept as is. Note: if the
-polynomial is freshly defined directly via <a class="reference internal" href="#polfromcsv">\PolFromCSV</a> its coefficients might still be internally in some
-format like <tt class="docutils literal">1.5e7</tt>; the macro will anyhow always first do the
-needed conversion to strict format <tt class="docutils literal">A/B[N]</tt>.</p>
-<p>Evaluations with polynomials treated by this can be much faster than
-with those handled by the non-starred variant
-<a class="reference internal" href="#polreducecoeffs-polname">\PolReduceCoeffs{polname}</a>: as the numerators and denominators
-remain smaller, this proves very beneficial in favorable cases
-(especially when the coefficients are decimal numbers) to the
-expansion speed of the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> macros used internally by
-<a class="reference internal" href="#polevalat">\PolEval</a>.</p>
-</blockquote>
-</div>
-<div class="section" id="polmakemonic-polname">
-<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
-done automatically, due to the case the original polynomial had integer
-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="#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
-coefficient is not modified.</blockquote>
-</div>
-</div>
-<div class="section" id="expandable-macros">
-<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="#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="#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="#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="#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="#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
-<a class="reference internal" href="#polgenfloatvariant-polname">\PolGenFloatVariant{polname}</a>), with already rounded
-coefficients. <a class="footnote-reference" href="#id24" id="id22">[2]</a> To use the <em>exact coefficients</em> with <em>exactly
-executed</em> additions and multiplications, just insert it in the float
-expression as in this example: <a class="footnote-reference" href="#id25" id="id23">[3]</a></p>
-<pre class="literal-block">
-\xintthefloatexpr 3.27*\xintexpr f(2.53)\relax^2\relax
-</pre>
-<p>The <tt class="docutils literal">f(2.53)</tt> is exactly computed then rounded at the time of
-getting raised to the power <tt class="docutils literal">2</tt>. Moving the <tt class="docutils literal">^2</tt> inside, that
-operation would also be treated exactly.</p>
-<table class="docutils footnote" frame="void" id="id24" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id22">[2]</a></td><td>Anyway each floating point operation starts by rounding its
-operands to the floating point precision.</td></tr>
-</tbody>
-</table>
-<table class="docutils footnote" frame="void" id="id25" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id23">[3]</a></td><td>The <tt class="docutils literal">\xintexpr</tt> here could be <tt class="docutils literal">\xinttheexpr</tt> but that
-would be less efficient. Cf. <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> documentation about
-nested expressions.</td></tr>
-</tbody>
-</table>
-</blockquote>
-</div>
-<div class="section" id="polfloateval-polname-at-fraction">
-<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="#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
-used if needed in the execution of <a class="reference internal" href="#poltypesetmonomialcmd">\PolTypesetMonomialCmd</a>,
-e.g. to insert a <tt class="docutils literal">\cdot</tt> in front of <tt class="docutils literal"><span class="pre">\PolVar^{\PolIndex}</span></tt> if
-the coefficient is not plus or minus one.</p>
-<p>The macro will execute <tt class="docutils literal">A</tt> if the coefficient has been found to be
-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="#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="#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="#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="#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
-with integer coefficients having no common prime divisor.</p>
-<p>See <a class="reference internal" href="#polmakeprimitive">\PolMakePrimitive</a>.</p>
-</blockquote>
-</div>
-<div class="section" id="poltoexpr-polname">
-<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">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id26">[4]</a></td><td>in a <tt class="docutils literal">\write</tt>, <tt class="docutils literal">\edef</tt>, or <tt class="docutils literal"><span class="pre">\csname...\endcsname</span></tt>, but
-not under <tt class="docutils literal"><span class="pre">\romannumeral-`0</span></tt>.</td></tr>
-</tbody>
-</table>
-<p>By default zero coefficients are skipped (issue <tt class="docutils literal">\poltoexpralltrue</tt> to
-get all of them in output).</p>
-<p>By default, no <tt class="docutils literal">+</tt> sign before negative coefficients, for
-compliance with Maple input format (but see
-<a class="reference internal" href="#poltoexprtermprefix-raw-coeff">\PolToExprTermPrefix{raw_coeff}</a>.) Also, like the default
-behaviour of <a class="reference internal" href="#poltypeset-polname">\PolTypeset{polname}</a>, does not print (for the non
-constant terms) coefficients equal to plus or minus one. The degree
-one monomial is output as <tt class="docutils literal">x</tt>, not <tt class="docutils literal">x^1</tt>. Complete customization is
-possible, see next macros.</p>
-<p>Of course <tt class="docutils literal">\PolToExpr{f}</tt> can be inserted in a <tt class="docutils literal">\poldef</tt>, as the
-latter expands token by token, hence will force complete expansion
-of <tt class="docutils literal">\PolToExpr{f}</tt>, but a simple <tt class="docutils literal">f(x)</tt> is more efficient for
-the identical result.</p>
-</blockquote>
-<div class="section" id="poltoexproneterm-raw-coeff-number">
-<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
-coefficients of absolute value <tt class="docutils literal">1</tt> not be printed explicitely
-(except of course for the constant term). Also by default, the
-monomial of degree one is <tt class="docutils literal">x</tt> not <tt class="docutils literal">x^1</tt>, and <tt class="docutils literal">x^0</tt> is skipped.</p>
-<p>For compatibility with Maple input requirements, by default a <tt class="docutils literal">*</tt>
-always precedes the <tt class="docutils literal">x^number</tt>, except if the coefficient is a one
-or a minus one. See <a class="reference internal" href="#poltoexprtimes">\PolToExprTimes</a>.</p>
-</blockquote>
-</div>
-<div class="section" id="poltoexpronetermstylea-raw-coeff-number">
-<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="#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">
-2*x^11/3+3*x^8/7-x^5-x^4/4-x^3-x^2/2-2*x+1
-</pre>
-<p>issue <tt class="docutils literal">\let\PolToExprOneTerm\PolToExprOneTermStyleB</tt> before usage of
-<tt class="docutils literal">\PolToExpr</tt>. Note that then <tt class="docutils literal">\PolToExprCmd</tt> isn't used at all.
-To revert to package default, issue
-<tt class="docutils literal">\let\PolToExprOneTerm\PolToExprOneTermStyleA</tt>.</p>
-<p>To suppress the <tt class="docutils literal">*</tt>'s, cf. <a class="reference internal" href="#poltoexprtimes">\PolToExprTimes</a>.</p>
-</blockquote>
-</div>
-<div class="section" id="poltoexprcmd-raw-coeff">
-<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
-equal to plus or minus one), and it defaults to
-<tt class="docutils literal"><span class="pre">\xintPRaw{\xintRawWithZeros{#1}}</span></tt>. One will have to redefine it
-to <tt class="docutils literal"><span class="pre">\xintIrr{#1}</span></tt> or to <tt class="docutils literal"><span class="pre">\xintPRaw{\xintIrr{#1}}</span></tt> to obtain in the
-output forcefully reduced coefficients.</blockquote>
-</div>
-<div class="section" id="poltoexprtermprefix-raw-coeff">
-<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="#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="#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
-<tt class="docutils literal">*</tt>. Redefine the macro to expand to nothing to get rid of it (but
-this will give output incompatible with some professional computer
-algebra software).</blockquote>
-</div>
-</div>
-<div class="section" id="id31">
-<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="#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
-point format.</p>
-<div class="admonition note">
-<p class="first admonition-title">Note</p>
-<p>It is not necessary to have issued
-<a class="reference internal" href="#polgenfloatvariant-polname">\PolGenFloatVariant{polname}</a>. The rounded coefficients are
-not easily recoverable from the <tt class="docutils literal">\xintfloatexpr</tt> polynomial
-function hence <tt class="docutils literal">\PolToFloatExprCmd</tt> operates from the <em>exact</em>
-coefficients anew.</p>
-<p class="last">Attention that both macros obey the prevailing float precision.
-If it is changed between those macro calls, then a mismatch
-exists between the coefficients as used in <tt class="docutils literal">\xintfloatexpr</tt> and
-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="#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="#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>
-<div class="admonition caution">
-<p class="first admonition-title">Caution!</p>
-<p>Currently (<a class="reference external" href="http://www.ctan.org/pkg/xint">xint</a> <tt class="docutils literal">1.3c</tt>) <tt class="docutils literal">\xintFloat{0}</tt> outputs <tt class="docutils literal">0.e0</tt>
-which is perfectly acceptable input for Python, but not for
-Maple. Thus, one should better leave the <a class="reference internal" href="#poltoexprallfalse">\poltoexprallfalse</a>
-toggle to its default <tt class="docutils literal">\iffalse</tt> state, if one intends to use
-the output in a Maple worksheet.</p>
-<p>But even then the zero polynomial will cause a problem. Workaround:</p>
-<pre class="literal-block">
-\renewcommand\PolToFloatExprCmd[1]{\xintiiifZero{#1}{0.0}{\xintFloat{#1}}}
-</pre>
-<p class="last">Usage of <tt class="docutils literal">\xintiiifZero</tt> and not <tt class="docutils literal">\xintifZero</tt> is only for
-optimization (I can't help it) because <tt class="docutils literal">#1</tt> is known to be
-in <tt class="docutils literal">xintfrac</tt> raw format.</p>
-</div>
-</blockquote>
-</div>
-</div>
-<div class="section" id="id35">
-<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="#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
-(the zero polynomial does give <tt class="docutils literal">{0/1[0]}</tt> and not an
-empty output.)</blockquote>
-</div>
-<div class="section" id="poltocsv-polname">
-<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="#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>
-<p>See <a class="reference internal" href="#poltosturm-polname-sturmname">\PolToSturm{polname}{sturmname}</a>.</p>
-</blockquote>
-</div>
-<div class="section" id="polsturmifzeroexactlyknown-sturmname-index-a-b">
-<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>
-<div class="admonition note">
-<p class="first admonition-title">Note</p>
-<p class="last"><tt class="docutils literal">index</tt> is allowed to be something like <tt class="docutils literal">1+2*3</tt> as it is fed
-to <tt class="docutils literal"><span class="pre">\the\numexpr...\relax</span></tt>.</p>
-</div>
-</blockquote>
-</div>
-<div class="section" id="polsturmisolatedzeroleft-sturmname-index">
-<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>
-<div class="admonition note">
-<p class="first admonition-title">Note</p>
-<p class="last">Of course, this is kept updated by macros such as
-<a class="reference internal" href="#polrefineinterval-n">\PolRefineInterval{sturmname}{index}</a>.</p>
-</div>
-<p>The value is pre-formatted using <a class="reference internal" href="#poldectostring">\PolDecTostring</a>.</p>
-</blockquote>
-</div>
-<div class="section" id="polsturmisolatedzeroright-sturmname-index">
-<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
-possibly refined afterwards.</p>
-<p>The value is pre-formatted using <a class="reference internal" href="#poldectostring">\PolDecTostring</a>.</p>
-</blockquote>
-</div>
-<div class="section" id="polsturmisolatedzeromultiplicity-sturmname-index">
-<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>
-<div class="admonition attention">
-<p class="first admonition-title">Attention!</p>
-<p class="last">A prior execution of <a class="reference internal" href="#id12">\PolSturmIsolateZeros*{sturmname}</a> is mandatory.</p>
-</div>
-<p>See <a class="reference internal" href="#the-degree-nine-polynomial-with-0-99-0-999-0-9999-as-triple-roots">The degree nine polynomial with 0.99, 0.999, 0.9999 as triple
-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="#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">&lt;sturmname&gt;_0</tt>, i.e. the number of distinct real roots of the
-polynomial originally used to create the Sturm chain via
-<a class="reference internal" href="#poltosturm-polname-sturmname">\PolToSturm{polname}{sturmname}</a>.</blockquote>
-<div class="admonition warning">
-<p class="first admonition-title">Warning</p>
-<p class="last">The next few macros counting roots, with or without multiplicities,
-less than or equal to some value, are under evaluation and may be
-removed from the package if their utility is judged to be not high
-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="#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
-number of fraction recognizable by the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> macros).</p>
-<div class="admonition attention">
-<p class="first admonition-title">Attention!</p>
-<p><a class="reference internal" href="#polsturmisolatezeros-sturmname">\PolSturmIsolateZeros{sturmname}</a> must have been executed
-beforehand.</p>
-<p class="last">And the argument is a <tt class="docutils literal">sturmname</tt>, not a <tt class="docutils literal">polname</tt> (this is
-why the macro contains Sturm in its name), simply to be reminded
-of the above constraint.</p>
-</div>
-</blockquote>
-</div>
-<div class="section" id="polsturmnbofrootsof-sturmname-lessthanorequaltoexpr-expression">
-<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
-given <tt class="docutils literal">expression</tt>.</p>
-<div class="admonition attention">
-<p class="first admonition-title">Attention!</p>
-<p class="last"><a class="reference internal" href="#polsturmisolatezeros-sturmname">\PolSturmIsolateZeros{sturmname}</a> must have been executed
-beforehand.</p>
-</div>
-</blockquote>
-</div>
-<div class="section" id="polsturmnbwithmultofrootsof-sturmname-lessthanorequalto-value">
-<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
-or equal to the given <tt class="docutils literal">value</tt>.</p>
-<div class="admonition attention">
-<p class="first admonition-title">Attention!</p>
-<p class="last"><a class="reference internal" href="#id12">\PolSturmIsolateZeros*{sturmname}</a> (or the double starred
-variant) must have been executed beforehand.</p>
-</div>
-</blockquote>
-</div>
-<div class="section" id="polsturmnbwithmultofrootsof-sturmname-lessthanorequaltoexpr-expression">
-<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>
-<div class="admonition attention">
-<p class="first admonition-title">Attention!</p>
-<p class="last"><a class="reference internal" href="#id12">\PolSturmIsolateZeros*{sturmname}</a> (or the double starred
-variant) must have been executed beforehand.</p>
-</div>
-</blockquote>
-</div>
-</div>
-<div class="section" id="polsturmnbofrationalroots-sturmname">
-<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">
-<p class="first admonition-title">Attention!</p>
-<p class="last"><a class="reference internal" href="#id14">\PolSturmIsolateZeros**{sturmname}</a> must have been executed
-beforehand.</p>
-</div>
-</blockquote>
-</div>
-<div class="section" id="polsturmnbofrationalrootswithmultiplicities-sturmname">
-<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">
-<p class="first admonition-title">Attention!</p>
-<p class="last"><a class="reference internal" href="#id14">\PolSturmIsolateZeros**{sturmname}</a> must have been executed
-beforehand.</p>
-</div>
-</blockquote>
-</div>
-<div class="section" id="polsturmrationalroot-sturmname-k">
-<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>
-<div class="admonition attention">
-<p class="first admonition-title">Attention!</p>
-<p class="last"><a class="reference internal" href="#id14">\PolSturmIsolateZeros**{sturmname}</a> must have been executed
-beforehand.</p>
-</div>
-</blockquote>
-</div>
-<div class="section" id="polsturmrationalrootindex-sturmname-k">
-<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
-<a class="reference internal" href="#polsturmrationalroot-sturmname-k">\PolSturmRationalRoot{sturmname}{k}</a> is equivalent to this
-nested call:</p>
-<pre class="literal-block">
-\PolSturmIsolatedZeroLeft{sturmname}{\PolSturmRationalRootIndex{sturmname}{k}}
-</pre>
-<div class="admonition attention">
-<p class="first admonition-title">Attention!</p>
-<p class="last"><a class="reference internal" href="#id14">\PolSturmIsolateZeros**{sturmname}</a> must have been executed
-beforehand.</p>
-</div>
-</blockquote>
-</div>
-<div class="section" id="polsturmrationalrootmultiplicity-sturmname-k">
-<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">
-<p class="first admonition-title">Attention!</p>
-<p class="last"><a class="reference internal" href="#id14">\PolSturmIsolateZeros**{sturmname}</a> must have been executed
-beforehand.</p>
-</div>
-</blockquote>
-</div>
-<div class="section" id="polintervalwidth-sturmname-index">
-<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="#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
-default for the former) and of <a class="reference internal" href="#polprintintervalsprintleftendpoint">\PolPrintIntervalsPrintLeftEndPoint</a>,
-<a class="reference internal" href="#polprintintervalsprintrightendpoint">\PolPrintIntervalsPrintRightEndPoint</a> (used in the default for the
-latter).</p>
-<div class="admonition attention">
-<p class="first admonition-title">Attention!</p>
-<p class="last">Some macros formerly mentioned here got removed at 0.7:
-<tt class="docutils literal">\PolPrintIntervalsTheEndPoint</tt>,
-<tt class="docutils literal"><span class="pre">\PolIfEndPointIsPositive{A}{B}</span></tt>,
-<tt class="docutils literal"><span class="pre">\PolIfEndPointIsNegative{A}{B}</span></tt>,
-<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="#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="#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="#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="#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
-used with arguments the Sturm chain name and interval index returned
-by <a class="reference internal" href="#polprintintervalsthesturmname">\PolPrintIntervalsTheSturmName</a> and
-<a class="reference internal" href="#polprintintervalstheindex">\PolPrintIntervalsTheIndex</a>.</blockquote>
-</div>
-<div class="section" id="polprintintervalstherightendpoint">
-<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="#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
-double-starred) variant of <a class="reference internal" href="#polsturmisolatezeros">\PolSturmIsolateZeros</a> was used earlier.</blockquote>
-</div>
-</div>
-<div class="section" id="poldectostring-decimal-number">
-<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
-the name <tt class="docutils literal">\xintDecToString</tt>, and the <tt class="docutils literal">polexpr</tt> macro is simply
-now an alias to it.</p>
-<p>For example
-<tt class="docutils literal"><span class="pre">\PolDecToString{123.456e-8}</span></tt> will expand to <tt class="docutils literal">0.00000123456</tt>
-and <tt class="docutils literal"><span class="pre">\PolDecToString{123.450e-8}</span></tt> to <tt class="docutils literal">0.00000123450</tt> which
-illustrates that trailing zeros are not trimmed. To trim trailing
-zeroes, one can use <tt class="docutils literal"><span class="pre">\PolDecToString{\xintREZ{#1}}</span></tt>.</p>
-<p>The precise behaviour of this macro may evolve in future releases of
-<a class="reference external" href="http://www.ctan.org/pkg/xint">xint</a>.</p>
-</blockquote>
-</div>
-</div>
-<div class="section" id="booleans-with-default-setting-as-indicated">
-<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="#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
-polynomials are defined.</p>
-<div class="admonition caution">
-<p class="first admonition-title">Caution!</p>
-<p class="last">The macro meanings as written to the log are to be considered
-unstable and undocumented internal structures.</p>
-</div>
-</blockquote>
-</div>
-<div class="section" id="poltypesetallfalse">
-<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="#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="#id157"><tt class="docutils literal">\polexprsetup</tt></a></h1>
-<blockquote>
-<p>Serves to customize the package. Currently only two keys are
-recognized:</p>
-<ul class="simple">
-<li><tt class="docutils literal">norr</tt>: the postfix that <a class="reference internal" href="#id14">\PolSturmIsolateZeros**{sturmname}</a>
-should append to <tt class="docutils literal">sturmname</tt> to declare the primitive polynomial
-obtained from original one after removal of all rational roots.
-The default value is <tt class="docutils literal">_norr</tt> (standing for “no rational roots”).</li>
-<li><tt class="docutils literal">sqfnorr</tt>: the postfix that <a class="reference internal" href="#id14">\PolSturmIsolateZeros**{sturmname}</a>
-should append to <tt class="docutils literal">sturmname</tt> to declare the primitive polynomial
-obtained from original one after removal of all rational roots and
-suppression of all multiplicities.
-The default value is <tt class="docutils literal">_sqf_norr</tt> (standing for “square-free with
-no rational roots”).</li>
-</ul>
-<p>The package executes <tt class="docutils literal">\polexprsetup{norr=_norr,
-sqfnorr=_sqf_norr}</tt> as default.</p>
-</blockquote>
-</div>
-<div class="section" id="technicalities">
-<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
-whole thing was already part of a macro argument, in such cases one
-can use <a class="reference internal" href="#id2">\PolDef{f}{P(x)}</a>
-rather. The colon in <tt class="docutils literal">:=</tt> may be active with no consequences.</p>
-</li>
-<li><p class="first">As a consequence of <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> addition and subtraction always using
-least common multiples for the denominators <a class="footnote-reference" href="#id37" id="id36">[5]</a>, user-chosen common
-denominators survive additions and multiplications. For example, this:</p>
-<pre class="literal-block">
-\poldef P(x):= 1/2 + 2/2*x + 3/2*x^3 + 4/2*x^4;
-\poldef Q(x):= 1/3 + (2/3)x + (3/3)x^3 + (4/3)x^4;
-\poldef PQ(x):= P(x)Q(x);
-</pre>
-<p>gives internally the polynomial:</p>
-<pre class="literal-block">
-1/6+4/6*x^1+4/6*x^2+6/6*x^3+20/6*x^4+16/6*x^5+9/6*x^6+24/6*x^7+16/6*x^8
-</pre>
-<p>where all coefficients have the same denominator 6. Notice though that
-<tt class="docutils literal">\PolToExpr{PQ}</tt> outputs the <tt class="docutils literal">6/6*x^3</tt> as <tt class="docutils literal">x^3</tt> because (by
-default) it recognizes and filters out coefficients equal to one or
-minus one (since release <tt class="docutils literal">0.3</tt>). One can use for example
-<tt class="docutils literal">\PolToCSV{PQ}</tt> to see the internally stored coefficients.</p>
-<table class="docutils footnote" frame="void" id="id37" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id36">[5]</a></td><td><p class="first last">prior to <tt class="docutils literal">0.4.1</tt>, <tt class="docutils literal">polexpr</tt> used to temporarily patch
-during the parsing of polynomials the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> macros. This
-patch was backported to <a class="reference external" href="http://www.ctan.org/pkg/xint">xint</a> at release <tt class="docutils literal">1.3</tt>.</p>
-</td></tr>
-</tbody>
-</table>
-</li>
-<li><p class="first"><a class="reference internal" href="#poldiff-polname-1-polname-2">\PolDiff{polname_1}{polname_2}</a> always applies <tt class="docutils literal">\xintIrr</tt> to the
-resulting coefficients, except that the <em>power of ten</em> part <tt class="docutils literal">[N]</tt>
-(for example an input in scientific notation such as <tt class="docutils literal">1.23e5</tt> gives
-<tt class="docutils literal">123/1[3]</tt> internally in xintfrac) is not taken into account in the
-reduction of the fraction. This is tentative and may change.</p>
-<p>Same remark for <a class="reference internal" href="#polantidiff-polname-1-polname-2">\PolAntiDiff{polname_1}{polname_2}</a>.</p>
-</li>
-<li><p class="first">Currently, the package stores all coefficients from index <tt class="docutils literal">0</tt> to
-index equal to the polynomial degree inside a single macro, as a list.
-This data structure is obviously very inefficient for polynomials of
-high degree and few coefficients (as an example with <tt class="docutils literal">\poldef
-<span class="pre">f(x):=x^1000</span> + x^500;</tt> the subsequent definition <tt class="docutils literal">\poldef <span class="pre">g(x):=</span>
-<span class="pre">f(x)^2;</span></tt> will do of the order of 1,000,000 multiplications and
-additions involvings only zeroes... which does take time). This
-may change in the future.</p>
-</li>
-<li><p class="first">As is to be expected internal structures of the package are barely
-documented and unstable. Don't use them.</p>
-</li>
-</ul>
-</div>
-<div class="section" id="change-log">
-<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">
-<li>The <a class="reference internal" href="#poldef">\poldef</a> parser itself,</li>
-<li>Differentiation and anti-differentiation,</li>
-<li>Euclidean division and GCDs,</li>
-<li>Various utilities such as <a class="reference internal" href="#polfromcsv">\PolFromCSV</a>,
-<a class="reference internal" href="#polmapcoeffs">\PolMapCoeffs</a>,
-<a class="reference internal" href="#poltocsv">\PolToCSV</a>, <a class="reference internal" href="#poltoexpr">\PolToExpr</a>, ...</li>
-</ul>
-<p>Only one-variable polynomials so far.</p>
-</li>
-<li><p class="first">v0.2 (2018/01/14)</p>
-<ul class="simple">
-<li>Fix: <tt class="docutils literal">&quot;README thinks \numexpr recognizes ^ operator&quot;</tt>.</li>
-<li>Convert README to reStructuredText markup.</li>
-<li>Move main documentation from README to separate <tt class="docutils literal">polexpr.txt</tt> file.</li>
-<li>Provide <tt class="docutils literal">polexpr.html</tt> as obtained via <a class="reference external" href="http://docutils.sourceforge.net/docs/index.html">DocUtils</a> <tt class="docutils literal">rst2html.py</tt>.</li>
-<li>Convert README to (CTAN compatible) Markdown markup.</li>
-</ul>
-<p>Due to lack of available time the test suite might not be extensive
-enough. Bug reports are very welcome!</p>
-</li>
-<li><p class="first">v0.3 (2018/01/17)</p>
-<ul>
-<li><p class="first">bug fixes:</p>
-<ul>
-<li><p class="first">the <tt class="docutils literal">0.1</tt> <a class="reference internal" href="#polevalat">\PolEval</a> accepted expressions for its second
-argument, but this was removed by mistake at <tt class="docutils literal">0.2</tt>. Restored.</p>
-<p><strong>Attention</strong>: at <tt class="docutils literal">0.4</tt> this has been reverted again, and
-<a class="reference internal" href="#polevalatexpr">\PolEval{P}\AtExpr{foo}</a> syntax is needed for
-using expressions in the second argument.</p>
-</li>
-</ul>
-</li>
-<li><p class="first">incompatible or breaking changes:</p>
-<ul class="simple">
-<li><a class="reference internal" href="#poltoexpr">\PolToExpr</a> now by default uses <em>descending</em>
-powers (it also treats differently coefficients equal to 1 or -1.)
-Use <a class="reference internal" href="#id30">\PolToExpr*</a> for <em>ascending</em> powers.</li>
-<li><a class="reference internal" href="#polevalat">\PolEval</a> reduced the output to smallest terms,
-but as this is costly with big fractions and not needed if e.g.
-wrapped in an <tt class="docutils literal">\xintRound</tt> or <tt class="docutils literal">\xintFloat</tt>, this step has been
-removed; the former meaning is available as <a class="reference internal" href="#polevalreducedat">\PolEvalReduced</a>.</li>
-</ul>
-</li>
-<li><p class="first">new (or newly documented) macros:</p>
-<ul class="simple">
-<li><a class="reference internal" href="#poltypesetcmd">\PolTypesetCmd</a></li>
-<li><a class="reference internal" href="#poltypesetcmdprefix">\PolTypesetCmdPrefix</a></li>
-<li><a class="reference internal" href="#poltypesetmonomialcmd">\PolTypesetMonomialCmd</a></li>
-<li><a class="reference internal" href="#polevalreducedat">\PolEvalReducedAt</a></li>
-<li><a class="reference internal" href="#poltofloatexpr">\PolToFloatExpr</a></li>
-<li><a class="reference internal" href="#poltoexproneterm">\PolToExprOneTerm</a></li>
-<li><a class="reference internal" href="#poltofloatexproneterm">\PolToFloatExprOneTerm</a></li>
-<li><a class="reference internal" href="#poltoexprcmd">\PolToExprCmd</a></li>
-<li><a class="reference internal" href="#id33">\PolToFloatExprCmd</a></li>
-<li><a class="reference internal" href="#poltoexprtermprefix">\PolToExprTermPrefix</a></li>
-<li><a class="reference internal" href="#poltoexprvar">\PolToExprVar</a></li>
-<li><a class="reference internal" href="#poltoexprtimes">\PolToExprTimes</a></li>
-</ul>
-</li>
-<li><p class="first">improvements:</p>
-<ul>
-<li><p class="first">documentation has a table of contents, internal hyperlinks,
-standardized signature notations and added explanations.</p>
-</li>
-<li><p class="first">one can do <tt class="docutils literal"><span class="pre">\PolLet{g}={f}</span></tt> or <tt class="docutils literal"><span class="pre">\PolLet{g}{f}</span></tt>.</p>
-</li>
-<li><p class="first"><tt class="docutils literal">\PolToExpr{f}</tt> is highly customizable.</p>
-</li>
-<li><p class="first"><a class="reference internal" href="#poldef">\poldef</a> and other defining macros prepare the polynomial
-functions for usage within <tt class="docutils literal">\xintthefloatexpr</tt> (or
-<tt class="docutils literal">\xintdeffloatvar</tt>). Coefficients are pre-rounded to the
-floating point precision. Indispensible for numerical algorithms,
-as exact fractions, even reduced, quickly become very big. See the
-documentation about how to use the exact polynomials also in
-floating point context.</p>
-<p><strong>Attention</strong>: this has been reverted at <tt class="docutils literal">0.4</tt>. The macro
-<a class="reference internal" href="#polgenfloatvariant">\PolGenFloatVariant</a> must be used for
-generation floating point polynomial functions.</p>
-</li>
-</ul>
-</li>
-</ul>
-</li>
-<li><p class="first">v0.3.1 (2018/01/18)</p>
-<p>Fixes two typos in example code included in the documentation.</p>
-</li>
-<li><p class="first">v0.4 (2018/02/16)</p>
-<ul>
-<li><p class="first">bug fixes:</p>
-<ul class="simple">
-<li>when Euclidean division gave a zero remainder, the internal
-representation of this zero polynomial could be faulty; this
-could cause mysterious bugs in conjunction with other package
-macros such as <a class="reference internal" href="#polmapcoeffs">\PolMapCoeffs</a>.</li>
-<li><a class="reference internal" href="#polgcd">\PolGCD</a> was buggy in case of first polynomial being
-of lesser degree than the second one.</li>
-</ul>
-</li>
-<li><p class="first">breaking changes:</p>
-<ul>
-<li><p class="first">formerly <a class="reference internal" href="#polevalat">\PolEval{P}\At{foo}</a> allowed <tt class="docutils literal">foo</tt> to
-be an expression, which was transparently handled via
-<tt class="docutils literal">\xinttheexpr</tt>. Now, <tt class="docutils literal">foo</tt> must be a fraction (or a macro
-expanding to such) in the format acceptable by <tt class="docutils literal">xintfrac.sty</tt>
-macros. Use <a class="reference internal" href="#polevalatexpr">\PolEval{P}\AtExpr{foo}</a> for more
-general arguments using expression syntax. E.g., if <tt class="docutils literal">foo</tt> is the
-name of a variable known to <tt class="docutils literal">\xintexpr</tt>.</p>
-<p>The same holds for <a class="reference internal" href="#polevalreducedat">\PolEvalReduced</a>
-and <a class="reference internal" href="#polfloatevalat">\PolFloatEval</a>.</p>
-</li>
-<li><p class="first">the <tt class="docutils literal">3.0</tt> automatic generation of floating point variants has
-been reverted. Not only do <em>not</em> the package macros automatically
-generate floating point variants of newly created polynomials,
-they actually make pre-existing such variant undefined.</p>
-<p>See <a class="reference internal" href="#polgenfloatvariant">\PolGenFloatVariant</a>.</p>
-</li>
-</ul>
-</li>
-<li><p class="first">new non-expandable macros:</p>
-<ul class="simple">
-<li><a class="reference internal" href="#polgenfloatvariant">\PolGenFloatVariant</a></li>
-<li><a class="reference internal" href="#polgloballet">\PolGlobalLet</a></li>
-<li><a class="reference internal" href="#poltypesetone">\PolTypesetOne</a></li>
-<li><a class="reference internal" href="#polquo">\PolQuo</a></li>
-<li><a class="reference internal" href="#polrem">\PolRem</a></li>
-<li><a class="reference internal" href="#poltosturm">\PolToSturm</a></li>
-<li><a class="reference internal" href="#id9">\PolToSturm*</a></li>
-<li><a class="reference internal" href="#polsettosturmchainsignchangesat">\PolSetToSturmChainSignChangesAt</a></li>
-<li><a class="reference internal" href="#polsettonbofzeroswithin">\PolSetToNbOfZerosWithin</a></li>
-<li><a class="reference internal" href="#polsturmisolatezeros">\PolSturmIsolateZeros</a></li>
-<li><a class="reference internal" href="#polrefineinterval">\PolRefineInterval*</a></li>
-<li><a class="reference internal" href="#polrefineinterval-n">\PolRefineInterval[N]</a></li>
-<li><a class="reference internal" href="#polensureintervallength">\PolEnsureIntervalLength</a></li>
-<li><a class="reference internal" href="#polensureintervallengths">\PolEnsureIntervalLengths</a></li>
-<li><a class="reference internal" href="#polprintintervals">\PolPrintIntervals</a></li>
-<li><a class="reference internal" href="#polprintintervalsprintexactzero">\PolPrintIntervalsPrintExactZero</a></li>
-<li><a class="reference internal" href="#polprintintervalsprintleftendpoint">\PolPrintIntervalsPrintLeftEndPoint</a></li>
-<li><a class="reference internal" href="#polprintintervalsprintrightendpoint">\PolPrintIntervalsPrintRightEndPoint</a></li>
-<li><a class="reference internal" href="#id20">\PolReduceCoeffs*</a></li>
-<li><a class="reference internal" href="#polmakemonic">\PolMakeMonic</a></li>
-</ul>
-</li>
-<li><p class="first">new expandable macros:</p>
-<ul class="simple">
-<li><a class="reference internal" href="#poltoexpronetermstylea">\PolToExprOneTermStyleA</a></li>
-<li><a class="reference internal" href="#polifcoeffisplusorminusone">\PolIfCoeffIsPlusOrMinusOne</a></li>
-<li><a class="reference internal" href="#polleadingcoeff">\PolLeadingCoeff</a></li>
-<li><a class="reference internal" href="#polsturmchainlength">\PolSturmChainLength</a></li>
-<li><a class="reference internal" href="#polsturmnbofisolatedzeros">\PolSturmNbOfIsolatedZeros</a></li>
-<li><a class="reference internal" href="#polsturmifzeroexactlyknown">\PolSturmIfZeroExactlyKnown</a></li>
-<li><a class="reference internal" href="#polsturmisolatedzeroleft">\PolSturmIsolatedZeroLeft</a></li>
-<li><a class="reference internal" href="#polsturmisolatedzeroright">\PolSturmIsolatedZeroRight</a></li>
-<li><tt class="docutils literal">\PolPrintIntervalsTheEndPoint</tt> (removed at 0.7)</li>
-<li><a class="reference internal" href="#polprintintervalstheindex">\PolPrintIntervalsTheIndex</a></li>
-<li><tt class="docutils literal">\PolIfEndPointIsPositive</tt> (removed at 0.7)</li>
-<li><tt class="docutils literal">\PolIfEndPointIsNegative</tt> (removed at 0.7)</li>
-<li><tt class="docutils literal">\PolIfEndPointIsZero</tt> (removed at 0.7)</li>
-<li><a class="reference internal" href="#polintervalwidth">\PolIntervalWidth</a></li>
-<li><a class="reference internal" href="#poldectostring">\PolDecToString</a></li>
-</ul>
-</li>
-<li><p class="first">improvements:</p>
-<p>The main new feature is implementation of the <a class="reference external" href="https://en.wikipedia.org/wiki/Sturm%27s_theorem">Sturm algorithm</a>
-for localization of the real roots of polynomials.</p>
-</li>
-</ul>
-</li>
-<li><p class="first">v0.4.1 (2018/03/01)</p>
-<p>Synced with xint 1.3.</p>
-</li>
-<li><p class="first">v0.4.2 (2018/03/03)</p>
-<p>Documentation fix.</p>
-</li>
-<li><p class="first">v0.5 (2018/04/08)</p>
-<ul class="simple">
-<li>bug fixes:<ul>
-<li><a class="reference internal" href="#polget-polname-fromarray-macro">\PolGet{polname}\fromarray\macro</a> crashed when <tt class="docutils literal">\macro</tt> was
-an <a class="reference external" href="http://www.ctan.org/pkg/xint">xinttools</a> array macro with no items. It now produces the zero
-polynomial.</li>
-</ul>
-</li>
-<li>breaking changes:<ul>
-<li><a class="reference internal" href="#poltosturm">\PolToSturm</a> creates primitive integer coefficients polynomials.
