texlive[58821] Master: polexpr (now generic)
commits+karl at tug.org
commits+karl at tug.org
Sat Apr 10 23:13:16 CEST 2021
Revision: 58821
http://tug.org/svn/texlive?view=revision&revision=58821
Author: karl
Date: 2021-04-10 23:13:16 +0200 (Sat, 10 Apr 2021)
Log Message:
-----------
polexpr (now generic)
Modified Paths:
--------------
trunk/Master/tlpkg/libexec/ctan2tds
Added Paths:
-----------
trunk/Master/texmf-dist/doc/generic/polexpr/
trunk/Master/texmf-dist/doc/generic/polexpr/README.md
trunk/Master/texmf-dist/doc/generic/polexpr/polexpr.html
trunk/Master/texmf-dist/tex/generic/polexpr/
trunk/Master/texmf-dist/tex/generic/polexpr/polexpr.sty
trunk/Master/texmf-dist/tex/generic/polexpr/polexprcore.tex
trunk/Master/texmf-dist/tex/generic/polexpr/polexprexpr.tex
trunk/Master/texmf-dist/tex/generic/polexpr/polexprsturm.tex
Removed Paths:
-------------
trunk/Master/texmf-dist/doc/latex/polexpr/
trunk/Master/texmf-dist/tex/latex/polexpr/
Added: trunk/Master/texmf-dist/doc/generic/polexpr/README.md
===================================================================
--- trunk/Master/texmf-dist/doc/generic/polexpr/README.md (rev 0)
+++ trunk/Master/texmf-dist/doc/generic/polexpr/README.md 2021-04-10 21:13:16 UTC (rev 58821)
@@ -0,0 +1,123 @@
+Package polexpr README
+======================
+
+Usage
+-----
+
+The package can be used with TeX based formats incorporating the
+e-TeX primitives. The `\expanded` primitive available generally
+since TeXLive 2019 is required.
+
+ \input polexpr.sty
+
+with Plain or other non-LaTeX macro formats, or:
+
+ \usepackage{polexpr}
+
+with the LaTeX macro format.
+
+The package currently requires xintexpr.sty `1.4d` or later.
+
+Abstract
+--------
+
+The package provides a parser `\poldef` of algebraic polynomial
+expressions. As it is based on
+[xintexpr](http://www.ctan.org/pkg/xint)
+the coefficients are allowed to be arbitrary rational numbers.
+
+Once defined, a polynomial is usable by its name either as a numerical
+function in `\xintexpr/\xinteval`, or for additional polynomial
+definitions, or as argument to the package macros. The localization of
+real roots to arbitrary precision as well as the determination of all
+rational roots is implemented via such macros.
+
+Since release `0.8`, polexpr extends the
+[xintexpr](http://www.ctan.org/pkg/xint) syntax to recognize
+polynomials as a new variable type (and not only as functions).
+Functionality which previously was implemented via macros such as the
+computation of a greatest common divisor is now available directly in
+`\xintexpr`, `\xinteval` or `\poldef` via infix or functional syntax.
+
+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 xintexpr 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 xintexpr 1.4. Requires it.
+- 0.8 (2021/03/29)
+ Complete refactoring of the package core for better integration with
+ and enhancement of xintexpr 1.4.
+
+Files of 0.8 release:
+
+- polexpr.sty, polexprcore.tex, polexprexpr.tex, polexprsturm.tex,
+- README.md,
+- polexpr.html (documentation)
+
+Acknowledgments
+---------------
+
+Thanks to Jürgen Gilg whose question about
+[xintexpr](http://www.ctan.org/pkg/xintexpr) 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.
+
+License
+-------
+
+Copyright (C) 2018-2021 Jean-François Burnol
+
+See documentation of package [xintexpr](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 files polexpr.sty, polexprcore.tex,
+polexprexpr.tex, polexprsturm.tex, this README.md and polexpr.html.
Property changes on: trunk/Master/texmf-dist/doc/generic/polexpr/README.md
___________________________________________________________________
Added: svn:eol-style
## -0,0 +1 ##
+native
\ No newline at end of property
Added: trunk/Master/texmf-dist/doc/generic/polexpr/polexpr.html
===================================================================
--- trunk/Master/texmf-dist/doc/generic/polexpr/polexpr.html (rev 0)
+++ trunk/Master/texmf-dist/doc/generic/polexpr/polexpr.html 2021-04-10 21:13:16 UTC (rev 58821)
@@ -0,0 +1,4086 @@
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+/* :License: Released under the terms of the `2-Clause BSD license`_, */
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+/* are permitted in any medium without royalty provided the copyright */
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+/* Copying and distribution of this file, with or without modification, */
+/* are permitted in any medium without royalty provided the copyright */
+/* notice and this notice are preserved. */
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+</head>
+<body>
+<div class="document" id="package-polexpr-documentation">
+<h1 class="title">Package polexpr documentation</h1>
+<p class="subtitle" id="id1">0.8 (2021/03/29)</p>
+
+<div class="contents topic" id="contents">
+<p class="topic-title">Contents</p>
+<ul class="simple">
+<li><p><a class="reference internal" href="#usage" id="id41">Usage</a></p></li>
+<li><p><a class="reference internal" href="#abstract" id="id42">Abstract</a></p></li>
+<li><p><a class="reference internal" href="#prerequisites" id="id43">Prerequisites</a></p></li>
+<li><p><a class="reference internal" href="#quick-syntax-overview" id="id44">Quick syntax overview</a></p></li>
+<li><p><a class="reference internal" href="#the-polexpr-0-8-extensions-to-the-xintexpr-syntax" id="id45">The polexpr <span class="docutils literal">0.8</span> extensions to the <span class="docutils literal">\xintexpr</span> syntax</a></p>
+<ul>
+<li><p><a class="reference internal" href="#warning-about-unstability-of-the-new-syntax" id="id46">Warning about unstability of the new syntax</a></p></li>
+<li><p><a class="reference internal" href="#infix-operators" id="id47">Infix operators <span class="docutils literal">+, <span class="pre">-,</span> *, /, **, ^</span></a></p></li>
+<li><p><a class="reference internal" href="#experimental-infix-operators" id="id48">Experimental infix operators <span class="docutils literal">//, /:</span></a></p></li>
+<li><p><a class="reference internal" href="#comparison-operators" id="id49">Comparison operators <span class="docutils literal"><, >, <=, >=, ==, !=</span></a></p></li>
+<li><p><a class="reference internal" href="#pol-nutple-expression" id="id50"><span class="docutils literal"><span class="pre">pol(<nutple</span> expression>)</span></a></p></li>
+<li><p><a class="reference internal" href="#xinteval-pol-expr" id="id51"><span class="docutils literal"><span class="pre">\xinteval{<pol.</span> <span class="pre">expr.>}</span></span></a></p></li>
+<li><p><a class="reference internal" href="#evalp-pol-expr-pol-expr" id="id52"><span class="docutils literal"><span class="pre">evalp(<pol.</span> <span class="pre">expr.>,</span> <pol. expr>)</span></a></p></li>
+<li><p><a class="reference internal" href="#deg-pol-expr" id="id53"><span class="docutils literal"><span class="pre">deg(<pol.</span> <span class="pre">expr.>)</span></span></a></p></li>
+<li><p><a class="reference internal" href="#coeffs-pol-expr" id="id54"><span class="docutils literal"><span class="pre">coeffs(<pol.</span> <span class="pre">expr.>)</span></span></a></p></li>
+<li><p><a class="reference internal" href="#coeff-pol-expr-num-expr" id="id55"><span class="docutils literal"><span class="pre">coeff(<pol.</span> <span class="pre">expr.>,</span> <num. <span class="pre">expr.>)</span></span></a></p></li>
+<li><p><a class="reference internal" href="#lcoeff-pol-expr" id="id56"><span class="docutils literal"><span class="pre">lcoeff(<pol.</span> <span class="pre">expr.>)</span></span></a></p></li>
+<li><p><a class="reference internal" href="#monicpart-pol-expr" id="id57"><span class="docutils literal"><span class="pre">monicpart(<pol.</span> <span class="pre">expr.>)</span></span></a></p></li>
+<li><p><a class="reference internal" href="#icontent-pol-expr" id="id58"><span class="docutils literal"><span class="pre">icontent(<pol.</span> <span class="pre">expr.>)</span></span></a></p></li>
+<li><p><a class="reference internal" href="#primpart-pol-expr" id="id59"><span class="docutils literal"><span class="pre">primpart(<pol.</span> <span class="pre">expr.>)</span></span></a></p></li>
+<li><p><a class="reference internal" href="#quorem-pol-expr-pol-expr" id="id60"><span class="docutils literal"><span class="pre">quorem(<pol.</span> <span class="pre">expr.>,</span> <pol. <span class="pre">expr.>)</span></span></a></p></li>
+<li><p><a class="reference internal" href="#quo-pol-expr-pol-expr" id="id61"><span class="docutils literal"><span class="pre">quo(<pol.</span> <span class="pre">expr.>,</span> <pol. <span class="pre">expr.>)</span></span></a></p></li>
+<li><p><a class="reference internal" href="#rem-pol-expr-pol-expr" id="id62"><span class="docutils literal"><span class="pre">rem(<pol.</span> <span class="pre">expr.>,</span> <pol. <span class="pre">expr.>)</span></span></a></p></li>
+<li><p><a class="reference internal" href="#prem-pol-expr-1-pol-expr-2" id="id63"><span class="docutils literal"><span class="pre">prem(<pol.</span> expr. 1>, <pol. expr. 2>)</span></a></p></li>
+<li><p><a class="reference internal" href="#divmod-pol-expr-1-pol-expr-2" id="id64"><span class="docutils literal"><span class="pre">divmod(<pol.</span> expr. 1>, <pol. expr. 2>)</span></a></p></li>
+<li><p><a class="reference internal" href="#mod-pol-expr-1-pol-expr-2" id="id65"><span class="docutils literal"><span class="pre">mod(<pol.</span> expr. 1>, <pol. expr. 2>)</span></a></p></li>
+<li><p><a class="reference internal" href="#polgcd-pol-expr-1-pol-expr-2" id="id66"><span class="docutils literal"><span class="pre">polgcd(<pol.</span> expr. 1>, <pol. expr. 2>, <span class="pre">...)</span></span></a></p></li>
+<li><p><a class="reference internal" href="#resultant-pol-expr-1-pol-expr-2" id="id67"><span class="docutils literal"><span class="pre">resultant(<pol.</span> expr. 1>, <pol. expr. 2>)</span></a></p></li>
+<li><p><a class="reference internal" href="#disc-pol-expr" id="id68"><span class="docutils literal"><span class="pre">disc(<pol.</span> <span class="pre">expr.>)</span></span></a></p></li>
+<li><p><a class="reference internal" href="#polpowmod-pol-expr-1-num-expr-pol-expr-2" id="id69"><span class="docutils literal"><span class="pre">polpowmod(<pol.</span> expr. 1>, <num. <span class="pre">expr.>,</span> <pol. expr. 2>)</span></a></p></li>
+<li><p><a class="reference internal" href="#rdcoeffs-pol-expr" id="id70"><span class="docutils literal"><span class="pre">rdcoeffs(<pol.</span> <span class="pre">expr.>)</span></span></a></p></li>
+<li><p><a class="reference internal" href="#rdzcoeffs-pol-expr" id="id71"><span class="docutils literal"><span class="pre">rdzcoeffs(<pol.</span> <span class="pre">expr.>)</span></span></a></p></li>
+<li><p><a class="reference internal" href="#diff1-pol-expr" id="id72"><span class="docutils literal"><span class="pre">diff1(<pol.</span> <span class="pre">expr.>)</span></span></a></p></li>
+<li><p><a class="reference internal" href="#diff2-pol-expr" id="id73"><span class="docutils literal"><span class="pre">diff2(<pol.</span> <span class="pre">expr.>)</span></span></a></p></li>
+<li><p><a class="reference internal" href="#diffn-pol-expr-p-num-expr-n" id="id74"><span class="docutils literal"><span class="pre">diffn(<pol.</span> expr. P>, <num. expr. n>)</span></a></p></li>
+<li><p><a class="reference internal" href="#antider-pol-expr-p" id="id75"><span class="docutils literal"><span class="pre">antider(<pol.</span> expr. P>)</span></a></p></li>
+<li><p><a class="reference internal" href="#intfrom-pol-expr-p-pol-expr-c" id="id76"><span class="docutils literal"><span class="pre">intfrom(<pol.</span> expr. P>, <pol. expr. c>)</span></a></p></li>
+<li><p><a class="reference internal" href="#integral-pol-expr-p-pol-expr-a-pol-expr-b" id="id77"><span class="docutils literal"><span class="pre">integral(<pol.</span> expr. P>, [<pol. expr. a>, <pol. expr. <span class="pre">b>])</span></span></a></p></li>
+</ul>
+</li>
+<li><p><a class="reference internal" href="#examples-of-localization-of-roots" id="id78">Examples of localization of roots</a></p>
+<ul>
+<li><p><a class="reference internal" href="#a-typical-example" id="id79">A typical example</a></p></li>
+<li><p><a class="reference internal" href="#a-degree-four-polynomial-with-nearby-roots" id="id80">A degree four polynomial with nearby roots</a></p></li>
+<li><p><a class="reference internal" href="#the-degree-nine-polynomial-with-0-99-0-999-0-9999-as-triple-roots" id="id81">The degree nine polynomial with 0.99, 0.999, 0.9999 as triple roots</a></p></li>
+<li><p><a class="reference internal" href="#a-degree-five-polynomial-with-three-rational-roots" id="id82">A degree five polynomial with three rational roots</a></p></li>
+<li><p><a class="reference internal" href="#a-mignotte-type-polynomial" id="id83">A Mignotte type polynomial</a></p></li>
+<li><p><a class="reference internal" href="#the-wilkinson-polynomial" id="id84">The Wilkinson polynomial</a></p></li>
+<li><p><a class="reference internal" href="#the-second-wilkinson-polynomial" id="id85">The second Wilkinson polynomial</a></p></li>
+<li><p><a class="reference internal" href="#the-degree-41-polynomial-with-2-1-9-1-8-0-0-1-1-9-2-as-roots" id="id86">The degree 41 polynomial with -2, -1.9, -1.8, ..., 0, 0.1, ..., 1.9, 2 as roots</a></p></li>
+<li><p><a class="reference internal" href="#roots-of-chebyshev-polynomials" id="id87">Roots of Chebyshev polynomials</a></p></li>
+</ul>
+</li>
+<li><p><a class="reference internal" href="#non-expandable-macros" id="id88">Non-expandable macros</a></p>
+<ul>
+<li><p><a class="reference internal" href="#poldef-polname-letter-expression-using-the-letter-as-indeterminate" id="id89"><span class="docutils literal">\poldef <span class="pre">polname(letter):=</span> expression using the letter as indeterminate;</span></a></p></li>
+<li><p><a class="reference internal" href="#poldef-letter-polname-expression-using-the-letter-as-indeterminate" id="id90"><span class="docutils literal"><span class="pre">\PolDef[letter]{polname}{expression</span> using the letter as indeterminate}</span></a></p></li>
+<li><p><a class="reference internal" href="#polgenfloatvariant-polname" id="id91"><span class="docutils literal">\PolGenFloatVariant{polname}</span></a></p></li>
+<li><p><a class="reference internal" href="#pollet-polname-2-polname-1" id="id92"><span class="docutils literal"><span class="pre">\PolLet{polname_2}={polname_1}</span></span></a></p></li>
+<li><p><a class="reference internal" href="#polgloballet-polname-2-polname-1" id="id93"><span class="docutils literal"><span class="pre">\PolGlobalLet{polname_2}={polname_1}</span></span></a></p></li>
+<li><p><a class="reference internal" href="#polassign-polname-toarray-macro" id="id94"><span class="docutils literal"><span class="pre">\PolAssign{polname}\toarray\macro</span></span></a></p></li>
+<li><p><a class="reference internal" href="#polget-polname-fromarray-macro" id="id95"><span class="docutils literal"><span class="pre">\PolGet{polname}\fromarray\macro</span></span></a></p></li>
+<li><p><a class="reference internal" href="#polfromcsv-polname-csv" id="id96"><span class="docutils literal"><span class="pre">\PolFromCSV{polname}{<csv>}</span></span></a></p></li>
+<li><p><a class="reference internal" href="#poltypeset-pol-expr" id="id97"><span class="docutils literal"><span class="pre">\PolTypeset{<pol.</span> <span class="pre">expr.>}</span></span></a></p>
+<ul>
+<li><p><a class="reference internal" href="#poltypesetcmd-raw-coeff" id="id98"><span class="docutils literal">\PolTypesetCmd{raw_coeff}</span></a></p></li>
+<li><p><a class="reference internal" href="#poltypesetone-raw-coeff" id="id99"><span class="docutils literal">\PolTypesetOne{raw_coeff}</span></a></p></li>
+<li><p><a class="reference internal" href="#id9" id="id100"><span class="docutils literal">\PolTypesetMonomialCmd</span></a></p></li>
+<li><p><a class="reference internal" href="#poltypesetcmdprefix-raw-coeff" id="id101"><span class="docutils literal">\PolTypesetCmdPrefix{raw_coeff}</span></a></p></li>
+</ul>
+</li>
+<li><p><a class="reference internal" href="#id11" id="id102"><span class="docutils literal"><span class="pre">\PolTypeset*{<pol.</span> <span class="pre">expr.>}</span></span></a></p></li>
+<li><p><a class="reference internal" href="#poldiff-polname-1-polname-2" id="id103"><span class="docutils literal"><span class="pre">\PolDiff{polname_1}{polname_2}</span></span></a></p></li>
+<li><p><a class="reference internal" href="#poldiff-n-polname-1-polname-2" id="id104"><span class="docutils literal"><span class="pre">\PolDiff[N]{polname_1}{polname_2}</span></span></a></p></li>
+<li><p><a class="reference internal" href="#polantidiff-polname-1-polname-2" id="id105"><span class="docutils literal"><span class="pre">\PolAntiDiff{polname_1}{polname_2}</span></span></a></p></li>
+<li><p><a class="reference internal" href="#polantidiff-n-polname-1-polname-2" id="id106"><span class="docutils literal"><span class="pre">\PolAntiDiff[N]{polname_1}{polname_2}</span></span></a></p></li>
+<li><p><a class="reference internal" href="#poldivide-polname-1-polname-2-polname-q-polname-r" id="id107"><span class="docutils literal"><span class="pre">\PolDivide{polname_1}{polname_2}{polname_Q}{polname_R}</span></span></a></p></li>
+<li><p><a class="reference internal" href="#polquo-polname-1-polname-2-polname-q" id="id108"><span class="docutils literal"><span class="pre">\PolQuo{polname_1}{polname_2}{polname_Q}</span></span></a></p></li>
+<li><p><a class="reference internal" href="#polrem-polname-1-polname-2-polname-r" id="id109"><span class="docutils literal"><span class="pre">\PolRem{polname_1}{polname_2}{polname_R}</span></span></a></p></li>
+<li><p><a class="reference internal" href="#polgcd-polname-1-polname-2-polname-gcd" id="id110"><span class="docutils literal"><span class="pre">\PolGCD{polname_1}{polname_2}{polname_GCD}</span></span></a></p></li>
+<li><p><a class="reference internal" href="#non-expandable-macros-related-to-the-root-localization-routines" id="id111">Non-expandable macros related to the root localization routines</a></p>
+<ul>
+<li><p><a class="reference internal" href="#poltosturm-polname-sturmname" id="id112"><span class="docutils literal"><span class="pre">\PolToSturm{polname}{sturmname}</span></span></a></p></li>
+<li><p><a class="reference internal" href="#id13" id="id113"><span class="docutils literal"><span class="pre">\PolToSturm*{polname}{sturmname}</span></span></a></p></li>
+<li><p><a class="reference internal" href="#polsettosturmchainsignchangesat-macro-sturmname-fraction" id="id114"><span class="docutils literal"><span class="pre">\PolSetToSturmChainSignChangesAt{\macro}{sturmname}{fraction}</span></span></a></p></li>
+<li><p><a class="reference internal" href="#polsettonbofzeroswithin-macro-sturmname-value-a-value-b" id="id115"><span class="docutils literal"><span class="pre">\PolSetToNbOfZerosWithin{\macro}{sturmname}{value_a}{value_b}</span></span></a></p></li>
+<li><p><a class="reference internal" href="#polsturmisolatezeros-sturmname" id="id116"><span class="docutils literal">\PolSturmIsolateZeros{sturmname}</span></a></p></li>
+<li><p><a class="reference internal" href="#id15" id="id117"><span class="docutils literal"><span class="pre">\PolSturmIsolateZeros*{sturmname}</span></span></a></p></li>
+<li><p><a class="reference internal" href="#id17" id="id118"><span class="docutils literal"><span class="pre">\PolSturmIsolateZeros**{sturmname}</span></span></a></p></li>
+<li><p><a class="reference internal" href="#polsturmisolatezerosandgetmultiplicities-sturmname" id="id119"><span class="docutils literal">\PolSturmIsolateZerosAndGetMultiplicities{sturmname}</span></a></p></li>
+<li><p><a class="reference internal" href="#polsturmisolatezerosgetmultiplicitiesandrationalroots-sturmname" id="id120"><span class="docutils literal">\PolSturmIsolateZerosGetMultiplicitiesAndRationalRoots{sturmname}</span></a></p></li>
+<li><p><a class="reference internal" href="#polsturmisolatezerosandfindrationalroots-sturmname" id="id121"><span class="docutils literal">\PolSturmIsolateZerosAndFindRationalRoots{sturmname}</span></a></p></li>
+<li><p><a class="reference internal" href="#polrefineinterval-sturmname-index" id="id122"><span class="docutils literal"><span class="pre">\PolRefineInterval*{sturmname}{index}</span></span></a></p></li>
+<li><p><a class="reference internal" href="#polrefineinterval-n-sturmname-index" id="id123"><span class="docutils literal"><span class="pre">\PolRefineInterval[N]{sturmname}{index}</span></span></a></p></li>
+<li><p><a class="reference internal" href="#polensureintervallength-sturmname-index-e" id="id124"><span class="docutils literal"><span class="pre">\PolEnsureIntervalLength{sturmname}{index}{E}</span></span></a></p></li>
+<li><p><a class="reference internal" href="#polensureintervallengths-sturmname-e" id="id125"><span class="docutils literal"><span class="pre">\PolEnsureIntervalLengths{sturmname}{E}</span></span></a></p></li>
+<li><p><a class="reference internal" href="#polprintintervals-varname-sturmname" id="id126"><span class="docutils literal"><span class="pre">\PolPrintIntervals[varname]{sturmname}</span></span></a></p>
+<ul>
+<li><p><a class="reference internal" href="#polprintintervalsnorealroots" id="id127"><span class="docutils literal">\PolPrintIntervalsNoRealRoots</span></a></p></li>
+<li><p><a class="reference internal" href="#polprintintervalsbeginenv" id="id128"><span class="docutils literal">\PolPrintIntervalsBeginEnv</span></a></p></li>
+<li><p><a class="reference internal" href="#polprintintervalsendenv" id="id129"><span class="docutils literal">\PolPrintIntervalsEndEnv</span></a></p></li>
+<li><p><a class="reference internal" href="#polprintintervalsknownroot" id="id130"><span class="docutils literal">\PolPrintIntervalsKnownRoot</span></a></p></li>
+<li><p><a class="reference internal" href="#polprintintervalsunknownroot" id="id131"><span class="docutils literal">\PolPrintIntervalsUnknownRoot</span></a></p></li>
+<li><p><a class="reference internal" href="#id18" id="id132"><span class="docutils literal">\PolPrintIntervalsPrintExactZero</span></a></p></li>
+<li><p><a class="reference internal" href="#id19" id="id133"><span class="docutils literal">\PolPrintIntervalsPrintLeftEndPoint</span></a></p></li>
+<li><p><a class="reference internal" href="#id20" id="id134"><span class="docutils literal">\PolPrintIntervalsPrintRightEndPoint</span></a></p></li>
+</ul>
+</li>
+<li><p><a class="reference internal" href="#id22" id="id135"><span class="docutils literal"><span class="pre">\PolPrintIntervals*[varname]{sturmname}</span></span></a></p>
+<ul>
+<li><p><a class="reference internal" href="#polprintintervalsprintmultiplicity" id="id136"><span class="docutils literal">\PolPrintIntervalsPrintMultiplicity</span></a></p></li>
+</ul>
+</li>
+</ul>
+</li>
+<li><p><a class="reference internal" href="#polmapcoeffs-macro-polname" id="id137"><span class="docutils literal"><span class="pre">\PolMapCoeffs{\macro}{polname}</span></span></a></p></li>
+<li><p><a class="reference internal" href="#polreducecoeffs-polname" id="id138"><span class="docutils literal">\PolReduceCoeffs{polname}</span></a></p></li>
+<li><p><a class="reference internal" href="#id24" id="id139"><span class="docutils literal"><span class="pre">\PolReduceCoeffs*{polname}</span></span></a></p></li>
+<li><p><a class="reference internal" href="#polmakemonic-polname" id="id140"><span class="docutils literal">\PolMakeMonic{polname}</span></a></p></li>
+<li><p><a class="reference internal" href="#polmakeprimitive-polname" id="id141"><span class="docutils literal">\PolMakePrimitive{polname}</span></a></p></li>
+</ul>
+</li>
+<li><p><a class="reference internal" href="#expandable-macros" id="id142">Expandable macros</a></p>
+<ul>
+<li><p><a class="reference internal" href="#poleval-polname-atexpr-numerical-expression" id="id143"><span class="docutils literal"><span class="pre">\PolEval{polname}\AtExpr{numerical</span> expression}</span></a></p></li>
+<li><p><a class="reference internal" href="#poleval-polname-at-fraction" id="id144"><span class="docutils literal"><span class="pre">\PolEval{polname}\At{fraction}</span></span></a></p></li>
+<li><p><a class="reference internal" href="#polevalreduced-polname-atexpr-numerical-expression" id="id145"><span class="docutils literal"><span class="pre">\PolEvalReduced{polname}\AtExpr{numerical</span> expression}</span></a></p></li>
+<li><p><a class="reference internal" href="#polevalreduced-polname-at-fraction" id="id146"><span class="docutils literal"><span class="pre">\PolEvalReduced{polname}\At{fraction}</span></span></a></p></li>
+<li><p><a class="reference internal" href="#polfloateval-polname-atexpr-numerical-expression" id="id147"><span class="docutils literal"><span class="pre">\PolFloatEval{polname}\AtExpr{numerical</span> expression}</span></a></p></li>
+<li><p><a class="reference internal" href="#polfloateval-polname-at-fraction" id="id148"><span class="docutils literal"><span class="pre">\PolFloatEval{polname}\At{fraction}</span></span></a></p></li>
+<li><p><a class="reference internal" href="#polifcoeffisplusorminusone-a-b" id="id149"><span class="docutils literal"><span class="pre">\PolIfCoeffIsPlusOrMinusOne{A}{B}</span></span></a></p></li>
+<li><p><a class="reference internal" href="#polleadingcoeff-polname" id="id150"><span class="docutils literal">\PolLeadingCoeff{polname}</span></a></p></li>
+<li><p><a class="reference internal" href="#polnthcoeff-polname-number" id="id151"><span class="docutils literal"><span class="pre">\PolNthCoeff{polname}{number}</span></span></a></p></li>
+<li><p><a class="reference internal" href="#poldegree-polname" id="id152"><span class="docutils literal">\PolDegree{polname}</span></a></p></li>
+<li><p><a class="reference internal" href="#policontent-polname" id="id153"><span class="docutils literal">\PolIContent{polname}</span></a></p></li>
+<li><p><a class="reference internal" href="#poltoexpr-pol-expr" id="id154"><span class="docutils literal"><span class="pre">\PolToExpr{<pol.</span> <span class="pre">expr.>}</span></span></a></p>
+<ul>
+<li><p><a class="reference internal" href="#id31" id="id155"><span class="docutils literal">\PolToExprVar</span></a></p></li>
+<li><p><a class="reference internal" href="#poltoexprinvar" id="id156"><span class="docutils literal">\PolToExprInVar</span></a></p></li>
+<li><p><a class="reference internal" href="#id32" id="id157"><span class="docutils literal">\PolToExprTimes</span></a></p></li>
+<li><p><a class="reference internal" href="#poltoexprcaret" id="id158"><span class="docutils literal">\PolToExprCaret</span></a></p></li>
+<li><p><a class="reference internal" href="#poltoexprcmd-raw-coeff" id="id159"><span class="docutils literal">\PolToExprCmd{raw_coeff}</span></a></p></li>
+<li><p><a class="reference internal" href="#poltoexproneterm-raw-coeff-number" id="id160"><span class="docutils literal"><span class="pre">\PolToExprOneTerm{raw_coeff}{number}</span></span></a></p></li>
+<li><p><a class="reference internal" href="#poltoexpronetermstylea-raw-coeff-number" id="id161"><span class="docutils literal"><span class="pre">\PolToExprOneTermStyleA{raw_coeff}{number}</span></span></a></p></li>
+<li><p><a class="reference internal" href="#poltoexpronetermstyleb-raw-coeff-number" id="id162"><span class="docutils literal"><span class="pre">\PolToExprOneTermStyleB{raw_coeff}{number}</span></span></a></p></li>
+<li><p><a class="reference internal" href="#poltoexprtermprefix-raw-coeff" id="id163"><span class="docutils literal">\PolToExprTermPrefix{raw_coeff}</span></a></p></li>
+</ul>
+</li>
+<li><p><a class="reference internal" href="#id34" id="id164"><span class="docutils literal"><span class="pre">\PolToExpr*{<pol.</span> <span class="pre">expr.>}</span></span></a></p></li>
+<li><p><a class="reference internal" href="#poltofloatexpr-pol-expr" id="id165"><span class="docutils literal"><span class="pre">\PolToFloatExpr{<pol.</span> <span class="pre">expr.>}</span></span></a></p>
+<ul>
+<li><p><a class="reference internal" href="#poltofloatexproneterm-raw-coeff-number" id="id166"><span class="docutils literal"><span class="pre">\PolToFloatExprOneTerm{raw_coeff}{number}</span></span></a></p></li>
+<li><p><a class="reference internal" href="#poltofloatexprcmd-raw-coeff" id="id167"><span class="docutils literal">\PolToFloatExprCmd{raw_coeff}</span></a></p></li>
+</ul>
+</li>
+<li><p><a class="reference internal" href="#id38" id="id168"><span class="docutils literal"><span class="pre">\PolToFloatExpr*{<pol.</span> <span class="pre">expr.>}</span></span></a></p></li>
+<li><p><a class="reference internal" href="#poltolist-polname" id="id169"><span class="docutils literal">\PolToList{polname}</span></a></p></li>
+<li><p><a class="reference internal" href="#poltocsv-polname" id="id170"><span class="docutils literal">\PolToCSV{polname}</span></a></p></li>
+<li><p><a class="reference internal" href="#expandable-macros-related-to-the-root-localization-routines" id="id171">Expandable macros related to the root localization routines</a></p>
+<ul>
+<li><p><a class="reference internal" href="#polsturmchainlength-sturmname" id="id172"><span class="docutils literal">\PolSturmChainLength{sturmname}</span></a></p></li>
+<li><p><a class="reference internal" href="#polsturmifzeroexactlyknown-sturmname-index-a-b" id="id173"><span class="docutils literal"><span class="pre">\PolSturmIfZeroExactlyKnown{sturmname}{index}{A}{B}</span></span></a></p></li>
+<li><p><a class="reference internal" href="#polsturmisolatedzeroleft-sturmname-index" id="id174"><span class="docutils literal"><span class="pre">\PolSturmIsolatedZeroLeft{sturmname}{index}</span></span></a></p></li>
+<li><p><a class="reference internal" href="#polsturmisolatedzeroright-sturmname-index" id="id175"><span class="docutils literal"><span class="pre">\PolSturmIsolatedZeroRight{sturmname}{index}</span></span></a></p></li>
+<li><p><a class="reference internal" href="#polsturmisolatedzeromultiplicity-sturmname-index" id="id176"><span class="docutils literal"><span class="pre">\PolSturmIsolatedZeroMultiplicity{sturmname}{index}</span></span></a></p></li>
+<li><p><a class="reference internal" href="#polsturmnbofisolatedzeros-sturmname" id="id177"><span class="docutils literal">\PolSturmNbOfIsolatedZeros{sturmname}</span></a></p></li>
+<li><p><a class="reference internal" href="#polsturmnbofrootsof-sturmname-lessthanorequalto-value" id="id178"><span class="docutils literal"><span class="pre">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualTo{value}</span></span></a></p></li>
+<li><p><a class="reference internal" href="#polsturmnbofrootsof-sturmname-lessthanorequaltoexpr-expression" id="id179"><span class="docutils literal"><span class="pre">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualToExpr{expression}</span></span></a></p></li>
+<li><p><a class="reference internal" href="#polsturmnbwithmultofrootsof-sturmname-lessthanorequalto-value" id="id180"><span class="docutils literal"><span class="pre">\PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualTo{value}</span></span></a></p></li>
+<li><p><a class="reference internal" href="#polsturmnbwithmultofrootsof-sturmname-lessthanorequaltoexpr-expression" id="id181"><span class="docutils literal"><span class="pre">\PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualToExpr{expression}</span></span></a></p></li>
+<li><p><a class="reference internal" href="#polsturmnbofrationalroots-sturmname" id="id182"><span class="docutils literal">\PolSturmNbOfRationalRoots{sturmname}</span></a></p></li>
+<li><p><a class="reference internal" href="#polsturmnbofrationalrootswithmultiplicities-sturmname" id="id183"><span class="docutils literal">\PolSturmNbOfRationalRootsWithMultiplicities{sturmname}</span></a></p></li>
+<li><p><a class="reference internal" href="#polsturmrationalroot-sturmname-k" id="id184"><span class="docutils literal"><span class="pre">\PolSturmRationalRoot{sturmname}{k}</span></span></a></p></li>
+<li><p><a class="reference internal" href="#polsturmrationalrootindex-sturmname-k" id="id185"><span class="docutils literal"><span class="pre">\PolSturmRationalRootIndex{sturmname}{k}</span></span></a></p></li>
+<li><p><a class="reference internal" href="#polsturmrationalrootmultiplicity-sturmname-k" id="id186"><span class="docutils literal"><span class="pre">\PolSturmRationalRootMultiplicity{sturmname}{k}</span></span></a></p></li>
+<li><p><a class="reference internal" href="#polintervalwidth-sturmname-index" id="id187"><span class="docutils literal"><span class="pre">\PolIntervalWidth{sturmname}{index}</span></span></a></p></li>
+</ul>
+</li>
+<li><p><a class="reference internal" href="#expandable-macros-for-use-within-execution-of-polprintintervals" id="id188">Expandable macros for use within execution of <span class="docutils literal">\PolPrintIntervals</span></a></p>
+<ul>
+<li><p><a class="reference internal" href="#polprintintervalsthevar" id="id189"><span class="docutils literal">\PolPrintIntervalsTheVar</span></a></p></li>
+<li><p><a class="reference internal" href="#polprintintervalstheindex" id="id190"><span class="docutils literal">\PolPrintIntervalsTheIndex</span></a></p></li>
+<li><p><a class="reference internal" href="#polprintintervalsthesturmname" id="id191"><span class="docutils literal">\PolPrintIntervalsTheSturmName</span></a></p></li>
+<li><p><a class="reference internal" href="#polprintintervalstheleftendpoint" id="id192"><span class="docutils literal">\PolPrintIntervalsTheLeftEndPoint</span></a></p></li>
+<li><p><a class="reference internal" href="#polprintintervalstherightendpoint" id="id193"><span class="docutils literal">\PolPrintIntervalsTheRightEndPoint</span></a></p></li>
+<li><p><a class="reference internal" href="#polprintintervalsthemultiplicity" id="id194"><span class="docutils literal">\PolPrintIntervalsTheMultiplicity</span></a></p></li>
+</ul>
+</li>
+</ul>
+</li>
+<li><p><a class="reference internal" href="#booleans-with-default-setting-as-indicated" id="id195">Booleans (with default setting as indicated)</a></p>
+<ul>
+<li><p><a class="reference internal" href="#xintverbosefalse" id="id196"><span class="docutils literal">\xintverbosefalse</span></a></p></li>
+<li><p><a class="reference internal" href="#polnewpolverbosefalse" id="id197"><span class="docutils literal">\polnewpolverbosefalse</span></a></p></li>
+<li><p><a class="reference internal" href="#poltypesetallfalse" id="id198"><span class="docutils literal">\poltypesetallfalse</span></a></p></li>
+<li><p><a class="reference internal" href="#poltoexprallfalse" id="id199"><span class="docutils literal">\poltoexprallfalse</span></a></p></li>
+</ul>
+</li>
+<li><p><a class="reference internal" href="#utilies" id="id200">Utilies</a></p>
+<ul>
+<li><p><a class="reference internal" href="#poldectostring-decimal-number" id="id201"><span class="docutils literal">\PolDecToString{decimal number}</span></a></p></li>
+<li><p><a class="reference internal" href="#polexprsetup" id="id202"><span class="docutils literal">\polexprsetup</span></a></p></li>
+</ul>
+</li>
+<li><p><a class="reference internal" href="#technicalities" id="id203">Technicalities</a></p></li>
+<li><p><a class="reference internal" href="#change-log" id="id204">CHANGE LOG</a></p></li>
+<li><p><a class="reference internal" href="#acknowledgments" id="id205">Acknowledgments</a></p></li>
+</ul>
+</div>
+<div class="section" id="usage">
+<h1><a class="toc-backref" href="#id41">Usage</a></h1>
+<p>The package can be used with TeX based formats incorporating the e-TeX
+primitives. The <span class="docutils literal">\expanded</span> primitive available generally since
+TeXLive 2019 is required.</p>
+<pre class="literal-block">\input polexpr.sty</pre>
+<p>with Plain or other non-LaTeX macro formats, or:</p>
+<pre class="literal-block">\usepackage{polexpr}</pre>
+<p>with the LaTeX macro format.</p>
+<p>The package requires <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> <span class="docutils literal">1.4d</span> or later.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Until <span class="docutils literal">0.8</span> the package only had a LaTeX interface. As a result,
+parts of this documentation may still give examples using LaTeX syntax such
+as <span class="docutils literal">\newcommand</span>. Please convert to the syntax appropriate to the
+TeX macro format used if needed.</p>
+</div>
+</div>
+<div class="section" id="abstract">
+<h1><a class="toc-backref" href="#id42">Abstract</a></h1>
+<p>The package provides a parser <span class="docutils literal">\poldef</span> of algebraic polynomial
+expressions. As it is based on <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a>
+the coefficients are allowed to be arbitrary rational numbers.</p>
+<p>Once defined, a polynomial is usable by its name either as a numerical
+function in <span class="docutils literal"><span class="pre">\xintexpr/\xinteval</span></span>, or for additional polynomial
+definitions, or as argument to the package macros. The localization of
+real roots to arbitrary precision as well as the determination of all
+rational roots is implemented via such macros.</p>
+<p>Since release <span class="docutils literal">0.8</span>, polexpr extends the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a>
+syntax to recognize
+polynomials as a new variable type (and not only as functions).
+Functionality which previously was implemented via macros such as the
+computation of a greatest common divisor is now available directly in
+<span class="docutils literal">\xintexpr</span>, <span class="docutils literal">\xinteval</span> or <span class="docutils literal">\poldef</span> via infix or functional
+syntax.</p>
+</div>
+<div class="section" id="prerequisites">
+<h1><a class="toc-backref" href="#id43">Prerequisites</a></h1>
+<ul>
+<li><p>The user must have some understanding of TeX as a macro-expansion
+based programming interface, and in particular of how <span class="docutils literal">\edef</span>
+differs from <span class="docutils literal">\def</span>: functionalities of the package as described in
+the <a class="reference internal" href="#expandable-macros">Expandable macros</a> section are suitable for usage in <span class="docutils literal">\edef</span>,
+<span class="docutils literal">\write</span> or <span class="docutils literal">\xinteval</span> context. At <span class="docutils literal">0.8</span> some of these
+macros have an even more convenient functional interface inside
+<span class="docutils literal">\xinteval</span>, as is described in a <a class="reference internal" href="#polexpr08">dedicated section</a>.</p>
+<p>Despite its name <span class="docutils literal">\poldef</span> is more to be seen as an <span class="docutils literal">\edef</span>
+although it does not define a TeX macro (at user level); and of course
+<span class="docutils literal">\edef</span> would do usually nothing on the typical input parsed by
+<span class="docutils literal">\poldef</span> which generally has no backslash in it: but if this input
+does contain macros, they will then be expanded fully and are supposed to
+produce recognizable syntax elements in this expansion only context.</p>
+<p>Note that the <span class="docutils literal">def</span> in <span class="docutils literal">\poldef</span> reminds us that the macro does
+some assignments hence is not usable in expandable only context. Its
+whole point is rather to define entities which, them, can then be used
+in the expandable only <span class="docutils literal">\xinteval</span> (or <span class="docutils literal">\poldef</span>) context.</p>
+</li>
+<li><p>The user must have some familiarity with <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> and in
+particular must know what <span class="docutils literal">\xintexpr</span>, <span class="docutils literal">\xinttheexpr</span>,
+<span class="docutils literal">\xinteval</span> and <span class="docutils literal">\xintfloatexpr</span>, <span class="docutils literal">\xintthefloatexpr</span>,
+<span class="docutils literal">\xintfloateval</span> mean and what are the good practices with them.</p></li>
+<li><p>The user will become quickly aware that exact computations with
+fractions easily lead to very big ones in very few steps; see
+<a class="reference internal" href="#polreducecoeffs-polname">\PolReduceCoeffs{polname}</a> in this context.</p></li>
+<li><p>Finally, it is mandatory to read the entire documentation before
+starting to use the package.</p></li>
+</ul>
+</div>
+<div class="section" id="quick-syntax-overview">
+<h1><a class="toc-backref" href="#id44">Quick syntax overview</a></h1>
+<p>The syntax is:</p>
+<pre class="literal-block">\poldef polname(x):= expression in variable x;</pre>
+<ul>
+<li><p>In place of <span class="docutils literal">x</span> an arbitrary <em>dummy variable</em> is authorized,
+i.e. per default one <span class="docutils literal">a, .., z, A, .., Z</span> (more letters can be declared
+under Unicode engines).</p></li>
+<li><p><span class="docutils literal">polname</span> consists of letters, digits, and the <span class="docutils literal">_</span> and <span class="docutils literal">'</span>
+characters. It <strong>must</strong> start with a letter: do not use the
+underscore <span class="docutils literal">_</span> as <em>first character</em> of a polynomial name (even
+if of catcode letter). No warning is emitted but dire consequences
+will result.</p>
+<div class="admonition hint">
+<p class="admonition-title">Hint</p>
+<p>The <span class="docutils literal">@</span> is usable too, independently of whether it is of catcode
+letter or other. This has always been the case, but was not
+documented by polexpr prior to <span class="docutils literal">0.8</span>, as the author has never
+found the time to provide some official guidelines on how to name
+temporary variables and the <span class="docutils literal">@</span> is used already as such internally
+to package; time has still not yet been found for <span class="docutils literal">0.8</span> to review
+the situation but it seems reasonable to recommend at any rate to
+restrict usage of <span class="docutils literal">@</span> to scratch variables of defined macros and
+to avoid using it to name document variable.</p>
+</div>
+</li>
+<li><p>The colon before the equality sign is optional and its catcode does
+not matter.</p></li>
+<li><p>The semi-colon at the end of the expression is mandatory. Its catcode
+does not matter if <span class="docutils literal">\poldef</span> is not used inside the argument of
+another macro.</p></li>
+</ul>
+<p>There is an alternative syntax</p>
+<pre class="literal-block">\PolDef[optional letter]{polname}{expression in the letter}</pre>
+<p>Its optional first argument defaults to <span class="docutils literal">x</span>.</p>
+<dl>
+<dt><span class="docutils literal">\poldef <span class="pre">f(x):=</span> 1 - x + quo(x^5,1 - x + x^2);</span></dt>
+<dd><p>defines polynomial <span class="docutils literal">f</span>. The indeterminate <span class="docutils literal">x</span> must be
+only submitted to algebraic operations.</p>
+<p>The <span class="docutils literal">quo()</span> function (new at <span class="docutils literal">0.8</span>) computes the euclidean
+division quotient.</p>
+</dd>
+</dl>
+<div class="admonition important">
+<p class="admonition-title">Important</p>
+<p>For backwards compatibility one can currently also use:</p>
+<pre class="literal-block">\poldef f(x):= 1 - x + x^5/(1 - x + x^2);</pre>
+<p>Due to precedence rules the first operand is <span class="docutils literal">x^5</span>, not of course
+<span class="docutils literal"><span class="pre">1-x+x^5</span></span>.</p>
+<p>Note that <span class="docutils literal"><span class="pre">(1-x^2)/(1-x)</span></span> produces <span class="docutils literal">1+x</span>
+but <span class="docutils literal"><span class="pre">(1/(1-x))*(1-x^2)</span></span> produces zero! One also has to be aware
+of some precedence rules, for example:</p>
+<pre class="literal-block">\poldef k(x):= (x-1)(x-2)(x-3)(x-4)/(x^2-5x+4);</pre>
+<p>does compute a degree 2 polynomial because the tacit multiplication
+ties more than the division operator.</p>
+<p>In short, it is safer to use the <span class="docutils literal">quo()</span> function which avoids
+surprises.</p>
+</div>
+<div class="admonition attention" id="warningtacit">
+<p class="admonition-title">Attention!</p>
+<p>Tacit multiplication means that
+<span class="docutils literal">1/2 x^2</span> skips the space and is treated like <span class="docutils literal"><span class="pre">1/(2*x^2)</span></span>.
+But then it gives zero!</p>
+<p>Thus one must use <span class="docutils literal">(1/2)x^2</span> or <span class="docutils literal">1/2*x^2</span> or
+<span class="docutils literal"><span class="pre">(1/2)*x^2</span></span> for disambiguation: <span class="docutils literal">x - 1/2*x^2 + <span class="pre">1/3*x^3...</span></span>. It is
+simpler to move the denominator to the right: <span class="docutils literal">x - x^2/2 + x^3/3 - ...</span>.</p>
+<p>It is worth noting that <span class="docutils literal"><span class="pre">1/2(x-1)(x-2)</span></span> suffers the same issue:
+<a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a>'s tacit multiplication always "ties more", hence this
+gets interpreted as <span class="docutils literal"><span class="pre">1/(2*(x-1)*(x-2))</span></span> which gives zero by
+polynomial division. Thus, use in such cases one of
+<span class="docutils literal"><span class="pre">(1/2)(x-1)(x-2)</span></span>, <span class="docutils literal"><span class="pre">1/2*(x-1)(x-2)</span></span> or <span class="docutils literal"><span class="pre">(x-1)(x-2)/2</span></span>.</p>
+</div>
+<div class="admonition warning">
+<p class="admonition-title">Warning</p>
+<p>The package does not currently know rational functions, but in order
+to leave open this as a future possibility, the usage of <span class="docutils literal">/</span> to stand
+for the
+euclidean quotient is <strong>deprecated</strong>.</p>
+<p>Please start using rather the <span class="docutils literal">quo()</span> function. It is possible
+that in a future major relase <span class="docutils literal">A/B</span> with <span class="docutils literal">B</span> a non-scalar will
+raise an error. Or, who knows, rational functions will be
+implemented sometime during the next decades, and then <span class="docutils literal">A/B</span> will
+naturally be the rational function.</p>
+</div>
+<div class="admonition important">
+<p class="admonition-title">Important</p>
+<p><span class="docutils literal">\poldef <span class="pre">P(x):=...;</span></span> defines <span class="docutils literal">P</span> both as a <em>function</em>,
+to be used as:</p>
+<pre class="literal-block">P(..numeric or even polynomial expression..)</pre>
+<p>and as a <em>variable</em> which can used inside polynomial expressions or
+as argument to some polynomial specific functions such as <span class="docutils literal">deg()</span>
+or <span class="docutils literal">polgcd()</span> <a class="footnote-reference brackets" href="#id3" id="id2">1</a>.</p>
+<dl class="footnote brackets">
+<dt class="label" id="id3"><span class="brackets"><a class="fn-backref" href="#id2">1</a></span></dt>
+<dd><p>Functional syntax accepts expressions as arguments; but the
+TeX <strong>macros</strong> described in the documentation, even the
+expandable ones, work only (there are a few exceptions to the
+general rule) with arguments being <em>names of declared
+polynomials</em>.</p>
+</dd>
+</dl>
+<p>One needs to have a clear understanding of the difference between
+<span class="docutils literal">P</span> used a function and <span class="docutils literal">P</span> used as a variable: if <span class="docutils literal">P</span> and
+<span class="docutils literal">Q</span> are both declared polynomials then:</p>
+<pre class="literal-block">(P+Q)(3)% <--- attention!</pre>
+<p>is currently evaluated as <span class="docutils literal"><span class="pre">(P+Q)*3</span></span>, because <span class="docutils literal">P+Q</span> is not known
+as a <em>function</em>, but <em>only as a variable of polynomial type</em>.
+Even worse:</p>
+<pre class="literal-block">(P)(3)% <--- attention!</pre>
+<p>will compute <span class="docutils literal">P*3</span>, because one can not in current <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> syntax
+enclose a function name in parentheses: consequently it is the variable
+which is used here. There is a <em>meager possibility</em> that in future
+some internal changes to <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> would let <span class="docutils literal"><span class="pre">(P)(3)</span></span> actually
+compute <span class="docutils literal">P(3)</span> and <span class="docutils literal"><span class="pre">(P+Q)(3)</span></span> compute <span class="docutils literal">P(3) + Q(3)</span>, but note
+that <span class="docutils literal"><span class="pre">(P)(P)</span></span> will then do <span class="docutils literal">P(P)</span> and not <span class="docutils literal">P*P</span>,
+the latter, current interpretation, looking more
+intuitive. Anyway, do not rely too extensively on tacit <span class="docutils literal">*</span> and use
+explicit <span class="docutils literal"><span class="pre">(P+Q)*(1+2)</span></span> if this is what is intended.</p>
+<p>As an alternative to explicit <span class="docutils literal"><span class="pre">P(3)+Q(3)</span></span> there is <span class="docutils literal">evalp(P+Q,3)</span>.</p>
+</div>
+<dl>
+<dt><span class="docutils literal"><span class="pre">\PolLet{g}={f}</span></span></dt>
+<dd><p>saves a copy of <span class="docutils literal">f</span> under name <span class="docutils literal">g</span>. Also usable without <span class="docutils literal">=</span>.</p>
+<p>Has exactly the same effect as <span class="docutils literal">\poldef <span class="pre">g(x):=f;</span></span> or <span class="docutils literal">\poldef <span class="pre">g(w):=f(w);</span></span>.</p>
+</dd>
+<dt><span class="docutils literal">\poldef <span class="pre">f(z):=</span> f^2;</span></dt>
+<dd><p>redefines <span class="docutils literal">f</span> in terms of itself. Prior to <span class="docutils literal">0.8</span> one needed
+the right hand side to be <span class="docutils literal"><span class="pre">f(z)^2</span></span>. Also, now <span class="docutils literal">sqr(f)</span> is
+possible (also <span class="docutils literal">sqr(f(x))</span> but not <span class="docutils literal"><span class="pre">sqr(f)(x)</span></span>).</p>
+</dd>
+</dl>
+<div class="admonition important">
+<p class="admonition-title">Important</p>
+<p>Note that <span class="docutils literal">f^2(z)</span> or <span class="docutils literal"><span class="pre">sqr(f)(z)</span></span> will give a logical but
+perhaps unexpected result: first <span class="docutils literal">f^2</span> is computed, then the
+opening parenthesis is seen which inserts a tacit multiplication
+<span class="docutils literal">*</span>, so in the end it is as if the input had been <span class="docutils literal">f^2 * z</span>.
+Although <span class="docutils literal">f</span> is both a variable and a function, <span class="docutils literal">f^2</span> is
+computed as a polynomial <em>variable</em> and ceases being a function.</p>
+</div>
+<dl>
+<dt><span class="docutils literal">\poldef <span class="pre">f(T):=</span> f(f);</span></dt>
+<dd><p>again modifies <span class="docutils literal">f</span>. Here it is used both as variable and as
+a function. Prior to <span class="docutils literal">0.8</span> it needed to be <span class="docutils literal">f(f(T))</span>.</p>
+</dd>
+<dt><span class="docutils literal">\poldef <span class="pre">k(z):=</span> <span class="pre">f-g(g^2)^2;</span></span></dt>
+<dd><p>if everybody followed, this should now define the zero polynomial...
