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+<!DOCTYPE html>
+<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en" lang="en">
+<head>
+<meta charset="utf-8"/>
+<meta name="generator" content="Docutils 0.16: http://docutils.sourceforge.net/" />
+<title>Package polexpr documentation</title>
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+/* Minimal style sheet for the HTML output of Docutils. */
+/* */
+/* :Author: Günter Milde, based on html4css1.css by David Goodger */
+/* :Id: $Id: minimal.css 8397 2019-09-20 11:09:34Z milde $ */
+/* :Copyright: © 2015 Günter Milde. */
+/* :License: Released under the terms of the `2-Clause BSD license`_, */
+/* in short: */
+/* */
+/* 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. */
+/* */
+/* This file is offered as-is, without any warranty. */
+/* */
+/* .. _2-Clause BSD license: http://www.spdx.org/licenses/BSD-2-Clause */
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+}
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+}
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+}
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+/* div.footer, */
+/* footer { border-top: 1px solid black; } */
+
+/* new HTML5 block elements: set display for older browsers */
+header, section, footer, aside, nav, main, article, figure {
+ display: block;
+}
+
+</style>
+<style type="text/css">
+
+/* CSS31_ style sheet for the output of Docutils HTML writers. */
+/* Rules for easy reading and pre-defined style variants. */
+/* */
+/* :Author: Günter Milde, based on html4css1.css by David Goodger */
+/* :Id: $Id: plain.css 8397 2019-09-20 11:09:34Z milde $ */
+/* :Copyright: © 2015 Günter Milde. */
+/* :License: Released under the terms of the `2-Clause BSD license`_, */
+/* in short: */
+/* */
+/* 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. */
+/* */
+/* This file is offered as-is, without any warranty. */
+/* */
+/* .. _2-Clause BSD license: http://www.spdx.org/licenses/BSD-2-Clause */
+/* .. _CSS3: http://www.w3.org/TR/CSS3 */
+
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+}
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+}
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+}
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+ margin-left: 5em;
+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+ border: 0;
+}
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+ border-bottom: thin solid;
+}
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+ content: "Table " counter(table) ": ";
+ font-weight: bold;
+}
+
+/* Explicit Markup Blocks */
+/* ====================== */
+
+/* Footnotes and Citations */
+/* ----------------------- */
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+/* line on the left */
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+ padding-left: 1ex;
+ border-left: solid;
+ border-left-width: thin;
+}
+
+/* Directives */
+/* ---------- */
+
+/* Body Elements */
+/* ~~~~~~~~~~~~~ */
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+img.align-left,
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+ float: left;
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+}
+.figure.align-right,
+figure.align-right,
+img.align-right,
+object.align-right {
+ display: block;
+ clear: right;
+ float: right;
+ margin-left: 1em;
+}
+/* Stop floating sidebars, images and figures at section level 1,2,3 */
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+
+/* Sidebar */
+
+/* Move right. In a layout with fixed margins, */
+/* it can be moved into the margin. */
+div.sidebar,
+aside.sidebar {
+ width: 30%;
+ max-width: 26em;
+ margin-left: 1em;
+ margin-right: -2%;
+ background-color: #ffffee;
+}
+
+/* Code */
+
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+
+/* Document Header and Footer */
+
+footer, header,
+div.footer, div.header {
+ font-size: smaller;
+ clear: both;
+ padding: 0.5em 2%;
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+ border: none;
+}
+
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+/* ============= */
+
+/* Emphasis */
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+/* strong */
+/* Interpreted Text */
+/* span.interpreted */
+/* Title Reference */
+/* cite */
+
+/* Inline Literals */
+/* possible values: normal, nowrap, pre, pre-wrap, pre-line */
+/* span.docutils.literal { white-space: pre-wrap; } */
+
+/* Hyperlink References */
+a { text-decoration: none; }
+
+/* External Targets */
+/* span.target.external */
+/* Internal Targets */
+/* span.target.internal */
+/* Footnote References */
+/* a.footnote-reference */
+/* Citation References */
+/* a.citation-reference */
+
+</style>
+</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">&lt;, &gt;, &lt;=, &gt;=, ==, !=</span></a></p></li>
+<li><p><a class="reference internal" href="#pol-nutple-expression" id="id50"><span class="docutils literal"><span class="pre">pol(&lt;nutple</span> expression&gt;)</span></a></p></li>