-This speeds up localization of roots via
-<a class="reference internal" href="#polsturmisolatezeros">\PolSturmIsolateZeros</a>. In case of user protests the author
-will make available again the code producing the bona fide Sturm
-polynomials as used formerly.</li>
-<li>polynomials created from <a class="reference internal" href="#polfromcsv">\PolFromCSV</a> or <a class="reference internal" href="#polget">\PolGet</a>
-get their coefficients normalized via <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a>'s <tt class="docutils literal">\xintRaw</tt>.</li>
-</ul>
-</li>
-<li>experimental change:<ul>
-<li>optional argument to <a class="reference internal" href="#polsturmisolatezeros">\PolSturmIsolateZeros</a> (see <a class="reference internal" href="#the-degree-41-polynomial-with-2-1-9-1-8-0-0-1-1-9-2-as-roots">The
-degree 41 polynomial with -2, -1.9, -1.8, ..., 0, 0.1, ..., 1.9, 2
-as roots</a> for usage). It will presumably be replaced in future by
-an interval specification.</li>
-</ul>
-</li>
-<li>new non-expandable macro:<ul>
-<li><a class="reference internal" href="#polmakeprimitive">\PolMakePrimitive</a></li>
-</ul>
-</li>
-<li>new expandable macro:<ul>
-<li><a class="reference internal" href="#policontent">\PolIContent</a></li>
-</ul>
-</li>
-</ul>
-</li>
-<li><p class="first">v0.5.1 (2018/04/22)</p>
-<ul class="simple">
-<li>new feature:<ul>
-<li>the character <tt class="docutils literal">'</tt> can be used in polynomial names.</li>
-</ul>
-</li>
-</ul>
-</li>
-<li><p class="first">v0.6 (2018/11/20)</p>
-<ul class="simple">
-<li>bugfix:<ul>
-<li>the starred variant <a class="reference internal" href="#id10">\PolToSturm*{polname}{sturmname}</a> was
-broken. On the occasion of the fix, its meaning has been modified,
-see its documentation.</li>
-<li>using <a class="reference internal" href="#poltosturm">\PolToSturm</a> with a constant polynomial
-caused a division by zero error.</li>
-</ul>
-</li>
-<li>new macro:<ul>
-<li><a class="reference internal" href="#id11">\PolSturmIsolateZeros*</a>
-acts like the <a class="reference internal" href="#polsturmisolatezeros">non-starred variant</a> then computes all the multiplicities.</li>
-</ul>
-</li>
-<li>new expandable macros:<ul>
-<li><a class="reference internal" href="#polsturmisolatedzeromultiplicity-sturmname-index">\PolSturmIsolatedZeroMultiplicity{sturmname}{index}</a></li>
-<li><a class="reference internal" href="#polsturmnbofrootsof-sturmname-lessthanorequalto-value">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualTo{value}</a></li>
-<li><a class="reference internal" href="#polsturmnbofrootsof-sturmname-lessthanorequaltoexpr-expression">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualToExpr{expression}</a></li>
-<li><a class="reference internal" href="#polsturmnbwithmultofrootsof-sturmname-lessthanorequalto-value">\PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualTo{value}</a></li>
-<li><a class="reference internal" href="#polsturmnbwithmultofrootsof-sturmname-lessthanorequaltoexpr-expression">\PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualToExpr{expression}</a></li>
-</ul>
-</li>
-</ul>
-</li>
-<li><p class="first">v0.7 (2018/12/08), v0.7.1 (bugfix), v0.7.2 (2nd bugfix) (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
-remains the same, some auxiliary macros for user-customization
-have been removed: <tt class="docutils literal">\PolPrintIntervalsTheEndPoint</tt>,
-<tt class="docutils literal"><span class="pre">\PolIfEndPointIsPositive{A}{B}</span></tt>,
-<tt class="docutils literal"><span class="pre">\PolIfEndPointIsNegative{A}{B}</span></tt>, and
-<tt class="docutils literal"><span class="pre">\PolIfEndPointIsZero{A}{B}</span></tt>.</li>
-</ul>
-</li>
-<li>bugfix:<ul>
-<li>it could happen that, contrarily to documentation, an interval
-computed by <a class="reference internal" href="#polsturmisolatezeros-sturmname">\PolSturmIsolateZeros{sturmname}</a> had zero as an
-endpoint,</li>
-<li><a class="reference internal" href="#polensureintervallength-sturmname-index-e">\PolEnsureIntervalLength{sturmname}{index}{E}</a> could under
-certain circumstances erroneously replace a non-zero root by
-zero,</li>
-<li><a class="reference internal" href="#polensureintervallengths-sturmname-e">\PolEnsureIntervalLengths{sturmname}{E}</a> crashed when used with
-a polynomial with no real roots, hence for which no isolation intervals
-existed (thanks to Thomas Söll for report).</li>
-</ul>
-</li>
-<li>new macros:<ul>
-<li><a class="reference internal" href="#id14">\PolSturmIsolateZeros**{sturmname}</a></li>
-<li><a class="reference internal" href="#polsturmisolatezerosgetmultiplicitiesandrationalroots-sturmname">\PolSturmIsolateZerosGetMultiplicitiesAndRationalRoots{sturmname}</a></li>
-<li><a class="reference internal" href="#polsturmisolatezerosandfindrationalroots-sturmname">\PolSturmIsolateZerosAndFindRationalRoots{sturmname}</a></li>
-<li><a class="reference internal" href="#polexprsetup">\polexprsetup</a></li>
-<li><a class="reference internal" href="#id18">\PolPrintIntervals*</a></li>
-<li><a class="reference internal" href="#polprintintervalsnorealroots">\PolPrintIntervalsNoRealRoots</a></li>
-<li><a class="reference internal" href="#polprintintervalsbeginenv">\PolPrintIntervalsBeginEnv</a></li>
-<li><a class="reference internal" href="#polprintintervalsendenv">\PolPrintIntervalsEndEnv</a></li>
-<li><a class="reference internal" href="#polprintintervalsknownroot">\PolPrintIntervalsKnownRoot</a></li>
-<li><a class="reference internal" href="#polprintintervalsunknownroot">\PolPrintIntervalsUnknownRoot</a></li>
-<li><a class="reference internal" href="#polprintintervalsprintmultiplicity">\PolPrintIntervalsPrintMultiplicity</a></li>
-</ul>
-</li>
-<li>new expandable macros:<ul>
-<li><a class="reference internal" href="#polsturmnbofrationalroots-sturmname">\PolSturmNbOfRationalRoots{sturmname}</a></li>
-<li><a class="reference internal" href="#polsturmnbofrationalrootswithmultiplicities-sturmname">\PolSturmNbOfRationalRootsWithMultiplicities{sturmname}</a></li>
-<li><a class="reference internal" href="#polsturmrationalroot-sturmname-k">\PolSturmRationalRoot{sturmname}{k}</a></li>
-<li><a class="reference internal" href="#polsturmrationalrootindex-sturmname-k">\PolSturmRationalRootIndex{sturmname}{k}</a></li>
-<li><a class="reference internal" href="#polsturmrationalrootmultiplicity-sturmname-k">\PolSturmRationalRootMultiplicity{sturmname}{k}</a></li>
-<li><a class="reference internal" href="#polprintintervalsthevar">\PolPrintIntervalsTheVar</a></li>
-<li><a class="reference internal" href="#polprintintervalsthesturmname">\PolPrintIntervalsTheSturmName</a></li>
-<li><a class="reference internal" href="#polprintintervalsthemultiplicity">\PolPrintIntervalsTheMultiplicity</a></li>
-</ul>
-</li>
-</ul>
-</li>
-<li><p class="first">v0.7.3 (2019/02/04)</p>
-<ul class="simple">
-<li>bugfix:<ul>
-<li>Debugging information not destined to user showed in log if root
-finding was done under <tt class="docutils literal">\xintverbosetrue</tt> regime.</li>
-<li><a class="reference internal" href="#polprintintervalsthevar">\PolPrintIntervalsTheVar</a> remained defined after
-<a class="reference internal" href="#polprintintervals">\PolPrintIntervals</a> but was left undefined after
-<a class="reference internal" href="#id18">\PolPrintIntervals*</a> (reported by Jürgen Gilg). Now remains
-defined in both cases, and <a class="reference internal" href="#polprintintervalsthesturmname">\PolPrintIntervalsTheSturmName</a>
-also.</li>
-<li>Polynomial names ending in digits caused errors (reported by Thomas
-Söll).</li>
-</ul>
-</li>
-</ul>
-</li>
-<li><p class="first">v0.7.4 (2019/02/12)</p>
-<ul class="simple">
-<li>bugfix:<ul>
-<li>20000000000 is too big for <tt class="docutils literal">\numexpr</tt>, shouldn't I know that?
-Thanks to Jürgen Gilg for report.</li>
-</ul>
-</li>
-</ul>
-</li>
-<li><p class="first">v0.7.5 (2020/01/31)</p>
-<p>Synced with xint 1.4. Requires it.</p>
-</li>
-</ul>
-</div>
-<div class="section" id="acknowledgments">
-<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
-concrete problems.</p>
-<p>Renewed thanks to them on occasion of the <tt class="docutils literal">0.6</tt> and <tt class="docutils literal">0.7</tt> releases for their
-continued interest.</p>
-<p>See README.md for the License.</p>
-</div>
-</div>
-</body>
-</html>
diff --git a/macros/latex/contrib/polexpr/polexpr.sty b/macros/latex/contrib/polexpr/polexpr.sty
deleted file mode 100644
index 30fef28914..0000000000
--- a/macros/latex/contrib/polexpr/polexpr.sty
+++ /dev/null
@@ -1,3164 +0,0 @@
-% author: Jean-François Burnol
-% License: LPPL 1.3c (author-maintained)
-\ProvidesPackage{polexpr}%
- [2020/01/31 v0.7.5 Polynomial expressions with rational coefficients (JFB)]%
-\RequirePackage{xintexpr}[2020/01/31]% xint 1.4
-\edef\POL@restorecatcodes % TODO: think better about what is reasonable here
- {\catcode`\noexpand\_ \the\catcode`\_ %
- \catcode`\noexpand\! \the\catcode`\! %
- \catcode`\noexpand\* \the\catcode`\* %
- \catcode`\noexpand\~ \the\catcode`\~ %
- \catcode`\noexpand\: \the\catcode`\: %
- \catcode0 \the\catcode0\relax}%
-\catcode`\_ 11 \catcode0 12 \catcode`\* 12
-\long\def\xint_stop_atfirstoftwo #1#2{ #1}% not yet in xint 1.3c
-\long\def\xint_stop_atsecondoftwo #1#2{ #2}%
-
-%% 0.7.5 VERY SERIOUS TROUBLES TO GET polexpr TO WORK WITH xintexpr 1.4
-
-%% I hesitated about incorporating it directly into xint 1.4
-%% Don't do this at home, only xint gurus are allowed.
-\let\POL@originalXINT_expr_redefinemacros\XINT_expr_redefinemacros
-\def\XINT_expr_redefinemacros
-{%
- \POL@originalXINT_expr_redefinemacros
- \POL@activateNEhook
-}%
-%% Using \def's and not \let's to get better readable trace
-%% in case I need to debug but this never happens
-\def\POL@activateNEhook@xint % done in a group
-{%
- \def\POL@NEhook@polfunc{\POL@NE@polfunc}%
-}%
-\def\POL@activateNEhook@pol
-{%
- \def\POL@NEhook@polfunc{\POL@NP@polfunc}%
-}%
-\def\POL@activateNEhook{\POL@activateNEhook@xint}%
-%
-%
-\catcode`~ 12
-\catcode`! 11
-\catcode`: 11
-% We drop consideration of \XINT_global matters
-% because we have other more urgent and arduous problems
-\def\POL@defpolfunc #1#2%
-{%
- \expandafter\POL@defpolfunc_a
- \csname XINT_#2_func_#1\expandafter\endcsname
- \csname XINT_#2_polfunc_#1\endcsname {#1}{#2}%
-}%
-\def\POL@defpolfunc_a #1#2#3#4%
-{%
- \protected % xintexpr 1.4 does things such as \expandafter\xintAdd\expanded
- \expandafter\def\expandafter#2\expandafter##\expandafter1\expandafter
- {%
- #2{##1}%
- }%
- \def#1##1##2##3%
- {%
- % put it directly at the correct level of bracing
- % don't worry for now about minimizing how many times ##3 is grabbed
- \expandafter##1\expandafter##2\expandafter{\expandafter
- {\romannumeral`^^@\POL@NEhook@polfunc{XINT_#4_polfunc_#3}#2{##3}}}%
- }%
-}%
-%
-\def\POL@polfunc@go #1#2#3{#2#3}% brace stripping intentional
-\def\POL@NEhook@polfunc{\POL@polfunc@go}% default for pure numerics
-%
-% Hook for expansion in \poldef
-\def\POL@NP@polfunc #1{%
-\def\POL@NP@polfunc ##1##2##3%
-{%
- \if0\expandafter\XINT:NE:hastilde\detokenize{##3}~!\relax
- \expandafter\XINT:NE:hashash\detokenize{##3}#1!\relax 0%
- \expandafter\POL@polfunc@go
- \else
- \expandafter\POL:NP:polfunc:p
- \fi {##1}{##2}{##3}%
-}}\expandafter\POL@NP@polfunc\string#%
-\def\POL:NP:polfunc:p #1#2#3%
-{%
- ~romannumeral~POL:usepolfunc:pol{#1}{#3}%
-}%
-\def\POL:usepolfunc:pol #1%#2%
-{%
-% Here we are in the core of \poldef and we really
-% need to get rid of some \expanded tokens so
-% we accept being exposed to \expanded but arrange to
-% remain invariant. Then we will try to speed up
-% polynomial composition (at this time the \#1
-% is a nested Horner type macro) by «pre-expanding»
-% the argument, but this means using the \POL@get
-% methods inside an \hbox
-%
-% \POL@applypolfunc will be defined \protected
-%
- \expandafter\xint_c_\expandafter\POL@applypolfunc
-% This will be \protected
- \csname#1\endcsname
-% #2% brace stripping is deliberate
-}%
-%
-% Hook for expansion in \xintexpr
-\def\POL@NE@polfunc #1{%
-\def\POL@NE@polfunc ##1##2##3%
-{%
- \if0\expandafter\XINT:NE:hastilde\detokenize{##3}~!\relax
- \expandafter\XINT:NE:hashash\detokenize{##3}#1!\relax 0%
- \expandafter\POL@polfunc@go
- \else
- \expandafter\POL:NE:polfunc:p
- \fi {##1}{##2}{##3}%
-}}\expandafter\POL@NE@polfunc\string#%
-\def\POL:NE:polfunc:p #1#2#3%
-{%
- ~romannumeral~POL:usepolfunc:xint{#1}{#3}%
-}%
-\def\POL:usepolfunc:xint #1%
-{%
-% This is done to overcome \protected and is useful
-% in case the polynomial function ends up nested
-% in some non-polynomial user declared function
-% as the latter (and other things) tries to pre-expand
-% its arguments (as they may be used multiple time)
-% using \expanded, but \#1 is protected.
-% And this works recursively. We are inside braces.
-% However we have a very big problem with constant
-% polynomial functions. We have to handle them
-% in a special way.
- -`0\csname#1\expandafter\endcsname\expanded
-}%
-\catcode`~ 13
-\catcode`: 12
-
-
-%% Start defining some \protected ones here
-\protected\def\POL@empty{}%
-\newif\ifPOL@pol
-\protected\def\POL@polglobaltrue {\global\let\ifPOL@pol\iftrue}%
-\protected\def\POL@polglobalfalse{\global\let\ifPOL@pol\iffalse}%
-
-
-%% Patch xintexpr to authorize ' in names (0.5.1)
-%% Adapted 0.7.5 to follow-up on xintexpr 1.4 internal changes
-%% (much simpler than previous stuff...)
-%% This allows ' as a character in a polynomial name (not initial one)
-\def\POL@XINT_expr_scanfunc_b #1%
-{%
- \ifcat \relax#1\xint_dothis{\iffalse{\fi}(_#1}\fi
- \if (#1\xint_dothis{\iffalse{\fi}(`}\fi
- \if 1\ifcat a#10\fi
- \ifnum\xint_c_ix<1\string#1 0\fi
- \if @#10\fi
- \if _#10\fi
- \if '#10\fi
- 1%
- \xint_dothis{\iffalse{\fi}(_#1}\fi
- \xint_orthat {#1\XINT_expr_scanfunc_a}%
-}%
-
-
-%% Activate polexp's modified xintexpr (only during definitions
-%% of polynomials)
-\def\POL@hackxintexpr {%
- \let\POL@originalXINT_expr_scanfunc_b\XINT_expr_scanfunc_b
- \let\XINT_expr_scanfunc_b\POL@XINT_expr_scanfunc_b
- \def\POL@activateNEhook{\POL@activateNEhook@pol}%
-}%
-\def\POL@restorexintexpr {%
- \let\XINT_expr_scanfunc_b\POL@originalXINT_expr_scanfunc_b
- \def\POL@activateNEhook{\POL@activateNEhook@xint}%
-}%
-
-
-%% AUXILIARIES
-\catcode`! 3
-%% added at 0.7
-\newcommand\polexprsetup[1]{\POL@setup_parsekeys #1,=!,\xint_bye}%
-\def\POL@setup_parsekeys #1=#2#3,{%
- \ifx!#2\expandafter\xint_bye\fi
- \csname POL@setup_setkey_\xint_zapspaces #1 \xint_gobble_i\endcsname
- \xint_firstoftwo
- {\PackageWarning{polexpr}{The \detokenize{#1} key is unknown! ignoring}}%
- {\xintZapLastSpaces{#2#3}}%
- \POL@setup_parsekeys
-}%
-\catcode`! 11
-\def\POL@setup_setkey_norr #1#2{\edef\POL@norr}%
-\def\POL@setup_setkey_sqfnorr #1#2{\edef\POL@sqfnorr}%
-\polexprsetup{norr=_norr, sqfnorr=_sqf_norr}
-
-\newcount\POL@count
-\newif\ifxintveryverbose
-\newif\ifpoltypesetall
-\newif\ifPOL@tosturm@makefirstprimitive
-\POL@tosturm@makefirstprimitivetrue
-\newif\ifPOL@isolz@nextwillneedrefine
-\newif\ifpoltoexprall
-%% the main exchange structure (stored in macros \POLuserpol@<name>)
-%% is: degree.\POL@empty{coeff0}{coeff1}....{coeffN}
-%% (degree=N except zero polynomial recognized from degree set to -1
-%% but it has always the {0/1[0]} coeff0.)
-\def\POL@ifZero#1{\expandafter\POL@ifZero@aux#1;}%
-\def\POL@ifZero@aux #1#2;{\if-#1\expandafter\xint_firstoftwo
- \else\expandafter\xint_secondoftwo
- \fi}%
-\def\POL@split#1.#2;#3#4% separates degree and list of coefficients
-% The \expandafter chain removes the \empty token
- {\def#3{#1}\expandafter\def\expandafter#4\expandafter{#2}}%
-%
-\def\POL@resultfromarray #1{% ATTENTION, **MUST** be executed with
-% \count@ set to 1 + degree (\count@ = 0 for zero polynomial)
-% Attention to the \protected here at 0.7.5
-% They are many all over the place
- \protected\edef\POL@result{\ifnum\count@>\z@
- \the\numexpr\count@-\@ne.\POL@empty
- \xintiloop [1+1]%
- \expandafter\POL@braceit\csname POL@array#1\xintiloopindex\endcsname
- \ifnum\xintiloopindex<\count@
- \repeat
-% Attention to this \protected\POL@empty
-% They are many all over the place
- \else-1.\POL@empty{0/1[0]}\fi}%
-}%
-\def\POL@braceit#1{{#1}}% needed as \xintiloopindex can not "see" through braces
-
-
-\newcommand\PolDef[3][x]{\poldef #2(#1):=#3;}%
-\def\poldef{\edef\POL@restoresemicolon{\catcode59=\the\catcode59\relax}%
- \catcode59 12 \POL@defpol}%
-\def\POL@defpol #1(#2)#3=#4;{%
- \POL@restoresemicolon
- \edef\POL@tmp{\ifxintverbose1\else0\fi}%
- \unless\ifxintveryverbose\xintverbosefalse\fi
- \POL@hackxintexpr
- \xintdeffunc __pol(#2):=0+(#4);% force conversion to raw if a constant
- \POL@restorexintexpr
- \if1\POL@tmp\xintverbosetrue\fi
- \edef\POL@polname{\xint_zapspaces #1 \xint_gobble_i}%
- \begingroup
- \setbox0\hbox{%
- \let\xintScalarAdd\xintAdd
- \let\xintScalarSub\xintSub
- \let\xintScalarMul\xintMul
- \let\xintScalarDiv\xintDiv
- \let\xintScalarPow\xintPow
- \let\xintScalarOpp\xintOpp
- \let\xintAdd\POL@add
- \let\xintMul\POL@mul
- \let\xintDiv\POL@div
- \let\xintPow\POL@pow
- \let\xintOpp\POL@opp
- \def\xintSub ##1##2{\xintAdd{##1}{\xintOpp{##2}}}%
- % \xintAdd{0} to get \POL@result defined even if numerical only expression
- % I could also test \ifPOL@pol, but this is anyhow small overhead
-% Attention that xintexpr 1.4 has braces all over the place
- \expandafter\xintAdd\expandafter{\expandafter0\expandafter}%
- \romannumeral0\csname XINT_expr_userfunc___pol\endcsname
- {\POL@polglobaltrue\protected\def\POL@result{1.\POL@empty{0/1[0]}{1/1[0]}}}%
- \expandafter}\expandafter
- \endgroup\expandafter
- \def\csname POLuserpol@\POL@polname\expandafter\endcsname
- \expandafter{\POL@result}%
- \expandafter\POL@newpol\expandafter{\POL@polname}%
-}%
-%
-
-
-\def\POL@newpol#1{%
-%% We must handle specially constant polynomials because they must
-%% be made to work expandably in \poldef of other polynomials due
-%% to complicated matters having to do with the \POL@ifpol conditional
- \ifnum\PolDegree{#1}<\@ne
- % non-zero constant
- % I am defining this one only for the Info message, no time now
- \expandafter\edef\csname XINT_expr_polfunc_#1\endcsname
- ##1{\PolNthCoeff{#1}{0}}%
- % No hooks here!
- \expandafter\edef\csname XINT_expr_func_#1\endcsname ##1##2##3%
- {##1##2{{\PolNthCoeff{#1}{0}}}}%
- \else
- % polynomial of degree at least 1. This means that mechanism
- % to get \POL@result will get activated and we must be very careful
- % to never \edef when the Horner macro will be converted to
- % a polynomial
- \POL@newpolhorner{#1}%
- \POL@defpolfunc{#1}{expr}%
- \fi
- \expandafter\let\csname XINT_flexpr_func_#1\endcsname\@undefined
- \ifxintverbose\POL@info{#1}\fi
-}%
-\def\POL@newfloatpol#1{%
-%% We must handle specially constant polynomials because they must
-%% be made to work expandably in \poldef of other polynomials due
-%% to complicated matters having to do with the \POL@ifpol conditional
- \ifnum\PolDegree{#1}<\@ne
- % non-zero constant
- % I am defining this one only for the Info message, no time now
- \expandafter\edef\csname XINT_flexpr_polfunc_#1\endcsname
- ##1{\PolNthCoeff{#1}{0}}%
- % No hooks here!
- \expandafter\edef\csname XINT_flexpr_func_#1\endcsname ##1##2##3%
- {##1##2{{\PolNthCoeff{#1}{0}}}}%
- \else
- % polynomial of degree at least 1. This means that mechanism
- % to get \POL@result will get activated and we must be very careful
- % to never \edef when the Horner macro will be converted to
- % a polynomial
- \POL@newfloatpolhorner{#1}%
- \POL@defpolfunc{#1}{flexpr}%
- \fi
- \ifxintverbose\POL@floatinfo{#1}\fi
-}%
-\def\POL@info #1{%
- \xintMessage {polexpr}{Info}%
- {Function #1 for the \string\xintexpr\space parser is
- associated to \string\XINT_expr_polfunc_#1\space
- whose meaning uses Horner scheme:
- \expandafter\meaning
- \csname XINT_expr_polfunc_#1\endcsname}%
-}%
-\def\POL@floatinfo #1{%
- \xintMessage {polexpr}{Info}%
- {Function #1 for the \string\xintfloatexpr\space parser is
- associated to \string\XINT_flexpr_polfunc_#1\space
- whose meaning uses Horner scheme:
- \expandafter\meaning
- \csname XINT_flexpr_polfunc_#1\endcsname}%
-}%
-%
-\def\POL@newpolhorner#1{%
- %% redefine function to expand by Horner scheme. Is this useful?
- %% perhaps bad idea for numerical evaluation of thing such as (1+x)^10?
-% note: I added {0/1[0]} item to zero polynomial also to facilitate this
- \expandafter\expandafter\expandafter\POL@split
- \csname POLuserpol@#1\endcsname;\POL@var@deg\POL@var@coeffs
- \edef\POL@var@coeffs{\xintRevWithBraces{\POL@var@coeffs}}%
- \begingroup
- \expandafter\POL@newpol@horner\POL@var@coeffs\relax
- \expandafter
- \endgroup
- \expandafter\def\csname XINT_expr_polfunc_#1\expandafter\endcsname
- \expandafter##\expandafter1\expandafter{\POL@tmp{##1}}%
-}%
-\def\POL@newfloatpolhorner#1{%
- %% redefine function to expand by Horner scheme. Is this useful?
- %% perhaps bad idea for numerical evaluation of thing such as (1+x)^10?
-% note: I added {0/1[0]} item to zero polynomial also to facilitate this
- \expandafter\expandafter\expandafter\POL@split
- \csname POLuserpol@#1\endcsname;\POL@var@deg\POL@var@coeffs
- \edef\POL@var@coeffs{\xintRevWithBraces{\POL@var@coeffs}}%
- \begingroup
- \expandafter\POL@newpol@floathorner\POL@var@coeffs\relax
- \expandafter
- \endgroup
- \expandafter\def\csname XINT_flexpr_polfunc_#1\expandafter\endcsname
- \expandafter##\expandafter1\expandafter{\POL@tmp{##1}}%
-}%
-\def\POL@newpol@horner#1{\let\xintAdd\relax\let\xintMul\relax
- \def\POL@tmp##1{#1}\POL@newpol@horner@loop.}%
-\def\POL@newpol@horner@loop.#1{%
- \if\relax#1\expandafter\xint_gob_til_dot\fi
- \edef\POL@tmp##1{\xintiiifZero{#1}
- {\@firstofone}{\xintAdd{#1}}%
- {\xintMul{##1}{\POL@tmp{##1}}}}%
- \POL@newpol@horner@loop.%
-}%
-\def\POL@newpol@floathorner#1{\let\XINTinFloatAdd\relax\let\XINTinFloatMul\relax
- \def\xintAdd{\XINTinFloatAdd}\def\xintMul{\XINTinFloatMul}%
- \edef\POL@tmp##1{\XINTinFloatdigits{#1}}%
- \POL@newpol@floathorner@loop.}%
-\def\POL@newpol@floathorner@loop.#1{%
- \if\relax#1\expandafter\xint_gob_til_dot\fi
- \edef\POL@tmp##1{\xintiiifZero{#1}
- {\@firstofone}{\xintAdd{\XINTinFloatdigits{#1}}}%
- {\xintMul{##1}{\POL@tmp{##1}}}}%
- \POL@newpol@floathorner@loop.%
-}%
-
-
-\newcommand\PolGenFloatVariant[1]{\POL@newfloatpol{#1}}%
-
-
-\newcommand\PolLet[2]{\if=\noexpand#2\expandafter\xint_firstoftwo
- \else\expandafter\xint_secondoftwo\fi
- \POL@@let\POL@let{#1}{#2}}%
-\def\POL@@let#1#2#3{\POL@let{#1}{#3}}%
-\def\POL@let#1#2{%
- \expandafter\let\csname POLuserpol@#1\expandafter\endcsname
- \csname POLuserpol@#2\endcsname
- \expandafter\let\csname XINT_expr_polfunc_#1\expandafter\endcsname
- \csname XINT_expr_polfunc_#2\endcsname
- \POL@defpolfunc{#1}{expr}%
- \ifxintverbose\POL@info{#1}\fi
-}%
-\newcommand\PolGlobalLet[2]{\begingroup
- \globaldefs\@ne
- \if=\noexpand#2\expandafter\xint_firstoftwo
- \else\expandafter\xint_secondoftwo\fi
-% do I need to check something here relative to \xintNewExpr?
- \POL@@globallet\POL@globallet {#1}{#2}}%
-\def\POL@@globallet#1#2#3{\POL@globallet{#1}{#3}}%
-\def\POL@globallet#1#2{\POL@let{#1}{#2}\endgroup}%
-
-\newcommand\PolAssign[1]{\def\POL@polname{#1}\POL@assign}% zap spaces in #1?
-\def\POL@assign#1\toarray#2{%
- \expandafter\expandafter\expandafter\POL@split
- \csname POLuserpol@\POL@polname\endcsname;\POL@var@deg\POL@var@coeffs
- \xintAssignArray\POL@var@coeffs\to#2%
- % modify \#200 macro to return 0/1[0] for out of range indices
- \@namedef{\xint_arrayname00}##1##2##3{%
- \@namedef{\xint_arrayname00}####1{%
- \ifnum####1>##1 \xint_dothis{ 0/1[0]}\fi
- \ifnum####1>\m@ne \xint_dothis
- {\expandafter\expandafter\expandafter##3%
- \csname##2####1\endcsname}\fi
- \unless\ifnum-####1>##1 \xint_dothis
- {\expandafter\expandafter\expandafter##3%
- \csname##2\the\numexpr##1+####1+\@ne\endcsname}\fi
- \xint_orthat{ 0/1[0]}}% space stops a \romannumeral0
- }%
- \csname\xint_arrayname00\expandafter\expandafter\expandafter\endcsname
- \expandafter\expandafter\expandafter
- {\csname\xint_arrayname0\expandafter\endcsname\expandafter}\expandafter
- {\xint_arrayname}{ }%
-}%
-
-
-\newcommand\PolGet{}%
-\def\PolGet#1#2\fromarray#3{%
- \begingroup % closed in \POL@getfromarray
- \POL@getfromarray{#1}{#3}%
- \POL@newpol{#1}%
-}%
-\def\POL@getfromarray#1#2{%
- \count@=#2{0} %<- intentional space
- \ifnum\count@=\z@
- \protected\def\POL@result{-1.\POL@empty{0/1[0]}}% 0.5 fix for empty array
- \else
- \xintloop
- \edef\POL@tmp{#2{\count@}}%
- \edef\POL@tmp{\xintRaw{\POL@tmp}}%
-% sadly xinttools (current 1.3a) arrays have no setters for individual items...
- \expandafter\let\csname POL@tmparray\the\count@\endcsname\POL@tmp
- \if0\xintiiSgn{\POL@tmp}%
- \advance\count@\m@ne
- \repeat
-% dans le cas particulier d'un array avec que des éléments nuls, \count@ est
-% ici devenu 0 et la boucle s'est arrêtée car #2{0} était au moins 1. De plus
-% \POL@tmparray1 est bien 0/1[0] donc ok pour polynôme nul dans \POL@result
- \count\tw@\count@
- \xintloop
-% on mouline tous les coeffs via \xintRaw
- \ifnum\count@>\@ne
- \advance\count@\m@ne
- \edef\POL@tmp{#2{\count@}}%
- \edef\POL@tmp{\xintRaw{\POL@tmp}}%
- \expandafter\let\csname POL@tmparray\the\count@\endcsname\POL@tmp
- \repeat
- \count@\count\tw@
- \def\POL@tmp##1.{{\csname POL@tmparray##1\endcsname}}%
- \protected\edef\POL@result{\the\numexpr\count@-\@ne.\POL@empty
- \xintiloop[1+1]%
- \expandafter\POL@tmp\xintiloopindex.%
- \ifnum\xintiloopindex<\count@
- \repeat}%
- \fi
- \expandafter
- \endgroup
- \expandafter
- \def\csname POLuserpol@#1\expandafter\endcsname
- \expandafter{\POL@result}%
-}%
-
-
-\newcommand\PolFromCSV[2]{%
- \begingroup % closed in \POL@getfromarray
- \xintAssignArray\xintCSVtoList{#2}\to\POL@arrayA
- \POL@getfromarray{#1}\POL@arrayA
- \POL@newpol{#1}%
-% semble un peu indirect et sous-optimal
-% mais je veux élaguer les coefficients nuls. Peut-être à revoir.
-}%
-
-
-\newcommand\PolTypesetCmdPrefix[1]{\xintiiifSgn{#1}{}{+}{+}}%
-\newcommand\PolTypesetCmd[1]{\xintifOne{\xintiiAbs{#1}}%
- {\ifnum\PolIndex=\z@\xintiiSgn{#1}\else
- \xintiiifSgn{#1}{-}{}{}\fi
- \let\PolIfCoeffIsPlusOrMinusOne\@firstoftwo}%
- {\PolTypesetOne{#1}%
- \let\PolIfCoeffIsPlusOrMinusOne\@secondoftwo}%
- }%
-\newcommand\PolTypesetOne{\xintSignedFrac}%
-\newcommand\PolTypesetMonomialCmd{%
- \ifcase\PolIndex\space
- %
- \or\PolVar
- \else\PolVar^{\PolIndex}%
- \fi
-}%
-\newcommand\PolTypeset{\@ifstar
- {\def\POL@ts@ascending{1}\POL@Typeset}%
- {\def\POL@ts@ascending{0}\POL@Typeset}%
-}%
-\newcommand\POL@Typeset[2][x]{% LaTeX \newcommand forces optional argument first
- \ensuremath{%
- \expandafter\expandafter\expandafter\POL@split
- \csname POLuserpol@#2\endcsname;\POL@var@deg\POL@var@coeffs
- \if\POL@ts@ascending1%
- \def\PolIndex{0}%
- \let\POL@ts@reverse\@firstofone
- \let\POL@@ne@or@m@ne\@ne
- \else
- \let\PolIndex\POL@var@deg
- \ifnum\PolIndex<\z@\def\PolIndex{0}\fi
- \let\POL@ts@reverse\xintRevWithBraces
- \let\POL@@ne@or@m@ne\m@ne
- \fi
- \def\PolVar{#1}%
- \ifnum\POL@var@deg<\z@
- \PolTypesetCmd{0/1[0]}\PolTypesetMonomialCmd
- \else
- \ifnum\POL@var@deg=\z@
- \expandafter\PolTypesetCmd\POL@var@coeffs\PolTypesetMonomialCmd
- \else
- \def\POL@ts@prefix##1{\let\POL@ts@prefix\PolTypesetCmdPrefix}%
- \expandafter\POL@ts@loop
- \romannumeral-`0\POL@ts@reverse{\POL@var@coeffs}\relax
- \fi
- \fi
- }%
-}%
-\def\POL@ts@loop{\ifpoltypesetall\expandafter\xint_firstoftwo
- \else\expandafter\xint_secondoftwo\fi
- {\POL@ts@nocheck}{\POL@ts@check}.%
-}%
-\def\POL@ts@check.#1{%
- \if\relax#1\expandafter\xint_gob_til_dot\fi
- \xintiiifZero{#1}%
- {}%
- {\POL@ts@prefix{#1}\PolTypesetCmd{#1}\PolTypesetMonomialCmd}%
- \edef\PolIndex{\the\numexpr\PolIndex+\POL@@ne@or@m@ne}\POL@ts@check.%
-}%
-\def\POL@ts@nocheck.#1{%
- \if\relax#1\expandafter\xint_gob_til_dot\fi
- \POL@ts@prefix{#1}\PolTypesetCmd{#1}\PolTypesetMonomialCmd
- \edef\PolIndex{\the\numexpr\PolIndex+\POL@@ne@or@m@ne}\POL@ts@nocheck.%
-}%
-
-
-\newcommand\PolMapCoeffs[2]{% #1 = macro, #2 = name
- \POL@mapcoeffs{#1}{#2}%
- \POL@newpol{#2}%
-}%
-\def\POL@mapcoeffs#1#2{%
- \begingroup
- \def\POL@mapcoeffs@macro{#1}%
- \expandafter\expandafter\expandafter\POL@split
- \csname POLuserpol@#2\endcsname;\POL@mapcoeffs@deg\POL@mapcoeffs@coeffs
-% ATTENTION à ne pas faire un \expandafter ici, car brace removal si 1 item
- \xintAssignArray\POL@mapcoeffs@coeffs\to\POL@arrayA
- \def\index{0}%
- \count@\z@
- \expandafter\POL@map@loop\expandafter.\POL@mapcoeffs@coeffs\relax
- \xintloop
-% this abuses that \POL@arrayA0 is never 0.
- \xintiiifZero{\csname POL@arrayA\the\count@\endcsname}%
- {\iftrue}%
- {\iffalse}%
- \advance\count@\m@ne
- \repeat
-% donc en sortie \count@ est 0 ssi pol nul.
