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authorNorbert Preining <norbert@preining.info>2019-09-02 13:46:59 +0900
committerNorbert Preining <norbert@preining.info>2019-09-02 13:46:59 +0900
commite0c6872cf40896c7be36b11dcc744620f10adf1d (patch)
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+<div class="document" id="package-polexpr-documentation">
+<h1 class="title">Package polexpr documentation</h1>
+<h2 class="subtitle" id="id1">0.7.4 (2019/02/12)</h2>
+
+<!-- comment: -*- fill-column: 72; mode: rst; -*- -->
+<div class="contents topic" id="contents">
+<p class="topic-title first">Contents</p>
+<ul class="simple">
+<li><a class="reference internal" href="#basic-syntax" id="id38">Basic syntax</a></li>
+<li><a class="reference internal" href="#examples-of-localization-of-roots" id="id39">Examples of localization of roots</a><ul>
+<li><a class="reference internal" href="#a-typical-example" id="id40">A typical example</a></li>
+<li><a class="reference internal" href="#a-degree-four-polynomial-with-nearby-roots" id="id41">A degree four polynomial with nearby roots</a></li>
+<li><a class="reference internal" href="#the-degree-nine-polynomial-with-0-99-0-999-0-9999-as-triple-roots" id="id42">The degree nine polynomial with 0.99, 0.999, 0.9999 as triple roots</a></li>
+<li><a class="reference internal" href="#a-degree-five-polynomial-with-three-rational-roots" id="id43">A degree five polynomial with three rational roots</a></li>
+<li><a class="reference internal" href="#a-mignotte-type-polynomial" id="id44">A Mignotte type polynomial</a></li>
+<li><a class="reference internal" href="#the-wilkinson-polynomial" id="id45">The Wilkinson polynomial</a></li>
+<li><a class="reference internal" href="#the-second-wilkinson-polynomial" id="id46">The second Wilkinson polynomial</a></li>
+<li><a class="reference internal" href="#the-degree-41-polynomial-with-2-1-9-1-8-0-0-1-1-9-2-as-roots" id="id47">The degree 41 polynomial with -2, -1.9, -1.8, ..., 0, 0.1, ..., 1.9, 2 as roots</a></li>
+<li><a class="reference internal" href="#roots-of-chebyshev-polynomials" id="id48">Roots of Chebyshev polynomials</a></li>
+</ul>
+</li>
+<li><a class="reference internal" href="#non-expandable-macros" id="id49">Non-expandable macros</a><ul>
+<li><a class="reference internal" href="#poldef-polname-letter-expression-in-letter" id="id50"><tt class="docutils literal">\poldef <span class="pre">polname(letter):=</span> expression in letter;</tt></a></li>
+<li><a class="reference internal" href="#poldef-letter-polname-expression-in-letter" id="id51"><tt class="docutils literal"><span class="pre">\PolDef[letter]{polname}{expression</span> in letter}</tt></a></li>
+<li><a class="reference internal" href="#polgenfloatvariant-polname" id="id52"><tt class="docutils literal">\PolGenFloatVariant{polname}</tt></a></li>
+<li><a class="reference internal" href="#pollet-polname-2-polname-1" id="id53"><tt class="docutils literal"><span class="pre">\PolLet{polname_2}={polname_1}</span></tt></a></li>
+<li><a class="reference internal" href="#polgloballet-polname-2-polname-1" id="id54"><tt class="docutils literal"><span class="pre">\PolGlobalLet{polname_2}={polname_1}</span></tt></a></li>
+<li><a class="reference internal" href="#polassign-polname-toarray-macro" id="id55"><tt class="docutils literal"><span class="pre">\PolAssign{polname}\toarray\macro</span></tt></a></li>
+<li><a class="reference internal" href="#polget-polname-fromarray-macro" id="id56"><tt class="docutils literal"><span class="pre">\PolGet{polname}\fromarray\macro</span></tt></a></li>
+<li><a class="reference internal" href="#polfromcsv-polname-csv" id="id57"><tt class="docutils literal"><span class="pre">\PolFromCSV{polname}{&lt;csv&gt;}</span></tt></a></li>
+<li><a class="reference internal" href="#poltypeset-polname" id="id58"><tt class="docutils literal">\PolTypeset{polname}</tt></a><ul>
+<li><a class="reference internal" href="#poltypesetcmd-raw-coeff" id="id59"><tt class="docutils literal">\PolTypesetCmd{raw_coeff}</tt></a></li>