+And <span class="docutils literal"><span class="pre">f-sqr(g(sqr(g)))</span></span> computes the same thing.</p>
+<p>We can check this in a typeset document like this:</p>
+<pre class="literal-block">\poldef f(x):= 1 - x + quo(x^5,1 - x + x^2);%
+\PolLet{g}={f}%
+\poldef f(z):= f^2;%
+\poldef f(T):= f(f);%
+\poldef k(w):= f-sqr(g(sqr(g)));%
+$$f(x) = \vcenter{\hsize10cm \PolTypeset{f}} $$
+$$g(z) = \PolTypeset{g} $$
+$$k(z) = \PolTypeset{k} $$
+\immediate\write128{f(x)=\PolToExpr{f}}% ah, here we see it also</pre>
+</dd>
+<dt><span class="docutils literal">\poldef <span class="pre">f'(x):=</span> diff1(f);</span></dt>
+<dd><p>(new at <span class="docutils literal">0.8</span>)</p>
+</dd>
+<dt><span class="docutils literal"><span class="pre">\PolDiff{f}{f'}</span></span></dt>
+<dd><p>Both set <span class="docutils literal">f'</span> (or any other chosen name) to the derivative
+of <span class="docutils literal">f</span>.</p>
+</dd>
+</dl>
+<div class="admonition important">
+<p class="admonition-title">Important</p>
+<p>This is not done automatically. If some new definition needs to use
+the derivative of some available polynomial, that derivative
+polynomial must have been previously defined: something such as
+<span class="docutils literal"><span class="pre">f'(3)^2</span></span> will not work without a prior definition of <span class="docutils literal">f'</span>.</p>
+<p>But one can now use <span class="docutils literal">diff1(f)</span> for on-the-spot construction with no
+permanent declaration, so here <span class="docutils literal"><span class="pre">evalp(diff1(f),3)^2</span></span>. And
+<span class="docutils literal"><span class="pre">diff1(f)^2</span></span> is same as <span class="docutils literal"><span class="pre">f'^2</span></span>, assuming here <span class="docutils literal">f'</span> was declared
+to be the derived polynomial.</p>
+<p>Notice that the name <span class="docutils literal">diff1()</span> is experimental and may change. Use
+<span class="docutils literal"><span class="pre">\PolDiff{f}{f'}</span></span> as the stable interface.</p>
+</div>
+<dl>
+<dt><span class="docutils literal">\PolTypeset{P}</span></dt>
+<dd><p>Typesets (switching to math mode if in text mode):</p>
+<pre class="literal-block">\poldef f(x):=(3+x)^5;%
+\PolDiff{f}{f'}\PolDiff{f'}{f''}\PolDiff{f''}{f'''}%
+$$f(z) = \PolTypeset[z]{f} $$
+$$f'(z) = \PolTypeset[z]{f'} $$
+$$f''(z) = \PolTypeset[z]{f''} $$
+$$f'''(z)= \PolTypeset[z]{f'''} $$</pre>
+<p>See <a class="reference internal" href="#poltypeset">the documentation</a> for the configurability
+via macros.</p>
+<p>Since <span class="docutils literal">0.8</span> <a class="reference internal" href="#poltypeset">\PolTypeset</a> accepts directly an
+expression, it does not have to be a pre-declared polynomial name:</p>
+<pre class="literal-block">\PolTypeset{mul(x-i,i=1..5)}</pre>
+</dd>
+<dt><span class="docutils literal">\PolToExpr{P}</span></dt>
+<dd><p>Expandably (contrarily to <a class="reference internal" href="#poltypeset">\PolTypeset</a>)
+produces <span class="docutils literal">c_n*x^n + ... + c_0</span> starting from the leading
+coefficient. The <span class="docutils literal">+</span> signs are omitted if followed by negative
+coefficients.</p>
+<p>This is useful for console or file output. This syntax is Maple and
+PSTricks <span class="docutils literal">\psplot[algebraic]</span> compatible; and also it is
+compatible with <span class="docutils literal">\poldef</span> input syntax, of course. See
+<a class="reference internal" href="#poltoexprcaret">\PolToExprCaret</a> for configuration of the <span class="docutils literal">^</span>, for example to
+use rather <span class="docutils literal">**</span> for Python syntax compliance.</p>
+<p>Changed at <span class="docutils literal">0.8</span>: the <span class="docutils literal">^</span> in output is by default of catcode 12
+so in a draft document one can use <span class="docutils literal">\PolToExpr{P}</span> inside the
+typesetting flow (without requiring math mode, where the <span class="docutils literal">*</span> would
+be funny and <span class="docutils literal">^12</span> would only put the <span class="docutils literal">1</span> as exponent anyhow;
+but arguably in text mode the <span class="docutils literal">+</span> and <span class="docutils literal">-</span> are not satisfactory
+for math, except sometimes in monospace typeface, and anyhow TeX is
+unable to break the expression across lines, barring special help).</p>
+<p>See <a class="reference internal" href="#poltoexpr-pol-expr">\PolToExpr{<pol. expr.>}</a> and related macros for customization.</p>
+<p>Extended at <span class="docutils literal">0.8</span> to accept as argument not only the name of a
+polynomial variable but more generally any polynomial expression.</p>
+</dd>
+</dl>
+</div>
+<div class="section" id="the-polexpr-0-8-extensions-to-the-xintexpr-syntax">
+<span id="polexpr08"></span><h1><a class="toc-backref" href="#id45">The polexpr <span class="docutils literal">0.8</span> extensions to the <span class="docutils literal">\xintexpr</span> syntax</a></h1>
+<p>All the syntax elements described in this section can be used in the
+<span class="docutils literal"><span class="pre">\xintexpr/\xinteval</span></span> context (where polynomials can be obtained from
+the <span class="docutils literal"><span class="pre">pol([])</span></span> constructor, once polexpr is loaded): their usage is
+not limited to only <span class="docutils literal">\poldef</span> context.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>If a variable <span class="docutils literal">myPol</span> defined via <span class="docutils literal">\xintdefvar</span> turns out
+to be a polynomial, the difference with those declared via <span class="docutils literal">\poldef</span>
+will be:</p>
+<ol class="arabic">
+<li><p><span class="docutils literal">myPol</span> is not usable as <em>function</em>, but only as a variable.
+Attention that <span class="docutils literal">f(x)</span> if <span class="docutils literal">f</span> is only a variable (even a
+polynomial one) will actually compute <span class="docutils literal">f * x</span>.</p></li>
+<li><p><span class="docutils literal">myPol</span> is not known to the polexpr package, hence for example the
+macros to achieve localization of its roots are unavailable.</p>
+<p>In a parallel universe I perhaps have implemented this expandably
+which means it could then be accessible with syntax such as
+<span class="docutils literal"><span class="pre">rightmostroot(pol([42,1,34,2,-8,1]))</span></span> but...</p>
+</li>
+</ol>
+</div>
+<div class="section" id="warning-about-unstability-of-the-new-syntax">
+<h2><a class="toc-backref" href="#id46">Warning about unstability of the new syntax</a></h2>
+<div class="admonition warning">
+<p class="admonition-title">Warning</p>
+<p>Consider the entirety of this section as <strong>UNSTABLE</strong> and
+<strong>EXPERIMENTAL</strong> (except perhaps regarding <span class="docutils literal">+</span>, <span class="docutils literal">-</span> and <span class="docutils literal">*</span>).</p>
+<p>And this applies even to items not explicitly flagged with one of
+<strong>unstable</strong>, <strong>Unstable</strong>, or <strong>UNSTABLE</strong> which only reflect that
+documentation was written over a period of time exceeding one minute,
+enough for the author mood changes to kick in.</p>
+<p>It is hard to find good names at the start of a life-long extension
+program of functionalities, and perhaps in future it will be
+preferred to rename everything or give to some functions other
+meanings. Such quasi-complete renamings happened already a few times
+during the week devoted to development.</p>
+</div>
+</div>
+<div class="section" id="infix-operators">
+<h2><a class="toc-backref" href="#id47">Infix operators <span class="docutils literal">+, <span class="pre">-,</span> *, /, **, ^</span></a></h2>
+<blockquote>
+<p>As has been explained in the <a class="reference internal" href="#quick-syntax-overview">Quick syntax overview</a> these infix
+operators have been made polynomial aware, not only in the
+<span class="docutils literal">\poldef</span> context, but generally in any <span class="docutils literal"><span class="pre">\xintexpr/\xinteval</span></span>
+context, inclusive of <span class="docutils literal">\xintdeffunc</span>.</p>
+<p>Conversely functions declared via <span class="docutils literal">\xintdeffunc</span> and making use of
+these operators will automatically be able to accept polynomials
+declared from <span class="docutils literal">\poldef</span> as variables.</p>
+<p>Usage of <span class="docutils literal">/</span> for euclidean division of polynomials is <strong>deprecated</strong>.
+Only in case of a scalar denominator is it to be considered stable.
+Please use rather <span class="docutils literal">quo()</span>.</p>
+</blockquote>
+<div class="admonition warning">
+<p class="admonition-title">Warning</p>
+<p>The <span class="docutils literal">pow(x,a)</span> function of <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> for <span class="docutils literal">x^a</span> with fractional
+<span class="docutils literal">a</span> will not (with current <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> <span class="docutils literal">1.4d</span>) accept a polynomial
+as first variable even if the second argument is an integer.</p>
+<p>It is possible (via <span class="docutils literal">\poormanloghack</span>) to instruct <span class="docutils literal">\xintexpr</span> to
+let <span class="docutils literal"><span class="pre">x**a</span></span> or <span class="docutils literal">x^a</span> be as <span class="docutils literal">pow(x,a)</span>. If this is done <span class="docutils literal">**</span>
+(resp. <span class="docutils literal">^</span>) will become unusable with polynomials (i.e. will create
+a low-level TeX error).</p>
+<p>And vice versa if polexpr gets loaded after the <span class="docutils literal">\poormanloghack</span>
+was used, <span class="docutils literal">**</span> and <span class="docutils literal">^</span> in <span class="docutils literal"><span class="pre">\xintexpr/\xinteval</span></span> will again only
+accept integer powers.</p>
+<p>Thus employ <span class="docutils literal">\poormanloghack</span> for at most one of <span class="docutils literal">**</span> or <span class="docutils literal">^</span>
+in order to keep one of them available for polynomials and integer
+powers.</p>
+</div>
+</div>
+<div class="section" id="experimental-infix-operators">
+<h2><a class="toc-backref" href="#id48">Experimental infix operators <span class="docutils literal">//, /:</span></a></h2>
+<blockquote>
+<p>Here is the tentative behaviour of <span class="docutils literal"><span class="pre">A//B</span></span> according to types:</p>
+<ul class="simple">
+<li><p><span class="docutils literal">A</span> non scalar and <span class="docutils literal">B</span> non scalar: euclidean quotient,</p></li>
+<li><p><span class="docutils literal">A</span> scalar and <span class="docutils literal">B</span> scalar: floored division,</p></li>
+<li><p><span class="docutils literal">A</span> scalar and <span class="docutils literal">B</span> non scalar: produces zero,</p></li>
+<li><p><span class="docutils literal">A</span> non scalar and <span class="docutils literal">B</span> scalar: coefficient per
+coefficient floored division.</p></li>
+</ul>
+<p>This is an <strong>experimental</strong> overloading of the <span class="docutils literal">//</span> and <span class="docutils literal">/:</span>
+from <span class="docutils literal">\xintexpr</span>.</p>
+<p>The behaviour in the last case, but not only, is to be considerd
+<strong>unstable</strong>. The alternative would be for <span class="docutils literal"><span class="pre">A//B</span></span> with <span class="docutils literal">B</span>
+scalar to act as <span class="docutils literal">quo(A,B)</span>. But, we have currently chosen to let
+<span class="docutils literal">//B</span> for a scalar <span class="docutils literal">B</span> act coefficient-wise on the numerator.
+Beware that it thus means it can be employed with the idea of doing
+euclidean division only by checking that <span class="docutils literal">B</span> is non-scalar.</p>
+<p>The <span class="docutils literal">/:</span> operator provides the associated remainder so always
+<span class="docutils literal">A</span> is reconstructed from <span class="docutils literal"><span class="pre">(A//B)*B</span> + <span class="pre">A/:B</span></span>.</p>
+<p>If <span class="docutils literal">:</span> is active character use <span class="docutils literal">/\string:</span> (it is safer to use
+<span class="docutils literal">/\string :</span> if it is not known if <span class="docutils literal">:</span> has catcode other, letter,
+or is active, but note that <span class="docutils literal">/:</span> is fine and needs no precaution if
+<span class="docutils literal">:</span> has catcode letter, it is only an active <span class="docutils literal">:</span> which is
+problematic, like for all other characters possibly used in an
+expression).</p>
+<blockquote>
+<p><strong>UNSTABLE</strong></p>
+<p>As explained above, there are (among other things) hesitations
+about behaviour with <span class="docutils literal">pol2</span> a scalar.</p>
+</blockquote>
+</blockquote>
+</div>
+<div class="section" id="comparison-operators">
+<h2><a class="toc-backref" href="#id49">Comparison operators <span class="docutils literal"><, >, <=, >=, ==, !=</span></a></h2>
+<blockquote>
+<p><strong>NOT YET IMPLEMENTED</strong></p>
+<p>As the internal representation by <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> and <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> of
+fractions does not currently require them to be in reduced terms,
+such operations would be a bit costly as they could not benefit from
+the <span class="docutils literal">\pdfstrcmp</span> engine primitive. In fact <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> does not use
+it yet anywhere, even for normalized pure integers, although it could
+speed up signifcantly certain aspects of core arithmetic.</p>
+<p>Equality of polynomials can currently be tested by computing the
+difference, which is a bit costly. And of course the <span class="docutils literal">deg()</span>
+function allows comparing degrees. In this context note the
+following syntax:</p>
+<pre class="literal-block">(deg(Q)) ?? { zero } { non-zero scalar } { non-scalar }</pre>
+<p>for branching.</p>
+</blockquote>
+</div>
+<div class="section" id="pol-nutple-expression">
+<h2><a class="toc-backref" href="#id50"><span class="docutils literal"><span class="pre">pol(<nutple</span> expression>)</span></a></h2>
+<blockquote>
+<p>This converts a nutple <span class="docutils literal"><span class="pre">[c0,c1,...,cN]</span></span> into the polynomial
+variable having these coefficients. Attention that the square
+brackets are <strong>mandatory</strong>, except of course if the argument is
+actually an expression producing such a "nutple".</p>
+<blockquote>
+<p>Currently, this process will not normalize the coefficients (such
+as reducing to lowest terms), it only trims out the leading zero
+coefficients.</p>
+</blockquote>
+<p>Inside <span class="docutils literal">\xintexpr</span>, this is the only (allowed) way to create ex
+nihilo a polynomial variable; inside <span class="docutils literal">\poldef</span> it is an alternative
+input syntax which is more efficient than typing <span class="docutils literal">c0 + c1 * x + c2 * x^2 + ...</span>.</p>
+</blockquote>
+<div class="admonition important">
+<p class="admonition-title">Important</p>
+<p>Whenever an expression with polynomials collapses to a constant, it
+becomes a scalar. There is currently no distinction during the
+parsing of expressions by <span class="docutils literal">\poldef</span>
+or <span class="docutils literal">\xintexpr</span> between constant polynomial variables and scalar
+variables.</p>
+<p>Naturally, <span class="docutils literal">\poldef</span> can be used to declare a constant polynomial
+<span class="docutils literal">P</span>, then <span class="docutils literal">P</span> can also be used as function having a value
+independent of argument, but as a variable, it is non-distinguishable
+from a scalar (of course functions such as <span class="docutils literal">deg()</span> tacitly
+consider scalars to be constant polynomials).</p>
+<p>Notice that we tend to use the vocable "variable" to refer to
+arbitrary expressions used as function arguments, without implying
+that we are actually referring to pre-declared variables in the sense
+of <span class="docutils literal">\xintdefvar</span>.</p>
+</div>
+</div>
+<div class="section" id="xinteval-pol-expr">
+<h2><a class="toc-backref" href="#id51"><span class="docutils literal"><span class="pre">\xinteval{<pol.</span> <span class="pre">expr.>}</span></span></a></h2>
+<blockquote>
+<p>This is documented here for lack of a better place: it evaluates the
+polynomial expression then outputs the "string" <span class="docutils literal"><span class="pre">pol([c0,</span> c1, <span class="pre">...,</span> cN])</span>
+if the degree <span class="docutils literal">N</span> is at least one (and the usual scalar output else).</p>
+<p>The "pol" word uses letter catcodes, which is actually mandatory for
+this output to be usable as input, but it does not make sense to use
+this inside <span class="docutils literal">\poldef</span> or <span class="docutils literal">\xintexpr</span> at it means basically
+executing <span class="docutils literal"><span class="pre">pol(coeffs(..expression..))</span></span> which is but a convoluted
+way to obtain the same result as <span class="docutils literal"><span class="pre">(..expression..)</span></span> (the
+parentheses delimiting the polynomial expression).</p>
+<p>For example, <span class="docutils literal"><span class="pre">\xinteval{(1+pol([0,1]))^10}</span></span> expands (in two steps)
+to:</p>
+<pre class="literal-block">pol([1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1])</pre>
+<p>You do need loading polexpr for this, else of course <span class="docutils literal"><span class="pre">pol([])</span></span>
+remains unknown to <span class="docutils literal">\xinteval{}</span> as well as the polynomial algebra !
+This example can also be done as
+<span class="docutils literal"><span class="pre">\xinteval{subs((1+x)^10,x=pol([0,1]))}</span></span>.</p>
+<p>I hesitated using as output the polynomial notation as produced by
+<a class="reference internal" href="#poltoexpr">\PolToExpr{}</a>, but finally opted for this.</p>
+</blockquote>
+</div>
+<div class="section" id="evalp-pol-expr-pol-expr">
+<h2><a class="toc-backref" href="#id52"><span class="docutils literal"><span class="pre">evalp(<pol.</span> <span class="pre">expr.>,</span> <pol. expr>)</span></a></h2>
+<blockquote>
+<p>Evaluates the first argument as a polynomial function of the
+second. Usually the second argument will be scalar, but this is not
+required:</p>
+<pre class="literal-block">\poldef K(x):= evalp(-3x^3-5x+1,-27x^4+5x-2);</pre>
+<p>If the first argument is an already declared polynomial <span class="docutils literal">P</span>, use
+rather the functional form <span class="docutils literal">P()</span> (which can accept a numerical as
+well as polynomial argument) as it is more efficient.</p>
+<p>One can also use <span class="docutils literal">subs()</span> syntax <a class="footnote-reference brackets" href="#id5" id="id4">2</a> (see <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> documentation):</p>
+<pre class="literal-block">\poldef K(x):= subs(-3y^3-5y+1, y = -27x^4+5x-2);</pre>
+<p>but the <span class="docutils literal">evalp()</span> will use a Horner evaluation scheme which is
+usually more efficient.</p>
+<dl class="footnote brackets">
+<dt class="label" id="id5"><span class="brackets"><a class="fn-backref" href="#id4">2</a></span></dt>
+<dd><p>by the way Maple uses the opposite, hence wrong, order
+<span class="docutils literal"><span class="pre">subs(x=...,</span> P)</span> but was written before computer science
+reached the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> heights. However it makes validating
+Maple results by polexpr sometimes cumbersome, but perhaps
+they will update it at some point.</p>
+</dd>
+</dl>
+<blockquote>
+<p><strong>name unstable</strong></p>
+<p><span class="docutils literal">poleval</span>? <span class="docutils literal">evalpol</span>? <span class="docutils literal">peval</span>? <span class="docutils literal">evalp</span>? <span class="docutils literal">value</span>?
+<span class="docutils literal">eval</span>? <span class="docutils literal">evalat</span>? <span class="docutils literal">eval1at2</span>? <span class="docutils literal">evalat2nd</span>?</p>
+<p>Life is so complicated when one asks questions. Not everybody does,
+though, as is amply demonstrated these days.</p>
+<p><strong>syntax unstable</strong></p>
+<p>I am hesitating about permuting the order of the arguments.</p>
+</blockquote>
+</blockquote>
+</div>
+<div class="section" id="deg-pol-expr">
+<h2><a class="toc-backref" href="#id53"><span class="docutils literal"><span class="pre">deg(<pol.</span> <span class="pre">expr.>)</span></span></a></h2>
+<blockquote>
+<p>Computes the degree.</p>
+</blockquote>
+<div class="admonition important">
+<p class="admonition-title">Important</p>
+<p>As <span class="docutils literal">\xintexpr</span> does not yet support infinities, the degree of
+the zero polynomial is <span class="docutils literal"><span class="pre">-1</span></span>. Beware that this breaks additivity
+of degrees, but <span class="docutils literal"><span class="pre">deg(P)<0</span></span> correctly detects the zero polynomial,
+and <span class="docutils literal"><span class="pre">deg(P)<=0</span></span> detects scalars.</p>
+</div>
+</div>
+<div class="section" id="coeffs-pol-expr">
+<h2><a class="toc-backref" href="#id54"><span class="docutils literal"><span class="pre">coeffs(<pol.</span> <span class="pre">expr.>)</span></span></a></h2>
+<blockquote>
+<p>Produces the nutple <span class="docutils literal"><span class="pre">[c0,c1,...,cN]</span></span> of coefficients. The highest
+degree coefficient is always non zero (except for the zero
+polynomial...).</p>
+<blockquote>
+<p><strong>name unstable</strong></p>
+<p>I am considering in particular using <span class="docutils literal">polcoeffs()</span> to avoid
+having to overload <span class="docutils literal">coeffs()</span> in future when matrix type
+will be added to <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a>.</p>
+</blockquote>
+</blockquote>
+</div>
+<div class="section" id="coeff-pol-expr-num-expr">
+<h2><a class="toc-backref" href="#id55"><span class="docutils literal"><span class="pre">coeff(<pol.</span> <span class="pre">expr.>,</span> <num. <span class="pre">expr.>)</span></span></a></h2>
+<blockquote>
+<p>As expected. Produces zero if <span class="docutils literal">n</span> is negative or higher than the
+degree.</p>
+<blockquote>
+<p><strong>name and syntax unstable</strong></p>
+<p>I am hesitating with <span class="docutils literal">coeff(n,pol)</span> syntax and also perhaps
+using <span class="docutils literal">polcoeff()</span> in order to avoid having to overload
+<span class="docutils literal">coeff()</span> when matrix type will be added to <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a>.</p>
+</blockquote>
+</blockquote>
+</div>
+<div class="section" id="lcoeff-pol-expr">
+<h2><a class="toc-backref" href="#id56"><span class="docutils literal"><span class="pre">lcoeff(<pol.</span> <span class="pre">expr.>)</span></span></a></h2>
+<blockquote>
+<p>The leading coefficient.</p>
+</blockquote>
+</div>
+<div class="section" id="monicpart-pol-expr">
+<h2><a class="toc-backref" href="#id57"><span class="docutils literal"><span class="pre">monicpart(<pol.</span> <span class="pre">expr.>)</span></span></a></h2>
+<blockquote>
+<p>Divides by the leading coefficient, except that <span class="docutils literal"><span class="pre">monicpart(0)==0</span></span>.</p>
+<blockquote>
+<p><strong>unstable</strong></p>
+<p>Currently the coefficients are reduced to lowest terms (contrarily
+to legacy behaviour of <a class="reference internal" href="#polmakemonic">\PolMakeMonic</a>), and
+additionally the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> <span class="docutils literal">\xintREZ</span> macro is applied which
+extracts powers of ten from numerator or denominator and stores
+them internally separately. This is generally beneficial to
+efficiency of multiplication.</p>
+</blockquote>
+</blockquote>
+</div>
+<div class="section" id="icontent-pol-expr">
+<h2><a class="toc-backref" href="#id58"><span class="docutils literal"><span class="pre">icontent(<pol.</span> <span class="pre">expr.>)</span></span></a></h2>
+<blockquote>
+<p>The gcd of the (possibly fractional) polynomial coefficients. It is
+always produced as an irreducible (non-negative) fraction. According
+to Gauss theorem the content of a product is the product of the
+contents.</p>
+<blockquote>
+<p><strong>name unstable</strong></p>
+<p>Some hesitation with using <span class="docutils literal">content()</span> rather.</p>
+</blockquote>
+</blockquote>
+</div>
+<div class="section" id="primpart-pol-expr">
+<h2><a class="toc-backref" href="#id59"><span class="docutils literal"><span class="pre">primpart(<pol.</span> <span class="pre">expr.>)</span></span></a></h2>
+<blockquote>
+<p>The quotient (except for the zero polynomial) by
+<span class="docutils literal"><span class="pre">icontent(<pol.</span> <span class="pre">expr.>)</span></span>. This is thus a polynomial with
+integer coefficients having <span class="docutils literal">1</span> as greatest common divisor. The
+sign of the leading coefficient is the same as in the original.</p>
+<p>And <span class="docutils literal"><span class="pre">primpart(0)==0</span></span>.</p>
+<p>The trailing zeros of the integer coefficients are extracted
+into a power of ten exponent part, in the internal representation.</p>
+</blockquote>
+</div>
+<div class="section" id="quorem-pol-expr-pol-expr">
+<h2><a class="toc-backref" href="#id60"><span class="docutils literal"><span class="pre">quorem(<pol.</span> <span class="pre">expr.>,</span> <pol. <span class="pre">expr.>)</span></span></a></h2>
+<blockquote>
+<p>Produces a nutple <span class="docutils literal">[Q,R]</span> with <span class="docutils literal">Q</span> the euclidean quotient and
+<span class="docutils literal">R</span> the remainder.</p>
+<blockquote>
+<p><strong>name unstable</strong></p>
+<p><span class="docutils literal">poldiv()</span>?</p>
+</blockquote>
+</blockquote>
+</div>
+<div class="section" id="quo-pol-expr-pol-expr">
+<h2><a class="toc-backref" href="#id61"><span class="docutils literal"><span class="pre">quo(<pol.</span> <span class="pre">expr.>,</span> <pol. <span class="pre">expr.>)</span></span></a></h2>
+<blockquote>
+<p>The euclidean quotient.</p>
+<p>The deprecated <span class="docutils literal">pol1/pol2</span> syntax computes the same polynomial.</p>
+</blockquote>
+</div>
+<div class="section" id="rem-pol-expr-pol-expr">
+<h2><a class="toc-backref" href="#id62"><span class="docutils literal"><span class="pre">rem(<pol.</span> <span class="pre">expr.>,</span> <pol. <span class="pre">expr.>)</span></span></a></h2>
+<blockquote>
+<p>The euclidean remainder. If <span class="docutils literal">pol2</span> is a (non-zero) scalar, this is
+zero.</p>
+<p>There is no infix operator associated to this, for lack of evident
+notation. Please advise.</p>
+<p><span class="docutils literal">/:</span> can be used if one is certain that <span class="docutils literal">pol2</span> is of
+degree at least one. But read the warning about it being unstable
+even in that case.</p>
+</blockquote>
+</div>
+<div class="section" id="prem-pol-expr-1-pol-expr-2">
+<span id="prem"></span><h2><a class="toc-backref" href="#id63"><span class="docutils literal"><span class="pre">prem(<pol.</span> expr. 1>, <pol. expr. 2>)</span></a></h2>
+<blockquote>
+<p>Produces a nutple <span class="docutils literal">[m, spR]</span> where <span class="docutils literal">spR</span> is the (special) pseudo
+Euclidean remainder. Its description is:</p>
+<ul>
+<li><p>the standard euclidean remainder <span class="docutils literal">R</span> is <span class="docutils literal">spR/m</span></p></li>
+<li><p><span class="docutils literal">m = b^f</span> with <span class="docutils literal">b</span> equal to the <strong>absolute value</strong> of the
+leading coefficient of <span class="docutils literal">pol2</span>,</p></li>
+<li><p><span class="docutils literal">f</span> is the number of non-zero coefficients in the euclidean
+quotient, if <span class="docutils literal"><span class="pre">deg(pol2)>0</span></span> (even if the remainder vanishes).</p>
+<p>If <span class="docutils literal">pol2</span> is a scalar however, the function outputs <span class="docutils literal">[1,0]</span>.</p>
+</li>
+</ul>
+<p>With these definitions one can show that if both <span class="docutils literal">pol1</span> and
+<span class="docutils literal">pol2</span> have integer coefficients, then this is also the case of
+<span class="docutils literal">spR</span>, which makes its interest (and also <span class="docutils literal">m*Q</span> has integer
+coefficients, with <span class="docutils literal">Q</span> the euclidean quotient, if <span class="docutils literal"><span class="pre">deg(pol2)>0</span></span>).
+Also, <span class="docutils literal">prem()</span> is computed faster than <span class="docutils literal">rem()</span> for such integer
+coefficients polynomials.</p>
+<div class="admonition hint">
+<p class="admonition-title">Hint</p>
+<p>If you want the euclidean quotient <span class="docutils literal">R</span> evaluated via <span class="docutils literal">spR/m</span>
+(which may be faster, even with non integer coefficients) use
+<span class="docutils literal"><span class="pre">subs(last(x)/first(x),x=prem(P,Q))</span></span> syntax as it avoids
+computing <span class="docutils literal">prem(P,Q)</span> twice. This does the trick both in
+<span class="docutils literal">\poldef</span> or in <span class="docutils literal">\xintdefvar</span>.</p>
+<p>However, as is explained in the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> documentation, using
+such syntax in an <span class="docutils literal">\xintdeffunc</span> is (a.t.t.o.w) illusory, due to
+technicalities of how <span class="docutils literal">subs()</span> gets converted into nested
+expandable macros. One needs an auxiliary function like this:</p>
+<pre class="literal-block">\xintdeffunc lastoverfirst(x):=last(x)/first(x);
+\xintdeffunc myR(x)=lastoverfirst(prem(x));</pre>
+<p>Then, <span class="docutils literal">myR(pol1,pol2)</span> will evaluate <span class="docutils literal">prem(pol1,pol2)</span> only
+once and compute a polynomial identical to the euclidean
+remainder (internal representations of coefficients may differ).</p>
+</div>
+<p>In this case of integer coefficients polynomials, the polexpr
+internal representation of the integer coefficients in the pseudo
+remainder will be with unit denominators only if that was already the
+case for those of <span class="docutils literal">pol1</span> and <span class="docutils literal">pol2</span> (no automatic reduction to
+lowest terms is made prior or after computation).</p>
+<p>Pay attention here that <span class="docutils literal">b</span> is the <strong>absolute value</strong> of the
+leading coefficient of <span class="docutils literal">pol2</span>. Thus the coefficients of the
+pseudo-remainder have the same signs as those of the standard
+remainder. This diverges from Maple's function with the same name.</p>
+</blockquote>
+</div>
+<div class="section" id="divmod-pol-expr-1-pol-expr-2">
+<h2><a class="toc-backref" href="#id64"><span class="docutils literal"><span class="pre">divmod(<pol.</span> expr. 1>, <pol. expr. 2>)</span></a></h2>
+<blockquote>
+<p>Overloads the scalar <span class="docutils literal">divmod()</span> and associates it with the
+experimental <span class="docutils literal">//</span> and <span class="docutils literal">/:</span> as extended to the polynomial type.</p>
+<p>In particular when both <span class="docutils literal">pol1</span> and <span class="docutils literal">pol2</span> are scalars, this is
+the usual <span class="docutils literal">divmod()</span> (as in Python) and for <span class="docutils literal">pol1</span> and <span class="docutils literal">pol2</span>
+non constant polynomials, this is the same as <span class="docutils literal">quorem()</span>.</p>
+<blockquote>
+<p><strong>Highly unstable</strong> overloading of <span class="docutils literal">\xinteval</span>'s <span class="docutils literal">divmod()</span>.</p>
+</blockquote>
+</blockquote>
+</div>
+<div class="section" id="mod-pol-expr-1-pol-expr-2">
+<h2><a class="toc-backref" href="#id65"><span class="docutils literal"><span class="pre">mod(<pol.</span> expr. 1>, <pol. expr. 2>)</span></a></h2>
+<blockquote>
+<p>The <span class="docutils literal">R</span> of the <span class="docutils literal">divmod()</span> output. Same as <span class="docutils literal">R</span> of <span class="docutils literal">quorem()</span>
+when the second argument <span class="docutils literal">pol2</span> is of degree at least one.</p>
+<blockquote>
+<p><strong>Highly unstable</strong> overloading of <span class="docutils literal">\xinteval</span>'s <span class="docutils literal">mod()</span>.</p>
+</blockquote>
+</blockquote>
+</div>
+<div class="section" id="polgcd-pol-expr-1-pol-expr-2">
+<h2><a class="toc-backref" href="#id66"><span class="docutils literal"><span class="pre">polgcd(<pol.</span> expr. 1>, <pol. expr. 2>, <span class="pre">...)</span></span></a></h2>
+<blockquote>
+<p>Evaluates to the greatest common polynomial divisor of all the
+polynomial inputs. The output is a <strong>primitive</strong> (in particular,
+with integer coefficients) polynomial. It is zero if and only if all
+inputs vanish.</p>
+<p>Attention, there must be either at least two polynomial variables, or
+alternatively, only one argument which then must be a bracketed list
+or some expression or variable evaluating to such a "nutple" whose
+items are polynomials (see the documentation of the scalar <span class="docutils literal">gcd()</span>
+in <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a>).</p>
+<blockquote>
+<p>The two variable case could (and was, during development) have been
+defined at user level like this:</p>
+<pre class="literal-block">\xintdeffunc polgcd_(P,Q):=
+ (deg(Q))??{P}{1}{polgcd_(Q,primpart(last(prem(P,Q))))};
+\xintdeffunc polgcd(P,Q):=polgcd_(primpart(P),primpart(Q));%</pre>
+<p>This is basically what is done internally for two polynomials, up
+to some internal optimizations.</p>
+</blockquote>
+<p><strong>UNSTABLE</strong></p>
+<p>I hesitate between returning a <em>primitive</em> or a <em>monic</em> polynomial.
+Maple returns a primitive polynomial if all inputs <a class="footnote-reference brackets" href="#id7" id="id6">3</a> have integer
+coefficients, else it returns a monic polynomial, but this is
+complicated technically for us to add such a check and would add
+serious overhead.</p>
+<p>Internally, computations are done using primitive
+integer-coefficients polynomials (as can be seen in the function
+template above). So I decided finally to output a primitive
+polynomial, as one can always apply <span class="docutils literal">monicpart()</span> to it.</p>
+<p>Attention that this is at odds with behaviour of the legacy
+<a class="reference internal" href="#polgcd">\PolGCD</a> (non expandable) macro.</p>
+<dl class="footnote brackets">
+<dt class="label" id="id7"><span class="brackets"><a class="fn-backref" href="#id6">3</a></span></dt>
+<dd><p>actually, only two polynomial arguments are allowed by Maple's
+<span class="docutils literal">gcd()</span> as far as I know.</p>
+</dd>
+</dl>
+</blockquote>
+</div>
+<div class="section" id="resultant-pol-expr-1-pol-expr-2">
+<h2><a class="toc-backref" href="#id67"><span class="docutils literal"><span class="pre">resultant(<pol.</span> expr. 1>, <pol. expr. 2>)</span></a></h2>
+<blockquote>
+<p>The resultant.</p>
+<blockquote>
+<p><strong>NOT YET IMPLEMENTED</strong></p>
+</blockquote>
+</blockquote>
+</div>
+<div class="section" id="disc-pol-expr">
+<h2><a class="toc-backref" href="#id68"><span class="docutils literal"><span class="pre">disc(<pol.</span> <span class="pre">expr.>)</span></span></a></h2>
+<blockquote>
+<p>The discriminant.</p>
+<blockquote>
+<p><strong>NOT YET IMPLEMENTED</strong></p>
+</blockquote>
+</blockquote>
+</div>
+<div class="section" id="polpowmod-pol-expr-1-num-expr-pol-expr-2">
+<h2><a class="toc-backref" href="#id69"><span class="docutils literal"><span class="pre">polpowmod(<pol.</span> expr. 1>, <num. <span class="pre">expr.>,</span> <pol. expr. 2>)</span></a></h2>
+<blockquote>
+<p>Modular exponentiation: <span class="docutils literal">mod(pol1^N, pol2)</span> in a more efficient
+manner than first computing <span class="docutils literal">pol1^N</span> then reducing modulo <span class="docutils literal">pol2</span>.</p>
+<p>Attention that this is using the <span class="docutils literal">mod()</span> operation, whose current
+experimental status is as follows:</p>
+<ul class="simple">
+<li><p>if <span class="docutils literal"><span class="pre">deg(pol2)>0</span></span>, the euclidean remainder operation,</p></li>
+<li><p>if <span class="docutils literal">pol2</span> is a scalar, coefficient-wise reduction modulo <span class="docutils literal">pol2</span>.</p></li>
+</ul>
+<p><strong>UNSTABLE</strong></p>
+<blockquote>
+<p>This is currently implemented at high level via <span class="docutils literal">\xintdeffunc</span> and
+recursive definitions, which were copied over from a scalar example
+in the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> manual:</p>
+<pre class="literal-block">\xintdeffunc polpowmod_(P, m, Q) :=
+ isone(m)?
+ % m=1: return P modulo Q
+ { mod(P,Q) }
+ % m > 1: test if odd or even and do recursive call
+ { odd(m)? { mod(P*sqr(polpowmod_(P, m//2, Q)), Q) }
+ { mod( sqr(polpowmod_(P, m//2, Q)), Q) }
+ }
+ ;%
+\xintdeffunc polpowmod(P, m, Q) := (m)?{polpowmod_(P, m, Q)}{1};%</pre>
+<p>Negative exponents are not currently implemented.</p>
+<p>For example:</p>
+<pre class="literal-block">\xinteval{subs(polpowmod(1+x,100,x^7),x=pol([0,1]))}
+\xinteval{subs(polpowmod(1+x,20,10), x=pol([0,1]))}</pre>
+<p>produce respectively:</p>
+<pre class="literal-block">pol([1, 100, 4950, 161700, 3921225, 75287520, 1192052400])
+pol([1, 0, 0, 0, 5, 4, 0, 0, 0, 0, 6, 0, 0, 0, 0, 4, 5, 0, 0, 0, 1])</pre>
+</blockquote>
+</blockquote>
+</div>
+<div class="section" id="rdcoeffs-pol-expr">
+<h2><a class="toc-backref" href="#id70"><span class="docutils literal"><span class="pre">rdcoeffs(<pol.</span> <span class="pre">expr.>)</span></span></a></h2>
+<blockquote>
+<p>This operates on the internal representation of the coefficients,
+reducing them to lowest terms.</p>
+<blockquote>
+<p><strong>name HIGHLY undecided</strong></p>
+</blockquote>
+</blockquote>
+</div>
+<div class="section" id="rdzcoeffs-pol-expr">
+<h2><a class="toc-backref" href="#id71"><span class="docutils literal"><span class="pre">rdzcoeffs(<pol.</span> <span class="pre">expr.>)</span></span></a></h2>
+<blockquote>
+<p>This operates on the internal representation of the coefficients,
+reducing them to lowest terms then extracting from numerator
+or denominator the maximal power of ten to store as a decimal
+exponent.</p>
+<p>This is sometimes favourable to more efficient polynomial algebra
+computations.</p>
+<blockquote>
+<p><strong>name HIGHLY undecided</strong></p>
+</blockquote>
+</blockquote>
+</div>
+<div class="section" id="diff1-pol-expr">
+<h2><a class="toc-backref" href="#id72"><span class="docutils literal"><span class="pre">diff1(<pol.</span> <span class="pre">expr.>)</span></span></a></h2>
+<blockquote>
+<p>The first derivative.</p>
+<blockquote>
+<p><strong>name UNSTABLE</strong></p>
+<p>This name may be used in future to be the partial derivative with
+respect to a first variable.</p>
+</blockquote>
+</blockquote>
+</div>
+<div class="section" id="diff2-pol-expr">
+<h2><a class="toc-backref" href="#id73"><span class="docutils literal"><span class="pre">diff2(<pol.</span> <span class="pre">expr.>)</span></span></a></h2>
+<blockquote>
+<p>The second derivative.</p>
+<blockquote>
+<p><strong>name UNSTABLE</strong></p>
+<p>This name may be used in future to be the partial derivative with
+respect to a second variable.</p>
+</blockquote>
+</blockquote>
+</div>
+<div class="section" id="diffn-pol-expr-p-num-expr-n">
+<h2><a class="toc-backref" href="#id74"><span class="docutils literal"><span class="pre">diffn(<pol.</span> expr. P>, <num. expr. n>)</span></a></h2>
+<blockquote>
+<p>The <span class="docutils literal">n</span>th derivative of <span class="docutils literal">P</span>. For <span class="docutils literal">n<0</span> computes iterated primitives
+vanishing at the origin.</p>
+<p>The coefficients are not reduced to lowest terms.</p>
+<blockquote>
+<p><strong>name and syntax UNSTABLE</strong></p>
+<p>I am also considering reversing the order of the arguments.</p>
+</blockquote>
+</blockquote>
+</div>
+<div class="section" id="antider-pol-expr-p">
+<h2><a class="toc-backref" href="#id75"><span class="docutils literal"><span class="pre">antider(<pol.</span> expr. P>)</span></a></h2>
+<blockquote>
+<p>The primitive of <span class="docutils literal">P</span> with no constant term. Same as <span class="docutils literal"><span class="pre">diffn(P,-1)</span></span>.</p>
+</blockquote>
+</div>
+<div class="section" id="intfrom-pol-expr-p-pol-expr-c">
+<h2><a class="toc-backref" href="#id76"><span class="docutils literal"><span class="pre">intfrom(<pol.</span> expr. P>, <pol. expr. c>)</span></a></h2>
+<blockquote>
+<p>The primitive of <span class="docutils literal">P</span> vanishing at <span class="docutils literal">c</span>, i.e. <span class="docutils literal">\int_c^x P(t)dt</span>.</p>
+<p>Also <span class="docutils literal">c</span> can be a polynomial... so if <span class="docutils literal">c</span> is monomial <span class="docutils literal">x</span>
+this will give zero!</p>
+<blockquote>
+<p><strong>UNSTABLE</strong></p>
+<p>Allowing general polynomial variable for <span class="docutils literal">c</span> adds a bit of
+overhead to the case of a pure scalar. So I am hesitating
+maintaining this feature whose interest appears dubious.</p>
+</blockquote>
+</blockquote>
+</div>
+<div class="section" id="integral-pol-expr-p-pol-expr-a-pol-expr-b">
+<h2><a class="toc-backref" href="#id77"><span class="docutils literal"><span class="pre">integral(<pol.</span> expr. P>, [<pol. expr. a>, <pol. expr. <span class="pre">b>])</span></span></a></h2>
+<blockquote>
+<p><span class="docutils literal">\int_a^b P(t)dt</span>.</p>
+<p>The brackets here are not denoting an optional argument
+but a <em>mandatory</em> nutple argument <span class="docutils literal">[a, b]</span> with <em>two items</em>.</p>
+<p><span class="docutils literal">a</span> and <span class="docutils literal">b</span> are not restricted to be scalars, they can be
+polynomials.</p>
+<blockquote>
+<p>To compute <span class="docutils literal"><span class="pre">\int_{x-1}^x</span> P(t)dt</span> it is more efficient to use
+<span class="docutils literal"><span class="pre">intfrom(x-1)</span></span>.</p>
+<p>Similary to compute <span class="docutils literal"><span class="pre">\int_x^{x+1}</span> P(t)dt</span>, use <span class="docutils literal"><span class="pre">-intfrom(x+1)</span></span>.</p>
+<p><strong>UNSTABLE</strong></p>
+<p>Am I right to allow general polynomials <span class="docutils literal">a</span> and <span class="docutils literal">b</span> hence add
+overhead to the pure scalar case ?</p>
+</blockquote>
+</blockquote>
+</div>
+</div>
+<div class="section" id="examples-of-localization-of-roots">
+<h1><a class="toc-backref" href="#id78">Examples of localization of roots</a></h1>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>As of <span class="docutils literal">0.8</span>, <span class="docutils literal">polexpr</span> is usable with Plain TeX and not only with
+LaTeX, the examples of this section have been converted to use a
+syntax which (at least at time of writing, March 2021) works in both.</p>
+<p>This is done in order for the examples to be easy to copy-paste to
+documents using either macro format.</p>
+</div>
+<ul>
+<li><p>To make printed decimal numbers more enjoyable than via
+<span class="docutils literal">\xintSignedFrac</span> (or <span class="docutils literal">\xintSignedFwOver</span> with Plain):</p>
+<pre class="literal-block">\def\PolTypesetOne#1{\PolDecToString{\xintREZ{#1}}}%</pre>
+<p><span class="docutils literal">\PolDecToString</span> will use decimal notation to incorporate the power
+of ten part; and the <span class="docutils literal">\xintREZ</span> 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">\def\PolToExprCmd#1{\PolDecToString{\xintREZ{#1}}}%</pre>
+<p>to modify output of <a class="reference internal" href="#poltoexpr-pol-expr">\PolToExpr{<pol. expr.>}</a>.</p>
+</li>
+<li><p>For extra info in log file use <span class="docutils literal">\xintverbosetrue</span>.</p></li>
+</ul>
+<div class="section" id="a-typical-example">
+<h2><a class="toc-backref" href="#id79">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}
+$$\PolSturmIsolatedZeroLeft{f}{2}<Z_2<\PolSturmIsolatedZeroRight{f}{2}$$
+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 $X_2$!</pre>
+</div>
+<div class="section" id="a-degree-four-polynomial-with-nearby-roots">
+<h2><a class="toc-backref" href="#id80">A degree four polynomial with nearby roots</a></h2>
+<p>Notice that this example is a bit outdated as <span class="docutils literal">0.7</span> release has
+added <span class="docutils literal"><span class="pre">\PolSturmIsolateZeros**{sturmname}</span></span> 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 < 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</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="#id81">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)
+\def\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}
+\def\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}</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 <span class="docutils literal">\PolSturmIsolateZeros**</span> 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 < 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</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 < 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</pre>
+</div>
+<div class="section" id="a-degree-five-polynomial-with-three-rational-roots">
+<h2><a class="toc-backref" href="#id82">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}
+ \def\PolTypesetCmdPrefix#1{\allowbreak\xintiiifSgn{#1}{}{+}{+}}%
+ $Q_0(x) = \PolTypeset{Q_0}$
+\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
+\def\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="#id83">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 < 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
+$$\PolSturmIsolatedZeroLeft{P}{2}<Z_2<\PolSturmIsolatedZeroRight{P}{2}$$</pre>
+<p>The last line produces:</p>
+<pre class="literal-block">0.09999900004999650028 < Z_2 < 0.09999900004999650029</pre>
+</div>
+<div class="section" id="the-wilkinson-polynomial">
+<h2><a class="toc-backref" href="#id84">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">%\xintverbosetrue % for the curious...
+
+\poldef f(x) := mul((x - i), i = 1..20);
+
+\def\PolTypesetCmdPrefix#1{\allowbreak\xintiiifSgn{#1}{}{+}{+}}%
+\def\PolTypesetOne#1{\xintDecToString{#1}}%
+
+\noindent\PolTypeset{f}
+
+\PolToSturm{f}{f}
+\PolSturmIsolateZeros{f}
+\PolPrintIntervals{f}
+
+% \vfill\eject
+
+% This page is commented out because it takes about 30s on a 2GHz CPU
+% \poldef g(x) := f(x) - 2**{-23} x**19;
+
+% \PolToSturm{g}{g}
+% \noindent\PolTypeset{g_0}% integer coefficient primitive polynomial
+
+% \PolSturmIsolateZeros{g}
+% \PolEnsureIntervalLengths{g}{-10}
+
+% \let\PolPrintIntervalsPrintMultiplicity\empty
+% \PolPrintIntervals*{g}</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, but the modified one <span class="docutils literal">f(x) - <span class="pre">2**-23</span> <span class="pre">x**19</span></span> takes about 20x longer.</p>
+<p>The Sturm chain polynomials
+have integer coefficients with up to 321 digits, whereas (surprisingly
+perhaps) those of the Sturm chain polynomials derived from <span class="docutils literal">f</span> 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
+<span class="docutils literal">f(x) - <span class="pre">2**-23</span> <span class="pre">x**19</span></span> 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="#id85">The second Wilkinson polynomial</a></h2>
+<pre class="literal-block">\poldef f(x) := mul(x - 2^-i, i = 1..20);
+
+%\PolTypeset{f}
+
+\PolToSturm{f}{f}
+\PolSturmIsolateZeros**{f}
+\PolPrintIntervals{f}</pre>
+<p>This takes more time than the polynomial with 1, 2, .., 20 as roots but
+less than the latter modified by the <span class="docutils literal"><span class="pre">2**-23</span></span> tiny change to one of its
+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 <span class="docutils literal">\xintIrr</span> 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="#id86">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 <span class="docutils literal">i/10</span> but this gives
+less efficient internal form for the coefficients (the <span class="docutils literal">10</span>'s end up
+in denominators).</p>
+<p>Using <span class="docutils literal">\PolToExpr{P}</span> after having done</p>
+<pre class="literal-block">\def\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 <span class="docutils literal">xint + polexpr</span>, 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="admonition-title">Note</p>
+<p>Release <span class="docutils literal">0.5</span> has <em>experimental</em> addition of optional argument
+<span class="docutils literal">E</span> to <span class="docutils literal">\PolSturmIsolateZeros</span>. It instructs to search roots only
+in interval <span class="docutils literal"><span class="pre">(-10^E,</span> 10^E)</span>. Important: the extremities are
+<em>assumed to not be roots</em>. In this example, the <span class="docutils literal">[1]</span> in
+<span class="docutils literal"><span class="pre">\PolSturmIsolateZeros[1]{S}</span></span> gives some speed gain; without it, it
+turns out in this case that <span class="docutils literal">polexpr</span> would have started with
+<span class="docutils literal"><span class="pre">(-10^6,</span> 10^6)</span> interval.</p>
+<p>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="#id87">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<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="#id88">Non-expandable macros</a></h1>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>At <span class="docutils literal">0.8</span> <span class="docutils literal">polexpr</span> is usable with Plain TeX and not only with
+LaTeX. Some examples given in this section may be using LaTeX syntax
+such as <span class="docutils literal">\renewcommand</span>. Convert to TeX primitives as appropriate
+if testing with a non LaTeX macro format.</p>
+</div>
+<div class="section" id="poldef-polname-letter-expression-using-the-letter-as-indeterminate">
+<span id="poldef"></span><h2><a class="toc-backref" href="#id89"><span class="docutils literal">\poldef <span class="pre">polname(letter):=</span> expression using the letter as indeterminate;</span></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, when used with argument the chosen <span class="docutils literal">polname</span>. Of
+course the <em>expression</em> can use other previously defined
+polynomials.</p>
+<p>Polynomial names must start with a letter and are constituted of
+letters, digits, underscores and the right tick <span class="docutils literal">'</span>.</p>
+<p>The whole <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> syntax is authorized:</p>
+<pre class="literal-block">\poldef mypol(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>Furthermore:</p>
+<ul class="simple">
+<li><p>a variable <span class="docutils literal">mypol</span> is defined which can be used in <span class="docutils literal">\poldef</span>
+as well as in <span class="docutils literal">\xinteval</span> for algebraic computations or as
+argument to polynomial aware functions,</p></li>
+<li><p>a function <span class="docutils literal">mypol()</span> is defined which can be used in <span class="docutils literal">\poldef</span>
+as well as in <span class="docutils literal">\xinteval</span>. It accepts there as argument scalars
+and also other polynomials (via their names, thanks to previous
+item).</p></li>
+</ul>
+<p>Notice that any function defined via <span class="docutils literal">\xintdeffunc</span> and using
+only algebraic operations (and ople indexing or slicing operations)
+should work fine in <span class="docutils literal"><span class="pre">\xintexpr/\xinteval</span></span> with such polynomial
+names as argument.</p>
+<p>In the case of a constant polynomial, the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> variable (not the
+internal data structure on which the package macros operate)
+associated to it is indistinguishable from a scalar, it is actually
+a scalar and has lost all traces from its origins as a polynomial
+(so for example can be used as argument to the <span class="docutils literal">cos()</span> function).