- \POL@resultfromarray A%
- \expandafter
- \endgroup
- \expandafter
- \def\csname POLuserpol@#2\expandafter\endcsname\expandafter{\POL@result}%
-}%
-\def\POL@map@loop.#1{\if\relax#1\expandafter\xint_gob_til_dot\fi
- \advance\count@\@ne
- \edef\POL@map@coeff{\POL@mapcoeffs@macro{#1}}%
- \expandafter
- \let\csname POL@arrayA\the\count@\endcsname\POL@map@coeff
- \edef\index{\the\numexpr\index+\@ne}%
- \POL@map@loop.}%
-\def\POL@xintIrr#1{\xintIrr{#1}[0]}%
-\newcommand\PolReduceCoeffs{\@ifstar\POL@sreducecoeffs\POL@reducecoeffs}%
-\def\POL@reducecoeffs#1{\PolMapCoeffs{\POL@xintIrr}{#1}}%
-\def\POL@sreducecoeffs#1{\PolMapCoeffs{\xintPIrr}{#1}}%
-
-
-%% EUCLIDEAN DIVISION
-\newcommand\PolDivide[4]{% #3=quotient, #4=remainder of #1 by #2
- \POL@divide{#1}{#2}%
- \expandafter\let\csname POLuserpol@#3\endcsname\POL@Q
- \POL@newpol{#3}%
- \expandafter\let\csname POLuserpol@#4\endcsname\POL@R
- \POL@newpol{#4}%
-}%
-\newcommand\PolQuo[3]{% #3=quotient of #1 by #2
- \POL@divide{#1}{#2}%
- \expandafter\let\csname POLuserpol@#3\endcsname\POL@Q
- \POL@newpol{#3}%
-}%
-\newcommand\PolRem[3]{% #3=remainder of #1 by #2
- \POL@divide{#1}{#2}%
- \expandafter\let\csname POLuserpol@#3\endcsname\POL@R
- \POL@newpol{#3}%
-}%
-\newcommand\POL@divide[2]{%
- \begingroup
- \let\xintScalarSub\xintSub
- \let\xintScalarAdd\xintAdd
- \let\xintScalarMul\xintMul
- \let\xintScalarDiv\xintDiv
- \expandafter\let\expandafter\POL@A\csname POLuserpol@#1\endcsname
- \expandafter\let\expandafter\POL@B\csname POLuserpol@#2\endcsname
- \POL@div@c
- \let\POL@Q\POL@result
- \ifnum\POL@degQ<\z@
- \let\POL@R\POL@A
- \else
- \count@\numexpr\POL@degR+\@ne\relax
- \POL@resultfromarray R%
- \let\POL@R\POL@result
- \fi
- \expandafter
- \endgroup
- \expandafter
- \def\csname POL@Q\expandafter\expandafter\expandafter\endcsname
- \expandafter\expandafter\expandafter{\expandafter\POL@Q\expandafter}%
- \expandafter
- \def\csname POL@R\expandafter\endcsname\expandafter{\POL@R}%
-}%
-
-
-%% GCD
-\newcommand\PolGCD[3]{% sets #3 to the (unitary) G.C.D. of #1 and #2
- \POL@GCD{#1}{#2}{#3}%
- \POL@newpol{#3}%
-}%
-\def\POL@GCD #1#2#3{%
- \begingroup
- \let\xintScalarSub\xintSub
- \let\xintScalarAdd\xintAdd
- \let\xintScalarMul\xintMul
- \let\xintScalarDiv\xintDiv
- \expandafter\let\expandafter\POL@A\csname POLuserpol@#1\endcsname
- \expandafter\let\expandafter\POL@B\csname POLuserpol@#2\endcsname
- \expandafter\POL@split\POL@A;\POL@degA\POL@polA
- \expandafter\POL@split\POL@B;\POL@degB\POL@polB
- \ifnum\POL@degA<\z@
- \expandafter\xint_firstoftwo\else\expandafter\xint_secondoftwo
- \fi
- {\ifnum\POL@degB<\z@
- \expandafter\xint_firstoftwo\else\expandafter\xint_secondoftwo
- \fi
- {\protected\def\POL@result{-1.\POL@empty{0/1[0]}}}%
- {\xintAssignArray\POL@polB\to\POL@arrayB
- \POL@normalize{B}%
- \POL@gcd@exit BA}}%
- {\ifnum\POL@degB<\z@
- \expandafter\xint_firstoftwo\else\expandafter\xint_secondoftwo
- \fi
- {\xintAssignArray\POL@polA\to\POL@arrayA
- \POL@normalize{A}%
- \POL@gcd@exit AB}%
- {\ifnum\POL@degA<\POL@degB\space
- \let\POL@tmp\POL@B\let\POL@B\POL@A\let\POL@A\POL@tmp
- \let\POL@tmp\POL@degB\let\POL@degB\POL@degA\let\POL@degA\POL@tmp
- \let\POL@tmp\POL@polB\let\POL@polB\POL@polA\let\POL@polA\POL@tmp
- \fi
- \xintAssignArray\POL@polA\to\POL@arrayA
- \xintAssignArray\POL@polB\to\POL@arrayB
- \POL@gcd AB%
- }}%
- \expandafter
- \endgroup
- \expandafter\def\csname POLuserpol@#3\expandafter\endcsname
- \expandafter{\POL@result}%
-}%
-\def\POL@normalize#1{%
- \expandafter\def\expandafter\POL@tmp\expandafter
- {\csname POL@array#1\csname POL@array#10\endcsname\endcsname}%
- \edef\POL@normalize@leading{\POL@tmp}%
- \expandafter\def\POL@tmp{1/1[0]}%
- \count@\csname POL@deg#1\endcsname\space
- \xintloop
- \ifnum\count@>\z@
- \expandafter\edef\csname POL@array#1\the\count@\endcsname
- {\xintIrr{\xintScalarDiv
- {\csname POL@array#1\the\count@\endcsname}%
- {\POL@normalize@leading}}[0]}%
- \advance\count@\m@ne
- \repeat
-}%
-\def\POL@gcd#1#2{%
- \POL@normalize{#2}%
- \edef\POL@degQ{\the\numexpr\csname POL@deg#1\endcsname
- -\csname POL@deg#2\endcsname}%
- \count@\numexpr\csname POL@deg#1\endcsname+\@ne\relax
- \count\tw@\numexpr\POL@degQ+\@ne\relax
- \xintloop
- \POL@gcd@getremainder@loopbody#1#2%
- \ifnum\count\tw@>\z@
- \repeat
- \expandafter\def\csname POL@array#10\endcsname{1}%
- \xintloop
- \xintiiifZero{\csname POL@array#1\the\count@\endcsname}%
- {\iftrue}%
- {\iffalse}%
- \advance\count@\m@ne
- \repeat
- \expandafter\edef\csname POL@deg#1\endcsname{\the\numexpr\count@-\@ne}%
- \ifnum\count@<\@ne
- \expandafter\POL@gcd@exit
- \else
- \expandafter\edef\csname POL@array#10\endcsname{\the\count@}%
- \expandafter\POL@gcd
- \fi{#2}{#1}%
-}%
-\def\POL@gcd@getremainder@loopbody#1#2{%
- \edef\POL@gcd@ratio{\csname POL@array#1\the\count@\endcsname}%
- \advance\count@\m@ne
- \advance\count\tw@\m@ne
- \count4 \count@
- \count6 \csname POL@deg#2\endcsname\space
- \xintloop
- \ifnum\count6>\z@
- \expandafter\edef\csname POL@array#1\the\count4\endcsname
- {\xintScalarSub
- {\csname POL@array#1\the\count4\endcsname}%
- {\xintScalarMul
- {\POL@gcd@ratio}%
- {\csname POL@array#2\the\count6\endcsname}}}%
- \advance\count4 \m@ne
- \advance\count6 \m@ne
- \repeat
-}%
-\def\POL@gcd@exit#1#2{%
- \count@\numexpr\csname POL@deg#1\endcsname+\@ne\relax
- \POL@resultfromarray #1%
-}%
-
-
-%% TODO: BEZOUT
-
-
-%% DIFFERENTIATION
-\def\POL@diff@loop@one #1/#2[#3]#4%
- {\xintIrr{\xintiiMul{#4}{#1}/#2[0]}[#3]}%
-\def\POL@diff#1{\POL@diff@loop1.}%
-\def\POL@diff@loop#1.#2{%
- \if\relax#2\expandafter\xint_gob_til_dot\fi
- {\expandafter\POL@diff@loop@one\romannumeral0\xintraw{#2}{#1}}%
- \expandafter\POL@diff@loop\the\numexpr#1+\@ne.%
-}%
-\newcommand\PolDiff[1][1]{%
- % optional parameter is how many times to derivate
- % first mandatory arg is name of polynomial function to derivate,
- % same name as in \NewPolExpr
- % second mandatory arg name of derivative
- \edef\POL@iterindex{\the\numexpr#1\relax}%
- \ifnum\POL@iterindex<\z@
- \expandafter\@firstoftwo
- \else
- \expandafter\@secondoftwo
- \fi
- {\PolAntiDiff[-\POL@iterindex]}{\POL@Diff}%
-}%
-\def\POL@Diff{%
- \ifcase\POL@iterindex\space
- \expandafter\POL@Diff@no
- \or\expandafter\POL@Diff@one
- \else\xint_afterfi{\POL@Iterate\POL@Diff@one}%
- \fi
-}%
-\def\POL@Diff@no #1#2{\POL@let{#2}{#1}}%
-\def\POL@Diff@one #1#2{\POL@Diff@@one {#1}{#2}\POL@newpol{#2}}%
-\def\POL@Diff@@one#1#2{%
- \expandafter\expandafter\expandafter\POL@split
- \csname POLuserpol@#1\endcsname;\POL@var@deg\POL@var@coeffs
- \ifnum\POL@var@deg<\@ne
- \@namedef{POLuserpol@#2}{-1.\POL@empty{0/1[0]}}%
- \else
- \edef\POL@var@coeffs{\expandafter\POL@diff\POL@var@coeffs\relax}%
- \expandafter\edef\csname POLuserpol@#2\endcsname
- {\the\numexpr\POL@var@deg-\@ne.\POL@empty\POL@var@coeffs}%
- \fi
-}%
-% lazy way but allows to share with AntiDiff
-\def\POL@Iterate#1#2#3{%
- \begingroup
- \xintverbosefalse
- #1{#2}{#3}%
- \xintloop
- \ifnum\POL@iterindex>\tw@
- #1{#3}{#3}%
- \edef\POL@iterindex{\the\numexpr\POL@iterindex-\@ne}%
- \repeat
- \expandafter
- \endgroup\expandafter
- \def\csname POLuserpol@#3\expandafter\endcsname
- \expandafter{\romannumeral`^^@\csname POLuserpol@#3\endcsname}%
- #1{#3}{#3}%
-}%
-
-
-%% ANTI-DIFFERENTIATION
-\def\POL@antidiff@loop@one #1/#2[#3]#4%
- {\xintIrr{#1/\xintiiMul{#4}{#2}[0]}[#3]}%
-\def\POL@antidiff{\POL@antidiff@loop1.}%
-\def\POL@antidiff@loop#1.#2{%
- \if\relax#2\expandafter\xint_gob_til_dot\fi
- {\expandafter\POL@antidiff@loop@one\romannumeral0\xintraw{#2}{#1}}%
- \expandafter\POL@antidiff@loop\the\numexpr#1+\@ne.%
-}%
-\newcommand\PolAntiDiff[1][1]{%
- % optional parameter is how many times to derivate
- % first mandatory arg is name of polynomial function to derivate,
- % same name as in \NewPolExpr
- % second mandatory arg name of derivative
- \edef\POL@iterindex{\the\numexpr#1\relax}%
- \ifnum\POL@iterindex<\z@
- \expandafter\@firstoftwo
- \else
- \expandafter\@secondoftwo
- \fi
- {\PolDiff[-\POL@iterindex]}{\POL@AntiDiff}%
-}%
-\def\POL@AntiDiff{%
- \ifcase\POL@iterindex\space
- \expandafter\POL@AntiDiff@no
- \or\expandafter\POL@AntiDiff@one
- \else\xint_afterfi{\POL@Iterate\POL@AntiDiff@one}%
- \fi
-}%
-\let\POL@AntiDiff@no\POL@Diff@no
-\def\POL@AntiDiff@one #1#2{\POL@AntiDiff@@one{#1}{#2}\POL@newpol{#2}}%
-\def\POL@AntiDiff@@one#1#2{%
- \expandafter\expandafter\expandafter\POL@split
- \csname POLuserpol@#1\endcsname;\POL@var@deg\POL@var@coeffs
- \ifnum\POL@var@deg<\z@
- \@namedef{POLuserpol@#2}{-1.\POL@empty{0/1[0]}}%
- \else
- \edef\POL@var@coeffs{\expandafter\POL@antidiff\POL@var@coeffs\relax}%
- \expandafter\edef\csname POLuserpol@#2\endcsname
- {\the\numexpr\POL@var@deg+\@ne.\POL@empty{0/1[0]}\POL@var@coeffs}%
- \fi
-}%
-
-%% IContent and \PolMakePrimitive (0.5)
-\def\POL@aux@mgcd@loop#1#2{%
- \if\relax#2\expandafter\POL@aux@mgcd@exit\fi
- \expandafter
- \POL@aux@mgcd@loop\romannumeral0\POL@aux@gcd#1.#2.%
-}%
-\def\POL@aux@mgcd@exit
- \expandafter
- \POL@aux@mgcd@loop\romannumeral0\POL@aux@gcd#1.\relax.{\xintiiabs{#1}}%
-\def\POL@aux@gcd#1.#2.{%
- \if0\xintiiSgn{#1}\expandafter\POL@aux@gcd@exit\fi
- \expandafter\POL@aux@gcd\romannumeral0\xintmod {#2}{#1}.#1.}%
-\def\POL@aux@gcd@exit
- \expandafter\POL@aux@gcd\romannumeral0\xintmod #1#2.#3.{{#1}}%
-
-\def\POL@icontent #1{\romannumeral0\expandafter
- \POL@aux@mgcd@loop\romannumeral`^^@#1\relax}%
-
-\newcommand\PolIContent[1]{\romannumeral0\expandafter
- \POL@aux@mgcd@loop\romannumeral`^^@\PolToList{#1}\relax}%
-
-
-\def\POL@makeprim@macro#1%
- {\xintREZ{\xintNum{\xintDiv{#1}{\POL@makeprim@icontent}}}}%
-\newcommand\PolMakePrimitive[1]{%
- % This does not need a full user declared polynomial on input, only
- % a \POLuserpol@name macro, but on output it is fully declared
- \edef\POL@makeprim@icontent{\PolIContent{#1}}%
- \PolMapCoeffs\POL@makeprim@macro{#1}%
-}%
-\def\POL@makeprimitive#1{%
- % Avoids declaring the polynomial, internal usage in \PolToSturm
- \edef\POL@makeprim@icontent{\PolIContent{#1}}%
- \POL@mapcoeffs\POL@makeprim@macro{#1}%
-}%
-
-
-%% Sturm Algorithm (polexpr 0.4)
-%% 0.5 uses primitive polynomials for faster evaluations afterwards
-%% 0.6 corrects misuse of \@ifstar! (mumble). \PolToSturm* was broken.
-%% 0.6's \PolToSturm* defines both normalized and unnormalized, the
-%% unnormalized using two underscores, so both are available
-%% Sole difference is that \PolToSturm* also declares them as
-%% user polynomials, whereas the non-starred only keeps the macros
-%% holding the coefficients in memory
-%% 0.6 fixes the case of a constant polynomial P which caused division
-%% by zero error from P'.
-\newcommand\PolToSturm{\@ifstar{\PolToSturm@@}{\PolToSturm@}}%
-\def\POL@aux@toint#1{\xintREZ{\xintNum{#1}}}% for polynomials with int. coeffs!
-%% Attention that some macros rely upon this one setting \POL@sturmname
-%% and \POL@sturm@N as it does
-\def\PolToSturm@#1#2{%
- \edef\POL@sturmname{#2}%
- % 0.6 uses 2 underscores (one before index, one after) to keep in memory
- % the unnormalized chain
- % This supposes #1 to be a genuine polynomial, not only a name with
- % a \POLuserpol@#1 macro
- \POL@let{\POL@sturmname _0_}{#1}%
- \ifnum\PolDegree{#1}=\z@
- \def\POL@sturm@N{0}%
- \POL@count\z@
- % if I applied the same as for positive degree, I should make it -1
- % if constant is negative. I also don't worry if polynomial is zero.
- \@namedef{POLuserpol@\POL@sturmname _0}{0.\POL@empty{1/1[0]}}%
- \else
- \ifPOL@tosturm@makefirstprimitive\POL@makeprimitive{\POL@sturmname _0_}\fi
- \POL@tosturm@dosturm
- \fi
- \expandafter
- \let\csname PolSturmChainLength_\POL@sturmname\endcsname\POL@sturm@N
- % declare the normalized ones as full-fledged polynomials
- % \POL@count\z@
- \xintloop
- \POL@newpol{\POL@sturmname _\the\POL@count}%
- \unless\ifnum\POL@sturm@N=\POL@count
- \advance\POL@count\@ne
- \repeat
-}%
-\def\PolToSturm@@#1#2{\PolToSturm@{#1}{#2}\POL@tosturm@declareunnormalized}%
-\def\POL@tosturm@declareunnormalized{%
- % optionally declare also the unnormalized ones
- \POL@count\z@
- \xintloop
- \POL@newpol{\POL@sturmname _\the\POL@count _}%
- \unless\ifnum\POL@sturm@N=\POL@count
- \advance\POL@count\@ne
- \repeat
-}%
-\def\POL@tosturm@dosturm{%
- \POL@Diff@@one{\POL@sturmname _0_}{\POL@sturmname _1_}%
- % re-utiliser \POL@varcoeffs directement?
- \POL@makeprimitive{\POL@sturmname _1_}% does not do \POL@newpol
- \POL@count\@ne
- \xintloop
- \POL@divide{\POL@sturmname _\the\numexpr\POL@count-\@ne\relax _}%
- {\POL@sturmname _\the\POL@count _}%
- \expandafter\POL@split\POL@R;\POL@degR\POL@polR
- \unless\ifnum\POL@degR=\m@ne
- \advance\POL@count\@ne
- \expandafter\let
- \csname POLuserpol@\POL@sturmname _\the\POL@count _\endcsname\POL@R
- \edef\POL@makeprim@icontent{-\POL@icontent\POL@polR}%
- % this avoids the \POL@newpol from \PolMapCoeffs
- \POL@mapcoeffs\POL@makeprim@macro{\POL@sturmname _\the\POL@count _}%
- \repeat
- \edef\POL@sturm@N{\the\POL@count}%
- % normalize (now always done even by starred variant)
- \ifnum\PolDegree{\POL@sturmname _\POL@sturm@N _}>\z@
- % \POL@count\POL@sturm@N\relax
- \xintloop
- \advance\POL@count\m@ne
- \POL@divide{\POL@sturmname _\the\POL@count _}%
- {\POL@sturmname _\POL@sturm@N _}%
- \expandafter
- \let\csname POLuserpol@\POL@sturmname _\the\POL@count\endcsname\POL@Q
- % quotient actually belongs to Z[X] and is primitive
- \POL@mapcoeffs\POL@aux@toint{\POL@sturmname _\the\POL@count}%
- \ifnum\POL@count>\z@
- \repeat
- \@namedef{POLuserpol@\POL@sturmname _\POL@sturm@N}{0.\POL@empty{1/1[0]}}%
- \else % they are already normalized
- \advance\POL@count\@ne % attention to include last one also
- \xintloop
- \advance\POL@count\m@ne
- \expandafter\let
- \csname POLuserpol@\POL@sturmname _\the\POL@count\expandafter\endcsname
- \csname POLuserpol@\POL@sturmname _\the\POL@count _\endcsname
- \ifnum\POL@count>\z@
- \repeat
- \fi
- % Back to \PolToSturm@, \POL@count holds 0
-}%
-\newcommand\PolSturmChainLength[1]
- {\romannumeral`^^@\csname PolSturmChainLength_#1\endcsname}%
-
-\newcommand\PolSetToSturmChainSignChangesAt[4][\global]{%
- \edef\POL@sturmchain@X{\xintREZ{#4}}%
- \edef\POL@sturmname{#3}%
- \edef\POL@sturmlength{\PolSturmChainLength{\POL@sturmname}}%
- \POL@sturmchain@getSV@at\POL@sturmchain@X
- #1\let#2\POL@sturmchain@SV
-}%
-\def\POL@sturmchain@getSV@at#1{% ATTENTION USES \POL@count
- \def\POL@sturmchain@SV{0}%
- \edef\POL@sturmchain@sign{\xintiiSgn{\POL@eval{\POL@sturmname _0}{#1}}}%
- \let\POL@isolz@lastsign\POL@sturmchain@sign
- \POL@count \z@
- \ifnum\POL@isolz@lastsign=\z@
- \edef\POL@isolz@lastsign
- {\xintiiSgn{\POL@eval{\POL@sturmname _1}{#1}}}%
- \POL@count \@ne
- \fi
- \xintloop
- \unless\ifnum\POL@sturmlength=\POL@count
- \advance\POL@count \@ne
- \edef\POL@isolz@newsign
- {\xintiiSgn{\POL@eval{\POL@sturmname _\the\POL@count}{#1}}}%
- \ifnum\POL@isolz@newsign=\numexpr-\POL@isolz@lastsign\relax
- \edef\POL@sturmchain@SV{\the\numexpr\POL@sturmchain@SV+\@ne}%
- \let\POL@isolz@lastsign=\POL@isolz@newsign
- \fi
- \repeat
-}%
-\newcommand\PolSetToNbOfZerosWithin[5][\global]{%
- \edef\POL@tmpA{\xintREZ{#4}}%
- \edef\POL@tmpB{\xintREZ{#5}}%
- \edef\POL@sturmname{#3}%
- \edef\POL@sturmlength{\PolSturmChainLength{\POL@sturmname}}%
- \POL@sturmchain@getSV@at\POL@tmpA
- \let\POL@SVA\POL@sturmchain@SV
- \POL@sturmchain@getSV@at\POL@tmpB
- \let\POL@SVB\POL@sturmchain@SV
- \ifnum\POL@SVA<\POL@SVB\space
- #1\edef#2{\the\numexpr\POL@SVB-\POL@SVA}%
- \else
- #1\edef#2{\the\numexpr\POL@SVA-\POL@SVB}%
- \fi
-}%
-
-
-% 0.6 added starred variant to count multiplicities
-% 0.7 added double starred variant to locate all rational roots
-\newcommand\PolSturmIsolateZeros{\@ifstar
- {\PolSturmIsolateZerosAndGetMultiplicities}%
- {\PolSturmIsolateZeros@}%
-}%
-\newcommand\PolSturmIsolateZerosAndGetMultiplicities{\@ifstar
- {\PolSturmIsolateZerosGetMultiplicitiesAndRationalRoots}%
- {\PolSturmIsolateZerosAndGetMultiplicities@}%
-}%
-% on aurait besoin de ça dans xint, mais il aurait un \xintRaw{#1} alors
-\def\POL@xintfrac@getNDE #1%
- {\expandafter\POL@xintfrac@getNDE@i\romannumeral`^^@#1}%
-\def\POL@xintfrac@getNDE@i #1/#2[#3]#4#5#6{\def#4{#1}\def#5{#2}\def#6{#3}}%
-\newcommand\PolSturmIsolateZerosGetMultiplicitiesAndRationalRoots[2][\empty]{%
- \PolSturmIsolateZerosAndFindRationalRoots[#1]{#2}%
- \ifnum\POL@isolz@NbOfRoots>\z@
- % get multiplicities of irrational (real) roots, if any
- \ifnum\POL@findrat@nbofirrroots>\z@
- \POL@findrat@getirrmult
- \fi
- \POL@isolzmult@defvar@M
- \fi
-}%
-% added at 0.7
-\newcommand\PolSturmIsolateZerosAndFindRationalRoots[2][\empty]{%
- % #1 optional E such that roots are searched in -10^E < x < 10^E
- % both -10^E and +10^E must not be roots!
- % #2 name of Sturm chain (already pre-computed)
- \edef\POL@sturmname{#2}%
- \edef\POL@sturm@N{\@nameuse{PolSturmChainLength_\POL@sturmname}}%
- % isolate the roots (detects case of constant polynomial)
- \PolSturmIsolateZeros@{\POL@sturmname}%
- \ifnum\POL@isolz@NbOfRoots=\z@
- % no real roots, define empty arrays nevertheless
- \begingroup\globaldefs\@ne
- \expandafter\xintAssignArray\expandafter\to\csname POL_ZM\POL@sturmname*\endcsname
- \expandafter\xintAssignArray\expandafter\to\csname POL_RI\POL@sturmname*\endcsname
- \endgroup
- \else
- % all we currently know is that multiplicities are at least one
- \begingroup\globaldefs\@ne
- \expandafter\POL@initarray\csname POL_ZM\POL@sturmname*\endcsname{1}%
- \endgroup
- % on ne va pas utiliser de Horner, mais des divisions par X - x, et ces
- % choses vont évoluer, ainsi que le coefficient dominant entier
- % (pour \POL@divide entre autres if faut des noms de user pol)
- \expandafter\let
- \csname POLuserpol@\POL@sturmname\POL@sqfnorr\expandafter\endcsname
- \csname POLuserpol@\POL@sturmname _0\endcsname
- \expandafter\let
- \csname POLuserpol@\POL@sturmname\POL@norr\expandafter\endcsname
- \csname POLuserpol@\POL@sturmname _0_\endcsname
- % attention formé avec\xintREZ d'où le \xintAbs pas \xintiiAbs
- % D and its exponent E will get updated along the way
- \edef\POL@findrat@D{\xintAbs{\PolLeadingCoeff{\POL@sturmname _0}}}%
- \POL@xintfrac@getNDE\POL@findrat@D\POL@findrat@Dint\POL@_\POL@findrat@Dexp
- \xintiiifOne{\POL@findrat@Dint}
- {\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
- \def\POL@findrat@index{1}%
- \POL@findrat@loop@secondpass@direct
- \else
- % we do a first pass scanning for "small" roots p/q (i.e. q < 1000)
- \def\POL@findrat@index{1}%
- \POL@findrat@loop@firstpass
- % and now we do the final pass finding them all
- \def\POL@findrat@index{1}%
- \POL@findrat@loop@secondpass
- \fi
- % declare the new polynomials
- \POL@newpol{\POL@sturmname\POL@sqfnorr}% without multiplicities
- \POL@newpol{\POL@sturmname\POL@norr}% with multiplicities
- % declare the array holding the interval indices for the rational roots
- \expandafter\POL@findrat@doRRarray\csname POL_RI\POL@sturmname*\endcsname
- \fi
-}%
-\def\POL@findrat@doRRarray#1{%
- % il faudrait un \xintAssignArray* qui fasse même expansion que \xintFor*
- \edef\POL@temp{%
- \xintiloop[1+1]
- \romannumeral0\csname POL_ZK\POL@sturmname*\xintiloopindex\endcsname
- \xintbracediloopindex % I should have named it \xintiloopbracedindex...
- {}%
- \ifnum\xintiloopindex<\POL@isolz@NbOfRoots\space
- \repeat }%
- \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%
- \endgroup
-}%
-\def\POL@findrat@loop@firstpass{%
- \PolSturmIfZeroExactlyKnown{\POL@sturmname}{\POL@findrat@index}%
- \POL@findrat@loop@decimal% get its multiplicity
- \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@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}%
- \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
- % j'ai beaucoup hésité néanmoins, car je pourrais faire \xintIrr ici,
- % mais attention aussi à l'interaction avec le \PolDecToString. Les racines
- % trouvées directement (qui peuvent être des nombres décimaux) sont elles
- % stockées comme fraction irréductibles (modulo action additionnelle de
- % \PolDecToString).
- \POL@xintfrac@getNDE
- {\xintIrr{\POL@xintexprGetVar{\POL@sturmname L_\POL@findrat@index}}[0]}%
- \POL@findrat@xN\POL@findrat@xD\POl@_
- % we can't move this to updatequotients because other branch will
- % need to do the division first anyhow
- \edef\POLuserpol@_findrat@oneterm{1.\POL@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
-}%
-% lacking from xint 1.3c, but \xintSgn has overhead, so we define ii version
-\def\xintiiifNeg{\romannumeral0\xintiiifneg }%
-\def\xintiiifneg #1%
-{%
- \ifcase \xintiiSgn{#1}
- \expandafter\xint_stop_atsecondoftwo
- \or\expandafter\xint_stop_atsecondoftwo
- \else\expandafter\xint_stop_atfirstoftwo
- \fi
-}%
-\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{%
- % attention that the width may have been already smaller than 10^{-6}
- \POL@get@IsoLeft@rawin
- \POL@get@IsoRight@rawin
- \edef\POL@findrat@localW
- {\the\numexpr-\expandafter\POL@findrat@getE
- % do I really need the \xintREZ?
- \romannumeral0\xintrez
- {\xintSub{\POL@IsoRight@rawin}{\POL@IsoLeft@rawin}}%
- }% at least 6, maybe larger
- \expandafter\POL@get@Int@aux
- \POL@IsoLeft@rawin\POL@IsoLeft@Int{-\POL@findrat@localW}%
- \expandafter\POL@get@Int@aux
- \POL@IsoRight@rawin\POL@IsoRight@Int{-\POL@findrat@localW}%
- % in case of odd, some waste here
- \edef\POL@findrat@halflocalW{\the\numexpr(\POL@findrat@localW+1)/2-1}%
- % Legendre Theorem will be used now but we separate a branch where
- % everything can be done with \numexpr
- \ifnum\POL@findrat@localW>9
- % not implemented yet by lazyness!
- % this root will be handled in second pass only
- \else
- \POL@findrat@gcdloop
- \fi
-}%
-\def\POL@findrat@gcdloop{%
- % 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@error@thisisimpossible
- \POL@findrat@gcdloop@p
-}%
-\def\POL@findrat@gcdloop@n{%
- \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}}%
- \POL@findrat@gcdloop@p
-}%
-\def\POL@findrat@gcdloop@p{%
- \edef\POL@findrat@gcdloop@Ap{\xintDec{\xintDouble\POL@IsoRight@Int}}%
- \edef\POL@findrat@gcdloop@A
- % at most 2e9: this is acceptable to \numexpr
- {2\romannumeral\xintreplicate\POL@findrat@localW{0}}%
- \xintAssign
- \xintiiDivision\POL@findrat@gcdloop@Ap\POL@findrat@gcdloop@A
- \to\POL@findrat@gcdloop@B\POL@findrat@gcdloop@An
- % on fait de la tambouille pour n'utiliser que \numexpr par la suite
- % le reste @An est < 2.10^9 au pire donc ok pour \numexpr
- % we will drop integral part in our updating P
- \let\POL@findrat@gcdloop@Binitial\POL@findrat@gcdloop@B
- \def\POL@findrat@gcdloop@B{0}% do as if B1 = 0
- \def\POL@findrat@gcdloop@Pp{1}% P0
- \def\POL@findrat@gcdloop@P{0}% P1
- \def\POL@findrat@gcdloop@Qp{0}% Q0
- \def\POL@findrat@gcdloop@Q{1}% Q1
- % A2=An can not be zero, as Ap (=A0) is odd and A (=A1=200...000) is even
- % first Binitial + P1/Q1 ( = Binitial) can not be root
- \let\POL@findrat@gcdloop@Ap\POL@findrat@gcdloop@A % A1
- \let\POL@findrat@gcdloop@A\POL@findrat@gcdloop@An % A2
- \def\next{\POL@findrat@gcdloop@update}%
- \def\POL@findrat@gcdloop@done{0}%
- \POL@findrat@gcdloop@body
-}%
-\def\POL@findrat@gcdloop@body{%
- % annoying that \numexpr has no divmod... use counts? but groups annoying
- \edef\POL@findrat@gcdloop@B
- {\the\numexpr(\POL@findrat@gcdloop@Ap+\POL@findrat@gcdloop@A/2)/%
- \POL@findrat@gcdloop@A - \@ne}%
- \edef\POL@findrat@gcdloop@An
- {\the\numexpr\POL@findrat@gcdloop@Ap-%
- \POL@findrat@gcdloop@B*\POL@findrat@gcdloop@A}%
- \edef\POL@findrat@gcdloop@Pn
- {\the\numexpr\POL@findrat@gcdloop@Pp+%
- \POL@findrat@gcdloop@B*\POL@findrat@gcdloop@P}%
- \edef\POL@findrat@gcdloop@Qn
- {\the\numexpr\POL@findrat@gcdloop@Qp+%
- \POL@findrat@gcdloop@B*\POL@findrat@gcdloop@Q}%
- \ifnum\expandafter\xintLength\expandafter{\POL@findrat@gcdloop@Qn}%
- >\POL@findrat@halflocalW\space
- \let\next\empty % no solution was found
- \else
- % with these conditions on denom, only candidates are by Legendre
- % theorem among the convergents as computed here
- \ifnum\POL@findrat@gcdloop@Qn>\POL@findrat@gcdloop@An\space
- % means that P/Q is in interval and is thus a candidate
- % it is automatically irreducible
- \edef\POL@findrat@x{\xintiiAdd
- {\xintiiMul{\POL@findrat@gcdloop@Qn}{\POL@findrat@gcdloop@Binitial}}%
- {\POL@findrat@gcdloop@Pn}/\POL@findrat@gcdloop@Qn[0]}%
- \POL@findrat@gcdloop@testit
- \if1\POL@findrat@gcdloop@done
- \let\next\empty % a solution was found
- \fi
- \fi
- \fi
- \next
-}%
-\def\POL@findrat@gcdloop@update{%
- \ifnum\POL@findrat@gcdloop@An>\z@
- \let\POL@findrat@gcdloop@Ap\POL@findrat@gcdloop@A
- \let\POL@findrat@gcdloop@A\POL@findrat@gcdloop@An
- \let\POL@findrat@gcdloop@Pp\POL@findrat@gcdloop@P
- \let\POL@findrat@gcdloop@P\POL@findrat@gcdloop@Pn
- \let\POL@findrat@gcdloop@Qp\POL@findrat@gcdloop@Q
- \let\POL@findrat@gcdloop@Q\POL@findrat@gcdloop@Qn
- \expandafter\POL@findrat@gcdloop@body
- \fi
-}%
-\def\POL@findrat@gcdloop@testit{%
- % zero should never occur here
- \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.\POL@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
- \ifnum\POL@degR=\m@ne % found a root
- \POL@findrat@loop@storeit
- \POL@findrat@loop@updatequotients
- \POL@findrat@loop@getmultiplicity % will continue updating the mult. one
- \def\POL@findrat@gcdloop@done{1}%
- \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}}}%
- \edef\POL@findrat@Rscaled{\xintMul{\POL@findrat@D}%
- {\POL@xintexprGetVar{\POL@sturmname R_\POL@findrat@index}}}%
- \xintiiifNeg{\POL@findrat@Lscaled}% using ii version is an abuse
- {% negative interval (right bound possibly zero!)
- % truncate towards zero (i.e. to the right) the left bound
- \edef\POL@findrat@Num{\xintNum{\POL@findrat@Lscaled}/1[0]}%
- % interval boundaries are not root hence in case that was exact
- % this will not be found as a root; check if in interval
- \xintifLt\POL@findrat@Num\POL@findrat@Rscaled
- \POL@findrat@loop@c
- {}% iterate
- }%
- {% positive interval (left bound possibly zero!)
- % truncate towards zero (i.e. to the left) the right bound
- \edef\POL@findrat@Num{\xintNum{\POL@findrat@Rscaled}/1[0]}%
- % check if in interval
- \xintifGt\POL@findrat@Num\POL@findrat@Lscaled
- \POL@findrat@loop@c
- {}% iterate
- }%
-}%
-\def\POL@findrat@loop@c{%
- % safer to do the edef as \POL@findrat@x used later in storeit
- \edef\POL@findrat@x{\xintIrr{\xintDiv\POL@findrat@Num\POL@findrat@D}[0]}%
- \POL@xintfrac@getNDE\POL@findrat@x\POL@findrat@xN\POL@findrat@xD\POL@_
- \edef\POLuserpol@_findrat@oneterm{1.\POL@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
- \ifnum\POL@degR=\m@ne % found a root
- \POL@findrat@loop@storeit
- \POL@findrat@loop@updatequotients
- \POL@findrat@loop@getmultiplicity % will continue updating the mult. one
- \fi
- % iterate
-}%
-\def\POL@findrat@loop@storeit{%
- % update storage, I can not use storeleftandright here (due to rawout etc...)
- \expandafter
- \xdef\csname POL_ZL\POL@sturmname*\POL@findrat@index\endcsname
- {\PolDecToString{\POL@findrat@x}}%
- \global\expandafter
- \let\csname POL_ZR\POL@sturmname*\POL@findrat@index\expandafter\endcsname
- \csname POL_ZL\POL@sturmname*\POL@findrat@index\endcsname
- \global\expandafter
- \let\csname POL_ZK\POL@sturmname*\POL@findrat@index\endcsname
- \xint_stop_atfirstoftwo
- \begingroup\xintglobaldefstrue
- % skip some overhead of \xintdefvar...
- \XINT_expr_defvar_one{\POL@sturmname L_\POL@findrat@index}%
- {\POL@findrat@x}%
- \XINT_expr_defvar_one{\POL@sturmname R_\POL@findrat@index}%
- {\POL@findrat@x}%
- \XINT_expr_defvar_one{\POL@sturmname Z_\POL@findrat@index _isknown}%
- {1}%
- \endgroup
-}%
-\def\POL@findrat@loop@updatequotients{%
- % attention last division must have been one testing vanishing of\POL@sqfnorr
- \expandafter\let\csname POLuserpol@\POL@sturmname\POL@sqfnorr\endcsname\POL@Q
- % quotient belongs to Z[X] and is primitive
- \POL@mapcoeffs\POL@aux@toint{\POL@sturmname\POL@sqfnorr}%
- % update the one with multiplicities
- \POL@divide{\POL@sturmname\POL@norr}{_findrat@oneterm}%
- \expandafter\let\csname POLuserpol@\POL@sturmname\POL@norr\endcsname\POL@Q
- \POL@mapcoeffs\POL@aux@toint{\POL@sturmname\POL@norr}
- % updating of \POL@findrat@D at end of execution of getmultiplicity
-}%
-\def\POL@findrat@loop@getmultiplicity{%
- % the one without multiplicity must not be divided again!