+<li><p><a class="reference internal" href="#xinteval-pol-expr" id="id51"><span class="docutils literal"><span class="pre">\xinteval{&lt;pol.</span> <span class="pre">expr.&gt;}</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(&lt;pol.</span> <span class="pre">expr.&gt;,</span> &lt;pol. expr&gt;)</span></a></p></li>
+<li><p><a class="reference internal" href="#deg-pol-expr" id="id53"><span class="docutils literal"><span class="pre">deg(&lt;pol.</span> <span class="pre">expr.&gt;)</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(&lt;pol.</span> <span class="pre">expr.&gt;)</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(&lt;pol.</span> <span class="pre">expr.&gt;,</span> &lt;num. <span class="pre">expr.&gt;)</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(&lt;pol.</span> <span class="pre">expr.&gt;)</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(&lt;pol.</span> <span class="pre">expr.&gt;)</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(&lt;pol.</span> <span class="pre">expr.&gt;)</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(&lt;pol.</span> <span class="pre">expr.&gt;)</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(&lt;pol.</span> <span class="pre">expr.&gt;,</span> &lt;pol. <span class="pre">expr.&gt;)</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(&lt;pol.</span> <span class="pre">expr.&gt;,</span> &lt;pol. <span class="pre">expr.&gt;)</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(&lt;pol.</span> <span class="pre">expr.&gt;,</span> &lt;pol. <span class="pre">expr.&gt;)</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(&lt;pol.</span> expr. 1&gt;, &lt;pol. expr. 2&gt;)</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(&lt;pol.</span> expr. 1&gt;, &lt;pol. expr. 2&gt;)</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(&lt;pol.</span> expr. 1&gt;, &lt;pol. expr. 2&gt;)</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(&lt;pol.</span> expr. 1&gt;, &lt;pol. expr. 2&gt;, <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(&lt;pol.</span> expr. 1&gt;, &lt;pol. expr. 2&gt;)</span></a></p></li>
+<li><p><a class="reference internal" href="#disc-pol-expr" id="id68"><span class="docutils literal"><span class="pre">disc(&lt;pol.</span> <span class="pre">expr.&gt;)</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(&lt;pol.</span> expr. 1&gt;, &lt;num. <span class="pre">expr.&gt;,</span> &lt;pol. expr. 2&gt;)</span></a></p></li>
+<li><p><a class="reference internal" href="#rdcoeffs-pol-expr" id="id70"><span class="docutils literal"><span class="pre">rdcoeffs(&lt;pol.</span> <span class="pre">expr.&gt;)</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(&lt;pol.</span> <span class="pre">expr.&gt;)</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(&lt;pol.</span> <span class="pre">expr.&gt;)</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(&lt;pol.</span> <span class="pre">expr.&gt;)</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(&lt;pol.</span> expr. P&gt;, &lt;num. expr. n&gt;)</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(&lt;pol.</span> expr. P&gt;)</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(&lt;pol.</span> expr. P&gt;, &lt;pol. expr. c&gt;)</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(&lt;pol.</span> expr. P&gt;, [&lt;pol. expr. a&gt;, &lt;pol. expr. <span class="pre">b&gt;])</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}{&lt;csv&gt;}</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{&lt;pol.</span> <span class="pre">expr.&gt;}</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*{&lt;pol.</span> <span class="pre">expr.&gt;}</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{&lt;pol.</span> <span class="pre">expr.&gt;}</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*{&lt;pol.</span> <span class="pre">expr.&gt;}</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{&lt;pol.</span> <span class="pre">expr.&gt;}</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*{&lt;pol.</span> <span class="pre">expr.&gt;}</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">&#64;</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">&#64;</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">&#64;</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 &quot;ties more&quot;, 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)% &lt;--- 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)% &lt;--- 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{&lt;pol. expr.&gt;}</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">&lt;, &gt;, &lt;=, &gt;=, ==, !=</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(&lt;nutple</span> expression&gt;)</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 &quot;nutple&quot;.</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 &quot;variable&quot; 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{&lt;pol.</span> <span class="pre">expr.&gt;}</span></span></a></h2>