+<strong>THIS MAY CHANGE</strong></p>
+<p>The <em>function</em> on the other hand remains a one-argument function,
+which simply has a constant value.</p>
+<div class="admonition attention">
+<p class="admonition-title">Attention!</p>
+<p>The function <span class="docutils literal">mypol()</span> is defined <strong>only</strong> for
+<span class="docutils literal"><span class="pre">\xintexpr/\xinteval</span></span>
+context. It will be unknown to <span class="docutils literal">\xintfloateval</span>.</p>
+<p>Worse, a
+previously existing floating point function 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 <span class="docutils literal">\poldef</span> or <span class="docutils literal">\PolDef</span> but
+as result of usage of the other package macros.</p>
+<p>See <a class="reference internal" href="#polgenfloatvariant-polname">\PolGenFloatVariant{polname}</a> to generate a <strong>function</strong>
+usable in <span class="docutils literal">\xintfloateval</span>. Such a function can only be
+used with scalar input, see next warning.</p>
+</div>
+<div class="admonition attention">
+<p class="admonition-title">Attention!</p>
+<p>Using the <strong>variable</strong> <span class="docutils literal">mypol</span> inside <span class="docutils literal">\xintfloateval</span> will
+generate low-level errors because the infix operators there are
+not polynomial-aware, and the polynomial specific functions such
+as <span class="docutils literal">deg()</span> are only defined for usage inside <span class="docutils literal">\xintexpr</span>.</p>
+<p>In short, currently polynomials defined via <span class="docutils literal">polexpr</span> can
+be used in floating point context only for numerical evaluations,
+via <strong>functions</strong> obtained from <a class="reference internal" href="#polgenfloatvariant-polname">\PolGenFloatVariant{polname}</a>
+usage.</p>
+<p>Changes to the original polynomial via package macros are not
+automatically mapped to the numerical floating point evaluator
+which must be manually updated as necessary when the original
+rational coefficient polynomial is modified.</p>
+<p><strong>THIS MAY CHANGE</strong></p>
+</div>
+<p>The original expression is lost after parsing, and in particular the
+package provides no way to typeset it (of course the package
+provides macros to typeset the computed polynomial). Typesetting
+the original expression has to be done manually, if needed.</p>
+</blockquote>
+</div>
+<div class="section" id="poldef-letter-polname-expression-using-the-letter-as-indeterminate">
+<span id="id8"></span><h2><a class="toc-backref" href="#id90"><span class="docutils literal"><span class="pre">\PolDef[letter]{polname}{expression</span> using the letter as indeterminate}</span></a></h2>
+<blockquote>
+<p>Does the same as <a class="reference internal" 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
+<span class="docutils literal">[letter]</span> optional argument, the variable is assumed to be <span class="docutils literal">x</span>.</p>
+</blockquote>
+</div>
+<div class="section" id="polgenfloatvariant-polname">
+<span id="polgenfloatvariant"></span><h2><a class="toc-backref" href="#id91"><span class="docutils literal">\PolGenFloatVariant{polname}</span></a></h2>
+<blockquote>
+<p>Makes the polynomial also usable in the <span class="docutils literal">\xintfloatexpr</span> 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-pol-expr">\PolToFloatExpr{<pol. expr.>}</a>.</p>
+<div class="admonition attention">
+<p class="admonition-title">Attention!</p>
+<p>Any operation, for example generating the derivative polynomial,
+or dividing two polynomials or using the <span class="docutils literal">\PolLet</span>, <strong>must</strong> be
+followed by explicit usage of <span class="docutils literal">\PolGenFloatVariant{polname}</span> if
+the new polynomial is to be used in <span class="docutils literal">\xintfloateval</span> <strong>as a
+function</strong>.</p>
+</div>
+</blockquote>
+</div>
+<div class="section" id="pollet-polname-2-polname-1">
+<span id="pollet"></span><h2><a class="toc-backref" href="#id92"><span class="docutils literal"><span class="pre">\PolLet{polname_2}={polname_1}</span></span></a></h2>
+<blockquote>
+<p>Makes a copy of the already defined polynomial <span class="docutils literal">polname_1</span> to a
+new one <span class="docutils literal">polname_2</span>. Same effect as
+<span class="docutils literal"><span class="pre">\PolDef{polname_2}{polname_1(x)}</span></span> but with less overhead. The
+<span class="docutils literal">=</span> is optional.</p>
+</blockquote>
+</div>
+<div class="section" id="polgloballet-polname-2-polname-1">
+<span id="polgloballet"></span><h2><a class="toc-backref" href="#id93"><span class="docutils literal"><span class="pre">\PolGlobalLet{polname_2}={polname_1}</span></span></a></h2>
+<blockquote>
+<p>Acts globally.</p>
+</blockquote>
+</div>
+<div class="section" id="polassign-polname-toarray-macro">
+<span id="polassign"></span><h2><a class="toc-backref" href="#id94"><span class="docutils literal"><span class="pre">\PolAssign{polname}\toarray\macro</span></span></a></h2>
+<blockquote>
+<p>Defines a one-argument expandable macro <span class="docutils literal"><span class="pre">\macro{#1}</span></span> which expands
+to the (raw) #1th polynomial coefficient.</p>
+<ul class="simple">
+<li><p>Attention, coefficients here are indexed starting at 1.</p></li>
+<li><p>With #1=-1, -2, ..., <span class="docutils literal"><span class="pre">\macro{#1}</span></span> returns leading coefficients.</p></li>
+<li><p>With #1=0, returns the number of coefficients, i.e. <span class="docutils literal">1 + deg f</span>
+for non-zero polynomials.</p></li>
+<li><p>Out-of-range #1's return <span class="docutils literal">0/1[0]</span>.</p></li>
+</ul>
+<p>See also <a class="reference internal" href="#polnthcoeff-polname-number">\PolNthCoeff{polname}{number}</a>. The main difference is that
+with <span class="docutils literal">\PolAssign</span>, <span class="docutils literal">\macro</span> is made a prefix to <span class="docutils literal">1 + deg f</span>
+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 <span class="docutils literal">Nth</span> coefficient, and due to
+expandability can not store it in a macro for future usage (of course,
+it can be an argument in an <span class="docutils literal">\edef</span>.) 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="#id95"><span class="docutils literal"><span class="pre">\PolGet{polname}\fromarray\macro</span></span></a></h2>
+<blockquote>
+<p>Does the converse operation to
+<span class="docutils literal"><span class="pre">\PolAssign{polname}\toarray\macro</span></span>. Each individual
+<span class="docutils literal">\macro{number}</span> gets expanded in an <span class="docutils literal">\edef</span> and then normalized
+via <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a>'s macro <span class="docutils literal">\xintRaw</span>.</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 <span class="docutils literal">f</span> as would have <span class="docutils literal">\poldef <span class="pre">f(x):=1-2x+5x^2-3x^3;</span></span>.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Prior to <span class="docutils literal">0.5</span>, coefficients were not normalized via
+<span class="docutils literal">\xintRaw</span> for internal storage.</p>
+</div>
+</blockquote>
+</div>
+<div class="section" id="polfromcsv-polname-csv">
+<span id="polfromcsv"></span><h2><a class="toc-backref" href="#id96"><span class="docutils literal"><span class="pre">\PolFromCSV{polname}{<csv>}</span></span></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 <span class="docutils literal">\edef</span> and then put into normalized
+form via <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a>'s macro <span class="docutils literal">\xintRaw</span>.</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="admonition-title">Note</p>
+<p>Prior to <span class="docutils literal">0.5</span>, coefficients were not normalized via
+<span class="docutils literal">\xintRaw</span> for internal storage.</p>
+</div>
+</blockquote>
+</div>
+<div class="section" id="poltypeset-pol-expr">
+<span id="poltypeset"></span><h2><a class="toc-backref" href="#id97"><span class="docutils literal"><span class="pre">\PolTypeset{<pol.</span> <span class="pre">expr.>}</span></span></a></h2>
+<blockquote>
+<p>Typesets in descending powers, switching to math mode if in text
+mode, after evaluating the polynomial expression:</p>
+<pre class="literal-block">\PolTypeset{mul(x-i,i=1..5)}% possible since polexpr 0.8</pre>
+<p>The letter used in the input expression is by default <span class="docutils literal">x</span>,
+but can be modified by a redefinition of <a class="reference internal" href="#poltoexprinvar">\PolToExprInVar</a>.</p>
+<p>It uses also by default the letter <span class="docutils literal">x</span> on output but this one can
+be changed via an optional argument:</p>
+<pre class="literal-block">\PolTypeset[z]{polname or polynomial expression}</pre>
+<p>By default zero coefficients are skipped (use <span class="docutils literal">\poltypesetalltrue</span>
+to get all of them in output).</p>
+<p>The following macros (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="#id98"><span class="docutils literal">\PolTypesetCmd{raw_coeff}</span></a></h3>
+<blockquote>
+<p>Checks if the coefficient is <span class="docutils literal">1</span> or <span class="docutils literal"><span class="pre">-1</span></span> and then skips printing
+the <span class="docutils literal">1</span>, 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="#id99"><span class="docutils literal">\PolTypesetOne{raw_coeff}</span></a></h3>
+<blockquote>
+<p>Defaults to <span class="docutils literal">\xintSignedFrac</span> (LaTeX) or <span class="docutils literal">\xintSignedFwOver</span>
+(else). But these <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> very old legacy macros are a bit
+annoyin as they insist in exhibiting a power of ten rather than
+using simpler decimal notation.</p>
+<p>As alternative one can do things such as:</p>
+<pre class="literal-block">\def\PolTypesetOne#1{\xintDecToString{\xintREZ{#1}}}
+% or with LaTeX+siunitx for example
+\renewcommand\PolTypesetOne[1]{\num{\xintPFloat[5]{#1}}}
+% (as \num of siunitx understands floating point notation)
+\renewcommand\PolTypesetOne[1]{\num{\xintRound{4}{#1}}}</pre>
+</blockquote>
+</div>
+<div class="section" id="id9">
+<span id="poltypesetmonomialcmd"></span><h3><a class="toc-backref" href="#id100"><span class="docutils literal">\PolTypesetMonomialCmd</span></a></h3>
+<blockquote>
+<p>This decides how a monomial (in variable <span class="docutils literal">\PolVar</span> and with
+exponent <span class="docutils literal">\PolIndex</span>) is to be printed. The default does nothing
+for the constant term, <span class="docutils literal">\PolVar</span> for the first degree and
+<span class="docutils literal"><span class="pre">\PolVar^{\PolIndex}</span></span> for higher degrees monomials. Beware that
+<span class="docutils literal">\PolIndex</span> expands to digit tokens and needs termination in
+<span class="docutils literal">\ifnum</span> tests.</p>
+</blockquote>
+</div>
+<div class="section" id="poltypesetcmdprefix-raw-coeff">
+<span id="poltypesetcmdprefix"></span><h3><a class="toc-backref" href="#id101"><span class="docutils literal">\PolTypesetCmdPrefix{raw_coeff}</span></a></h3>
+<blockquote>
+<p>Expands to a <span class="docutils literal">+</span> if the <span class="docutils literal">raw_coeff</span> is zero or positive, and to
+nothing if <span class="docutils literal">raw_coeff</span> is negative, as in latter case the
+<span class="docutils literal">\xintSignedFrac</span> (or <span class="docutils literal">\xintSignedFwOver</span>) used by
+<a class="reference internal" href="#poltypesetcmd-raw-coeff">\PolTypesetCmd{raw_coeff}</a> will put the <span class="docutils literal">-</span> 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.</p>
+</blockquote>
+</div>
+</div>
+<div class="section" id="id11">
+<span id="id10"></span><h2><a class="toc-backref" href="#id102"><span class="docutils literal"><span class="pre">\PolTypeset*{<pol.</span> <span class="pre">expr.>}</span></span></a></h2>
+<blockquote>
+<p>Typesets in ascending powers. Use e.g. <span class="docutils literal">[h]</span> optional argument
+(after the <span class="docutils literal">*</span>) to use letter <span class="docutils literal">h</span> rather than <span class="docutils literal">x</span>.</p>
+<p>Extended at <span class="docutils literal">0.8</span> to accept general expressions and not only
+polynomial names. Redefine <a class="reference internal" href="#poltoexprinvar">\PolToExprInVar</a> to use in the
+expression another letter than default <span class="docutils literal">x</span>.</p>
+</blockquote>
+</div>
+<div class="section" id="poldiff-polname-1-polname-2">
+<span id="poldiff"></span><h2><a class="toc-backref" href="#id103"><span class="docutils literal"><span class="pre">\PolDiff{polname_1}{polname_2}</span></span></a></h2>
+<blockquote>
+<p>This sets <span class="docutils literal">polname_2</span> to the first derivative of <span class="docutils literal">polname_1</span>. It
+is allowed to issue <span class="docutils literal"><span class="pre">\PolDiff{f}{f}</span></span>, effectively replacing <span class="docutils literal">f</span>
+by <span class="docutils literal">f'</span>.</p>
+<p>Coefficients of the result <span class="docutils literal">polname_2</span> 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="#id104"><span class="docutils literal"><span class="pre">\PolDiff[N]{polname_1}{polname_2}</span></span></a></h2>
+<blockquote>
+<p>This sets <span class="docutils literal">polname_2</span> to the <span class="docutils literal">N</span>-th derivative of <span class="docutils literal">polname_1</span>.
+Identical arguments is allowed. With <span class="docutils literal">N=0</span>, same effect as
+<span class="docutils literal"><span class="pre">\PolLet{polname_2}={polname_1}</span></span>. With negative <span class="docutils literal">N</span>, switches to
+using <span class="docutils literal">\PolAntiDiff</span>.</p>
+</blockquote>
+</div>
+<div class="section" id="polantidiff-polname-1-polname-2">
+<span id="polantidiff"></span><h2><a class="toc-backref" href="#id105"><span class="docutils literal"><span class="pre">\PolAntiDiff{polname_1}{polname_2}</span></span></a></h2>
+<blockquote>
+<p>This sets <span class="docutils literal">polname_2</span> to the primitive of <span class="docutils literal">polname_1</span> vanishing
+at zero.</p>
+<p>Coefficients of the result <span class="docutils literal">polname_2</span> 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="#id106"><span class="docutils literal"><span class="pre">\PolAntiDiff[N]{polname_1}{polname_2}</span></span></a></h2>
+<blockquote>
+<p>This sets <span class="docutils literal">polname_2</span> to the result of <span class="docutils literal">N</span> successive integrations on
+<span class="docutils literal">polname_1</span>. With negative <span class="docutils literal">N</span>, it switches to using <span class="docutils literal">\PolDiff</span>.</p>
+</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="#id107"><span class="docutils literal"><span class="pre">\PolDivide{polname_1}{polname_2}{polname_Q}{polname_R}</span></span></a></h2>
+<blockquote>
+<p>This sets <span class="docutils literal">polname_Q</span> and <span class="docutils literal">polname_R</span> to be the quotient and
+remainder in the Euclidean division of <span class="docutils literal">polname_1</span> by
+<span class="docutils literal">polname_2</span>.</p>
+</blockquote>
+</div>
+<div class="section" id="polquo-polname-1-polname-2-polname-q">
+<span id="polquo"></span><h2><a class="toc-backref" href="#id108"><span class="docutils literal"><span class="pre">\PolQuo{polname_1}{polname_2}{polname_Q}</span></span></a></h2>
+<blockquote>
+<p>This sets <span class="docutils literal">polname_Q</span> to be the quotient in the Euclidean division
+of <span class="docutils literal">polname_1</span> by <span class="docutils literal">polname_2</span>.</p>
+</blockquote>
+</div>
+<div class="section" id="polrem-polname-1-polname-2-polname-r">
+<span id="polrem"></span><h2><a class="toc-backref" href="#id109"><span class="docutils literal"><span class="pre">\PolRem{polname_1}{polname_2}{polname_R}</span></span></a></h2>
+<blockquote>
+<p>This sets <span class="docutils literal">polname_R</span> to be the remainder in the Euclidean division
+of <span class="docutils literal">polname_1</span> by <span class="docutils literal">polname_2</span>.</p>
+</blockquote>
+</div>
+<div class="section" id="polgcd-polname-1-polname-2-polname-gcd">
+<span id="polgcd"></span><h2><a class="toc-backref" href="#id110"><span class="docutils literal"><span class="pre">\PolGCD{polname_1}{polname_2}{polname_GCD}</span></span></a></h2>
+<blockquote>
+<p>This sets <span class="docutils literal">polname_GCD</span> to be the (monic) GCD of the two first
+polynomials. It is a unitary polynomial except if both <span class="docutils literal">polname_1</span>
+and <span class="docutils literal">polname_2</span> vanish, then <span class="docutils literal">polname_GCD</span> is the zero
+polynomial.</p>
+</blockquote>
+</div>
+<div class="section" id="non-expandable-macros-related-to-the-root-localization-routines">
+<h2><a class="toc-backref" href="#id111">Non-expandable macros related to the root localization routines</a></h2>
+<div class="section" id="poltosturm-polname-sturmname">
+<span id="poltosturm"></span><h3><a class="toc-backref" href="#id112"><span class="docutils literal"><span class="pre">\PolToSturm{polname}{sturmname}</span></span></a></h3>
+<blockquote>
+<p>With <span class="docutils literal">polname</span> being for example <span class="docutils literal">P</span>, the macro starts by
+computing polynomials <span class="docutils literal">P</span> and <span class="docutils literal">P'</span>, then computes the (opposite
+of the) remainder in euclidean division, iteratively.</p>
+<p>The last non-zero remainder <span class="docutils literal">P_N_</span> (where <span class="docutils literal">N</span> is obtainable as
+<a class="reference internal" href="#polsturmchainlength-sturmname">\PolSturmChainLength{sturmname}</a>) is up to a factor
+the GCD of <span class="docutils literal">P</span> and <span class="docutils literal">P'</span> hence it is a constant if and only if
+<span class="docutils literal">P</span> is square-free.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<ul class="simple">
+<li><p>Since <span class="docutils literal">0.5</span> 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 <span class="docutils literal">1</span> or <span class="docutils literal"><span class="pre">-1</span></span>.</p></li>
+<li><p>After this normalization to primitive polynomials, they are
+stored internally as <span class="docutils literal">sturmname_k_</span>, <span class="docutils literal">k=0,1, ...</span>.</p></li>
+<li><p>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="#id12">PolToSturm*</a>.</p></li>
+</ul>
+</div>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>It is perfectly allowed to use the polynomial name as Sturm chain name:
+<span class="docutils literal"><span class="pre">\PolToSturm{f}(f}</span></span>.</p>
+</div>
+<p>The macro then declares <span class="docutils literal">sturmname_0</span>, <span class="docutils literal">sturmname_1</span>, ..., which are
+the (non-declared) <span class="docutils literal">sturmname_k_</span> divided by the last one. Division is
+not done if this last one is the constant <span class="docutils literal">1</span> or <span class="docutils literal"><span class="pre">-1</span></span>, 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 <span class="docutils literal">sturmname_0</span> has exactly the same real and complex roots as
+polynomial <span class="docutils literal">polname</span>, but with each root now of multiplicity one:
+i.e. it is the "square-free part" of original polynomial <span class="docutils literal">polname</span>.</p>
+<p>Notice that <span class="docutils literal">sturmname_1</span> isn't necessarily the derivative of
+<span class="docutils literal">sturmname_0</span> due to the various normalizations.</p>
+<p>The polynomials <span class="docutils literal">sturmname_k</span> main utility is for the execution of
+<a class="reference internal" href="#polsturmisolatezeros-sturmname">\PolSturmIsolateZeros{sturmname}</a>. Be careful not to use these
+names <span class="docutils literal">sturmname_0</span>, <span class="docutils literal">sturmname_1</span>, etc... for defining other
+polynomials after having done <span class="docutils literal"><span class="pre">\PolToSturm{polname}{sturmname}</span></span> and
+before executing <span class="docutils literal">\PolSturmIsolateZeros{sturmname}</span> 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="id13">
+<span id="id12"></span><h3><a class="toc-backref" href="#id113"><span class="docutils literal"><span class="pre">\PolToSturm*{polname}{sturmname}</span></span></a></h3>
+<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 <span class="docutils literal">sturmname_k_</span>, <span class="docutils literal">k=0,1, <span class="pre">...,</span> N</span>, with
+<span class="docutils literal">N</span> being <a class="reference internal" href="#polsturmchainlength-sturmname">\PolSturmChainLength{sturmname}</a>.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>This behaviour was modified at <span class="docutils literal">0.6</span>, anyhow the macro was
+broken at <span class="docutils literal">0.5</span>.</p>
+</div>
+<div class="admonition hint">
+<p class="admonition-title">Hint</p>
+<p>The square-free part of <span class="docutils literal">polname</span> is <span class="docutils literal">sturmname_0</span>, and their
+quotient is the polynomial with name
+<span class="docutils literal">sturname_\PolSturmChainLength{sturmname}_</span>. 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 <span class="docutils literal">f</span> as
+a product <span class="docutils literal">c f_1 f_2 f_3 ...</span> of a constant and square-free (primitive)
+polynomials, where each <span class="docutils literal">f_i</span> divides its predecessor.</p>
+</div>
+</blockquote>
+</div>
+<div class="section" id="polsettosturmchainsignchangesat-macro-sturmname-fraction">
+<span id="polsettosturmchainsignchangesat"></span><h3><a class="toc-backref" href="#id114"><span class="docutils literal"><span class="pre">\PolSetToSturmChainSignChangesAt{\macro}{sturmname}{fraction}</span></span></a></h3>
+<blockquote>
+<p>Sets macro <span class="docutils literal">\macro</span> to the number of sign changes in the Sturm
+chain with name prefix <span class="docutils literal">sturmname</span>, at location <span class="docutils literal">fraction</span>
+(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="admonition-title">Note</p>
+<p>The author was lazy and did not provide rather an expandable
+variant, where one would do <span class="docutils literal"><span class="pre">\edef\macro{\PolNbOf...}</span></span>.</p>
+<p>This will presumably get added in a future release.</p>
+<p>After some hesitation it was decided the macro would by default
+act globally. To make the scope of its macro definition local,
+use <span class="docutils literal">[\empty]</span> as extra optional argument.</p>
+</div>
+</blockquote>
+</div>
+<div class="section" id="polsettonbofzeroswithin-macro-sturmname-value-a-value-b">
+<span id="polsettonbofzeroswithin"></span><h3><a class="toc-backref" href="#id115"><span class="docutils literal"><span class="pre">\PolSetToNbOfZerosWithin{\macro}{sturmname}{value_a}{value_b}</span></span></a></h3>
+<blockquote>
+<p>Applies the <a class="reference external" href="https://en.wikipedia.org/wiki/Sturm%27s_theorem">Sturm Theorem</a> to set <span class="docutils literal">\macro</span> to the exact number
+of <strong>distinct</strong> roots of <span class="docutils literal">sturmname_0</span> in the interval <span class="docutils literal">(value_a, value_b]</span> (the macro first re-orders the value for <span class="docutils literal">value_a <= value_b</span> to hold).</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The author was lazy and did not provide rather an expandable
+variant, where one would do <span class="docutils literal"><span class="pre">\edef\macro{\PolNbOf...}</span></span>.</p>
+<p>This will presumably get added in future.</p>
+<p>After some hesitation it was decided the macro would by default
+act globally. To make the scope of its macro definition local,
+use <span class="docutils literal">[\empty]</span> 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 <span class="docutils literal">\numexpr</span>) 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="#id14">\PolSturmIsolateZeros*</a>.</p>
+</blockquote>
+</div>
+<div class="section" id="polsturmisolatezeros-sturmname">
+<span id="polsturmisolatezeros"></span><h3><a class="toc-backref" href="#id116"><span class="docutils literal">\PolSturmIsolateZeros{sturmname}</span></a></h3>
+<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="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 <span class="docutils literal"><span class="pre">\PolToSturm{f}{f}</span></span> is perfectly legal (the
+<span class="docutils literal">sturmname</span> can be same as the <span class="docutils literal">polname</span>): it defines
+polynomials <span class="docutils literal">f_0</span>, <span class="docutils literal">f_1</span>, ... having <span class="docutils literal">f</span> has name prefix.</p>
+<p>Such a prior call
+to <span class="docutils literal">\PolToSturm</span> must have been made at any rate for
+<span class="docutils literal">\PolSturmIsolateZeros</span> 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><p>singleton <span class="docutils literal">{a}</span>: then <span class="docutils literal">a</span> is a root, (necessarily a decimal
+number, but not all such decimal numbers are exactly identified yet).</p></li>
+<li><p>open intervals <span class="docutils literal">(a,b)</span>: then there is exactly one root <span class="docutils literal">z</span>
+such that <span class="docutils literal">a < z < b</span>, and the end points are guaranteed to not
+be roots.</p></li>
+</ul>
+<p>The interval boundaries are decimal numbers, originating
+in iterated decimal subdivision from initial intervals
+<span class="docutils literal"><span class="pre">(-10^E,</span> 0)</span> and <span class="docutils literal">(0, 10^E)</span> with <span class="docutils literal">E</span> 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 <span class="docutils literal">(a/10^f, <span class="pre">(a+1)/10^f)</span></span> with <span class="docutils literal">a</span> an integer, which is
+neither <span class="docutils literal">0</span> nor <span class="docutils literal"><span class="pre">-1</span></span>. Hence either <span class="docutils literal">a</span> and <span class="docutils literal">a+1</span> 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="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>See <a class="reference internal" href="#polrefineinterval-sturmname-index">\PolRefineInterval*{sturmname}{index}</a> which addresses
+this issue.</p>
+</div>
+<p>The interval boundaries (and exactly found roots) are made available
+for future computations in <span class="docutils literal">\xintexpr</span>-essions or polynomial
+definitions as variables <span class="docutils literal"><sturmname>L_1</span>,
+<span class="docutils literal"><sturmname>L_2</span>, etc..., for the left end-points and
+<span class="docutils literal"><sturmname>R_1</span>, <span class="docutils literal"><sturmname>R_2</span>, ..., for the right
+end-points.</p>
+<p>Thus for example, if <span class="docutils literal">sturmname</span> is <span class="docutils literal">f</span>, one can use the
+<a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> variables <span class="docutils literal">fL_1</span>, <span class="docutils literal">fL_2</span>, ... 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 <span class="docutils literal">fZ_1_isknown</span>
+will have value <span class="docutils literal">1</span> if the root in the first interval is known,
+and <span class="docutils literal">0</span> 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="admonition-title">Important</p>
+<p>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 <span class="docutils literal">k</span>-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 <span class="docutils literal">value</span> or <span class="docutils literal">expression</span>, and the same but with
+multiplicities.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The current polexpr implementation defines 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 described above with <strong>global scpe</strong>. On the
+other hand the Sturm sequence polynomials do obey the current scope.</p>
+</div>
+<div class="admonition note">
+<p class="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>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="id15">
+<span id="id14"></span><h3><a class="toc-backref" href="#id117"><span class="docutils literal"><span class="pre">\PolSturmIsolateZeros*{sturmname}</span></span></a></h3>
+<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 <span class="docutils literal">index</span>-th
+interval (intervals are enumerated from left to right, with index
+starting at <span class="docutils literal">1</span>).</p>
+<p>Furthermore, if for example the <span class="docutils literal">sturmname</span> is <span class="docutils literal">f</span>, <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a>
+variables <span class="docutils literal">fM_1</span>, <span class="docutils literal">fM_2</span>... hold the multiplicities thus
+computed.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>It is <strong>not</strong> necessary to have executed the <a class="reference internal" href="#id12">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="id17">
+<span id="id16"></span><h3><a class="toc-backref" href="#id118"><span class="docutils literal"><span class="pre">\PolSturmIsolateZeros**{sturmname}</span></span></a></h3>
+<blockquote>
+<p>The macro does the same as <a class="reference internal" href="#id15">\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="admonition-title">Note</p>
+<p>After execution of this macro, a root is "known" if and only if
+it is rational.</p>
+</div>
+<p>Furthermore, primitive polynomial <span class="docutils literal">sturmname_sqf_norr</span> is created
+to match the (square-free) <span class="docutils literal">sturmname_0</span> 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 <span class="docutils literal">sturmname_norr</span> is <span class="docutils literal">sturmname_0_</span> 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><h3><a class="toc-backref" href="#id119"><span class="docutils literal">\PolSturmIsolateZerosAndGetMultiplicities{sturmname}</span></a></h3>
+<blockquote>
+<p>This is another name for <a class="reference internal" href="#id15">\PolSturmIsolateZeros*{sturmname}</a>.</p>
+</blockquote>
+</div>
+<div class="section" id="polsturmisolatezerosgetmultiplicitiesandrationalroots-sturmname">
+<span id="polsturmisolatezerosgetmultiplicitiesandrationalroots"></span><h3><a class="toc-backref" href="#id120"><span class="docutils literal">\PolSturmIsolateZerosGetMultiplicitiesAndRationalRoots{sturmname}</span></a></h3>
+<blockquote>
+<p>This is another name for <a class="reference internal" href="#id17">\PolSturmIsolateZeros**{sturmname}</a>.</p>
+</blockquote>
+</div>
+<div class="section" id="polsturmisolatezerosandfindrationalroots-sturmname">
+<h3><a class="toc-backref" href="#id121"><span class="docutils literal">\PolSturmIsolateZerosAndFindRationalRoots{sturmname}</span></a></h3>
+<blockquote>
+<p>This works exactly like <a class="reference internal" href="#id17">\PolSturmIsolateZeros**{sturmname}</a>
+(inclusive of declaring the polynomials <span class="docutils literal">sturmname_sqf_norr</span> and
+<span class="docutils literal">sturmname_norr</span> 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="admonition-title">Note</p>
+<p>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="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
+<span class="docutils literal">sturmnameM_1</span>, <span class="docutils literal">sturmnameM_2</span>, ... 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="#id21">\PolPrintIntervals*</a> for example will be
+erroneous for the intervals with irrational roots.</p>
+<p>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><h3><a class="toc-backref" href="#id122"><span class="docutils literal"><span class="pre">\PolRefineInterval*{sturmname}{index}</span></span></a></h3>
+<blockquote>
+<p>The <span class="docutils literal">index</span>-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 <span class="docutils literal">k</span>th root is then
+strictly separated from the other roots.</p>
+</blockquote>
+</div>
+<div class="section" id="polrefineinterval-n-sturmname-index">
+<span id="polrefineinterval-n"></span><h3><a class="toc-backref" href="#id123"><span class="docutils literal"><span class="pre">\PolRefineInterval[N]{sturmname}{index}</span></span></a></h3>
+<blockquote>
+<p>The <span class="docutils literal">index</span>-th interval (starting count at one) is further
+subdivided once, reducing its length by a factor of 10. This is done
+<span class="docutils literal">N</span> times if the optional argument <span class="docutils literal">[N]</span> is present.</p>
+</blockquote>
+</div>
+<div class="section" id="polensureintervallength-sturmname-index-e">
+<span id="polensureintervallength"></span><h3><a class="toc-backref" href="#id124"><span class="docutils literal"><span class="pre">\PolEnsureIntervalLength{sturmname}{index}{E}</span></span></a></h3>
+<blockquote>
+<p>The <span class="docutils literal">index</span>-th interval is subdivided until its length becomes at
+most <span class="docutils literal">10^E</span>. This means (for <span class="docutils literal">E<0</span>) that the first <span class="docutils literal"><span class="pre">-E</span></span> digits
+after decimal mark of the <span class="docutils literal">k</span>th root will then be known exactly.</p>
+</blockquote>
+</div>
+<div class="section" id="polensureintervallengths-sturmname-e">
+<span id="polensureintervallengths"></span><h3><a class="toc-backref" href="#id125"><span class="docutils literal"><span class="pre">\PolEnsureIntervalLengths{sturmname}{E}</span></span></a></h3>
+<blockquote>
+<p>The intervals as obtained from <span class="docutils literal">\PolSturmIsolateZeros</span> are (if
+necessary) subdivided further by (base 10) dichotomy in order for
+each of them to have length at most <span class="docutils literal">10^E</span> (length will be shorter
+than <span class="docutils literal">10^E</span> 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
+<span class="docutils literal"><span class="pre">-E</span></span> digits (for <span class="docutils literal">E<0</span>) after decimal mark.</p>
+</blockquote>
+</div>
+<div class="section" id="polprintintervals-varname-sturmname">
+<span id="polprintintervals"></span><h3><a class="toc-backref" href="#id126"><span class="docutils literal"><span class="pre">\PolPrintIntervals[varname]{sturmname}</span></span></a></h3>
+<blockquote>
+<p>This is a convenience macro which prints the bounds for the roots
+<span class="docutils literal">Z_1</span>, <span class="docutils literal">Z_2</span>, ... (the optional argument <span class="docutils literal">varname</span> allows to
+specify a replacement for the default <span class="docutils literal">Z</span>). This will be done (by
+default) in a
+math mode <span class="docutils literal">array</span>, one interval per row, and pattern <span class="docutils literal">rcccl</span>,
+where the second and fourth column hold the <span class="docutils literal"><</span> sign, except when
+the interval reduces to a singleton, which means the root is known
+exactly.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The explanations here and in this section are for LaTeX. With
+other TeX macro formats, the LaTeX syntax such as for example
+<span class="docutils literal"><span class="pre">\begin{array}{rcccl}</span></span> which appears in the documentation here
+is actually replaced with quasi-equivalent direct use of TeX
+primitives.</p>
+</div>
+<p>See next macros which govern its output.</p>
+</blockquote>
+<div class="section" id="polprintintervalsnorealroots">
+<h4><a class="toc-backref" href="#id127"><span class="docutils literal">\PolPrintIntervalsNoRealRoots</span></a></h4>
+<blockquote>
+<p>Executed in place of an <span class="docutils literal">array</span> environment, when there are no
+real roots. Default definition:</p>
+<pre class="literal-block">\newcommand\PolPrintIntervalsNoRealRoots{}</pre>
+</blockquote>
+</div>
+<div class="section" id="polprintintervalsbeginenv">
+<h4><a class="toc-backref" href="#id128"><span class="docutils literal">\PolPrintIntervalsBeginEnv</span></a></h4>
+<blockquote>
+<p>Default definition:</p>
+<pre class="literal-block">\newcommand\PolPrintIntervalsBeginEnv{\[\begin{array}{rcccl}}</pre>
+</blockquote>
+</div>
+<div class="section" id="polprintintervalsendenv">
+<h4><a class="toc-backref" href="#id129"><span class="docutils literal">\PolPrintIntervalsEndEnv</span></a></h4>
+<blockquote>
+<p>Default definition:</p>
+<pre class="literal-block">\newcommand\PolPrintIntervalsEndEnv{\end{array}\]}</pre>
+</blockquote>
+</div>
+<div class="section" id="polprintintervalsknownroot">
+<h4><a class="toc-backref" href="#id130"><span class="docutils literal">\PolPrintIntervalsKnownRoot</span></a></h4>
+<blockquote>
+<p>Default definition:</p>
+<pre class="literal-block">\newcommand\PolPrintIntervalsKnownRoot{%
+ &&\PolPrintIntervalsTheVar_{\PolPrintIntervalsTheIndex}%
+ &=&\PolPrintIntervalsPrintExactZero
+}</pre>
+</blockquote>
+</div>
+<div class="section" id="polprintintervalsunknownroot">
+<h4><a class="toc-backref" href="#id131"><span class="docutils literal">\PolPrintIntervalsUnknownRoot</span></a></h4>
+<blockquote>
+<p>Default definition:</p>
+<pre class="literal-block">\newcommand\PolPrintIntervalsUnknownRoot{%
+ \PolPrintIntervalsPrintLeftEndPoint&<&%
+ \PolPrintIntervalsTheVar_{\PolPrintIntervalsTheIndex}&<&%
+ \PolPrintIntervalsPrintRightEndPoint
+}</pre>
+</blockquote>
+</div>
+<div class="section" id="id18">
+<span id="polprintintervalsprintexactzero"></span><h4><a class="toc-backref" href="#id132"><span class="docutils literal">\PolPrintIntervalsPrintExactZero</span></a></h4>
+<blockquote>
+<p>Default definition:</p>
+<pre class="literal-block">\newcommand\PolPrintIntervalsPrintExactZero{\PolPrintIntervalsTheLeftEndPoint}</pre>
+</blockquote>
+</div>
+<div class="section" id="id19">
+<span id="polprintintervalsprintleftendpoint"></span><h4><a class="toc-backref" href="#id133"><span class="docutils literal">\PolPrintIntervalsPrintLeftEndPoint</span></a></h4>
+<blockquote>
+<p>Default definition:</p>
+<pre class="literal-block">\newcommand\PolPrintIntervalsPrintLeftEndPoint{\PolPrintIntervalsTheLeftEndPoint}</pre>
+</blockquote>
+</div>
+<div class="section" id="id20">
+<span id="polprintintervalsprintrightendpoint"></span><h4><a class="toc-backref" href="#id134"><span class="docutils literal">\PolPrintIntervalsPrintRightEndPoint</span></a></h4>
+<blockquote>
+<p>Default definition is:</p>
+<pre class="literal-block">\newcommand\PolPrintIntervalsPrintRightEndPoint{\PolPrintIntervalsTheRightEndPoint}</pre>
+</blockquote>
+</div>
+</div>
+<div class="section" id="id22">
+<span id="id21"></span><h3><a class="toc-backref" href="#id135"><span class="docutils literal"><span class="pre">\PolPrintIntervals*[varname]{sturmname}</span></span></a></h3>
+<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
+<span class="docutils literal"><span class="pre">\POL@@PrintIntervals...</span></span> macros. We only reproduce here one
+definition:</p>
+<pre class="literal-block">\newcommand\POL@@PrintIntervalsPrintExactZero{%
+ \displaystyle
+ \xintSignedFrac{\PolPrintIntervalsTheLeftEndPoint}%
+}%</pre>
+<p>Multiplicities are printed using this auxiliary macro:</p>
+</blockquote>
+<div class="section" id="polprintintervalsprintmultiplicity">
+<h4><a class="toc-backref" href="#id136"><span class="docutils literal">\PolPrintIntervalsPrintMultiplicity</span></a></h4>
+<blockquote>
+<p>whose default definition is:</p>
+<pre class="literal-block">\newcommand\PolPrintIntervalsPrintMultiplicity{(\mbox{mult. }\PolPrintIntervalsTheMultiplicity)}</pre>
+</blockquote>
+</div>
+</div>
+</div>
+<div class="section" id="polmapcoeffs-macro-polname">
+<span id="polmapcoeffs"></span><h2><a class="toc-backref" href="#id137"><span class="docutils literal"><span class="pre">\PolMapCoeffs{\macro}{polname}</span></span></a></h2>
+<blockquote>
+<p>It modifies ('in-place': original coefficients get lost) each
+coefficient of the defined polynomial via the <em>expandable</em> macro
+<span class="docutils literal">\macro</span>. The degree is adjusted as necessary if some leading
+coefficients vanish after the operation. In replacement text of
+<span class="docutils literal">\macro</span>, <span class="docutils literal">\index</span> expands to the coefficient index (which is
+defined to be zero for the constant term).</p>
+<p>Notice that <span class="docutils literal">\macro</span> will have to handle inputs of the shape
+<span class="docutils literal">A/B[N]</span> (<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 <span class="docutils literal"><span class="pre">\xintSqr{\index}</span></span>) to replace <span class="docutils literal">n</span>-th coefficient
+<span class="docutils literal">f_n</span> by <span class="docutils literal">f_n*n^2</span>.</p>
+</blockquote>
+</div>
+<div class="section" id="polreducecoeffs-polname">
+<span id="polreducecoeffs"></span><h2><a class="toc-backref" href="#id138"><span class="docutils literal">\PolReduceCoeffs{polname}</span></a></h2>
+<blockquote>
+<p>About the same as <span class="docutils literal"><span class="pre">\PolMapCoeffs{\xintIrr}{polname}</span></span> (but
+maintaining a <span class="docutils literal">[0]</span> 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'.</p>
+</blockquote>
+</div>
+<div class="section" id="id24">
+<span id="id23"></span><h2><a class="toc-backref" href="#id139"><span class="docutils literal"><span class="pre">\PolReduceCoeffs*{polname}</span></span></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 <span class="docutils literal">A/B[N]</span>, then <span class="docutils literal">A/B</span> is reduced to
+smallest terms, but the <span class="docutils literal">10^N</span> 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 <span class="docutils literal">1.5e7</span>; the macro will anyhow always first do the
+needed conversion to strict format <span class="docutils literal">A/B[N]</span>.</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="#id140"><span class="docutils literal">\PolMakeMonic{polname}</span></a></h2>
+<blockquote>
+<p>Divides by the leading coefficient. It is recommended to execute
+<a class="reference internal" href="#id24">\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.</p>
+</blockquote>
+</div>
+<div class="section" id="polmakeprimitive-polname">
+<span id="polmakeprimitive"></span><h2><a class="toc-backref" href="#id141"><span class="docutils literal">\PolMakePrimitive{polname}</span></a></h2>
+<blockquote>
+<p>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.</p>
+</blockquote>
+</div>
+</div>
+<div class="section" id="expandable-macros">
+<h1><a class="toc-backref" href="#id142">Expandable macros</a></h1>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>At <span class="docutils literal">0.8</span> <span class="docutils literal">polexpr</span> is usable with Plain TeX and not only with
+LaTeX. Some examples given in this section may be using LaTeX syntax
+such as <span class="docutils literal">\renewcommand</span>. Convert to TeX primitives as appropriate
+if testing with a non LaTeX macro format.</p>
+</div>
+<p>All these macros expand completely in two steps except <span class="docutils literal">\PolToExpr</span>
+and <span class="docutils literal">\PolToFloatExpr</span> (and their auxiliaries) which need a
+<span class="docutils literal">\write</span>, <span class="docutils literal">\edef</span> or a <span class="docutils literal"><span class="pre">\csname...\endcsname</span></span> context.</p>
+<div class="section" id="poleval-polname-atexpr-numerical-expression">
+<span id="polevalatexpr"></span><h2><a class="toc-backref" href="#id143"><span class="docutils literal"><span class="pre">\PolEval{polname}\AtExpr{numerical</span> expression}</span></a></h2>
+<blockquote>
+<p>It boils down to
+<span class="docutils literal">\xinttheexpr polname(numerical <span class="pre">expression)\relax</span></span>.</p>
+</blockquote>
+</div>
+<div class="section" id="poleval-polname-at-fraction">
+<span id="polevalat"></span><h2><a class="toc-backref" href="#id144"><span class="docutils literal"><span class="pre">\PolEval{polname}\At{fraction}</span></span></a></h2>
+<blockquote>
+<p>Evaluates the polynomial at value <span class="docutils literal">fraction</span> 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.</p>
+</blockquote>
+</div>
+<div class="section" id="polevalreduced-polname-atexpr-numerical-expression">
+<span id="polevalreducedatexpr"></span><h2><a class="toc-backref" href="#id145"><span class="docutils literal"><span class="pre">\PolEvalReduced{polname}\AtExpr{numerical</span> expression}</span></a></h2>
+<blockquote>
+<p>Boils down to <span class="docutils literal">\xinttheexpr reduce(polname(numerical <span class="pre">expression))\relax</span></span>.</p>
+</blockquote>
+</div>
+<div class="section" id="polevalreduced-polname-at-fraction">
+<span id="polevalreducedat"></span><h2><a class="toc-backref" href="#id146"><span class="docutils literal"><span class="pre">\PolEvalReduced{polname}\At{fraction}</span></span></a></h2>
+<blockquote>
+<p>Evaluates the polynomial at value <span class="docutils literal">fraction</span> 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.</p>
+</blockquote>
+</div>
+<div class="section" id="polfloateval-polname-atexpr-numerical-expression">
+<span id="polfloatevalatexpr"></span><h2><a class="toc-backref" href="#id147"><span class="docutils literal"><span class="pre">\PolFloatEval{polname}\AtExpr{numerical</span> expression}</span></a></h2>
+<blockquote>
+<p>Boils down to <span class="docutils literal">\xintthefloatexpr polname(numerical <span class="pre">expression)\relax</span></span>.</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 brackets" href="#id27" id="id25">4</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 brackets" href="#id28" id="id26">5</a></p>
+<pre class="literal-block">\xintthefloatexpr 3.27*\xintexpr f(2.53)\relax^2\relax</pre>
+<p>The <span class="docutils literal">f(2.53)</span> is exactly computed then rounded at the time of
+getting raised to the power <span class="docutils literal">2</span>. Moving the <span class="docutils literal">^2</span> inside, that
+operation would also be treated exactly.</p>
+<dl class="footnote brackets">
+<dt class="label" id="id27"><span class="brackets"><a class="fn-backref" href="#id25">4</a></span></dt>
+<dd><p>Anyway each floating point operation starts by rounding its
+operands to the floating point precision.</p>
+</dd>
+<dt class="label" id="id28"><span class="brackets"><a class="fn-backref" href="#id26">5</a></span></dt>
+<dd><p>The <span class="docutils literal">\xintexpr</span> here could be <span class="docutils literal">\xinttheexpr</span> but that
+would be less efficient. Cf. <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> documentation about
+nested expressions.</p>
+</dd>
+</dl>
+</blockquote>
+</div>
+<div class="section" id="polfloateval-polname-at-fraction">
+<span id="polfloatevalat"></span><h2><a class="toc-backref" href="#id148"><span class="docutils literal"><span class="pre">\PolFloatEval{polname}\At{fraction}</span></span></a></h2>
+<blockquote>
+<p>Evaluates the polynomial at value <span class="docutils literal">fraction</span> 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.</p>
+</blockquote>
+</div>
+<div class="section" id="polifcoeffisplusorminusone-a-b">
+<span id="polifcoeffisplusorminusone"></span><h2><a class="toc-backref" href="#id149"><span class="docutils literal"><span class="pre">\PolIfCoeffIsPlusOrMinusOne{A}{B}</span></span></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 <span class="docutils literal">\cdot</span> in front of <span class="docutils literal"><span class="pre">\PolVar^{\PolIndex}</span></span> if
+the coefficient is not plus or minus one.</p>
+<p>The macro will execute <span class="docutils literal">A</span> if the coefficient has been found to be
+plus or minus one, and <span class="docutils literal">B</span> if not.</p>
+</blockquote>
+</div>
+<div class="section" id="polleadingcoeff-polname">
+<span id="polleadingcoeff"></span><h2><a class="toc-backref" href="#id150"><span class="docutils literal">\PolLeadingCoeff{polname}</span></a></h2>
+<blockquote>
+<p>Expands to the leading coefficient.</p>
+</blockquote>
+</div>
+<div class="section" id="polnthcoeff-polname-number">
+<span id="polnthcoeff"></span><h2><a class="toc-backref" href="#id151"><span class="docutils literal"><span class="pre">\PolNthCoeff{polname}{number}</span></span></a></h2>
+<blockquote>
+<p>It expands to the raw <span class="docutils literal">N</span>-th coefficient (<span class="docutils literal">0/1[0]</span> if the index
+number is out of range). With <span class="docutils literal"><span class="pre">N=-1</span></span>, <span class="docutils literal"><span class="pre">-2</span></span>, ... expands to the
+leading coefficients.</p>
+</blockquote>
+</div>
+<div class="section" id="poldegree-polname">
+<span id="poldegree"></span><h2><a class="toc-backref" href="#id152"><span class="docutils literal">\PolDegree{polname}</span></a></h2>
+<blockquote>
+<p>It expands to the degree. This is <span class="docutils literal"><span class="pre">-1</span></span> if zero polynomial but this
+may change in future. Should it then expand to <span class="docutils literal"><span class="pre">-\infty</span></span> ?</p>
+</blockquote>
+</div>
+<div class="section" id="policontent-polname">
+<span id="policontent"></span><h2><a class="toc-backref" href="#id153"><span class="docutils literal">\PolIContent{polname}</span></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-pol-expr">
+<span id="poltoexpr"></span><h2><a class="toc-backref" href="#id154"><span class="docutils literal"><span class="pre">\PolToExpr{<pol.</span> <span class="pre">expr.>}</span></span></a></h2>
+<blockquote>
+<p>Produces expandably <a class="footnote-reference brackets" href="#id30" id="id29">6</a> the string <span class="docutils literal"><span class="pre">coeff_N*x^N+...</span></span>, i.e. the
+polynomial is using descending powers.</p>
+<dl class="footnote brackets">
+<dt class="label" id="id30"><span class="brackets"><a class="fn-backref" href="#id29">6</a></span></dt>
+<dd><p>requires exhaustive expansion, for example as triggered by
+<span class="docutils literal">\write</span> or <span class="docutils literal">\edef</span>.</p>
+</dd>
+</dl>
+<p>Since <span class="docutils literal">0.8</span> the input is not restricted to be a polynomial name but
+is allowed to be an arbitrary expression (where by default the
+letter <span class="docutils literal">x</span> is recognized as the indeterminate; see
+<a class="reference internal" href="#poltoexprinvar">\PolToExprInVar</a>).</p>
+<p>The default output (which also by default uses the letter <span class="docutils literal">x</span> and is
+completely configurable, see in particular <a class="reference internal" href="#poltoexprvar">\PolToExprVar</a>) is
+compatible with both</p>
+<ul class="simple">
+<li><p>the Maple's input format,</p></li>
+<li><p>and the PSTricks <span class="docutils literal">\psplot[algebraic]</span> input format.</p></li>
+</ul>
+<p>Attention that it is not compatible with Python, but see
+<a class="reference internal" href="#poltoexprcaret">\PolToExprCaret</a> in this regard.</p>
+<p>It has the following characteristics:</p>
+<ul class="simple">
+<li><p>vanishing coefficients are skipped (issue <span class="docutils literal">\poltoexpralltrue</span> to
+override this and produce output such as <span class="docutils literal">x^3+0*x^2+0*x^1+0</span>),</p></li>
+<li><p>negative coefficients are not prefixed by a <span class="docutils literal">+</span> sign (else,
+Maple would not be happy),</p></li>
+<li><p>coefficients numerically equal to <span class="docutils literal">1</span> (or <span class="docutils literal"><span class="pre">-1</span></span>) are present
+only via their sign,</p></li>
+<li><p>the letter <span class="docutils literal">x</span> is used and the degree one monomial is output as
+<span class="docutils literal">x</span>, not as <span class="docutils literal">x^1</span>.</p></li>
+<li><p>(<span class="docutils literal">0.8</span>) the caret <span class="docutils literal">^</span> is of catcode 12. This means that one
+can for convenience typeset in regular text mode, for example
+using <span class="docutils literal">\texttt</span> (in LaTeX). But TeX will not know how to break
+the expression across end-of-lines anyhow. Formerly <span class="docutils literal">^</span> was
+suitable for math mode but as the exponent is not braced this
+worked only for polynomials of degrees at most 9. Anyhow this
+is not supposed to be a typesetting macro.</p></li>
+</ul>
+<p>Complete customization is possible, see the next macros. Any user
+redefinition must maintain the expandability property.</p>
+</blockquote>
+<div class="section" id="id31">
+<span id="poltoexprvar"></span><h3><a class="toc-backref" href="#id155"><span class="docutils literal">\PolToExprVar</span></a></h3>
+<blockquote>
+<p>Defaults to <span class="docutils literal">x</span>. The letter used in input.</p>
+</blockquote>
+</div>
+<div class="section" id="poltoexprinvar">
+<h3><a class="toc-backref" href="#id156"><span class="docutils literal">\PolToExprInVar</span></a></h3>
+<blockquote>
+<p>Defaults to <span class="docutils literal">x</span>: the letter used as the polynomial indeterminate.</p>
+<p>Recall that declared polynomials are more efficiently used in
+algebraic expressions without the <span class="docutils literal">(x)</span>, i.e. <span class="docutils literal">P*Q</span> is better
+than <span class="docutils literal"><span class="pre">P(x)*Q(x)</span></span>. Thus the input, even if an expression, does not
+have to contain any <span class="docutils literal">x</span>.</p>
+<p>(new with <span class="docutils literal">0.8</span>)</p>
+</blockquote>
+</div>
+<div class="section" id="id32">
+<span id="poltoexprtimes"></span><h3><a class="toc-backref" href="#id157"><span class="docutils literal">\PolToExprTimes</span></a></h3>
+<blockquote>
+<p>Defaults to <span class="docutils literal">*</span>.</p>
+</blockquote>
+</div>
+<div class="section" id="poltoexprcaret">
+<h3><a class="toc-backref" href="#id158"><span class="docutils literal">\PolToExprCaret</span></a></h3>
+<blockquote>
+<p>Defaults to <span class="docutils literal">^</span> of catcode 12. Set it to
+expand to <span class="docutils literal">**</span> for Python compatible output.</p>
+<p>(new with <span class="docutils literal">0.8</span>)</p>
+</blockquote>
+</div>
+<div class="section" id="poltoexprcmd-raw-coeff">
+<span id="poltoexprcmd"></span><h3><a class="toc-backref" href="#id159"><span class="docutils literal">\PolToExprCmd{raw_coeff}</span></a></h3>
+<blockquote>
+<p>Defaults to <span class="docutils literal"><span class="pre">\xintPRaw{\xintRawWithZeros{#1}}</span></span>.</p>
+<p>This means that the coefficient value is printed-out as a fraction
+<span class="docutils literal">a/b</span>, skipping the <span class="docutils literal">/b</span> part if <span class="docutils literal">b</span> turns out to be one.</p>
+<p>Configure it to be <span class="docutils literal"><span class="pre">\xintPRaw{\xintIrr{#1}}</span></span> if the fractions
+must be in irreducible terms.</p>
+<p>An alternative is <span class="docutils literal"><span class="pre">\xintDecToString{\xintREZ{#1}}</span></span> which uses
+integer or decimal fixed point format such as <span class="docutils literal">23.0071</span> if the
+internal representation of the number only has a power of ten as
+denominator (the effect of <span class="docutils literal">\xintREZ</span> here is to remove trailing
+decimal zeros). The behaviour of <span class="docutils literal">\xintDecToString</span> is not yet
+stable for other cases, and for example at time of writing no
+attempt is made to identify inputs having a finite decimal expansion
+so for example <span class="docutils literal">23.007/2</span> or <span class="docutils literal">23.007/25</span> can appear in output
+and not their finite decimal expansion with no denominator.</p>
+</blockquote>
+</div>
+<div class="section" id="poltoexproneterm-raw-coeff-number">
+<span id="poltoexproneterm"></span><h3><a class="toc-backref" href="#id160"><span class="docutils literal"><span class="pre">\PolToExprOneTerm{raw_coeff}{number}</span></span></a></h3>
+<blockquote>
+<p>This is the macro which from the coefficient and the exponent
+produces the corresponding term in output, such as <span class="docutils literal">2/3*x^7</span>.</p>
+<p>For its default definition, see the source code. It uses
+<a class="reference internal" href="#poltoexprcmd">\PolToExprCmd</a>, <a class="reference internal" href="#poltoexprtimes">\PolToExprTimes</a>, <a class="reference internal" href="#poltoexprvar">\PolToExprVar</a> and
+<a class="reference internal" href="#poltoexprcaret">\PolToExprCaret</a>.</p>
+</blockquote>
+</div>
+<div class="section" id="poltoexpronetermstylea-raw-coeff-number">
+<span id="poltoexpronetermstylea"></span><h3><a class="toc-backref" href="#id161"><span class="docutils literal"><span class="pre">\PolToExprOneTermStyleA{raw_coeff}{number}</span></span></a></h3>
+<blockquote>
+<p>This holds the default package meaning of <span class="docutils literal">\PolToExprOneTerm</span>.</p>
+</blockquote>
+</div>
+<div class="section" id="poltoexpronetermstyleb-raw-coeff-number">
+<span id="poltoexpronetermstyleb"></span><h3><a class="toc-backref" href="#id162"><span class="docutils literal"><span class="pre">\PolToExprOneTermStyleB{raw_coeff}{number}</span></span></a></h3>
+<blockquote>
+<p>This holds an alternative meaning, which puts the fractional part of
+a coefficient after the monomial, i.e. like this:</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><a class="reference internal" href="#poltoexprcmd">\PolToExprCmd</a> isn't used at all in this style. But
+<a class="reference internal" href="#poltoexprtimes">\PolToExprTimes</a>, <a class="reference internal" href="#poltoexprvar">\PolToExprVar</a> and <a class="reference internal" href="#poltoexprcaret">\PolToExprCaret</a> are obeyed.</p>
+<p>To activate it use <span class="docutils literal">\let\PolToExprOneTerm\PolToExprOneTermStyleB</span>.