- % check if we have remaining multiplicity
- \POL@divide{\POL@sturmname\POL@norr}{_findrat@oneterm}%
- \expandafter\POL@split\POL@R;\POL@degR\POL@polR
- \ifnum\POL@degR=\m@ne % yes
- \expandafter\let\csname POLuserpol@\POL@sturmname\POL@norr\endcsname\POL@Q
- \POL@mapcoeffs\POL@aux@toint{\POL@sturmname\POL@norr}%
- \expandafter
- \xdef
- \csname POL_ZM\POL@sturmname*\POL@findrat@index\endcsname
- {\the\numexpr
- \csname POL_ZM\POL@sturmname*\POL@findrat@index\endcsname+\@ne}%
- \expandafter\POL@findrat@loop@getmultiplicity
- \else
- % done with multiplicity for this rational root, update stuff
- \edef\POL@findrat@nbofirrroots
- {\the\numexpr\POL@findrat@nbofirrroots-\@ne}%
- \@namedef{POL@IfMultIsKnown\POL@findrat@index}{\xint_firstoftwo}%
- \edef\POL@findrat@D{\xintAbs{\PolLeadingCoeff{\POL@sturmname\POL@sqfnorr}}}%
- \POL@xintfrac@getNDE\POL@findrat@D\POL@findrat@Dint\POL@_\POL@findrat@Dexp
- \xintiiifOne{\POL@findrat@Dint}
- {\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}}%
- \fi
-}%
-\def\POL@findrat@getirrmult{%
- % first get the GCD of remaining pol with its derivative
- \POL@divide{\POL@sturmname\POL@norr}{\POL@sturmname\POL@sqfnorr}%
- \expandafter\let
- % attention au _ (cf. grosse astuce pour \POL@isolzmult@loop)
- \csname POLuserpol@@_1\POL@sturmname _\endcsname\POL@Q
- \ifnum\PolDegree{@_1\POL@sturmname _}>\z@
- % il reste des multiplicités (mais peut-être pour des racines complexes)
- % (ou pour des racines en-dehors de l'intervalle optionnel)
- % attention recyclage ici de \POL@isolzmult@loop qui dépend de
- % la grosse astuce avec \@gobble
- \POL@makeprimitive{@_1\POL@sturmname _}%
- \let\POL@originalsturmname\POL@sturmname
- % trick to get isolzmult@loop to define @@lastGCD to @_1sturmname_
- % because it will do \POL@sturmname _\POL@sturm@N _
- \edef\POL@sturmname{@_1\POL@sturmname}%
- \let\POL@sturm@N\@gobble% !
- \let\POL@isolz@NbOfRoots@with_unknown_mult\POL@findrat@nbofirrroots
- \POL@tosturm@makefirstprimitivefalse
- \POL@isolzmult@loop
- \POL@tosturm@makefirstprimitivetrue
- \let\POL@sturmname\POL@originalsturmname
- \fi
-}%
-
-
-\newcommand\PolSturmIsolateZerosAndGetMultiplicities@[2][\empty]{%
- % #1 optional E such that roots are searched in -10^E < x < 10^E
- % both -10^E and +10^E must not be roots!
- % #2 name of Sturm chain (already pre-computed)
- \edef\POL@sturmname{#2}%
- \edef\POL@sturm@N{\@nameuse{PolSturmChainLength_\POL@sturmname}}%
- % isolate the roots (detects case of constant polynomial)
- \PolSturmIsolateZeros@{\POL@sturmname}%
- \ifnum\POL@isolz@NbOfRoots=\z@
- % no roots, define empty array nevertheless
- \begingroup\globaldefs\@ne
- \expandafter\xintAssignArray\expandafter\to\csname POL_ZM\POL@sturmname*\endcsname
- \endgroup
- \else
- % all we currently know is that multiplicities are at least one
- \begingroup\globaldefs\@ne
- \expandafter\POL@initarray\csname POL_ZM\POL@sturmname*\endcsname{1}%
- \endgroup
- % check if GCD had positive degree (hence some roots, maybe complex, have
- % multiplicity)
- \ifnum\PolDegree{\POL@sturmname _\POL@sturm@N _}>\z@
- % scratch array of flags to signal known multiplicities
- \POL@initarray\POL@IfMultIsKnown\xint_secondoftwo
- % this count has utility for the case there are other roots
- % either complex or outside interval (in case of optional argument)
- \let\POL@isolz@NbOfRoots@with_unknown_mult\POL@isolz@NbOfRoots
- % store Sturm chain name, it is needed and altered in isolzmult@loop
- \let\POL@originalsturmname\POL@sturmname
- \POL@tosturm@makefirstprimitivefalse
- \POL@isolzmult@loop
- \POL@tosturm@makefirstprimitivetrue
- \let\POL@sturmname\POL@originalsturmname
- \fi
- \POL@isolzmult@defvar@M
- \fi
-}%
-\def\POL@isolzmult@defvar@M{%
- % Attention that is used not only in ...GetMultiplicities@ but also
- % in FindRationalRoots
- \begingroup\xintglobaldefstrue
- % added at 0.7
- \let\x\POL@isolz@NbOfRoots
- \xintloop
- % skip some overhead of \xintdefvar...
- \XINT_expr_defvar_one{\POL@sturmname M_\x}%
- {\csname POL_ZM\POL@sturmname*\x\endcsname}%
- \edef\x{\the\numexpr\x-\@ne}%
- \ifnum\x>\z@
- \repeat
- \endgroup
-}%
-\def\POL@isolzmult@loop{%
- % we are here only if last iteration gave a new GCD still of degree > 0
- % \POL@sturm@N is the one from last iteration
- % Attention to not use \POL@sturmname directly in first arg. of \PolToSturm
- % Attention that we need for the case of known roots also to have the last
- % GCD (with its multiplicities) known as a genuine polynomial
- % - because of usage of \POL@eval in @isknown branch
- % - because \PolToSturm@ does a \POL@let which would be anomalous
- % if the extended structure is not existing
- \edef\POL@isolzmult@lastGCD{\POL@sturmname _\POL@sturm@N _}%
- \edef\POL@isolzmult@newsturmname{@_1\POL@sturmname}%
- \POL@newpol{\POL@isolzmult@lastGCD}%
- \PolToSturm@{\POL@isolzmult@lastGCD}{\POL@isolzmult@newsturmname}%
- % now both \POL@sturmname and \POL@sturm@N have changed
- \edef\POL@isolzmult@newGCDdegree{\PolDegree{\POL@sturmname _\POL@sturm@N _}}%
- \let\POL@isolzmult@index\POL@isolz@NbOfRoots
- \xintloop
- % ATTENTION that this executes macros which also modifies \POL@sturmname!
- % (but not \POL@sturm@N)
- \POL@isolzmult@doone
- \edef\POL@isolzmult@index{\the\numexpr\POL@isolzmult@index-\@ne}%
- \if1\ifnum\POL@isolz@NbOfRoots@with_unknown_mult=\z@ 0\fi
- \ifnum\POL@isolzmult@index=\z@ 0\fi 1%
- \repeat
- \let\POL@sturmname\POL@isolzmult@newsturmname
- \if1\ifnum\POL@isolz@NbOfRoots@with_unknown_mult=\z@ 0\fi
- % (if new GCD is constant, time to abort)
- \ifnum\POL@isolzmult@newGCDdegree=\z@ 0\fi 1%
- \expandafter\POL@isolzmult@loop
- \fi
-}%
-\def\POL@isolzmult@doone{%
- \csname POL@IfMultIsKnown\POL@isolzmult@index\endcsname
- {}% nothing to do
- {\POL@SturmIfZeroExactlyKnown{\POL@originalsturmname}%
- {\POL@isolzmult@index}%
- \POL@isolzmult@loop@isknown
- \POL@isolzmult@loop@isnotknown
- \POL@isolzmult@loop@sharedbody
- }%
-}%
-\def\POL@isolzmult@loop@isknown{%
- \xintifZero
- % attention that \POL@eval requires a declared polynomial
- {\POL@eval{\POL@isolzmult@lastGCD}%
- {\POL@xintexprGetVar{\POL@originalsturmname L_\POL@isolzmult@index}}}%
- {\let\POL@isolzmult@haszero\@ne}%
- {\let\POL@isolzmult@haszero\z@}%
-}%
-\def\POL@isolzmult@loop@isnotknown{%
- \edef\POL@isolzmult@loop@A
- {\POL@xintexprGetVar{\POL@originalsturmname L_\POL@isolzmult@index}}
- \edef\POL@isolzmult@loop@B
- {\POL@xintexprGetVar{\POL@originalsturmname
- R_\POL@isolzmult@index}}
- % attention that \PolSetToNbOfZerosWithin sets \POL@sturmname to 2nd argument
- \PolSetToNbOfZerosWithin
- \POL@isolzmult@haszero % nb of zeros A < x <= B, here 0 or 1
- \POL@isolzmult@newsturmname
- \POL@isolzmult@loop@A
- \POL@isolzmult@loop@B
-}%
-\def\POL@isolzmult@loop@sharedbody{%
- \ifnum\POL@isolzmult@haszero>\z@
- \expandafter
- \xdef
- \csname POL_ZM\POL@originalsturmname*\POL@isolzmult@index\endcsname
- {\the\numexpr
- \csname POL_ZM\POL@originalsturmname
- *\POL@isolzmult@index\endcsname+\@ne}%
- \else
- % multiplicity now known, no need to check this index in future
- \@namedef{POL@IfMultIsKnown\POL@isolzmult@index}{\xint_firstoftwo}%
- \edef\POL@isolz@NbOfRoots@with_unknown_mult
- {\the\numexpr\POL@isolz@NbOfRoots@with_unknown_mult-\@ne}%
- \fi
-}%
-
-
-\newcommand\PolSturmIsolateZeros@[2][\empty]{%
- % #1 optional E such that roots are searched in -10^E < x < 10^E
- % both -10^E and +10^E must not be roots!
- % #2 name of Sturm chain (already pre-computed from a given polynomial)
- % For reasons I have forgotten (no time now) this code **must** be used
- % with a *normalized* Sturm chain.
- \edef\POL@sturmname{#2}%
- \edef\POL@sturmlength{\PolSturmChainLength{#2}}%
- % attention to constant polynomial, we must redefine the arrays then
- \ifnum\POL@sturmlength>\z@
- \ifx\empty#1\relax
- \POL@isolz@getsignchanges@plusinf
- \POL@isolz@getsignchanges@minusinf
- \else
- \edef\POL@isolz@E{\the\numexpr\xint_zapspaces #1 \xint_gobble_i\relax}%
- \POL@sturmchain@getSV@at{1[\POL@isolz@E]}%
- \let\POL@isolz@plusinf@SV \POL@sturmchain@SV
- \let\POL@isolz@plusinf@sign\POL@sturmchain@sign
- \POL@sturmchain@getSV@at{-1[\POL@isolz@E]}%
- \let\POL@isolz@minusinf@SV \POL@sturmchain@SV
- \let\POL@isolz@minusinf@sign\POL@sturmchain@sign
- \ifnum\POL@isolz@plusinf@sign=\z@
- \PackageError{polexpr}%
-{The polynomial #2 vanishes at set upper bound 10^\POL@isolz@E}%
-{Compile again with a bigger exponent in source. (X to abort).}%
- \fi
- \ifnum\POL@isolz@minusinf@sign=\z@
- \PackageError{polexpr}%
-{The polynomial #2 vanishes at set lower bound -10^\POL@isolz@E}%
-{Compile again with a bigger exponent in source. (X to abort).}%
- \fi
- \fi
- \edef\POL@isolz@NbOfRoots
- {\the\numexpr\POL@isolz@minusinf@SV-\POL@isolz@plusinf@SV}%
- \else
- % constant polynomial
- \def\POL@isolz@NbOfRoots{0}%
- \fi
- \ifnum\POL@isolz@NbOfRoots=\z@
- \begingroup\globaldefs\@ne
- \expandafter\xintAssignArray\expandafter\to\csname POL_ZL#2*\endcsname
- \expandafter\xintAssignArray\expandafter\to\csname POL_ZR#2*\endcsname
- \expandafter\xintAssignArray\expandafter\to\csname POL_ZK#2*\endcsname
- \endgroup
- \else
- \begingroup\globaldefs\@ne
- \expandafter\POL@initarray\csname POL_ZL#2*\endcsname{0}%
- \expandafter\POL@initarray\csname POL_ZR#2*\endcsname{0}%
- \expandafter\POL@initarray\csname POL_ZK#2*\endcsname
- \xint_stop_atsecondoftwo
- \endgroup
- \ifx\empty#1\relax\expandafter\POL@isolz@getaprioribound\fi
- \expandafter\POL@isolz@main
- \fi
-}%
-\def\POL@initarray#1#2{%
-% ATTENTION, if only one item, \xintAssignArray UNBRACES IT
-% so we use an \empty trick to avoid that. Maybe considered a bug of xinttools?
- \expandafter\xintAssignArray\expandafter\empty
- \romannumeral\xintreplicate{\POL@isolz@NbOfRoots}{{#2}}\to#1%
-}%
-\def\POL@isolz@getsignchanges@plusinf{%
- % Count number of sign changes at plus infinity in Sturm sequence
- \def\POL@isolz@plusinf@SV{0}%
- \edef\POL@isolz@lastsign{\xintiiSgn{\PolLeadingCoeff{\POL@sturmname _0}}}%
- \let\POL@isolz@plusinf@sign\POL@isolz@lastsign
- \POL@count\@ne
- \xintloop
- \edef\POL@isolz@newsign
- {\xintiiSgn{\PolLeadingCoeff{\POL@sturmname _\the\POL@count}}}%
- \unless\ifnum\POL@isolz@newsign=\POL@isolz@lastsign
- \edef\POL@isolz@plusinf@SV{\the\numexpr\POL@isolz@plusinf@SV+\@ne}%
- \fi
- \let\POL@isolz@lastsign=\POL@isolz@newsign
- \ifnum\POL@sturmlength>\POL@count
- \advance\POL@count\@ne
- \repeat
-}%
-\def\POL@isolz@getsignchanges@minusinf{%
- % Count number of sign changes at minus infinity in Sturm sequence
- \def\POL@isolz@minusinf@SV{0}%
- \edef\POL@isolz@lastsign{\xintiiSgn{\PolLeadingCoeff{\POL@sturmname _0}}}%
- \ifodd\PolDegree{\POL@sturmname _0}
- \edef\POL@isolz@lastsign{\xintiiOpp{\POL@isolz@lastsign}}%
- \fi
- \let\POL@isolz@minusinf@sign\POL@isolz@lastsign
- \POL@count\@ne
- \xintloop
- \edef\POL@isolz@newsign
- {\xintiiSgn{\PolLeadingCoeff{\POL@sturmname _\the\POL@count}}}%
- \ifodd\PolDegree{\POL@sturmname _\the\POL@count}
- \edef\POL@isolz@newsign{\xintiiOpp{\POL@isolz@newsign}}%
- \fi
- \unless\ifnum\POL@isolz@newsign=\POL@isolz@lastsign
- \edef\POL@isolz@minusinf@SV{\the\numexpr\POL@isolz@minusinf@SV+\@ne}%
- \fi
- \let\POL@isolz@lastsign=\POL@isolz@newsign
- \ifnum\POL@sturmlength>\POL@count
- \advance\POL@count\@ne
- \repeat
-}%
-% utility macro for a priori bound on root decimal exponent, via Float Rounding
-\def\POL@isolz@updateE #1e#2;%
- {\unless\ifnum#2<\POL@isolz@E\space\edef\POL@isolz@E{\the\numexpr#2+\@ne}\fi}%
-\def\POL@isolz@getaprioribound{%
- \PolAssign{\POL@sturmname _0}\toarray\POL@arrayA
- \edef\POL@isolz@leading{\POL@arrayA{\POL@arrayA{0}}}%
- \POL@count\z@
- \xintloop
- \advance\POL@count\@ne
- \ifnum\POL@arrayA{0}>\POL@count
- \expandafter\edef\csname POL@arrayA\the\POL@count\endcsname
- {\xintDiv{\POL@arrayA\POL@count}\POL@isolz@leading}%
- \repeat
- \def\POL@isolz@E{1}% WE SEEK SMALLEST E SUCH HAT -10^E < roots < +10^E
- \advance\POL@count\m@ne
- \xintloop
- \ifnum\POL@count>\z@
- \expandafter\POL@isolz@updateE
- % use floating point to get decimal exponent
- \romannumeral0\xintfloat[4]% should I use with [2] rather? (should work)
- {\xintAdd{1/1[0]}{\xintAbs{\POL@arrayA\POL@count}}};%
- \advance\POL@count\m@ne
- \repeat
- % \ifxintverbose\xintMessage{polexpr}{Info}%
- % {Roots a priori bounded in absolute value by 10 to the \POL@isolz@E.}%
- % \fi
-}%
-\def\POL@IsoRight@raw{\POL@IsoRight@Int/1[\POL@isolz@E]}%
-\def\POL@IsoLeft@raw {\POL@IsoLeft@Int/1[\POL@isolz@E]}%
-\def\POL@IsoRight@rawout{%
- \ifnum\POL@IsoRightSign=\z@\expandafter\xintREZ\fi\POL@IsoRight@raw
-}%
-\def\POL@IsoLeft@rawout{%
- \ifnum\POL@IsoRightSign=\z@
- \expandafter\xint_firstoftwo\else\expandafter\xint_secondoftwo
- \fi{\xintREZ\POL@IsoRight@raw}%
- {\POL@IsoLeft@Int/1[\POL@isolz@E]}%
-}%
-\def\POL@isolz@main {%
-% NOTE 2018/02/16. THIS WILL PRESUMABLY BE RE-ORGANIZED IN FUTURE TO DO
-% FIRST POSITIVE ROOTS THEN NEGATIVE ROOTS VIA CHANGE OF VARIABLE TO OPPOSITE.
- \global\POL@isolz@nextwillneedrefinefalse
- \def\POL@IsoRight@Int{0}%
- \POL@sturmchain@getSV@at\POL@IsoRight@raw
- \let\POL@IsoRightSV \POL@sturmchain@SV
- \let\POL@IsoRightSign\POL@sturmchain@sign
- \let\POL@IsoAtZeroSV \POL@IsoRightSV
- \let\POL@IsoAtZeroSign\POL@IsoRightSign
- \ifnum\POL@IsoAtZeroSign=\z@
- \xdef\POL@isolz@IntervalIndex
- {\the\numexpr\POL@isolz@minusinf@SV-\POL@IsoRightSV}%
- \POL@refine@storeleftandright % store zero root, \POL@IsoRightSign is zero
- \edef\POL@IsoRightSV{\the\numexpr\POL@IsoRightSV+\@ne}%
-% subtlety here if original polynomial had multiplicities, but ok. I checked!
- \edef\POL@IsoRightSign % evaluated twice, but that's not so bad
- {\xintiiOpp{\xintiiSgn{\POL@eval{\POL@sturmname _1}{0/1[0]}}}}%
- \fi
- \def\POL@IsoLeft@Int{-1}% -10^E isn't a root!
- \let\POL@IsoLeftSV \POL@isolz@minusinf@SV
- \let\POL@IsoLeftSign\POL@isolz@minusinf@sign
- % \POL@IsoRight@SV was modified if zero is a root
- \edef\POL@isolz@NbOfNegRoots{\the\numexpr\POL@IsoLeftSV-\POL@IsoRightSV}%
- \gdef\POL@isolz@IntervalIndex{0}%
- \let\POL@isolz@@E\POL@isolz@E
- \ifnum\POL@isolz@NbOfNegRoots>\z@
-% refactored at 0.7 to fix cases leading to an intervals with zero as end-point
- \POL@isolz@findroots@neg
- \fi
- \let\POL@isolz@E\POL@isolz@@E
- \def\POL@IsoLeft@Int{0}%
- \let\POL@IsoLeftSV \POL@IsoAtZeroSV % véritable SV en zéro
- \let\POL@IsoLeftSign\POL@IsoAtZeroSign% véritable signe en zéro
- \ifnum\POL@IsoLeftSign=\z@
- \xdef\POL@isolz@IntervalIndex{\the\numexpr\POL@isolz@IntervalIndex+\@ne}%
- \fi
- \let\POL@@IsoRightSV \POL@isolz@plusinf@SV
- \let\POL@@IsoRightSign\POL@isolz@plusinf@sign % 10^E not a root!
- \edef\POL@isolz@NbOfPosRoots
- {\the\numexpr\POL@IsoLeftSV-\POL@@IsoRightSV}% attention @@
- \ifnum\POL@isolz@NbOfPosRoots>\z@
- % always do that to avoid zero as end-point whether it is a root or not
- \global\POL@isolz@nextwillneedrefinetrue
- \POL@isolz@findroots@pos
- \fi
-}%
-\def\POL@isolz@findroots@neg{%
- \def\POL@IsoRight@Int{-1}%
- \POL@isolz@findnextzeroboundeddecade@neg
- \def\POL@IsoLeft@Int{-10}%
- \let\POL@@IsoRightSign\POL@IsoRightSign % a zero there is possible
- \let\POL@@IsoRightSV \POL@IsoRightSV
- % this will do possibly recursive \POL@isolz@check's
- \POL@isolz@explorenexteightsubdecades@neg
- \ifnum\POL@isolz@IntervalIndex<\POL@isolz@NbOfNegRoots\space
- % above did not explore -2, -1 for this optimization (SV known at Right)
- \def\POL@IsoRight@Int{-1}%
- \let\POL@IsoRightSign\POL@@IsoRightSign
- \let\POL@IsoRightSV \POL@@IsoRightSV
- \POL@isolz@check
- \ifnum\POL@isolz@IntervalIndex<\POL@isolz@NbOfNegRoots\space
- \def\POL@IsoLeft@Int{-1}%
- \let\POL@IsoLeftSign\POL@@IsoRightSign
- \let\POL@IsoLeftSV \POL@@IsoRightSV
- % I don't like being inside TeX conditionals
- \expandafter\expandafter\expandafter\POL@isolz@findroots@neg
- \fi
- \fi
-}%
-\def\POL@isolz@findnextzeroboundeddecade@neg{%
- \xintloop
- \edef\POL@isolz@E{\the\numexpr\POL@isolz@E-\@ne}%
- \POL@sturmchain@getSV@at\POL@IsoRight@raw
- \let\POL@IsoRightSV \POL@sturmchain@SV
- \let\POL@IsoRightSign\POL@sturmchain@sign
- % would an \ifx test be quicker? (to be checked)
- \ifnum\POL@IsoRightSV=\POL@IsoLeftSV\space
- % no roots in-between, iterate
- \repeat
-}%
-\def\POL@isolz@explorenexteightsubdecades@neg{%
- \xintloop
- \edef\POL@IsoRight@Int{\the\numexpr\POL@IsoLeft@Int+\@ne}%
- % we could arguably do a more efficient dichotomy here
- \POL@sturmchain@getSV@at\POL@IsoRight@raw
- \let\POL@IsoRightSV \POL@sturmchain@SV
- \let\POL@IsoRightSign\POL@sturmchain@sign
- \POL@isolz@check % may recurse if multiple roots are to be found
- \ifnum\POL@isolz@IntervalIndex=\POL@isolz@NbOfNegRoots\space
- \expandafter\xintbreakloop
- \fi
- \let\POL@IsoLeft@Int\POL@IsoRight@Int
- \let\POL@IsoLeftSign\POL@IsoRightSign
- \let\POL@IsoLeftSV\POL@IsoRightSV
- \ifnum\POL@IsoRight@Int < -\tw@
- \repeat
-}%
-\def\POL@isolz@findroots@pos{%
- % remark (2018/12/08), this needs some refactoring, I hardly understand
- % the logic and it hides most into the recursion done by \POL@isolz@check
- % It would probably make more sense to proceed like done for the negative
- % but here finding the largest roots first.
- \def\POL@IsoRight@Int{1}%
- \POL@isolz@findnextzeroboundeddecade@pos
- \unless\ifnum\POL@IsoRightSV=\POL@IsoLeftSV\space
- % this actually explores the whole of some interval (0, 10^{e-1}]
- % in a context where some roots are known to be in (10^{e-1}, 10^{e}]
- % and none are larger
- \POL@isolz@check % will recurse inside groups if needed with modified E
- \fi
- % we know get the roots in the last 9 decades from 10^{e-1} to 10^{e}
- % we should arguably do a more efficient dichotomy here
- \def\POL@IsoLeft@Int{1}%
- \let\POL@IsoLeftSV\POL@IsoRightSV
- \let\POL@IsoLeftSign\POL@IsoRightSign
- \xintloop
- \edef\POL@IsoRight@Int{\the\numexpr\POL@IsoLeft@Int+\@ne}%
- \POL@sturmchain@getSV@at\POL@IsoRight@raw
- \let\POL@IsoRightSV \POL@sturmchain@SV
- \let\POL@IsoRightSign\POL@sturmchain@sign
- \POL@isolz@check % recurses in needed
- \let\POL@IsoLeft@Int\POL@IsoRight@Int
- \let\POL@IsoLeftSign\POL@IsoRightSign
- \let\POL@IsoLeftSV\POL@IsoRightSV
- \ifnum\POL@isolz@IntervalIndex=\POL@isolz@NbOfRoots\space
- \expandafter\xintbreakloop
- \fi
- \ifnum\POL@IsoLeft@Int < \xint_c_ix
- \repeat
- \ifnum\POL@isolz@IntervalIndex<\POL@isolz@NbOfRoots\space
- % get now the last, rightmost, root (or roots)
- \def\POL@IsoRight@Int{10}%
- \let\POL@IsoRightSign\POL@@IsoRightSign
- \let\POL@IsoRightSV\POL@@IsoRightSV
- \POL@isolz@check
- \fi
-}%
-\def\POL@isolz@findnextzeroboundeddecade@pos{%
- \xintloop
- \edef\POL@isolz@E{\the\numexpr\POL@isolz@E-\@ne}%
- \POL@sturmchain@getSV@at\POL@IsoRight@raw
- \let\POL@IsoRightSV \POL@sturmchain@SV
- \let\POL@IsoRightSign\POL@sturmchain@sign
- \ifnum\POL@IsoRightSV=\POL@@IsoRightSV\space
- \let\POL@@IsoRightSign\POL@IsoRightSign % root here possible!
- \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
- \ifcase\numexpr\POL@IsoLeftSV-\POL@IsoRightSV\relax
- % no root in ]left, right]
- \global\POL@isolz@nextwillneedrefinefalse
- \or
- % exactly one root in ]left, right]
- \xdef\POL@isolz@IntervalIndex{\the\numexpr\POL@isolz@IntervalIndex+\@ne}%
- \ifnum\POL@IsoRightSign=\z@
- % if right boundary is a root, ignore previous flag
- \global\POL@isolz@nextwillneedrefinefalse
- \fi
- % if left boundary is known to have been a root we refine interval
- \ifPOL@isolz@nextwillneedrefine
- \expandafter\expandafter\expandafter\POL@isolz@refine
- \else
- % \POL@IsoRightSign is zero iff root now exactly known
- \POL@refine@storeleftandright
- \ifnum\POL@IsoRightSign=\z@
- \global\POL@isolz@nextwillneedrefinetrue
- \fi
- \fi
- \else
- % more than one root, we need to recurse
- \expandafter\POL@isolz@recursedeeper
- \fi
-}%
-\def\POL@isolz@recursedeeper{%
-% NOTE 2018/02/16. I SHOULD DO A REAL BINARY DICHOTOMY HERE WHICH ON AVERAGE
-% SHOULD BRING SOME GAIN (LIKE WHAT IS ALREADY DONE FOR THE "refine" MACROS.
-% THUS IN FUTURE THIS MIGHT BE REFACTORED.
-\begingroup
- \edef\POL@isolz@E{\the\numexpr\POL@isolz@E-\@ne}%
- \edef\POL@@IsoRight@Int{\xintDSL{\POL@IsoRight@Int}}%
- \let\POL@@IsoRightSign \POL@IsoRightSign
- \let\POL@@IsoRightSV \POL@IsoRightSV
- \edef\POL@IsoLeft@Int {\xintDSL{\POL@IsoLeft@Int}}%
- \xintiloop[1+1]
- \edef\POL@IsoRight@Int{\xintInc{\POL@IsoLeft@Int}}%
- \POL@sturmchain@getSV@at\POL@IsoRight@raw
- \let\POL@IsoRightSV \POL@sturmchain@SV
- \let\POL@IsoRightSign\POL@sturmchain@sign
- \POL@isolz@check
- \let\POL@IsoLeft@Int\POL@IsoRight@Int
- \let\POL@IsoLeftSV\POL@IsoRightSV
- \let\POL@IsoLeftSign\POL@IsoRightSign% not used, actually
- \ifnum\POL@IsoLeftSV=\POL@@IsoRightSV\space
- \expandafter\xintbreakiloop
- \fi
- \ifnum\xintiloopindex < \xint_c_ix
- \repeat
- \let\POL@IsoRight@Int\POL@@IsoRight@Int
- \let\POL@IsoRightSign\POL@@IsoRightSign
- \let\POL@IsoRightSV \POL@@IsoRightSV
- % if we exited the loop via breakiloop this is superfluous
- % but it only costs one \ifnum
- \POL@isolz@check
-\endgroup
-}%
-\def\POL@isolz@refine{%
- % starting point is first root = left < unique second root < right
- % even if we hit exactly via refinement second root, we set flag false as
- % processing will continue with original right end-point, which isn't a root
- \global\POL@isolz@nextwillneedrefinefalse
-\begingroup
- \let\POL@@IsoRightSign\POL@IsoRightSign % already evaluated
- \xintloop
- \edef\POL@isolz@E{\the\numexpr\POL@isolz@E-\@ne}%
- \edef\POL@IsoLeft@Int {\xintDSL{\POL@IsoLeft@Int}}%
- \edef\POL@IsoRight@Int{\xintInc{\POL@IsoLeft@Int}}%
- \edef\POL@IsoRightSign
- {\xintiiSgn{\POL@eval{\POL@sturmname _0}{\POL@IsoRight@raw}}}%
- \ifnum\POL@IsoRightSign=\POL@@IsoRightSign\space
- \repeat
- % now second root has been separated from the one at left end point
-% we update the storage of the root at left for it to have the same number
-% of digits in mantissa. No, I decided not to do that to avoid complications.
- % \begingroup
- % \let\POL@IsoRight@Int\POL@IsoLeft@Int
- % \def\POL@IsoRightSign{0}%
- % \edef\POL@isolz@IntervalIndex{\the\numexpr\POL@isolz@IntervalIndex-\@ne}%
- % \POL@refine@storeleftandright
- % \endgroup
- \edef\POL@@IsoRight@Int{\xintDSL{\xintInc{\xintDSR{\POL@IsoLeft@Int}}}}%
- \let\POL@IsoLeft@Int\POL@IsoRight@Int
- \let\POL@IsoLeftSign\POL@IsoRightSign
- \ifnum\POL@IsoRightSign=\z@ % check if new Left is actually a root
- \else
- \edef\POL@IsoRight@Int{\xintDec{\POL@@IsoRight@Int}}%
- \edef\POL@IsoRightSign
- {\xintiiSgn{\POL@eval{\POL@sturmname _0}{\POL@IsoRight@raw}}}%
- \ifnum\POL@IsoRightSign=\POL@@IsoRightSign\space
- \POL@refine@doonce % we need to locate in interval (1, 9) in local scale
- \else
- \let\POL@IsoLeft@Int\POL@IsoRight@Int
- \ifnum\POL@IsoRightSign=\z@
- \def\POL@IsoLeftSign{0}%
- \else
- \let\POL@IsoRight@Int\POL@@IsoRight@Int
- % the IsoRightSign is now wrong but here we don't care
- \fi\fi
- \fi
- % on exit, exact root found iff \POL@IsoRightSign is zero
- \POL@refine@storeleftandright
-\endgroup
-}%
-\def\POL@refine@doonce{% if exact root is found, always in IsoRight on exit
-% NOTE: FUTURE REFACTORING WILL GET RID OF \xintiiAdd WHICH ARE A BIT COSTLY
-% BUT BASICALLY NEEDED TO HANDLE BOTH NEGATIVE AND POSITIVE HERE.
-% I WILL RE-ORGANIZE THE WHOLE THING IN FUTURE TO GET ROOTS STARTING FROM
-% THE ORIGIN AND SIMPLY RE-LABEL THE NEGATIVE ONE AT THE END. 2018/02/16.
- \let\POL@@IsoRight@Int\POL@IsoRight@Int % 9
- \let\POL@@IsoRightSign\POL@IsoRightSign
- \edef\POL@IsoRight@Int{\xintiiAdd{4}{\POL@IsoLeft@Int}}% 5
- \edef\POL@IsoRightSign
- {\xintiiSgn{\POL@eval{\POL@sturmname _0}{\POL@IsoRight@raw}}}%
- \ifnum\POL@IsoRightSign=\POL@IsoLeftSign\space
- \let\POL@IsoLeft@Int\POL@IsoRight@Int % 5
- \edef\POL@IsoRight@Int{\xintiiAdd{2}{\POL@IsoLeft@Int}}%
- \edef\POL@IsoRightSign
- {\xintiiSgn{\POL@eval{\POL@sturmname _0}{\POL@IsoRight@raw}}}%
- \ifnum\POL@IsoRightSign=\POL@IsoLeftSign\space
- \let\POL@IsoLeft@Int\POL@IsoRight@Int % 7
- \edef\POL@IsoRight@Int{\xintInc{\POL@IsoLeft@Int}}%
- \edef\POL@IsoRightSign
- {\xintiiSgn{\POL@eval{\POL@sturmname _0}{\POL@IsoRight@raw}}}%
- \ifnum\POL@IsoRightSign=\POL@IsoLeftSign\space
- \let\POL@IsoLeft@Int\POL@IsoRight@Int % 8
- \let\POL@IsoRight@Int\POL@@IsoRight@Int % 9
- \let\POL@IsoRightSign\POL@@IsoRightSign % opposite of one at left
- \fi % else 7, 8 with possible root at 8
- \else
- \ifnum\POL@IsoRightSign=\z@
- \let\POL@IsoLeft@Int\POL@IsoRight@Int % root at 7
- \def\POL@IsoLeftSign{0}%
- \else
- \let\POL@@IsoRight@Int\POL@IsoRight@Int % 7
- \edef\POL@IsoRight@Int{\xintInc{\POL@IsoLeft@Int}}% 6
- \edef\POL@IsoRightSign
- {\xintiiSgn{\POL@eval{\POL@sturmname _0}{\POL@IsoRight@raw}}}%
- \ifnum\POL@IsoRightSign=\POL@IsoLeftSign\space
- \let\POL@IsoLeft@Int\POL@IsoRight@Int % 6
- \let\POL@IsoRight@Int\POL@@IsoRight@Int % 7
- \let\POL@IsoRightSign\POL@@IsoRightSign
- \fi % else 5, 6 with possible root at 6
- \fi\fi
- \else
- \ifnum\POL@IsoRightSign=\z@
- \let\POL@IsoLeft@Int\POL@IsoRight@Int % root at 5
- \def\POL@IsoLeftSign{0}%
- \else
- \let\POL@@IsoRight@Int\POL@IsoRight@Int % 5
- \edef\POL@IsoRight@Int{\xintiiAdd{2}{\POL@IsoLeft@Int}}%
- \edef\POL@IsoRightSign
- {\xintiiSgn{\POL@eval{\POL@sturmname _0}{\POL@IsoRight@raw}}}%
- \ifnum\POL@IsoRightSign=\POL@IsoLeftSign\space
- \let\POL@IsoLeft@Int\POL@IsoRight@Int % 3
- \edef\POL@IsoRight@Int{\xintInc{\POL@IsoLeft@Int}}% 4
- \edef\POL@IsoRightSign
- {\xintiiSgn{\POL@eval{\POL@sturmname _0}{\POL@IsoRight@raw}}}%
- \ifnum\POL@IsoRightSign=\POL@IsoLeftSign\space
- \let\POL@IsoLeft@Int\POL@IsoRight@Int % 4
- \let\POL@IsoRight@Int\POL@@IsoRight@Int % 5
- \let\POL@IsoRightSign\POL@@IsoRightSign
- \fi % else 3, 4 with possible root at 4
- \else
- \ifnum\POL@IsoRightSign=\z@
- \let\POL@IsoLeft@Int\POL@IsoRight@Int % root at 3
- \def\POL@IsoLeftSign{0}%
- \else
- \let\POL@@IsoRight@Int\POL@IsoRight@Int % 3
- \edef\POL@IsoRight@Int{\xintInc{\POL@IsoLeft@Int}}% 2
- \edef\POL@IsoRightSign
- {\xintiiSgn{\POL@eval{\POL@sturmname _0}{\POL@IsoRight@raw}}}%
- \ifnum\POL@IsoRightSign=\POL@IsoLeftSign\space
- \let\POL@IsoLeft@Int\POL@IsoRight@Int % 2
- \let\POL@IsoRight@Int\POL@@IsoRight@Int % 3
- \let\POL@IsoRightSign\POL@@IsoRightSign
- \fi % else 1, 2 with possible root at 2
- \fi\fi
- \fi\fi
-}%
-\def\POL@refine@storeleftandright{%
- \expandafter
- \xdef\csname POL_ZL\POL@sturmname*\POL@isolz@IntervalIndex\endcsname
- {\PolDecToString{\POL@IsoLeft@rawout}}%
- \expandafter
- \xdef\csname POL_ZR\POL@sturmname*\POL@isolz@IntervalIndex\endcsname
- {\PolDecToString{\POL@IsoRight@rawout}}%
- % added at 0.6
- \ifnum\POL@IsoRightSign=\z@
- \global
- \expandafter
- \let\csname POL_ZK\POL@sturmname*\POL@isolz@IntervalIndex\endcsname
- \xint_stop_atfirstoftwo
- \fi
- \begingroup\xintglobaldefstrue
- % skip some overhead of \xintdefvar...