+<blockquote>
+<p>This is documented here for lack of a better place: it evaluates the
+polynomial expression then outputs the &quot;string&quot; <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 &quot;pol&quot; 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(&lt;pol.</span> <span class="pre">expr.&gt;,</span> &lt;pol. expr&gt;)</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(&lt;pol.</span> <span class="pre">expr.&gt;)</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)&lt;0</span></span> correctly detects the zero polynomial,
+and <span class="docutils literal"><span class="pre">deg(P)&lt;=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(&lt;pol.</span> <span class="pre">expr.&gt;)</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(&lt;pol.</span> <span class="pre">expr.&gt;,</span> &lt;num. <span class="pre">expr.&gt;)</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(&lt;pol.</span> <span class="pre">expr.&gt;)</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(&lt;pol.</span> <span class="pre">expr.&gt;)</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(&lt;pol.</span> <span class="pre">expr.&gt;)</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(&lt;pol.</span> <span class="pre">expr.&gt;)</span></span></a></h2>
+<blockquote>
+<p>The quotient (except for the zero polynomial) by
+<span class="docutils literal"><span class="pre">icontent(&lt;pol.</span> <span class="pre">expr.&gt;)</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(&lt;pol.</span> <span class="pre">expr.&gt;,</span> &lt;pol. <span class="pre">expr.&gt;)</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(&lt;pol.</span> <span class="pre">expr.&gt;,</span> &lt;pol. <span class="pre">expr.&gt;)</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(&lt;pol.</span> <span class="pre">expr.&gt;,</span> &lt;pol. <span class="pre">expr.&gt;)</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(&lt;pol.</span> expr. 1&gt;, &lt;pol. expr. 2&gt;)</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)&gt;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)&gt;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(&lt;pol.</span> expr. 1&gt;, &lt;pol. expr. 2&gt;)</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(&lt;pol.</span> expr. 1&gt;, &lt;pol. expr. 2&gt;)</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(&lt;pol.</span> expr. 1&gt;, &lt;pol. expr. 2&gt;, <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 &quot;nutple&quot; 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(&lt;pol.</span> expr. 1&gt;, &lt;pol. expr. 2&gt;)</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(&lt;pol.</span> <span class="pre">expr.&gt;)</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(&lt;pol.</span> expr. 1&gt;, &lt;num. <span class="pre">expr.&gt;,</span> &lt;pol. expr. 2&gt;)</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)&gt;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 &gt; 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(&lt;pol.</span> <span class="pre">expr.&gt;)</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(&lt;pol.</span> <span class="pre">expr.&gt;)</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(&lt;pol.</span> <span class="pre">expr.&gt;)</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(&lt;pol.</span> <span class="pre">expr.&gt;)</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(&lt;pol.</span> expr. P&gt;, &lt;num. expr. n&gt;)</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&lt;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(&lt;pol.</span> expr. P&gt;)</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(&lt;pol.</span> expr. P&gt;, &lt;pol. expr. c&gt;)</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(&lt;pol.</span> expr. P&gt;, [&lt;pol. expr. a&gt;, &lt;pol. expr. <span class="pre">b&gt;])</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{&lt;pol. expr.&gt;}</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}
+$$\PolSturmIsolatedZeroLeft{f}{2}&lt;Z_2&lt;\PolSturmIsolatedZeroRight{f}{2}$$
+And here is the first root with twenty digits after decimal mark:
+\PolEnsureIntervalLength{f}{1}{-20}
+$$\PolSturmIsolatedZeroLeft{f}{1}&lt;Z_1&lt;\PolSturmIsolatedZeroRight{f}{1}$$
+The first element of the Sturm chain has degree $\PolDegree{f_0}$. As
+this is the original degreee $\PolDegree{f}$ we know that $f$ is square free.
+Its derivative is up to a constant \PolTypeset{f_1} (in this example
+it is identical with it).
+\PolToSturm{f_1}{f_1}\PolSturmIsolateZeros{f_1}%
+The derivative has \PolSturmNbOfIsolatedZeros{f_1} distinct real
+roots:
+\PolPrintIntervals[W]{f_1}
+\PolEnsureIntervalLengths{f_1}{-10}%
+Here they are with ten digits after decimal mark:
+\PolPrintIntervals[W]{f_1}
+\PolDiff{f_1}{f''}
+\PolToSturm{f''}{f''}
+\PolSturmIsolateZeros{f''}
+The second derivative is \PolTypeset{f''}.