+To revert to the package default behaviour, issue
+<span class="docutils literal">\let\PolToExprOneTerm\PolToExprOneTermStyleA</span>.</p>
+</blockquote>
+</div>
+<div class="section" id="poltoexprtermprefix-raw-coeff">
+<span id="poltoexprtermprefix"></span><h3><a class="toc-backref" href="#id163"><span class="docutils literal">\PolToExprTermPrefix{raw_coeff}</span></a></h3>
+<blockquote>
+<p>It receives as argument the coefficient. Its default behaviour is
+to produce a <span class="docutils literal">+</span> if the coefficient is positive, which will thus
+serve to separate the monomials in the output. This is to match
+the default for <a class="reference internal" href="#poltoexprcmd-raw-coeff">\PolToExprCmd{raw_coeff}</a> which in case of a
+positive coefficient does not output an explicit <span class="docutils literal">+</span> prefix.</p>
+</blockquote>
+</div>
+</div>
+<div class="section" id="id34">
+<span id="id33"></span><h2><a class="toc-backref" href="#id164"><span class="docutils literal"><span class="pre">\PolToExpr*{<pol.</span> <span class="pre">expr.>}</span></span></a></h2>
+<blockquote>
+<p>Ascending powers: <span class="docutils literal"><span class="pre">coeff_0+coeff_1*x+coeff_2*x^2+...</span></span>.</p>
+<p>Extended at <span class="docutils literal">0.8</span> to accept general expressions as input.</p>
+<p>Customizable with the same macros as for
+<a class="reference internal" href="#poltoexpr-pol-expr">\PolToExpr{<pol. expr.>}</a>.</p>
+</blockquote>
+</div>
+<div class="section" id="poltofloatexpr-pol-expr">
+<span id="poltofloatexpr"></span><h2><a class="toc-backref" href="#id165"><span class="docutils literal"><span class="pre">\PolToFloatExpr{<pol.</span> <span class="pre">expr.>}</span></span></a></h2>
+<blockquote>
+<p>Similar to <a class="reference internal" href="#poltoexpr-pol-expr">\PolToExpr{<pol. expr.>}</a> but using <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="admonition-title">Note</p>
+<p>This is unrelated to <a class="reference internal" href="#polgenfloatvariant-polname">\PolGenFloatVariant{polname}</a>:
+<a class="reference internal" href="#poltofloatexprcmd-raw-coeff">\PolToFloatExprCmd{raw_coeff}</a> operates on the <em>exact</em>
+coefficients anew (and may thus produce something else than
+the coefficients of the polynomial function acting
+in <span class="docutils literal">\xintfloateval</span> if the floating point precision was changed
+in between).</p>
+</div>
+<p>Extended at <span class="docutils literal">0.8</span> to accept general expressions as input.</p>
+</blockquote>
+<div class="section" id="poltofloatexproneterm-raw-coeff-number">
+<span id="poltofloatexproneterm"></span><h3><a class="toc-backref" href="#id166"><span class="docutils literal"><span class="pre">\PolToFloatExprOneTerm{raw_coeff}{number}</span></span></a></h3>
+<blockquote>
+<p>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.</p>
+</blockquote>
+</div>
+<div class="section" id="poltofloatexprcmd-raw-coeff">
+<span id="id36"></span><h3><a class="toc-backref" href="#id167"><span class="docutils literal">\PolToFloatExprCmd{raw_coeff}</span></a></h3>
+<blockquote>
+<p>The one-argument macro used by <span class="docutils literal">\PolToFloatExprOneTerm</span>.
+It defaults to <span class="docutils literal"><span class="pre">\xintFloat{#1}</span></span>.</p>
+<div class="admonition caution">
+<p class="admonition-title">Caution!</p>
+<p>Currently <span class="docutils literal">\xintFloat{0}</span> outputs <span class="docutils literal">0.e0</span>
+which is perfectly acceptable input for Python, but not for
+Maple. Thus, one should better leave the <span class="docutils literal">\\ifpoltoexprall</span> TeX
+Boolean to its default <a class="reference internal" href="#poltoexprallfalse">\poltoexprallfalse</a>, 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>Usage of <span class="docutils literal">\xintiiifZero</span> and not <span class="docutils literal">\xintifZero</span> is only for
+optimization (I can't help it) because <span class="docutils literal">#1</span> is known to be
+in <span class="docutils literal">xintfrac</span> raw format.</p>
+</div>
+</blockquote>
+</div>
+</div>
+<div class="section" id="id38">
+<span id="id37"></span><h2><a class="toc-backref" href="#id168"><span class="docutils literal"><span class="pre">\PolToFloatExpr*{<pol.</span> <span class="pre">expr.>}</span></span></a></h2>
+<blockquote>
+<p>Ascending powers.</p>
+<p>Extended at <span class="docutils literal">0.8</span> to accept general expressions as input.</p>
+</blockquote>
+</div>
+<div class="section" id="poltolist-polname">
+<span id="poltolist"></span><h2><a class="toc-backref" href="#id169"><span class="docutils literal">\PolToList{polname}</span></a></h2>
+<blockquote>
+<p>Expands to <span class="docutils literal"><span class="pre">{coeff_0}{coeff_1}...{coeff_N}</span></span> with <span class="docutils literal">N</span> = degree, and
+<span class="docutils literal">coeff_N</span> the leading coefficient
+(the zero polynomial does give <span class="docutils literal">{0/1[0]}</span> and not an
+empty output.)</p>
+</blockquote>
+</div>
+<div class="section" id="poltocsv-polname">
+<span id="poltocsv"></span><h2><a class="toc-backref" href="#id170"><span class="docutils literal">\PolToCSV{polname}</span></a></h2>
+<blockquote>
+<p>Expands to <span class="docutils literal">coeff_0, coeff_1, coeff_2, <span class="pre">.....,</span> coeff_N</span>, starting
+with constant term and ending with leading coefficient. Converse
+to <a class="reference internal" href="#polfromcsv-polname-csv">\PolFromCSV</a>.</p>
+</blockquote>
+</div>
+<div class="section" id="expandable-macros-related-to-the-root-localization-routines">
+<h2><a class="toc-backref" href="#id171">Expandable macros related to the root localization routines</a></h2>
+<div class="section" id="polsturmchainlength-sturmname">
+<span id="polsturmchainlength"></span><h3><a class="toc-backref" href="#id172"><span class="docutils literal">\PolSturmChainLength{sturmname}</span></a></h3>
+<blockquote>
+<p>Returns the integer <span class="docutils literal">N</span> such that <span class="docutils literal">sturmname_N</span> is the last one
+in the Sturm chain <span class="docutils literal">sturmname_0</span>, <span class="docutils literal">sturmname_1</span>, ...</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><h3><a class="toc-backref" href="#id173"><span class="docutils literal"><span class="pre">\PolSturmIfZeroExactlyKnown{sturmname}{index}{A}{B}</span></span></a></h3>
+<blockquote>
+<p>Executes <span class="docutils literal">A</span> if the <span class="docutils literal">index</span>-th interval reduces to a singleton,
+i.e. the root is known exactly, else <span class="docutils literal">B</span>.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p><span class="docutils literal">index</span> is allowed to be something like <span class="docutils literal">1+2*3</span> as it is fed
+to <span class="docutils literal"><span class="pre">\the\numexpr...\relax</span></span>.</p>
+</div>
+</blockquote>
+</div>
+<div class="section" id="polsturmisolatedzeroleft-sturmname-index">
+<span id="polsturmisolatedzeroleft"></span><h3><a class="toc-backref" href="#id174"><span class="docutils literal"><span class="pre">\PolSturmIsolatedZeroLeft{sturmname}{index}</span></span></a></h3>
+<blockquote>
+<p>Expands to the left end-point for the <span class="docutils literal">index</span>-th interval, as
+computed by some earlier <a class="reference internal" href="#polsturmisolatezeros-sturmname">\PolSturmIsolateZeros{sturmname}</a>.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>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><h3><a class="toc-backref" href="#id175"><span class="docutils literal"><span class="pre">\PolSturmIsolatedZeroRight{sturmname}{index}</span></span></a></h3>
+<blockquote>
+<p>Expands to the right end-point for the <span class="docutils literal">index</span>-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><h3><a class="toc-backref" href="#id176"><span class="docutils literal"><span class="pre">\PolSturmIsolatedZeroMultiplicity{sturmname}{index}</span></span></a></h3>
+<blockquote>
+<p>Expands to the multiplicity of the unique root contained in the
+<span class="docutils literal">index</span>-th interval.</p>
+<div class="admonition attention">
+<p class="admonition-title">Attention!</p>
+<p>A prior execution of <a class="reference internal" href="#id15">\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><h3><a class="toc-backref" href="#id177"><span class="docutils literal">\PolSturmNbOfIsolatedZeros{sturmname}</span></a></h3>
+<blockquote>
+<p>Expands to the number of real roots of the polynomial
+<span class="docutils literal"><sturmname>_0</span>, 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>.</p>
+</blockquote>
+<div class="admonition warning">
+<p class="admonition-title">Warning</p>
+<p>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>
+<div class="section" id="polsturmnbofrootsof-sturmname-lessthanorequalto-value">
+<h3><a class="toc-backref" href="#id178"><span class="docutils literal"><span class="pre">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualTo{value}</span></span></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 <span class="docutils literal">value</span> (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="admonition-title">Attention!</p>
+<p><a class="reference internal" href="#polsturmisolatezeros-sturmname">\PolSturmIsolateZeros{sturmname}</a> must have been executed
+beforehand.</p>
+<p>And the argument is a <span class="docutils literal">sturmname</span>, not a <span class="docutils literal">polname</span> (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="#id179"><span class="docutils literal"><span class="pre">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualToExpr{expression}</span></span></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 <span class="docutils literal">expression</span>.</p>
+<div class="admonition attention">
+<p class="admonition-title">Attention!</p>
+<p><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="#id180"><span class="docutils literal"><span class="pre">\PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualTo{value}</span></span></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 <span class="docutils literal">value</span>.</p>
+<div class="admonition attention">
+<p class="admonition-title">Attention!</p>
+<p><a class="reference internal" href="#id15">\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="#id181"><span class="docutils literal"><span class="pre">\PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualToExpr{expression}</span></span></a></h3>
+<blockquote>
+<p>Expands to the total number of roots (counted with multiplicities)
+which are less than or equal to the given <span class="docutils literal">expression</span>.</p>
+<div class="admonition attention">
+<p class="admonition-title">Attention!</p>
+<p><a class="reference internal" href="#id15">\PolSturmIsolateZeros*{sturmname}</a> (or the double starred
+variant) must have been executed beforehand.</p>
+</div>
+</blockquote>
+</div>
+<div class="section" id="polsturmnbofrationalroots-sturmname">
+<h3><a class="toc-backref" href="#id182"><span class="docutils literal">\PolSturmNbOfRationalRoots{sturmname}</span></a></h3>
+<blockquote>
+<p>Expands to the number of rational roots (without multiplicities).</p>
+<div class="admonition attention">
+<p class="admonition-title">Attention!</p>
+<p><a class="reference internal" href="#id17">\PolSturmIsolateZeros**{sturmname}</a> must have been executed
+beforehand.</p>
+</div>
+</blockquote>
+</div>
+<div class="section" id="polsturmnbofrationalrootswithmultiplicities-sturmname">
+<h3><a class="toc-backref" href="#id183"><span class="docutils literal">\PolSturmNbOfRationalRootsWithMultiplicities{sturmname}</span></a></h3>
+<blockquote>
+<p>Expands to the number of rational roots (counted with multiplicities).</p>
+<div class="admonition attention">
+<p class="admonition-title">Attention!</p>
+<p><a class="reference internal" href="#id17">\PolSturmIsolateZeros**{sturmname}</a> must have been executed
+beforehand.</p>
+</div>
+</blockquote>
+</div>
+<div class="section" id="polsturmrationalroot-sturmname-k">
+<h3><a class="toc-backref" href="#id184"><span class="docutils literal"><span class="pre">\PolSturmRationalRoot{sturmname}{k}</span></span></a></h3>
+<blockquote>
+<p>Expands to the <span class="docutils literal">k</span>th rational root (they are ordered and indexed
+starting at 1 for the most negative).</p>
+<div class="admonition attention">
+<p class="admonition-title">Attention!</p>
+<p><a class="reference internal" href="#id17">\PolSturmIsolateZeros**{sturmname}</a> must have been executed
+beforehand.</p>
+</div>
+</blockquote>
+</div>
+<div class="section" id="polsturmrationalrootindex-sturmname-k">
+<h3><a class="toc-backref" href="#id185"><span class="docutils literal"><span class="pre">\PolSturmRationalRootIndex{sturmname}{k}</span></span></a></h3>
+<blockquote>
+<p>Expands to <span class="docutils literal">index</span> of the <span class="docutils literal">k</span>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="admonition-title">Attention!</p>
+<p><a class="reference internal" href="#id17">\PolSturmIsolateZeros**{sturmname}</a> must have been executed
+beforehand.</p>
+</div>
+</blockquote>
+</div>
+<div class="section" id="polsturmrationalrootmultiplicity-sturmname-k">
+<h3><a class="toc-backref" href="#id186"><span class="docutils literal"><span class="pre">\PolSturmRationalRootMultiplicity{sturmname}{k}</span></span></a></h3>
+<blockquote>
+<p>Expands to the multiplicity of the <span class="docutils literal">k</span>th rational root.</p>
+<div class="admonition attention">
+<p class="admonition-title">Attention!</p>
+<p><a class="reference internal" href="#id17">\PolSturmIsolateZeros**{sturmname}</a> must have been executed
+beforehand.</p>
+</div>
+</blockquote>
+</div>
+<div class="section" id="polintervalwidth-sturmname-index">
+<span id="polintervalwidth"></span><h3><a class="toc-backref" href="#id187"><span class="docutils literal"><span class="pre">\PolIntervalWidth{sturmname}{index}</span></span></a></h3>
+<blockquote>
+<p>The <span class="docutils literal">10^E</span> width of the current <span class="docutils literal">index</span>-th root localization
+interval. Output is in <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> raw <span class="docutils literal">1/1[E]</span> format (if not zero).</p>
+</blockquote>
+</div>
+</div>
+<div class="section" id="expandable-macros-for-use-within-execution-of-polprintintervals">
+<h2><a class="toc-backref" href="#id188">Expandable macros for use within execution of <span class="docutils literal">\PolPrintIntervals</span></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="admonition-title">Attention!</p>
+<p>Some macros formerly mentioned here got removed at 0.7:
+<span class="docutils literal">\PolPrintIntervalsTheEndPoint</span>,
+<span class="docutils literal"><span class="pre">\PolIfEndPointIsPositive{A}{B}</span></span>,
+<span class="docutils literal"><span class="pre">\PolIfEndPointIsNegative{A}{B}</span></span>,
+<span class="docutils literal"><span class="pre">\PolIfEndPointIsZero{A}{B}</span></span>.</p>
+</div>
+<div class="section" id="polprintintervalsthevar">
+<h3><a class="toc-backref" href="#id189"><span class="docutils literal">\PolPrintIntervalsTheVar</span></a></h3>
+<blockquote>
+<p>Expands to the name (default <span class="docutils literal">Z</span>) used for representing the roots,
+which was passed as optional argument <span class="docutils literal">varname</span> to
+<a class="reference internal" href="#polprintintervals-varname-sturmname">\PolPrintIntervals[varname]{sturmname}</a>.</p>
+</blockquote>
+</div>
+<div class="section" id="polprintintervalstheindex">
+<h3><a class="toc-backref" href="#id190"><span class="docutils literal">\PolPrintIntervalsTheIndex</span></a></h3>
+<blockquote>
+<p>Expands to the index of the considered interval (indexing starting
+at 1 for the leftmost interval).</p>
+</blockquote>
+</div>
+<div class="section" id="polprintintervalsthesturmname">
+<h3><a class="toc-backref" href="#id191"><span class="docutils literal">\PolPrintIntervalsTheSturmName</span></a></h3>
+<blockquote>
+<p>Expands to the argument which was passed as <span class="docutils literal">sturmname</span> to
+<a class="reference internal" href="#polprintintervals-varname-sturmname">\PolPrintIntervals[varname]{sturmname}</a>.</p>
+</blockquote>
+</div>
+<div class="section" id="polprintintervalstheleftendpoint">
+<h3><a class="toc-backref" href="#id192"><span class="docutils literal">\PolPrintIntervalsTheLeftEndPoint</span></a></h3>
+<blockquote>
+<p>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>.</p>
+</blockquote>
+</div>
+<div class="section" id="polprintintervalstherightendpoint">
+<h3><a class="toc-backref" href="#id193"><span class="docutils literal">\PolPrintIntervalsTheRightEndPoint</span></a></h3>
+<blockquote>
+<p>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.</p>
+</blockquote>
+</div>
+<div class="section" id="polprintintervalsthemultiplicity">
+<h3><a class="toc-backref" href="#id194"><span class="docutils literal">\PolPrintIntervalsTheMultiplicity</span></a></h3>
+<blockquote>
+<p>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.</p>
+</blockquote>
+</div>
+</div>
+</div>
+<div class="section" id="booleans-with-default-setting-as-indicated">
+<h1><a class="toc-backref" href="#id195">Booleans (with default setting as indicated)</a></h1>
+<div class="section" id="xintverbosefalse">
+<h2><a class="toc-backref" href="#id196"><span class="docutils literal">\xintverbosefalse</span></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
+<span class="docutils literal">true</span> triggers the writing of information to the log when new
+polynomial or scalar variables are defined.</p>
+<div class="admonition caution">
+<p class="admonition-title">Caution!</p>
+<p>The macro and variable meanings as written to the log are to be
+considered unstable and undocumented internal structures.</p>
+</div>
+</blockquote>
+</div>
+<div class="section" id="polnewpolverbosefalse">
+<h2><a class="toc-backref" href="#id197"><span class="docutils literal">\polnewpolverbosefalse</span></a></h2>
+<blockquote>
+<p>When <span class="docutils literal">\poldef</span> is used, both a variable and a function are
+defined. The default <span class="docutils literal">\polnewpolverbosefalse</span> setting suppresses
+the print-out to the log and terminal of the function macro meaning,
+as it only duplicates the information contained in the variable
+which is already printed out to the log and terminal.</p>
+<p>However <a class="reference internal" href="#polgenfloatvariant-polname">\PolGenFloatVariant{polname}</a> does still print out the
+information relative to the polynomial function it defines for use in
+<span class="docutils literal">\xintfloateval{}</span> as there is no float polynomial variable, only the
+function, and it is the only way to see its rounded coefficients
+(<span class="docutils literal">\xintverbosefalse</span> suppresses also that info).</p>
+<p>If set to <span class="docutils literal">true</span>, it overrides in both cases
+<span class="docutils literal">\xintverbosefalse</span>. The setting only affects polynomial
+declarations. Scalar variables such as those holding information on
+roots obey only the <span class="docutils literal"><span class="pre">\xintverbose...</span></span> setting.</p>
+<p>(new with <span class="docutils literal">0.8</span>)</p>
+</blockquote>
+</div>
+<div class="section" id="poltypesetallfalse">
+<h2><a class="toc-backref" href="#id198"><span class="docutils literal">\poltypesetallfalse</span></a></h2>
+<blockquote>
+<p>If <span class="docutils literal">true</span>, <a class="reference internal" href="#poltypeset">\PolTypeset</a> will also typeset the vanishing
+coefficients.</p>
+</blockquote>
+</div>
+<div class="section" id="poltoexprallfalse">
+<h2><a class="toc-backref" href="#id199"><span class="docutils literal">\poltoexprallfalse</span></a></h2>
+<blockquote>
+<p>If <span class="docutils literal">true</span>, <a class="reference internal" href="#poltoexpr-pol-expr">\PolToExpr{<pol. expr.>}</a> and <a class="reference internal" href="#poltofloatexpr-pol-expr">\PolToFloatExpr{<pol. expr.>}</a> will
+also include the vanishing coefficients in their outputs.</p>
+</blockquote>
+</div>
+</div>
+<div class="section" id="utilies">
+<h1><a class="toc-backref" href="#id200">Utilies</a></h1>
+<div class="section" id="poldectostring-decimal-number">
+<span id="poldectostring"></span><h2><a class="toc-backref" href="#id201"><span class="docutils literal">\PolDecToString{decimal number}</span></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 <span class="docutils literal">1.3</span> of <span class="docutils literal">2018/03/01</span>) under
+the name <span class="docutils literal">\xintDecToString</span>, and the <span class="docutils literal">polexpr</span> macro is simply
+now an alias to it.</p>
+<p>For example
+<span class="docutils literal"><span class="pre">\PolDecToString{123.456e-8}</span></span> will expand to <span class="docutils literal">0.00000123456</span>
+and <span class="docutils literal"><span class="pre">\PolDecToString{123.450e-8}</span></span> to <span class="docutils literal">0.00000123450</span> which
+illustrates that trailing zeros are not trimmed. To trim trailing
+zeroes, one can use <span class="docutils literal"><span class="pre">\PolDecToString{\xintREZ{#1}}</span></span>.</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">xintexpr</a>.</p>
+</blockquote>
+</div>
+<div class="section" id="polexprsetup">
+<h2><a class="toc-backref" href="#id202"><span class="docutils literal">\polexprsetup</span></a></h2>
+<blockquote>
+<p>Serves to customize the package. Currently only two keys are
+recognized:</p>
+<ul class="simple">
+<li><p><span class="docutils literal">norr</span>: the postfix that <a class="reference internal" href="#id17">\PolSturmIsolateZeros**{sturmname}</a>
+should append to <span class="docutils literal">sturmname</span> to declare the primitive polynomial
+obtained from original one after removal of all rational roots.
+The default value is <span class="docutils literal">_norr</span> (standing for “no rational roots”).</p></li>
+<li><p><span class="docutils literal">sqfnorr</span>: the postfix that <a class="reference internal" href="#id17">\PolSturmIsolateZeros**{sturmname}</a>
+should append to <span class="docutils literal">sturmname</span> to declare the primitive polynomial
+obtained from original one after removal of all rational roots and
+suppression of all multiplicities.
+The default value is <span class="docutils literal">_sqf_norr</span> (standing for “square-free with
+no rational roots”).</p></li>
+</ul>
+<p>The package executes <span class="docutils literal">\polexprsetup{norr=_norr, sqfnorr=_sqf_norr}</span> as default.</p>
+</blockquote>
+</div>
+</div>
+<div class="section" id="technicalities">
+<h1><a class="toc-backref" href="#id203">Technicalities</a></h1>
+<ul>
+<li><p>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="#id8">\PolDef{f}{P(x)}</a>
+rather. The colon in <span class="docutils literal">:=</span> may be active with no consequences.</p></li>
+<li><p>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 brackets" href="#id40" id="id39">7</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*Q;</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
+<span class="docutils literal">\PolToExpr{PQ}</span> outputs the <span class="docutils literal">6/6*x^3</span> as <span class="docutils literal">x^3</span> because (by
+default) it recognizes and filters out coefficients equal to one or
+minus one (since release <span class="docutils literal">0.3</span>). One can use for example
+<span class="docutils literal">\PolToCSV{PQ}</span> to see the internally stored coefficients.</p>
+<dl class="footnote brackets">
+<dt class="label" id="id40"><span class="brackets"><a class="fn-backref" href="#id39">7</a></span></dt>
+<dd><p>prior to <span class="docutils literal">0.4.1</span>, <span class="docutils literal">polexpr</span> 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">xintexpr</a> at release <span class="docutils literal">1.3</span>.</p>
+</dd>
+</dl>
+</li>
+<li><p><a class="reference internal" href="#poldiff-polname-1-polname-2">\PolDiff{polname_1}{polname_2}</a> always applies <span class="docutils literal">\xintIrr</span> to the
+resulting coefficients, except that the <em>power of ten</em> part <span class="docutils literal">[N]</span>
+(for example an input in scientific notation such as <span class="docutils literal">1.23e5</span> gives
+<span class="docutils literal">123/1[3]</span> 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>Currently, the package stores all coefficients from index <span class="docutils literal">0</span> 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 <span class="docutils literal">\poldef <span class="pre">f(x):=x^1000</span> + x^500;</span> the subsequent definition <span class="docutils literal">\poldef <span class="pre">g(x):=</span> <span class="pre">f(x)^2;</span></span> 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>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="#id204">CHANGE LOG</a></h1>
+<ul>
+<li><p>v0.1 (2018/01/11): initial release. Features:</p>
+<ul class="simple">
+<li><p>The <a class="reference internal" href="#poldef">\poldef</a> parser itself,</p></li>
+<li><p>Differentiation and anti-differentiation,</p></li>
+<li><p>Euclidean division and GCDs,</p></li>
+<li><p>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>, ...</p></li>
+</ul>
+<p>Only one-variable polynomials so far.</p>
+</li>
+<li><p>v0.2 (2018/01/14)</p>
+<ul class="simple">
+<li><p>Fix: <span class="docutils literal">"README thinks \numexpr recognizes ^ operator"</span>.</p></li>
+<li><p>Convert README to reStructuredText markup.</p></li>
+<li><p>Move main documentation from README to separate <span class="docutils literal">polexpr.txt</span> file.</p></li>
+<li><p>Provide <span class="docutils literal">polexpr.html</span> as obtained via <a class="reference external" href="http://docutils.sourceforge.net/docs/index.html">DocUtils</a> <span class="docutils literal">rst2html.py</span>.</p></li>
+<li><p>Convert README to (CTAN compatible) Markdown markup.</p></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>v0.3 (2018/01/17)</p>
+<ul>
+<li><p>bug fixes:</p>
+<ul>
+<li><p>the <span class="docutils literal">0.1</span> <a class="reference internal" href="#polevalat">\PolEval</a> accepted expressions for its second
+argument, but this was removed by mistake at <span class="docutils literal">0.2</span>. Restored.</p>
+<p><strong>Attention</strong>: at <span class="docutils literal">0.4</span> 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>incompatible or breaking changes:</p>
+<ul class="simple">
+<li><p><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="#id33">\PolToExpr*</a> for <em>ascending</em> powers.</p></li>
+<li><p><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 <span class="docutils literal">\xintRound</span> or <span class="docutils literal">\xintFloat</span>, this step has been
+removed; the former meaning is available as <a class="reference internal" href="#polevalreducedat">\PolEvalReduced</a>.</p></li>
+</ul>
+</li>
+<li><p>new (or newly documented) macros:</p>
+<ul class="simple">
+<li><p><a class="reference internal" href="#poltypesetcmd">\PolTypesetCmd</a></p></li>
+<li><p><a class="reference internal" href="#poltypesetcmdprefix">\PolTypesetCmdPrefix</a></p></li>
+<li><p><a class="reference internal" href="#poltypesetmonomialcmd">\PolTypesetMonomialCmd</a></p></li>
+<li><p><a class="reference internal" href="#polevalreducedat">\PolEvalReducedAt</a></p></li>
+<li><p><a class="reference internal" href="#poltofloatexpr">\PolToFloatExpr</a></p></li>
+<li><p><a class="reference internal" href="#poltoexproneterm">\PolToExprOneTerm</a></p></li>
+<li><p><a class="reference internal" href="#poltofloatexproneterm">\PolToFloatExprOneTerm</a></p></li>
+<li><p><a class="reference internal" href="#poltoexprcmd">\PolToExprCmd</a></p></li>
+<li><p><a class="reference internal" href="#id36">\PolToFloatExprCmd</a></p></li>
+<li><p><a class="reference internal" href="#poltoexprtermprefix">\PolToExprTermPrefix</a></p></li>
+<li><p><a class="reference internal" href="#poltoexprvar">\PolToExprVar</a></p></li>
+<li><p><a class="reference internal" href="#poltoexprtimes">\PolToExprTimes</a></p></li>
+</ul>
+</li>
+<li><p>improvements:</p>
+<ul>
+<li><p>documentation has a table of contents, internal hyperlinks,
+standardized signature notations and added explanations.</p></li>
+<li><p>one can do <span class="docutils literal"><span class="pre">\PolLet{g}={f}</span></span> or <span class="docutils literal"><span class="pre">\PolLet{g}{f}</span></span>.</p></li>
+<li><p><span class="docutils literal">\PolToExpr{f}</span> is highly customizable.</p></li>
+<li><p><a class="reference internal" href="#poldef">\poldef</a> and other defining macros prepare the polynomial
+functions for usage within <span class="docutils literal">\xintthefloatexpr</span> (or
+<span class="docutils literal">\xintdeffloatvar</span>). 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 <span class="docutils literal">0.4</span>. 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>v0.3.1 (2018/01/18)</p>
+<p>Fixes two typos in example code included in the documentation.</p>
+</li>
+<li><p>v0.4 (2018/02/16)</p>
+<ul>
+<li><p>bug fixes:</p>
+<ul class="simple">
+<li><p>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>.</p></li>
+<li><p><a class="reference internal" href="#polgcd">\PolGCD</a> was buggy in case of first polynomial being
+of lesser degree than the second one.</p></li>
+</ul>
+</li>
+<li><p>breaking changes:</p>
+<ul>
+<li><p>formerly <a class="reference internal" href="#polevalat">\PolEval{P}\At{foo}</a> allowed <span class="docutils literal">foo</span> to
+be an expression, which was transparently handled via
+<span class="docutils literal">\xinttheexpr</span>. Now, <span class="docutils literal">foo</span> must be a fraction (or a macro
+expanding to such) in the format acceptable by <span class="docutils literal">xintfrac.sty</span>
+macros. Use <a class="reference internal" href="#polevalatexpr">\PolEval{P}\AtExpr{foo}</a> for more
+general arguments using expression syntax. E.g., if <span class="docutils literal">foo</span> is the
+name of a variable known to <span class="docutils literal">\xintexpr</span>.</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>the <span class="docutils literal">3.0</span> 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>new non-expandable macros:</p>
+<ul class="simple">
+<li><p><a class="reference internal" href="#polgenfloatvariant">\PolGenFloatVariant</a></p></li>
+<li><p><a class="reference internal" href="#polgloballet">\PolGlobalLet</a></p></li>
+<li><p><a class="reference internal" href="#poltypesetone">\PolTypesetOne</a></p></li>
+<li><p><a class="reference internal" href="#polquo">\PolQuo</a></p></li>
+<li><p><a class="reference internal" href="#polrem">\PolRem</a></p></li>
+<li><p><a class="reference internal" href="#poltosturm">\PolToSturm</a></p></li>
+<li><p><a class="reference internal" href="#id12">\PolToSturm*</a></p></li>
+<li><p><a class="reference internal" href="#polsettosturmchainsignchangesat">\PolSetToSturmChainSignChangesAt</a></p></li>
+<li><p><a class="reference internal" href="#polsettonbofzeroswithin">\PolSetToNbOfZerosWithin</a></p></li>
+<li><p><a class="reference internal" href="#polsturmisolatezeros">\PolSturmIsolateZeros</a></p></li>
+<li><p><a class="reference internal" href="#polrefineinterval">\PolRefineInterval*</a></p></li>
+<li><p><a class="reference internal" href="#polrefineinterval-n">\PolRefineInterval[N]</a></p></li>
+<li><p><a class="reference internal" href="#polensureintervallength">\PolEnsureIntervalLength</a></p></li>
+<li><p><a class="reference internal" href="#polensureintervallengths">\PolEnsureIntervalLengths</a></p></li>
+<li><p><a class="reference internal" href="#polprintintervals">\PolPrintIntervals</a></p></li>
+<li><p><a class="reference internal" href="#polprintintervalsprintexactzero">\PolPrintIntervalsPrintExactZero</a></p></li>
+<li><p><a class="reference internal" href="#polprintintervalsprintleftendpoint">\PolPrintIntervalsPrintLeftEndPoint</a></p></li>
+<li><p><a class="reference internal" href="#polprintintervalsprintrightendpoint">\PolPrintIntervalsPrintRightEndPoint</a></p></li>
+<li><p><a class="reference internal" href="#id23">\PolReduceCoeffs*</a></p></li>
+<li><p><a class="reference internal" href="#polmakemonic">\PolMakeMonic</a></p></li>
+</ul>
+</li>
+<li><p>new expandable macros:</p>
+<ul class="simple">
+<li><p><a class="reference internal" href="#poltoexpronetermstylea">\PolToExprOneTermStyleA</a></p></li>
+<li><p><a class="reference internal" href="#polifcoeffisplusorminusone">\PolIfCoeffIsPlusOrMinusOne</a></p></li>
+<li><p><a class="reference internal" href="#polleadingcoeff">\PolLeadingCoeff</a></p></li>
+<li><p><a class="reference internal" href="#polsturmchainlength">\PolSturmChainLength</a></p></li>
+<li><p><a class="reference internal" href="#polsturmnbofisolatedzeros">\PolSturmNbOfIsolatedZeros</a></p></li>
+<li><p><a class="reference internal" href="#polsturmifzeroexactlyknown">\PolSturmIfZeroExactlyKnown</a></p></li>
+<li><p><a class="reference internal" href="#polsturmisolatedzeroleft">\PolSturmIsolatedZeroLeft</a></p></li>
+<li><p><a class="reference internal" href="#polsturmisolatedzeroright">\PolSturmIsolatedZeroRight</a></p></li>
+<li><p><span class="docutils literal">\PolPrintIntervalsTheEndPoint</span> (removed at 0.7)</p></li>
+<li><p><a class="reference internal" href="#polprintintervalstheindex">\PolPrintIntervalsTheIndex</a></p></li>
+<li><p><span class="docutils literal">\PolIfEndPointIsPositive</span> (removed at 0.7)</p></li>
+<li><p><span class="docutils literal">\PolIfEndPointIsNegative</span> (removed at 0.7)</p></li>
+<li><p><span class="docutils literal">\PolIfEndPointIsZero</span> (removed at 0.7)</p></li>
+<li><p><a class="reference internal" href="#polintervalwidth">\PolIntervalWidth</a></p></li>
+<li><p><a class="reference internal" href="#poldectostring">\PolDecToString</a></p></li>
+</ul>
+</li>
+<li><p>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>v0.4.1 (2018/03/01)</p>
+<p>Synced with xint 1.3.</p>
+</li>
+<li><p>v0.4.2 (2018/03/03)</p>
+<p>Documentation fix.</p>
+</li>
+<li><p>v0.5 (2018/04/08)</p>
+<ul class="simple">
+<li><p>bug fixes:</p>
+<ul>
+<li><p><a class="reference internal" href="#polget-polname-fromarray-macro">\PolGet{polname}\fromarray\macro</a> crashed when <span class="docutils literal">\macro</span> 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.</p></li>
+</ul>
+</li>
+<li><p>breaking changes:</p>
+<ul>
+<li><p><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.</p></li>
+<li><p>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 <span class="docutils literal">\xintRaw</span>.</p></li>
+</ul>
+</li>
+<li><p>experimental change:</p>
+<ul>
+<li><p>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.</p></li>
+</ul>
+</li>
+<li><p>new non-expandable macro:</p>
+<ul>
+<li><p><a class="reference internal" href="#polmakeprimitive">\PolMakePrimitive</a></p></li>
+</ul>
+</li>
+<li><p>new expandable macro:</p>
+<ul>
+<li><p><a class="reference internal" href="#policontent">\PolIContent</a></p></li>
+</ul>
+</li>
+</ul>
+</li>
+<li><p>v0.5.1 (2018/04/22)</p>
+<ul class="simple">
+<li><p>new feature:</p>
+<ul>
+<li><p>the character <span class="docutils literal">'</span> can be used in polynomial names.</p></li>
+</ul>
+</li>
+</ul>
+</li>
+<li><p>v0.6 (2018/11/20)</p>
+<ul class="simple">
+<li><p>bugfix:</p>
+<ul>
+<li><p>the starred variant <a class="reference internal" href="#id13">\PolToSturm*{polname}{sturmname}</a> was
+broken. On the occasion of the fix, its meaning has been modified,
+see its documentation.</p></li>
+<li><p>using <a class="reference internal" href="#poltosturm">\PolToSturm</a> with a constant polynomial
+caused a division by zero error.</p></li>
+</ul>
+</li>
+<li><p>new macro:</p>
+<ul>
+<li><p><a class="reference internal" href="#id14">\PolSturmIsolateZeros*</a>
+acts like the <a class="reference internal" href="#polsturmisolatezeros">non-starred variant</a> then computes all the multiplicities.</p></li>
+</ul>
+</li>
+<li><p>new expandable macros:</p>
+<ul>
+<li><p><a class="reference internal" href="#polsturmisolatedzeromultiplicity-sturmname-index">\PolSturmIsolatedZeroMultiplicity{sturmname}{index}</a></p></li>
+<li><p><a class="reference internal" href="#polsturmnbofrootsof-sturmname-lessthanorequalto-value">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualTo{value}</a></p></li>
+<li><p><a class="reference internal" href="#polsturmnbofrootsof-sturmname-lessthanorequaltoexpr-expression">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualToExpr{expression}</a></p></li>
+<li><p><a class="reference internal" href="#polsturmnbwithmultofrootsof-sturmname-lessthanorequalto-value">\PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualTo{value}</a></p></li>
+<li><p><a class="reference internal" href="#polsturmnbwithmultofrootsof-sturmname-lessthanorequaltoexpr-expression">\PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualToExpr{expression}</a></p></li>
+</ul>
+</li>
+</ul>
+</li>
+<li><p>v0.7 (2018/12/08), v0.7.1 (bugfix), v0.7.2 (2nd bugfix) (2018/12/09)</p>
+<ul class="simple">
+<li><p>breaking changes:</p>
+<ul>
+<li><p>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: <span class="docutils literal">\PolPrintIntervalsTheEndPoint</span>,
+<span class="docutils literal"><span class="pre">\PolIfEndPointIsPositive{A}{B}</span></span>,
+<span class="docutils literal"><span class="pre">\PolIfEndPointIsNegative{A}{B}</span></span>, and
+<span class="docutils literal"><span class="pre">\PolIfEndPointIsZero{A}{B}</span></span>.</p></li>
+</ul>
+</li>
+<li><p>bugfix:</p>
+<ul>
+<li><p>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,</p></li>
+<li><p><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,</p></li>
+<li><p><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).</p></li>
+</ul>
+</li>
+<li><p>new macros:</p>
+<ul>
+<li><p><a class="reference internal" href="#id17">\PolSturmIsolateZeros**{sturmname}</a></p></li>
+<li><p><a class="reference internal" href="#polsturmisolatezerosgetmultiplicitiesandrationalroots-sturmname">\PolSturmIsolateZerosGetMultiplicitiesAndRationalRoots{sturmname}</a></p></li>
+<li><p><a class="reference internal" href="#polsturmisolatezerosandfindrationalroots-sturmname">\PolSturmIsolateZerosAndFindRationalRoots{sturmname}</a></p></li>
+<li><p><a class="reference internal" href="#polexprsetup">\polexprsetup</a></p></li>
+<li><p><a class="reference internal" href="#id21">\PolPrintIntervals*</a></p></li>
+<li><p><a class="reference internal" href="#polprintintervalsnorealroots">\PolPrintIntervalsNoRealRoots</a></p></li>
+<li><p><a class="reference internal" href="#polprintintervalsbeginenv">\PolPrintIntervalsBeginEnv</a></p></li>
+<li><p><a class="reference internal" href="#polprintintervalsendenv">\PolPrintIntervalsEndEnv</a></p></li>
+<li><p><a class="reference internal" href="#polprintintervalsknownroot">\PolPrintIntervalsKnownRoot</a></p></li>
+<li><p><a class="reference internal" href="#polprintintervalsunknownroot">\PolPrintIntervalsUnknownRoot</a></p></li>
+<li><p><a class="reference internal" href="#polprintintervalsprintmultiplicity">\PolPrintIntervalsPrintMultiplicity</a></p></li>
+</ul>
+</li>
+<li><p>new expandable macros:</p>
+<ul>
+<li><p><a class="reference internal" href="#polsturmnbofrationalroots-sturmname">\PolSturmNbOfRationalRoots{sturmname}</a></p></li>
+<li><p><a class="reference internal" href="#polsturmnbofrationalrootswithmultiplicities-sturmname">\PolSturmNbOfRationalRootsWithMultiplicities{sturmname}</a></p></li>
+<li><p><a class="reference internal" href="#polsturmrationalroot-sturmname-k">\PolSturmRationalRoot{sturmname}{k}</a></p></li>
+<li><p><a class="reference internal" href="#polsturmrationalrootindex-sturmname-k">\PolSturmRationalRootIndex{sturmname}{k}</a></p></li>
+<li><p><a class="reference internal" href="#polsturmrationalrootmultiplicity-sturmname-k">\PolSturmRationalRootMultiplicity{sturmname}{k}</a></p></li>
+<li><p><a class="reference internal" href="#polprintintervalsthevar">\PolPrintIntervalsTheVar</a></p></li>
+<li><p><a class="reference internal" href="#polprintintervalsthesturmname">\PolPrintIntervalsTheSturmName</a></p></li>
+<li><p><a class="reference internal" href="#polprintintervalsthemultiplicity">\PolPrintIntervalsTheMultiplicity</a></p></li>
+</ul>
+</li>
+</ul>
+</li>
+<li><p>v0.7.3 (2019/02/04)</p>
+<ul class="simple">
+<li><p>bugfix:</p>
+<ul>
+<li><p>Debugging information not destined to user showed in log if root
+finding was done under <span class="docutils literal">\xintverbosetrue</span> regime.</p></li>
+<li><p><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="#id21">\PolPrintIntervals*</a> (reported by Jürgen Gilg). Now remains
+defined in both cases, and <a class="reference internal" href="#polprintintervalsthesturmname">\PolPrintIntervalsTheSturmName</a>
+also.</p></li>
+<li><p>Polynomial names ending in digits caused errors (reported by Thomas
+Söll).</p></li>
+</ul>
+</li>
+</ul>
+</li>
+<li><p>v0.7.4 (2019/02/12)</p>
+<ul class="simple">
+<li><p>bugfix:</p>
+<ul>
+<li><p>20000000000 is too big for <span class="docutils literal">\numexpr</span>, shouldn't I know that?
+Thanks to Jürgen Gilg for report.</p></li>
+</ul>
+</li>
+</ul>
+</li>
+<li><p>v0.7.5 (2020/01/31)</p>
+<p>Synced with xintexpr 1.4. Requires it.</p>
+</li>
+<li><p>v0.8 (2021/03/29)</p>
+<p>Synced with xintexpr 1.4d. Requires it.</p>
+<ul class="simple">
+<li><p>breaking changes:</p>
+<ul>
+<li><p>As the usability of character <span class="docutils literal">'</span> in names has been extended
+from <span class="docutils literal">\poldef</span> to also generally <span class="docutils literal">\xintexpr</span>, <span class="docutils literal">\xintdefvar</span>,
+and <span class="docutils literal">\xintdeffunc</span>, it breaks there the infix operators
+<span class="docutils literal">'and'</span>, <span class="docutils literal">'or'</span>, <span class="docutils literal">'xor'</span> and <span class="docutils literal">'mod'</span>. See the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a>
+documentation for the <span class="docutils literal">&&</span>, <span class="docutils literal">||</span>, <span class="docutils literal">xor()</span> and <span class="docutils literal">/:</span>
+alternatives.</p></li>
+<li><p><a class="reference internal" href="#poltoexpr">\PolToExpr</a> by default uses a catcode 12
+<span class="docutils literal">^</span>. See its documentation and the new configuration
+<a class="reference internal" href="#poltoexprcaret">\PolToExprCaret</a>.</p></li>
+</ul>
+</li>
+<li><p>deprecated:</p>
+<ul>
+<li><p>Usage of <span class="docutils literal">P/Q</span> for the euclidean quotient of two polynomials is
+deprecated. Start using <span class="docutils literal">quo(P,Q)</span> in its place.</p></li>
+</ul>
+</li>
+<li><p>bugfix:</p>
+<ul>
+<li><p>The <span class="docutils literal">\xintglobaldefstrue</span> setting was obeyed only partially
+by the polexpr macros defining polynomials.</p></li>
+<li><p>The <span class="docutils literal">\xintexpr</span> variables storing the values of the extremities
+of the intervals as found by <a class="reference internal" href="#polsturmisolatezeros">\PolSturmIsolateZeros</a> were not updated at 0.7.5 to the
+xintexpr 1.4 format and thus caused low-level TeX errors if used.</p></li>
+<li><p>Attempting to use in <span class="docutils literal">\poldef</span> a function previously declared
+via <span class="docutils literal">\xintdeffunc</span> which made usage of the indexing or slicing
+"ople" syntax typically caused <span class="docutils literal">TeX capacity exceeded</span> error.
+Indeed 0.7.5 only partially made polexpr able to cope with the
+extended possibilities for xintexpr 1.4 user-declared functions.
+Hopefully <span class="docutils literal">0.8</span> achieves full functionality in this context.</p></li>
+</ul>
+</li>
+<li><p>new macros:</p>
+<ul>
+<li><p><a class="reference internal" href="#polnewpolverbosefalse">\polnewpolverbosefalse</a></p></li>
+<li><p><a class="reference internal" href="#poltoexprcaret">\PolToExprCaret</a></p></li>
+<li><p><a class="reference internal" href="#poltoexprinvar">\PolToExprInVar</a></p></li>
+<li><p>alongside the major new functionalities described in the next item
+<a class="reference internal" href="#poltypeset">\PolTypeset</a> and <a class="reference internal" href="#poltoexpr">\PolToExpr</a> have
+been enhanced to accept as argument a general expression and not
+only a pre-declared polynomial name.</p></li>
+</ul>
+</li>
+<li><p>new features:</p>
+<ul>
+<li><p>The package is usable under Plain and probably most any TeX format,
+and not only under LaTeX.</p></li>
+<li><p>The core of the package has been rewritten entirely in order to
+start letting <span class="docutils literal">\xintexpr</span> recognize a polynomial type as a genuine
+variable. This has allowed:</p>
+<ul>
+<li><p>to solve the reduced inter-operability problems between polexpr
+and <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> which arose as consequences to the deep <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> <span class="docutils literal">1.4</span>
+evolution,</p></li>
+<li><p>to make available most of the functionality associated to
+expandable macros directly in the <span class="docutils literal">\xinteval</span> syntax as
+operators or functions,</p></li>
+<li><p>to provide (expandable) functional interface in <span class="docutils literal">\xinteval</span> to
+features previously available only via (for some, non-expandable)
+macro interface such as gcd computations.</p></li>
+</ul>
+</li>
+</ul>
+</li>
+</ul>
+<p>See the updated <a class="reference internal" href="#quick-syntax-overview">Quick syntax overview</a> and then <a class="reference internal" href="#polexpr08">the extended syntax
+description</a>.</p>
+</li>
+</ul>
+</div>
+<div class="section" id="acknowledgments">
+<h1><a class="toc-backref" href="#id205">Acknowledgments</a></h1>
+<p>Thanks to Jürgen Gilg whose question about <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</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 <span class="docutils literal">0.6</span>, <span class="docutils literal">0.7</span>, and <span class="docutils literal">0.8</span>
+releases for their continued interest.</p>
+<p>See README.md for the License.</p>
+</div>
+</div>
+</body>
+</html>
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+native
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===================================================================
--- trunk/Master/texmf-dist/tex/generic/polexpr/polexpr.sty (rev 0)
+++ trunk/Master/texmf-dist/tex/generic/polexpr/polexpr.sty 2021-04-10 21:13:16 UTC (rev 58821)
@@ -0,0 +1,1057 @@
+% author: Jean-François Burnol
+% License: LPPL 1.3c (author-maintained)
+% Usage: \input polexpr.sty (Plain or other macro formats)
+% or \usepackage{polexpr} (LaTeX macro format)
+% polexpr.sty (this file) inputs:
+% polexprcore.tex
+% polexprexpr.tex
+% polexprsturm.tex
+\begingroup\catcode61\catcode48\catcode32=10\relax%
+ \catcode13=5 % ^^M
+ \endlinechar=13 %
+ \catcode123=1 % {
+ \catcode125=2 % }
+ \catcode64=11 % @
+ \catcode35=6 % #
+ \catcode44=12 % ,
+ \catcode45=12 % -
+ \catcode46=12 % .
+ \catcode58=12 % :
+ \def\z {\endgroup}%
+ \expandafter\let\expandafter\x\csname ver at polexpr.sty\endcsname
+ \expandafter\let\expandafter\w\csname ver at xintexpr.sty\endcsname
+ \expandafter
+ \ifx\csname PackageInfo\endcsname\relax
+ \def\y#1#2{\immediate\write-1{Package #1 Info: #2.}}%
+ \else
+ \def\y#1#2{\PackageInfo{#1}{#2}}%
+ \fi
+ \expandafter
+ % I don't think engine exists providing \expanded but not \numexpr
+ \ifx\csname expanded\endcsname\relax
+ \y{polexpr}{\expanded not available, aborting input}%
+ \aftergroup\endinput
+ \else
+ \ifx\x\relax % plain-TeX, first loading of polexpr.sty
+ \ifx\w\relax % but xintexpr.sty not yet loaded.
+ \expandafter\def\expandafter\z\expandafter
+ {\z\input xintexpr.sty\relax}%
+ \fi
+ \else
+ \def\empty {}%
+ \ifx\x\empty % LaTeX, first loading,
+ % variable is initialized, but \ProvidesPackage not yet seen
+ \ifx\w\relax % xintexpr.sty not yet loaded.
+ \expandafter\def\expandafter\z\expandafter
+ {\z\RequirePackage{xintexpr}[2021/02/20]}%
+ \fi
+ \else
+ \aftergroup\endinput % polexpr already loaded.