- \XINT_expr_defvar_one{\POL@sturmname L_\POL@isolz@IntervalIndex}%
- {\POL@IsoLeft@rawout}%
- \XINT_expr_defvar_one{\POL@sturmname R_\POL@isolz@IntervalIndex}%
- {\POL@IsoRight@rawout}%
- % added at 0.7
- \XINT_expr_defvar_one{\POL@sturmname Z_\POL@isolz@IntervalIndex _isknown}%
- {\ifnum\POL@IsoRightSign=\z@ 1\else 0\fi}%
- \endgroup
-}%
-%% \PolRefineInterval
-\def\POL@xintexprGetVar#1{\csname XINT_expr_varvalue_#1\endcsname}%
-% attention, also used by \POL@findrat@loop@a
-\def\POL@get@IsoLeft@rawin{%
- \edef\POL@IsoLeft@rawin
- {\POL@xintexprGetVar{\POL@sturmname L_\POL@isolz@IntervalIndex}}%
-}%
-% attention, also used by \POL@findrat@loop@a
-\def\POL@get@IsoRight@rawin{%
- \edef\POL@IsoRight@rawin
- {\POL@xintexprGetVar{\POL@sturmname R_\POL@isolz@IntervalIndex}}%
-}%
-% attention, also used by \POL@findrat@loop@a
-\def\POL@get@Int@aux #1/1[#2]#3#4{\edef#3{\xintDSH{#4-#2}{#1}}}%
-\def\POL@get@IsoLeft@Int{%
- \expandafter\POL@get@Int@aux\POL@IsoLeft@rawin\POL@IsoLeft@Int\POL@isolz@E
-}%
-\newcommand\PolRefineInterval{\@ifstar\POL@srefine@start\POL@refine@start}%
-\newcommand\POL@refine@start[3][1]{%
- \edef\POL@isolz@IntervalIndex{\the\numexpr#3}%
- \edef\POL@sturmname{#2}%
- \expandafter\POL@refine@sharedbody\expandafter
- {\expandafter\POL@refine@loop\expandafter{\the\numexpr#1}}%
-}%
-\def\POL@srefine@start#1#2{%
- \edef\POL@isolz@IntervalIndex{\the\numexpr#2}%
- \edef\POL@sturmname{#1}%
- \POL@refine@sharedbody
- {\let\POL@refine@left@next\POL@refine@main % we want to recurse if needed
- \let\POL@refine@right@next\POL@refine@main % we want to recurse if needed
- \POL@refine@main}%
-}%
-\def\POL@refine@sharedbody#1{%
- \POL@get@IsoLeft@rawin
- \edef\POL@IsoLeftSign
- {\xintiiSgn{\POL@eval{\POL@sturmname _0}{\POL@IsoLeft@rawin}}}%
- \ifnum\POL@IsoLeftSign=\z@
- % do nothing if that interval was already a singleton
- \else
- % else both end-points are not roots and there is a single one in-between
- \POL@get@IsoRight@rawin
- \edef\POL@IsoRightSign{\the\numexpr-\POL@IsoLeftSign}%
- \edef\POL@isolz@E{\expandafter\POL@refine@getE
- % je pense que le xintrez ici est superflu
- \romannumeral0\xintrez{\xintSub{\POL@IsoRight@rawin}{\POL@IsoLeft@rawin}}}%
- \POL@get@IsoLeft@Int
- \edef\POL@IsoRight@Int{\xintInc{\POL@IsoLeft@Int}}%
- #1%
- \POL@refine@storeleftandright % \POL@IsoRightSign not zero
- \fi
-}%
-\def\POL@refine@loop#1{%
- \let\POL@refine@left@next \@empty % no recursion at end sub-intervals
- \let\POL@refine@right@next\@empty
- \xintiloop[1+1]
- \POL@refine@main
- \ifnum\POL@IsoRightSign=\z@
- \expandafter\xintbreakiloop
- \fi
- \ifnum\xintiloopindex<#1
- \repeat
-}%
-\def\POL@refine@main{%
- \edef\POL@isolz@E{\the\numexpr\POL@isolz@E-\@ne}%
- \edef\POL@IsoLeft@Int{\xintDSL{\POL@IsoLeft@Int}}%
- \edef\POL@IsoRight@Int{\xintDSL{\POL@IsoRight@Int}}%
- \let\POL@@IsoRight@Int\POL@IsoRight@Int
- \let\POL@@IsoRightSign\POL@IsoRightSign
- \edef\POL@IsoRight@Int{\xintInc{\POL@IsoLeft@Int}}%
- \edef\POL@IsoRightSign
- {\xintiiSgn{\POL@eval{\POL@sturmname _0}{\POL@IsoRight@raw}}}%
- \ifnum\POL@IsoRightSign=\z@
- \let\POL@IsoLeft@Int\POL@IsoRight@Int % root at 1
- \def\POL@IsoLeftSign{0}%
- \let\POL@next\@empty
- \else
- \ifnum\POL@IsoRightSign=\POL@@IsoRightSign\space
- \let\POL@next\POL@refine@left@next % may be \@empty or \POL@refine@main for recursion
- \let\POL@refine@right@next\@empty
- \else
- \let\POL@IsoLeft@Int\POL@IsoRight@Int
- \edef\POL@IsoRight@Int{\xintDec{\POL@@IsoRight@Int}}%
- \edef\POL@IsoRightSign
- {\xintiiSgn{\POL@eval{\POL@sturmname _0}{\POL@IsoRight@raw}}}%
- \ifnum\POL@IsoRightSign=\z@
- \let\POL@IsoLeft@Int\POL@IsoRight@Int % root at 9
- \def\POL@IsoLeftSign{0}%
- \let\POL@next\@empty
- \else
- \ifnum\POL@IsoRightSign=\POL@@IsoRightSign\space
- \let\POL@next\POL@refine@doonce
- \else
- \let\POL@IsoLeft@Int\POL@IsoRight@Int
- \let\POL@IsoRight@Int\POL@@IsoRight@Int
- \let\POL@IsoRightSign\POL@@IsoRightSign
- \let\POL@next\POL@refine@right@next
- \let\POL@refine@left@next\@empty
- \fi
- \fi
- \fi\fi
- \POL@next
-}%
-% lacking pre-defined xintfrac macro here (such as an \xintRawExponent)
-\def\POL@refine@getE#1[#2]{#2}% \xintREZ already applied, for safety
-
-
-\newcommand\PolIntervalWidth[2]{%
-% le \xintRez est à cause des E positifs, car trailing zéros explicites
-% si je travaillais à partir des variables xintexpr directement ne devrait
-% pas être nécessaire, mais trop fragile par rapport à chgt internes possibles
- \romannumeral0\xintrez{\xintSub{\@nameuse{POL_ZR#1*}{#2}}%
- {\@nameuse{POL_ZL#1*}{#2}}}
-}%
-
-
-\newcommand\PolEnsureIntervalLengths[2]{% #1 = Sturm chain name,
- % localize roots in intervals of length at most 10^{#2}
- \edef\POL@sturmname{#1}%
- \edef\POL@ensure@targetE{\the\numexpr#2}%
- \edef\POL@nbofroots{\csname POL_ZL\POL@sturmname*0\endcsname}%
- \ifnum\POL@nbofroots>\z@
- \expandafter\POL@ensureintervallengths
- \fi
-}%
-\def\POL@ensureintervallengths{%
- \POL@count\z@
- % \POL@count used by \POL@sturmchain@getSV@at but latter not used
- \xintloop
- \advance\POL@count\@ne
- \edef\POL@isolz@IntervalIndex{\the\POL@count}%
- \POL@ensure@one
- \ifnum\POL@nbofroots>\POL@count
- \repeat
-}%
-\newcommand\PolEnsureIntervalLength[3]{% #1 = Sturm chain name,
- % #2 = index of interval
- % localize roots in intervals of length at most 10^{#3}
- \edef\POL@sturmname{#1}%
- \edef\POL@ensure@targetE{\the\numexpr#3}%
- \edef\POL@isolz@IntervalIndex{\the\numexpr#2}%
-% peut-être autoriser -1, -2, ... ?
- \ifnum\POL@isolz@IntervalIndex>\z@
-% 0.7, add this safeguard but attention means this structure must be in place
- \ifnum\csname POL_ZL\POL@sturmname*0\endcsname>\z@
-% je ne fais pas les \expandafter mais je préfèrerai ne pas être à l'intérieur
- \POL@ensure@one
- \fi
- \fi
-}%
-\def\POL@ensure@one{%
- \POL@get@IsoLeft@rawin
- \POL@get@IsoRight@rawin
- \edef\POL@ensure@delta{\xintREZ{\xintSub{\POL@IsoRight@rawin}{\POL@IsoLeft@rawin}}}%
- \xintiiifZero{\POL@ensure@delta}
- {}
- {\edef\POL@isolz@E{\expandafter\POL@refine@getE\POL@ensure@delta}%
- \POL@get@IsoLeft@Int
- \edef\POL@IsoRight@Int{\xintInc{\POL@IsoLeft@Int}}%
- \ifnum\POL@isolz@E>\POL@ensure@targetE\space
- \edef\POL@IsoLeftSign
- {\xintiiSgn{\POL@eval{\POL@sturmname _0}{\POL@IsoLeft@raw}}}%
- % at start left and right are not roots, and values of opposite signs
- % \edef\POL@IsoRightSign{\the\numexpr-\POL@IsoLeftSign}%
- \xintloop
- \POL@ensure@Eloopbody % decreases E by one at each iteration
- % if separation level is still too coarse we recurse at deeper level
- \ifnum\POL@isolz@E>\POL@ensure@targetE\space
- \repeat
- % will check if right is at a zero, it needs \POL@IsoRightSign set up
- \POL@refine@storeleftandright
- \fi
- }%
-}%
-\def\POL@ensure@Eloopbody {%
- \edef\POL@isolz@E{\the\numexpr\POL@isolz@E-\@ne}%
- \edef\POL@IsoLeft@Int{\xintDSL{\POL@IsoLeft@Int}}%
- % this will loop at most ten times
- \xintloop
- \edef\POL@IsoRight@Int{\xintInc{\POL@IsoLeft@Int}}%
- \edef\POL@IsoRightSign
- {\xintiiSgn{\POL@eval{\POL@sturmname _0}{\POL@IsoRight@raw}}}%
- % if we have found a zero at right boundary the \ifnum test will fail
- % and we exit the loop
- % else we exit the loop if sign at right boundary is opposite of
- % sign at left boundary (the latter is +1 or -1, never 0)
- % this is a bit wasteful if we go ten times to the right, because
- % we know that there the sign will be opposite, evaluation was superfluous
- \ifnum\POL@IsoLeftSign=\POL@IsoRightSign\space
- \let\POL@IsoLeft@Int\POL@IsoRight@Int
- \repeat
- % check for case when we exited the inner loop because we actually
- % found a zero, then we force exit from the main (E decreasing) loop
- \ifnum\POL@IsoRightSign=\z@
- \expandafter\xintbreakloop
- \fi
-}%
-
-
-\catcode`_ 8
-\newcommand\PolPrintIntervals
- {\@ifstar{\PolPrintIntervals@@}{\PolPrintIntervals@}}%
-\newcommand\PolPrintIntervals@@{%
- \begingroup
- \def\POL@AfterPrintIntervals{\endgroup}%
- \def\arraystretch{2}%
- \let\PolPrintIntervalsPrintExactZero\POL@@PrintIntervalsPrintExactZero
- \let\PolPrintIntervalsUnknownRoot\POL@@PrintIntervalsUnknownRoot
- \let\PolPrintIntervalsKnownRoot\POL@@PrintIntervalsKnownRoot
- \def\PolPrintIntervalsBeginEnv{\[\begin{array}{cl}}%\]
- \def\PolPrintIntervalsEndEnv{\end{array}\]}%
- \PolPrintIntervals@
-}%
-\newcommand\PolPrintIntervals@[2][Z]{\POL@PrintIntervals{#1}{#2}}%
-\newcommand\POL@PrintIntervals[2]{%
- \def\PolPrintIntervalsTheVar{#1}%
- \def\PolPrintIntervalsTheSturmName{#2}%
- \ifnum\@nameuse{POL_ZL#2*}{0}=\z@
- \PolPrintIntervalsNoRealRoots
- \else
- \gdef\PolPrintIntervalsTheIndex{1}%
- \POL@PrintIntervals@DoDefs
- \begingroup\edef\POL@tmp{\endgroup
- \unexpanded\expandafter{\PolPrintIntervalsBeginEnv}%
- \unexpanded\expandafter{\POL@PrintIntervals@Loop}%
- \unexpanded\expandafter{\PolPrintIntervalsEndEnv}%
- }\POL@tmp
- \fi
- \POL@AfterPrintIntervals
- \def\PolPrintIntervalsTheVar{#1}%
- \def\PolPrintIntervalsTheSturmName{#2}%
-}%
-\let\POL@AfterPrintIntervals\@empty
-\newcommand\PolPrintIntervalsNoRealRoots{}%
-\newcommand\PolPrintIntervalsBeginEnv{\[\begin{array}{rcccl}}%
-\newcommand\PolPrintIntervalsEndEnv{\end{array}\]}%
-\newcommand\PolPrintIntervalsKnownRoot{%
- &&\PolPrintIntervalsTheVar_{\PolPrintIntervalsTheIndex}%
- &=&\PolPrintIntervalsPrintExactZero
-}%
-\newcommand\PolPrintIntervalsUnknownRoot{%
- \PolPrintIntervalsPrintLeftEndPoint&<&%
- \PolPrintIntervalsTheVar_{\PolPrintIntervalsTheIndex}&<&%
- \PolPrintIntervalsPrintRightEndPoint
-}%
-\newcommand\PolPrintIntervalsPrintExactZero {\PolPrintIntervalsTheLeftEndPoint}%
-\newcommand\PolPrintIntervalsPrintLeftEndPoint {\PolPrintIntervalsTheLeftEndPoint}%
-\newcommand\PolPrintIntervalsPrintRightEndPoint{\PolPrintIntervalsTheRightEndPoint}%
-\newcommand\PolPrintIntervalsPrintMultiplicity{(\mbox{mult. }\PolPrintIntervalsTheMultiplicity)}%
-%
-\newcommand\POL@@PrintIntervalsKnownRoot{%
- \PolPrintIntervalsPrintMultiplicity&%
- \PolPrintIntervalsTheVar_{\PolPrintIntervalsTheIndex}=%
- \PolPrintIntervalsPrintExactZero
-}%
-\newcommand\POL@@PrintIntervalsPrintExactZero{%
- \displaystyle
- \xintSignedFrac{\PolPrintIntervalsTheLeftEndPoint}%
-}%
-\newcommand\POL@@PrintIntervalsUnknownRoot{%
- \PolPrintIntervalsPrintMultiplicity&%
- \xintifSgn{\PolPrintIntervalsTheLeftEndPoint}%
- {\xintifSgn{\PolPrintIntervalsTheRightEndPoint}
- {\PolPrintIntervalsTheVar_{\PolPrintIntervalsTheIndex}=%
- \PolPrintIntervalsPrintRightEndPoint\dots}%
- {0>\PolPrintIntervalsTheVar_{\PolPrintIntervalsTheIndex}>%
- \PolPrintIntervalsPrintLeftEndPoint}%
- {\PolErrorThisShouldNotHappenPleaseReportToAuthorA}}%
- {\xintifSgn{\PolPrintIntervalsTheRightEndPoint}
- {\PolErrorThisShouldNotHappenPleaseReportToAuthorB}%
- {\PolErrorThisShouldNotHappenPleaseReportToAuthorC}%
- {0<\PolPrintIntervalsTheVar_{\PolPrintIntervalsTheIndex}<%
- \PolPrintIntervalsPrintRightEndPoint}}%
- {\xintifSgn{\PolPrintIntervalsTheRightEndPoint}
- {\PolErrorThisShouldNotHappenPleaseReportToAuthorD}%
- {\PolErrorThisShouldNotHappenPleaseReportToAuthorE}%
- {\PolPrintIntervalsTheVar_{\PolPrintIntervalsTheIndex}=%
- \PolPrintIntervalsPrintLeftEndPoint\dots}}%
-}%
-%
-\catcode`_ 11
-\def\POL@PrintIntervals@Loop{%
- \POL@SturmIfZeroExactlyKnown\PolPrintIntervalsTheSturmName
- \PolPrintIntervalsTheIndex
- \PolPrintIntervalsKnownRoot
- \PolPrintIntervalsUnknownRoot
- \xdef\PolPrintIntervalsTheIndex{\the\numexpr\PolPrintIntervalsTheIndex+\@ne}%
- \unless\ifnum\PolPrintIntervalsTheIndex>
- \@nameuse{POL_ZL\PolPrintIntervalsTheSturmName*0}
- \POL@PrintIntervals@DoDefs
- \xint_afterfi{\\\POL@PrintIntervals@Loop}%
- \fi
-}%
-\def\POL@PrintIntervals@DoDefs{%
- \xdef\PolPrintIntervalsTheLeftEndPoint{%
- \csname POL_ZL\PolPrintIntervalsTheSturmName*\PolPrintIntervalsTheIndex
- \endcsname
- }%
- \xdef\PolPrintIntervalsTheRightEndPoint{%
- \csname POL_ZR\PolPrintIntervalsTheSturmName*\PolPrintIntervalsTheIndex
- \endcsname
- }%
- \xdef\PolPrintIntervalsTheMultiplicity{%
- \ifcsname POL_ZM\PolPrintIntervalsTheSturmName*\PolPrintIntervalsTheIndex
- \endcsname
- \csname POL_ZM\PolPrintIntervalsTheSturmName*\PolPrintIntervalsTheIndex
- \endcsname
- \else
- ?% or use 0 ?
- \fi
- }%
-}%
-
-
-\newcommand\PolSturmIfZeroExactlyKnown[2]{% #1 = sturmname, #2=index
- \romannumeral0\csname POL_ZK#1*\endcsname{#2}%
-}%
-\newcommand\POL@SturmIfZeroExactlyKnown[2]{% #1 = sturmname, #2=index
- \romannumeral0\csname POL_ZK#1*\the\numexpr#2\endcsname
-}%
-\newcommand\PolSturmIsolatedZeroMultiplicity[2]{%
- \romannumeral`^^@\csname POL_ZM#1*\endcsname{#2}%
-}%
-\newcommand\PolSturmIsolatedZeroLeft[2]{%
- \romannumeral`^^@\csname POL_ZL#1*\endcsname{#2}%
-}%
-\newcommand\PolSturmIsolatedZeroRight[2]{%
- \romannumeral`^^@\csname POL_ZR#1*\endcsname{#2}%
-}%
-\newcommand\PolSturmNbOfIsolatedZeros[1]{%
- \romannumeral`^^@\csname POL_ZL#1*0\endcsname
-}%
-\newcommand\PolSturmRationalRoot[2]{%
- \romannumeral`^^@\csname POL_ZL#1*%
- \csname POL_RI#1*\endcsname{#2}\endcsname
-}%
-\newcommand\PolSturmRationalRootIndex[2]{%
- \romannumeral`^^@\csname POL_RI#1*\endcsname{#2}%
-}%
-\newcommand\PolSturmRationalRootMultiplicity[2]{%
- \romannumeral`^^@\csname POL_ZM#1%
- *\csname POL_RI#1*\endcsname{#2}\endcsname
-}%
-\newcommand\PolSturmNbOfRationalRoots[1]{%
- \romannumeral`^^@\csname POL_RI#1*0\endcsname
-}%
-\newcommand\PolSturmNbOfRationalRootsWithMultiplicities[1]{%
-% means the \POL@norr must not have been changed in-between...
- \the\numexpr\PolDegree{#1}-\PolDegree{#1\POL@norr}\relax
-}%
-
-
-\let\PolDecToString\xintDecToString
-
-
-\newcommand\PolMakeMonic[1]{%
- \edef\POL@leadingcoeff{\PolLeadingCoeff{#1}}%
- \edef\POL@leadingcoeff@inverse{\xintDiv{1/1[0]}{\POL@leadingcoeff}}%
- \PolMapCoeffs{\xintMul{\POL@leadingcoeff@inverse}}{#1}%
-}%
-
-
-%% CORE ALGEBRA MACROS
-%% We do this non-expandably, but in a nestable way... this is the whole
-%% point because \xintdeffunc as used by \poldef creates a big nested macro.
-%% The idea is to execute it with another meaning given to \xintAdd etc..,
-%% so that it operates on "polynomials". This is a mixture of expandable
-%% and non-expandable techniques.
-%%
-%% And it was complicated to let it work with xintexpr 1.4
-%%
-\def\POL@get#1#2#3{%
- \relax %!! part de la tambouille pour fonctionner en xint 1.4
- \POL@polglobalfalse
- \begingroup
- \protected\def\POL@result{#3}%
- #3%
- \expandafter
- \endgroup
- \expandafter\def\expandafter#1\expandafter{\POL@result}%
- \unless\ifPOL@pol
- % avoid expanding more than twice #3
- % #3 must be purely numerical or at least compatible with \edef
- % this is why at 0.7.5 I had to handle especially constant
- % polynomial functions to remove any protection from them
- % (because the protection triggers the COMPOSITION when
- % the polynomial is found as argument of another one and
- % this is not expandable)
- \edef#1{#3}%
- \xintiiifZero{#1}%
- {\def#1{-1.\POL@empty{0/1[0]}}}%
- {\edef#1{0.\POL@empty{#1}}}%
- \fi
- #2%
-}%
-
-%% COMPOSITION
-%% This did not exist before 0.7.5 and is part of its adaptation to xint 1.4
-%% We thus took up this opportunity to speed up substantially composition.
-%% Very serious difficulties with constant polynomials. Had to handle them
-%% especially.
-%% OK, that was really tough, but advantage now is that composition
-%% at 0.7.5 should be more efficient than before. However when polynomials
-%% become big via composition, coefficients also are big and the time
-%% taken by arithmetic dominates. No time to test really, though, relieved
-%% I can release xint 1.4 at last. My basic polexpr test suite passes,
-%% but it goes back already to old releases.
-\protected\def\POL@applypolfunc#1#2%
-{%
-% This #2 may be also invoing \POL@applypolfunc...
- \POL@get\POL@A\POL@applypolfunc@b#2#1%
-}%
-\def\POL@applypolfunc@b #1%
-{%
-% and now the have our Horner scheme nested macro
-% which hopefully will do its job with \POL@add, \POL@mul etc...
- \POL@polglobalfalse
- \expandafter#1\expanded
- {{\POL@polglobaltrue\protected\def\noexpand\POL@result{\POL@A}}}%
- \unless\ifPOL@pol
- \odef\POL@result{#1{0}}%
- \xintiiifZero{\POL@result}%
- {\protected\def\POL@result{-1.\POL@empty{0/1[0]}}}%
- {\protected\edef\POL@result{0.\POL@empty{\POL@result}}}%
- \fi
-}%
-
-%% ADDITION
-\def\POL@add {\POL@get\POL@A\POL@add@b}%
-\def\POL@add@b{\POL@get\POL@B\POL@add@c}%
-\def\POL@add@c{%
- \POL@polglobaltrue
- \POL@ifZero\POL@A
- {\let\POL@result\POL@B}%
- {\POL@ifZero\POL@B
- {\let\POL@result\POL@A}%
- {\POL@@add}}%
-}%
-\def\POL@@add{%
- \expandafter\POL@split\POL@A;\POL@degA\POL@polA
- \expandafter\POL@split\POL@B;\POL@degB\POL@polB
- \ifnum\POL@degA>\POL@degB\relax
- \xintAssignArray\POL@polA\to\POL@arrayA
- \xintAssignArray\POL@polB\to\POL@arrayB
- \else
- \xintAssignArray\POL@polB\to\POL@arrayA
- \xintAssignArray\POL@polA\to\POL@arrayB
- \let\POL@tmp\POL@degB\let\POL@degB\POL@degA\let\POL@degA\POL@tmp
- \fi
- \count@\z@
- \xintloop
- \advance\count@\@ne
- \expandafter\edef\csname POL@arrayA\the\count@\endcsname
- {\xintScalarAdd{\@nameuse{POL@arrayA\the\count@}}%
- {\@nameuse{POL@arrayB\the\count@}}}%
- \unless\ifnum\POL@degB<\count@
- \repeat
- \count@\@nameuse{POL@arrayA0} % 1+\POL@degA
- % trim zero leading coefficients (we could check for equal degrees,
- % but would not bring much as anyhow loop exists immediately if not)
- \xintloop
- % this abuses that \POL@arrayA0 is never zero
- \xintiiifZero{\@nameuse{POL@arrayA\the\count@}}%
- {\iftrue}%
- {\iffalse}%
- \advance\count@\m@ne
- \repeat
- \POL@resultfromarray A% attention that \POL@arrayA0 not updated
-}%
-
-%% MULTIPLICATION
-\def\POL@mul {\POL@get\POL@A\POL@mul@b}%
-\def\POL@mul@b{\POL@get\POL@B\POL@mul@c}%
-\def\POL@mul@c{%
- \POL@polglobaltrue
- \POL@ifZero\POL@A
- {\protected\def\POL@result{-1.\POL@empty{0/1[0]}}}%
- {\POL@ifZero\POL@B
- {\protected\def\POL@result{-1.\POL@empty{0/1[0]}}}%
- {\POL@@mul}}%
-}%
-\def\POL@@mul{%
- \expandafter\POL@split\POL@A;\POL@degA\POL@polA
- \expandafter\POL@split\POL@B;\POL@degB\POL@polB
- \ifnum\POL@degA>\POL@degB\relax
- \xintAssignArray\POL@polA\to\POL@arrayA
- \xintAssignArray\POL@polB\to\POL@arrayB
- \else
- \xintAssignArray\POL@polB\to\POL@arrayA
- \xintAssignArray\POL@polA\to\POL@arrayB
- \let\POL@tmp\POL@degB
- \let\POL@degB\POL@degA
- \let\POL@degA\POL@tmp
- \fi
- \count@\z@
- \xintloop
- \POL@@mul@phaseIloopbody
- \unless\ifnum\POL@degB<\count@
- \repeat
- \xintloop
- \unless\ifnum\POL@degA<\count@ % car attention au cas de mêmes degrés
- \POL@@mul@phaseIIloopbody
- \repeat
- \edef\POL@degC{\the\numexpr\POL@degA+\POL@degB}%
- \xintloop
- \unless\ifnum\POL@degC<\count@
- \POL@@mul@phaseIIIloopbody
- \repeat
- %\count@\the\numexpr\POL@degC+\@ne\relax % never zero polynomial here
- \POL@resultfromarray C%
-}%
-\def\POL@@mul@phaseIloopbody{%
- \advance\count@\@ne
- \def\POL@tmp{0[0]}%
- \count\tw@\z@
- \xintloop
- \advance\count\tw@\@ne
- \edef\POL@tmp{%
- \xintScalarAdd
- {\POL@tmp}%
- {\xintScalarMul
- {\@nameuse{POL@arrayA\the\count\tw@}}%
- {\@nameuse{POL@arrayB\the\numexpr\count@+\@ne-\count\tw@}}%
- }%
- }%
- \ifnum\count\tw@<\count@
- \repeat
- \expandafter\let\csname POL@arrayC\the\count@\endcsname\POL@tmp
-}%
-\def\POL@@mul@phaseIIloopbody{%
- \advance\count@\@ne
- \def\POL@tmp{0[0]}%
- \count\tw@\count@
- \advance\count\tw@-\@nameuse{POL@arrayB0} %
- \xintloop
- \ifnum\count\tw@<\count@
- \advance\count\tw@\@ne
- \edef\POL@tmp{%
- \xintScalarAdd
- {\POL@tmp}%
- {\xintScalarMul
- {\@nameuse{POL@arrayA\the\count\tw@}}%
- {\@nameuse{POL@arrayB\the\numexpr\count@+\@ne-\count\tw@}}%
- }%
- }%
- \repeat
- \expandafter\let\csname POL@arrayC\the\count@\endcsname\POL@tmp
-}%
-\def\POL@@mul@phaseIIIloopbody{%
- \advance\count@\@ne
- \def\POL@tmp{0[0]}%
- \count\tw@\count@
- \advance\count\tw@-\@nameuse{POL@arrayB0} %
- \xintloop
- \advance\count\tw@\@ne
- \edef\POL@tmp{%
- \xintScalarAdd{\POL@tmp}%
- {\xintScalarMul
- {\@nameuse{POL@arrayA\the\count\tw@}}%
- {\@nameuse{POL@arrayB\the\numexpr\count@+\@ne-\count\tw@}}%
- }%
- }%
- \ifnum\@nameuse{POL@arrayA0}>\count\tw@
- \repeat
- \expandafter\let\csname POL@arrayC\the\count@\endcsname\POL@tmp
-}%
-
-%% POWERS (SCALAR EXPONENT...)
-\def\POL@pow #1#2{%
- \POL@polglobalfalse
- \begingroup
- \protected\def\POL@result{#1}%
- #1%
- \expandafter
- \endgroup
- \expandafter\def\expandafter\POL@A\expandafter{\POL@result}%
- \unless\ifPOL@pol
- \edef\POL@A{\xintScalarPow{#1}{#2}}% no error check
- \xintiiifZero{\POL@A}%
- {\protected\def\POL@result{-1.\POL@empty{0/1[0]}}}%
- {\protected\edef\POL@result{0.\POL@empty{\POL@A}}}%
- \else
- \edef\POL@B{\numexpr\xintNum{#2}\relax}% no check on exponent >= 0
- \ifcase\POL@B
- \protected\def\POL@result{0.\POL@empty{1/1[0]}}%
- \or
- \let\POL@result\POL@A
- \else
- \POL@@pow@check
- \fi
- \fi
- \POL@polglobaltrue
-}%
-\def\POL@@pow@check {%
-% no problem here with leftover tokens!
-% should I have used that I-don't-care technique more elsewhere?
- \ifnum\@ne>\POL@A
- % polynomial is a constant, must get rid of dot and \empty (\POL@empty)
- \edef\POL@A{\expandafter\xintScalarPow\romannumeral`^^@%
- \expandafter\xint_gob_til_dot\POL@A{\POL@B}}%
- \xintiiifZero{\POL@A}%
- {\protected\def\POL@result{-1.\POL@empty{0/1[0]}}}%
- {\protected\edef\POL@result{0.\POL@empty{\POL@A}}}%
- \else
- \ifnum\@ne=\POL@A
- % perhaps a constant times X, check constant term
- \xintiiifZero
- {\expandafter\xint_firstoftwo\romannumeral`^^@%
- \expandafter\xint_gob_til_dot\POL@A}
- {\protected\edef\POL@result
- {\the\POL@B.% here at least 2.