+It has \PolSturmNbOfIsolatedZeros{f''} distinct real
+roots:
+\PolPrintIntervals[X]{f''}
+Here is the positive one with 20 digits after decimal mark:
+\PolEnsureIntervalLength{f''}{2}{-20}%
+$$X_2 = \PolSturmIsolatedZeroLeft{f''}{2}\dots$$
+The more mathematically advanced among our dear readers will be able
+to give the exact value for $X_2$!</pre>
+</div>
+<div class="section" id="a-degree-four-polynomial-with-nearby-roots">
+<h2><a class="toc-backref" href="#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 &lt; Z_1 &lt; 1.1, 1.10 &lt; Z_2 &lt; 1.11, 1.110 &lt; Z_3 &lt; 1.111, and 1.111 &lt; Z_4 &lt; 1.112
+% but the above bounds do not allow minimizing separation between roots
+% so we refine:
+\PolRefineInterval*{Q}{1}
+\PolRefineInterval*{Q}{2}
+\PolRefineInterval*{Q}{3}
+\PolRefineInterval*{Q}{4}
+\PolPrintIntervals{Q}
+% reports 1.05 &lt; Z_1 &lt; 1.06, 1.105 &lt; Z_2 &lt; 1.106, 1.1105 &lt; Z_3 &lt; 1.1106,
+% and 1.11105 &lt; Z_4 &lt; 1.11106.
+\PolEnsureIntervalLengths{Q}{-6}
+\PolPrintIntervals{Q}
+% of course finds here all roots exactly</pre>
+</div>
+<div class="section" id="the-degree-nine-polynomial-with-0-99-0-999-0-9999-as-triple-roots">
+<h2><a class="toc-backref" href="#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 = &quot;sturmname&quot;
+\xintFor* ##1 in {\xintSeq{1}{\PolSturmNbOfIsolatedZeros{#1}}}\do{%
+ The multiplicity is \PolSturmIsolatedZeroMultiplicity{#1}{##1}
+ \PolSturmIfZeroExactlyKnown{#1}{##1}%
+ {at the root $x=\PolSturmIsolatedZeroLeft{#1}{##1}$}
+ {for the root such that
+ $\PolSturmIsolatedZeroLeft{#1}{##1}&lt;x&lt;\PolSturmIsolatedZeroRight{#1}{##1}$}
+ \par
+}}%
+\PolDef{f}{(x-0.99)^3(x-0.999)^3(x-0.9999)^3}
+\def\PolTypesetOne#1{\PolDecToString{\xintREZ{#1}}}
+\PolTypeset{f}\par
+\PolToSturm{f}{f}% it is allowed to use &quot;polname&quot; as &quot;sturmname&quot; too
+\PolSturmIsolateZerosAndGetMultiplicities{f}% use the &quot;sturmname&quot; here
+% or \PolSturmIsolateZeros*{f} which is exactly the same, but shorter..
+
+\showmultiplicities{f}</pre>
+<p>In this example, the output will look like this (but using math mode):</p>
+<pre class="literal-block">x^9 - 8.9667x^8 + 35.73400293x^7 - 83.070418400109x^6 + 124.143648875193123x^5
+- 123.683070924326075877x^4 + 82.149260397553075617891x^3
+- 35.07602992699900159127007x^2 + 8.7364078733314648368671733x
+- 0.967100824643585986488103299
+
+The multiplicity is 3 at the root x = 0.99
+The multiplicity is 3 at the root x = 0.999
+The multiplicity is 3 at the root x = 0.9999</pre>
+<p>On first pass, these rational roots were found (due to their relative
+magnitudes, using <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 &lt; x &lt; 0.99
+The multiplicity is 1 for the root such that 0.9991 &lt; x &lt; 0.9992
+The multiplicity is 1 for the root such that 0.9997 &lt; x &lt; 0.9998</pre>
+<p>Which means that the multiplicity-3 roots each became a real and a pair of
+complex ones. Let's see them better:</p>
+<pre class="literal-block">\PolEnsureIntervalLengths{g}{-10}
+
+\showmultiplicities{g}</pre>
+<p>which produces:</p>
+<pre class="literal-block">The multiplicity is 1 for the root such that 0.9899888032 &lt; x &lt; 0.9899888033
+The multiplicity is 1 for the root such that 0.9991447980 &lt; x &lt; 0.9991447981
+The multiplicity is 1 for the root such that 0.9997663986 &lt; x &lt; 0.9997663987</pre>
+</div>
+<div class="section" id="a-degree-five-polynomial-with-three-rational-roots">
+<h2><a class="toc-backref" href="#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 &lt; Z_1 &lt; -1, 0.09 &lt; Z_2 &lt; 0.10, 0.1 &lt; Z_3 &lt; 0.2, 1 &lt; Z_4 &lt; 2
+Let us refine the second and third intervals to separate the corresponding
+roots:
+\PolRefineInterval*{P}{2}% will refine to 0.0999990 &lt; Z_2 &lt; 0.0999991
+\PolRefineInterval*{P}{3}% will refine to 0.100001 &lt; Z_3 &lt; 0.100002
+\PolPrintIntervals{P}%
+Let us now get to know all roots with 10 digits after decimal mark:
+\PolEnsureIntervalLengths{P}{-10}%
+\PolPrintIntervals{P}% now all roots are known 10 decimal digits after mark
+Finally, we display 20 digits of the second root:
+\PolEnsureIntervalLength{P}{2}{-20}% makes Z_2 known with 20 digits after mark
+$$\PolSturmIsolatedZeroLeft{P}{2}&lt;Z_2&lt;\PolSturmIsolatedZeroRight{P}{2}$$</pre>
+<p>The last line produces:</p>
+<pre class="literal-block">0.09999900004999650028 &lt; Z_2 &lt; 0.09999900004999650029</pre>
+</div>
+<div class="section" id="the-wilkinson-polynomial">
+<h2><a class="toc-backref" href="#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&lt;15
+\advance\mycount 1
+\repeat
+
+$$T_{15} = \PolTypeset[X]{T_15}$$
+\PolToSturm{T_15}{T_15}
+\PolSturmIsolateZeros{T_15}
+\PolEnsureIntervalLengths{T_15}{-10}
+\PolPrintIntervals{T_15}</pre>
+</div>
+</div>
+<div class="section" id="non-expandable-macros">
+<h1><a class="toc-backref" href="#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{&lt;pol. expr.&gt;}</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}{&lt;csv&gt;}</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{&lt;pol.</span> <span class="pre">expr.&gt;}</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*{&lt;pol.</span> <span class="pre">expr.&gt;}</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 &quot;square-free part&quot; 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 &lt;= 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 &lt; z &lt; 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">&lt;sturmname&gt;L_1</span>,
+<span class="docutils literal">&lt;sturmname&gt;L_2</span>, etc..., for the left end-points and
+<span class="docutils literal">&lt;sturmname&gt;R_1</span>, <span class="docutils literal">&lt;sturmname&gt;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 &quot;known&quot; 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&lt;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&lt;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">&lt;</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{%
+ &amp;&amp;\PolPrintIntervalsTheVar_{\PolPrintIntervalsTheIndex}%
+ &amp;=&amp;\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&amp;&lt;&amp;%
+ \PolPrintIntervalsTheVar_{\PolPrintIntervalsTheIndex}&amp;&lt;&amp;%
+ \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&#64;&#64;PrintIntervals...</span></span> macros. We only reproduce here one
+definition:</p>
+<pre class="literal-block">\newcommand\POL&#64;&#64;PrintIntervalsPrintExactZero{%
+ \displaystyle
+ \xintSignedFrac{\PolPrintIntervalsTheLeftEndPoint}%
+}%</pre>
+<p>Multiplicities are printed using this auxiliary macro:</p>
+</blockquote>
+<div class="section" id="polprintintervalsprintmultiplicity">
+<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{&lt;pol.</span> <span class="pre">expr.&gt;}</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*{&lt;pol.</span> <span class="pre">expr.&gt;}</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{&lt;pol. expr.&gt;}</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{&lt;pol.</span> <span class="pre">expr.&gt;}</span></span></a></h2>
+<blockquote>
+<p>Similar to <a class="reference internal" href="#poltoexpr-pol-expr">\PolToExpr{&lt;pol. expr.&gt;}</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*{&lt;pol.</span> <span class="pre">expr.&gt;}</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">&lt;sturmname&gt;_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{&lt;pol. expr.&gt;}</a> and <a class="reference internal" href="#poltofloatexpr-pol-expr">\PolToFloatExpr{&lt;pol. expr.&gt;}</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">&quot;README thinks \numexpr recognizes ^ operator&quot;</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">&amp;&amp;</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
+&quot;ople&quot; 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>