+ \fi
+ \fi
+ \fi
+\z%
+\XINTsetupcatcodes%
+\XINT_providespackage
+\ProvidesPackage{polexpr}%
+ [2021/03/29 v0.8 Polynomial expressions with rational coefficients (JFB)]%
+\begingroup
+ \def\x#1/#2/#3 #4\xint:{#1#2#3}%
+ \ifnum\expandafter\x\expanded{\csname ver at xintexpr.sty\endcsname}\xint:
+ <20210220 % actually 20200131 (xint 1.4) is presumably ok
+ \immediate\write128{! Package polexpr error: xintexpr too old, aborting input}%
+ \else\expandafter\xint_gobble_i
+ \fi
+\endinput\endgroup
+\let\PolDecToString\xintDecToString
+\long\def\POL at ifstar#1#2%
+{%
+ \begingroup\def\@tempa{#1}\def\@tempb{#2}%
+ \futurelet\@let at token\POL@@ifstar
+}%
+\def\POL@@ifstar
+{%
+ \xint_firstofone{\ifx} \@let at token\def\next{\POL@@again\POL@@ifstar}\else
+ \ifx*\@let at token\def\next##1{\expandafter\endgroup\@tempa}\else
+ \def\next{\expandafter\endgroup\@tempb}\fi\fi\next
+}%
+\xint_firstofone{\def\POL@@again#1} {\futurelet\@let at token#1}%
+\long\def\POL at chkopt#1[#2]%
+{%
+ \begingroup\def\@tempa{#1}\def\@tempb{#1[#2]}%
+ \futurelet\@let at token\POL@@ifopt
+}%
+\def\POL@@ifopt
+{%
+ \xint_firstofone{\ifx} \@let at token\def\next{\POL@@again\POL@@ifopt}\else
+ \ifx[\@let at token\def\next{\expandafter\endgroup\@tempa}\else %]
+ \def\next{\expandafter\endgroup\@tempb}\fi\fi\next
+}%
+% \polexprsetup added at 0.7
+\catcode`! 3
+\def\polexprsetup#1{\POL at setup_parsekeys #1,=!,\xint_bye}%
+\def\POL at setup_parsekeys #1=#2#3,{%
+ \ifx!#2\expandafter\xint_bye\fi
+ \csname POL at setup_setkey_\xint_zapspaces #1 \xint_gobble_i\endcsname
+ \xint_firstoftwo
+ {\PackageWarning{polexpr}{The \detokenize{#1} key is unknown! ignoring}}%
+ {\xintZapLastSpaces{#2#3}}%
+ \POL at setup_parsekeys
+}%
+\def\POL at setup_setkey_norr #1#2{\edef\POL at norr}%
+\def\POL at setup_setkey_sqfnorr #1#2{\edef\POL at sqfnorr}%
+\polexprsetup{norr=_norr, sqfnorr=_sqf_norr}
+\catcode`! 11 % special catcode for ! as used in xintexpr.sty
+%
+\newif\ifxintveryverbose
+\newif\ifpolnewpolverbose
+\newif\ifpoltypesetall
+\newif\ifpoltoexprall
+%%
+%% Main data format for non-expandable manipulations
+%%
+%% The main exchange structure is:
+%% N.\empty{coeff0}{coeff1}....{coeffN}
+%% It is stored in macros \POLuserpol@<name of polynomial>
+%% The \empty is basically there to avoid brace-stripping
+%% in some grabbing contexts (maybe I should revisit this)
+%%
+%% The zero polynomial is stored as -1.\empty{0/1[0]}
+%% Degree zero polynomials are 0.\empty{numeric value}
+%%
+%% Depending on input path the numeric values coeff0, coeff1, ...., coeffN
+%% may have been or not already converted into A/B[n] format.
+%% As a rule, computations are not followed with reducing the fractions
+%% to smallest terms; the innocent may be unaware that computing
+%% with fractions quickly give gigantic numbers. There is \PolReduceCoeffs
+%% to do that.
+%%
+%% This base structure is maintained at 0.8 for legacy reasons but perhaps I
+%% need to revisit this. A characteristic of the package so far is that it
+%% thus stores and manipulate polynomials basically as the complete sequence
+%% of coefficients, (using the xintfrac "zero" for missing coefficients) which
+%% means that it will handle poorly polynomials of high degrees such as X^500.
+%%
+%% Test if zero
+\def\POL at ifZero#1{\expandafter\POL at ifZero@aux#1;}%
+\def\POL at ifZero@aux #1#2;{\if-#1\expandafter\xint_firstoftwo
+ \else\expandafter\xint_secondoftwo
+ \fi}%
+%% Split into degree and coefficients
+% The \expandafter chain removes the \empty token
+\def\POL at split#1.#2;#3#4%
+ {\def#3{#1}\expandafter\def\expandafter#4\expandafter{#2}}%
+%% Define from values stored in a "macros-array"
+\def\POL at resultfromarray #1{%
+ \edef\POL at result{\ifnum\count@>\z@
+ \the\numexpr\count at -\@ne.\noexpand\empty
+ \xintiloop [1+1]%
+ \expandafter\POL at braceit\csname POL at array#1\xintiloopindex\endcsname
+ \ifnum\xintiloopindex<\count@
+ \repeat
+ \else-1.\noexpand\empty{0/1[0]}\fi}%
+}%
+\def\POL at braceit#1{{#1}}% needed as \xintiloopindex can not "see" through braces
+%%
+%% Conversion between legacy data storage and the one used for the
+%% the novel polexpr 0.8 notion of \xintexpr polynomial variables
+%%
+%% The 0.8 expandable implementation of core algebra is also manipulating
+%% the complete list of coefficients. The internal data structure is
+%% (this is the numeric leaf in xintexpr ople terminology) currently:
+%% PN.{coeff0}{coeff1}....{coeffN}
+%% where the P letter identifies the polynomial type.
+%% Here the degree N is *always* at least 1: if some evaluation ends
+%% up in a constant polynomial it will always be output as a genuine
+%% scalar numeric variable, as a rule in in A/B[n] format
+%%
+%% This is not definitive and I need to think about it more (in particular
+%% in the distant perspective of supporting multi-variable polynomials).
+%% However modifying this will be costly labor at this stage.
+%%
+\input polexprcore.tex\relax % load expandable algebra
+\def\POL at vartolegacy #1% \romannumeral\POL at vartolegacy ... \xint:
+{%
+ \if 0#1\xint_dothis\POL at vartolegacy@zero\fi
+ \if P#1\xint_dothis\POL at vartolegacy@pol\fi
+ \xint_orthat\POL at vartolegacy@scalar #1%
+}%
+\def\POL at vartolegacy@zero #1\xint:{\xint_c_ -1.\empty{0/1[0]}}%
+\def\POL at vartolegacy@scalar #1\xint:{\xint_c_ 0.\empty{#1}}%
+\def\POL at vartolegacy@pol P#1.#2\xint:{\xint_c_ #1.\empty#2}%
+%
+\def\POL at tovar#1{\romannumeral\expandafter\expandafter\expandafter
+ \POL at legacytovar\csname POLuserpol@#1\endcsname}%
+\def\POL at legacytovar #1.% \romannumeral\POL at legacytovar N.\empty{c0}...
+{%
+ \ifnum #1<\xint_c_i\xint_dothis\POL at legacytovar@scalar\fi
+ \xint_orthat\POL at legacytovar@pol #1.%
+}%
+\def\POL at legacytovar@scalar #1.\empty#2{\xint_c_ #2}%
+\def\POL at legacytovar@pol #1.\empty{\xint_c_ P#1.}%
+%%
+%% Extend \xintexpr (\xintdefvar, \xintdeffunc) to recognize the new
+%% polynomial type
+%%
+%% **** It does NOT apply to \xintfloatexpr context
+%%
+\input polexprexpr.tex\relax
+%%
+%% \poldef
+%%
+\def\PolDef{\POL at chkopt\POL at oPolDef[x]}%
+\def\POL at oPolDef[#1]#2#3{\poldef #2(#1):=#3;}%
+\def\poldef{\edef\POL at restoresemicolon{\catcode59=\the\catcode59\relax}%
+ \catcode59 12 \POL at defpol}%
+\def\POL at defpol #1(#2)#3=#4;{%
+ \POL at restoresemicolon
+ \edef\POL at polname{\xint_zapspaces #1 \xint_gobble_i}%
+\begingroup
+ \unless\ifxintveryverbose\xintverbosefalse\fi
+ %% RADICAL CHANGE AT 0.8:
+ %% we define a **variable** not a **function**
+ %% ever since polexpr initial version, a function was defined and
+ %% the associated macros was then deconstructed in further analysis
+ %% via non-expandable approach. At 0.8 the polynomial algebra has
+ %% been implemented expandably allowing direct plug-in into \xintexpr
+ \xintdefvar __pol = subs(#4,#2=qraw({{P1.{0/1[0]}{1/1[0]}}}));%
+ \expandafter
+\endgroup
+ \expandafter\def\expandafter\POL at result\expandafter
+ {\romannumeral0\expandafter\xint_stop_atfirstofone
+ \romannumeral0\csname XINT_expr_varvalue___pol\endcsname}%
+ \XINT_global\expandafter\def\csname POLuserpol@\POL at polname\expandafter\endcsname
+ \expandafter{\romannumeral\expandafter\POL at vartolegacy\POL at result\xint:}%
+ \expandafter\POL at newpol\expandafter{\POL at polname}%
+}%
+\def\POL at newpol#1{%
+ % 0.7.5 had some complicated special handling of constant
+ % polynomials, but these are complications of the past
+ % First a variable usable in \poldef but not in \xintexpr for arithmetic
+ % only for special dedicated functions such as coeff(), deg()
+ % (when they will be implemented). In \poldef, composition of polynomials
+ % in P(Q) syntax will be more efficient than P(Q(x)).
+ % This will use \XINT_global and obey \xintverbose... setting
+ \XINT_expr_defvar_one{#1}{{\POL at tovar{#1}}}%
+ % Second a function usable not only in \poldef but also in \xintexpr
+ % Will use \XINT_global
+ \POL at newpolhorner{#1}%
+ \POL at defpolfunc{#1}{expr}%
+ \XINT_global\expandafter\let\csname XINT_flexpr_func_#1\endcsname\@undefined
+ \ifpolnewpolverbose\POL at info{#1}\fi
+}%
+\def\POL at newfloatpol#1{%
+ \POL at newfloatpolhorner{#1}%
+ \POL at defpolfunc{#1}{flexpr}%
+ \ifpolnewpolverbose\POL at floatinfo{#1}%
+ \else
+ \ifxintverbose\POL at floatinfo{#1}\fi
+ \fi
+}%
+\def\POL at info #1{%
+ \xintMessage {polexpr}{Info}%
+ {Function #1 for the \string\xintexpr\space parser is
+ \ifxintglobaldefs(globally) \fi
+ associated to \string\XINT_expr_polfunc_#1\space
+ with meaning:
+ \expandafter\meaning
+ \csname XINT_expr_polfunc_#1\endcsname}%
+}%
+\def\POL at floatinfo #1{%
+ \xintMessage {polexpr}{Info}%
+ {Function #1 for the \string\xintfloatexpr\space parser is
+ \ifxintglobaldefs(globally) \fi
+ associated to \string\XINT_flexpr_polfunc_#1\space
+ with meaning:
+ \expandafter\meaning
+ \csname XINT_flexpr_polfunc_#1\endcsname}%
+}%
+%
+\def\POL at newpolhorner#1{%
+ \expandafter\expandafter\expandafter\POL at split
+ \csname POLuserpol@#1\endcsname;\POL at var@deg\POL at var@coeffs
+ \edef\POL at var@coeffs{\xintRevWithBraces{\POL at var@coeffs}}%
+ \begingroup
+ \expandafter\POL at newpol@horner\POL at var@coeffs\relax
+ \expandafter
+ \endgroup
+ \expandafter\XINT_global
+ \expandafter\def\csname XINT_expr_polfunc_#1\expandafter\endcsname
+ \expandafter##\expandafter1\expandafter{\POL at tmp{##1}}%
+}%
+\def\POL at 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 at split
+ \csname POLuserpol@#1\endcsname;\POL at var@deg\POL at var@coeffs
+ \edef\POL at var@coeffs{\xintRevWithBraces{\POL at var@coeffs}}%
+ \begingroup
+ \expandafter\POL at newpol@floathorner\POL at var@coeffs\relax
+ \expandafter
+ \endgroup
+ \expandafter\def\csname XINT_flexpr_polfunc_#1\expandafter\endcsname
+ \expandafter##\expandafter1\expandafter{\POL at tmp{##1}}%
+}%
+\def\POL at newpol@horner#1{\let\xintPolAdd\relax\let\xintPolMul\relax
+ \def\POL at tmp##1{#1}\POL at newpol@horner at loop.}%
+\def\POL at newpol@horner at loop.#1{%
+ \if\relax#1\expandafter\xint_gob_til_dot\fi
+ \edef\POL at tmp##1{\xintiiifZero{#1}
+ {\xint_firstofone}{\xintPolAdd{#1}}%
+ {\xintPolMul{##1}{\POL at tmp{##1}}}}%
+ \POL at newpol@horner at loop.%
+}%
+\def\POL at newpol@floathorner#1{\let\XINTinFloatAdd\relax\let\XINTinFloatMul\relax
+ \edef\POL at tmp##1{\XINTinFloatdigits{#1}}%
+ \POL at newpol@floathorner at loop.}%
+\def\POL at newpol@floathorner at loop.#1{%
+ \if\relax#1\expandafter\xint_gob_til_dot\fi
+ \edef\POL at tmp##1{\xintiiifZero{#1}
+ {\xint_firstofone}{\XINTinFloatAdd{\XINTinFloatdigits{#1}}}%
+ {\XINTinFloatMul{##1}{\POL at tmp{##1}}}}%
+ \POL at newpol@floathorner at loop.%
+}%
+%%
+%% Non-expandable polynomial manipulations
+%%
+\def\PolGenFloatVariant#1{\POL at newfloatpol{#1}}%
+%
+\def\PolLet#1#2{\if=\noexpand#2\expandafter\xint_firstoftwo
+ \else\expandafter\xint_secondoftwo\fi
+ \POL@@let\POL at let{#1}{#2}}%
+\def\POL@@let#1#2#3{\POL at let{#1}{#3}}%
+\def\POL at let#1#2{%
+ \XINT_global
+ \expandafter\let\csname POLuserpol@#1\expandafter\endcsname
+ \csname POLuserpol@#2\endcsname
+ \XINT_expr_defvar_one{#1}{{\POL at tovar{#1}}}%
+ \XINT_global
+ \expandafter\let\csname XINT_expr_polfunc_#1\expandafter\endcsname
+ \csname XINT_expr_polfunc_#2\endcsname
+ \POL at defpolfunc{#1}{expr}%
+ \ifpolnewpolverbose\POL at info{#1}\fi
+}%
+\def\PolGlobalLet#1#2{\begingroup\xintglobaldefstrue\PolLet{#1}{#2}\endgroup}
+%
+\def\PolAssign#1{\def\POL at polname{#1}\POL at assign}% zap spaces in #1?
+\def\POL at assign#1\toarray#2{%
+ \expandafter\expandafter\expandafter\POL at split
+ \csname POLuserpol@\POL at polname\endcsname;\POL at var@deg\POL at var@coeffs
+ \xintAssignArray\POL at 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 at 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}{ }%
+}%
+\def\PolGet{}%
+\def\PolGet#1#2\fromarray#3{%
+ \begingroup % closed in \POL at getfromarray
+ \POL at getfromarray{#1}{#3}%
+ \POL at newpol{#1}%
+}%
+\def\POL at getfromarray#1#2{%
+ \count@=#2{0} %<- intentional space
+ \ifnum\count@=\z@
+ \def\POL at result{-1.\empty{0/1[0]}}% 0.5 fix for empty array
+ \else
+ \xintloop
+ \edef\POL at tmp{#2{\count@}}%
+ \edef\POL at tmp{\xintRaw{\POL at tmp}}%
+% sadly xinttools (current 1.3a) arrays have no setters for individual items...
+ \expandafter\let\csname POL at tmparray\the\count@\endcsname\POL at tmp
+ \if0\xintiiSgn{\POL at tmp}%
+ \advance\count@\m at ne
+ \repeat
+ \count\tw@\count@
+ \xintloop
+ \ifnum\count@>\@ne
+ \advance\count@\m at ne
+ \edef\POL at tmp{#2{\count@}}%
+ \edef\POL at tmp{\xintRaw{\POL at tmp}}%
+ \expandafter\let\csname POL at tmparray\the\count@\endcsname\POL at tmp
+ \repeat
+ \count@\count\tw@
+ \def\POL at tmp##1.{{\csname POL at tmparray##1\endcsname}}%
+ \edef\POL at result{\the\numexpr\count at -\@ne.\noexpand\empty
+ \xintiloop[1+1]%
+ \expandafter\POL at tmp\xintiloopindex.%
+ \ifnum\xintiloopindex<\count@
+ \repeat}%
+ \fi
+ \expandafter
+ \endgroup
+ \expandafter
+ \XINT_global
+ \expandafter
+ \def\csname POLuserpol@#1\expandafter\endcsname
+ \expandafter{\POL at result}%
+}%
+%
+\def\PolFromCSV#1#2{%
+ \begingroup % closed in \POL at getfromarray
+ \xintAssignArray\xintCSVtoList{#2}\to\POL at arrayA
+ \POL at getfromarray{#1}\POL at arrayA
+ \POL at newpol{#1}%
+}%
+%
+\def\PolMapCoeffs#1#2{% #1 = macro, #2 = name
+ \POL at mapcoeffs{#1}{#2}%
+ \POL at newpol{#2}%
+}%
+\def\POL at mapcoeffs#1#2{%
+ \begingroup
+ \def\POL at mapcoeffs@macro{#1}%
+ \expandafter\expandafter\expandafter\POL at split
+ \csname POLuserpol@#2\endcsname;\POL at mapcoeffs@deg\POL at mapcoeffs@coeffs
+% ATTENTION à ne pas faire un \expandafter ici, car brace removal si 1 item
+ \xintAssignArray\POL at mapcoeffs@coeffs\to\POL at arrayA
+ \def\index{0}%
+ \count@\z@
+ \expandafter\POL at map@loop\expandafter.\POL at mapcoeffs@coeffs\relax
+ \xintloop
+% this abuses that \POL at arrayA0 is never 0.
+ \xintiiifZero{\csname POL at arrayA\the\count@\endcsname}%
+ {\iftrue}%
+ {\iffalse}%
+ \advance\count@\m at ne
+ \repeat
+% donc en sortie \count@ est 0 ssi pol nul.
+ \POL at resultfromarray A%
+ \expandafter
+ \endgroup
+ \expandafter
+ \XINT_global
+ \expandafter
+ \def\csname POLuserpol@#2\expandafter\endcsname\expandafter{\POL at result}%
+}%
+\def\POL at map@loop.#1{\if\relax#1\expandafter\xint_gob_til_dot\fi
+ \advance\count@\@ne
+ \edef\POL at map@coeff{\POL at mapcoeffs@macro{#1}}%
+ \expandafter
+ \let\csname POL at arrayA\the\count@\endcsname\POL at map@coeff
+ \edef\index{\the\numexpr\index+\@ne}%
+ \POL at map@loop.}%
+%
+\def\POL at xintIrr#1{\xintIrr{#1}[0]}%
+\def\PolReduceCoeffs{\POL at ifstar\POL at sreducecoeffs\POL at reducecoeffs}%
+\def\POL at reducecoeffs#1{\PolMapCoeffs{\POL at xintIrr}{#1}}%
+\def\POL at sreducecoeffs#1{\PolMapCoeffs{\xintPIrr}{#1}}%
+%
+\def\PolMakeMonic#1{%
+ \edef\POL at leadingcoeff{\PolLeadingCoeff{#1}}%
+ \edef\POL at leadingcoeff@inverse{\xintDiv{1/1[0]}{\POL at leadingcoeff}}%
+ \PolMapCoeffs{\xintMul{\POL at leadingcoeff@inverse}}{#1}%
+}%
+%
+%% \PolMakePrimitive (0.5)
+% This uses expandable \PolIContent
+% Note: the integer coefficients stored in A/1[n] form with
+% A not having trailing zeroes, due to usage of \xintREZ here.
+\def\POL at makeprim@macro#1%
+ {\xintREZ{\xintNum{\xintDiv{#1}{\POL at makeprim@icontent}}}}%
+\def\PolMakePrimitive#1{%
+ % This does not need a full user declared polynomial on input, only
+ % a \POLuserpol at name macro, but on output it is fully declared
+ \edef\POL at makeprim@icontent{\PolIContent{#1}}%
+ \PolMapCoeffs\POL at makeprim@macro{#1}%
+}%
+\def\POL at makeprimitive#1{%
+ % Avoids declaring the polynomial, internal usage in \PolToSturm
+ \edef\POL at makeprim@icontent{\PolIContent{#1}}%
+ \POL at mapcoeffs\POL at makeprim@macro{#1}%
+}%
+%
+%% Euclidean division
+% now based on the expandable routine from polexprcore.tex
+%
+\def\PolDivide#1#2#3#4{% #3=quotient, #4=remainder of #1 by #2
+ \POL at divide{#1}{#2}%
+ \XINT_global\expandafter\let\csname POLuserpol@#3\endcsname\POL at Q
+ \POL at newpol{#3}%
+ \XINT_global\expandafter\let\csname POLuserpol@#4\endcsname\POL at R
+ \POL at newpol{#4}%
+}%
+\def\PolQuo#1#2#3{% #3=quotient of #1 by #2
+ \POL at divide{#1}{#2}%
+ \XINT_global\expandafter\let\csname POLuserpol@#3\endcsname\POL at Q
+ \POL at newpol{#3}%
+}%
+\def\PolRem#1#2#3{% #3=remainder of #1 by #2
+ \POL at divide{#1}{#2}%
+ \XINT_global\expandafter\let\csname POLuserpol@#3\endcsname\POL at R
+ \POL at newpol{#3}%
+}%
+\def\POL at divide#1#2{%
+ % much simpler at 0.8 thanks to our expandable macros
+ \xintAssign\xintPolQuoRem{\POL at tovar{#1}}{\POL at tovar{#2}}\to\POL at Q\POL at R
+ \odef\POL at Q{\romannumeral\expandafter\POL at vartolegacy\POL at Q\xint:}%
+ \odef\POL at R{\romannumeral\expandafter\POL at vartolegacy\POL at R\xint:}%
+}%
+%% Euclidean special pseudo-remainder
+\def\POL at getprem#1#2{%
+ \let\POL at Q\undefined % trap errors in Sturm code update to use \POL at prem
+ % this was simpler before I converted \xintPolPRem into returning a tuple...
+ \odef\POL at R{\romannumeral\expandafter\POL at vartolegacy
+ \romannumeral0\expandafter\xint_stop_atsecondoftwo
+ \romannumeral`&&@\xintPolPRem{\POL at tovar{#1}}{\POL at tovar{#2}}%
+ \xint:}%
+}%
+%
+%%%%%%%%%%%%
+%%
+%% Things are currenly implemented twice : here the legacy macros
+%% such as GCD or Diff, and in polexprcore.tex the expandable
+%% support macros for the \xinteval interface.
+%%
+%% Soon, I will probably remove all legacy code (like I did already
+%% for division) and make the user macros simple wrappers to the
+%% expandable code.
+%%
+%% But for 0.8 release, I preferred not to yet, as I did not have
+%% really the time to compare speed. Usage of the "special
+%% pseudo euclidean remainder" (expandable) code in Sturm chain
+%% construction proved very beneficial as it divided by 3 the
+%% \PolToSturm execution time on the Wilkinson perturbed type 1
+%% example in the documentation.
+%%
+%%%%%%%%%%%%
+%
+%% GCD
+%
+% It seems I didn't even use here the (now deleted) macros implementing
+% division, and I redid here what was needed: this code, which I leave
+% standing as I have other priorities, does not use the \POL at divide !
+%
+\def\PolGCD#1#2#3{% sets #3 to the (unitary) G.C.D. of #1 and #2
+ \POL at GCD{#1}{#2}{#3}%
+ \POL at newpol{#3}%
+}%
+\def\POL at GCD #1#2#3{%
+ \begingroup
+ \expandafter\let\expandafter\POL at A\csname POLuserpol@#1\endcsname
+ \expandafter\let\expandafter\POL at B\csname POLuserpol@#2\endcsname
+ \expandafter\POL at split\POL at A;\POL at degA\POL at polA
+ \expandafter\POL at split\POL at B;\POL at degB\POL at polB
+ \ifnum\POL at degA<\z@
+ \expandafter\xint_firstoftwo\else\expandafter\xint_secondoftwo
+ \fi
+ {\ifnum\POL at degB<\z@
+ \expandafter\xint_firstoftwo\else\expandafter\xint_secondoftwo
+ \fi
+ {\def\POL at result{-1.\empty{0/1[0]}}}%
+ {\xintAssignArray\POL at polB\to\POL at arrayB
+ \POL at normalize{B}%
+ \POL at gcd@exit BA}}%
+ {\ifnum\POL at degB<\z@
+ \expandafter\xint_firstoftwo\else\expandafter\xint_secondoftwo
+ \fi
+ {\xintAssignArray\POL at polA\to\POL at arrayA
+ \POL at normalize{A}%
+ \POL at gcd@exit AB}%
+ {\ifnum\POL at degA<\POL at degB\space
+ \let\POL at tmp\POL at B\let\POL at B\POL at A\let\POL at A\POL at tmp
+ \let\POL at tmp\POL at degB\let\POL at degB\POL at degA\let\POL at degA\POL at tmp
+ \let\POL at tmp\POL at polB\let\POL at polB\POL at polA\let\POL at polA\POL at tmp
+ \fi
+ \xintAssignArray\POL at polA\to\POL at arrayA
+ \xintAssignArray\POL at polB\to\POL at arrayB
+ \POL at gcd AB%
+ }}%
+ \expandafter
+ \endgroup
+ \expandafter
+ \XINT_global
+ \expandafter\def\csname POLuserpol@#3\expandafter\endcsname
+ \expandafter{\POL at result}%
+}%
+\def\POL at normalize#1{%
+ \expandafter\def\expandafter\POL at tmp\expandafter
+ {\csname POL at array#1\csname POL at array#10\endcsname\endcsname}%
+ \edef\POL at normalize@leading{\POL at tmp}%
+ \expandafter\def\POL at tmp{1/1[0]}%
+ \count@\csname POL at deg#1\endcsname\space
+ \xintloop
+ \ifnum\count@>\z@
+ \expandafter\edef\csname POL at array#1\the\count@\endcsname
+ {\xintIrr{\xintDiv
+ {\csname POL at array#1\the\count@\endcsname}%
+ {\POL at normalize@leading}}[0]}%
+ \advance\count@\m at ne
+ \repeat
+}%
+\def\POL at gcd#1#2{%
+ \POL at normalize{#2}%
+ \edef\POL at degQ{\the\numexpr\csname POL at deg#1\endcsname
+ -\csname POL at deg#2\endcsname}%
+ \count@\numexpr\csname POL at deg#1\endcsname+\@ne\relax
+ \count\tw@\numexpr\POL at degQ+\@ne\relax
+ \xintloop
+ \POL at gcd@getremainder at loopbody#1#2%
+ \ifnum\count\tw@>\z@
+ \repeat
+ \expandafter\def\csname POL at array#10\endcsname{1}%
+ \xintloop
+ \xintiiifZero{\csname POL at array#1\the\count@\endcsname}%
+ {\iftrue}%
+ {\iffalse}%
+ \advance\count@\m at ne
+ \repeat
+ \expandafter\edef\csname POL at deg#1\endcsname{\the\numexpr\count at -\@ne}%
+ \ifnum\count@<\@ne
+ \expandafter\POL at gcd@exit
+ \else
+ \expandafter\edef\csname POL at array#10\endcsname{\the\count@}%
+ \expandafter\POL at gcd
+ \fi{#2}{#1}%
+}%
+\def\POL at gcd@getremainder at loopbody#1#2{%
+ \edef\POL at gcd@ratio{\csname POL at array#1\the\count@\endcsname}%
+ \advance\count@\m at ne
+ \advance\count\tw@\m at ne
+ \count4 \count@
+ \count6 \csname POL at deg#2\endcsname\space
+ \xintloop
+ \ifnum\count6>\z@
+ \expandafter\edef\csname POL at array#1\the\count4\endcsname
+ {\xintSub
+ {\csname POL at array#1\the\count4\endcsname}%
+ {\xintMul
+ {\POL at gcd@ratio}%
+ {\csname POL at array#2\the\count6\endcsname}}}%
+ \advance\count4 \m at ne
+ \advance\count6 \m at ne
+ \repeat
+}%
+\def\POL at gcd@exit#1#2{%
+ \count@\numexpr\csname POL at deg#1\endcsname+\@ne\relax
+ \POL at resultfromarray #1%
+}%
+%
+%% DIFFERENTIATION
+%
+\def\POL at diff@loop at one #1/#2[#3]#4%
+ {\xintIrr{\xintiiMul{#4}{#1}/#2[0]}[#3]}%
+\def\POL at diff#1{\POL at diff@loop1.}%
+\def\POL at diff@loop#1.#2{%
+ \if\relax#2\expandafter\xint_gob_til_dot\fi
+ {\expandafter\POL at diff@loop at one\romannumeral0\xintraw{#2}{#1}}%
+ \expandafter\POL at diff@loop\the\numexpr#1+\@ne.%
+}%
+\def\PolDiff{\POL at chkopt\POL at oPolDiff[1]}%
+\def\POL at oPolDiff[#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 at iterindex{\the\numexpr#1\relax}%
+ \ifnum\POL at iterindex<\z@
+ \expandafter\xint_firstoftwo
+ \else
+ \expandafter\xint_secondoftwo
+ \fi
+ {\PolAntiDiff[-\POL at iterindex]}{\POL at Diff}%
+}%
+\def\POL at Diff{%
+ \ifcase\POL at iterindex\space
+ \expandafter\POL at Diff@no
+ \or\expandafter\POL at Diff@one
+ \else\xint_afterfi{\POL at Iterate\POL at Diff@one}%
+ \fi
+}%
+\def\POL at Diff@no #1#2{\POL at let{#2}{#1}}%
+\def\POL at Diff@one #1#2{\POL at Diff@@one {#1}{#2}\POL at newpol{#2}}%
+\def\POL at Diff@@one#1#2{%
+ \expandafter\expandafter\expandafter\POL at split
+ \csname POLuserpol@#1\endcsname;\POL at var@deg\POL at var@coeffs
+ \ifnum\POL at var@deg<\@ne
+ \XINT_global\@namedef{POLuserpol@#2}{-1.\empty{0/1[0]}}%
+ \else
+ \edef\POL at var@coeffs{\expandafter\POL at diff\POL at var@coeffs\relax}%
+ \XINT_global\expandafter\edef\csname POLuserpol@#2\endcsname
+ {\the\numexpr\POL at var@deg-\@ne.\noexpand\empty\POL at var@coeffs}%
+ \fi
+}%
+% lazy way but allows to share with AntiDiff
+\def\POL at Iterate#1#2#3{%
+ \begingroup
+ \xintverbosefalse
+ #1{#2}{#3}%
+ \xintloop
+ \ifnum\POL at iterindex>\tw@
+ #1{#3}{#3}%
+ \edef\POL at iterindex{\the\numexpr\POL at iterindex-\@ne}%
+ \repeat
+ \expandafter
+ \endgroup\expandafter
+ \XINT_global
+ \expandafter
+ \def\csname POLuserpol@#3\expandafter\endcsname
+ \expandafter{\romannumeral`&&@\csname POLuserpol@#3\endcsname}%
+ #1{#3}{#3}%
+}%
+%
+%% ANTI-DIFFERENTIATION
+%
+\def\POL at antidiff@loop at one #1/#2[#3]#4%
+ {\xintIrr{#1/\xintiiMul{#4}{#2}[0]}[#3]}%
+\def\POL at antidiff{\POL at antidiff@loop1.}%
+\def\POL at antidiff@loop#1.#2{%
+ \if\relax#2\expandafter\xint_gob_til_dot\fi
+ {\expandafter\POL at antidiff@loop at one\romannumeral0\xintraw{#2}{#1}}%
+ \expandafter\POL at antidiff@loop\the\numexpr#1+\@ne.%
+}%
+\def\PolAntiDiff{\POL at chkopt\POL at oPolAntiDiff[1]}%
+\def\POL at oPolAntiDiff[#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 at iterindex{\the\numexpr#1\relax}%
+ \ifnum\POL at iterindex<\z@
+ \expandafter\xint_firstoftwo
+ \else
+ \expandafter\xint_secondoftwo
+ \fi
+ {\PolDiff[-\POL at iterindex]}{\POL at AntiDiff}%
+}%
+\def\POL at AntiDiff{%
+ \ifcase\POL at iterindex\space
+ \expandafter\POL at AntiDiff@no
+ \or\expandafter\POL at AntiDiff@one
+ \else\xint_afterfi{\POL at Iterate\POL at AntiDiff@one}%
+ \fi
+}%
+\let\POL at AntiDiff@no\POL at Diff@no
+\def\POL at AntiDiff@one #1#2{\POL at AntiDiff@@one{#1}{#2}\POL at newpol{#2}}%
+\def\POL at AntiDiff@@one#1#2{%
+ \expandafter\expandafter\expandafter\POL at split
+ \csname POLuserpol@#1\endcsname;\POL at var@deg\POL at var@coeffs
+ \ifnum\POL at var@deg<\z@
+ \XINT_global\@namedef{POLuserpol@#2}{-1.\empty{0/1[0]}}%
+ \else
+ \edef\POL at var@coeffs{\expandafter\POL at antidiff\POL at var@coeffs\relax}%
+ \XINT_global\expandafter\edef\csname POLuserpol@#2\endcsname
+ {\the\numexpr\POL at var@deg+\@ne.\noexpand\empty{0/1[0]}\POL at var@coeffs}%
+ \fi
+}%
+%
+%%
+%% Localization of roots
+%%
+% this is big. It provides also output macros, of both expandable and
+% non-expandable type
+\input polexprsturm.tex\relax
+%
+%
+%% Non-expandable output macros
+%
+\def\PolTypesetCmdPrefix#1{\xintiiifSgn{#1}{}{+}{+}}%
+\def\PolTypesetCmd#1{\xintifOne{\xintiiAbs{#1}}%
+ {\ifnum\PolIndex=\z@\xintiiSgn{#1}\else
+ \xintiiifSgn{#1}{-}{}{}\fi
+ \let\PolIfCoeffIsPlusOrMinusOne\xint_firstoftwo}%
+ {\PolTypesetOne{#1}%
+ \let\PolIfCoeffIsPlusOrMinusOne\xint_secondoftwo}%
+ }%
+\ifdefined\frac
+\def\PolTypesetOne{\xintSignedFrac}%
+\else
+\def\PolTypesetOne{\xintSignedFwOver}%
+\fi
+\catcode`^ 7
+\def\PolTypesetMonomialCmd{%
+ \ifcase\PolIndex\space
+ %
+ \or\PolVar
+ \else\PolVar^{\PolIndex}%
+ \fi
+}%
+\catcode`^ 11 % normal xint catcode
+\def\PolTypeset{\POL at ifstar
+ {\def\POL at ts@ascending{1}\POL at Typeset}%
+ {\def\POL at ts@ascending{0}\POL at Typeset}%
+}%
+%%
+%% \PolTypeset
+%%
+%% extended at 0.8 to handle arbitrary expressions on input
+%%
+\def\POL at Typeset{\POL at chkopt\POL at oPOL@Typeset[x]}%
+\def\POL at oPOL@Typeset[#1]#2{%
+ \ifmmode\let\POL at endtypeset\empty\else$\def\POL at endtypeset{$}\fi
+ \ifcsname POLuserpol@#2\endcsname
+ \expandafter\expandafter\expandafter\POL at split
+ \csname POLuserpol@#2\endcsname;\POL at var@deg\POL at var@coeffs
+ \else
+ \xintAssign\expandafter\xint_firstofone\romannumeral0\xintbareeval
+ subs((deg(x),coeffs(x)),x=subs(#2,\PolToExprInVar=pol([0,1])))\relax
+ \to\POL at var@deg\POL at var@coeffs
+ \fi
+ \if\POL at ts@ascending1%
+ \def\PolIndex{0}%
+ \let\POL at ts@reverse\xint_firstofone
+ \let\POL@@ne at or@m at ne\@ne
+ \else
+ \let\PolIndex\POL at var@deg
+ \ifnum\PolIndex<\z@\def\PolIndex{0}\fi
+ \let\POL at ts@reverse\xintRevWithBraces
+ \let\POL@@ne at or@m at ne\m at ne
+ \fi
+ \def\PolVar{#1}%
+ \ifnum\POL at var@deg<\z@
+ \PolTypesetCmd{0/1[0]}\PolTypesetMonomialCmd
+ \else
+ \ifnum\POL at var@deg=\z@
+ \expandafter\PolTypesetCmd\POL at var@coeffs\PolTypesetMonomialCmd
+ \else
+ \def\POL at ts@prefix##1{\let\POL at ts@prefix\PolTypesetCmdPrefix}%
+ \expandafter\POL at ts@loop
+ \romannumeral-`0\POL at ts@reverse{\POL at var@coeffs}\relax
+ \fi
+ \fi
+ \POL at endtypeset
+}%
+\def\POL at ts@loop{\ifpoltypesetall\expandafter\xint_firstoftwo
+ \else\expandafter\xint_secondoftwo\fi
+ {\POL at ts@nocheck}{\POL at ts@check}.%
+}%
+\def\POL at ts@check.#1{%
+ \if\relax#1\expandafter\xint_gob_til_dot\fi
+ \xintiiifZero{#1}%
+ {}%
+ {\POL at ts@prefix{#1}\PolTypesetCmd{#1}\PolTypesetMonomialCmd}%
+ \edef\PolIndex{\the\numexpr\PolIndex+\POL@@ne at or@m at ne}\POL at ts@check.%
+}%
+\def\POL at ts@nocheck.#1{%
+ \if\relax#1\expandafter\xint_gob_til_dot\fi
+ \POL at ts@prefix{#1}\PolTypesetCmd{#1}\PolTypesetMonomialCmd
+ \edef\PolIndex{\the\numexpr\PolIndex+\POL@@ne at or@m at ne}\POL at ts@nocheck.%
+}%
+%
+%%
+%% Expandable output macros (legacy)
+%%
+\def\POL at eval@fork#1\At#2#3\krof{#2}%
+\def\PolEval#1#2#3{\romannumeral`&&@\POL at eval@fork
+ #2\PolEvalAt
+ \At\PolEvalAtExpr\krof {#1}{#3}%
+}%
+\def\PolEvalAt#1#2{%
+ \xintpraw{\csname XINT_expr_polfunc_#1\endcsname{#2}}%
+}%
+\def\POL at eval#1#2{%
+ \csname XINT_expr_polfunc_#1\endcsname{#2}%
+}%
+\def\PolEvalAtExpr#1#2{\xinttheexpr #1(#2)\relax}%
+%
+\def\PolEvalReduced#1#2#3{\romannumeral`&&@\POL at eval@fork
+ #2\PolEvalReducedAt
+ \At\PolEvalReducedAtExpr\krof {#1}{#3}%
+}%
+\def\PolEvalReducedAt#1#2{%
+ \xintpraw % in order not to print denominator if the latter equals 1
+ {\xintIrr{\csname XINT_expr_polfunc_#1\endcsname{#2}}[0]}%
+}%
+\def\PolEvalReducedAtExpr#1#2{%
+ \xintpraw
+ {\expandafter\xintIrr\romannumeral`&&@\xintthebareeval#1(#2)\relax[0]}%
+}%
+%
+\def\PolFloatEval#1#2#3{\romannumeral`&&@\POL at eval@fork
+ #2\PolFloatEvalAt
+ \At\PolFloatEvalAtExpr\krof {#1}{#3}%
+}%
+\def\PolFloatEvalAt#1#2{%
+ \xintpfloat{\csname XINT_flexpr_polfunc_#1\endcsname{#2}}%
+}%
+\def\PolFloatEvalAtExpr#1#2{\xintthefloatexpr #1(#2)\relax}%
+\def\PolLeadingCoeff#1{%
+ \romannumeral`&&@\expandafter\expandafter\expandafter\xintlastitem
+ \expandafter\expandafter\expandafter
+ {\csname POLuserpol@#1\endcsname}%
+}%
+%
+\def\PolNthCoeff#1#2{\romannumeral`&&@%
+ \expandafter\POL at 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 at 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?
+\def\PolDegree#1{\romannumeral`&&@\expandafter\expandafter\expandafter
+ \POL at degree\csname POLuserpol@#1\endcsname;}%
+\def\POL at degree #1.#2;{#1}%
+%
+\def\PolToList#1{\romannumeral`&&@\expandafter\expandafter\expandafter
+ \xint_gob_til_dot\csname POLuserpol@#1\endcsname}%
+%
+\def\PolToCSV#1{\romannumeral0\xintlistwithsep{, }{\PolToList{#1}}}%
+%
+% \PolIContent (0.5)
+% Why did I call this IContent and not Content? Ah, I see Maple terminology
+% But I realize now I misread the Maple doc, its icontent() is the gcd of
+% all coeffs of a multivariate polynomial. Whereas content(,) second argument
+% specifies which variable to consider expression as being univariate in it
+%
+\def\POL at icontent#1{\romannumeral0\expandafter\XINT_fgcd_out
+ \romannumeral0\expandafter\XINT_fgcdof\romannumeral`&&@#1^}%
+% Since xintexpr 1.4d, \xintGCDof always outputs an irreducible fraction A/B.
+% (with B=1 if A/B integer).
+\def\PolIContent#1{\xintGCDof{\PolToList{#1}}}%
+%
+\def\PolToExprCmd#1{\xintPRaw{\xintRawWithZeros{#1}}}%
+\def\PolToFloatExprCmd#1{\xintFloat{#1}}%
+% \def\PolTypesetCmdPrefix#1{\xintiiifSgn{#1}{}{+}{+}}%
+\let\PolToExprTermPrefix\PolTypesetCmdPrefix
+\def\PolToExprOneTermStyleA#1#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\PolToExprCaret\xintiiAbs{#2}%
+ \fi
+}%
+\let\PolToExprOneTerm\PolToExprOneTermStyleA
+\def\PolToExprOneTermStyleB#1#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\PolToExprCaret\xintiiAbs{#2}%
+ \fi
+ \xintiiifOne{\xintDenominator{#1}}{}{/\xintDenominator{#1}}%
+}%
+\def\PolToFloatExprOneTerm#1#2{%
+ \ifnum#2=\z@
+ \PolToFloatExprCmd{#1}%
+ \else
+ \PolToFloatExprCmd{#1}\PolToExprTimes
+ \fi
+ \ifcase\xintiiAbs{#2} %<-- space here mandatory
+ \or\PolToExprVar
+ \else\PolToExprVar\PolToExprCaret\xintiiAbs{#2}%
+ \fi
+}%
+\def\PolToExprTimes{*}%
+\def\PolToExprVar{x}%
+\def\PolToExprInVar{x}%
+\edef\PolToExprCaret{\string ^}%
+%%
+%% \PolToExpr
+%%
+%% extended at 0.8 to handle arbitrary expressions on input
+%%
+\def\PolToExpr#1{%
+ \if*\noexpand#1\expandafter\xint_firstoftwo\else
+ \expandafter\xint_secondoftwo\fi
+ \PolToExprAscending\PolToExprDescending{#1}}%
+\def\PolToFloatExpr#1{%
+ \if*\noexpand#1\expandafter\xint_firstoftwo\else
+ \expandafter\xint_secondoftwo\fi
+ \PolToFloatExprAscending\PolToFloatExprDescending{#1}}%
+\def\PolToExpr at getit#1%
+{%
+ \ifcsname XINT_expr_varvalue_#1\endcsname
+ \csname XINT_expr_varvalue_#1\expandafter\endcsname
+ \else
+ \expandafter\xint_firstofone\romannumeral0%
+ \xintbareeval subs(#1,\PolToExprInVar=pol([0,1]))\expandafter\relax
+ \fi
+}%
+\def\PolToExprAscending#1#2{%
+ \expandafter\POL at toexpr\romannumeral0\PolToExpr at getit{#2}%
+ \PolToExprOneTerm\POL at toexprA
+}%
+\def\PolToFloatExprAscending#1#2{%
+ \expandafter\POL at toexpr\romannumeral0\PolToExpr at getit{#2}%
+ \PolToFloatExprOneTerm\POL at toexprA
+}%
+\def\PolToExprDescending#1{%
+ \expandafter\POL at toexpr\romannumeral0\PolToExpr at getit{#1}%
+ \PolToExprOneTerm\POL at toexprD
+}%
+\def\PolToFloatExprDescending#1{%
+ \expandafter\POL at toexpr\romannumeral0\PolToExpr at getit{#1}%
+ \PolToFloatExprOneTerm\POL at toexprD
+}%
+\def\POL at toexpr#1#2#3{\POL at toexpr@fork#3#2#1\relax}%
+\def\POL at toexpr@fork #1#2#3{%
+ \POL_Pfork
+ #3\POL at toexpr@pol
+ P\POL at toexpr@cst
+ \krof #1#2#3%
+}%
+\def\POL at toexpr@cst#1#2#3\relax{#2{#3}{0}}%
+\def\POL at toexpr@pol#1#2P#3.{#1{#3}#2\empty}%
+% now back to legacy pre 0.8 code
+\def\POL at toexprA #1#2\empty#3{%
+ \ifpoltoexprall\expandafter\POL at toexprall@b
+ \else\expandafter\POL at toexpr@b
+ \fi {#3}#2{0}1.%
+}%
+\def\POL at toexprD #1#2#3\relax{% #3 has \empty to prevent brace removal
+ \expandafter\POL at toexprD@a\expandafter#2%
+ \the\numexpr #1\expandafter.\romannumeral0\xintrevwithbraces{#3}\relax
+}%
+\def\POL at toexprD@a #1#2.#3{%
+ \ifpoltoexprall\expandafter\POL at toexprall@b
+ \else\expandafter\POL at toexpr@b
+ \fi{#3}#1{-#2}\the\numexpr\@ne+-#2.%
+}%
+\def\POL at toexpr@b #1#2#3{%
+ \xintiiifZero{#1}%
+ {\expandafter\POL at toexpr@loop\expandafter\POL at toexpr@b}%
+ {#2{#1}{#3}%
+ \expandafter\POL at toexpr@loop\expandafter\POL at toexpr@c}%
+ \expandafter#2%
+}%
+\def\POL at toexpr@c #1#2#3{%
+ \xintiiifZero{#1}%
+ {}%
+ {\PolToExprTermPrefix{#1}#2{#1}{#3}}%
+ \expandafter\POL at toexpr@loop\expandafter\POL at toexpr@c
+ \expandafter#2%
+}%
+\def\POL at toexprall@b #1#2#3{%
+ #2{#1}{#3}%
+ \expandafter\POL at toexpr@loop\expandafter\POL at toexprall@c
+ \expandafter#2%
+}%
+\def\POL at toexprall@c #1#2#3{%
+ \PolToExprTermPrefix{#1}#2{#1}{#3}%
+ \expandafter\POL at toexpr@loop\expandafter\POL at toexprall@c
+ \expandafter#2%
+}%
+\def\POL at toexpr@loop#1#2#3.#4{%
+ \if\relax#4\expandafter\xint_gob_til_dot\fi
+ #1{#4}#2{#3}\the\numexpr\@ne+#3.%
+}%
+\XINT_restorecatcodes_endinput%
Property changes on: trunk/Master/texmf-dist/tex/generic/polexpr/polexpr.sty
___________________________________________________________________
Added: svn:eol-style
## -0,0 +1 ##
+native
\ No newline at end of property
Added: trunk/Master/texmf-dist/tex/generic/polexpr/polexprcore.tex
===================================================================
--- trunk/Master/texmf-dist/tex/generic/polexpr/polexprcore.tex (rev 0)
+++ trunk/Master/texmf-dist/tex/generic/polexpr/polexprcore.tex 2021-04-10 21:13:16 UTC (rev 58821)
@@ -0,0 +1,1366 @@
+%% This file polexprcore.tex is part of the polexpr package (0.8, 2021/03/29)
+%% Core routines to match infix operators +, -, *, //, /:, ^, ** and some
+%% functions
+%% The atoms representing polynomials inside \xintexpr are
+%% - for constants: a numeric value (indistinguishable. from scalars)
+%% - for degree at least 1: P<degree>.{c0}{c1}....{cN} with N = degree
+%% Auxiliaries
+\long\def\POL_Pfork #1P#2#3\krof{#2}%
+\long\def\POL_PPfork #1PP#2#3\krof{#2}%
+\long\def\POL_zeroPfork #10P#2#3\krof{#2}%
+\long\def\POL_secondofthree#1#2#3{#2}%
+% \long\def\POL_Apply:x #1#2%
+% {%
+% \POL_apply:x_loop {#1}#2%
+% \xint_Bye\xint_Bye\xint_Bye\xint_Bye
+% \xint_Bye\xint_Bye\xint_Bye\xint_Bye\xint_bye
+% }%
+\long\def\POL_bBye#1\xint_Bye{}%
+\long\def\POL_apply:x_loop #1#2#3#4#5#6#7#8#9%
+{%
+ \POL_bBye #2\xint_Bye{#1{#2}}%
+ \POL_bBye #3\xint_Bye{#1{#3}}%
+ \POL_bBye #4\xint_Bye{#1{#4}}%
+ \POL_bBye #5\xint_Bye{#1{#5}}%
+ \POL_bBye #6\xint_Bye{#1{#6}}%
+ \POL_bBye #7\xint_Bye{#1{#7}}%
+ \POL_bBye #8\xint_Bye{#1{#8}}%
+ \POL_bBye #9\xint_Bye{#1{#9}}%
+ \POL_apply:x_loop {#1}%
+}%
+\long\def\POL_apply:x_iloop #1#2#3#4#5#6#7#8#9%
+{%
+ \POL_bBye #2\xint_Bye{#10{#2}}%
+ \POL_bBye #3\xint_Bye{#11{#3}}%
+ \POL_bBye #4\xint_Bye{#12{#4}}%
+ \POL_bBye #5\xint_Bye{#13{#5}}%
+ \POL_bBye #6\xint_Bye{#14{#6}}%
+ \POL_bBye #7\xint_Bye{#15{#7}}%
+ \POL_bBye #8\xint_Bye{#16{#8}}%
+ \POL_bBye #9\xint_Bye{#17{#9}}%
+ \POL_apply:x_iloop_a#1%
+}%
+\def\POL_apply:x_iloop_a#1#2.%
+{%
+ \expandafter\POL_apply:x_iloop
+ \expandafter{\expandafter#1\the\numexpr\xint_c_viii+#1.}%
+}%
+\long\def\POL_apply:x_iloop #1#2#3#4#5#6#7#8#9%
+{%
+ \POL_bBye #2\xint_Bye{#10{#2}}%
+ \POL_bBye #3\xint_Bye{#11{#3}}%
+ \POL_bBye #4\xint_Bye{#12{#4}}%
+ \POL_bBye #5\xint_Bye{#13{#5}}%
+ \POL_bBye #6\xint_Bye{#14{#6}}%
+ \POL_bBye #7\xint_Bye{#15{#7}}%
+ \POL_bBye #8\xint_Bye{#16{#8}}%
+ \POL_bBye #9\xint_Bye{#17{#9}}%
+ \POL_apply:x_iloop_a#1%
+}%
+\def\POL_apply:x_iloop_a#1#2.%
+{%
+ \expandafter\POL_apply:x_iloop
+ \expandafter{\expandafter#1\the\numexpr\xint_c_viii+#1.}%
+}%
+%%
+%% ADDITION
+%%
+\def\xintPolAdd #1%
+{%
+ \expanded\expandafter\POL_add_in\romannumeral`&&@#1\xint:
+}%
+\def\POL_add_in #1\xint:#2%
+{%
+ {%
+ \expandafter\POL_add_fork
+% Fragile but this macro is not public anyhow and won't get arbitrary input
+% At odds with systematic \xint: style further down
+ \romannumeral`&&@#2\xint_bye\xint_bye\xint_bye\xint_bye\empty
+ #1\xint_bye\xint_bye\xint_bye\xint_bye\empty
+ \empty
+ }%
+}%
+% Careful that first means "first here" i.e. the original second argument,
+% and vice versa
+\def\POL_add_fork #1#2\empty#3%
+{%
+ \POL_PPfork
+ #1#3{\POL_add_a}%
+ #1P{\POL_add_second_is_scalar}%
+ #3P{\POL_add_first_is_scalar}%
+ PP{\POL_add_both_are_scalar}%
+ \krof #1#2\empty#3%
+}%
+\def\POL_add_first_is_scalar #1\xint_bye#2\empty#3.#4%
+{%
+ #3.{\xintAdd{#1}{#4}}%
+}%
+\def\POL_add_second_is_scalar #1.#2#3\empty#4\xint_bye#5\empty\empty
+{%
+ #1.{\xintAdd{#2}{#4}}#3%
+}%
+\def\POL_add_both_are_scalar #1\xint_bye#2\empty#3\xint_bye#4\empty\empty
+{%
+ \xintAdd{#1}{#3}%
+}%
+\def\POL_add_a P#1.#2#3#4#5\empty P#6.#7#8#9%
+{%
+ \expandafter\POL_add_b
+ \expanded\bgroup\unexpanded{#1.#6.}%
+ \xint_bye #2\POL_add_Eb\xint_bye
+ \xint_bye #7\POL_add_Fb\xint_bye {\xintAdd{#2}{#7}}%
+ \xint_bye #3\POL_add_Ec\xint_bye
+ \xint_bye #8\POL_add_Fc\xint_bye {\xintAdd{#3}{#8}}%
+ \xint_bye #4\POL_add_Ed\xint_bye
+ \xint_bye #9\POL_add_Fd\xint_bye {\xintAdd{#4}{#9}}%
+ \POL_add_A #5\empty
+}%
+\def\POL_add_b #1.#2.%
+{%
+ \ifnum#1=#2 \expandafter\POL_add_c
+ \else
+ \ifnum#1>#2 P#1.\else P#2.\fi
+ \fi
+}%
+% No brace stripping possible, because constant polynomials are really
+% represented by scalars in all those internal contexts, so real
+% polynomials have at least two coefficients
+\def\POL_add_c #1\empty
+{%
+ \expandafter\POL_add_d
+ \romannumeral0\XINT_revwbr_loop {}%
+ #1\xint:\xint:\xint:\xint:%
+ \xint:\xint:\xint:\xint:\xint_bye
+ \xint_bye
+}%
+% Attention, reused in various other locations. It is all f-expandable.