- \POL@empty
- \romannumeral\xintreplicate{\POL@B}{{0/1[0]}}%
- {\xintScalarPow
- {\expandafter\xint_secondoftwo\romannumeral`^^@%
- \expandafter\xint_gob_til_dot\POL@A}%
- {\POL@B}}}}%
- {\POL@@pow}% not constant times X, use general recursion
- \else
- \POL@@pow% general recursion
- \fi\fi
-}%
-\def\POL@@pow@recurse#1#2{%
- \begingroup
- #1%
- \expandafter
- \endgroup
- \expandafter\def\expandafter\POL@A\expandafter{\POL@result}%
- \edef\POL@B{\numexpr\xintNum{#2}\relax}%
- \ifcase\POL@B
- \POL@thisshouldneverhappen
- \or
- \let\POL@result\POL@A
- \else
- \expandafter\POL@@pow
- \fi
-}%
-\def\POL@@pow {%
- \let\POL@pow@exp\POL@B
- \let\POL@B\POL@A
- \POL@@mul
- \let\POL@sqA\POL@result
- \ifodd\POL@pow@exp\space
- \expandafter\POL@@pow@odd
- \the\numexpr(\POL@pow@exp+\@ne)/\tw@-\@ne\expandafter.%
- \else
- \expandafter\POL@@pow@even
- \the\numexpr(\POL@pow@exp+\@ne)/\tw@-\@ne\expandafter.%
- \fi
-}%
-\def\POL@@pow@even#1.{%
- \expandafter\POL@@pow@recurse\expandafter
- {\expandafter\def\expandafter\POL@result\expandafter{\POL@sqA}}%
- {#1}%
-}%
-\def\POL@@pow@odd#1.{%
- \expandafter\POL@@pow@odd@i\expandafter{\POL@A}{#1}%
-}%
-\def\POL@@pow@odd@i #1#2{%
- \expandafter\POL@@pow@recurse\expandafter
- {\expandafter\def\expandafter\POL@result\expandafter{\POL@sqA}}%
- {#2}%
- \expandafter\POL@mul\expandafter
- {\expandafter\def\expandafter\POL@result\expandafter
- {\POL@result}\POL@polglobaltrue}%
- {\protected\def\POL@result{#1}\POL@polglobaltrue}%
-}%
-
-%% DIVISION
-%% no check on divisor being non-zero
-\def\POL@div {\POL@get\POL@A\POL@div@b}%
-\def\POL@div@b{\POL@get\POL@B\POL@div@c}%
-\def\POL@div@c{%
- \POL@polglobaltrue
- \expandafter\POL@split\POL@A;\POL@degA\POL@polA
- \expandafter\POL@split\POL@B;\POL@degB\POL@polB
- \ifnum\POL@degA<\POL@degB\space
- \@namedef{POL@arrayQ1}{0/1[0]}%
- \def\POL@degQ{-1}%
- \else
- \xintAssignArray\POL@polA\to\POL@arrayR
- \xintAssignArray\POL@polB\to\POL@arrayB
- \POL@@div
- \fi
- \count@\numexpr\POL@degQ+\@ne\relax
- \POL@resultfromarray Q%
-}%
-\def\POL@@div{%
- \xintAssignArray\POL@polA\to\POL@arrayR
- \xintAssignArray\POL@polB\to\POL@arrayB
- \edef\POL@B@leading{\csname POL@arrayB\the\numexpr\POL@degB+\@ne\endcsname}%
- \edef\POL@degQ{\the\numexpr\POL@degA-\POL@degB}%
- \count@\numexpr\POL@degA+\@ne\relax
- \count\tw@\numexpr\POL@degQ+\@ne\relax
- \xintloop
- \POL@@div@loopbody
- \ifnum\count\tw@>\z@
- \repeat
- %%\expandafter\def\csname POL@arrayR0\endcsname{1}%
- \xintloop
- \xintiiifZero{\csname POL@arrayR\the\count@\endcsname}%
- {\iftrue}%
- {\iffalse}%
- \advance\count@\m@ne
- \repeat
- \edef\POL@degR{\the\numexpr\count@-\@ne}%
-}%
-\def\POL@@div@loopbody{%
- \edef\POL@@div@ratio{%
- \xintScalarDiv{\csname POL@arrayR\the\count@\endcsname}%
- {\POL@B@leading}}%
- \expandafter\let\csname POL@arrayQ\the\count\tw@\endcsname
- \POL@@div@ratio
- \advance\count@\m@ne
- \advance\count\tw@\m@ne
- \count4 \count@
- \count6 \POL@degB\space
- \xintloop
- \ifnum\count6>\z@
- \expandafter\edef\csname POL@arrayR\the\count4\endcsname
- {\xintScalarSub
- {\csname POL@arrayR\the\count4\endcsname}%
- {\xintScalarMul
- {\POL@@div@ratio}%
- {\csname POL@arrayB\the\count6\endcsname}}}%
- \advance\count4 \m@ne
- \advance\count6 \m@ne
- \repeat
-}%
-
-%% MINUS SIGN AS UNARY OPERATOR
-\def\POL@opp #1{%
- \POL@polglobalfalse
- \begingroup
- \protected\def\POL@result{#1}%
- #1%
- \expandafter
- \endgroup
- \expandafter\def\expandafter\POL@A\expandafter{\POL@result}%
- \unless\ifPOL@pol
- \edef\POL@A{\xintScalarOpp{#1}}%
- \xintiiifZero{\POL@A}%
- {\protected\def\POL@result{-1.\POL@empty{0/1[0]}}}%
- {\protected\edef\POL@result{0.\POL@empty{\POL@A}}}%
- \else
- \edef\POL@B{0.\POL@empty{-1/1[0]}}%
- \POL@@mul
- \fi
- \POL@polglobaltrue
-}%
-
-
-%% EXPANDABLE MACROS
-\def\POL@eval@fork#1\At#2#3\krof{#2}%
-\newcommand\PolEval[3]{\romannumeral`^^@\POL@eval@fork
- #2\PolEvalAt
- \At\PolEvalAtExpr\krof {#1}{#3}%
-}%
-\newcommand\PolEvalAt[2]
- {\xintpraw{\csname XINT_expr_polfunc_#1\endcsname{#2}}}%
-\newcommand\POL@eval[2]
- {\csname XINT_expr_polfunc_#1\endcsname{#2}}%
-\newcommand\PolEvalAtExpr[2]{\xinttheexpr #1(#2)\relax}%
-%
-\newcommand\PolEvalReduced[3]{\romannumeral`^^@\POL@eval@fork
- #2\PolEvalReducedAt
- \At\PolEvalReducedAtExpr\krof {#1}{#3}%
-}%
-\newcommand\PolEvalReducedAt[2]{%
- \xintpraw % in order not to print denominator if the latter equals 1
- {\xintIrr{\csname XINT_expr_polfunc_#1\endcsname{#2}}[0]}%
-}%
-\newcommand\PolEvalReducedAtExpr[2]{%
- \xintpraw
- {\expandafter\xintIrr\romannumeral`^^@\xintthebareeval#1(#2)\relax[0]}%
-}%
-%
-\newcommand\PolFloatEval[3]{\romannumeral`^^@\POL@eval@fork
- #2\PolFloatEvalAt
- \At\PolFloatEvalAtExpr\krof {#1}{#3}%
-}%
-\newcommand\PolFloatEvalAt[2]
- {\xintpfloat{\csname XINT_flexpr_polfunc_#1\endcsname{#2}}}%
-\newcommand\PolFloatEvalAtExpr[2]{\xintthefloatexpr #1(#2)\relax}%
-
-
-\newcommand\PolSturmIntervalIndex[3]{\the\numexpr\POL@eval@fork
- #2\PolSturmIntervalIndexAt
- \At\PolSturmIntervalIndexAtExpr\krof {#1}{#3}%
-}%
-\newcommand\PolSturmIntervalIndexAtExpr[2]
- {\PolSturmIntervalIndexAt{#1}{\xinttheexpr#2\relax}}%
-\newcommand\PolSturmIntervalIndexAt[2]
- {\expandafter\POL@sturm@index@at\romannumeral`^^@#2!{#1}\xint_bye\relax}%
-\def\POL@sturm@index@at#1!#2%
-{%
- \expandafter\POL@sturm@index@at@iloop
- \romannumeral`^^@\PolSturmNbOfIsolatedZeros{#2}!{#2}{#1}%
-}%
-% implementation is sub-optimal as it should use some kind of binary tree
-% search rather than comparing to the intervals from right to left as here
-\def\POL@sturm@index@at@iloop #1!%
-{%
- \ifnum #1=\z@ 0\expandafter\xint_bye\fi
- \POL@sturm@index@at@iloop@a #1!%
-}%
-\def\POL@sturm@index@at@iloop@a #1!#2#3%
-{% #1 = index, #2 = sturmname, #3 value
- \PolSturmIfZeroExactlyKnown{#2}{#1}
- {\xintifCmp{#3}{\POL@xintexprGetVar{#2L_#1}}%
- {}%
- {#1\xint_bye}%
- {0\xint_bye}%
- }%
- {\xintifGt{#3}{\POL@xintexprGetVar{#2L_#1}}%
- {\xintifLt{#3}{\POL@xintexprGetVar{#2R_#1}}%
- {#1\xint_bye}%
- {0\xint_bye}%
- }%
- {}%
- }%
- % catcode of ! is 11 in polexpr.sty
- \expandafter\POL@sturm@index@at@iloop\the\numexpr#1-\@ne !{#2}{#3}%
-}%
-
-
-\def\POL@leq@fork#1\LessThanOrEqualTo#2#3\krof{#2}%
-\newcommand\PolSturmNbOfRootsOf[3]{\romannumeral`^^@\POL@leq@fork
- #2\PolNbOfRootsLessThanOrEqualTo
- \LessThanOrEqualTo\PolNbOfRootsLessThanOrEqualToExpr\krof {#1}{#3}%
-}%
-\newcommand\PolNbOfRootsLessThanOrEqualToExpr[2]
- {\PolNbOfRootsLessThanOrEqualTo{#1}{\xinttheexpr#2\relax}}%
-\newcommand\PolNbOfRootsLessThanOrEqualTo[1]{%
- \ifnum\PolSturmNbOfIsolatedZeros{#1}=\z@
- \expandafter\xint_firstofthree\expandafter0%
- \else
- \expandafter\PolNbOfRootsLessThanOrEqualTo@%
- \fi {#1}%
-}%
-\def\PolNbOfRootsLessThanOrEqualTo@ #1#2%
-{%
- \expandafter\POL@nbofrootsleq@prep\romannumeral`^^@#2!{#1}%
-}%
-\def\POL@nbofrootsleq@prep#1!#2%
-{%
- \expandafter\POL@nbofrootsleq@iloop\expandafter 1\expandafter !%
- \romannumeral0\xintsgn{\POL@eval{#2_0}{#1}}!%
- #1!{#2}%
-}%
-\def\POL@nbofrootsleq@iloop#1!#2!#3!#4%
-{% #1 = index, #2 = sign of evaluation at value, #3 = value, #4 = sturmname
- \xintifCmp{#3}{\POL@xintexprGetVar{#4L_#1}}%
- {\POL@nbofrootsleq@return #1-\@ne !}%
- {\POL@nbofrootsleq@return
- \PolSturmIfZeroExactlyKnown{#4}{#1}{#1}{#1-\@ne}!%
- }%
- % in third branch we are sure that if root is exactly known
- % the test \xintifLt will be negative
- {\xintifLt{#3}{\POL@xintexprGetVar{#4R_#1}}%
- {\POL@nbofrootsleq@return
- #1\ifnum#2=\xintSgn{\POL@eval{#4_0}{\POL@xintexprGetVar{#4L_#1}}}
- -\@ne\fi !%
- }%
- {\ifnum#1=\PolSturmNbOfIsolatedZeros{#4}
- \expandafter\POL@nbofrootsleq@rightmost
- \fi \expandafter\POL@nbofrootsleq@iloop \the\numexpr\@ne+%
- }%
- }%
- #1!#2!#3!{#4}%
-}%
-\def\POL@nbofrootsleq@return #1!#2!#3!#4!#5{\the\numexpr #1\relax}%
-\def\POL@nbofrootsleq@rightmost\expandafter\POL@nbofrootsleq@iloop
- \the\numexpr\@ne+#1!#2!#3!#4{#1}%
-
-
-\newcommand\PolSturmNbWithMultOfRootsOf[3]
-{\the\numexpr0\POL@leq@fork
- #2\PolNbWithMultOfRootsLessThanOrEqualTo
- \LessThanOrEqualTo\PolNbWithMultOfRootsLessThanOrEqualToExpr\krof {#1}{#3}%
-}%
-\newcommand\PolNbWithMultOfRootsLessThanOrEqualToExpr[2]
- {\PolNbWithMultOfRootsLessThanOrEqualTo{#1}{\xinttheexpr#2\relax}}%
-\newcommand\PolNbWithMultOfRootsLessThanOrEqualTo[1]{%
- \ifnum\PolSturmNbOfIsolatedZeros{#1}=\z@
- \expandafter\POL@nbwmofroots@noroots
- \else
- \expandafter\PolNbWithMultOfRootsLessThanOrEqualTo@%
- \fi {#1}%
-}%
-\def\POL@nbwmofroots@noroots#1#2{\relax}%
-\def\PolNbWithMultOfRootsLessThanOrEqualTo@ #1#2%
-{%
- \expandafter\POL@nbwmofrootsleq@prep\romannumeral`^^@#2!{#1}%
-}%
-\def\POL@nbwmofrootsleq@prep#1!#2%
-{%
- \expandafter\POL@nbwmofrootsleq@iloop\expandafter 1\expandafter !%
- \romannumeral0\xintsgn{\POL@eval{#2_0}{#1}}!%
- #1!{#2}%
-}%
-\def\POL@nbwmofrootsleq@iloop#1!#2!#3!#4%
-{% #1 = index, #2 = sign of evaluation at value, #3 = value, #4 = sturmname
- \xintifCmp{#3}{\POL@xintexprGetVar{#4L_#1}}%
- {\POL@nbwmofrootsleq@return !}%
- {\POL@nbwmofrootsleq@return
- \PolSturmIfZeroExactlyKnown{#4}{#1}%
- {+\PolSturmIsolatedZeroMultiplicity{#4}{#1}}{}!%
- }%
- % in third branch we are sure that if root is exactly known
- % the test \xintifLt will be negative
- {\xintifLt{#3}{\POL@xintexprGetVar{#4R_#1}}%
- {\POL@nbwmofrootsleq@return
- \unless
- \ifnum#2=\xintSgn{\POL@eval{#4_0}{\POL@xintexprGetVar{#4L_#1}}}
- +\PolSturmIsolatedZeroMultiplicity{#4}{#1}\fi !%
- }%
- {+\PolSturmIsolatedZeroMultiplicity{#4}{#1}%
- \ifnum#1=\PolSturmNbOfIsolatedZeros{#4}
- \expandafter\POL@nbwmofrootsleq@return\expandafter !%
- \fi
- \expandafter\POL@nbwmofrootsleq@iloop \the\numexpr\@ne+%
- }%
- }%
- #1!#2!#3!{#4}%
-}%
-\def\POL@nbwmofrootsleq@return #1!#2!#3!#4!#5{#1\relax}%
-
-
-\newcommand\PolLeadingCoeff[1]{%
- \romannumeral`^^@\expandafter\expandafter\expandafter\xintlastitem
- \expandafter\expandafter\expandafter
- {\csname POLuserpol@#1\endcsname}%
-}%
-%
-\newcommand\PolNthCoeff[2]{\romannumeral`^^@%
- \expandafter\POL@nthcoeff
- \romannumeral0\xintnthelt{\ifnum\numexpr#2<\z@#2\else(#2)+1\fi}%
- {\expandafter\expandafter\expandafter
- \xint_gob_til_dot\csname POLuserpol@#1\endcsname}@%
-}%
-\def\POL@nthcoeff#1@{\if @#1@\expandafter\xint_firstoftwo
- \else\expandafter\xint_secondoftwo\fi
- {0/1[0]}{#1}}%
-%
-% returns -1 for zero polynomial for context of numerical expression
-% should it return -\infty?
-\newcommand\PolDegree[1]{\romannumeral`^^@\expandafter\expandafter\expandafter
- \POL@degree\csname POLuserpol@#1\endcsname;}%
-\def\POL@degree #1.#2;{#1}%
-%
-\newcommand\PolToList[1]{\romannumeral`^^@\expandafter\expandafter\expandafter
- \xint_gob_til_dot\csname POLuserpol@#1\endcsname}%
-%
-\newcommand\PolToCSV[1]{\romannumeral0\xintlistwithsep{, }{\PolToList{#1}}}%
-
-
-\newcommand\PolToExprCmd[1]{\xintPRaw{\xintRawWithZeros{#1}}}%
-\newcommand\PolToFloatExprCmd[1]{\xintFloat{#1}}%
-\let\PolToExprTermPrefix\PolTypesetCmdPrefix
-\newcommand\PolToExprOneTermStyleA[2]{%
- \ifnum#2=\z@
- \PolToExprCmd{#1}%
- \else
- \xintifOne{\xintiiAbs{#1}}
- {\xintiiifSgn{#1}{-}{}{}}% + from \PolToExprTermPrefix
- {\PolToExprCmd{#1}\PolToExprTimes}%
- \fi
- \ifcase\xintiiAbs{#2} %<-- space here mandatory
- \or\PolToExprVar
- \else\PolToExprVar^\xintiiAbs{#2}%
- \fi
-}%
-\let\PolToExprOneTerm\PolToExprOneTermStyleA
-\newcommand\PolToExprOneTermStyleB[2]{%
- \ifnum#2=\z@
- \xintNumerator{#1}%
- \else
- \xintifOne{\xintiiAbs{\xintNumerator{#1}}}
- {\xintiiifSgn{#1}{-}{}{}}% + from \PolToExprTermPrefix
- {\xintNumerator{#1}\PolToExprTimes}%
- \fi
- \ifcase\xintiiAbs{#2} %<-- space here mandatory
- \or\PolToExprVar
- \else\PolToExprVar^\xintiiAbs{#2}%
- \fi
- \xintiiifOne{\xintDenominator{#1}}{}{/\xintDenominator{#1}}%
-}%
-\newcommand\PolToFloatExprOneTerm[2]{%
- \ifnum#2=\z@
- \PolToFloatExprCmd{#1}%
- \else
- \PolToFloatExprCmd{#1}\PolToExprTimes
- \fi
- \ifcase\xintiiAbs{#2} %<-- space here mandatory
- \or\PolToExprVar
- \else\PolToExprVar^\xintiiAbs{#2}%
- \fi
-}%
-\newcommand\PolToExprTimes{*}%
-\newcommand\PolToExprVar{x}%
-\newcommand\PolToExpr[1]{%
- \if*\noexpand#1\expandafter\xint_firstoftwo\else
- \expandafter\xint_secondoftwo\fi
- \PolToExprAscending\PolToExprDescending{#1}}%
-\newcommand\PolToFloatExpr[1]{%
- \if*\noexpand#1\expandafter\xint_firstoftwo\else
- \expandafter\xint_secondoftwo\fi
- \PolToFloatExprAscending\PolToFloatExprDescending{#1}}%
-\newcommand\PolToExprAscending[2]{%
- \expandafter\POL@toexpr\csname POLuserpol@#2\endcsname
- \PolToExprOneTerm\POL@toexprA}%
-\newcommand\PolToFloatExprAscending[2]{%
- \expandafter\POL@toexpr\csname POLuserpol@#2\endcsname
- \PolToFloatExprOneTerm\POL@toexprA}%
-\newcommand\PolToExprDescending[1]{%
- \expandafter\POL@toexpr\csname POLuserpol@#1\endcsname
- \PolToExprOneTerm\POL@toexprD}%
-\newcommand\PolToFloatExprDescending[1]{%
- \expandafter\POL@toexpr\csname POLuserpol@#1\endcsname
- \PolToFloatExprOneTerm\POL@toexprD}%
-%
-\def\POL@toexpr#1#2#3{\expandafter\POL@toexpr@
- \expandafter#3\expandafter#2#1\relax}%
-\def\POL@toexpr@#1#2#3.{%
- \ifnum#3<\z@
- #2{0/1[0]}{0}\expandafter\xint_gobble_v
- \else
- \expandafter#1%
- \fi {#3}#2}%
-%
-\def\POL@toexprA #1#2\POL@empty#3{%
- \ifpoltoexprall\expandafter\POL@toexprall@b
- \else\expandafter\POL@toexpr@b
- \fi {#3}#2{0}1.%
-}%
-\def\POL@toexprD #1#2#3\relax{% #3 has \empty (\POL@empty) to prevent brace removal
- \expandafter\POL@toexprD@a\expandafter#2%
- \the\numexpr #1\expandafter.\romannumeral0\xintrevwithbraces{#3}\relax
-}%
-\def\POL@toexprD@a #1#2.#3{%
- \ifpoltoexprall\expandafter\POL@toexprall@b
- \else\expandafter\POL@toexpr@b
- \fi{#3}#1{-#2}\the\numexpr\@ne+-#2.%
-}%
-\def\POL@toexpr@b #1#2#3{%
- \xintiiifZero{#1}%
- {\expandafter\POL@toexpr@loop\expandafter\POL@toexpr@b}%
- {#2{#1}{#3}%
- \expandafter\POL@toexpr@loop\expandafter\POL@toexpr@c}%
- \expandafter#2%
-}%
-\def\POL@toexpr@c #1#2#3{%
- \xintiiifZero{#1}%
- {}%
- {\PolToExprTermPrefix{#1}#2{#1}{#3}}%
- \expandafter\POL@toexpr@loop\expandafter\POL@toexpr@c
- \expandafter#2%
-}%
-\def\POL@toexprall@b #1#2#3{%
- #2{#1}{#3}%
- \expandafter\POL@toexpr@loop\expandafter\POL@toexprall@c
- \expandafter#2%
-}%
-\def\POL@toexprall@c #1#2#3{%
- \PolToExprTermPrefix{#1}#2{#1}{#3}%
- \expandafter\POL@toexpr@loop\expandafter\POL@toexprall@c
- \expandafter#2%
-}%
-\def\POL@toexpr@loop#1#2#3.#4{%
- \if\relax#4\expandafter\xint_gob_til_dot\fi
- #1{#4}#2{#3}\the\numexpr\@ne+#3.%
-}%
-
-
-\POL@restorecatcodes
-\endinput
diff --git a/macros/latex/contrib/polexpr/polexpr.txt b/macros/latex/contrib/polexpr/polexpr.txt
deleted file mode 100644
index 898375926b..0000000000
--- a/macros/latex/contrib/polexpr/polexpr.txt
+++ /dev/null
@@ -1,2598 +0,0 @@
-.. comment: -*- fill-column: 72; mode: rst; -*-
-
-===============================
- Package polexpr documentation
-===============================
-
-0.7.5 (2020/01/31)
-==================
-
-.. contents::
-
-Basic syntax
-------------
-
-The syntax is::
-
- \poldef polname(x):= expression in variable x;
-
-where:
-
-- in place of ``x`` an arbitrary *dummy variable* is authorized,
- i.e. per default any of ``[a-z|A-Z]`` (more letters can be declared
- under Unicode engines.)
-
-- ``polname`` consists of letters, digits, and the ``_`` and
- ``'`` characters. It must start with a letter.
-
-.. attention::
-
- The ``'`` is authorized since ``0.5.1``. As a result some constructs
- recognized by the ``\xintexpr`` parser, such as ``var1 'and' var2``
- will get misinterpreted and cause errors. However these constructs
- are unlikely to be frequently needed in polynomial expressions, and
- the ``\xintexpr`` syntax offers alternatives, so it was deemed a
- small evil. Of course the ``\xintexpr`` parser is modified only
- temporarily during execution of ``\poldef``.
-
-One can also issue::
-
- \PolDef{polname}{expression in variable x}
-
-which admits an optional first argument to modify the variable letter
-from its default ``x``.
-
-``\poldef f(x):= 1-x+x^2;``
- defines polynomial ``f``. Polynomial names must start with a
- letter and may contain letters, digits, underscores and the right
- tick character. The
- variable must be a single letter. The colon character is optional.
- The semi-colon at end of expression is mandatory.
-
-``\PolDef{f}{1-x+x^2}``
- does the same as ``\poldef f(x):= 1-x+x^2;`` To use another letter
- than ``x`` in the expression, one must pass it as an extra optional
- argument to ``\PolDef``. Useful if the semi-colon has been assigned
- some non-standard catcode by some package.
-
-``\PolLet{g}={f}``
- saves a copy of ``f`` under name ``g``. Also usable without ``=``.
-
-``\poldef f(z):= f(z)^2;``
- redefines ``f`` in terms of itself.
-
-``\poldef f(T):= f(f(T));``
- again redefines ``f`` in terms of its (new) self.
-
-``\poldef k(z):= f(z)-g(g(z)^2)^2;``
- should now define the zero polynomial... Let's check:
- ``\[ k(z) = \PolTypeset[z]{k} \]``
-
-``\PolDiff{f}{f'}``
- sets ``f'`` to the derivative of ``f``. The name doesn't have to be
- ``f'`` (in fact the ``'`` is licit only since ``0.5.1``).
-
-.. important::
-
- This is not done automatically. If some new definition needs to use
- the derivative of some available polynomial, that derivative
- polynomial must have been defined via ``\PolDiff``: something like
- ``T'(x)^2`` will not work without a prior ``\PolDiff{T}{T'}``.
-
-``\PolDiff{f'}{f''}``
- obtains second derivative.
-
-``\PolDiff[3]{f}{f'''}``
- computes the third derivative.
-
-::
-
- $f(z) = \PolTypeset[z]{f} $\newline
- $f'(z) = \PolTypeset[z]{f'} $\newline
- $f''(z) = \PolTypeset[z]{f''} $\newline
- $f'''(z)= \PolTypeset[z]{f'''} $\par
-
-.. important::
-
- The package does not currently know rational functions: ``/`` in
- a parsed polynomial expression does the Euclidean quotient::
-
- (1-x^2)/(1-x)
-
- does give ``1+x`` but ::
-
- (1/(1-x))*(1-x^2)
-
- evaluates to zero. This will work as expected::
-
- \poldef k(x):= (x-1)(x-2)(x-3)(x-4)/(x^2-5x+4);
-
-.. _warningtacit:
-
-.. attention::
-
- ``1/2 x^2`` skips the space and is treated like ``1/(2*x^2)`` because
- of the tacit multiplication rules of \xintexpr. But this means it
- gives zero! Thus one must use ``(1/2)x^2`` or ``1/2*x^2`` or
- ``(1/2)*x^2`` for disambiguation: ``x - 1/2*x^2 + 1/3*x^3...``. It is
- even simpler to move the denominator to the right: ``x - x^2/2 +
- x^3/3 - ...``.
-
- It is worth noting that ``1/2(x-1)(x-2)`` suffers the same issue:
- xint_ tacit multiplication always "ties more", hence this gets
- interpreted as ``1/(2*(x-1)*(x-2))`` which gives zero by polynomial
- division. Thus, use one of ``(1/2)(x-1)(x-2)``, ``1/2*(x-1)(x-2)`` or
- ``(x-1)(x-2)/2``.
-
-After::
-
- \poldef f_1(x):= 25(x-1)(x^2-2)(x-3)(x-4)(x-5);%
- \poldef f_2(x):= 37(x-1)(x^2-2)(x-6)(x-7)(x-8);%
-
-the macro call ``\PolGCD{f_1}{f_2}{k}`` sets ``k`` to the (unitary) GCD of
-``f_1`` and ``f_2`` (hence to the expansion of ``(x-1)(x^2-2)``.)
-
-``\PolToExpr{k}``
- will (expandably) give in this case ``x^3-x^2-2*x+2``. This is
- useful for console or file output (the syntax is Maple- and
- PSTricks-compatible; the letter used in output can be
- (non-expandably) changed via a redefinition of `\\PolToExprVar`_.)
-
-``\PolToExpr*{k}``
- gives ascending powers: ``2-2*x-x^2+x^3``.
-
-Examples of localization of roots
----------------------------------
-
-- To make printed decimal numbers more enjoyable than via
- ``\xintSignedFrac``::
-
- \renewcommand\PolTypesetOne[1]{\PolDecToString{\xintREZ{#1}}}%
-
- ``\PolDecToString`` will use decimal notation to incorporate the power
- of ten part; and the ``\xintREZ`` will have the effect to suppress
- trailing zeros if present in raw numerator (if those digits end up
- after decimal mark.) Notice that the above are expandable macros and
- that one can also do::
-
- \renewcommand\PolToExprCmd[1]{\PolDecToString{\xintREZ{#1}}}%
-
- to modify output of `\\PolToExpr{polname}`_.
-
-- For extra info in log file use ``\xintverbosetrue``.
-
-- Only for some of these examples is the output included here.
-
-
-A typical example
-~~~~~~~~~~~~~~~~~
-
-In this example the polynomial is square-free.
-
-::
-
- \poldef f(x) := x^7 - x^6 - 2x + 1;
-
- \PolToSturm{f}{f}
- \PolSturmIsolateZeros{f}
- The \PolTypeset{f} polynomial has \PolSturmNbOfIsolatedZeros{f} distinct real
- roots which are located in the following intervals:
- \PolPrintIntervals{f}
- Here is the second root with ten more decimal digits:
- \PolRefineInterval[10]{f}{2}
- \[\PolSturmIsolatedZeroLeft{f}{2}<Z_2<\PolSturmIsolatedZeroRight{f}{2}\]
- And here is the first root with twenty digits after decimal mark:
- \PolEnsureIntervalLength{f}{1}{-20}
- \[\PolSturmIsolatedZeroLeft{f}{1}<Z_1<\PolSturmIsolatedZeroRight{f}{1}\]
- The first element of the Sturm chain has degree $\PolDegree{f_0}$. As
- this is the original degreee $\PolDegree{f}$ we know that $f$ is square free.
- Its derivative is up to a constant \PolTypeset{f_1} (in this example
- it is identical with it).
- \PolToSturm{f_1}{f_1}\PolSturmIsolateZeros{f_1}%
- The derivative has \PolSturmNbOfIsolatedZeros{f_1} distinct real
- roots:
- \PolPrintIntervals[W]{f_1}
- \PolEnsureIntervalLengths{f_1}{-10}%
- Here they are with ten digits after decimal mark:
- \PolPrintIntervals[W]{f_1}
- \PolDiff{f_1}{f''}
- \PolToSturm{f''}{f''}
- \PolSturmIsolateZeros{f''}
- The second derivative is \PolTypeset{f''}.
- It has \PolSturmNbOfIsolatedZeros{f''} distinct real
- roots:
- \PolPrintIntervals[X]{f''}
- Here is the positive one with 20 digits after decimal mark:
- \PolEnsureIntervalLength{f''}{2}{-20}%
- \[X_2 = \PolSturmIsolatedZeroLeft{f''}{2}\dots\]
- The more mathematically advanced among our dear readers will be able
- to give the exact value for $X_2$!
-
-A degree four polynomial with nearby roots
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
-Notice that this example is a bit outdated as ``0.7`` release has
-added ``\PolSturmIsolateZeros**{sturmname}`` which would find exactly
-the roots. The steps here retain their interest when one is interested
-in finding isolating intervals for example to prepare some demonstration
-of dichotomy method.
-
-
-::
-
- \PolDef{Q}{(x-1.050001)(x-1.105001)(x-1.110501)(x-1.111051)}
- \PolTypeset{Q}
- \PolToSturm{Q}{Q} % it is allowed to use same prefix for Sturm chain
- \PolSturmIsolateZeros{Q}
- \PolPrintIntervals{Q}
- % reports 1.0 < Z_1 < 1.1, 1.10 < Z_2 < 1.11, 1.110 < Z_3 < 1.111, and 1.111 < Z_4 < 1.112
- % but the above bounds do not allow minimizing separation between roots
- % so we refine:
- \PolRefineInterval*{Q}{1}
- \PolRefineInterval*{Q}{2}
- \PolRefineInterval*{Q}{3}
- \PolRefineInterval*{Q}{4}
- \PolPrintIntervals{Q}
- % reports 1.05 < Z_1 < 1.06, 1.105 < Z_2 < 1.106, 1.1105 < Z_3 < 1.1106,
- % and 1.11105 < Z_4 < 1.11106.
- \PolEnsureIntervalLengths{Q}{-6}
- \PolPrintIntervals{Q}
- % of course finds here all roots exactly
-
-
-The degree nine polynomial with 0.99, 0.999, 0.9999 as triple roots
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
-::
-
- % define a user command (xinttools is loaded automatically by polexpr)
- \newcommand\showmultiplicities[1]{% #1 = "sturmname"
- \xintFor* ##1 in {\xintSeq{1}{\PolSturmNbOfIsolatedZeros{#1}}}\do{%
- The multiplicity is \PolSturmIsolatedZeroMultiplicity{#1}{##1}
- \PolSturmIfZeroExactlyKnown{#1}{##1}%
- {at the root $x=\PolSturmIsolatedZeroLeft{#1}{##1}$}
- {for the root such that
- $\PolSturmIsolatedZeroLeft{#1}{##1}<x<\PolSturmIsolatedZeroRight{#1}{##1}$}
- \par
- }}%
- \PolDef{f}{(x-0.99)^3(x-0.999)^3(x-0.9999)^3}
- \renewcommand\PolTypesetOne[1]{\PolDecToString{\xintREZ{#1}}}
- \PolTypeset{f}\par
- \PolToSturm{f}{f}% it is allowed to use "polname" as "sturmname" too
- \PolSturmIsolateZerosAndGetMultiplicities{f}% use the "sturmname" here
- % or \PolSturmIsolateZeros*{f} which is exactly the same, but shorter..
-
- \showmultiplicities{f}
-
-In this example, the output will look like this (but using math mode)::
-
- x^9 - 8.9667x^8 + 35.73400293x^7 - 83.070418400109x^6 + 124.143648875193123x^5
- - 123.683070924326075877x^4 + 82.149260397553075617891x^3
- - 35.07602992699900159127007x^2 + 8.7364078733314648368671733x
- - 0.967100824643585986488103299
-
- The multiplicity is 3 at the root x = 0.99
- The multiplicity is 3 at the root x = 0.999
- The multiplicity is 3 at the root x = 0.9999
-
-On first pass, these rational roots were found (due to their relative
-magnitudes, using ``\PolSturmIsolateZeros**`` was not needed here). But
-multiplicity computation works also with (decimal) roots not yet
-identified or with non-decimal or irrational roots.
-
-It is fun to modify only a tiny bit the polynomial and see if polexpr
-survives::
-
- \PolDef{g}{f(x)+1e-27}
- \PolTypeset{g}\par
- \PolToSturm{g}{g}
- \PolSturmIsolateZeros*{g}
-
- \showmultiplicities{g}
-
-This produces::
-
- x^9 - 8.9667x^8 + 35.73400293x^7 - 83.070418400109x^6 + 124.143648875193123x^5
- - 123.683070924326075877x^4 + 82.149260397553075617891x^3
- - 35.07602992699900159127007x^2 + 8.7364078733314648368671733x
- - 0.967100824643585986488103298
-
- The multiplicity is 1 for the root such that 0.98 < x < 0.99
- The multiplicity is 1 for the root such that 0.9991 < x < 0.9992
- The multiplicity is 1 for the root such that 0.9997 < x < 0.9998
-
-Which means that the multiplicity-3 roots each became a real and a pair of
-complex ones. Let's see them better::
-
- \PolEnsureIntervalLengths{g}{-10}
-
- \showmultiplicities{g}
-
-which produces::
-
- The multiplicity is 1 for the root such that 0.9899888032 < x < 0.9899888033
- The multiplicity is 1 for the root such that 0.9991447980 < x < 0.9991447981
- The multiplicity is 1 for the root such that 0.9997663986 < x < 0.9997663987
-
-A degree five polynomial with three rational roots
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
-::
-
- \poldef Q(x) := 1581755751184441 x^5
- -14907697165025339 x^4
- +48415668972339336 x^3
- -63952057791306264 x^2
- +46833913221154895 x
- -49044360626280925;
-
- \PolToSturm{Q}{Q}
- %\begin{flushleft}
- \renewcommand\PolTypesetCmdPrefix[1]{\allowbreak\xintiiifSgn{#1}{}{+}{+}}%
- $Q_0(x) = \PolTypeset{Q_0}$
- %\end{flushleft}
- \PolSturmIsolateZeros**{Q}
- \PolPrintIntervals{Q}
-
- $Q_{norr}(x) = \PolTypeset{Q_norr}$
-
-Here, all real roots are rational::
-
- Z_1 = 833719/265381
- Z_2 = 165707065/52746197
- Z_3 = 355/113
-
- Q_norr(x) = x^2 + 1
-
-And let's get their decimal expansion too::
-
- % print decimal expansion of the found roots
- \renewcommand\PolPrintIntervalsPrintExactZero
- {\xintTrunc{20}{\PolPrintIntervalsTheLeftEndPoint}\dots}
- \PolPrintIntervals{Q}
-
- Z_1 = 3.14159265358107777120...
- Z_2 = 3.14159265358979340254...
- Z_3 = 3.14159292035398230088...
-
-
-A Mignotte type polynomial
-~~~~~~~~~~~~~~~~~~~~~~~~~~
-
-::
-
- \PolDef{P}{x^10 - (10x-1)^2}%
- \PolTypeset{P} % prints it in expanded form
- \PolToSturm{P}{P} % we can use same prefix for Sturm chain
- \PolSturmIsolateZeros{P} % finds 4 real roots
- This polynomial has \PolSturmNbOfIsolatedZeros{P} distinct real roots:
- \PolPrintIntervals{P}%
- % 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
- \PolRefineInterval*{P}{3}% will refine to 0.100001 < Z_3 < 0.100002
- \PolPrintIntervals{P}%
- Let us now get to know all roots with 10 digits after decimal mark:
- \PolEnsureIntervalLengths{P}{-10}%
- \PolPrintIntervals{P}% now all roots are known 10 decimal digits after mark
- Finally, we display 20 digits of the second root:
- \PolEnsureIntervalLength{P}{2}{-20}% makes Z_2 known with 20 digits after mark
- \[\PolSturmIsolatedZeroLeft{P}{2}<Z_2<\PolSturmIsolatedZeroRight{P}{2}\]
-
-The last line produces::
-
- 0.09999900004999650028 < Z_2 < 0.09999900004999650029
-
-
-The Wilkinson polynomial
-~~~~~~~~~~~~~~~~~~~~~~~~
-
-See `Wilkinson polynomial`_.
-
-::
-
- \documentclass{article}
- \usepackage{polexpr}
- \begin{document}
- %\xintverbosetrue % for the curious...
-
- \poldef f(x) := mul((x - i), i = 1..20);
-
- \renewcommand\PolTypesetCmdPrefix[1]{\allowbreak\xintiiifSgn{#1}{}{+}{+}}%
- \renewcommand\PolTypesetOne[1]{\xintDecToString{#1}}%
-
- \noindent\PolTypeset{f}
-
- \PolToSturm{f}{f}
- \PolSturmIsolateZeros{f}
- \PolPrintIntervals{f}
-
- \clearpage
-
- \poldef g(x) := f(x) - 2**{-23} x**19;
-
- % be patient!
- \PolToSturm{g}{g}
- \noindent\PolTypeset{g_0}% integer coefficient primitive polynomial
-
- \PolSturmIsolateZeros{g}
- \PolEnsureIntervalLengths{g}{-10}
-
- \renewcommand\PolPrintIntervalsPrintMultiplicity{}
- \PolPrintIntervals*{g}
-
- \end{document}
-
-
-The first polynomial::
-
- f(x) = x**20
- - 210 x**19
- + 20615 x**18
- - 1256850 x**17
- + 53327946 x**16
- - 1672280820 x**15
- + 40171771630 x**14
- - 756111184500 x**13
- + 11310276995381 x**12
- - 135585182899530 x**11
- + 1307535010540395 x**10
- - 10142299865511450 x**9
- + 63030812099294896 x**8
- - 311333643161390640 x**7
- + 1206647803780373360 x**6
- - 3599979517947607200 x**5
- + 8037811822645051776 x**4
- - 12870931245150988800 x**3
- + 13803759753640704000 x**2
- - 8752948036761600000 x
- + 2432902008176640000
-
-is handled fast enough (a few seconds), but the modified one ``f(x) -
-2**-23 x**19`` takes about 20x longer (the Sturm chain polynomials
-have integer coefficients with up to 321 digits, whereas (surprisingly
-perhaps) those of the Sturm chain polynomials derived from ``f`` never
-have more than 21 digits ...).
-
-Once the Sturm chain is computed and the zeros isolated, obtaining their
-decimal digits is relatively faster. Here is for the ten real roots of
-``f(x) - 2**-23 x**19`` as computed by the code above::
-
- Z_1 = 0.9999999999...
- Z_2 = 2.0000000000...
- Z_3 = 2.9999999999...
- Z_4 = 4.0000000002...
- Z_5 = 4.9999999275...
- Z_6 = 6.0000069439...
- Z_7 = 6.9996972339...
- Z_8 = 8.0072676034...
- 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
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
-::
-
- \PolDef{P}{mul((x-i*1e-1), i=-20..20)}% i/10 is same but less efficient
-
-In the defining expression we could have used ``i/10`` but this gives
-less efficient internal form for the coefficients (the ``10``'s end up
-in denominators). Using ``\PolToExpr{P}`` after having done
-
-::
-
- \renewcommand\PolToExprCmd[1]{\PolDecToString{\xintREZ{#1}}}
-
-we get this expanded form::
-
- x^41
- -28.7*x^39
- +375.7117*x^37
- -2975.11006*x^35
- +15935.28150578*x^33
- -61167.527674162*x^31
- +173944.259366417394*x^29
- -373686.963560544648*x^27
- +613012.0665016658846445*x^25
- -771182.31133138163125495*x^23
- +743263.86672885754888959569*x^21
- -545609.076599482896371978698*x^19
- +301748.325708943677229642930528*x^17
- -123655.8987669450434698869844544*x^15
- +36666.1782054884005855608205864192*x^13
- -7607.85821367459445649518380016128*x^11
- +1053.15135918687298508885950223794176*x^9
- -90.6380005918141132650786081964032*x^7
- +4.33701563847327366842552218288128*x^5
- -0.0944770968420804735498178265088*x^3
- +0.00059190121813899276854174416896*x
-
-which shows coefficients with up to 36 significant digits...
-
-Stress test: not a hard challenge to ``xint + polexpr``, but be a bit patient!
-
-::
-
- \PolDef{P}{mul((x-i*1e-1), i=-20..20)}%
- \PolToSturm{P}{S} % dutifully computes S_0, ..., S_{41}
- % the [1] optional argument limits the search to interval (-10,10)
- \PolSturmIsolateZeros[1]{S} % finds *exactly* (but a bit slowly) all 41 roots!
- \PolPrintIntervals{S} % nice, isn't it?