+<li><a class="reference internal" href="#poltypesetone-raw-coeff" id="id60"><tt class="docutils literal">\PolTypesetOne{raw_coeff}</tt></a></li>
+<li><a class="reference internal" href="#id6" id="id61"><tt class="docutils literal">\PolTypesetMonomialCmd</tt></a></li>
+<li><a class="reference internal" href="#poltypesetcmdprefix-raw-coeff" id="id62"><tt class="docutils literal">\PolTypesetCmdPrefix{raw_coeff}</tt></a></li>
+</ul>
+</li>
+<li><a class="reference internal" href="#id8" id="id63"><tt class="docutils literal"><span class="pre">\PolTypeset*{polname}</span></tt></a></li>
+<li><a class="reference internal" href="#poldiff-polname-1-polname-2" id="id64"><tt class="docutils literal"><span class="pre">\PolDiff{polname_1}{polname_2}</span></tt></a></li>
+<li><a class="reference internal" href="#poldiff-n-polname-1-polname-2" id="id65"><tt class="docutils literal"><span class="pre">\PolDiff[N]{polname_1}{polname_2}</span></tt></a></li>
+<li><a class="reference internal" href="#polantidiff-polname-1-polname-2" id="id66"><tt class="docutils literal"><span class="pre">\PolAntiDiff{polname_1}{polname_2}</span></tt></a></li>
+<li><a class="reference internal" href="#polantidiff-n-polname-1-polname-2" id="id67"><tt class="docutils literal"><span class="pre">\PolAntiDiff[N]{polname_1}{polname_2}</span></tt></a></li>
+<li><a class="reference internal" href="#poldivide-polname-1-polname-2-polname-q-polname-r" id="id68"><tt class="docutils literal"><span class="pre">\PolDivide{polname_1}{polname_2}{polname_Q}{polname_R}</span></tt></a></li>
+<li><a class="reference internal" href="#polquo-polname-1-polname-2-polname-q" id="id69"><tt class="docutils literal"><span class="pre">\PolQuo{polname_1}{polname_2}{polname_Q}</span></tt></a></li>
+<li><a class="reference internal" href="#polrem-polname-1-polname-2-polname-r" id="id70"><tt class="docutils literal"><span class="pre">\PolRem{polname_1}{polname_2}{polname_R}</span></tt></a></li>
+<li><a class="reference internal" href="#polgcd-polname-1-polname-2-polname-gcd" id="id71"><tt class="docutils literal"><span class="pre">\PolGCD{polname_1}{polname_2}{polname_GCD}</span></tt></a></li>
+<li><a class="reference internal" href="#poltosturm-polname-sturmname" id="id72"><tt class="docutils literal"><span class="pre">\PolToSturm{polname}{sturmname}</span></tt></a></li>
+<li><a class="reference internal" href="#id10" id="id73"><tt class="docutils literal"><span class="pre">\PolToSturm*{polname}{sturmname}</span></tt></a></li>
+<li><a class="reference internal" href="#polsettosturmchainsignchangesat-macro-sturmname-fraction" id="id74"><tt class="docutils literal"><span class="pre">\PolSetToSturmChainSignChangesAt{\macro}{sturmname}{fraction}</span></tt></a></li>
+<li><a class="reference internal" href="#polsettonbofzeroswithin-macro-sturmname-value-a-value-b" id="id75"><tt class="docutils literal"><span class="pre">\PolSetToNbOfZerosWithin{\macro}{sturmname}{value_a}{value_b}</span></tt></a></li>
+<li><a class="reference internal" href="#polsturmisolatezeros-sturmname" id="id76"><tt class="docutils literal">\PolSturmIsolateZeros{sturmname}</tt></a></li>
+<li><a class="reference internal" href="#id12" id="id77"><tt class="docutils literal"><span class="pre">\PolSturmIsolateZeros*{sturmname}</span></tt></a></li>
+<li><a class="reference internal" href="#id14" id="id78"><tt class="docutils literal"><span class="pre">\PolSturmIsolateZeros**{sturmname}</span></tt></a></li>
+<li><a class="reference internal" href="#polsturmisolatezerosandgetmultiplicities-sturmname" id="id79"><tt class="docutils literal">\PolSturmIsolateZerosAndGetMultiplicities{sturmname}</tt></a></li>
+<li><a class="reference internal" href="#polsturmisolatezerosgetmultiplicitiesandrationalroots-sturmname" id="id80"><tt class="docutils literal">\PolSturmIsolateZerosGetMultiplicitiesAndRationalRoots{sturmname}</tt></a></li>
+<li><a class="reference internal" href="#polsturmisolatezerosandfindrationalroots-sturmname" id="id81"><tt class="docutils literal">\PolSturmIsolateZerosAndFindRationalRoots{sturmname}</tt></a></li>