+\def\POL_add_d #1%
+{%
+% abuse of \XINT_Sgn internals compatible to #1 being \xint_bye
+ \if0\XINT_Sgn#1\xint:
+ \xint_dothis\POL_add_d
+ \fi
+ \xint_orthat{\POL_add_e {#1}}%
+}%
+\def\POL_add_e #1%
+{%
+ \xint_bye#1\POL_add_e_zero\xint_bye \POL_add_f\empty{#1}%
+}%
+\def\POL_add_e_zero\xint_bye\POL_add_f\empty #1{0/1[0]}%
+% #1 starts with \empty to avoid brace stripping.
+\def\POL_add_f #1\xint_bye
+{%
+ \expandafter\POL_add_g
+ \the\numexpr
+ \xintLength{#1}-\xint_c_ii\expandafter.%
+ \romannumeral0\expandafter
+ \XINT_revwbr_loop\expandafter {\expandafter}%
+ #1\xint:\xint:\xint:\xint:%
+ \xint:\xint:\xint:\xint:\xint_bye
+}%
+\def\POL_add_g #1.%
+{%
+ \ifnum#1=\xint_c_\expandafter\POL_add_h\fi
+ P#1.%
+}%
+\def\POL_add_h P0.#1{#1}%
+% Attention reused in \POL_mul_d and \POL_quorem_c
+\def\POL_add_A #1#2#3#4#5\empty#6#7#8#9%
+{%
+ \xint_bye #1\POL_add_Ea\xint_bye
+ \xint_bye #6\POL_add_Fa\xint_bye {\xintAdd{#1}{#6}}%
+ \xint_bye #2\POL_add_Eb\xint_bye
+ \xint_bye #7\POL_add_Fb\xint_bye {\xintAdd{#2}{#7}}%
+ \xint_bye #3\POL_add_Ec\xint_bye
+ \xint_bye #8\POL_add_Fc\xint_bye {\xintAdd{#3}{#8}}%
+ \xint_bye #4\POL_add_Ed\xint_bye
+ \xint_bye #9\POL_add_Fd\xint_bye {\xintAdd{#4}{#9}}%
+ \POL_add_A #5\empty
+}%
+\def\POL_add_Ea\xint_bye
+ \xint_bye #1\POL_add_Fa\xint_bye #2\xint_bye\xint_bye
+ \POL_add_Eb\xint_bye\xint_bye#3\POL_add_Fb\xint_bye #4\xint_bye\xint_bye
+ \POL_add_Ec\xint_bye\xint_bye#5\POL_add_Fc\xint_bye #6\xint_bye\xint_bye
+ \POL_add_Ed\xint_bye\xint_bye#7\POL_add_Fd\xint_bye #8%
+ \POL_add_A#9\empty
+{%
+ \xint_bye #1\POL_add_G\xint_bye{#1}%
+ \xint_bye #3\POL_add_G\xint_bye{#3}%
+ \xint_bye #5\POL_add_G\xint_bye{#5}%
+ \xint_bye #7\POL_add_G\xint_bye{#7}%
+ \iffalse{\fi}%
+}%
+\def\POL_add_G#1\empty{\iffalse{\fi}}%
+\def\POL_add_Fa\xint_bye #1%
+ \xint_bye #2\POL_add_Eb \xint_bye
+ \xint_bye\xint_bye\POL_add_Fb\xint_bye #3%
+ \xint_bye #4\POL_add_Ec \xint_bye
+ \xint_bye\xint_bye\POL_add_Fc\xint_bye #5%
+ \xint_bye #6\POL_add_Ed #7\POL_add_A
+ #8\empty#9\empty
+{%
+ \expandafter\xint_bye\POL_secondofthree #1%
+ \POL_add_G\xint_bye{\POL_secondofthree#1}%
+ \xint_bye #2\POL_add_G\xint_bye{#2}%
+ \xint_bye #4\POL_add_G\xint_bye{#4}%
+ \xint_bye #6\POL_add_G\xint_bye{#6}%
+ \iffalse{\fi}#8\empty%
+}%
+\def\POL_add_Eb\xint_bye
+ \xint_bye #1\POL_add_Fb\xint_bye #2\xint_bye\xint_bye
+ \POL_add_Ec\xint_bye\xint_bye#3\POL_add_Fc\xint_bye #4\xint_bye\xint_bye
+ \POL_add_Ed\xint_bye\xint_bye#5\POL_add_Fd\xint_bye #6%
+ \POL_add_A#7\empty
+{%
+ \xint_bye #1\POL_add_G\xint_bye{#1}%
+ \xint_bye #3\POL_add_G\xint_bye{#3}%
+ \xint_bye #5\POL_add_G\xint_bye{#5}%
+ \iffalse{\fi}%
+}%
+\def\POL_add_Fb\xint_bye #1%
+ \xint_bye #2\POL_add_Ec \xint_bye
+ \xint_bye\xint_bye\POL_add_Fc\xint_bye #3%
+ \xint_bye #4\POL_add_Ed #5\POL_add_A
+ #6\empty#7\empty
+{%
+ \expandafter\xint_bye\POL_secondofthree #1%
+ \POL_add_G\xint_bye{\POL_secondofthree#1}%
+ \xint_bye #2\POL_add_G\xint_bye{#2}%
+ \xint_bye #4\POL_add_G\xint_bye{#4}%
+ \iffalse{\fi}#6\empty
+}%
+\def\POL_add_Ec\xint_bye
+ \xint_bye #1\POL_add_Fc\xint_bye #2\xint_bye\xint_bye
+ \POL_add_Ed\xint_bye\xint_bye#3\POL_add_Fd\xint_bye #4%
+ \POL_add_A#5\empty
+{%
+ \xint_bye #1\POL_add_G\xint_bye{#1}%
+ \xint_bye #3\POL_add_G\xint_bye{#3}%
+ \iffalse{\fi}%
+}%
+\def\POL_add_Fc\xint_bye #1\xint_bye #2\POL_add_Ed #3\POL_add_A
+ #4\empty#5\empty
+{%
+ \expandafter\xint_bye\POL_secondofthree #1%
+ \POL_add_G\xint_bye{\POL_secondofthree#1}%
+ \xint_bye #2\POL_add_G\xint_bye{#2}%
+ \iffalse{\fi}#4\empty
+}%
+\def\POL_add_Ed\xint_bye\xint_bye#1\POL_add_Fd\xint_bye#2\POL_add_A#3\empty
+{%
+ \xint_bye #1\POL_add_G\xint_bye{#1}%
+ \iffalse{\fi}%
+}%
+\def\POL_add_Fd\xint_bye#1\POL_add_A #2\empty#3\empty
+{%
+ \expandafter\xint_bye \POL_secondofthree #1%
+ \POL_add_G\xint_bye{\POL_secondofthree#1}%
+ \iffalse{\fi}#2\empty
+}%
+%%
+%% OPPOSITE
+%%
+\def\xintPolOpp #1%
+{%
+ \expanded{%
+ \expandafter\POL_opp_fork\romannumeral`&&@#1%
+ \xint_Bye\xint_Bye\xint_Bye\xint_Bye
+ \xint_Bye\xint_Bye\xint_Bye\xint_Bye\xint_bye
+ }%
+}%
+\def\POL_opp_fork #1%
+{%
+ \if P#1\xint_dothis\POL_opp_a\fi
+ \xint_orthat\POL_opp_scalar #1%
+}%
+\def\POL_opp_scalar #1\xint_Bye#2\xint_bye
+{%
+ \XINT_Opp #1%
+}%
+\def\POL_opp_a #1.%
+{%
+ #1.\POL_apply:x_loop{\XINT_Opp}%
+}%
+%%
+%% SUBTRACTION
+%%
+\def\xintPolSub #1%
+{%
+ \expanded\expandafter\POL at sub\romannumeral`&&@#1\xint:
+}%
+\def\POL at sub #1\xint:#2%
+{%
+ {%
+ \expandafter
+ \POL_add_fork\expanded{%
+ \expandafter\POL_opp_fork \romannumeral`&&@#2%
+ \xint_Bye\xint_Bye\xint_Bye\xint_Bye
+ \xint_Bye\xint_Bye\xint_Bye\xint_Bye\xint_bye
+ }%
+ \xint_bye\xint_bye\xint_bye\xint_bye\empty
+ #1\xint_bye\xint_bye\xint_bye\xint_bye\empty
+ \empty
+ }%
+}%
+%%
+%% MULTIPLICATION
+%%
+\def\xintPolSqr #1%
+{%
+ \expanded\expandafter\POL_sqr_in\romannumeral`&&@#1\xint:
+}%
+\def\POL_sqr_in #1\xint:
+{%
+ {%
+ \expandafter\POL_mul_fork
+ #1\xint_bye
+ #1\xint_Bye\xint_Bye\xint_Bye\xint_Bye
+ \xint_Bye\xint_Bye\xint_Bye\xint_Bye\xint_bye
+ }%
+}%
+%
+\def\xintPolMul #1%
+{%
+ \expanded\expandafter\POL_mul_in\romannumeral`&&@#1\xint:
+}%
+\def\POL_mul_in #1\xint:#2%
+{%
+ {%
+ \expandafter\POL_mul_fork
+ \romannumeral`&&@#2\xint_bye
+ #1\xint_Bye\xint_Bye\xint_Bye\xint_Bye
+ \xint_Bye\xint_Bye\xint_Bye\xint_Bye\xint_bye
+ }%
+}%
+\def\POL_mul_fork #1#2\xint_bye#3%
+{%
+ \POL_PPfork
+ #1#3{\POL_mul_a}%
+ #1P{\POL_mul_second_is_scalar}%
+ #3P{\POL_mul_first_is_scalar}%
+ PP{\POL_mul_both_are_scalar}%
+ \krof #1#2\xint_bye#3%
+}%
+\def\POL_mul_both_are_scalar #1\xint_bye#2\xint_Bye#3\xint_bye
+{%
+ \xintMul{#1}{#2}%
+}%
+\def\POL_mul_second_is_scalar #1\xint_bye#2\xint_Bye
+{%
+ \POL_mul_first_is_scalar #2\xint_bye#1\xint_Bye
+}%
+\def\POL_mul_first_is_scalar #1%
+{%
+ \xint_gob_til_zero#1\POL_mul_zero0\POL_mul_scalar #1%
+}%
+\def\POL_mul_zero0\POL_mul_scalar #1\xint_bye#2\xint_bye{0/1[0]}%
+\def\POL_mul_scalar #1\xint_bye P#2.%
+{%
+ P#2.\POL_apply:x_loop{\xintMul{#1}}%
+}%
+\def\POL_mul_a P#1.#2#3P#4.#5\xint_bye
+{%
+ P\the\numexpr#1+#4.%
+ \expandafter\POL_mul_b
+ \expanded{\POL_apply:x_loop{\xintMul{#2}}#5\xint_bye}%
+ \xint:
+ #3\empty#5\xint_bye
+}%
+\def\POL_mul_b #1{{#1}\POL_mul_c\empty}%
+\def\POL_mul_c #1\xint:#2%
+{%
+ \xint_bye#2\POL_mul_E\xint_bye
+ \expandafter\POL_mul_d\expandafter{#1}{#2}%
+}%
+\def\POL_mul_d #1#2#3\empty#4\xint_bye
+{%
+ \expandafter\POL_mul_b
+ \expanded\bgroup
+ \expandafter\POL_add_A
+ \expanded{\POL_apply:x_loop{\xintMul{#2}}#4\xint_bye}%
+ \xint_bye\xint_bye\xint_bye\xint_bye\empty
+ #1\xint_bye\xint_bye\xint_bye\xint_bye\empty
+ \xint:
+ #3\empty#4\xint_bye
+}%
+\def\POL_mul_E\xint_bye
+ \expandafter\POL_mul_d\expandafter#1#2\xint_bye
+% This #1 starts with \empty
+{%
+ #1%
+}%
+%%
+%% POWERS
+%%
+\def\xintPolPow #1%
+{%
+ \expanded\expandafter\POL_pow_in\romannumeral`&&@#1\xint:
+}%
+\def\POL_pow_in #1\xint:#2%
+{%
+ {%
+ \expandafter\POL_pow_fork\the\numexpr \xintNum{#2}.#1\empty
+ }%
+}%
+\def\POL_pow_fork #1#2.%
+{%
+ \xint_UDzerominusfork
+ #1-\POL_pow_zero
+ 0#1\POL_pow_neg
+ 0-\POL_pow_pos
+ \krof
+ #1#2.%
+}%
+\def\POL_pow_zero #1\empty{1/1[0]}%
+\def\POL_pow_neg #1.#2%
+{%
+ \POL_Pfork #2{\POL_pow_neg_pol}P{\POL_pow_scalar}\krof #1.#2%
+}%
+\def\POL_pow_pos #1.#2%
+{%
+ \POL_Pfork #2{\POL_pow_a}P{\POL_pow_scalar}\krof #1.#2%
+}%
+\def\POL_pow_scalar #1.#2\empty
+{%
+ \xintPow{#2}{#1}%
+}%
+\def\POL_pow_neg_pol #1.#2\empty
+{%
+ \romannumeral0\XINT_signalcondition{InvalidOperation}%
+ {Not supported: polynomial to negative power #1}{}{1/1[0]}%
+}%
+\def\POL_pow_a #1.%
+{%
+% trailing \empty will disappear in expanded context (old comment)
+ \ifnum#1=\xint_c_i\xint_afterfi\xint_gob_til_dot\fi
+ \expandafter\POL_pow_b \the\numexpr#1-\xint_c_i.%
+}%
+\def\POL_pow_b #1.%
+{%
+ \ifodd #1 \xint_dothis{\expandafter\POL_pow_even}\fi
+ \xint_orthat{\expandafter\POL_pow_odd}\the\numexpr#1/\xint_c_ii.%
+}%
+\def\POL_pow_even #1.#2\empty
+{%
+ \expandafter\POL_pow_a
+ \expanded{\unexpanded{#1.}%
+ \POL_mul_a#2\xint_bye
+ #2\xint_Bye\xint_Bye\xint_Bye\xint_Bye
+ \xint_Bye\xint_Bye\xint_Bye\xint_Bye\xint_bye
+ }\empty
+}%
+\def\POL_pow_odd #1.#2\empty
+{%
+ \expanded
+ {\unexpanded{\POL_mul_a #2\xint_bye}%
+ \expandafter\POL_pow_a
+ \expanded{\unexpanded{#1.}%
+ \POL_mul_a#2\xint_bye
+ #2\xint_Bye\xint_Bye\xint_Bye\xint_Bye
+ \xint_Bye\xint_Bye\xint_Bye\xint_Bye\xint_bye
+ }%
+ \empty
+ }%
+ \xint_Bye\xint_Bye\xint_Bye\xint_Bye
+ \xint_Bye\xint_Bye\xint_Bye\xint_Bye\xint_bye
+}%
+%%
+%% DIVISION
+%%
+%
+% / is deprecated for polynomial Euclidean division
+%
+\def\xintPolQuo #1%
+{%
+ \romannumeral0\expandafter\xint_stop_atfirstoftwo
+ \expanded\expandafter\POL_quorem_in\romannumeral`&&@#1\xint:
+}%
+% there is no operator, for lack of obvious best notation
+\def\xintPolRem #1%
+{%
+ \romannumeral0\expandafter\xint_stop_atsecondoftwo
+ \expanded\expandafter\POL_quorem_in\romannumeral`&&@#1\xint:
+}%
+% //
+\def\xintPolDivModQ #1%
+{%
+ \romannumeral0\expandafter\xint_stop_atfirstoftwo
+ \expanded\expandafter\POL_divmod_in\romannumeral`&&@#1\xint:
+}%
+% /:
+\def\xintPolDivModR #1%
+{%
+ \romannumeral0\expandafter\xint_stop_atsecondoftwo
+ \expanded\expandafter\POL_divmod_in\romannumeral`&&@#1\xint:
+}%
+% "divmod" will apply coefficient per coefficient when divisor is scalar
+% I have found it convenient to treat constant polynomials
+% as really being scalars. But I need perhaps to think more about it.
+\def\xintPolDivMod #1%
+{%
+ \expanded\expandafter\POL_divmod_in\romannumeral`&&@#1\xint:
+}%
+% the euclidean division
+\def\xintPolQuoRem #1%
+{%
+ \expanded\expandafter\POL_quorem_in\romannumeral`&&@#1\xint:
+}%
+\def\POL_quorem_in #1\xint:#2%
+{%
+ {%
+ \expandafter\POL_quorem_fork
+ \romannumeral`&&@#2\xint_bye#1\xint_bye
+ }%
+}%
+% the overloading of divmod which does euclidean division if divisor is not a scalar
+\def\POL_divmod_in #1\xint:#2%
+{%
+ {%
+ \expandafter\POL_divmod_fork
+ \romannumeral`&&@#2\xint_bye#1\xint_bye
+ }%
+}%
+% "first" and "second" refer to the actual positions, permuted compared
+% to original arguments
+\def\POL_quorem_fork #1#2\xint_bye#3%
+{%
+ \POL_PPfork
+ #1#3{\POL_quorem_a}% both polynomials -> {eucl. quotient}{remainder}
+ #1P{\POL_quorem_second_is_scalar}% -> {zero quotient}{scalar}
+ #3P{\POL_quorem_first_is_scalar}% -> {polynomial/scalar}{zero}
+ PP{\POL_quorem_both_are_scalar}% -> {scalar/scalar}{zero}
+ \krof #1#2\xint_bye#3%
+}%
+\def\POL_quorem_first_is_scalar #1\xint_bye#2\xint_bye
+{%
+ {\expandafter\POL_quorem_first_is_scalar_i\expandafter
+ {\romannumeral0\xintinv{#1}}%
+ #2\xint_Bye\xint_Bye\xint_Bye\xint_Bye
+ \xint_Bye\xint_Bye\xint_Bye\xint_Bye\xint_bye}{0/1[0]}%
+}%
+\def\POL_quorem_first_is_scalar_i #1#2.%
+{%
+ #2.\POL_apply:x_loop{\xintMul{#1}}%
+}%
+% #2 was initial first argument and is scalar
+\def\POL_quorem_second_is_scalar #1\xint_bye#2\xint_bye
+{%
+ {0/1[0]}{#2}%
+}%
+\def\POL_quorem_both_are_scalar #1\xint_bye#2\xint_bye
+{%
+ {\xintDiv{#2}{#1}}{0/1[0]}%
+}%
+% attention that "first", "second" refer to the actual arguments positions
+\def\POL_divmod_fork #1#2\xint_bye#3%
+{%
+ \POL_PPfork
+ #1#3{\POL_quorem_a}% both polynomials -> {eucl. quotient}{remainder}
+ #1P{\POL_quorem_second_is_scalar}% -> {zero quotient}{scalar}
+ #3P{\POL_divmod_first_is_scalar}% -> {per coeff//scalar}{per coeff/:scalar}
+ PP{\POL_divmod_both_are_scalar}% -> {s1//s2}{s1/:s2}
+ \krof #1#2\xint_bye#3%
+}%
+\def\POL_divmod_both_are_scalar #1\xint_bye#2\xint_bye
+{%
+ \xintDivMod{#2}{#1}%
+}%
+\def\POL_divmod_first_is_scalar #1\xint_bye #2.#3\xint_bye
+{%
+ \expandafter\POL_divmod_first_is_scalar_a
+ \expanded{\unexpanded{{#1}}\expandafter}%
+ \romannumeral0\XINT_revwbr_loop {}%
+ #3\xint:\xint:\xint:\xint:%
+ \xint:\xint:\xint:\xint:\xint_bye
+ \xint_Bye\xint_Bye\xint_Bye\xint_Bye
+ \xint_Bye\xint_Bye\xint_Bye\xint_Bye\xint_bye
+ \iffalse{\fi}%
+ \xint:
+}%
+\long\def\POL_exchange_args#1#2#3{#1{#3}{#2}}%
+\def\POL_divmod_first_is_scalar_a #1%
+{%
+ \expandafter\POL_divmod_first_is_scalar_b
+ \expanded\bgroup
+ \POL_apply:x_loop{\POL_exchange_args\xintDivMod{#1}}%
+}%
+% attention re-use of \POL_add_d
+\def\POL_divmod_first_is_scalar_b #1\xint:
+{%
+ {\expandafter\POL_add_d\expanded{%
+ \POL_apply:x_loop{\expandafter\xint_firstoftwo\xint_firstofone}%
+ #1\xint_Bye\xint_Bye\xint_Bye\xint_Bye
+ \xint_Bye\xint_Bye\xint_Bye\xint_Bye\xint_bye}\xint_bye}%
+ {\expandafter\POL_add_d\expanded{%
+ \POL_apply:x_loop{\expandafter\xint_secondoftwo\xint_firstofone}%
+ #1\xint_Bye\xint_Bye\xint_Bye\xint_Bye
+ \xint_Bye\xint_Bye\xint_Bye\xint_Bye\xint_bye}\xint_bye}%
+}%
+\def\POL_quorem_a P#1.#2\xint_bye P#3.#4\xint_bye
+{%
+ \ifnum#1>#3 \xint_dothis{\POL_quorem_easy #3.}\fi
+ \xint_orthat
+ {\expandafter\POL_quorem_EQuo
+ \expanded\bgroup
+ \expandafter\POL_quorem_b\the\numexpr#3-#1\expandafter.%
+ \expanded\bgroup
+ \xintRevWithBraces
+ }%
+ {#2}%
+ \noexpand\xint_Bye
+ \xint:
+ \expandafter\POL_placemark_loop
+ \the\numexpr#1-\xint_c_vii\expandafter.%
+ \romannumeral0\xintrevwithbraces{#4}%
+% This added {1} is related to termination clean-up (a bit annoying) process
+ {1}%
+ \the\numexpr#3-#1.%
+ \iffalse{\fi}%
+}%
+\def\POL_quorem_easy #1.#2\xintrevwithbraces#3#4.#5#6%
+{%
+ {0/1[0]}{P#1.#3}%
+}%
+\def\POL_placemark_loop #1#2.%
+{%
+ \xint_gob_til_minus#1\POL_placemark_loop_end-%
+ \expandafter\POL_placemark_step\the\numexpr#1#2-\xint_c_viii.%
+}%
+\def\POL_placemark_step #1.#2#3#4#5#6#7#8#9%
+{%
+ {#2}{#3}{#4}{#5}{#6}{#7}{#8}{#9}\POL_placemark_loop#1.%
+}%
+\def\POL_placemark_loop_end-%
+ \expandafter\POL_placemark_step\the\numexpr-#1-\xint_c_viii.%
+{%
+ \csname POL_placemark_end#1\endcsname
+}%
+\expandafter\def\csname POL_placemark_end1\endcsname
+ #1#2#3#4#5#6#7{{#1}{#2}{#3}{#4}{#5}{#6}{#7}\noexpand\xint_bye\xint:}%
+\expandafter\def\csname POL_placemark_end2\endcsname
+ #1#2#3#4#5#6{{#1}{#2}{#3}{#4}{#5}{#6}\noexpand\xint_bye\xint:}%
+\expandafter\def\csname POL_placemark_end3\endcsname
+ #1#2#3#4#5{{#1}{#2}{#3}{#4}{#5}\noexpand\xint_bye\xint:}%
+\expandafter\def\csname POL_placemark_end4\endcsname
+ #1#2#3#4{{#1}{#2}{#3}{#4}\noexpand\xint_bye\xint:}%
+\expandafter\def\csname POL_placemark_end5\endcsname
+ #1#2#3{{#1}{#2}{#3}\noexpand\xint_bye\xint:}%
+\expandafter\def\csname POL_placemark_end6\endcsname
+ #1#2{{#1}{#2}\noexpand\xint_bye\xint:}%
+\expandafter\def\csname POL_placemark_end7\endcsname
+ #1{{#1}\noexpand\xint_bye\xint:}%
+\expandafter\def\csname POL_placemark_end8\endcsname
+ {\noexpand\xint_bye\xint:}%
+\def\POL_quorem_b #1.#2#3\xint:#4#5\xint:#6%
+{%
+% \xintDiv FG computes F/G
+ \expandafter\POL_quorem_c\romannumeral0\xintdiv{\XINT_Opp#4}{#2}.%
+ #1.{#2}#3\xint:
+% there is already \xint_Bye at ends of #3
+ #3\xint_Bye\xint_Bye\xint_Bye
+ \xint_Bye\xint_Bye\xint_Bye\xint_Bye\xint_bye
+% this terminates the \expanded from \POL_apply:x_loop
+ \iffalse{\fi}%
+ \xint_bye\xint_bye\xint_bye\xint_bye\empty
+ #5\xint_bye\xint_bye\xint_bye\empty
+% a \iffalse{\fi} will get inserted by \POL_add_A here
+ {#6}\xint_bye\xint:
+}%
+\def\POL_quorem_c #1.#2.#3\xint:%
+{%
+ {\XINT_Opp#1}%
+ \expandafter\POL_quorem_d\the\numexpr#2-\xint_c_i\expandafter.%
+ \expanded\bgroup
+ \unexpanded{#3}\xint:
+ \expandafter\POL_add_A
+ \expanded\bgroup
+ \POL_apply:x_loop{\xintMul{#1}}%
+}%
+\def\POL_quorem_d #1#2.%
+{%
+ \xint_gob_til_minus#1\POL_quorem_E-%
+ \POL_quorem_b #1#2.%
+}%
+\def\POL_quorem_E-\POL_quorem_b-1.#1\xint:#2\xint_bye\xint:#3.%
+{%
+% this terminates the \POL_quorem_a \expanded
+ \iffalse{\fi}\xint:#3.%
+% recycling some termination code from addition
+ {\expandafter\POL_quorem_ERem_fix\expanded{\POL_add_d#2\xint_bye}}%
+}%
+\def\POL_quorem_ERem_fix #1%
+{%
+ \if P#1\expandafter\POL_quorem_ERem_fix_a\fi 0/1[0]%
+}%
+\def\POL_quorem_ERem_fix_a 0/1[0]#1.#2%
+{%
+ \ifcase #1
+ \or
+ \expandafter\xint_firstofone
+ \else
+ P\the\numexpr#1-\xint_c_i.%
+ \fi
+}%
+\def\POL_quorem_EQuo#1\xint:#2.%
+{%
+ {\ifnum#2=\xint_c_
+ #1%
+ \else
+ P#2.\romannumeral0\XINT_revwbr_loop {}%
+ #1\xint:\xint:\xint:\xint:%
+ \xint:\xint:\xint:\xint:\xint_bye
+ \fi}%
+}%
+\def\xintPolPRem #1%
+{%
+ \expanded\expandafter\POL_prem_in\romannumeral`&&@#1\xint:
+}%
+\def\POL_prem_in #1\xint:#2%
+{%
+ \bgroup
+ \expandafter\POL_prem_fork
+ \romannumeral`&&@#2\xint:#1\xint:
+ \POL_prem_end
+}%
+\def\POL_prem_fork #1#2\xint:#3%
+{%
+ \POL_PPfork
+ #1#3{\POL_prem_a}% both polynomials
+ #1P{\POL_prem_second_is_scalar}% -> scalar
+ #3P{\POL_prem_first_is_scalar}% -> zero
+ PP{\POL_prem_both_are_scalar}% -> zero
+ \krof #1#2\xint:#3%
+}%
+\def\POL_prem_first_is_scalar #1\xint:#2\xint:\POL_prem_end
+{%
+ \iffalse{\fi}{1/1[0]}{0/1[0]}%
+}%
+\def\POL_prem_second_is_scalar #1\xint:#2\xint:\POL_prem_end
+{%
+ \iffalse{\fi}{1/1[0]}{#2}%
+}%
+\def\POL_prem_both_are_scalar #1\xint:#2\xint:\POL_prem_end
+{%
+ \iffalse{\fi}{1/1[0]}{0/1[0]}%
+}%
+\def\POL_prem_a P#1.#2\xint: P#3.#4\xint:
+{%
+ \ifnum#1>#3 \xint_dothis{\POL_prem_easy #3.}\fi
+ \xint_orthat
+ {\expandafter\POL_prem_b\the\numexpr#3-#1\expandafter.%
+ \expanded\bgroup
+ \xintRevWithBraces
+ }%
+ {#2}%
+ \noexpand\xint_Bye
+ \xint:
+ \expandafter\POL_placeBye_loop
+ \the\numexpr#1-\xint_c_vii\expandafter.%
+ \romannumeral0\xintrevwithbraces{#4}%
+ {1/1[0]}%
+ \iffalse{\fi}%
+}%
+\def\POL_prem_easy #1.#2\xintrevwithbraces#3#4\POL_prem_end
+{%
+ \iffalse{\fi}{1/1[0]}{P#1.#3}%
+}%
+\def\POL_placeBye_loop #1#2.%
+{%
+ \xint_gob_til_minus#1\POL_placeBye_loop_end-%
+ \expandafter\POL_placeBye_step\the\numexpr#1#2-\xint_c_viii.%
+}%
+\def\POL_placeBye_step #1.#2#3#4#5#6#7#8#9%
+{%
+ {#2}{#3}{#4}{#5}{#6}{#7}{#8}{#9}\POL_placeBye_loop#1.%
+}%
+\def\POL_placeBye_loop_end-%
+ \expandafter\POL_placeBye_step\the\numexpr-#1-\xint_c_viii.%
+{%
+ \csname POL_placeBye_end#1\endcsname
+}%
+\expandafter\def\csname POL_placeBye_end1\endcsname
+ #1#2#3#4#5#6#7{{#1}{#2}{#3}{#4}{#5}{#6}{#7}\noexpand\xint_Bye\xint:{1}}%
+\expandafter\def\csname POL_placeBye_end2\endcsname
+ #1#2#3#4#5#6{{#1}{#2}{#3}{#4}{#5}{#6}\noexpand\xint_Bye\xint:{1}}%
+\expandafter\def\csname POL_placeBye_end3\endcsname
+ #1#2#3#4#5{{#1}{#2}{#3}{#4}{#5}\noexpand\xint_Bye\xint:{1}}%
+\expandafter\def\csname POL_placeBye_end4\endcsname
+ #1#2#3#4{{#1}{#2}{#3}{#4}\noexpand\xint_Bye\xint:{1}}%
+\expandafter\def\csname POL_placeBye_end5\endcsname
+ #1#2#3{{#1}{#2}{#3}\noexpand\xint_Bye\xint:{1}}%
+\expandafter\def\csname POL_placeBye_end6\endcsname
+ #1#2{{#1}{#2}\noexpand\xint_Bye\xint:{1}}%
+\expandafter\def\csname POL_placeBye_end7\endcsname
+ #1{{#1}\noexpand\xint_Bye\xint:{1}}%
+\expandafter\def\csname POL_placeBye_end8\endcsname
+ {\noexpand\xint_Bye\xint:{1}}%
+\def\POL_prem_b_skip#1#2\unexpanded#3#4#5\xint_Bye#6\xint:#7#8#9%
+{%
+ \iffalse{\fi\expandafter}\xint_gobble_i#5#1%
+}%
+\def\POL_prem_b #1.#2#3\xint:#4#5\xint:#6#7%
+{%
+ \expandafter\POL_prem_c\the\numexpr#1-\xint_c_i\expandafter.%
+ \expanded\bgroup
+ \unexpanded{{#2}#3}\xint:
+ \if0\XINT_Sgn#4\xint:\xint_afterfi
+ {\expandafter\POL_prem_b_skip\expandafter
+ {\expandafter{\romannumeral0\xintmul{#6}{#7}}\xint_Bye\xint:{#6}}%
+ }%
+ \fi
+ \expandafter\POL_add_A
+ \expanded\bgroup
+ \expanded{\noexpand\POL_apply:x_loop{\noexpand\xintMul
+ {\if1\XINT_Sgn#2\xint:\expandafter\XINT_Opp\fi#4}}}%
+% there is already \xint_Bye at ends of #3
+ #3\xint_Bye\xint_Bye\xint_Bye
+ \xint_Bye\xint_Bye\xint_Bye\xint_Bye\xint_bye
+% separator for \POL_add_A
+ \unexpanded{\xint_bye\xint_bye\xint_bye\xint_bye\empty}%
+% there is already \xint_Bye at ends of #5
+ \expanded{\noexpand\POL_apply:x_loop{\noexpand\xintMul{\XINT_Abs#2}}}%
+ #5\xint_Bye\xint_Bye\xint_Bye
+ \xint_Bye\xint_Bye\xint_Bye\xint_Bye\xint_bye
+ \unexpanded{\xint_bye\xint_bye\xint_bye\xint_bye\empty}%
+% a \iffalse{\fi} will get inserted by \POL_add_A exit routine and if will
+% terminate the \expanded triggered here after \POL_prem_c
+% what is next will have already have been expanded
+ {\xintMul{\xintMul{\XINT_Abs#2}{#6}}{#7}}\noexpand\xint_Bye\xint:
+ {\xintMul{\XINT_Abs#2}{#6}}%
+% This terminates the \expanded following \POL_add_A
+ \iffalse{\fi}%
+}%
+\def\POL_prem_c #1%
+{%
+ \xint_gob_til_minus#1\POL_prem_E_-\POL_prem_b#1%
+}%
+% attention that #2 here has a two dummies at end
+% advantage is that \POL_add_a will always think it is non scalar
+\def\POL_prem_E_-\POL_prem_b-1.#1\xint:#2\xint_Bye\xint:#3%
+{%
+ \expandafter\POL_prem_E\expanded{\POL_add_d#2\xint_bye}%
+}%
+\def\POL_prem_E #1%
+{%
+ \if P#1\expandafter\POL_prem_E_i
+ \else\expandafter\POL_prem_E_zero
+ \fi #1%
+}%
+\def\POL_prem_E_zero #1\POL_prem_end{\iffalse{\fi}{#1}{0/1[0]}}%
+\def\POL_prem_E_i P#1.%
+{%
+ \ifnum #1>\xint_c_i\POL_prem_E_ii#1.\fi
+ \POL_prem_E_iii%
+}%
+\def\POL_prem_E_iii#1\POL_prem_end{\iffalse{\fi}#1}%
+\def\POL_prem_E_ii#1.#2\POL_prem_E_iii#3%
+ {#2{#3}{P\the\numexpr#1-\xint_c_i\iffalse}\fi.}%
+\def\POL_prem_end{\iffalse{{\fi}}}%
+%%
+%% SUPPORT FOR FUNCTIONAL INTERFACE
+%%
+% should I do a qpol([]) ?, i.e. without testing for leading zeros, hence
+% would be faster ? but advantage would arise only for very high degree
+% pol([]) this one checks for zeros in the right most coeffs
+\def\xintPolPol#1{\romannumeral`&&@\expandafter\POL_add_d
+ \romannumeral0\expandafter\XINT_revwbr_loop\expandafter
+ {\expandafter}%
+ \romannumeral`&&@#1\xint:\xint:\xint:\xint:
+ \xint:\xint:\xint:\xint:\xint_bye
+ \xint_bye
+}%
+% attention to not overwrite macro names (there is a legacy \PolEvalAt)
+\def\xintPolEvalAt#1#2%
+{%
+% generally, #2 will be scalar, but we allow also a polynomial here
+% should I test for #2 being the monomial, hence handle it very quickly?
+ \romannumeral`&&@\expandafter\POL_evalat_in\romannumeral`&&@#2\xint:
+ #1\xint:\xint:\xint:\xint:
+ \xint:\xint:\xint:\xint:\xint_bye\xint:
+}%
+\def\POL_evalat_in #1\xint:
+{%
+ \expandafter\POL_evalat_fork\expanded{\unexpanded{#1\xint:}\expandafter}%
+ \romannumeral`&&@%
+}%
+\def\POL_evalat_fork #1\xint:#2%
+{%
+ \POL_Pfork
+ #2{\POL_evalat_pol}%
+ P{\POL_evalat_cst}%
+ \krof #1\xint:#2%
+}%
+\def\POL_evalat_cst #1\xint: #2\xint:#3\xint_bye\xint:{#2}%
+\def\POL_evalat_pol #1\xint: P#2.%
+{%
+ \expanded{\unexpanded{\POL_evalat_a#1\xint:}\expandafter}%
+ \romannumeral0\XINT_revwbr_loop{}%
+}%
+\def\POL_evalat_a#1\xint:#2%
+{%
+ \POL_evalat_loop#2\xint:#1\xint:
+}%
+\def\POL_evalat_loop#1\xint:#2\xint:#3%
+{%
+ \xint_gob_til_xint:#3\POL_evalat_E\xint:
+% I have dropped here my old strict \xintFoo = \romannumeral0\xintfoo style
+% ATTENTION! We must allow evaluating at a polynomial expression
+ \expandafter\POL_evalat_loop
+ \romannumeral`&&@\xintPolAdd{#3}{\xintPolMul{#2}{#1}}\xint:#2\xint:
+}%
+\def\POL_evalat_E\xint:\expandafter\POL_evalat_loop
+ \romannumeral`&&@\xintPolAdd #1#2\xint:#3\xint:
+{%
+ \xint_thirdofthree#2%
+}%
+%
+\def\xintPolDeg#1%
+{%
+ \romannumeral`&&@\expandafter\POL_deg_fork\romannumeral`&&@#1\xint:
+}%
+\def\POL_deg_fork #1%
+{%
+ \POL_zeroPfork
+ #1P{\POL_deg_zero}%
+ 0#1{\POL_deg_pol}%
+ 0P{\POL_deg_cst}%
+ \krof #1%
+}%
+% usual hesitations about using or not raw frac format
+\def\POL_deg_zero#1\xint:{-1}%
+\def\POL_deg_cst #1\xint:{0}%
+\def\POL_deg_pol P#1.#2\xint:{#1}%
+%
+\def\xintPolCoeffs#1%
+{%
+ \romannumeral`&&@\expandafter\POL_coeffs_fork\romannumeral`&&@#1\xint:
+}%
+\def\POL_coeffs_fork #1%
+{%
+ \POL_Pfork
+ #1\POL_coeffs_pol
+ P\POL_coeffs_cst
+ \krof #1%
+}%
+% usual hesitations about using or not raw frac format
+\def\POL_coeffs_cst #1\xint:{{#1}}%
+% no brace stripping possible, at least two coefficients
+% annoying that we had to put this delimiter \xint:
+\def\POL_coeffs_pol P#1.#2\xint:{#2}%
+%
+\def\xintPolCoeff#1#2%
+{%
+ \romannumeral`&&@\expandafter\POL_coeff_fork
+ \the\numexpr\xintNum{#2}\expandafter.%
+ \romannumeral`&&@#1\xint:
+}%
+\def\POL_coeff_fork #1.#2%
+{%
+ \POL_Pfork
+ #2\POL_coeff_pol
+ P\POL_coeff_cst
+ \krof #1.#2%
+}%
+\def\POL_coeff_cst#1%
+{%
+ \xint_UDzerofork
+ #1\POL_coeff_itself
+ 0\POL_coeff_zero
+ \krof #1%
+}%
+\def\POL_coeff_itself#1.#2\xint:{#2}%
+\def\POL_coeff_zero#1\xint:{0/1[0]}%
+\def\POL_coeff_pol #1.P#2.%
+{%
+ \ifnum#1<\xint_c_\xint_dothis\POL_coeff_zero\fi
+ \ifnum#1>#2 \xint_dothis\POL_coeff_zero\fi
+ \xint_orthat\POL_coeff_a{#1}%
+}%
+\def\POL_coeff_a#1{\expandafter\POL_coeff_b\romannumeral\xintgobble{#1}}%
+\def\POL_coeff_b#1#2\xint:{#1}%
+%
+\def\xintPolLCoeff#1%
+{%
+ \romannumeral`&&@\expandafter\POL_lcoeff_fork
+ \romannumeral`&&@#1\xint:
+}%
+\def\POL_lcoeff_fork #1%
+{%
+ \POL_Pfork
+ #1\POL_lcoeff_pol
+ P\POL_lcoeff_cst
+ \krof #1%
+}%
+\def\POL_lcoeff_cst#1\xint:{#1}%
+\def\POL_lcoeff_pol P#1.%
+{%
+ \expandafter\POL_lcoeff_a\romannumeral\xintgobble{#1}%
+}%
+\def\POL_lcoeff_a#1\xint:{#1}%
+%
+\def\xintPolMonicPart#1%
+{%
+ \romannumeral`&&@\expandafter\POL_monicpart_fork
+ \romannumeral`&&@#1\xint:
+}%
+\def\POL_monicpart_fork #1%
+{%
+ \POL_Pfork
+ #1\POL_monicpart_pol
+ P\POL_monicpart_cst
+ \krof #1%
+}%
+% monicpart(0) must be 0 to avoid breaking algorithms
+\def\POL_monicpart_cst#1#2\xint:{\if#10\xint_dothis0\fi\xint_orthat1/1[0]}%
+\def\POL_monicpart_pol P#1.#2\xint:%
+{%
+ \expanded{%
+ P#1.%
+ \expandafter\POL_monicpart_a\romannumeral\xintgobble{#1}%
+ #2#2\xint_Bye\xint_Bye\xint_Bye\xint_Bye
+ \xint_Bye\xint_Bye\xint_Bye\xint_Bye\xint_bye
+ }%
+}%
+\def\POL at DivByFirstAndIrrAndREZ#1#2{\xintREZ{\xintIrr{\xintDiv{#2}{#1}}}}%
+\def\POL_monicpart_a#1%
+{%
+ \POL_apply:x_loop{\POL at DivByFirstAndIrrAndREZ{#1}}%
+}%
+%
+\def\xintPolIContent#1%
+{%
+ \romannumeral`&&@\expandafter\POL_icontent_fork
+ \romannumeral`&&@#1^%
+}%
+\def\POL_icontent_fork #1%
+{%
+ \POL_Pfork
+ #1\POL_icontent_pol
+ P\POL_icontent_cst
+ \krof #1%
+}%
+\def\POL_icontent_cst #1^{\xintIrr{\xintAbs{#1}}[0]}%
+\def\POL_icontent_pol P#1.%
+{%
+% 1.4d xintfrac \XINT_fgcdof much saner than 1.4 version !
+% \XINT_fgcd_out does \xintIrr
+ \expandafter\XINT_fgcd_out\romannumeral0\XINT_fgcdof
+}%
+%
+\def\xintPolPrimPart#1%
+{%
+ \romannumeral`&&@\expandafter\POL_primpart_fork
+ \romannumeral`&&@#1\xint:
+}%
+\def\POL_primpart_fork #1%
+{%
+ \POL_Pfork
+ #1\POL_primpart_pol
+ P\POL_primpart_cst
+ \krof #1%
+}%
+\def\POL_primpart_cst#1#2\xint:{\if#10\xint_dothis0\fi\xint_orthat1/1[0]}%
+\def\POL_primpart_pol P#1.#2\xint:%
+{%
+ \expanded{%
+ P#1.\expandafter\POL_primpart_a
+ \romannumeral0\expandafter\XINT_fgcd_out
+ \romannumeral0\XINT_fgcdof#2^\xint:
+ #2\xint_Bye\xint_Bye\xint_Bye\xint_Bye
+ \xint_Bye\xint_Bye\xint_Bye\xint_Bye\xint_bye
+ }%
+}%
+% cf legacy \POL at makeprim@macro
+\def\POL at DivByFirstAndNumAndREZ#1#2{\xintREZ{\xintNum{\xintDiv{#2}{#1}}}}%
+\def\POL_primpart_a#1\xint:{\POL_apply:x_loop{\POL at DivByFirstAndNumAndREZ{#1}}}%
+%
+\def\xintPolRedCoeffs#1%
+{%
+ \romannumeral`&&@\expandafter\POL_redcoeffs_fork
+ \romannumeral`&&@#1\xint:
+}%
+\def\POL_redcoeffs_fork #1%
+{%
+ \POL_Pfork
+ #1\POL_redcoeffs_pol
+ P\POL_redcoeffs_cst
+ \krof #1%
+}%
+\def\POL_redcoeffs_cst#1\xint:{\xintIrr{#1}[0]}%
+\def\POL_redcoeffs_pol P#1.#2\xint:%
+{%
+ \expanded{%
+ P#1.\POL_apply:x_loop\POL at xintIrr
+ #2\xint_Bye\xint_Bye\xint_Bye\xint_Bye
+ \xint_Bye\xint_Bye\xint_Bye\xint_Bye\xint_bye
+ }%
+}%
+%
+\def\xintPolSRedCoeffs#1%
+{%
+ \romannumeral`&&@\expandafter\POL_sredcoeffs_fork
+ \romannumeral`&&@#1\xint:
+}%
+\def\POL_sredcoeffs_fork #1%
+{%
+ \POL_Pfork
+ #1\POL_sredcoeffs_pol
+ P\POL_sredcoeffs_cst
+ \krof #1%
+}%
+\def\POL_sredcoeffs_cst#1\xint:{\xintREZ{\xintIrr{#1}[0]}}%
+\def\POL_sredcoeffs_pol P#1.#2\xint:%
+{%
+ \expanded{%
+ P#1.\POL_apply:x_loop\POL at xintIrrAndREZ
+ #2\xint_Bye\xint_Bye\xint_Bye\xint_Bye
+ \xint_Bye\xint_Bye\xint_Bye\xint_Bye\xint_bye
+ }%
+}%
+\def\POL at xintIrrAndREZ#1{\xintREZ{\xintIrr{#1}[0]}}%
+%
+\def\xintPolDiffOne#1%
+{%
+ \romannumeral`&&@\expandafter\POL_diffone_fork
+ \romannumeral`&&@#1\xint:
+}%
+\def\POL_diffone_fork #1%
+{%
+ \POL_Pfork
+ #1\POL_diffone_pol
+ P\POL_diffone_cst
+ \krof #1%
+}%
+\def\POL_diffone_cst#1\xint:{0/1[0]}%
+\def\POL_diffone_pol P#1.#2#3\xint:%
+{%
+ \expanded{%
+ \ifnum#1=\xint_c_i #3%
+ \else
+ P\the\numexpr#1-\xint_c_i.%
+ \POL_apply:x_iloop{\POL_diffone_diff1.}%
+ #3\xint_Bye\xint_Bye\xint_Bye\xint_Bye
+ \xint_Bye\xint_Bye\xint_Bye\xint_Bye\xint_bye
+ \fi
+ }%
+}%
+\def\POL_diffone_diff#1.#2#3{\xintMul{#1+#2}{#3}}%
+%
+\def\xintPolAntiOne#1%
+{%
+ \romannumeral`&&@\expandafter\POL_antione_fork
+ \romannumeral`&&@#1\xint:
+}%
+\def\POL_antione_fork #1%
+{%
+ \POL_Pfork
+ #1\POL_antione_pol
+ P\POL_antione_cst
+ \krof #1%
+}%
+\def\POL_antione_cst#1%
+{%
+ \xint_gob_til_zero#1\POL_antione_zero0\POL_antione_cst_i#1%
+}%
+\def\POL_antione_cst_i#1\xint:{P1.{0/1[O]}{#1}}%
+\def\POL_antione_zero#1\xint:{0/1[0]}%
+\def\POL_antione_pol P#1.#2\xint:%
+{%
+ \expanded{%
+ P\the\numexpr#1+\xint_c_i.{0/1[0]}%
+ \POL_apply:x_iloop{\POL_antione_anti1.}%
+ #2\xint_Bye\xint_Bye\xint_Bye\xint_Bye
+ \xint_Bye\xint_Bye\xint_Bye\xint_Bye\xint_bye
+ }%
+}%
+\def\POL_antione_anti#1.#2#3{\xintDiv{#3}{#1+#2}}%
+%
+% #2 can be a polynomial
+\def\xintPolIntFrom#1%#2%
+{%
+ \romannumeral`&&@\expandafter\POL_intfrom_a\expandafter
+ {\romannumeral`&&@\xintPolAntiOne{#1}}%
+}%
+\def\POL_intfrom_a #1#2%
+{%
+ \xintPolSub{#1}{\xintPolEvalAt{#1}{#2}}%
+}%
+%
+\def\xintPolIntegral#1#2%
+{%
+ \romannumeral`&&@\expandafter\POL_integral_a\expanded
+ {\xintPolAntiOne{#1}\xint:#2\xint:}%
+}%
+\def\POL_integral_a #1\xint:#2#3\xint:
+{%
+ \xintPolSub{\xintPolEvalAt{#1}{#3}}{\xintPolEvalAt{#1}{#2}}%
+}%
+%
+\def\xintPolDiffTwo#1%
+{%
+ \romannumeral`&&@\expandafter\POL_difftwo_fork
+ \romannumeral`&&@#1\xint:
+}%
+\def\POL_difftwo_fork #1%
+{%
+ \POL_Pfork
+ #1\POL_difftwo_pol
+ P\POL_difftwo_cst
+ \krof #1%
+}%
+\def\POL_difftwo_cst#1\xint:{0/1[0]}%
+\def\POL_difftwo_pol P#1.%
+{%
+ \ifcase #1
+ \or \expandafter\POL_difftwo_zeroout
+ \or \expandafter\POL_difftwo_cstout
+ \else\expandafter\POL_difftwo_polout
+ \fi #1.%
+}%
+\def\POL_difftwo_zeroout#1\xint:{0/1[0]}%
+\def\POL_difftwo_cstout 2.#1#2#3\xint:{\xintMul{2}{#3}}%
+\def\POL_difftwo_polout #1.#2#3#4\xint:%
+{%
+ \expanded{%
+ P\the\numexpr#1-\xint_c_ii.%
+ \POL_apply:x_iloop{\POL_difftwo_diff2.}%
+ #4\xint_Bye\xint_Bye\xint_Bye\xint_Bye
+ \xint_Bye\xint_Bye\xint_Bye\xint_Bye\xint_bye
+ }%
+}%
+\def\POL_difftwo_diff#1.#2#3{\xintMul{\the\numexpr(#1+#2)*(#1+#2-\xint_c_i)\relax}{#3}}%
+%
+\def\POL_diffone_iter_fork #1%
+{%
+ \POL_Pfork
+ #1\POL_diffone_iter_pol
+ P\POL_diffone_iter_cst
+ \krof #1%
+}%
+\def\POL_diffone_iter_cst#1\xint:{0/1[0]\xint:}%
+\def\POL_diffone_iter_pol P#1.#2#3\xint:%
+{%
+ \expanded{%
+ \ifnum#1=\xint_c_i #3%
+ \else
+ P\the\numexpr#1-\xint_c_i.%
+ \POL_apply:x_iloop{\POL_diffone_diff1.}%
+ #3\xint_Bye\xint_Bye\xint_Bye\xint_Bye
+ \xint_Bye\xint_Bye\xint_Bye\xint_Bye\xint_bye
+ \fi
+ }\xint:
+}%
+%
+\def\POL_antione_iter_fork #1%
+{%
+ \POL_Pfork
+ #1\POL_antione_iter_pol
+ P\POL_antione_iter_cst
+ \krof #1%
+}%
+\def\POL_antione_iter_cst#1%
+{%
+ \xint_gob_til_zero#1\POL_antione_iter_zero0\POL_antione_iter_cst_i#1%
+}%
+\def\POL_antione_iter_cst_i#1\xint:{P1.{0/1[O]}{#1}\xint:}%
+\def\POL_antione_iter_zero#1\xint:{0/1[0]\xint:}%
+\def\POL_antione_iter_pol P#1.#2\xint:%
+{%
+ \expanded{%
+ P\the\numexpr#1+\xint_c_i.{0/1[0]}%
+ \POL_apply:x_iloop{\POL_antione_anti1.}%
+ #2\xint_Bye\xint_Bye\xint_Bye\xint_Bye
+ \xint_Bye\xint_Bye\xint_Bye\xint_Bye\xint_bye
+ }\xint:
+}%
+%
+\def\xintPolDiffN#1#2%
+{%
+ \romannumeral`&&@\expandafter\POL_diffn_fork
+ \the\numexpr\xintNum{#2}\expandafter.%
+ \romannumeral`&&@#1\xint:
+}%
+\def\POL_diffn_fork #1%
+{%
+ \xint_UDzerominusfork
+ #1-\POL_diffn_none
+ 0#1\POL_diffn_anti
+ 0-\POL_diffn_diff
+ \krof #1%
+}%
+\def\POL_diffn_none0.#1\xint:{#1}%
+\def\POL_diffn_diff#1.%#2\xint:%
+{%
+ \ifnum#1>\xint_c_i
+ \expandafter\POL_diffn_diff\the\numexpr#1-\xint_c_i\expandafter.%
+ \romannumeral`&&@\expandafter\POL_diffone_iter_fork
+ \else
+ \expandafter\POL_diffone_fork
+ \fi
+}%
+\def\POL_diffn_anti#1.%#2\xint:%
+{%
+ \ifnum#1<-\xint_c_i
+ \expandafter\POL_diffn_anti\the\numexpr#1+\xint_c_i\expandafter.%
+ \romannumeral`&&@\expandafter\POL_antione_iter_fork
+ \else
+ \expandafter\POL_antione_fork
+ \fi
+}%
+%
+% Support for (multi-variable) polgcd
+%
+\def\xintPolGCDof #1%
+{%
+ \romannumeral`&&@\expandafter\POL_polgcdof\romannumeral`&&@#1^%
+}%
+\def\XINT_PolGCDof{\romannumeral`&&@\POL_polgcdof}%
+\def\POL_polgcdof #1%
+{%
+ \romannumeral`&&@\expandafter
+ \POL_polgcdof_chkempty\romannumeral`&&@#1\xint:
+}%
+\def\POL_polgcdof_chkempty #1%
+{%
+ \xint_gob_til_^#1\POL_polgcdof_empty ^\POL_polgcdof_in #1%
+}%
+\def\POL_polgcdof_empty #1\xint:{1/1[0]}% hesitation
+\def\POL_polgcdof_in #1\xint:
+{%
+ \expandafter\POL_polgcdof_loop
+ \romannumeral`&&@\xintPolPrimPart{#1}\xint:
+}%
+\def\POL_polgcdof_loop #1\xint:#2%
+{%
+ \expandafter\POL_polgcdof_chkend\romannumeral`&&@#2\xint:#1\xint:\xint:
+}%
+\def\POL_polgcdof_chkend #1%
+{%
+ \xint_gob_til_^#1\POL_polgcdof_end ^\POL_polgcdof_loop_pair #1%
+}%
+% hesitation with returning a monic polynomial
+%\def\POL_polgcdof_end #1\xint:#2\xint:\xint:{\xintPolMonicPart{#2}}%
+\def\POL_polgcdof_end #1\xint:#2\xint:\xint:{#2}%
+\def\POL_polgcdof_loop_pair #1\xint:
+{%
+ \expandafter\POL_polgcdof_loop
+ \romannumeral`&&@\expandafter\POL_polgcd_pair
+ \romannumeral`&&@\xintPolPrimPart{#1}\xint:
+}%
+% MEMO comme le #2 sera au début le pgcd accumulé il sera souvent de plus
+% petit degré donc il y aura souvent un premier mod "easy" un peu inutile
+% J'hésite à faire une permutation avant de lancer le polgcd_pair
+\def\POL_polgcd_pair#1\xint:#2\xint:
+{%
+ \xintiiifSgn {\xintPolDeg {#1}}%
+ {#2}%
+ {1}%
+ {\expandafter\POL_polgcd_pair
+ \romannumeral`&&@\xintPolPrimPart
+ {\expandafter\xint_secondoftwo
+ \romannumeral`&&@\xintPolPRem {#2}{#1}}\xint:
+ #1\xint:
+ }%
+}%
+%
+\endinput
Property changes on: trunk/Master/texmf-dist/tex/generic/polexpr/polexprcore.tex
___________________________________________________________________
Added: svn:eol-style
## -0,0 +1 ##
+native
\ No newline at end of property
Added: trunk/Master/texmf-dist/tex/generic/polexpr/polexprexpr.tex
===================================================================
--- trunk/Master/texmf-dist/tex/generic/polexpr/polexprexpr.tex (rev 0)
+++ trunk/Master/texmf-dist/tex/generic/polexpr/polexprexpr.tex 2021-04-10 21:13:16 UTC (rev 58821)
@@ -0,0 +1,179 @@
+%% This file polexprexpr.tex is part of the polexpr package (0.8, 2021/03/29)
+%% Extending \xintexpr syntax:
+%%
+%% 1. Authorize ' in variable and function names
+%% This currently breaks infix operators 'and', 'or', 'xor', 'mod'
+%% hence forces usage everywhere of &&, ||, /: and xor() syntax
+%% (if : is active then use /\string : input syntax!)