-
-.. note::
-
- Release ``0.5`` has *experimental* addition of optional argument
- ``E`` to ``\PolSturmIsolateZeros``. It instructs to search roots only
- in interval ``(-10^E, 10^E)``. Important: the extremities are
- *assumed to not be roots*. In this example, the ``[1]`` in
- ``\PolSturmIsolateZeros[1]{S}`` gives some speed gain; without it, it
- turns out in this case that ``polexpr`` would have started with
- ``(-10^6, 10^6)`` interval.
-
- Please note that this will probably get replaced in future by the
- specification of a general interval. Do not rely on meaning of this
- optional argument keeping the same.
-
-Roots of Chebyshev polynomials
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
-::
-
- \newcount\mycount
- \poldef T_0(x) := 1;
- \poldef T_1(x) := x;
- \mycount 2
- \xintloop
- \poldef T_\the\mycount(x) :=
- 2x*T_\the\numexpr\mycount-1(x)
- - T_\the\numexpr\mycount-2(x);
- \ifnum\mycount<15
- \advance\mycount 1
- \repeat
-
- \[T_{15} = \PolTypeset[X]{T_15}\]
- \PolToSturm{T_15}{T_15}
- \PolSturmIsolateZeros{T_15}
- \PolEnsureIntervalLengths{T_15}{-10}
- \PolPrintIntervals{T_15}
-
-
-Non-expandable macros
----------------------
-
-.. _poldef;:
-
-``\poldef polname(letter):= expression in letter;``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- This evaluates the *polynomial expression* and stores the coefficients
- in a private structure accessible later via other package macros,
- under the user-chosen ``polname``. Of course the *expression* can
- use other previously defined polynomials. Names must start with a
- letter and are constituted of letters, digits, underscores and
- (since ``0.5.1``) the right tick ``'``.
- The whole xintexpr_ syntax is authorized::
-
- \poldef sin(z) := add((-1)^i z^(2i+1)/(2i+1)!, i = 0..10);
-
- With fractional coefficients, beware the `tacit multiplication issue
- <warningtacit_>`_.
-
- As a side effect the function ``polname()`` is recognized as a
- genuine ``\xintexpr...\relax`` function for (exact) numerical
- evaluation (or within an ``\xintdefvar`` assignment.) It computes
- values not according to the original expression but via the Horner
- scheme corresponding to the polynomial coefficients.
-
- .. attention::
-
- Release ``0.3`` also did the necessary set-up to let the
- polynomial be known to the ``\xintfloatexpr`` (or
- ``\xintdeffloatvar``) parser.
-
- Since ``0.4`` this isn't done automatically. Even more, a
- previously existing floating point variant of the same name will
- be let undefined again, to avoid hard to debug mismatches between
- exact and floating point polynomials. This also applies when the
- polynomial is produced not via ``\poldef`` or ``\PolDef`` but as
- a product of the other package macros.
-
- See `\\PolGenFloatVariant{polname}`_.
-
- The original expression is lost after parsing, and in particular
- the package provides no way to typeset it. This has to be done
- manually, if needed.
-
-.. _PolDef:
-
-``\PolDef[letter]{polname}{expression in letter}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- Does the same as `\\poldef <poldef;>`_ in an undelimited macro
- format (thus avoiding potential problems with the catcode of the
- semi-colon in presence of some packages.) In absence of the
- ``[letter]`` optional argument, the variable is assumed to be ``x``.
-
-.. _PolGenFloatVariant:
-
-``\PolGenFloatVariant{polname}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- Makes the polynomial also usable in the ``\xintfloatexpr`` parser.
- It will therein evaluates via an Horner scheme with coefficients
- already pre-rounded to the float precision.
-
- See also `\\PolToFloatExpr{polname}`_.
-
- .. attention::
-
- Release ``0.3`` did this automatically on ``\PolDef`` and
- ``\poldef`` but this was removed at ``0.4`` for optimization.
-
- Any operation, for example generating the derivative polynomial,
- or dividing two polynomials or using the ``\PolLet``, **must** be
- followed by explicit usage of ``\PolGenFloatVariant{polname}`` if
- the new polynomial is to be used in ``\xintfloatexpr`` or alike
- context.
-
-.. _PolLet:
-
-``\PolLet{polname_2}={polname_1}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- Makes a copy of the already defined polynomial ``polname_1`` to a
- new one ``polname_2``. Same effect as
- ``\PolDef{polname_2}{polname_1(x)}`` but with less overhead. The
- ``=`` is optional.
-
-.. _PolGlobalLet:
-
-``\PolGlobalLet{polname_2}={polname_1}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- Acts globally.
-
-.. _PolAssign:
-
-``\PolAssign{polname}\toarray\macro``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- Defines a one-argument expandable macro ``\macro{#1}`` which expands
- to the (raw) #1th polynomial coefficient.
-
- - Attention, coefficients here are indexed starting at 1.
-
- - With #1=-1, -2, ..., ``\macro{#1}`` returns leading coefficients.
-
- - With #1=0, returns the number of coefficients, i.e. ``1 + deg f``
- for non-zero polynomials.
-
- - Out-of-range #1's return ``0/1[0]``.
-
- See also `\\PolNthCoeff{polname}{number}`_. The main difference is that
- with ``\PolAssign``, ``\macro`` is made a prefix to ``1 + deg f``
- already defined (hidden to user) macros holding individually the
- coefficients but `\\PolNthCoeff{polname}{number}`_ does each time the job
- to expandably recover the ``Nth`` coefficient, and due to
- expandability can not store it in a macro for future usage (of course,
- it can be an argument in an ``\edef``.) The other difference
- is the shift by one in indexing, mentioned above (negative
- indices act the same in both.)
-
-.. _PolGet:
-
-``\PolGet{polname}\fromarray\macro``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- Does the converse operation to
- ``\PolAssign{polname}\toarray\macro``. Each individual
- ``\macro{number}`` gets expanded in an ``\edef`` and then normalized
- via xintfrac_\ 's macro ``\xintRaw``.
-
- The leading zeros are removed from the polynomial.
-
- (contrived) Example::
-
- \xintAssignArray{1}{-2}{5}{-3}\to\foo
- \PolGet{f}\fromarray\foo
-
- This will define ``f`` as would have ``\poldef f(x):=1-2x+5x^2-3x^3;``.
-
- .. note::
-
- Prior to ``0.5``, coefficients were not normalized via
- ``\xintRaw`` for internal storage.
-
-.. _PolFromCSV:
-
-``\PolFromCSV{polname}{<csv>}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- Defines a polynomial directly from the comma separated list of values
- (or a macro expanding to such a list) of its coefficients, the *first
- item* gives the constant term, the *last item* gives the leading
- coefficient, except if zero, then it is dropped (iteratively). List
- items are each expanded in an ``\edef`` and then put into normalized
- form via xintfrac_\ 's macro ``\xintRaw``.
-
- As leading zero coefficients are removed::
-
- \PolFromCSV{f}{0, 0, 0, 0, 0, 0, 0, 0, 0, 0}
-
- defines the zero polynomial, which holds only one coefficient.
-
- See also expandable macro `\\PolToCSV <\\PolToCSV{polname}_>`_.
-
- .. note::
-
- Prior to ``0.5``, coefficients were not normalized via
- ``\xintRaw`` for internal storage.
-
-.. _PolTypeset:
-
-``\PolTypeset{polname}``
-~~~~~~~~~~~~~~~~~~~~~~~~
-
- Typesets in descending powers in math mode. It uses letter ``x`` but
- this can be changed via an optional argument::
-
- \PolTypeset[z]{polname}
-
- By default zero coefficients are skipped (issue ``\poltypesetalltrue``
- to get all of them in output).
-
- These commands (whose meanings will be found in the package code)
- can be re-defined for customization. Their default definitions are
- expandable, but this is not a requirement.
-
-.. _PolTypesetCmd:
-
-``\PolTypesetCmd{raw_coeff}``
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
- Checks if the coefficient is ``1`` or ``-1`` and then skips printing
- the ``1``, except for the constant term. Also it sets conditional
- `\\PolIfCoeffIsPlusOrMinusOne{A}{B}`_.
-
- The actual printing of the coefficients, when not equal to plus or
- minus one is handled by `\\PolTypesetOne{raw_coeff}`_.
-
-.. _PolTypesetOne:
-
-``\PolTypesetOne{raw_coeff}``
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
- The default is ``\xintSignedFrac`` but this macro is annoying as it
- insists to use a power of ten, and not decimal notation.
-
- One can do things such as for example: [#]_
-
- ::
-
- \renewcommand\PolTypesetOne[1]{\num{\xintPFloat[5]{#1}}}
- \renewcommand\PolTypesetOne[1]{\num{\xintRound{4}{#1}}}
-
- where e.g. we used the ``\num`` macro of ``siunitx`` as it
- understands floating point notation.
-
- .. [#] the difference in the syntaxes of ``\xintPFloat`` and
- ``\xintRound`` is explained from the fact that
- ``\xintPFloat`` by default uses the prevailing precision
- hence the extra argument like here ``5`` is an optional one.
-
- One can also give a try to using `\\PolDecToString{decimal number}`_
- which uses decimal notation (at least for the numerator part).
-
-.. _PolTypesetMonomialCmd:
-
-``\PolTypesetMonomialCmd``
-^^^^^^^^^^^^^^^^^^^^^^^^^^
-
- This decides how a monomial (in variable ``\PolVar`` and with
- exponent ``\PolIndex``) is to be printed. The default does nothing
- for the constant term, ``\PolVar`` for the first degree and
- ``\PolVar^{\PolIndex}`` for higher degrees monomials. Beware that
- ``\PolIndex`` expands to digit tokens and needs termination in
- ``\ifnum`` tests.
-
-.. _PolTypesetCmdPrefix:
-
-``\PolTypesetCmdPrefix{raw_coeff}``
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
- Expands to a ``+`` if the ``raw_coeff`` is zero or positive, and to
- nothing if ``raw_coeff`` is negative, as in latter case the
- ``\xintSignedFrac`` used by `\\PolTypesetCmd{raw_coeff}`_ will put
- the ``-`` sign in front of the fraction (if it is a fraction) and
- this will thus serve as separator in the typeset formula. Not used
- for the first term.
-
-.. _PolTypeset*:
-
-``\PolTypeset*{polname}``
-~~~~~~~~~~~~~~~~~~~~~~~~~
-
- Typesets in ascending powers. Use e.g. ``[h]`` optional argument
- (after the ``*``) to use letter ``h`` rather than ``x``.
-
-.. _PolDiff:
-
-``\PolDiff{polname_1}{polname_2}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- This sets ``polname_2`` to the first derivative of ``polname_1``. It
- is allowed to issue ``\PolDiff{f}{f}``, effectively replacing ``f``
- by ``f'``.
-
- Coefficients of the result ``polname_2`` are irreducible fractions
- (see `Technicalities`_ for the whole story.)
-
-.. _PolDiff[N]:
-
-``\PolDiff[N]{polname_1}{polname_2}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- This sets ``polname_2`` to the ``N``-th derivative of ``polname_1``.
- Identical arguments is allowed. With ``N=0``, same effect as
- ``\PolLet{polname_2}={polname_1}``. With negative ``N``, switches to
- using ``\PolAntiDiff``.
-
-.. _PolAntiDiff:
-
-``\PolAntiDiff{polname_1}{polname_2}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- This sets ``polname_2`` to the primitive of ``polname_1`` vanishing
- at zero.
-
- Coefficients of the result ``polname_2`` are irreducible fractions
- (see `Technicalities`_ for the whole story.)
-
-.. _PolAntiDiff[N]:
-
-``\PolAntiDiff[N]{polname_1}{polname_2}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- This sets ``polname_2`` to the result of ``N`` successive integrations on
- ``polname_1``. With negative ``N``, it switches to using ``\PolDiff``.
-
-.. _PolDivide:
-
-``\PolDivide{polname_1}{polname_2}{polname_Q}{polname_R}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- This sets ``polname_Q`` and ``polname_R`` to be the quotient and
- remainder in the Euclidean division of ``polname_1`` by
- ``polname_2``.
-
-.. _PolQuo:
-
-``\PolQuo{polname_1}{polname_2}{polname_Q}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- This sets ``polname_Q`` to be the quotient in the Euclidean division
- of ``polname_1`` by ``polname_2``.
-
-.. _PolRem:
-
-``\PolRem{polname_1}{polname_2}{polname_R}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- This sets ``polname_R`` to be the remainder in the Euclidean division
- of ``polname_1`` by ``polname_2``.
-
-.. _PolGCD:
-
-``\PolGCD{polname_1}{polname_2}{polname_GCD}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- This sets ``polname_GCD`` to be the (monic) GCD of the two first
- polynomials. It is a unitary polynomial except if both ``polname_1``
- and ``polname_2`` vanish, then ``polname_GCD`` is the zero
- polynomial.
-
-.. ``\PolIGCD{polname_1}{polname_2}{polname_iGCD}``
- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- **NOT YET**
-
- This **assumes** that the two polynomials have integer coefficients.
- It then computes the greatest common divisor in the integer
- polynomial ring, normalized to have a positive leading coefficient
- (if the inputs are not both zero).
-
- ``\PolIContent{polname}``
- ~~~~~~~~~~~~~~~~~~~~~~~~~
-
- **NOT YET**
-
- This computes a positive rational number such that dividing the
- polynomial with it returns an integer coefficients polynomial with
- no common factor among the coefficients.
-
-.. _PolToSturm:
-
-``\PolToSturm{polname}{sturmname}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- With ``polname`` being for example ``P``, the macro starts by
- computing polynomials ``P`` and ``P'``, then computes the (opposite
- of the) remainder in euclidean division, iteratively.
-
- The last non-zero remainder ``P_N_`` (where ``N`` is obtainable as
- `\\PolSturmChainLength{sturmname}`_) is up to a factor
- the GCD of ``P`` and ``P'`` hence it is a constant if and only if
- ``P`` is square-free.
-
- .. note::
-
- - Since ``0.5`` all these polynomials are divided by their rational
- content, so they have integer coefficients with no common factor,
- and the last one if a constant is either ``1`` or ``-1``.
-
- - After this normalization to primitive polynomials, they are
- stored internally as ``sturmname_k_``, ``k=0,1, ...``.
-
- - These polynomials are used internally only. To keep them as
- genuine declared polynomials also after the macro call, use the
- starred variant `PolToSturm*`_.
-
- .. note::
-
- It is perfectly allowed to use the polynomial name as Sturm chain name:
- ``\PolToSturm{f}(f}``.
-
- The macro then declares ``sturmname_0``, ``sturmname_1``, ..., which are
- the (non-declared) ``sturmname_k_`` divided by the last one. Division is
- not done if this last one is the constant ``1`` or ``-1``, i.e. if the
- original polynomial was square-free. These polynomials are primitive
- polynomials too, i.e. with integer coefficients having no common factor.
-
- Thus ``sturmname_0`` has exactly the same real and complex roots as
- polynomial ``polname``, but with each root now of multiplicity one:
- i.e. it is the "square-free part" of original polynomial ``polname``.
-
- Notice that ``sturmname_1`` isn't necessarily the derivative of
- ``sturmname_0`` due to the various normalizations.
-
- 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
- before executing ``\PolSturmIsolateZeros{sturmname}`` else the
- latter will behave erroneously.
-
- `\\PolSturmChainLength{sturmname}`_ gives the index of the last
- element of the Sturm chain.
-
-.. _PolToSturm*:
-
-``\PolToSturm*{polname}{sturmname}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- Does the same as `un-starred version <PolToSturm_>`_ and additionally it
- keeps for user usage the memory of the *un-normalized* Sturm chain
- polynomials ``sturmname_k_``, ``k=0,1, ..., N``, with
- ``N`` being `\\PolSturmChainLength{sturmname}`_.
-
- .. note::
-
- This behaviour was modified at ``0.6``, anyhow the macro was
- broken at ``0.5``.
-
- .. hint::
-
- The square-free part of ``polname`` is ``sturmname_0``, and their
- quotient is the polynomial with name
- ``sturname_\PolSturmChainLength{sturmname}_``. It thus easy to
- set-up a loop iteratively computing the latter until the last one
- is a constant, thus obtaining the decomposition of an ``f`` as
- a product ``c f_1 f_2 f_3 ...`` of a constant and square-free (primitive)
- polynomials, where each ``f_i`` divides its predecessor.
-
-.. _PolSetToSturmChainSignChangesAt:
-
-``\PolSetToSturmChainSignChangesAt{\macro}{sturmname}{fraction}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- Sets macro ``\macro`` to the number of sign changes in the Sturm
- chain with name prefix ``sturmname``, at location ``fraction``
- (which must be in format as acceptable by the xintfrac_ macros.)
-
- .. note::
-
- The author was lazy and did not provide rather an expandable
- variant, where one would do ``\edef\macro{\PolNbOf...}``.
-
- This will presumably get added in a future release.
-
- After some hesitation it was decided the macro would by default
- act globally. To make the scope of its macro definition local,
- use ``[\empty]`` as extra optional argument.
-
-.. _PolSetToNbOfZerosWithin:
-
-``\PolSetToNbOfZerosWithin{\macro}{sturmname}{value_a}{value_b}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- Applies the `Sturm Theorem`_ to set ``\macro`` to the exact number
- of **distinct** roots of ``sturmname_0`` in the interval ``(value_a,
- value_b]`` (the macro first re-orders the value for ``value_a <=
- value_b`` to hold).
-
- .. note::
-
- The author was lazy and did not provide rather an expandable
- variant, where one would do ``\edef\macro{\PolNbOf...}``.
-
- This will presumably get added in future.
-
- After some hesitation it was decided the macro would by default
- act globally. To make the scope of its macro definition local,
- use ``[\empty]`` as extra optional argument.
-
- See also the expandable
- `\\PolSturmNbOfRootsOf{sturmname}\\LessThanOrEqualTo{value}`_, from
- which it is immediate (with ``\numexpr``) to create an expandable
- variant of this macro. However the difference is that this macro
- requires only `\\PolToSturm <PolToSturm_>`_ to have been executed,
- whereas the expandable variant requires prior execution of
- `\\PolSturmIsolateZeros <PolSturmIsolateZeros_>`_.
-
- See also the expandable
- `\\PolSturmNbWithMultOfRootsOf{sturmname}\\LessThanOrEqualTo{value}`_
- which requires prior execution of
- `\\PolSturmIsolateZeros* <PolSturmIsolateZeros*_>`_.
-
-
-.. _PolSturmIsolateZeros:
-
-``\PolSturmIsolateZeros{sturmname}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- The macros locates, using `Sturm theorem`_, as many disjoint
- intervals as there are (real) roots.
-
- .. important::
-
- The Sturm chain must have been produced by an earlier
- `\\PolToSturm{polname}{sturmname}`_.
-
- Why does this macro ask for argument the name of Sturm chain,
- rather than the name of a polynomial? well this is mainly for
- legacy reason, and because it is accompanied by other macros for
- which it is simpler to assume the argument will be the name of an
- already computed Sturm chain.
-
- Notice that ``\PolToSturm{f}{f}`` is perfectly legal (the
- ``sturmname`` can be same as the ``polname``): it defines
- polynomials ``f_0``, ``f_1``, ... having ``f`` has name prefix.
-
- Such a prior call
- to ``\PolToSturm`` must have been made at any rate for
- ``\PolSturmIsolateZeros`` to be usable.
-
- After its execution they are two types of such intervals (stored in
- memory and accessible via macros or xintexpr_ variables, see below):
-
- - singleton ``{a}``: then ``a`` is a root, (necessarily a decimal
- number, but not all such decimal numbers are exactly identified yet).
-
- - open intervals ``(a,b)``: then there is exactly one root ``z``
- such that ``a < z < b``, and the end points are guaranteed to not
- be roots.
-
- The interval boundaries are decimal numbers, originating
- in iterated decimal subdivision from initial intervals
- ``(-10^E, 0)`` and ``(0, 10^E)`` with ``E`` chosen initially large
- enough so that all roots are enclosed; if zero is a root it is always
- identified as such. The non-singleton intervals are of the
- type ``(a/10^f, (a+1)/10^f)`` with ``a`` an integer, which is
- neither ``0`` nor ``-1``. Hence either ``a`` and ``a+1`` are both positive
- or they are both negative.
-
- One does not *a priori* know what will be the lengths of these
- intervals (except that they are always powers of ten), they
- vary depending on how many digits two successive roots have in
- common in their respective decimal expansions.
-
- .. important::
-
- If some two consecutive intervals share an end-point, no
- information is yet gained about the separation between the two
- roots which could at this stage be arbitrarily small.
-
- See `\\PolRefineInterval*{sturmname}{index}`_ which addresses
- this issue.
-
- .. This procedure is covariant
- with the independent variable ``x`` becoming ``-x``.
- Hmm, pas sûr et trop fatigué
-
- The interval boundaries (and exactly found roots) are made available
- for future computations in ``\xintexpr``-essions or polynomial
- definitions as variables ``<sturmname>L_1``,
- ``<sturmname>L_2``, etc..., for the left end-points and
- ``<sturmname>R_1``, ``<sturmname>R_2``, ..., for the right
- end-points.
-
- Thus for example, if ``sturmname`` is ``f``, one can use the
- xintexpr_ variables ``fL_1``, ``fL_2``, ... to refer in expressions
- to the left end-points (or to the exact root, if left and right end
- points coincide). Additionally, xintexpr_ variable ``fZ_1_isknown``
- will have value ``1`` if the root in the first interval is known,
- and ``0`` otherwise. And similarly for the other intervals.
-
- Also, macros `\\PolSturmIsolatedZeroLeft{sturmname}{index}`_ and
- `\\PolSturmIsolatedZeroRight{sturmname}{index}`_ are provided which
- expand to these same values, written in decimal notation (i.e.
- pre-processed by `\\PolDecToString <PolDecToString_>`_.) And there
- is also `\\PolSturmIfZeroExactlyKnown{sturmname}{index}{A}{B}`_.
-
- .. important::
-
- Trailing zeroes in the stored decimal numbers accessible via the
- macros are significant: they are also present in the decimal
- expansion of the exact root.
-
- These variables and macros are automatically updated when one next
- uses macros such as `\\PolRefineInterval*{sturmname}{index}`_.
-
- The start of decimal expansion of a positive ``k``-th root is given
- by `\\PolSturmIsolatedZeroLeft{sturmname}{k}
- <PolSturmIsolatedZeroLeft_>`_, and for a negative root it is given
- by `\PolSturmIsolatedZeroRight{sturmname}{k}
- <PolSturmIsolatedZeroRight_>`_. These two decimal
- numbers are either both zero or both of the same sign.
-
- The number of distinct roots is obtainable expandably as
- `\\PolSturmNbOfIsolatedZeros{sturmname}`_.
-
- Furthermore
- `\\PolSturmNbOfRootsOf{sturmname}\\LessThanOrEqualTo{value}`_ and
- `\\PolSturmNbOfRootsOf{sturmname}\\LessThanOrEqualToExpr{expression}`_.
- will expandably compute respectively the number of real roots at
- most equal to ``value`` or ``expression``, and the same but with
- multiplicities.
-
- .. note::
-
- In the current implementation the xintexpr_ variables
- and xinttools_ arrays are globally defined. On the
- other hand the Sturm sequence polynomials obey the current scope.
-
- .. note::
-
- As all computations are done *exactly* there can be no errors...
- apart those due to bad coding by author. The results are exact
- bounds for the mathematically exact real roots.
-
- Future releases will perhaps also provide macros based on Newton
- or Regula Falsi methods. Exact computations with such methods
- lead however quickly to very big fractions, and this forces usage
- of some rounding scheme for the abscissas if computation times
- are to remain reasonable. This raises issues of its own, which
- are studied in numerical mathematics.
-
-.. _PolSturmIsolateZeros*:
-
-``\PolSturmIsolateZeros*{sturmname}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- The macro does the same as `\\PolSturmIsolateZeros{sturmname}`_ and
- then in addition it does the extra work to determine all
- multiplicities (of the real roots):
- after executing this macro,
- `\\PolSturmIsolatedZeroMultiplicity{sturmname}{index}`_ will expand
- to the multiplicity of the root located in the ``index``\ -th
- interval (intervals are enumerated from left to right, with index
- starting at ``1``).
-
- Furthermore, if for example the ``sturmname`` is ``f``, xintexpr_
- variables ``fM_1``, ``fM_2``... hold the multiplicities thus
- computed.
-
- .. note::
-
- It is **not** necessary to have executed the `PolToSturm*`_ starred
- variant, as the non-starred variant keeps internally the memory of the
- original GCD (and even of the full non-normalized original Sturm
- chain), even though it does not make the declarations as *user-level*
- genuine polynomials.
-
- See `The degree nine polynomial with 0.99, 0.999, 0.9999 as triple
- roots`_ for an example.
-
-.. _PolSturmIsolateZeros**:
-
-``\PolSturmIsolateZeros**{sturmname}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- The macro does the same as `\\PolSturmIsolateZeros*{sturmname}`_ and
- in addition it does the extra work to determine all the *rational*
- roots.
-
- .. note::
-
- After execution of this macro, a root is "known" if and only if
- it is rational.
-
- Furthermore, primitive polynomial ``sturmname_sqf_norr`` is created
- to match the (square-free) ``sturmname_0`` from which all rational
- roots have been removed (see `\\polexprsetup`_ for customizing this
- name). The number of distinct rational roots is thus the difference
- between the degrees of these two polynomials (see also
- `\\PolSturmNbOfRationalRoots{sturmname}`_).
-
- And ``sturmname_norr`` is ``sturmname_0_`` from which all rational
- roots have been removed (see `\\polexprsetup`_), i.e. it contains
- the irrational roots of the original polynomial, with the same
- multiplicities.
-
- See `A degree five polynomial with three rational
- roots`_ for an example.
-
-.. _PolSturmIsolateZerosAndGetMultiplicities:
-
-``\PolSturmIsolateZerosAndGetMultiplicities{sturmname}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- This is another name for `\\PolSturmIsolateZeros*{sturmname}`_.
-
-.. _PolSturmIsolateZerosGetMultiplicitiesAndRationalRoots:
-
-``\PolSturmIsolateZerosGetMultiplicitiesAndRationalRoots{sturmname}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- This is another name for `\\PolSturmIsolateZeros**{sturmname}`_.
-
-
-``\PolSturmIsolateZerosAndFindRationalRoots{sturmname}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- This works exactly like `\\PolSturmIsolateZeros**{sturmname}`_
- (inclusive of declaring the polynomials ``sturmname_sqf_norr`` and
- ``sturmname_norr`` with no rational roots) except that it does *not*
- compute the multiplicities of the *non-rational* roots.
-
- .. note::
-
- There is no macro to find the rational roots but not compute
- their multiplicities at the same time.
-
- .. attention::
-
- This macro does *not* define xintexpr_ variables
- ``sturmnameM_1``, ``sturmnameM_2``, ... holding the
- multiplicities and it leaves the multiplicity array (whose accessor
- is `\\PolSturmIsolatedZeroMultiplicity{sturmname}{index}`_) into
- a broken state, as all non-rational roots will supposedly have
- multiplicity one. This means that the output of
- `\\PolPrintIntervals* <PolPrintIntervals*_>`_ for example will be
- erroneous for the intervals with irrational roots.
-
- I decided to document it because finding multiplicities of the
- non rational roots is somewhat costly, and one may be interested
- only into finding the rational roots (of course random
- polynomials with integer coefficients will not have *any*
- rational root anyhow).
-
-
-.. _PolRefineInterval*:
-
-``\PolRefineInterval*{sturmname}{index}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- The ``index``\ -th interval (starting indexing at one) is further
- subdivided as many times as is necessary in order for the newer
- interval to have both its end-points distinct from the end-points of
- the original interval. This means that the ``k``\ th root is then
- strictly separated from the other roots.
-
-.. _PolRefineInterval[N]:
-
-``\PolRefineInterval[N]{sturmname}{index}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- The ``index``\ -th interval (starting count at one) is further
- subdivided once, reducing its length by a factor of 10. This is done
- ``N`` times if the optional argument ``[N]`` is present.
-
-.. _PolEnsureIntervalLength:
-
-``\PolEnsureIntervalLength{sturmname}{index}{E}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- The ``index``\ -th interval is subdivided until its length becomes at
- most ``10^E``. This means (for ``E<0``) that the first ``-E`` digits
- after decimal mark of the ``k``\ th root will then be known exactly.
-
-.. _PolEnsureIntervalLengths:
-
-``\PolEnsureIntervalLengths{sturmname}{E}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- The intervals as obtained from ``\PolSturmIsolateZeros`` are (if
- necessary) subdivided further by (base 10) dichotomy in order for
- each of them to have length at most ``10^E`` (length will be shorter
- than ``10^E`` in output only if it did not change or became zero.)
-
- This means that decimal expansions of all roots will be known with
- ``-E`` digits (for ``E<0``) after decimal mark.
-
-.. _PolPrintIntervals:
-
-``\PolPrintIntervals[varname]{sturmname}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- This is a convenience macro which prints the bounds for the roots
- ``Z_1``, ``Z_2``, ... (the optional argument ``varname`` allows to
- specify a replacement for the default ``Z``). This will be done (by
- default) in a
- math mode ``array``, one interval per row, and pattern ``rcccl``,
- where the second and fourth column hold the ``<`` sign, except when
- the interval reduces to a singleton, which means the root is known
- exactly.
-
- .. attention::
-
- This macro was refactored at 0.7, its default output remained
- identical but the ways to customize it got completely
- modified.
-
- See next macros which govern its output.
-
-``\PolPrintIntervalsNoRealRoots``
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
- Executed in place of an ``array`` environment, when there are no
- real roots. Default definition::
-
- \newcommand\PolPrintIntervalsNoRealRoots{}
-
-``\PolPrintIntervalsBeginEnv``
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
- Default definition::
-
- \newcommand\PolPrintIntervalsBeginEnv{\[\begin{array}{rcccl}}
-
-``\PolPrintIntervalsEndEnv``
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
- Default definition::
-
- \newcommand\PolPrintIntervalsEndEnv{\end{array}\]}
-
-``\PolPrintIntervalsKnownRoot``
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
- Default definition::
-
- \newcommand\PolPrintIntervalsKnownRoot{%
- &&\PolPrintIntervalsTheVar_{\PolPrintIntervalsTheIndex}%
- &=&\PolPrintIntervalsPrintExactZero
- }
-
-``\PolPrintIntervalsUnknownRoot``
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
- Default definition::
-
- \newcommand\PolPrintIntervalsUnknownRoot{%
- \PolPrintIntervalsPrintLeftEndPoint&<&%
- \PolPrintIntervalsTheVar_{\PolPrintIntervalsTheIndex}&<&%
- \PolPrintIntervalsPrintRightEndPoint
- }
-
-
-.. _PolPrintIntervalsPrintExactZero:
-
-``\PolPrintIntervalsPrintExactZero``
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
- Default definition::
-
- \newcommand\PolPrintIntervalsPrintExactZero{\PolPrintIntervalsTheLeftEndPoint}
-
-
-.. _PolPrintIntervalsPrintLeftEndPoint:
-
-``\PolPrintIntervalsPrintLeftEndPoint``
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
- Default definition::
-
- \newcommand\PolPrintIntervalsPrintLeftEndPoint{\PolPrintIntervalsTheLeftEndPoint}
-
-.. _PolPrintIntervalsPrintRightEndPoint:
-
-``\PolPrintIntervalsPrintRightEndPoint``
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
- Default definition is::
-
- \newcommand\PolPrintIntervalsPrintRightEndPoint{\PolPrintIntervalsTheRightEndPoint}
-
-.. _PolPrintIntervals*:
-
-``\PolPrintIntervals*[varname]{sturmname}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- This starred variant produces an alternative output (which
- displays the root multiplicity), and is provided as an
- example of customization.
-
- As replacement for `\\PolPrintIntervalsKnownRoot`_,
- `\\PolPrintIntervalsPrintExactZero`_,
- `\\PolPrintIntervalsUnknownRoot`_ it uses its own
- ``\POL@@PrintIntervals...`` macros. We only reproduce here one
- definition::
-
- \newcommand\POL@@PrintIntervalsPrintExactZero{%
- \displaystyle
- \xintSignedFrac{\PolPrintIntervalsTheLeftEndPoint}%
- }%
-
- Multiplicities are printed using this auxiliary macro:
-
-``\PolPrintIntervalsPrintMultiplicity``
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
- whose default definition is::
-
- \newcommand\PolPrintIntervalsPrintMultiplicity{(\mbox{mult. }\PolPrintIntervalsTheMultiplicity)}
-
-
-.. _PolMapCoeffs:
-
-``\PolMapCoeffs{\macro}{polname}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- It modifies ('in-place': original coefficients get lost) each
- coefficient of the defined polynomial via the *expandable* macro
- ``\macro``. The degree is adjusted as necessary if some leading
- coefficients vanish after the operation. In replacement text of
- ``\macro``, ``\index`` expands to the coefficient index (which is
- defined to be zero for the constant term).
-
- Notice that ``\macro`` will have to handle inputs of the shape
- ``A/B[N]`` (xintfrac_ internal notation). This means that it probably
- will have to be expressed in terms of macros from xintfrac_ package.
-
- Example::
-
- \def\foo#1{\xintMul{#1}{\the\numexpr\index*\index\relax}}
-
- (or with ``\xintSqr{\index}``) to replace ``n``-th coefficient
- ``f_n`` by ``f_n*n^2``.
-
-.. _PolReduceCoeffs:
-
-``\PolReduceCoeffs{polname}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- About the same as ``\PolMapCoeffs{\xintIrr}{polname}`` (but
- maintaining a ``[0]`` postfix for speedier xintfrac_ parsing when
- polynomial function is used for computations.) This is a
- one-argument macro, working 'in-place'.
-
-.. _PolReduceCoeffs*:
-
-``\PolReduceCoeffs*{polname}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- This starred variant leaves un-touched the decimal exponent in the
- internal representation of the fractional coefficients, i.e. if a
- coefficient is internally ``A/B[N]``, then ``A/B`` is reduced to
- smallest terms, but the ``10^N`` part is kept as is. Note: if the
- polynomial is freshly defined directly via `\\PolFromCSV
- <PolFromCSV_>`_ its coefficients might still be internally in some
- format like ``1.5e7``; the macro will anyhow always first do the
- needed conversion to strict format ``A/B[N]``.
-
- Evaluations with polynomials treated by this can be much faster than
- with those handled by the non-starred variant
- `\\PolReduceCoeffs{polname}`_: as the numerators and denominators
- remain smaller, this proves very beneficial in favorable cases
- (especially when the coefficients are decimal numbers) to the
- expansion speed of the xintfrac_ macros used internally by
- `\\PolEval <PolEvalAt_>`_.
-
-.. _PolMakeMonic:
-
-``\PolMakeMonic{polname}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- Divides by the leading coefficient. It is recommended to execute
- `\\PolReduceCoeffs*{polname}`_ immediately afterwards. This is not
- done automatically, due to the case the original polynomial had integer
- coefficients and we want to keep the leading one as common
- denominator.
-
-.. _PolMakePrimitive:
-
-``\PolMakePrimitive{polname}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- Divides by the integer content see (`\\PolIContent
- <PolIContent_>`_). This thus produces a polynomial with integer
- coefficients having no common factor. The sign of the leading
- coefficient is not modified.
-
-Expandable macros
------------------
-
-All these macros expand completely in two steps except ``\PolToExpr``
-and ``\PolToFloatExpr`` (and their auxiliaries) which need a
-``\write``, ``\edef`` or a ``\csname...\endcsname`` context.
-
-.. _PolEvalAtExpr:
-
-``\PolEval{polname}\AtExpr{numerical expression}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- It boils down to
- ``\xinttheexpr polname(numerical expression)\relax``.
-
-.. _PolEvalAt:
-
-``\PolEval{polname}\At{fraction}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- Evaluates the polynomial at value ``fraction`` which must be in (or
- expand to) a format acceptable to the xintfrac_ macros.
-
-.. _PolEvalReducedAtExpr:
-
-``\PolEvalReduced{polname}\AtExpr{numerical expression}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- Boils down to ``\xinttheexpr reduce(polname(numerical expression))\relax``.
-
-.. _PolEvalReducedAt:
-
-``\PolEvalReduced{polname}\At{fraction}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- Evaluates the polynomial at value ``fraction`` which must be in (or
- expand to) a format acceptable to the xintfrac_ macros, and produce
- an irreducible fraction.
-
-.. _PolFloatEvalAtExpr:
-
-``\PolFloatEval{polname}\AtExpr{numerical expression}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- Boils down to ``\xintthefloatexpr polname(numerical expression)\relax``.
-
- This is done via a Horner Scheme (see `\\poldef <poldef;_>`_ and
- `\\PolGenFloatVariant{polname}`_), with already rounded
- coefficients. [#]_ To use the *exact coefficients* with *exactly
- executed* additions and multiplications, just insert it in the float
- expression as in this example: [#]_
-
- ::
-
- \xintthefloatexpr 3.27*\xintexpr f(2.53)\relax^2\relax
-
- The ``f(2.53)`` is exactly computed then rounded at the time of
- getting raised to the power ``2``. Moving the ``^2`` inside, that
- operation would also be treated exactly.
-
-
- .. [#] Anyway each floating point operation starts by rounding its
- operands to the floating point precision.
-
- .. [#] The ``\xintexpr`` here could be ``\xinttheexpr`` but that
- would be less efficient. Cf. xintexpr_ documentation about
- nested expressions.