+%%
+%% 2. Map infix operators to the polexprcore macros
+%%
+%% Overloading of infix operators must be done even outside of \poldef's
+%% scope else functions declared via \xintdeffunc would not be usable in
+%% \poldef as they would be using the xintfrac macros unaware of polynomials
+%%
+%% The overloading of // and /: is experimental.
+%%
+%% 3. Support for the polynomial functions to work in \xintdeffunc
+%%
+%% 4. Support macros for the new functions acting on polynomial variables
+%
+% 1.
+\def\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}%
+}%
+% 2.
+% the minus sign as prefix
+\def\POL_tmp #1#2%
+{%
+ \expandafter\def\csname XINT_expr_exec_#1\endcsname##1##2##3% \XINT_expr_exec_<op><level>
+ {%
+ \expandafter ##1\expandafter ##2\expandafter
+ {\romannumeral`&&@\XINT:NEhook:f:one:from:one{\romannumeral`&&@#2##3}}%
+ }%
+}%
+\POL_tmp{-xii} \xintPolOpp
+\POL_tmp{-xiv} \xintPolOpp
+\POL_tmp{-xvi} \xintPolOpp
+\POL_tmp{-xviii}\xintPolOpp
+% infix operators
+\def\POL_tmp #1#2%
+{%
+ \expandafter\def\csname XINT_expr_exec_#1\endcsname##1##2##3##4% \XINT_expr_exec_<op>
+ {%
+ \expandafter##2\expandafter##3\expandafter
+ {\romannumeral`&&@\XINT:NEhook:f:one:from:two{\romannumeral`&&@#2##1##4}}%
+ }%
+}%
+\POL_tmp + \xintPolAdd
+\POL_tmp - \xintPolSub
+\POL_tmp * \xintPolMul
+\POL_tmp / \xintPolQuo
+% there is no infix operator mapped to \xintPolRem
+% for lack of notation: perhaps /; ? advices welcome
+\POL_tmp{//}\xintPolDivModQ
+\POL_tmp{/:}\xintPolDivModR
+\POL_tmp ^ \xintPolPow
+\expandafter\let\csname XINT_expr_op_**\expandafter\endcsname
+ \csname XINT_expr_op_^\endcsname
+% 3.
+% Matches with "mysterious stuff" section of xintexpr source code
+\let\POL:NEhook:polfunc\expandafter
+\toks0\expandafter{\XINT_expr_redefinemacros}%
+\toks2 {\let\POL:NEhook:polfunc\POL:NE:polfunc}%
+\edef\XINT_expr_redefinemacros{\the\toks0 \the\toks2}%
+\catcode`~ 12
+\def\POL at defpolfunc #1#2%
+{%
+ \expandafter\POL at defpolfunc_a
+ \csname XINT_#2_func_#1\expandafter\endcsname
+ \csname XINT_#2_polfunc_#1\endcsname
+}%
+\def\POL at defpolfunc_a #1#2%
+{%
+ \XINT_global
+ \def#1##1##2##3%
+ {%
+ \expandafter##1\expandafter##2\expandafter{%
+ \romannumeral`&&@\POL:NEhook:polfunc{\romannumeral`&&@#2##3}}%
+ }%
+}%
+\def\POL:NE:polfunc #1{%
+\def\POL:NE:polfunc ##1%
+{%
+ \if0\XINT:NE:hastilde ##1~!\relax % this ! of catcode 11
+ \XINT:NE:hashash ##1#1!\relax 0\else
+ \xint_dothis\POL:NE:polfunc_a\fi
+ \xint_orthat\POL:NE:polfunc_b
+ ##1&&A%
+}}\expandafter\POL:NE:polfunc\string#%
+\def\POL:NE:polfunc_a\romannumeral`&&@#1#2&&A%
+{%
+% If we are here #2 was not braced; \string is done with \escapechar126
+ \expandafter{\expanded{~romannumeral~POL:NE:usepolfunc%
+ {\expandafter\xint_gobble_i\string#1}}#2}%
+}%
+\def\POL:NE:polfunc_b#1{%
+\def\POL:NE:polfunc_b\romannumeral`&&@##1##2&&A%
+{%
+ \expandafter{%
+ \romannumeral`&&@%
+ \if0\XINT:NE:hastilde ##2~!\relax
+ \XINT:NE:hashash ##2#1!\relax 0\else
+ \POL:NE:polfunc_c\fi
+ ##1{##2}}%
+}}\expandafter\POL:NE:polfunc_b\string#%
+% In this case the \expandafter inserted by \POL:NE:usepolfunc
+% expansion will be superfluous
+\def\POL:NE:polfunc_c#1#2% #1=\fi
+{%
+ \expanded{#1~romannumeral~POL:NE:usepolfunc%
+ {\expandafter\xint_gobble_i\string#2}}%
+}%
+% This \expandafter is in case there is an \expanded after that due
+% to some slicing constructs
+% Call: \romannumeral\POL:NE:usepolfunc
+\def\POL:NE:usepolfunc#1{`&&@\csname#1\expandafter\endcsname}%
+\catcode`~ 3 % its normal catcode in xint bundle
+% 4.
+\def\POL_tmp #1#2#3%
+{%
+ \expandafter\def\csname XINT_expr_func_#1\endcsname##1##2##3%
+ {%
+ \expandafter ##1\expandafter ##2\expandafter
+ {%
+ \romannumeral`&&@#2{\romannumeral`&&@#3##3}%
+ }%
+ }%
+}%
+\POL_tmp {sqr} \XINT:NEhook:f:one:from:one \xintPolSqr
+\POL_tmp {pol} \XINT:NEhook:f:one:from:one \xintPolPol
+\POL_tmp {deg} \XINT:NEhook:f:one:from:one \xintPolDeg
+\POL_tmp {coeffs} \XINT:NEhook:f:one:from:one \xintPolCoeffs
+\POL_tmp {coeff} \XINT:NEhook:f:one:from:two \xintPolCoeff
+\POL_tmp {lcoeff} \XINT:NEhook:f:one:from:one \xintPolLCoeff
+\POL_tmp {monicpart} \XINT:NEhook:f:one:from:one \xintPolMonicPart
+\POL_tmp {icontent} \XINT:NEhook:f:one:from:one \xintPolIContent
+\POL_tmp {primpart} \XINT:NEhook:f:one:from:one \xintPolPrimPart
+\POL_tmp {rdcoeffs} \XINT:NEhook:f:one:from:one \xintPolRedCoeffs
+\POL_tmp {rdzcoeffs} \XINT:NEhook:f:one:from:one \xintPolSRedCoeffs
+\POL_tmp {diff1} \XINT:NEhook:f:one:from:one \xintPolDiffOne
+\POL_tmp {diff2} \XINT:NEhook:f:one:from:one \xintPolDiffTwo
+\POL_tmp {diffn} \XINT:NEhook:f:one:from:two \xintPolDiffN
+\POL_tmp {antider} \XINT:NEhook:f:one:from:one \xintPolAntiOne
+\POL_tmp {integral} \XINT:NEhook:f:one:from:two \xintPolIntegral
+\POL_tmp {quorem} \XINT:NEhook:f:one:from:two \xintPolQuoRem
+\POL_tmp {quo} \XINT:NEhook:f:one:from:two \xintPolQuo
+\POL_tmp {rem} \XINT:NEhook:f:one:from:two \xintPolRem
+\POL_tmp {prem} \XINT:NEhook:f:one:from:two \xintPolPRem
+\POL_tmp {divmod} \XINT:NEhook:f:one:from:two \xintPolDivMod
+\POL_tmp {mod} \XINT:NEhook:f:one:from:two \xintPolDivModR
+\POL_tmp {evalp} \XINT:NEhook:f:one:from:two \xintPolEvalAt
+\def\XINT_expr_func_polgcd #1#2#3%
+{%
+ \expandafter #1\expandafter #2\expandafter{\expandafter
+ {\romannumeral`&&@\XINT:NEhook:f:from:delim:u\XINT_PolGCDof#3^}}%
+}%
+% this is provisory
+\xintdeffunc polpowmod_(P, m, Q) :=
+ isone(m)?
+ % m=1: return P modulo Q
+ { mod(P,Q) }
+ % m > 1: test if odd or even and do recursive call
+ { odd(m)? { mod(P*sqr(polpowmod_(P, m//2, Q)), Q) }
+ { mod( sqr(polpowmod_(P, m//2, Q)), Q) }
+ }
+ ;%
+\xintdeffunc polpowmod(P, m, Q) := (m)?{polpowmod_(P, m, Q)}{1};%
+%
+\endinput
Property changes on: trunk/Master/texmf-dist/tex/generic/polexpr/polexprexpr.tex
___________________________________________________________________
Added: svn:eol-style
## -0,0 +1 ##
+native
\ No newline at end of property
Added: trunk/Master/texmf-dist/tex/generic/polexpr/polexprsturm.tex
===================================================================
--- trunk/Master/texmf-dist/tex/generic/polexpr/polexprsturm.tex (rev 0)
+++ trunk/Master/texmf-dist/tex/generic/polexpr/polexprsturm.tex 2021-04-10 21:13:16 UTC (rev 58821)
@@ -0,0 +1,1775 @@
+%% This file polexprsturm.tex is part of the polexpr package (0.8, 2021/03/29)
+%% 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'.
+%% 0.8 - fixes 0.7.5 failure to have updated to xint 1.4 format the defined
+%% \xintexpr variables holding the localization intervals extremities
+%% - also, it uses the prem() in computing the Sturm chain, for a 3X
+%% speed gain in the case of the "perturbed" first Wilkinson example
+%%
+\newcount\POL at count
+\newif\ifPOL at tosturm@makefirstprimitive\POL at tosturm@makefirstprimitivetrue
+\newif\ifPOL at isolz@nextwillneedrefine
+%%
+\def\PolToSturm{\POL at ifstar{\PolToSturm@@}{\PolToSturm@}}%
+\def\POL at aux@toint#1{\xintREZ{\xintNum{#1}}}% for polynomials with int. coeffs!
+%% Attention that some macros rely upon this one setting \POL at sturmname
+%% and \POL at sturm@N as it does
+\def\PolToSturm@#1#2{%
+ \edef\POL at 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 at let{\POL at sturmname _0_}{#1}%
+ \ifnum\PolDegree{#1}=\z@
+ \def\POL at sturm@N{0}%
+ \POL at 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.
+ \XINT_global\@namedef{POLuserpol@\POL at sturmname _0}{0.\empty{1/1[0]}}%
+ \else
+ \ifPOL at tosturm@makefirstprimitive\POL at makeprimitive{\POL at sturmname _0_}\fi
+ \POL at tosturm@dosturm
+ \fi
+ \expandafter
+ \let\csname PolSturmChainLength_\POL at sturmname\endcsname\POL at sturm@N
+ % declare the normalized ones as full-fledged polynomials
+ % \POL at count\z@
+ \xintloop
+ \POL at newpol{\POL at sturmname _\the\POL at count}%
+ \unless\ifnum\POL at sturm@N=\POL at count
+ \advance\POL at count\@ne
+ \repeat
+}%
+\def\PolToSturm@@#1#2{\PolToSturm@{#1}{#2}\POL at tosturm@declareunnormalized}%
+\def\POL at tosturm@declareunnormalized{%
+ % optionally declare also the unnormalized ones
+ \POL at count\z@
+ \xintloop
+ \POL at newpol{\POL at sturmname _\the\POL at count _}%
+ \unless\ifnum\POL at sturm@N=\POL at count
+ \advance\POL at count\@ne
+ \repeat
+}%
+\def\POL at tosturm@dosturm{%
+ \POL at Diff@@one{\POL at sturmname _0_}{\POL at sturmname _1_}%
+ % re-utiliser \POL at varcoeffs directement?
+ \POL at makeprimitive{\POL at sturmname _1_}% does not do \POL at newpol
+ \POL at count\@ne
+ \xintloop
+ % prior to 0.8, code was using here \POL at divide
+ \POL at getprem{\POL at sturmname _\the\numexpr\POL at count-\@ne\relax _}%
+ {\POL at sturmname _\the\POL at count _}%
+ \expandafter\POL at split\POL at R;\POL at degR\POL at polR
+ \unless\ifnum\POL at degR=\m at ne
+ \advance\POL at count\@ne
+ \XINT_global\expandafter\let
+ \csname POLuserpol@\POL at sturmname _\the\POL at count _\endcsname\POL at R
+ \edef\POL at makeprim@icontent{-\POL at icontent\POL at polR}%
+ % this avoids the \POL at newpol from \PolMapCoeffs
+ \POL at mapcoeffs\POL at makeprim@macro{\POL at sturmname _\the\POL at count _}%
+ \repeat
+ \edef\POL at sturm@N{\the\POL at count}%
+ % normalize (now always done even by starred variant)
+ \ifnum\PolDegree{\POL at sturmname _\POL at sturm@N _}>\z@
+ % \POL at count\POL at sturm@N\relax
+ \xintloop
+ \advance\POL at count\m at ne
+ \POL at divide{\POL at sturmname _\the\POL at count _}%
+ {\POL at sturmname _\POL at sturm@N _}%
+ \XINT_global\expandafter
+ \let\csname POLuserpol@\POL at sturmname _\the\POL at count\endcsname\POL at Q
+ % quotient actually belongs to Z[X] and is primitive
+ \POL at mapcoeffs\POL at aux@toint{\POL at sturmname _\the\POL at count}%
+ \ifnum\POL at count>\z@
+ \repeat
+ \XINT_global\@namedef{POLuserpol@\POL at sturmname _\POL at sturm@N}{0.\empty{1/1[0]}}%
+ \else % they are already normalized
+ \advance\POL at count\@ne % attention to include last one also
+ \xintloop
+ \advance\POL at count\m at ne
+ \XINT_global\expandafter\let
+ \csname POLuserpol@\POL at sturmname _\the\POL at count\expandafter\endcsname
+ \csname POLuserpol@\POL at sturmname _\the\POL at count _\endcsname
+ \ifnum\POL at count>\z@
+ \repeat
+ \fi
+ % Back to \PolToSturm@, \POL at count holds 0
+}%
+\def\PolSturmChainLength#1{%
+ \romannumeral`&&@\csname PolSturmChainLength_#1\endcsname
+}%
+\def\PolSetToSturmChainSignChangesAt{%
+ \POL at chkopt\POL at oPolSetToSturmChainSignChangesAt[\global]%
+}%
+\def\POL at oPolSetToSturmChainSignChangesAt[#1]#2#3#4{%
+ \edef\POL at sturmchain@X{\xintREZ{#4}}%
+ \edef\POL at sturmname{#3}%
+ \edef\POL at sturmlength{\PolSturmChainLength{\POL at sturmname}}%
+ \POL at sturmchain@getSV at at\POL at sturmchain@X
+ #1\let#2\POL at sturmchain@SV
+}%
+\def\POL at sturmchain@getSV at at#1{% ATTENTION USES \POL at count
+ \def\POL at sturmchain@SV{0}%
+ \edef\POL at sturmchain@sign{\xintiiSgn{\POL at eval{\POL at sturmname _0}{#1}}}%
+ \let\POL at isolz@lastsign\POL at sturmchain@sign
+ \POL at count \z@
+ \ifnum\POL at isolz@lastsign=\z@
+ \edef\POL at isolz@lastsign
+ {\xintiiSgn{\POL at eval{\POL at sturmname _1}{#1}}}%
+ \POL at count \@ne
+ \fi
+ \xintloop
+ \unless\ifnum\POL at sturmlength=\POL at count
+ \advance\POL at count \@ne
+ \edef\POL at isolz@newsign
+ {\xintiiSgn{\POL at eval{\POL at sturmname _\the\POL at count}{#1}}}%
+ \ifnum\POL at isolz@newsign=\numexpr-\POL at isolz@lastsign\relax
+ \edef\POL at sturmchain@SV{\the\numexpr\POL at sturmchain@SV+\@ne}%
+ \let\POL at isolz@lastsign=\POL at isolz@newsign
+ \fi
+ \repeat
+}%
+\def\PolSetToNbOfZerosWithin{%
+ \POL at chkopt\POL at oPolSetToNbOfZerosWithin[\global]%
+}%
+\def\POL at oPolSetToNbOfZerosWithin[#1]#2#3#4#5{%
+ \edef\POL at tmpA{\xintREZ{#4}}%
+ \edef\POL at tmpB{\xintREZ{#5}}%
+ \edef\POL at sturmname{#3}%
+ \edef\POL at sturmlength{\PolSturmChainLength{\POL at sturmname}}%
+ \POL at sturmchain@getSV at at\POL at tmpA
+ \let\POL at SVA\POL at sturmchain@SV
+ \POL at sturmchain@getSV at at\POL at tmpB
+ \let\POL at SVB\POL at sturmchain@SV
+ \ifnum\POL at SVA<\POL at SVB\space
+ #1\edef#2{\the\numexpr\POL at SVB-\POL at SVA}%
+ \else
+ #1\edef#2{\the\numexpr\POL at SVA-\POL at SVB}%
+ \fi
+}%
+% 0.6 added starred variant to count multiplicities
+% 0.7 added double starred variant to locate all rational roots
+\def\PolSturmIsolateZeros{\POL at ifstar
+ {\PolSturmIsolateZerosAndGetMultiplicities}%
+ {\PolSturmIsolateZeros@}%
+}%
+\def\PolSturmIsolateZerosAndGetMultiplicities{\POL at ifstar
+ {\PolSturmIsolateZerosGetMultiplicitiesAndRationalRoots}%
+ {\PolSturmIsolateZerosAndGetMultiplicities@}%
+}%
+% on aurait besoin de ça dans xint, mais il aurait un \xintRaw{#1} alors
+\def\POL at xintfrac@getNDE #1%
+ {\expandafter\POL at xintfrac@getNDE at i\romannumeral`&&@#1}%
+\def\POL at xintfrac@getNDE at i #1/#2[#3]#4#5#6{\def#4{#1}\def#5{#2}\def#6{#3}}%
+%
+\def\PolSturmIsolateZerosGetMultiplicitiesAndRationalRoots{%
+ \POL at chkopt\POL at oPolSturmIsolateZerosGetMultiplicitiesAndRationalRoots[\empty]%
+}%
+\def\POL at oPolSturmIsolateZerosGetMultiplicitiesAndRationalRoots[#1]#2{%
+ \PolSturmIsolateZerosAndFindRationalRoots[#1]{#2}%
+ \ifnum\POL at isolz@NbOfRoots>\z@
+ % get multiplicities of irrational (real) roots, if any
+ \ifnum\POL at findrat@nbofirrroots>\z@
+ \POL at findrat@getirrmult
+ \fi
+ \POL at isolzmult@defvar at M
+ \fi
+}%
+% added at 0.7
+\def\PolSturmIsolateZerosAndFindRationalRoots{%
+ \POL at chkopt\POL at oPolSturmIsolateZerosAndFindRationalRoots[\empty]%
+}%
+\def\POL at oPolSturmIsolateZerosAndFindRationalRoots[#1]#2{%
+ % #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 at sturmname{#2}%
+ \edef\POL at sturm@N{\@nameuse{PolSturmChainLength_\POL at sturmname}}%
+ % isolate the roots (detects case of constant polynomial)
+ \PolSturmIsolateZeros@{\POL at sturmname}%
+ \ifnum\POL at isolz@NbOfRoots=\z@
+ % no real roots, define empty arrays nevertheless
+ \begingroup\globaldefs\@ne
+ \expandafter\xintAssignArray\expandafter\to\csname POL_ZM\POL at sturmname*\endcsname
+ \expandafter\xintAssignArray\expandafter\to\csname POL_RI\POL at sturmname*\endcsname
+ \endgroup
+ \else
+ % all we currently know is that multiplicities are at least one
+ \begingroup\globaldefs\@ne
+ \expandafter\POL at initarray\csname POL_ZM\POL at 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 at divide entre autres if faut des noms de user pol)
+ \XINT_global
+ \expandafter\let
+ \csname POLuserpol@\POL at sturmname\POL at sqfnorr\expandafter\endcsname
+ \csname POLuserpol@\POL at sturmname _0\endcsname
+ \XINT_global
+ \expandafter\let
+ \csname POLuserpol@\POL at sturmname\POL at norr\expandafter\endcsname
+ \csname POLuserpol@\POL at 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 at findrat@D{\xintAbs{\PolLeadingCoeff{\POL at sturmname _0}}}%
+ \POL at xintfrac@getNDE\POL at findrat@D\POL at findrat@Dint\POL at _\POL at findrat@Dexp
+ \xintiiifOne{\POL at findrat@Dint}
+ {\let\POL at findrat@E\POL at findrat@Dexp} % aussi ok pour 1[0]
+ {\edef\POL at findrat@E{\the\numexpr\xintLen{\POL at findrat@Dint}%
+ +\POL at findrat@Dexp}}%
+% ATTENTION QUE LA CONVENTION DE SIGNE POUR \POL at findrat@E EST OPPOSÉE À CELLE
+% POUR LE CODE PLUS ANCIEN FAISANT "REFINE"
+ \POL at initarray\POL at IfMultIsKnown\xint_secondoftwo
+ \let\POL at findrat@nbofirrroots\POL at isolz@NbOfRoots
+ % find all rational roots, and their multiplicities,
+ % factor them out in passing from original (Sturm root) polynomial
+ \ifnum\POL at findrat@E<7
+ \def\POL at findrat@index{1}%
+ \POL at findrat@loop at secondpass@direct
+ \else
+ % we do a first pass scanning for "small" roots p/q (i.e. q < 1000)
+ \def\POL at findrat@index{1}%
+ \POL at findrat@loop at firstpass
+ % and now we do the final pass finding them all
+ \def\POL at findrat@index{1}%
+ \POL at findrat@loop at secondpass
+ \fi
+ % declare the new polynomials
+ \POL at newpol{\POL at sturmname\POL at sqfnorr}% without multiplicities
+ \POL at newpol{\POL at sturmname\POL at norr}% with multiplicities
+ % declare the array holding the interval indices for the rational roots
+ \expandafter\POL at findrat@doRRarray\csname POL_RI\POL at sturmname*\endcsname
+ \fi
+}%
+\def\POL at findrat@doRRarray#1{%
+ % il faudrait un \xintAssignArray* qui fasse même expansion que \xintFor*
+ \edef\POL at temp{%
+ \xintiloop[1+1]
+ \romannumeral0\csname POL_ZK\POL at sturmname*\xintiloopindex\endcsname
+ \xintbracediloopindex % I should have named it \xintiloopbracedindex...
+ {}%
+ \ifnum\xintiloopindex<\POL at 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 at temp\to#1%
+ \endgroup
+}%
+\def\POL at findrat@loop at firstpass{%
+ \PolSturmIfZeroExactlyKnown{\POL at sturmname}{\POL at findrat@index}%
+ \POL at findrat@loop at decimal% get its multiplicity
+ \POL at findrat@loop at aa % refine interval and check
+ \edef\POL at findrat@index{\the\numexpr\POL at findrat@index+\@ne}%
+ \ifnum\POL at findrat@index>\POL at isolz@NbOfRoots
+ \else
+ \expandafter\POL at findrat@loop at firstpass
+ \fi
+}%
+\def\POL at findrat@loop at aa{%
+ % we do a first pass to identify roots with denominators < 1000
+ \PolEnsureIntervalLength{\POL at sturmname}{\POL at findrat@index}{-6}%
+ % attention that perhaps now the root is known!
+ \PolSturmIfZeroExactlyKnown{\POL at sturmname}{\POL at findrat@index}%
+ \POL at findrat@loop at decimal
+ \POL at findrat@loop at a
+}%
+\def\POL at findrat@loop at 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 at xintfrac@getNDE
+ {\xintIrr{\POL at xintexprGetVar{\POL at sturmname L_\POL at findrat@index}}[0]}%
+ \POL at findrat@xN\POL at findrat@xD\POl at _
+ % we can't move this to updatequotients because other branch will
+ % need to do the division first anyhow
+ \edef\POLuserpol at _findrat@oneterm{1.\noexpand\empty
+ {\xintiiOpp\POL at findrat@xN/1[0]}{\POL at findrat@xD/1[0]}}%
+ \POL at divide{\POL at sturmname\POL at sqfnorr}{_findrat at oneterm}% the one without mult.
+ %\expandafter\POL at split\POL at R;\POL at degR\POL at polR
+ \POL at findrat@loop at updatequotients
+ \POL at findrat@loop at 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 at findrat@getE #1/1[#2]{#2}% /1 as it should be there.
+% so an error will arise if not but cf \POL at refine@getE where I did not put it
+\def\POL at findrat@loop at a{%
+ % attention that the width may have been already smaller than 10^{-6}
+ \POL at get@IsoLeft at rawin
+ \POL at get@IsoRight at rawin
+ \edef\POL at findrat@localW
+ {\the\numexpr-\expandafter\POL at findrat@getE
+ % do I really need the \xintREZ?
+ \romannumeral0\xintrez
+ {\xintSub{\POL at IsoRight@rawin}{\POL at IsoLeft@rawin}}%
+ }% at least 6, maybe larger
+ \expandafter\POL at get@Int at aux
+ \POL at IsoLeft@rawin\POL at IsoLeft@Int{-\POL at findrat@localW}%
+ \expandafter\POL at get@Int at aux
+ \POL at IsoRight@rawin\POL at IsoRight@Int{-\POL at findrat@localW}%
+ % in case of odd, some waste here
+ \edef\POL at findrat@halflocalW{\the\numexpr(\POL at 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 at findrat@localW>9
+ % not implemented yet by lazyness!
+ % this root will be handled in second pass only
+ \else
+ \POL at findrat@gcdloop
+ \fi
+}%
+\def\POL at findrat@gcdloop{%
+ % we must be careful with sign
+ % but we are certain no extremity is a root
+ \let\POL at findrat@ifnegative\xint_secondoftwo
+ \xintiiifSgn\POL at IsoLeft@Int
+ \POL at findrat@gcdloop at n
+ \POL at error@thisisimpossible
+ \POL at findrat@gcdloop at p
+}%
+\def\POL at findrat@gcdloop at n{%
+ \let\POL at findrat@ifnegative\xint_firstoftwo
+ \let\POL at temp\POL at IsoRight@Int
+ \edef\POL at IsoRight@Int{\xintiiOpp{\POL at IsoLeft@Int}}%
+ \edef\POL at IsoLeft@Int{\xintiiOpp{\POL at temp}}%
+ \POL at findrat@gcdloop at p
+}%
+\def\POL at findrat@gcdloop at p{%
+ \edef\POL at findrat@gcdloop at Ap{\xintDec{\xintDouble\POL at IsoRight@Int}}%
+ \edef\POL at findrat@gcdloop at A
+ % at most 2e9: this is acceptable to \numexpr
+ {2\romannumeral\xintreplicate\POL at findrat@localW{0}}%
+ \xintAssign
+ \xintiiDivision\POL at findrat@gcdloop at Ap\POL at findrat@gcdloop at A
+ \to\POL at findrat@gcdloop at B\POL at findrat@gcdloop at 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 at findrat@gcdloop at Binitial\POL at findrat@gcdloop at B
+ \def\POL at findrat@gcdloop at B{0}% do as if B1 = 0
+ \def\POL at findrat@gcdloop at Pp{1}% P0
+ \def\POL at findrat@gcdloop at P{0}% P1
+ \def\POL at findrat@gcdloop at Qp{0}% Q0
+ \def\POL at findrat@gcdloop at 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 at findrat@gcdloop at Ap\POL at findrat@gcdloop at A % A1
+ \let\POL at findrat@gcdloop at A\POL at findrat@gcdloop at An % A2
+ \def\next{\POL at findrat@gcdloop at update}%
+ \def\POL at findrat@gcdloop at done{0}%
+ \POL at findrat@gcdloop at body
+}%
+\def\POL at findrat@gcdloop at body{%
+ % annoying that \numexpr has no divmod... use counts? but groups annoying
+ \edef\POL at findrat@gcdloop at B
+ {\the\numexpr(\POL at findrat@gcdloop at Ap+\POL at findrat@gcdloop at A/2)/%
+ \POL at findrat@gcdloop at A - \@ne}%
+ \edef\POL at findrat@gcdloop at An
+ {\the\numexpr\POL at findrat@gcdloop at Ap-%
+ \POL at findrat@gcdloop at B*\POL at findrat@gcdloop at A}%
+ \edef\POL at findrat@gcdloop at Pn
+ {\the\numexpr\POL at findrat@gcdloop at Pp+%
+ \POL at findrat@gcdloop at B*\POL at findrat@gcdloop at P}%
+ \edef\POL at findrat@gcdloop at Qn
+ {\the\numexpr\POL at findrat@gcdloop at Qp+%
+ \POL at findrat@gcdloop at B*\POL at findrat@gcdloop at Q}%
+ \ifnum\expandafter\xintLength\expandafter{\POL at findrat@gcdloop at Qn}%
+ >\POL at 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 at findrat@gcdloop at Qn>\POL at findrat@gcdloop at An\space
+ % means that P/Q is in interval and is thus a candidate
+ % it is automatically irreducible
+ \edef\POL at findrat@x{\xintiiAdd
+ {\xintiiMul{\POL at findrat@gcdloop at Qn}{\POL at findrat@gcdloop at Binitial}}%
+ {\POL at findrat@gcdloop at Pn}/\POL at findrat@gcdloop at Qn[0]}%
+ \POL at findrat@gcdloop at testit
+ \if1\POL at findrat@gcdloop at done
+ \let\next\empty % a solution was found
+ \fi
+ \fi
+ \fi
+ \next
+}%
+\def\POL at findrat@gcdloop at update{%
+ \ifnum\POL at findrat@gcdloop at An>\z@
+ \let\POL at findrat@gcdloop at Ap\POL at findrat@gcdloop at A
+ \let\POL at findrat@gcdloop at A\POL at findrat@gcdloop at An
+ \let\POL at findrat@gcdloop at Pp\POL at findrat@gcdloop at P
+ \let\POL at findrat@gcdloop at P\POL at findrat@gcdloop at Pn
+ \let\POL at findrat@gcdloop at Qp\POL at findrat@gcdloop at Q
+ \let\POL at findrat@gcdloop at Q\POL at findrat@gcdloop at Qn
+ \expandafter\POL at findrat@gcdloop at body
+ \fi
+}%
+\def\POL at findrat@gcdloop at testit{%
+ % zero should never occur here
+ \POL at findrat@ifnegative{\edef\POL at findrat@x{-\POL at findrat@x}}{}%
+ \POL at xintfrac@getNDE\POL at findrat@x\POL at findrat@xN\POL at findrat@xD\POL at _
+ \edef\POLuserpol at _findrat@oneterm{1.\noexpand\empty
+ {\xintiiOpp{\POL at findrat@xN}/1[0]}{\POL at findrat@xD/1[0]}}%
+ \POL at divide{\POL at sturmname\POL at sqfnorr}{_findrat at oneterm}% the one without mult.
+ \expandafter\POL at split\POL at R;\POL at degR\POL at polR
+ \ifnum\POL at degR=\m at ne % found a root
+ \POL at findrat@loop at storeit
+ \POL at findrat@loop at updatequotients
+ \POL at findrat@loop at getmultiplicity % will continue updating the mult. one
+ \def\POL at findrat@gcdloop at done{1}%
+ \fi
+}%
+% This is second phase
+\def\POL at findrat@loop at secondpass{%
+ \PolSturmIfZeroExactlyKnown{\POL at sturmname}{\POL at findrat@index}%
+ {}% nothing more to be done, already stored
+ \POL at findrat@loop at bb % refine interval and check
+ \edef\POL at findrat@index{\the\numexpr\POL at findrat@index+\@ne}%
+ \ifnum\POL at findrat@index>\POL at isolz@NbOfRoots
+ \else
+ \expandafter\POL at findrat@loop at secondpass
+ \fi
+}%
+\def\POL at findrat@loop at secondpass@direct{%
+ \PolSturmIfZeroExactlyKnown{\POL at sturmname}{\POL at findrat@index}%
+ \POL at findrat@loop at decimal
+ \POL at findrat@loop at bb
+ \edef\POL at findrat@index{\the\numexpr\POL at findrat@index+\@ne}%
+ \ifnum\POL at findrat@index>\POL at isolz@NbOfRoots
+ \else
+ \expandafter\POL at findrat@loop at secondpass@direct
+ \fi
+}%
+\def\POL at findrat@loop at bb{%
+ \PolEnsureIntervalLength{\POL at sturmname}{\POL at findrat@index}{-\POL at findrat@E}%
+ % ATTENTION THAT PERHAPS NOW THE ROOT IS KNOWN!
+ \PolSturmIfZeroExactlyKnown{\POL at sturmname}{\POL at findrat@index}%
+ \POL at findrat@loop at decimal
+ \POL at findrat@loop at b
+}%
+\def\POL at findrat@loop at b{%
+ \edef\POL at findrat@Lscaled{\xintMul{\POL at findrat@D}%
+ {\POL at xintexprGetVar{\POL at sturmname L_\POL at findrat@index}}}%
+ \edef\POL at findrat@Rscaled{\xintMul{\POL at findrat@D}%
+ {\POL at xintexprGetVar{\POL at sturmname R_\POL at findrat@index}}}%
+ \xintiiifNeg{\POL at 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 at findrat@Num{\xintNum{\POL at 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 at findrat@Num\POL at findrat@Rscaled
+ \POL at findrat@loop at c
+ {}% iterate
+ }%
+ {% positive interval (left bound possibly zero!)
+ % truncate towards zero (i.e. to the left) the right bound
+ \edef\POL at findrat@Num{\xintNum{\POL at findrat@Rscaled}/1[0]}%
+ % check if in interval
+ \xintifGt\POL at findrat@Num\POL at findrat@Lscaled
+ \POL at findrat@loop at c
+ {}% iterate
+ }%
+}%
+\def\POL at findrat@loop at c{%
+ % safer to do the edef as \POL at findrat@x used later in storeit
+ \edef\POL at findrat@x{\xintIrr{\xintDiv\POL at findrat@Num\POL at findrat@D}[0]}%
+ \POL at xintfrac@getNDE\POL at findrat@x\POL at findrat@xN\POL at findrat@xD\POL at _
+ \edef\POLuserpol at _findrat@oneterm{1.\noexpand\empty
+ {\xintiiOpp{\POL at findrat@xN}/1[0]}{\POL at findrat@xD/1[0]}}%
+ \POL at divide{\POL at sturmname\POL at sqfnorr}{_findrat at oneterm}% the one without mult.
+ \expandafter\POL at split\POL at R;\POL at degR\POL at polR
+ \ifnum\POL at degR=\m at ne % found a root
+ \POL at findrat@loop at storeit
+ \POL at findrat@loop at updatequotients
+ \POL at findrat@loop at getmultiplicity % will continue updating the mult. one
+ \fi
+ % iterate
+}%
+\def\POL at findrat@loop at storeit{%
+ % update storage, I can not use storeleftandright here (due to rawout etc...)
+ \expandafter
+ \xdef\csname POL_ZL\POL at sturmname*\POL at findrat@index\endcsname
+ {\PolDecToString{\POL at findrat@x}}%
+ \global\expandafter
+ \let\csname POL_ZR\POL at sturmname*\POL at findrat@index\expandafter\endcsname
+ \csname POL_ZL\POL at sturmname*\POL at findrat@index\endcsname
+ \global\expandafter
+ \let\csname POL_ZK\POL at sturmname*\POL at findrat@index\endcsname
+ \xint_stop_atfirstoftwo
+ \begingroup\xintglobaldefstrue
+ % skip some overhead of \xintdefvar...
+ % BUT attention to changes in xint 1.4 internal format !
+ \XINT_expr_defvar_one{\POL at sturmname L_\POL at findrat@index}%
+ {{\POL at findrat@x}}%
+ \XINT_expr_defvar_one{\POL at sturmname R_\POL at findrat@index}%
+ {{\POL at findrat@x}}%
+ \XINT_expr_defvar_one{\POL at sturmname Z_\POL at findrat@index _isknown}%
+ {{1}}%
+ \endgroup
+}%
+\def\POL at findrat@loop at updatequotients{%
+ % attention last division must have been one testing vanishing of\POL at sqfnorr
+ \XINT_global\expandafter\let\csname POLuserpol@\POL at sturmname\POL at sqfnorr\endcsname\POL at Q
+ % quotient belongs to Z[X] and is primitive
+ \POL at mapcoeffs\POL at aux@toint{\POL at sturmname\POL at sqfnorr}%
+ % update the one with multiplicities
+ \POL at divide{\POL at sturmname\POL at norr}{_findrat at oneterm}%
+ \XINT_global\expandafter\let\csname POLuserpol@\POL at sturmname\POL at norr\endcsname\POL at Q
+ \POL at mapcoeffs\POL at aux@toint{\POL at sturmname\POL at norr}
+ % updating of \POL at findrat@D at end of execution of getmultiplicity
+}%
+\def\POL at findrat@loop at getmultiplicity{%
+ % the one without multiplicity must not be divided again!
+ % check if we have remaining multiplicity
+ \POL at divide{\POL at sturmname\POL at norr}{_findrat at oneterm}%
+ \expandafter\POL at split\POL at R;\POL at degR\POL at polR
+ \ifnum\POL at degR=\m at ne % yes
+ \XINT_global\expandafter\let\csname POLuserpol@\POL at sturmname\POL at norr\endcsname\POL at Q
+ \POL at mapcoeffs\POL at aux@toint{\POL at sturmname\POL at norr}%
+ \expandafter
+ \xdef
+ \csname POL_ZM\POL at sturmname*\POL at findrat@index\endcsname
+ {\the\numexpr
+ \csname POL_ZM\POL at sturmname*\POL at findrat@index\endcsname+\@ne}%
+ \expandafter\POL at findrat@loop at getmultiplicity
+ \else
+ % done with multiplicity for this rational root, update stuff
+ \edef\POL at findrat@nbofirrroots
+ {\the\numexpr\POL at findrat@nbofirrroots-\@ne}%
+ \@namedef{POL at IfMultIsKnown\POL at findrat@index}{\xint_firstoftwo}%
+ \edef\POL at findrat@D{\xintAbs{\PolLeadingCoeff{\POL at sturmname\POL at sqfnorr}}}%
+ \POL at xintfrac@getNDE\POL at findrat@D\POL at findrat@Dint\POL at _\POL at findrat@Dexp
+ \xintiiifOne{\POL at findrat@Dint}
+ {\let\POL at findrat@E\POL at findrat@Dexp} % aussi ok pour 1[0]
+ {\edef\POL at findrat@E{\the\numexpr\xintLen{\POL at findrat@Dint}%
+ +\POL at findrat@Dexp}}%
+ \fi
+}%
+\def\POL at findrat@getirrmult{%
+ % first get the GCD of remaining pol with its derivative
+ \POL at divide{\POL at sturmname\POL at norr}{\POL at sturmname\POL at sqfnorr}%
+ \expandafter\let
+ % attention au _ (cf. grosse astuce pour \POL at isolzmult@loop)
+ \csname POLuserpol@@_1\POL at sturmname _\endcsname\POL at Q
+ \ifnum\PolDegree{@_1\POL at 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 at isolzmult@loop qui dépend de
+ % la grosse astuce avec \@gobble
+ \POL at makeprimitive{@_1\POL at sturmname _}%
+ \let\POL at originalsturmname\POL at sturmname
+ % trick to get isolzmult at loop to define @@lastGCD to @_1sturmname_
+ % because it will do \POL at sturmname _\POL at sturm@N _
+ \edef\POL at sturmname{@_1\POL at sturmname}%
+ \let\POL at sturm@N\@gobble% !
+ \let\POL at isolz@NbOfRoots at with_unknown_mult\POL at findrat@nbofirrroots
+ \POL at tosturm@makefirstprimitivefalse
+\expanded{\unexpanded{%
+ \unless\ifxintveryverbose\xintverbosefalse\polnewpolverbosefalse\fi
+ \POL at isolzmult@loop
+}\ifxintverbose\noexpand\xintverbosetrue\fi
+ \ifpolnewpolverbose\noexpand\polnewpolverbosetrue\fi}%
+ \POL at tosturm@makefirstprimitivetrue
+ \let\POL at sturmname\POL at originalsturmname
+ \fi
+}%
+\def\PolSturmIsolateZerosAndGetMultiplicities@{%
+ \POL at chkopt\POL at oPolSturmIsolateZerosAndGetMultiplicities@[\empty]%
+}%
+\def\POL at oPolSturmIsolateZerosAndGetMultiplicities@[#1]#2{%
+ % #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 at sturmname{#2}%
+ \edef\POL at sturm@N{\@nameuse{PolSturmChainLength_\POL at sturmname}}%
+ % isolate the roots (detects case of constant polynomial)
+ \PolSturmIsolateZeros@{\POL at sturmname}%
+ \ifnum\POL at isolz@NbOfRoots=\z@
+ % no roots, define empty array nevertheless
+ \begingroup\globaldefs\@ne
+ \expandafter\xintAssignArray\expandafter\to\csname POL_ZM\POL at sturmname*\endcsname
+ \endgroup
+ \else
+ % all we currently know is that multiplicities are at least one
+ \begingroup\globaldefs\@ne
+ \expandafter\POL at initarray\csname POL_ZM\POL at sturmname*\endcsname{1}%
+ \endgroup
+ % check if GCD had positive degree (hence some roots, maybe complex, have
+ % multiplicity)
+ \ifnum\PolDegree{\POL at sturmname _\POL at sturm@N _}>\z@
+ % scratch array of flags to signal known multiplicities
+ \POL at initarray\POL at 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 at isolz@NbOfRoots at with_unknown_mult\POL at isolz@NbOfRoots
+ % store Sturm chain name, it is needed and altered in isolzmult at loop
+ \let\POL at originalsturmname\POL at sturmname
+ \POL at tosturm@makefirstprimitivefalse
+\expanded{\unexpanded{%
+ \unless\ifxintveryverbose\xintverbosefalse\polnewpolverbosefalse\fi
+ \POL at isolzmult@loop
+}\ifxintverbose\noexpand\xintverbosetrue\fi
+ \ifpolnewpolverbose\noexpand\polnewpolverbosetrue\fi}%
+ \POL at tosturm@makefirstprimitivetrue
+ \let\POL at sturmname\POL at originalsturmname
+ \fi
+ \POL at isolzmult@defvar at M
+ \fi
+}%
+\def\POL at isolzmult@defvar at M{%
+ % Attention that is used not only in ...GetMultiplicities@ but also
+ % in FindRationalRoots
+ \begingroup\xintglobaldefstrue
+ % added at 0.7
+ \let\x\POL at isolz@NbOfRoots
+ \xintloop
+ % skip some overhead of \xintdefvar...
+ % ATTENTION to xint 1.4 internal changes !
+ \XINT_expr_defvar_one{\POL at sturmname M_\x}%
+ {{\csname POL_ZM\POL at sturmname*\x\endcsname}}%
+ \edef\x{\the\numexpr\x-\@ne}%
+ \ifnum\x>\z@
+ \repeat
+ \endgroup
+}%
+\def\POL at isolzmult@loop{%
+ % we are here only if last iteration gave a new GCD still of degree > 0
+ % \POL at sturm@N is the one from last iteration
+ % Attention to not use \POL at 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 at eval in @isknown branch
+ % - because \PolToSturm@ does a \POL at let which would be anomalous
+ % if the extended structure is not existing
+ \edef\POL at isolzmult@lastGCD{\POL at sturmname _\POL at sturm@N _}%
+ \edef\POL at isolzmult@newsturmname{@_1\POL at sturmname}%
+ \POL at newpol{\POL at isolzmult@lastGCD}%
+ \PolToSturm@{\POL at isolzmult@lastGCD}{\POL at isolzmult@newsturmname}%
+ % now both \POL at sturmname and \POL at sturm@N have changed
+ \edef\POL at isolzmult@newGCDdegree{\PolDegree{\POL at sturmname _\POL at sturm@N _}}%
+ \let\POL at isolzmult@index\POL at isolz@NbOfRoots
+ \xintloop
+ % ATTENTION that this executes macros which also modifies \POL at sturmname!