-
-.. _PolFloatEvalAt:
-
-``\PolFloatEval{polname}\At{fraction}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- Evaluates the polynomial at value ``fraction`` which must be in (or
- expand to) a format acceptable to the xintfrac_ macros, and produces
- a floating point number.
-
-.. _PolIfCoeffIsPlusOrMinusOne:
-
-``\PolIfCoeffIsPlusOrMinusOne{A}{B}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- This macro is a priori undefined.
-
- It is defined via the default `\\PolTypesetCmd{raw_coeff}`_ to be
- used if needed in the execution of `\\PolTypesetMonomialCmd`_,
- e.g. to insert a ``\cdot`` in front of ``\PolVar^{\PolIndex}`` if
- the coefficient is not plus or minus one.
-
- The macro will execute ``A`` if the coefficient has been found to be
- plus or minus one, and ``B`` if not.
-
-.. _PolLeadingCoeff:
-
-``\PolLeadingCoeff{polname}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- Expands to the leading coefficient.
-
-.. _PolNthCoeff:
-
-``\PolNthCoeff{polname}{number}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- It expands to the raw ``N``-th coefficient (``0/1[0]`` if the index
- number is out of range). With ``N=-1``, ``-2``, ... expands to the
- leading coefficients.
-
-.. _PolDegree:
-
-``\PolDegree{polname}``
-~~~~~~~~~~~~~~~~~~~~~~~
-
- It expands to the degree. This is ``-1`` if zero polynomial but this
- may change in future. Should it then expand to ``-\infty`` ?
-
-.. _PolIContent:
-
-``\PolIContent{polname}``
-~~~~~~~~~~~~~~~~~~~~~~~~~
-
- It expands to the contents of the polynomial, i.e. to the positive
- fraction such that dividing by this fraction produces a polynomial
- with integer coefficients having no common prime divisor.
-
- See `\\PolMakePrimitive <PolMakePrimitive_>`_.
-
-.. _PolToExpr:
-
-``\PolToExpr{polname}``
-~~~~~~~~~~~~~~~~~~~~~~~
-
- Expands [#]_ to ``coeff_N*x^N+...`` (descending powers.)
-
- .. [#] in a ``\write``, ``\edef``, or ``\csname...\endcsname``, but
- not under ``\romannumeral-`0``.
-
- By default zero coefficients are skipped (issue ``\poltoexpralltrue`` to
- get all of them in output).
-
- By default, no ``+`` sign before negative coefficients, for
- compliance with Maple input format (but see
- `\\PolToExprTermPrefix{raw_coeff}`_.) Also, like the default
- behaviour of `\\PolTypeset{polname}`_, does not print (for the non
- constant terms) coefficients equal to plus or minus one. The degree
- one monomial is output as ``x``, not ``x^1``. Complete customization is
- possible, see next macros.
-
- Of course ``\PolToExpr{f}`` can be inserted in a ``\poldef``, as the
- latter expands token by token, hence will force complete expansion
- of ``\PolToExpr{f}``, but a simple ``f(x)`` is more efficient for
- the identical result.
-
-.. _PolToExprOneTerm:
-
-``\PolToExprOneTerm{raw_coeff}{number}``
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
- This two argument expandable command takes care of the monomial and
- its coefficient. The default definition is done in order for
- coefficients of absolute value ``1`` not be printed explicitely
- (except of course for the constant term). Also by default, the
- monomial of degree one is ``x`` not ``x^1``, and ``x^0`` is skipped.
-
- For compatibility with Maple input requirements, by default a ``*``
- always precedes the ``x^number``, except if the coefficient is a one
- or a minus one. See `\\PolToExprTimes`_.
-
-.. _PolToExprOneTermStyleA:
-
-``\PolToExprOneTermStyleA{raw_coeff}{number}``
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
- Holds the default package meaning of
- `\\PolToExprOneTerm{raw_coeff}{number}`_.
-
-.. _PolToExprOneTermStyleB:
-
-``\PolToExprOneTermStyleB{raw_coeff}{number}``
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
- For output in this style::
-
- 2*x^11/3+3*x^8/7-x^5-x^4/4-x^3-x^2/2-2*x+1
-
- issue ``\let\PolToExprOneTerm\PolToExprOneTermStyleB`` before usage of
- ``\PolToExpr``. Note that then ``\PolToExprCmd`` isn't used at all.
- To revert to package default, issue
- ``\let\PolToExprOneTerm\PolToExprOneTermStyleA``.
-
- To suppress the ``*``'s, cf. `\\PolToExprTimes`_.
-
-.. _PolToExprCmd:
-
-``\PolToExprCmd{raw_coeff}``
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
- It is the one-argument macro used by the package definition of
- ``\PolToExprOneTerm`` for the coefficients themselves (when not
- equal to plus or minus one), and it defaults to
- ``\xintPRaw{\xintRawWithZeros{#1}}``. One will have to redefine it
- to ``\xintIrr{#1}`` or to ``\xintPRaw{\xintIrr{#1}}`` to obtain in the
- output forcefully reduced coefficients.
-
-.. _PolToExprTermPrefix:
-
-``\PolToExprTermPrefix{raw_coeff}``
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
- Defined identically as `\\PolTypesetCmdPrefix{raw_coeff}`_. It
- prefixes with a plus sign for non-negative coefficients, because
- they don't carry one by themselves.
-
-.. _PolToExprVar:
-
-``\PolToExprVar``
-^^^^^^^^^^^^^^^^^
-
- 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 ``x``.
-
-.. _PolToExprTimes:
-
-``\PolToExprTimes``
-^^^^^^^^^^^^^^^^^^^
-
- This expands to the symbol used for multiplication of an
- ``x^{number}`` by the corresponding coefficient. The default is
- ``*``. Redefine the macro to expand to nothing to get rid of it (but
- this will give output incompatible with some professional computer
- algebra software).
-
-.. _PolToExpr*:
-
-``\PolToExpr*{polname}``
-~~~~~~~~~~~~~~~~~~~~~~~~
-
- Expands to ``coeff_0+coeff_1*x+coeff_2*x^2+...`` (ascending powers).
- Customizable like `\\PolToExpr{polname}`_ via the same macros.
-
-.. _PolToFloatExpr:
-
-``\PolToFloatExpr{polname}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- Similar to `\\PolToExpr{polname}`_ but uses `\\PolToFloatExprCmd
- <\\PolToFloatExprCmd{raw_coeff}>`_
- which by default rounds and converts the coefficients to floating
- point format.
-
- .. note::
-
- It is not necessary to have issued
- `\\PolGenFloatVariant{polname}`_. The rounded coefficients are
- not easily recoverable from the ``\xintfloatexpr`` polynomial
- function hence ``\PolToFloatExprCmd`` operates from the *exact*
- coefficients anew.
-
- Attention that both macros obey the prevailing float precision.
- If it is changed between those macro calls, then a mismatch
- exists between the coefficients as used in ``\xintfloatexpr`` and
- those output by ``\PolToFloatExpr{polname}``.
-
-.. _PolToFloatExprOneTerm:
-
-``\PolToFloatExprOneTerm{raw_coeff}{number}``
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
- Similar to `\\PolToExprOneTerm
- <\\PolToExprOneTerm{raw_coeff}{number}>`_. But does not treat
- especially coefficients equal to plus or minus one.
-
-.. _PolToFloatExprCmd:
-
-``\PolToFloatExprCmd{raw_coeff}``
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
- It is the one-argument macro used by ``\PolToFloatExprOneTerm``.
- Its package definition is ``\xintFloat{#1}``.
-
- .. caution::
-
- Currently (xint_ ``1.3c``) ``\xintFloat{0}`` outputs ``0.e0``
- which is perfectly acceptable input for Python, but not for
- Maple. Thus, one should better leave the `\\poltoexprallfalse`_
- toggle to its default ``\iffalse`` state, if one intends to use
- the output in a Maple worksheet.
-
- But even then the zero polynomial will cause a problem. Workaround::
-
- \renewcommand\PolToFloatExprCmd[1]{\xintiiifZero{#1}{0.0}{\xintFloat{#1}}}
-
- Usage of ``\xintiiifZero`` and not ``\xintifZero`` is only for
- optimization (I can't help it) because ``#1`` is known to be
- in ``xintfrac`` raw format.
-
-.. _PolToFloatExpr*:
-
-``\PolToFloatExpr*{polname}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- Typesets in ascending powers.
-
-.. _PolToList:
-
-``\PolToList{polname}``
-~~~~~~~~~~~~~~~~~~~~~~~
-
- Expands to ``{coeff_0}{coeff_1}...{coeff_N}`` with ``N`` = degree, and
- ``coeff_N`` the leading coefficient
- (the zero polynomial does give ``{0/1[0]}`` and not an
- empty output.)
-
-.. _PolToCSV:
-
-``\PolToCSV{polname}``
-~~~~~~~~~~~~~~~~~~~~~~
-
- Expands to ``coeff_0, coeff_1, coeff_2, ....., coeff_N``, starting
- with constant term and ending with leading coefficient. Converse
- to `\\PolFromCSV <\\PolFromCSV{polname}{\<csv\>}_>`_.
-
-.. _PolSturmChainLength:
-
-``\PolSturmChainLength{sturmname}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- Returns the integer ``N`` such that ``sturmname_N`` is the last one
- in the Sturm chain ``sturmname_0``, ``sturmname_1``, ...
-
- See `\\PolToSturm{polname}{sturmname}`_.
-
-.. _PolSturmIfZeroExactlyKnown:
-
-``\PolSturmIfZeroExactlyKnown{sturmname}{index}{A}{B}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- Executes ``A`` if the ``index``\ -th interval reduces to a singleton,
- i.e. the root is known exactly, else ``B``.
-
- .. note::
-
- ``index`` is allowed to be something like ``1+2*3`` as it is fed
- to ``\the\numexpr...\relax``.
-
-.. _PolSturmIsolatedZeroLeft:
-
-``\PolSturmIsolatedZeroLeft{sturmname}{index}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- Expands to the left end-point for the ``index``\ -th interval, as
- computed by some earlier `\\PolSturmIsolateZeros{sturmname}`_.
-
- .. note::
-
- Of course, this is kept updated by macros such as
- `\\PolRefineInterval{sturmname}{index} <PolRefineInterval[N]_>`_.
-
- The value is pre-formatted using `\\PolDecTostring
- <PolDecToString_>`_.
-
-.. _PolSturmIsolatedZeroRight:
-
-``\PolSturmIsolatedZeroRight{sturmname}{index}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- Expands to the right end-point for the ``index``\ -th interval as
- computed by some earlier `\\PolSturmIsolateZeros{sturmname}`_ and
- possibly refined afterwards.
-
- The value is pre-formatted using `\\PolDecTostring
- <PolDecToString_>`_.
-
-.. _PolSturmIsolatedZeroMultiplicity:
-
-``\PolSturmIsolatedZeroMultiplicity{sturmname}{index}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- Expands to the multiplicity of the unique root contained in the
- ``index``\ -th interval.
-
- .. attention::
-
- A prior execution of `\\PolSturmIsolateZeros*{sturmname}`_ is mandatory.
-
- See `The degree nine polynomial with 0.99, 0.999, 0.9999 as triple
- roots`_ for an example of use.
-
-.. _PolSturmNbOfIsolatedZeros:
-
-``\PolSturmNbOfIsolatedZeros{sturmname}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- Expands to the number of real roots of the polynomial
- ``<sturmname>_0``, i.e. the number of distinct real roots of the
- polynomial originally used to create the Sturm chain via
- `\\PolToSturm{polname}{sturmname}`_.
-
-.. warning::
-
- The next few macros counting roots, with or without multiplicities,
- less than or equal to some value, are under evaluation and may be
- removed from the package if their utility is judged to be not high
- enough. They can be re-coded at user level on the basis of the other
- documented package macros anyway.
-
-``\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualTo{value}``
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
- Expands to the number of distinct roots (of the polynomial used to
- create the Sturm chain) less than or equal to the ``value`` (i.e. a
- number of fraction recognizable by the xintfrac_ macros).
-
- .. attention::
-
- `\\PolSturmIsolateZeros{sturmname}`_ must have been executed
- beforehand.
-
- And the argument is a ``sturmname``, not a ``polname`` (this is
- why the macro contains Sturm in its name), simply to be reminded
- of the above constraint.
-
-``\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualToExpr{expression}``
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
- Expands to the number of distinct roots (of the polynomial
- used to create the Sturm chain) which are less than or equal to the
- given ``expression``.
-
- .. attention::
-
- `\\PolSturmIsolateZeros{sturmname}`_ must have been executed
- beforehand.
-
-``\PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualTo{value}``
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
- Expands to the number counted with multiplicities of the roots (of
- the polynomial used to create the Sturm chain) which are less than
- or equal to the given ``value``.
-
- .. attention::
-
- `\\PolSturmIsolateZeros*{sturmname}`_ (or the double starred
- variant) must have been executed beforehand.
-
-``\PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualToExpr{expression}``
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
- Expands to the total number of roots (counted with multiplicities)
- which are less than or equal to the given ``expression``.
-
- .. attention::
-
- `\\PolSturmIsolateZeros*{sturmname}`_ (or the double starred
- variant) must have been executed beforehand.
-
-``\PolSturmNbOfRationalRoots{sturmname}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- Expands to the number of rational roots (without multiplicities).
-
- .. attention::
-
- `\\PolSturmIsolateZeros**{sturmname}`_ must have been executed
- beforehand.
-
-``\PolSturmNbOfRationalRootsWithMultiplicities{sturmname}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- Expands to the number of rational roots (counted with multiplicities).
-
- .. attention::
-
- `\\PolSturmIsolateZeros**{sturmname}`_ must have been executed
- beforehand.
-
-``\PolSturmRationalRoot{sturmname}{k}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- Expands to the ``k``\ th rational root (they are ordered and indexed
- starting at 1 for the most negative).
-
- .. attention::
-
- `\\PolSturmIsolateZeros**{sturmname}`_ must have been executed
- beforehand.
-
-``\PolSturmRationalRootIndex{sturmname}{k}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- Expands to ``index`` of the ``k``\ th rational root as part of the
- ordered real roots (without multiplicities). I.e., above macro
- `\\PolSturmRationalRoot{sturmname}{k}`_ is equivalent to this
- nested call::
-
- \PolSturmIsolatedZeroLeft{sturmname}{\PolSturmRationalRootIndex{sturmname}{k}}
-
- .. attention::
-
- `\\PolSturmIsolateZeros**{sturmname}`_ must have been executed
- beforehand.
-
-``\PolSturmRationalRootMultiplicity{sturmname}{k}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- Expands to the multiplicity of the ``k``\ th rational root.
-
- .. attention::
-
- `\\PolSturmIsolateZeros**{sturmname}`_ must have been executed
- beforehand.
-
-.. _PolIntervalWidth:
-
-``\PolIntervalWidth{sturmname}{index}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- The ``10^E`` width of the current ``index``\ -th root localization
- interval. Output is in xintfrac_ raw ``1/1[E]`` format (if not zero).
-
-Expandable macros for use within execution of ``\PolPrintIntervals``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
-These macros are for usage within custom user redefinitions of
-`\\PolPrintIntervalsKnownRoot`_, `\\PolPrintIntervalsUnknownRoot`_, or
-in redefinitions of `\PolPrintIntervalsPrintExactZero`_ (used in the
-default for the former) and of `\\PolPrintIntervalsPrintLeftEndPoint`_,
-`\\PolPrintIntervalsPrintRightEndPoint`_ (used in the default for the
-latter).
-
-.. attention::
-
- Some macros formerly mentioned here got removed at 0.7:
- ``\PolPrintIntervalsTheEndPoint``,
- ``\PolIfEndPointIsPositive{A}{B}``,
- ``\PolIfEndPointIsNegative{A}{B}``,
- ``\PolIfEndPointIsZero{A}{B}``.
-
-``\PolPrintIntervalsTheVar``
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
- Expands to the name (default ``Z``) used for representing the roots,
- which was passed as optional argument ``varname`` to
- `\\PolPrintIntervals[varname]{sturmname}`_.
-
-``\PolPrintIntervalsTheIndex``
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
- Expands to the index of the considered interval (indexing starting
- at 1 for the leftmost interval).
-
-``\PolPrintIntervalsTheSturmName``
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
- Expands to the argument which was passed as ``sturmname`` to
- `\\PolPrintIntervals[varname]{sturmname}`_.
-
-``\PolPrintIntervalsTheLeftEndPoint``
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
- The left end point of the interval, as would be produced by
- `\\PolSturmIsolatedZeroLeft <PolSturmIsolatedZeroLeft_>`_ if it was
- used with arguments the Sturm chain name and interval index returned
- by `\\PolPrintIntervalsTheSturmName`_ and
- `\\PolPrintIntervalsTheIndex`_.
-
-``\PolPrintIntervalsTheRightEndPoint``
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
- The right end point of the interval, as would be produced by
- `\\\PolSturmIsolatedZeroRight <PolSturmIsolatedZeroRight_>`_ for
- this Sturm chain name and index.
-
-``\PolPrintIntervalsTheMultiplicity``
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
- The multiplicity of the unique root within the interval of index
- `\\PolPrintIntervalsTheIndex`_. Makes sense only if the starred (or
- double-starred) variant of `\\PolSturmIsolateZeros
- <PolSturmIsolateZeros_>`_ was used earlier.
-
-.. _PolDecToString:
-
-``\PolDecToString{decimal number}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
- This is a utility macro to print decimal numbers. It has been
- backported to xintfrac_ (release ``1.3`` of ``2018/03/01``) under
- the name ``\xintDecToString``, and the ``polexpr`` macro is simply
- now an alias to it.
-
- For example
- ``\PolDecToString{123.456e-8}`` will expand to ``0.00000123456``
- and ``\PolDecToString{123.450e-8}`` to ``0.00000123450`` which
- illustrates that trailing zeros are not trimmed. To trim trailing
- zeroes, one can use ``\PolDecToString{\xintREZ{#1}}``.
-
- The precise behaviour of this macro may evolve in future releases of
- xint_.
-
-Booleans (with default setting as indicated)
---------------------------------------------
-
-``\xintverbosefalse``
-~~~~~~~~~~~~~~~~~~~~~
-
- This is actually an xintexpr_ configuration. Setting it to
- ``true`` triggers the writing of information to the log when new
- polynomials are defined.
-
- .. caution::
-
- The macro meanings as written to the log are to be considered
- unstable and undocumented internal structures.
-
-``\poltypesetallfalse``
-~~~~~~~~~~~~~~~~~~~~~~~
-
- If ``true``, `\\PolTypeset{polname}`_ will also typeset the vanishing
- coefficients.
-
-
-``\poltoexprallfalse``
-~~~~~~~~~~~~~~~~~~~~~~
-
- If ``true``, `\\PolToExpr{polname}`_ and `\\PolToFloatExpr{polname}`_ will
- also include the vanishing coefficients in their outputs.
-
-``\polexprsetup``
------------------
-
- Serves to customize the package. Currently only two keys are
- recognized:
-
- - ``norr``: the postfix that `\\PolSturmIsolateZeros**{sturmname}`_
- should append to ``sturmname`` to declare the primitive polynomial
- obtained from original one after removal of all rational roots.
- The default value is ``_norr`` (standing for “no rational roots”).
-
- - ``sqfnorr``: the postfix that `\\PolSturmIsolateZeros**{sturmname}`_
- should append to ``sturmname`` to declare the primitive polynomial
- obtained from original one after removal of all rational roots and
- suppression of all multiplicities.
- The default value is ``_sqf_norr`` (standing for “square-free with
- no rational roots”).
-
- The package executes ``\polexprsetup{norr=_norr,
- sqfnorr=_sqf_norr}`` as default.
-
-Technicalities
---------------
-
-- The catcode of the semi-colon is reset temporarily by `\\poldef
- <poldef;_>`_ macro in case some other package (for example the French
- babel module) may have made it active. This will fail though if the
- whole thing was already part of a macro argument, in such cases one
- can use `\\PolDef{f}{P(x)} <PolDef_>`_
- rather. The colon in ``:=`` may be active with no consequences.
-
-- As a consequence of xintfrac_ addition and subtraction always using
- least common multiples for the denominators [#]_, user-chosen common
- denominators survive additions and multiplications. For example, this::
-
- \poldef P(x):= 1/2 + 2/2*x + 3/2*x^3 + 4/2*x^4;
- \poldef Q(x):= 1/3 + (2/3)x + (3/3)x^3 + (4/3)x^4;
- \poldef PQ(x):= P(x)Q(x);
-
- gives internally the polynomial::
-
- 1/6+4/6*x^1+4/6*x^2+6/6*x^3+20/6*x^4+16/6*x^5+9/6*x^6+24/6*x^7+16/6*x^8
-
- where all coefficients have the same denominator 6. Notice though that
- ``\PolToExpr{PQ}`` outputs the ``6/6*x^3`` as ``x^3`` because (by
- default) it recognizes and filters out coefficients equal to one or
- minus one (since release ``0.3``). One can use for example
- ``\PolToCSV{PQ}`` to see the internally stored coefficients.
-
- .. [#] prior to ``0.4.1``, ``polexpr`` used to temporarily patch
- during the parsing of polynomials the xintfrac_ macros. This
- patch was backported to xint_ at release ``1.3``.
-
-- `\\PolDiff{polname_1}{polname_2}`_ always applies ``\xintIrr`` to the
- resulting coefficients, except that the *power of ten* part ``[N]``
- (for example an input in scientific notation such as ``1.23e5`` gives
- ``123/1[3]`` internally in xintfrac) is not taken into account in the
- reduction of the fraction. This is tentative and may change.
-
- Same remark for `\\PolAntiDiff{polname_1}{polname_2}`_.
-
-- Currently, the package stores all coefficients from index ``0`` to
- index equal to the polynomial degree inside a single macro, as a list.
- This data structure is obviously very inefficient for polynomials of
- high degree and few coefficients (as an example with ``\poldef
- f(x):=x^1000 + x^500;`` the subsequent definition ``\poldef g(x):=
- f(x)^2;`` will do of the order of 1,000,000 multiplications and
- additions involvings only zeroes... which does take time). This
- may change in the future.
-
-- As is to be expected internal structures of the package are barely
- documented and unstable. Don't use them.
-
-
-CHANGE LOG
-----------
-
-- v0.1 (2018/01/11): initial release. Features:
-
- * The `\\poldef <poldef;_>`_ parser itself,
- * Differentiation and anti-differentiation,
- * Euclidean division and GCDs,
- * Various utilities such as `\\PolFromCSV <PolFromCSV_>`_,
- `\\PolMapCoeffs <PolMapCoeffs_>`_,
- `\\PolToCSV <PolToCSV_>`_, `\\PolToExpr <PolToExpr_>`_, ...
-
- Only one-variable polynomials so far.
-
-- v0.2 (2018/01/14)
-
- * Fix: ``"README thinks \numexpr recognizes ^ operator"``.
- * Convert README to reStructuredText markup.
- * Move main documentation from README to separate ``polexpr.txt`` file.
- * Provide ``polexpr.html`` as obtained via DocUtils_ ``rst2html.py``.
- * Convert README to (CTAN compatible) Markdown markup.
-
- Due to lack of available time the test suite might not be extensive
- enough. Bug reports are very welcome!
-
-- v0.3 (2018/01/17)
-
- * bug fixes:
-
- - the ``0.1`` `\\PolEval <PolEvalAt_>`_ accepted expressions for its second
- argument, but this was removed by mistake at ``0.2``. Restored.
-
- **Attention**: at ``0.4`` this has been reverted again, and
- `\\PolEval{P}\\AtExpr{foo} <PolEvalAtExpr_>`_ syntax is needed for
- using expressions in the second argument.
- * incompatible or breaking changes:
-
- - `\\PolToExpr <PolToExpr_>`_ now by default uses *descending*
- powers (it also treats differently coefficients equal to 1 or -1.)
- Use `\\PolToExpr* <PolToExpr*_>`_ for *ascending* powers.
- - `\\PolEval <PolEvalAt_>`_ reduced the output to smallest terms,
- but as this is costly with big fractions and not needed if e.g.
- wrapped in an ``\xintRound`` or ``\xintFloat``, this step has been
- removed; the former meaning is available as `\\PolEvalReduced
- <PolEvalReducedAt_>`_.
- * new (or newly documented) macros:
-
- - `\\PolTypesetCmd <PolTypesetCmd_>`_
- - `\\PolTypesetCmdPrefix <PolTypesetCmdPrefix_>`_
- - `\\PolTypesetMonomialCmd <PolTypesetMonomialCmd_>`_
- - `\\PolEvalReducedAt <PolEvalReducedAt_>`_
- - `\\PolToFloatExpr <PolToFloatExpr_>`_
- - `\\PolToExprOneTerm <PolToExprOneTerm_>`_
- - `\\PolToFloatExprOneTerm <PolToFloatExprOneTerm_>`_
- - `\\PolToExprCmd <PolToExprCmd_>`_
- - `\\PolToFloatExprCmd <PolToFloatExprCmd_>`_
- - `\\PolToExprTermPrefix <PolToExprTermPrefix_>`_
- - `\\PolToExprVar <PolToExprVar_>`_
- - `\\PolToExprTimes <PolToExprTimes_>`_
- * improvements:
-
- - documentation has a table of contents, internal hyperlinks,
- standardized signature notations and added explanations.
- - one can do ``\PolLet{g}={f}`` or ``\PolLet{g}{f}``.
- - ``\PolToExpr{f}`` is highly customizable.
- - `\\poldef <poldef;_>`_ and other defining macros prepare the polynomial
- functions for usage within ``\xintthefloatexpr`` (or
- ``\xintdeffloatvar``). Coefficients are pre-rounded to the
- floating point precision. Indispensible for numerical algorithms,
- as exact fractions, even reduced, quickly become very big. See the
- documentation about how to use the exact polynomials also in
- floating point context.
-
- **Attention**: this has been reverted at ``0.4``. The macro
- `\\PolGenFloatVariant <PolGenFloatVariant_>`_ must be used for
- generation floating point polynomial functions.
-
-- v0.3.1 (2018/01/18)
-
- Fixes two typos in example code included in the documentation.
-
-- v0.4 (2018/02/16)
-
- * bug fixes:
-
- - when Euclidean division gave a zero remainder, the internal
- representation of this zero polynomial could be faulty; this
- could cause mysterious bugs in conjunction with other package
- macros such as `\\PolMapCoeffs <PolMapCoeffs_>`_.
- - `\\PolGCD <PolGCD_>`_ was buggy in case of first polynomial being
- of lesser degree than the second one.
- * breaking changes:
-
- - formerly `\\PolEval{P}\\At{foo} <PolEvalAt_>`_ allowed ``foo`` to
- be an expression, which was transparently handled via
- ``\xinttheexpr``. Now, ``foo`` must be a fraction (or a macro
- expanding to such) in the format acceptable by ``xintfrac.sty``
- macros. Use `\\PolEval{P}\\AtExpr{foo} <PolEvalAtExpr_>`_ for more
- general arguments using expression syntax. E.g., if ``foo`` is the
- name of a variable known to ``\xintexpr``.
-
- The same holds for `\\PolEvalReduced <PolEvalReducedAt_>`_
- and `\\PolFloatEval <PolFloatEvalAt_>`_.
- - the ``3.0`` automatic generation of floating point variants has
- been reverted. Not only do *not* the package macros automatically
- generate floating point variants of newly created polynomials,
- they actually make pre-existing such variant undefined.
-
- See `\\PolGenFloatVariant <PolGenFloatVariant_>`_.
- * new non-expandable macros:
-
- - `\\PolGenFloatVariant <PolGenFloatVariant_>`_
- - `\\PolGlobalLet <PolGlobalLet_>`_
- - `\\PolTypesetOne <PolTypesetOne_>`_
- - `\\PolQuo <PolQuo_>`_
- - `\\PolRem <PolRem_>`_
- - `\\PolToSturm <PolToSturm_>`_
- - `\\PolToSturm\* <PolToSturm*_>`_
- - `\\PolSetToSturmChainSignChangesAt <PolSetToSturmChainSignChangesAt_>`_
- - `\\PolSetToNbOfZerosWithin <PolSetToNbOfZerosWithin_>`_
- - `\\PolSturmIsolateZeros <PolSturmIsolateZeros_>`_
- - `\\PolRefineInterval* <PolRefineInterval*_>`_
- - `\\PolRefineInterval[N] <PolRefineInterval[N]_>`_
- - `\\PolEnsureIntervalLength <PolEnsureIntervalLength_>`_
- - `\\PolEnsureIntervalLengths <PolEnsureIntervalLengths_>`_
- - `\\PolPrintIntervals <PolPrintIntervals_>`_
- - `\\PolPrintIntervalsPrintExactZero <PolPrintIntervalsPrintExactZero_>`_
- - `\\PolPrintIntervalsPrintLeftEndPoint <PolPrintIntervalsPrintLeftEndPoint_>`_
- - `\\PolPrintIntervalsPrintRightEndPoint <PolPrintIntervalsPrintRightEndPoint_>`_
- - `\\PolReduceCoeffs* <PolReduceCoeffs*_>`_
- - `\\PolMakeMonic <PolMakeMonic_>`_
- * new expandable macros:
-
- - `\\PolToExprOneTermStyleA <PolToExprOneTermStyleA_>`_
- - `\\PolIfCoeffIsPlusOrMinusOne <PolIfCoeffIsPlusOrMinusOne_>`_
- - `\\PolLeadingCoeff <PolLeadingCoeff_>`_
- - `\\PolSturmChainLength <PolSturmChainLength_>`_
- - `\\PolSturmNbOfIsolatedZeros <PolSturmNbOfIsolatedZeros_>`_
- - `\\PolSturmIfZeroExactlyKnown <PolSturmIfZeroExactlyKnown_>`_
- - `\\PolSturmIsolatedZeroLeft <PolSturmIsolatedZeroLeft_>`_
- - `\\PolSturmIsolatedZeroRight <PolSturmIsolatedZeroRight_>`_
- - ``\PolPrintIntervalsTheEndPoint`` (removed at 0.7)
- - `\\PolPrintIntervalsTheIndex`_
- - ``\PolIfEndPointIsPositive`` (removed at 0.7)
- - ``\PolIfEndPointIsNegative`` (removed at 0.7)
- - ``\PolIfEndPointIsZero`` (removed at 0.7)
- - `\\PolIntervalWidth <PolIntervalWidth_>`_
- - `\\PolDecToString <PolDecToString_>`_
- * improvements:
-
- The main new feature is implementation of the `Sturm algorithm`_
- for localization of the real roots of polynomials.
-
-- v0.4.1 (2018/03/01)
-
- Synced with xint 1.3.
-
-- v0.4.2 (2018/03/03)
-
- Documentation fix.
-
-- v0.5 (2018/04/08)
-
- * bug fixes:
-
- - `\\PolGet{polname}\\fromarray\\macro`_ crashed when ``\macro`` was
- an xinttools_ array macro with no items. It now produces the zero
- polynomial.
- * breaking changes:
-
- - `\\PolToSturm`_ creates primitive integer coefficients polynomials.
- This speeds up localization of roots via
- `\\PolSturmIsolateZeros`_. In case of user protests the author
- will make available again the code producing the bona fide Sturm
- polynomials as used formerly.
- - polynomials created from `\\PolFromCSV`_ or `\\PolGet <PolGet_>`_
- get their coefficients normalized via xintfrac_\ 's ``\xintRaw``.
- * experimental change:
-
- - optional argument to `\\PolSturmIsolateZeros`_ (see `The
- degree 41 polynomial with -2, -1.9, -1.8, ..., 0, 0.1, ..., 1.9, 2
- as roots`_ for usage). It will presumably be replaced in future by
- an interval specification.
- * new non-expandable macro:
-
- - `\\PolMakePrimitive`_
- * new expandable macro:
-
- - `\\PolIContent`_
-
-- v0.5.1 (2018/04/22)
-
- * new feature:
-
- - the character ``'`` can be used in polynomial names.
-
-- v0.6 (2018/11/20)
-
- * bugfix:
-
- - the starred variant `\\PolToSturm*{polname}{sturmname}`_ was
- broken. On the occasion of the fix, its meaning has been modified,
- see its documentation.
-
- - using `\\PolToSturm <PolToSturm_>`_ with a constant polynomial
- caused a division by zero error.
-
- * new macro:
-
- - `\\PolSturmIsolateZeros* <PolSturmIsolateZeros*_>`_
- acts like the `non-starred variant
- <PolSturmIsolateZeros_>`_ then computes all the multiplicities.
-
- * new expandable macros:
-
- - `\\PolSturmIsolatedZeroMultiplicity{sturmname}{index}`_
- - `\\PolSturmNbOfRootsOf{sturmname}\\LessThanOrEqualTo{value}`_
- - `\\PolSturmNbOfRootsOf{sturmname}\\LessThanOrEqualToExpr{expression}`_
- - `\\PolSturmNbWithMultOfRootsOf{sturmname}\\LessThanOrEqualTo{value}`_
- - `\\PolSturmNbWithMultOfRootsOf{sturmname}\\LessThanOrEqualToExpr{expression}`_
-
-- v0.7 (2018/12/08), v0.7.1 (bugfix), v0.7.2 (2nd bugfix) (2018/12/09)
-
- * breaking changes:
-
- - although `\\PolPrintIntervals[varname]{sturmname}`_ default output
- remains the same, some auxiliary macros for user-customization
- have been removed: ``\PolPrintIntervalsTheEndPoint``,
- ``\PolIfEndPointIsPositive{A}{B}``,
- ``\PolIfEndPointIsNegative{A}{B}``, and
- ``\PolIfEndPointIsZero{A}{B}``.
-
- * bugfix:
-
- - it could happen that, contrarily to documentation, an interval
- computed by `\\PolSturmIsolateZeros{sturmname}`_ had zero as an
- endpoint,
- - `\\PolEnsureIntervalLength{sturmname}{index}{E}`_ could under
- certain circumstances erroneously replace a non-zero root by
- zero,
- - `\\PolEnsureIntervalLengths{sturmname}{E}`_ crashed when used with
- a polynomial with no real roots, hence for which no isolation intervals
- existed (thanks to Thomas Söll for report).
-
- * new macros:
-
- - `\\PolSturmIsolateZeros**{sturmname}`_
- - `\\PolSturmIsolateZerosGetMultiplicitiesAndRationalRoots{sturmname}`_
- - `\\PolSturmIsolateZerosAndFindRationalRoots{sturmname}`_
- - `\\polexprsetup`_
- - `\\PolPrintIntervals* <PolPrintIntervals*_>`_
- - `\\PolPrintIntervalsNoRealRoots`_
- - `\\PolPrintIntervalsBeginEnv`_
- - `\\PolPrintIntervalsEndEnv`_
- - `\\PolPrintIntervalsKnownRoot`_
- - `\\PolPrintIntervalsUnknownRoot`_
- - `\\PolPrintIntervalsPrintMultiplicity`_
-
- * new expandable macros:
-
- - `\\PolSturmNbOfRationalRoots{sturmname}`_
- - `\\PolSturmNbOfRationalRootsWithMultiplicities{sturmname}`_
- - `\\PolSturmRationalRoot{sturmname}{k}`_
- - `\\PolSturmRationalRootIndex{sturmname}{k}`_
- - `\\PolSturmRationalRootMultiplicity{sturmname}{k}`_
- - `\\PolPrintIntervalsTheVar`_
- - `\\PolPrintIntervalsTheSturmName`_
- - `\\PolPrintIntervalsTheMultiplicity`_
-
-- v0.7.3 (2019/02/04)
-
- * bugfix:
-
- - Debugging information not destined to user showed in log if root
- finding was done under ``\xintverbosetrue`` regime.
- - `\\PolPrintIntervalsTheVar`_ remained defined after
- `\\PolPrintIntervals`_ but was left undefined after
- `\\PolPrintIntervals*`_ (reported by Jürgen Gilg). Now remains
- defined in both cases, and `\\PolPrintIntervalsTheSturmName`_
- also.
- - Polynomial names ending in digits caused errors (reported by Thomas
- Söll).
-
-- v0.7.4 (2019/02/12)
-
- * bugfix:
-
- - 20000000000 is too big for ``\numexpr``, shouldn't I know that?
- Thanks to Jürgen Gilg for report.
-
-- v0.7.5 (2020/01/31)
-
- Synced with xint 1.4. Requires it.
-
-
-Acknowledgments
----------------
-
-Thanks to Jürgen Gilg whose question about xint_ usage for
-differentiating polynomials was the initial trigger leading to this
-package, and to Jürgen Gilg and Thomas Söll for testing it on some
-concrete problems.
-
-Renewed thanks to them on occasion of the ``0.6`` and ``0.7`` releases for their
-continued interest.
-
-See README.md for the License.
-
-.. _xinttools:
-.. _xintfrac:
-.. _xintexpr:
-.. _xint: http://www.ctan.org/pkg/xint
-
-.. _Wilkinson polynomial: https://en.wikipedia.org/wiki/Wilkinson%27s_polynomial
-
-.. _Sturm algorithm:
-.. _Sturm Theorem: https://en.wikipedia.org/wiki/Sturm%27s_theorem
-
-.. _DocUtils: http://docutils.sourceforge.net/docs/index.html