+ % (but not \POL at sturm@N)
+ \POL at isolzmult@doone
+ \edef\POL at isolzmult@index{\the\numexpr\POL at isolzmult@index-\@ne}%
+ \if1\ifnum\POL at isolz@NbOfRoots at with_unknown_mult=\z@ 0\fi
+ \ifnum\POL at isolzmult@index=\z@ 0\fi 1%
+ \repeat
+ \let\POL at sturmname\POL at isolzmult@newsturmname
+ \if1\ifnum\POL at isolz@NbOfRoots at with_unknown_mult=\z@ 0\fi
+ % (if new GCD is constant, time to abort)
+ \ifnum\POL at isolzmult@newGCDdegree=\z@ 0\fi 1%
+ \expandafter\POL at isolzmult@loop
+ \fi
+}%
+\def\POL at isolzmult@doone{%
+ \csname POL at IfMultIsKnown\POL at isolzmult@index\endcsname
+ {}% nothing to do
+ {\POL at SturmIfZeroExactlyKnown{\POL at originalsturmname}%
+ {\POL at isolzmult@index}%
+ \POL at isolzmult@loop at isknown
+ \POL at isolzmult@loop at isnotknown
+ \POL at isolzmult@loop at sharedbody
+ }%
+}%
+\def\POL at isolzmult@loop at isknown{%
+ \xintifZero
+ % attention that \POL at eval requires a declared polynomial
+ {\POL at eval{\POL at isolzmult@lastGCD}%
+ {\POL at xintexprGetVar{\POL at originalsturmname L_\POL at isolzmult@index}}}%
+ {\let\POL at isolzmult@haszero\@ne}%
+ {\let\POL at isolzmult@haszero\z@}%
+}%
+\def\POL at isolzmult@loop at isnotknown{%
+ \edef\POL at isolzmult@loop at A
+ {\POL at xintexprGetVar{\POL at originalsturmname L_\POL at isolzmult@index}}
+ \edef\POL at isolzmult@loop at B
+ {\POL at xintexprGetVar{\POL at originalsturmname
+ R_\POL at isolzmult@index}}
+ % attention that \PolSetToNbOfZerosWithin sets \POL at sturmname to 2nd argument
+ \PolSetToNbOfZerosWithin
+ \POL at isolzmult@haszero % nb of zeros A < x <= B, here 0 or 1
+ \POL at isolzmult@newsturmname
+ \POL at isolzmult@loop at A
+ \POL at isolzmult@loop at B
+}%
+\def\POL at isolzmult@loop at sharedbody{%
+ \ifnum\POL at isolzmult@haszero>\z@
+ \expandafter
+ \xdef
+ \csname POL_ZM\POL at originalsturmname*\POL at isolzmult@index\endcsname
+ {\the\numexpr
+ \csname POL_ZM\POL at originalsturmname
+ *\POL at isolzmult@index\endcsname+\@ne}%
+ \else
+ % multiplicity now known, no need to check this index in future
+ \@namedef{POL at IfMultIsKnown\POL at isolzmult@index}{\xint_firstoftwo}%
+ \edef\POL at isolz@NbOfRoots at with_unknown_mult
+ {\the\numexpr\POL at isolz@NbOfRoots at with_unknown_mult-\@ne}%
+ \fi
+}%
+\def\PolSturmIsolateZeros@{%
+ \POL at chkopt\POL at oPolSturmIsolateZeros@[\empty]%
+}%
+\def\POL at oPolSturmIsolateZeros@[#1]#2{%
+ % #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 at sturmname{#2}%
+ \edef\POL at sturmlength{\PolSturmChainLength{#2}}%
+ % attention to constant polynomial, we must redefine the arrays then
+ \ifnum\POL at sturmlength>\z@
+ \ifx\empty#1\relax
+ \POL at isolz@getsignchanges at plusinf
+ \POL at isolz@getsignchanges at minusinf
+ \else
+ \edef\POL at isolz@E{\the\numexpr\xint_zapspaces #1 \xint_gobble_i\relax}%
+ \POL at sturmchain@getSV at at{1[\POL at isolz@E]}%
+ \let\POL at isolz@plusinf at SV \POL at sturmchain@SV
+ \let\POL at isolz@plusinf at sign\POL at sturmchain@sign
+ \POL at sturmchain@getSV at at{-1[\POL at isolz@E]}%
+ \let\POL at isolz@minusinf at SV \POL at sturmchain@SV
+ \let\POL at isolz@minusinf at sign\POL at sturmchain@sign
+ \ifnum\POL at isolz@plusinf at sign=\z@
+ \PackageError{polexpr}%
+{The polynomial #2 vanishes at set upper bound 10^\POL at isolz@E}%
+{Compile again with a bigger exponent in source. (X to abort).}%
+ \fi
+ \ifnum\POL at isolz@minusinf at sign=\z@
+ \PackageError{polexpr}%
+{The polynomial #2 vanishes at set lower bound -10^\POL at isolz@E}%
+{Compile again with a bigger exponent in source. (X to abort).}%
+ \fi
+ \fi
+ \edef\POL at isolz@NbOfRoots
+ {\the\numexpr\POL at isolz@minusinf at SV-\POL at isolz@plusinf at SV}%
+ \else
+ % constant polynomial
+ \def\POL at isolz@NbOfRoots{0}%
+ \fi
+ \ifnum\POL at 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 at initarray\csname POL_ZL#2*\endcsname{0}%
+ \expandafter\POL at initarray\csname POL_ZR#2*\endcsname{0}%
+ \expandafter\POL at initarray\csname POL_ZK#2*\endcsname
+ \xint_stop_atsecondoftwo
+ \endgroup
+ \ifx\empty#1\relax\expandafter\POL at isolz@getaprioribound\fi
+ \expandafter\POL at isolz@main
+ \fi
+}%
+\def\POL at 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 at isolz@NbOfRoots}{{#2}}\to#1%
+}%
+\def\POL at isolz@getsignchanges at plusinf{%
+ % Count number of sign changes at plus infinity in Sturm sequence
+ \def\POL at isolz@plusinf at SV{0}%
+ \edef\POL at isolz@lastsign{\xintiiSgn{\PolLeadingCoeff{\POL at sturmname _0}}}%
+ \let\POL at isolz@plusinf at sign\POL at isolz@lastsign
+ \POL at count\@ne
+ \xintloop
+ \edef\POL at isolz@newsign
+ {\xintiiSgn{\PolLeadingCoeff{\POL at sturmname _\the\POL at count}}}%
+ \unless\ifnum\POL at isolz@newsign=\POL at isolz@lastsign
+ \edef\POL at isolz@plusinf at SV{\the\numexpr\POL at isolz@plusinf at SV+\@ne}%
+ \fi
+ \let\POL at isolz@lastsign=\POL at isolz@newsign
+ \ifnum\POL at sturmlength>\POL at count
+ \advance\POL at count\@ne
+ \repeat
+}%
+\def\POL at isolz@getsignchanges at minusinf{%
+ % Count number of sign changes at minus infinity in Sturm sequence
+ \def\POL at isolz@minusinf at SV{0}%
+ \edef\POL at isolz@lastsign{\xintiiSgn{\PolLeadingCoeff{\POL at sturmname _0}}}%
+ \ifodd\PolDegree{\POL at sturmname _0}
+ \edef\POL at isolz@lastsign{\xintiiOpp{\POL at isolz@lastsign}}%
+ \fi
+ \let\POL at isolz@minusinf at sign\POL at isolz@lastsign
+ \POL at count\@ne
+ \xintloop
+ \edef\POL at isolz@newsign
+ {\xintiiSgn{\PolLeadingCoeff{\POL at sturmname _\the\POL at count}}}%
+ \ifodd\PolDegree{\POL at sturmname _\the\POL at count}
+ \edef\POL at isolz@newsign{\xintiiOpp{\POL at isolz@newsign}}%
+ \fi
+ \unless\ifnum\POL at isolz@newsign=\POL at isolz@lastsign
+ \edef\POL at isolz@minusinf at SV{\the\numexpr\POL at isolz@minusinf at SV+\@ne}%
+ \fi
+ \let\POL at isolz@lastsign=\POL at isolz@newsign
+ \ifnum\POL at sturmlength>\POL at count
+ \advance\POL at count\@ne
+ \repeat
+}%
+% utility macro for a priori bound on root decimal exponent, via Float Rounding
+\def\POL at isolz@updateE #1e#2;%
+ {\unless\ifnum#2<\POL at isolz@E\space\edef\POL at isolz@E{\the\numexpr#2+\@ne}\fi}%
+\def\POL at isolz@getaprioribound{%
+ \PolAssign{\POL at sturmname _0}\toarray\POL at arrayA
+ \edef\POL at isolz@leading{\POL at arrayA{\POL at arrayA{0}}}%
+ \POL at count\z@
+ \xintloop
+ \advance\POL at count\@ne
+ \ifnum\POL at arrayA{0}>\POL at count
+ \expandafter\edef\csname POL at arrayA\the\POL at count\endcsname
+ {\xintDiv{\POL at arrayA\POL at count}\POL at isolz@leading}%
+ \repeat
+ \def\POL at isolz@E{1}% WE SEEK SMALLEST E SUCH HAT -10^E < roots < +10^E
+ \advance\POL at count\m at ne
+ \xintloop
+ \ifnum\POL at count>\z@
+ \expandafter\POL at 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 at arrayA\POL at count}}};%
+ \advance\POL at count\m at ne
+ \repeat
+ % \ifxintverbose\xintMessage{polexpr}{Info}%
+ % {Roots a priori bounded in absolute value by 10 to the \POL at isolz@E.}%
+ % \fi
+}%
+\def\POL at IsoRight@raw{\POL at IsoRight@Int/1[\POL at isolz@E]}%
+\def\POL at IsoLeft@raw {\POL at IsoLeft@Int/1[\POL at isolz@E]}%
+\def\POL at IsoRight@rawout{%
+ \ifnum\POL at IsoRightSign=\z@\expandafter\xintREZ\fi\POL at IsoRight@raw
+}%
+\def\POL at IsoLeft@rawout{%
+ \ifnum\POL at IsoRightSign=\z@
+ \expandafter\xint_firstoftwo\else\expandafter\xint_secondoftwo
+ \fi{\xintREZ\POL at IsoRight@raw}%
+ {\POL at IsoLeft@Int/1[\POL at isolz@E]}%
+}%
+\def\POL at 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 at isolz@nextwillneedrefinefalse
+ \def\POL at IsoRight@Int{0}%
+ \POL at sturmchain@getSV at at\POL at IsoRight@raw
+ \let\POL at IsoRightSV \POL at sturmchain@SV
+ \let\POL at IsoRightSign\POL at sturmchain@sign
+ \let\POL at IsoAtZeroSV \POL at IsoRightSV
+ \let\POL at IsoAtZeroSign\POL at IsoRightSign
+ \ifnum\POL at IsoAtZeroSign=\z@
+ \xdef\POL at isolz@IntervalIndex
+ {\the\numexpr\POL at isolz@minusinf at SV-\POL at IsoRightSV}%
+ \POL at refine@storeleftandright % store zero root, \POL at IsoRightSign is zero
+ \edef\POL at IsoRightSV{\the\numexpr\POL at IsoRightSV+\@ne}%
+% subtlety here if original polynomial had multiplicities, but ok. I checked!
+ \edef\POL at IsoRightSign % evaluated twice, but that's not so bad
+ {\xintiiOpp{\xintiiSgn{\POL at eval{\POL at sturmname _1}{0/1[0]}}}}%
+ \fi
+ \def\POL at IsoLeft@Int{-1}% -10^E isn't a root!
+ \let\POL at IsoLeftSV \POL at isolz@minusinf at SV
+ \let\POL at IsoLeftSign\POL at isolz@minusinf at sign
+ % \POL at IsoRight@SV was modified if zero is a root
+ \edef\POL at isolz@NbOfNegRoots{\the\numexpr\POL at IsoLeftSV-\POL at IsoRightSV}%
+ \gdef\POL at isolz@IntervalIndex{0}%
+ \let\POL at isolz@@E\POL at isolz@E
+ \ifnum\POL at isolz@NbOfNegRoots>\z@
+% refactored at 0.7 to fix cases leading to an intervals with zero as end-point
+ \POL at isolz@findroots at neg
+ \fi
+ \let\POL at isolz@E\POL at isolz@@E
+ \def\POL at IsoLeft@Int{0}%
+ \let\POL at IsoLeftSV \POL at IsoAtZeroSV % véritable SV en zéro
+ \let\POL at IsoLeftSign\POL at IsoAtZeroSign% véritable signe en zéro
+ \ifnum\POL at IsoLeftSign=\z@
+ \xdef\POL at isolz@IntervalIndex{\the\numexpr\POL at isolz@IntervalIndex+\@ne}%
+ \fi
+ \let\POL@@IsoRightSV \POL at isolz@plusinf at SV
+ \let\POL@@IsoRightSign\POL at isolz@plusinf at sign % 10^E not a root!
+ \edef\POL at isolz@NbOfPosRoots
+ {\the\numexpr\POL at IsoLeftSV-\POL@@IsoRightSV}% attention @@
+ \ifnum\POL at isolz@NbOfPosRoots>\z@
+ % always do that to avoid zero as end-point whether it is a root or not
+ \global\POL at isolz@nextwillneedrefinetrue
+ \POL at isolz@findroots at pos
+ \fi
+}%
+\def\POL at isolz@findroots at neg{%
+ \def\POL at IsoRight@Int{-1}%
+ \POL at isolz@findnextzeroboundeddecade at neg
+ \def\POL at IsoLeft@Int{-10}%
+ \let\POL@@IsoRightSign\POL at IsoRightSign % a zero there is possible
+ \let\POL@@IsoRightSV \POL at IsoRightSV
+ % this will do possibly recursive \POL at isolz@check's
+ \POL at isolz@explorenexteightsubdecades at neg
+ \ifnum\POL at isolz@IntervalIndex<\POL at isolz@NbOfNegRoots\space
+ % above did not explore -2, -1 for this optimization (SV known at Right)
+ \def\POL at IsoRight@Int{-1}%
+ \let\POL at IsoRightSign\POL@@IsoRightSign
+ \let\POL at IsoRightSV \POL@@IsoRightSV
+ \POL at isolz@check
+ \ifnum\POL at isolz@IntervalIndex<\POL at isolz@NbOfNegRoots\space
+ \def\POL at IsoLeft@Int{-1}%
+ \let\POL at IsoLeftSign\POL@@IsoRightSign
+ \let\POL at IsoLeftSV \POL@@IsoRightSV
+ % I don't like being inside TeX conditionals
+ \expandafter\expandafter\expandafter\POL at isolz@findroots at neg
+ \fi
+ \fi
+}%
+\def\POL at isolz@findnextzeroboundeddecade at neg{%
+ \xintloop
+ \edef\POL at isolz@E{\the\numexpr\POL at isolz@E-\@ne}%
+ \POL at sturmchain@getSV at at\POL at IsoRight@raw
+ \let\POL at IsoRightSV \POL at sturmchain@SV
+ \let\POL at IsoRightSign\POL at sturmchain@sign
+ % would an \ifx test be quicker? (to be checked)
+ \ifnum\POL at IsoRightSV=\POL at IsoLeftSV\space
+ % no roots in-between, iterate
+ \repeat
+}%
+\def\POL at isolz@explorenexteightsubdecades at neg{%
+ \xintloop
+ \edef\POL at IsoRight@Int{\the\numexpr\POL at IsoLeft@Int+\@ne}%
+ % we could arguably do a more efficient dichotomy here
+ \POL at sturmchain@getSV at at\POL at IsoRight@raw
+ \let\POL at IsoRightSV \POL at sturmchain@SV
+ \let\POL at IsoRightSign\POL at sturmchain@sign
+ \POL at isolz@check % may recurse if multiple roots are to be found
+ \ifnum\POL at isolz@IntervalIndex=\POL at isolz@NbOfNegRoots\space
+ \expandafter\xintbreakloop
+ \fi
+ \let\POL at IsoLeft@Int\POL at IsoRight@Int
+ \let\POL at IsoLeftSign\POL at IsoRightSign
+ \let\POL at IsoLeftSV\POL at IsoRightSV
+ \ifnum\POL at IsoRight@Int < -\tw@
+ \repeat
+}%
+\def\POL at isolz@findroots at pos{%
+ % remark (2018/12/08), this needs some refactoring, I hardly understand
+ % the logic and it hides most into the recursion done by \POL at isolz@check
+ % It would probably make more sense to proceed like done for the negative
+ % but here finding the largest roots first.
+ \def\POL at IsoRight@Int{1}%
+ \POL at isolz@findnextzeroboundeddecade at pos
+ \unless\ifnum\POL at IsoRightSV=\POL at 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 at 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 at IsoLeft@Int{1}%
+ \let\POL at IsoLeftSV\POL at IsoRightSV
+ \let\POL at IsoLeftSign\POL at IsoRightSign
+ \xintloop
+ \edef\POL at IsoRight@Int{\the\numexpr\POL at IsoLeft@Int+\@ne}%
+ \POL at sturmchain@getSV at at\POL at IsoRight@raw
+ \let\POL at IsoRightSV \POL at sturmchain@SV
+ \let\POL at IsoRightSign\POL at sturmchain@sign
+ \POL at isolz@check % recurses in needed
+ \let\POL at IsoLeft@Int\POL at IsoRight@Int
+ \let\POL at IsoLeftSign\POL at IsoRightSign
+ \let\POL at IsoLeftSV\POL at IsoRightSV
+ \ifnum\POL at isolz@IntervalIndex=\POL at isolz@NbOfRoots\space
+ \expandafter\xintbreakloop
+ \fi
+ \ifnum\POL at IsoLeft@Int < \xint_c_ix
+ \repeat
+ \ifnum\POL at isolz@IntervalIndex<\POL at isolz@NbOfRoots\space
+ % get now the last, rightmost, root (or roots)
+ \def\POL at IsoRight@Int{10}%
+ \let\POL at IsoRightSign\POL@@IsoRightSign
+ \let\POL at IsoRightSV\POL@@IsoRightSV
+ \POL at isolz@check
+ \fi
+}%
+\def\POL at isolz@findnextzeroboundeddecade at pos{%
+ \xintloop
+ \edef\POL at isolz@E{\the\numexpr\POL at isolz@E-\@ne}%
+ \POL at sturmchain@getSV at at\POL at IsoRight@raw
+ \let\POL at IsoRightSV \POL at sturmchain@SV
+ \let\POL at IsoRightSign\POL at sturmchain@sign
+ \ifnum\POL at IsoRightSV=\POL@@IsoRightSV\space
+ \let\POL@@IsoRightSign\POL at IsoRightSign % root here possible!
+ \repeat
+}%
+\def\POL at isolz@check{% \POL at IsoRightSign must be ready for use here
+% \ifxintverbose
+% \xintMessage{polexpr}{Info}%
+% {\the\numexpr\POL at IsoLeftSV-\POL at IsoRightSV\relax\space roots
+% in (\POL at IsoLeft@raw,\POL at IsoRight@raw] (E = \POL at isolz@E)}%
+% \fi
+ \ifcase\numexpr\POL at IsoLeftSV-\POL at IsoRightSV\relax
+ % no root in ]left, right]
+ \global\POL at isolz@nextwillneedrefinefalse
+ \or
+ % exactly one root in ]left, right]
+ \xdef\POL at isolz@IntervalIndex{\the\numexpr\POL at isolz@IntervalIndex+\@ne}%
+ \ifnum\POL at IsoRightSign=\z@
+ % if right boundary is a root, ignore previous flag
+ \global\POL at isolz@nextwillneedrefinefalse
+ \fi
+ % if left boundary is known to have been a root we refine interval
+ \ifPOL at isolz@nextwillneedrefine
+ \expandafter\expandafter\expandafter\POL at isolz@refine
+ \else
+ % \POL at IsoRightSign is zero iff root now exactly known
+ \POL at refine@storeleftandright
+ \ifnum\POL at IsoRightSign=\z@
+ \global\POL at isolz@nextwillneedrefinetrue
+ \fi
+ \fi
+ \else
+ % more than one root, we need to recurse
+ \expandafter\POL at isolz@recursedeeper
+ \fi
+}%
+\def\POL at 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 at isolz@E{\the\numexpr\POL at isolz@E-\@ne}%
+ \edef\POL@@IsoRight at Int{\xintDSL{\POL at IsoRight@Int}}%
+ \let\POL@@IsoRightSign \POL at IsoRightSign
+ \let\POL@@IsoRightSV \POL at IsoRightSV
+ \edef\POL at IsoLeft@Int {\xintDSL{\POL at IsoLeft@Int}}%
+ \xintiloop[1+1]
+ \edef\POL at IsoRight@Int{\xintInc{\POL at IsoLeft@Int}}%
+ \POL at sturmchain@getSV at at\POL at IsoRight@raw
+ \let\POL at IsoRightSV \POL at sturmchain@SV
+ \let\POL at IsoRightSign\POL at sturmchain@sign
+ \POL at isolz@check
+ \let\POL at IsoLeft@Int\POL at IsoRight@Int
+ \let\POL at IsoLeftSV\POL at IsoRightSV
+ \let\POL at IsoLeftSign\POL at IsoRightSign% not used, actually
+ \ifnum\POL at IsoLeftSV=\POL@@IsoRightSV\space
+ \expandafter\xintbreakiloop
+ \fi
+ \ifnum\xintiloopindex < \xint_c_ix
+ \repeat
+ \let\POL at IsoRight@Int\POL@@IsoRight at Int
+ \let\POL at IsoRightSign\POL@@IsoRightSign
+ \let\POL at IsoRightSV \POL@@IsoRightSV
+ % if we exited the loop via breakiloop this is superfluous
+ % but it only costs one \ifnum
+ \POL at isolz@check
+\endgroup
+}%
+\def\POL at 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 at isolz@nextwillneedrefinefalse
+\begingroup
+ \let\POL@@IsoRightSign\POL at IsoRightSign % already evaluated
+ \xintloop
+ \edef\POL at isolz@E{\the\numexpr\POL at isolz@E-\@ne}%
+ \edef\POL at IsoLeft@Int {\xintDSL{\POL at IsoLeft@Int}}%
+ \edef\POL at IsoRight@Int{\xintInc{\POL at IsoLeft@Int}}%
+ \edef\POL at IsoRightSign
+ {\xintiiSgn{\POL at eval{\POL at sturmname _0}{\POL at IsoRight@raw}}}%
+ \ifnum\POL at 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 at IsoRight@Int\POL at IsoLeft@Int
+ % \def\POL at IsoRightSign{0}%
+ % \edef\POL at isolz@IntervalIndex{\the\numexpr\POL at isolz@IntervalIndex-\@ne}%
+ % \POL at refine@storeleftandright
+ % \endgroup
+ \edef\POL@@IsoRight at Int{\xintDSL{\xintInc{\xintDSR{\POL at IsoLeft@Int}}}}%
+ \let\POL at IsoLeft@Int\POL at IsoRight@Int
+ \let\POL at IsoLeftSign\POL at IsoRightSign
+ \ifnum\POL at IsoRightSign=\z@ % check if new Left is actually a root
+ \else
+ \edef\POL at IsoRight@Int{\xintDec{\POL@@IsoRight at Int}}%
+ \edef\POL at IsoRightSign
+ {\xintiiSgn{\POL at eval{\POL at sturmname _0}{\POL at IsoRight@raw}}}%
+ \ifnum\POL at IsoRightSign=\POL@@IsoRightSign\space
+ \POL at refine@doonce % we need to locate in interval (1, 9) in local scale
+ \else
+ \let\POL at IsoLeft@Int\POL at IsoRight@Int
+ \ifnum\POL at IsoRightSign=\z@
+ \def\POL at IsoLeftSign{0}%
+ \else
+ \let\POL at IsoRight@Int\POL@@IsoRight at Int
+ % the IsoRightSign is now wrong but here we don't care
+ \fi\fi
+ \fi
+ % on exit, exact root found iff \POL at IsoRightSign is zero
+ \POL at refine@storeleftandright
+\endgroup
+}%
+\def\POL at 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 at Int\POL at IsoRight@Int % 9
+ \let\POL@@IsoRightSign\POL at IsoRightSign
+ \edef\POL at IsoRight@Int{\xintiiAdd{4}{\POL at IsoLeft@Int}}% 5
+ \edef\POL at IsoRightSign
+ {\xintiiSgn{\POL at eval{\POL at sturmname _0}{\POL at IsoRight@raw}}}%
+ \ifnum\POL at IsoRightSign=\POL at IsoLeftSign\space
+ \let\POL at IsoLeft@Int\POL at IsoRight@Int % 5
+ \edef\POL at IsoRight@Int{\xintiiAdd{2}{\POL at IsoLeft@Int}}%
+ \edef\POL at IsoRightSign
+ {\xintiiSgn{\POL at eval{\POL at sturmname _0}{\POL at IsoRight@raw}}}%
+ \ifnum\POL at IsoRightSign=\POL at IsoLeftSign\space
+ \let\POL at IsoLeft@Int\POL at IsoRight@Int % 7
+ \edef\POL at IsoRight@Int{\xintInc{\POL at IsoLeft@Int}}%
+ \edef\POL at IsoRightSign
+ {\xintiiSgn{\POL at eval{\POL at sturmname _0}{\POL at IsoRight@raw}}}%
+ \ifnum\POL at IsoRightSign=\POL at IsoLeftSign\space
+ \let\POL at IsoLeft@Int\POL at IsoRight@Int % 8
+ \let\POL at IsoRight@Int\POL@@IsoRight at Int % 9
+ \let\POL at IsoRightSign\POL@@IsoRightSign % opposite of one at left
+ \fi % else 7, 8 with possible root at 8
+ \else
+ \ifnum\POL at IsoRightSign=\z@
+ \let\POL at IsoLeft@Int\POL at IsoRight@Int % root at 7
+ \def\POL at IsoLeftSign{0}%
+ \else
+ \let\POL@@IsoRight at Int\POL at IsoRight@Int % 7
+ \edef\POL at IsoRight@Int{\xintInc{\POL at IsoLeft@Int}}% 6
+ \edef\POL at IsoRightSign
+ {\xintiiSgn{\POL at eval{\POL at sturmname _0}{\POL at IsoRight@raw}}}%
+ \ifnum\POL at IsoRightSign=\POL at IsoLeftSign\space
+ \let\POL at IsoLeft@Int\POL at IsoRight@Int % 6
+ \let\POL at IsoRight@Int\POL@@IsoRight at Int % 7
+ \let\POL at IsoRightSign\POL@@IsoRightSign
+ \fi % else 5, 6 with possible root at 6
+ \fi\fi
+ \else
+ \ifnum\POL at IsoRightSign=\z@
+ \let\POL at IsoLeft@Int\POL at IsoRight@Int % root at 5
+ \def\POL at IsoLeftSign{0}%
+ \else
+ \let\POL@@IsoRight at Int\POL at IsoRight@Int % 5
+ \edef\POL at IsoRight@Int{\xintiiAdd{2}{\POL at IsoLeft@Int}}%
+ \edef\POL at IsoRightSign
+ {\xintiiSgn{\POL at eval{\POL at sturmname _0}{\POL at IsoRight@raw}}}%
+ \ifnum\POL at IsoRightSign=\POL at IsoLeftSign\space
+ \let\POL at IsoLeft@Int\POL at IsoRight@Int % 3
+ \edef\POL at IsoRight@Int{\xintInc{\POL at IsoLeft@Int}}% 4
+ \edef\POL at IsoRightSign
+ {\xintiiSgn{\POL at eval{\POL at sturmname _0}{\POL at IsoRight@raw}}}%
+ \ifnum\POL at IsoRightSign=\POL at IsoLeftSign\space
+ \let\POL at IsoLeft@Int\POL at IsoRight@Int % 4
+ \let\POL at IsoRight@Int\POL@@IsoRight at Int % 5
+ \let\POL at IsoRightSign\POL@@IsoRightSign
+ \fi % else 3, 4 with possible root at 4
+ \else
+ \ifnum\POL at IsoRightSign=\z@
+ \let\POL at IsoLeft@Int\POL at IsoRight@Int % root at 3
+ \def\POL at IsoLeftSign{0}%
+ \else
+ \let\POL@@IsoRight at Int\POL at IsoRight@Int % 3
+ \edef\POL at IsoRight@Int{\xintInc{\POL at IsoLeft@Int}}% 2
+ \edef\POL at IsoRightSign
+ {\xintiiSgn{\POL at eval{\POL at sturmname _0}{\POL at IsoRight@raw}}}%
+ \ifnum\POL at IsoRightSign=\POL at IsoLeftSign\space
+ \let\POL at IsoLeft@Int\POL at IsoRight@Int % 2
+ \let\POL at IsoRight@Int\POL@@IsoRight at Int % 3
+ \let\POL at IsoRightSign\POL@@IsoRightSign
+ \fi % else 1, 2 with possible root at 2
+ \fi\fi
+ \fi\fi
+}%
+\def\POL at refine@storeleftandright{%
+ \expandafter
+ \xdef\csname POL_ZL\POL at sturmname*\POL at isolz@IntervalIndex\endcsname
+ {\PolDecToString{\POL at IsoLeft@rawout}}%
+ \expandafter
+ \xdef\csname POL_ZR\POL at sturmname*\POL at isolz@IntervalIndex\endcsname
+ {\PolDecToString{\POL at IsoRight@rawout}}%
+ % added at 0.6
+ \ifnum\POL at IsoRightSign=\z@
+ \global
+ \expandafter
+ \let\csname POL_ZK\POL at sturmname*\POL at isolz@IntervalIndex\endcsname
+ \xint_stop_atfirstoftwo
+ \fi
+ \begingroup\xintglobaldefstrue
+ % skip some overhead of \xintdefvar...
+ % Let me repeat: ATTENTION to change of internal format at xint 1.4
+ \XINT_expr_defvar_one{\POL at sturmname L_\POL at isolz@IntervalIndex}%
+ {{\POL at IsoLeft@rawout}}%
+ \XINT_expr_defvar_one{\POL at sturmname R_\POL at isolz@IntervalIndex}%
+ {{\POL at IsoRight@rawout}}%
+ % added at 0.7
+ \XINT_expr_defvar_one{\POL at sturmname Z_\POL at isolz@IntervalIndex _isknown}%
+ {{\ifnum\POL at IsoRightSign=\z@ 1\else 0\fi}}%
+ \endgroup
+}%
+%% \PolRefineInterval
+%% ATTENTION TO xint 1.4 INTERNAL CHANGES
+\def\POL at xintexprGetVar#1{\expandafter\expandafter\expandafter\xint_firstofone
+ \csname XINT_expr_varvalue_#1\endcsname}%
+% attention, also used by \POL at findrat@loop at a
+\def\POL at get@IsoLeft at rawin{%
+ \edef\POL at IsoLeft@rawin
+ {\POL at xintexprGetVar{\POL at sturmname L_\POL at isolz@IntervalIndex}}%
+}%
+% attention, also used by \POL at findrat@loop at a
+\def\POL at get@IsoRight at rawin{%
+ \edef\POL at IsoRight@rawin
+ {\POL at xintexprGetVar{\POL at sturmname R_\POL at isolz@IntervalIndex}}%
+}%
+% attention, also used by \POL at findrat@loop at a
+\def\POL at get@Int at aux #1/1[#2]#3#4{\edef#3{\xintDSH{#4-#2}{#1}}}%
+\def\POL at get@IsoLeft at Int{%
+ \expandafter\POL at get@Int at aux\POL at IsoLeft@rawin\POL at IsoLeft@Int\POL at isolz@E
+}%
+\def\PolRefineInterval{\POL at ifstar\POL at srefine@start\POL at refine@start}%
+\def\POL at refine@start{%
+ \POL at chkopt\POL at oPOL@refine at start[1]%
+}%
+\def\POL at oPOL@refine at start[#1]#2#3{%
+ \edef\POL at isolz@IntervalIndex{\the\numexpr#3}%
+ \edef\POL at sturmname{#2}%
+ \expandafter\POL at refine@sharedbody\expandafter
+ {\expandafter\POL at refine@loop\expandafter{\the\numexpr#1}}%
+}%
+\def\POL at srefine@start#1#2{%
+ \edef\POL at isolz@IntervalIndex{\the\numexpr#2}%
+ \edef\POL at sturmname{#1}%
+ \POL at refine@sharedbody
+ {\let\POL at refine@left at next\POL at refine@main % we want to recurse if needed
+ \let\POL at refine@right at next\POL at refine@main % we want to recurse if needed
+ \POL at refine@main}%
+}%
+\def\POL at refine@sharedbody#1{%
+ \POL at get@IsoLeft at rawin
+ \edef\POL at IsoLeftSign
+ {\xintiiSgn{\POL at eval{\POL at sturmname _0}{\POL at IsoLeft@rawin}}}%
+ \ifnum\POL at 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 at get@IsoRight at rawin
+ \edef\POL at IsoRightSign{\the\numexpr-\POL at IsoLeftSign}%
+ \edef\POL at isolz@E{\expandafter\POL at refine@getE
+ % je pense que le xintrez ici est superflu
+ \romannumeral0\xintrez{\xintSub{\POL at IsoRight@rawin}{\POL at IsoLeft@rawin}}}%
+ \POL at get@IsoLeft at Int
+ \edef\POL at IsoRight@Int{\xintInc{\POL at IsoLeft@Int}}%
+ #1%
+ \POL at refine@storeleftandright % \POL at IsoRightSign not zero
+ \fi
+}%
+\def\POL at refine@loop#1{%
+ \let\POL at refine@left at next \empty % no recursion at end sub-intervals
+ \let\POL at refine@right at next\empty
+ \xintiloop[1+1]
+ \POL at refine@main
+ \ifnum\POL at IsoRightSign=\z@
+ \expandafter\xintbreakiloop
+ \fi
+ \ifnum\xintiloopindex<#1
+ \repeat
+}%
+\def\POL at refine@main{%
+ \edef\POL at isolz@E{\the\numexpr\POL at isolz@E-\@ne}%
+ \edef\POL at IsoLeft@Int{\xintDSL{\POL at IsoLeft@Int}}%
+ \edef\POL at IsoRight@Int{\xintDSL{\POL at IsoRight@Int}}%
+ \let\POL@@IsoRight at Int\POL at IsoRight@Int
+ \let\POL@@IsoRightSign\POL at IsoRightSign
+ \edef\POL at IsoRight@Int{\xintInc{\POL at IsoLeft@Int}}%
+ \edef\POL at IsoRightSign
+ {\xintiiSgn{\POL at eval{\POL at sturmname _0}{\POL at IsoRight@raw}}}%
+ \ifnum\POL at IsoRightSign=\z@
+ \let\POL at IsoLeft@Int\POL at IsoRight@Int % root at 1
+ \def\POL at IsoLeftSign{0}%
+ \let\POL at next\empty
+ \else
+ \ifnum\POL at IsoRightSign=\POL@@IsoRightSign\space
+ \let\POL at next\POL at refine@left at next % may be \empty or \POL at refine@main for recursion
+ \let\POL at refine@right at next\empty
+ \else
+ \let\POL at IsoLeft@Int\POL at IsoRight@Int
+ \edef\POL at IsoRight@Int{\xintDec{\POL@@IsoRight at Int}}%
+ \edef\POL at IsoRightSign
+ {\xintiiSgn{\POL at eval{\POL at sturmname _0}{\POL at IsoRight@raw}}}%
+ \ifnum\POL at IsoRightSign=\z@
+ \let\POL at IsoLeft@Int\POL at IsoRight@Int % root at 9
+ \def\POL at IsoLeftSign{0}%
+ \let\POL at next\empty
+ \else
+ \ifnum\POL at IsoRightSign=\POL@@IsoRightSign\space
+ \let\POL at next\POL at refine@doonce
+ \else
+ \let\POL at IsoLeft@Int\POL at IsoRight@Int
+ \let\POL at IsoRight@Int\POL@@IsoRight at Int
+ \let\POL at IsoRightSign\POL@@IsoRightSign
+ \let\POL at next\POL at refine@right at next
+ \let\POL at refine@left at next\empty
+ \fi
+ \fi
+ \fi\fi
+ \POL at next
+}%
+% lacking pre-defined xintfrac macro here (such as an \xintRawExponent)
+\def\POL at refine@getE#1[#2]{#2}% \xintREZ already applied, for safety
+%
+%
+\def\PolIntervalWidth#1#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}}}
+}%
+\def\PolEnsureIntervalLengths#1#2{% #1 = Sturm chain name,
+ % localize roots in intervals of length at most 10^{#2}
+ \edef\POL at sturmname{#1}%
+ \edef\POL at ensure@targetE{\the\numexpr#2}%
+ \edef\POL at nbofroots{\csname POL_ZL\POL at sturmname*0\endcsname}%
+ \ifnum\POL at nbofroots>\z@
+ \expandafter\POL at ensureintervallengths
+ \fi
+}%
+\def\POL at ensureintervallengths{%
+ \POL at count\z@
+ % \POL at count used by \POL at sturmchain@getSV at at but latter not used
+ \xintloop
+ \advance\POL at count\@ne
+ \edef\POL at isolz@IntervalIndex{\the\POL at count}%
+ \POL at ensure@one
+ \ifnum\POL at nbofroots>\POL at count
+ \repeat
+}%
+\def\PolEnsureIntervalLength#1#2#3{% #1 = Sturm chain name,
+ % #2 = index of interval
+ % localize roots in intervals of length at most 10^{#3}
+ \edef\POL at sturmname{#1}%
+ \edef\POL at ensure@targetE{\the\numexpr#3}%
+ \edef\POL at isolz@IntervalIndex{\the\numexpr#2}%
+% peut-être autoriser -1, -2, ... ?
+ \ifnum\POL at isolz@IntervalIndex>\z@
+% 0.7, add this safeguard but attention means this structure must be in place
+ \ifnum\csname POL_ZL\POL at sturmname*0\endcsname>\z@
+% je ne fais pas les \expandafter mais je préfèrerais ne pas être à l'intérieur
+ \POL at ensure@one
+ \fi
+ \fi
+}%
+\def\POL at ensure@one{%
+ \POL at get@IsoLeft at rawin
+ \POL at get@IsoRight at rawin
+ \edef\POL at ensure@delta{\xintREZ{\xintSub{\POL at IsoRight@rawin}{\POL at IsoLeft@rawin}}}%
+ \xintiiifZero{\POL at ensure@delta}
+ {}
+ {\edef\POL at isolz@E{\expandafter\POL at refine@getE\POL at ensure@delta}%
+ \POL at get@IsoLeft at Int
+ \edef\POL at IsoRight@Int{\xintInc{\POL at IsoLeft@Int}}%
+ \ifnum\POL at isolz@E>\POL at ensure@targetE\space
+ \edef\POL at IsoLeftSign
+ {\xintiiSgn{\POL at eval{\POL at sturmname _0}{\POL at IsoLeft@raw}}}%
+ % at start left and right are not roots, and values of opposite signs
+ % \edef\POL at IsoRightSign{\the\numexpr-\POL at IsoLeftSign}%
+ \xintloop
+ \POL at ensure@Eloopbody % decreases E by one at each iteration
+ % if separation level is still too coarse we recurse at deeper level
+ \ifnum\POL at isolz@E>\POL at ensure@targetE\space
+ \repeat
+ % will check if right is at a zero, it needs \POL at IsoRightSign set up
+ \POL at refine@storeleftandright
+ \fi
+ }%
+}%
+\def\POL at ensure@Eloopbody {%
+ \edef\POL at isolz@E{\the\numexpr\POL at isolz@E-\@ne}%
+ \edef\POL at IsoLeft@Int{\xintDSL{\POL at IsoLeft@Int}}%
+ % this will loop at most ten times
+ \xintloop
+ \edef\POL at IsoRight@Int{\xintInc{\POL at IsoLeft@Int}}%
+ \edef\POL at IsoRightSign
+ {\xintiiSgn{\POL at eval{\POL at sturmname _0}{\POL at 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 at IsoLeftSign=\POL at IsoRightSign\space
+ \let\POL at IsoLeft@Int\POL at 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 at IsoRightSign=\z@
+ \expandafter\xintbreakloop
+ \fi
+}%
+%
+%% \PolPrintIntervals
+\catcode`_ 8
+\catcode`& 4
+\def\PolPrintIntervals{\POL at ifstar{\PolPrintIntervals@@}{\PolPrintIntervals@}}%
+% As explained in the docs, this is an example of customization so is not
+% itself customizable, apart from redefining it entirely!
+\def\PolPrintIntervals@@{%
+ \begingroup
+ \def\POL at AfterPrintIntervals{\endgroup}%
+ \let\PolPrintIntervalsPrintExactZero\POL@@PrintIntervalsPrintExactZero
+ \let\PolPrintIntervalsUnknownRoot\POL@@PrintIntervalsUnknownRoot
+ \let\PolPrintIntervalsKnownRoot\POL@@PrintIntervalsKnownRoot
+\ifdefined\array
+ \def\arraystretch{2}%
+ \def\PolPrintIntervalsBeginEnv{\[\begin{array}{cl}}%\]
+ \def\PolPrintIntervalsEndEnv{\end{array}\]}%
+\else
+ \def\PolPrintIntervalsBeginEnv{$$\tabskip0pt plus 1000pt minus 1000pt
+ \halign to\displaywidth\bgroup
+ \hfil\vrule height 2\ht\strutbox
+ depth 2\dp\strutbox
+ width \z@
+ $####$\tabskip6pt&$####$\hfil
+ \tabskip0pt plus 1000pt minus 1000pt\cr}%$$
+ \def\PolPrintIntervalsEndEnv{\crcr\egroup$$}%$$
+\fi
+ \PolPrintIntervals@
+}%
+\def\PolPrintIntervals@{%
+ \POL at chkopt\POL at oPolPrintIntervals@[Z]%
+}%
+\def\POL at oPolPrintIntervals@[#1]#2{%
+ \def\PolPrintIntervalsTheVar{#1}%
+ \def\PolPrintIntervalsTheSturmName{#2}%
+ \ifnum\@nameuse{POL_ZL#2*}{0}=\z@
+ \PolPrintIntervalsNoRealRoots
+ \else
+ \gdef\PolPrintIntervalsTheIndex{1}%
+ \POL at PrintIntervals@DoDefs
+ \begingroup\edef\POL at tmp{\endgroup
+ \unexpanded\expandafter{\PolPrintIntervalsBeginEnv}%
+ \unexpanded\expandafter{\POL at PrintIntervals@Loop}%
+ \unexpanded\expandafter{\PolPrintIntervalsEndEnv}%
+ }\POL at tmp
+ \fi
+ \POL at AfterPrintIntervals
+ \def\PolPrintIntervalsTheVar{#1}%
+ \def\PolPrintIntervalsTheSturmName{#2}%
+}%
+\let\POL at AfterPrintIntervals\empty
+\let\PolPrintIntervalsNoRealRoots\empty
+\def\PolPrintIntervalsArrayStretch{1}%
+\ifdefined\array
+ \def\PolPrintIntervalsBeginEnv{\[\begin{array}{rcccl}}%
+ \def\PolPrintIntervalsEndEnv{\end{array}\]}%
+\else
+ \def\PolPrintIntervalsBeginEnv
+ {$$\tabskip 0pt plus 1000pt minus 1000pt
+ \halign to\displaywidth\bgroup
+ \hfil\vrule height\PolPrintIntervalsArrayStretch\ht\strutbox
+ depth \PolPrintIntervalsArrayStretch\dp\strutbox
+ width \z@
+ $##$\tabskip 6pt &\hfil $##$\hfil &\hfil $##$\hfil &\hfil $##$\hfil &$##$\hfil
+ \tabskip 0pt plus 1000pt minus 1000pt \cr
+ }%$$
+ \def\PolPrintIntervalsEndEnv{\crcr\egroup$$}%$$
+\fi
+\def\PolPrintIntervalsKnownRoot{%
+ &&\PolPrintIntervalsTheVar_{\PolPrintIntervalsTheIndex}%
+ &=&\PolPrintIntervalsPrintExactZero
+}%
+\def\PolPrintIntervalsUnknownRoot{%
+ \PolPrintIntervalsPrintLeftEndPoint&<&%
+ \PolPrintIntervalsTheVar_{\PolPrintIntervalsTheIndex}&<&%
+ \PolPrintIntervalsPrintRightEndPoint
+}%
+\def\PolPrintIntervalsPrintExactZero {\PolPrintIntervalsTheLeftEndPoint}%
+\def\PolPrintIntervalsPrintLeftEndPoint {\PolPrintIntervalsTheLeftEndPoint}%
+\def\PolPrintIntervalsPrintRightEndPoint{\PolPrintIntervalsTheRightEndPoint}%
+%
+\ifdefined\mbox
+\def\PolPrintIntervalsPrintMultiplicity{(\mbox{mult. }\PolPrintIntervalsTheMultiplicity)}%
+\else
+\def\PolPrintIntervalsPrintMultiplicity{(\hbox{mult. }\PolPrintIntervalsTheMultiplicity)}%
+\fi
+%
+\def\POL@@PrintIntervalsKnownRoot{%
+ \PolPrintIntervalsPrintMultiplicity&%
+ \PolPrintIntervalsTheVar_{\PolPrintIntervalsTheIndex}=%
+ \PolPrintIntervalsPrintExactZero
+}%
+\ifdefined\frac
+\def\POL@@PrintIntervalsPrintExactZero{%
+ \displaystyle
+ \xintSignedFrac{\PolPrintIntervalsTheLeftEndPoint}%
+}%
+\else
+\def\POL@@PrintIntervalsPrintExactZero{%
+ \displaystyle
+ \xintSignedFwOver{\PolPrintIntervalsTheLeftEndPoint}%
+}%
+\fi
+\def\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`& 7
+\catcode`_ 11
+\def\POL at PrintIntervals@Loop#1{%
+\def\POL at PrintIntervals@Loop{%
+ \POL at SturmIfZeroExactlyKnown\PolPrintIntervalsTheSturmName
+ \PolPrintIntervalsTheIndex
+ \PolPrintIntervalsKnownRoot
+ \PolPrintIntervalsUnknownRoot
+ \xdef\PolPrintIntervalsTheIndex{\the\numexpr\PolPrintIntervalsTheIndex+\@ne}%
+ \unless\ifnum\PolPrintIntervalsTheIndex>
+ \@nameuse{POL_ZL\PolPrintIntervalsTheSturmName*0}
+ \POL at PrintIntervals@DoDefs
+ \xint_afterfi{#1\POL at PrintIntervals@Loop}%
+ \fi
+}}%
+\ifdefined\array\POL at PrintIntervals@Loop{\\}\else\POL at PrintIntervals@Loop{\cr}\fi
+\def\POL at 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
+ }%
+}%
+%
+%% Expandable interface
+%
+\def\PolSturmIfZeroExactlyKnown#1#2{% #1 = sturmname, #2=index
+ \romannumeral0\csname POL_ZK#1*\endcsname{#2}%
+}%
+\def\POL at SturmIfZeroExactlyKnown#1#2{% #1 = sturmname, #2=index
+ \romannumeral0\csname POL_ZK#1*\the\numexpr#2\endcsname
+}%
+\def\PolSturmIsolatedZeroMultiplicity#1#2{%
+ \romannumeral`&&@\csname POL_ZM#1*\endcsname{#2}%
+}%
+\def\PolSturmIsolatedZeroLeft#1#2{%
+ \romannumeral`&&@\csname POL_ZL#1*\endcsname{#2}%
+}%
+\def\PolSturmIsolatedZeroRight#1#2{%
+ \romannumeral`&&@\csname POL_ZR#1*\endcsname{#2}%
+}%
+\def\PolSturmNbOfIsolatedZeros#1{%
+ \romannumeral`&&@\csname POL_ZL#1*0\endcsname
+}%
+\def\PolSturmRationalRoot#1#2{%
+ \romannumeral`&&@\csname POL_ZL#1*%
+ \csname POL_RI#1*\endcsname{#2}\endcsname
+}%
+\def\PolSturmRationalRootIndex#1#2{%
+ \romannumeral`&&@\csname POL_RI#1*\endcsname{#2}%
+}%
+\def\PolSturmRationalRootMultiplicity#1#2{%
+ \romannumeral`&&@\csname POL_ZM#1%
+ *\csname POL_RI#1*\endcsname{#2}\endcsname
+}%
+\def\PolSturmNbOfRationalRoots#1{%
+ \romannumeral`&&@\csname POL_RI#1*0\endcsname
+}%
+\def\PolSturmNbOfRationalRootsWithMultiplicities#1{%
+% means the \POL at norr must not have been changed in-between...
+ \the\numexpr\PolDegree{#1}-\PolDegree{#1\POL at norr}\relax
+}%
+\def\PolSturmIntervalIndex#1#2#3{\the\numexpr\POL at eval@fork
+ #2\PolSturmIntervalIndexAt
+ \At\PolSturmIntervalIndexAtExpr\krof {#1}{#3}%
+}%
+\def\PolSturmIntervalIndexAtExpr#1#2{%
+ \PolSturmIntervalIndexAt{#1}{\xinttheexpr#2\relax}%
+}%
+% ! is of catcode 11 in all of polexpr
+\def\PolSturmIntervalIndexAt#1#2{%
+ \expandafter\POL at sturm@index at at\romannumeral`&&@#2!{#1}\xint_bye\relax
+}%
+\def\POL at sturm@index at at#1!#2%
+{%
+ \expandafter\POL at sturm@index at 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 at sturm@index at at@iloop #1!%
+{%
+ \ifnum #1=\z@ 0\expandafter\xint_bye\fi
+ \POL at sturm@index at at@iloop at a #1!%
+}%
+\def\POL at sturm@index at at@iloop at a #1!#2#3%
+{% #1 = index, #2 = sturmname, #3 value
+ \PolSturmIfZeroExactlyKnown{#2}{#1}
+ {\xintifCmp{#3}{\POL at xintexprGetVar{#2L_#1}}%
+ {}%
+ {#1\xint_bye}%
+ {0\xint_bye}%
+ }%
+ {\xintifGt{#3}{\POL at xintexprGetVar{#2L_#1}}%
+ {\xintifLt{#3}{\POL at xintexprGetVar{#2R_#1}}%
+ {#1\xint_bye}%
+ {0\xint_bye}%
+ }%
+ {}%
+ }%
+ % attention that catcode of ! is 11 in polexpr.sty
+ \expandafter\POL at sturm@index at at@iloop\the\numexpr#1-\@ne !{#2}{#3}%
+}%
+%
+\def\POL at leq@fork#1\LessThanOrEqualTo#2#3\krof{#2}%
+\def\PolSturmNbOfRootsOf#1#2#3{\romannumeral`&&@\POL at leq@fork
+ #2\PolNbOfRootsLessThanOrEqualTo
+ \LessThanOrEqualTo\PolNbOfRootsLessThanOrEqualToExpr\krof {#1}{#3}%
+}%
+\def\PolNbOfRootsLessThanOrEqualToExpr#1#2
+ {\PolNbOfRootsLessThanOrEqualTo{#1}{\xinttheexpr#2\relax}}%
+\def\PolNbOfRootsLessThanOrEqualTo#1{%
+ \ifnum\PolSturmNbOfIsolatedZeros{#1}=\z@
+ \expandafter\xint_firstofthree\expandafter0%
+ \else
+ \expandafter\PolNbOfRootsLessThanOrEqualTo@%
+ \fi {#1}%
+}%
+\def\PolNbOfRootsLessThanOrEqualTo@ #1#2%
+{%
+ \expandafter\POL at nbofrootsleq@prep\romannumeral`&&@#2!{#1}%
+}%
+\def\POL at nbofrootsleq@prep#1!#2%
+{%
+ \expandafter\POL at nbofrootsleq@iloop\expandafter 1\expandafter !%
+ \romannumeral0\xintsgn{\POL at eval{#2_0}{#1}}!%
+ #1!{#2}%
+}%
+\def\POL at nbofrootsleq@iloop#1!#2!#3!#4%
+{% #1 = index, #2 = sign of evaluation at value, #3 = value, #4 = sturmname
+ \xintifCmp{#3}{\POL at xintexprGetVar{#4L_#1}}%
+ {\POL at nbofrootsleq@return #1-\@ne !}%
+ {\POL at 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 at xintexprGetVar{#4R_#1}}%
+ {\POL at nbofrootsleq@return
+ #1\ifnum#2=\xintSgn{\POL at eval{#4_0}{\POL at xintexprGetVar{#4L_#1}}}
+ -\@ne\fi !%
+ }%
+ {\ifnum#1=\PolSturmNbOfIsolatedZeros{#4}
+ \expandafter\POL at nbofrootsleq@rightmost
+ \fi \expandafter\POL at nbofrootsleq@iloop \the\numexpr\@ne+%
+ }%
+ }%
+ #1!#2!#3!{#4}%
+}%
+\def\POL at nbofrootsleq@return #1!#2!#3!#4!#5{\the\numexpr #1\relax}%
+\def\POL at nbofrootsleq@rightmost\expandafter\POL at nbofrootsleq@iloop
+ \the\numexpr\@ne+#1!#2!#3!#4{#1}%
+%
+\def\PolSturmNbWithMultOfRootsOf#1#2#3{%
+ \the\numexpr0\POL at leq@fork
+ #2\PolNbWithMultOfRootsLessThanOrEqualTo
+ \LessThanOrEqualTo\PolNbWithMultOfRootsLessThanOrEqualToExpr\krof {#1}{#3}%
+}%
+\def\PolNbWithMultOfRootsLessThanOrEqualToExpr#1#2{%
+ \PolNbWithMultOfRootsLessThanOrEqualTo{#1}{\xinttheexpr#2\relax}%
+}%
+\def\PolNbWithMultOfRootsLessThanOrEqualTo#1{%
+ \ifnum\PolSturmNbOfIsolatedZeros{#1}=\z@
+ \expandafter\POL at nbwmofroots@noroots
+ \else
+ \expandafter\PolNbWithMultOfRootsLessThanOrEqualTo@%
+ \fi {#1}%
+}%
+\def\POL at nbwmofroots@noroots#1#2{\relax}%
+\def\PolNbWithMultOfRootsLessThanOrEqualTo@ #1#2%
+{%
+ \expandafter\POL at nbwmofrootsleq@prep\romannumeral`&&@#2!{#1}%
+}%
+\def\POL at nbwmofrootsleq@prep#1!#2%
+{%
+ \expandafter\POL at nbwmofrootsleq@iloop\expandafter 1\expandafter !%
+ \romannumeral0\xintsgn{\POL at eval{#2_0}{#1}}!%
+ #1!{#2}%
+}%
+\def\POL at nbwmofrootsleq@iloop#1!#2!#3!#4%
+{% #1 = index, #2 = sign of evaluation at value, #3 = value, #4 = sturmname
+ \xintifCmp{#3}{\POL at xintexprGetVar{#4L_#1}}%
+ {\POL at nbwmofrootsleq@return !}%
+ {\POL at 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 at xintexprGetVar{#4R_#1}}%
+ {\POL at nbwmofrootsleq@return
+ \unless
+ \ifnum#2=\xintSgn{\POL at eval{#4_0}{\POL at xintexprGetVar{#4L_#1}}}
+ +\PolSturmIsolatedZeroMultiplicity{#4}{#1}\fi !%
+ }%
+ {+\PolSturmIsolatedZeroMultiplicity{#4}{#1}%
+ \ifnum#1=\PolSturmNbOfIsolatedZeros{#4}
+ \expandafter\POL at nbwmofrootsleq@return\expandafter !%
+ \fi
+ \expandafter\POL at nbwmofrootsleq@iloop \the\numexpr\@ne+%
+ }%
+ }%
+ #1!#2!#3!{#4}%
+}%
+\def\POL at nbwmofrootsleq@return #1!#2!#3!#4!#5{#1\relax}%
+\endinput
Property changes on: trunk/Master/texmf-dist/tex/generic/polexpr/polexprsturm.tex
___________________________________________________________________
Added: svn:eol-style
## -0,0 +1 ##
+native
\ No newline at end of property
Modified: trunk/Master/tlpkg/libexec/ctan2tds
===================================================================
--- trunk/Master/tlpkg/libexec/ctan2tds 2021-04-10 21:11:28 UTC (rev 58820)
+++ trunk/Master/tlpkg/libexec/ctan2tds 2021-04-10 21:13:16 UTC (rev 58821)
@@ -2130,6 +2130,7 @@
'plnfss', '\.tex|\.pfd',
'plstmary', '^stmary.tex',
'pmx', 'pmx\.tex|' . $standardtex,
+ 'polexpr', 'polexpr.*\.tex|' . $standardtex,
'poormanlog', 'poormanlog\.tex|' . $standardtex,
'poster-mac', 'poster\.tex|' . $standardtex,
'poetrytex', 'poetrytex\.sty$', # not -style
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