+<li><a class="reference internal" href="#polrefineinterval-sturmname-index" id="id82"><tt class="docutils literal"><span class="pre">\PolRefineInterval*{sturmname}{index}</span></tt></a></li>
+<li><a class="reference internal" href="#polrefineinterval-n-sturmname-index" id="id83"><tt class="docutils literal"><span class="pre">\PolRefineInterval[N]{sturmname}{index}</span></tt></a></li>
+<li><a class="reference internal" href="#polensureintervallength-sturmname-index-e" id="id84"><tt class="docutils literal"><span class="pre">\PolEnsureIntervalLength{sturmname}{index}{E}</span></tt></a></li>
+<li><a class="reference internal" href="#polensureintervallengths-sturmname-e" id="id85"><tt class="docutils literal"><span class="pre">\PolEnsureIntervalLengths{sturmname}{E}</span></tt></a></li>
+<li><a class="reference internal" href="#polprintintervals-varname-sturmname" id="id86"><tt class="docutils literal"><span class="pre">\PolPrintIntervals[varname]{sturmname}</span></tt></a><ul>
+<li><a class="reference internal" href="#polprintintervalsnorealroots" id="id87"><tt class="docutils literal">\PolPrintIntervalsNoRealRoots</tt></a></li>
+<li><a class="reference internal" href="#polprintintervalsbeginenv" id="id88"><tt class="docutils literal">\PolPrintIntervalsBeginEnv</tt></a></li>
+<li><a class="reference internal" href="#polprintintervalsendenv" id="id89"><tt class="docutils literal">\PolPrintIntervalsEndEnv</tt></a></li>
+<li><a class="reference internal" href="#polprintintervalsknownroot" id="id90"><tt class="docutils literal">\PolPrintIntervalsKnownRoot</tt></a></li>
+<li><a class="reference internal" href="#polprintintervalsunknownroot" id="id91"><tt class="docutils literal">\PolPrintIntervalsUnknownRoot</tt></a></li>
+<li><a class="reference internal" href="#id15" id="id92"><tt class="docutils literal">\PolPrintIntervalsPrintExactZero</tt></a></li>
+<li><a class="reference internal" href="#id16" id="id93"><tt class="docutils literal">\PolPrintIntervalsPrintLeftEndPoint</tt></a></li>
+<li><a class="reference internal" href="#id17" id="id94"><tt class="docutils literal">\PolPrintIntervalsPrintRightEndPoint</tt></a></li>
+</ul>
+</li>
+<li><a class="reference internal" href="#id19" id="id95"><tt class="docutils literal"><span class="pre">\PolPrintIntervals*[varname]{sturmname}</span></tt></a><ul>
+<li><a class="reference internal" href="#polprintintervalsprintmultiplicity" id="id96"><tt class="docutils literal">\PolPrintIntervalsPrintMultiplicity</tt></a></li>
+</ul>
+</li>
+<li><a class="reference internal" href="#polmapcoeffs-macro-polname" id="id97"><tt class="docutils literal"><span class="pre">\PolMapCoeffs{\macro}{polname}</span></tt></a></li>
+<li><a class="reference internal" href="#polreducecoeffs-polname" id="id98"><tt class="docutils literal">\PolReduceCoeffs{polname}</tt></a></li>
+<li><a class="reference internal" href="#id21" id="id99"><tt class="docutils literal"><span class="pre">\PolReduceCoeffs*{polname}</span></tt></a></li>
+<li><a class="reference internal" href="#polmakemonic-polname" id="id100"><tt class="docutils literal">\PolMakeMonic{polname}</tt></a></li>
+<li><a class="reference internal" href="#polmakeprimitive-polname" id="id101"><tt class="docutils literal">\PolMakePrimitive{polname}</tt></a></li>
+</ul>
+</li>
+<li><a class="reference internal" href="#expandable-macros" id="id102">Expandable macros</a><ul>
+<li><a class="reference internal" href="#poleval-polname-atexpr-numerical-expression" id="id103"><tt class="docutils literal"><span class="pre">\PolEval{polname}\AtExpr{numerical</span> expression}</tt></a></li>
+<li><a class="reference internal" href="#poleval-polname-at-fraction" id="id104"><tt class="docutils literal"><span class="pre">\PolEval{polname}\At{fraction}</span></tt></a></li>
+<li><a class="reference internal" href="#polevalreduced-polname-atexpr-numerical-expression" id="id105"><tt class="docutils literal"><span class="pre">\PolEvalReduced{polname}\AtExpr{numerical</span> expression}</tt></a></li>
+<li><a class="reference internal" href="#polevalreduced-polname-at-fraction" id="id106"><tt class="docutils literal"><span class="pre">\PolEvalReduced{polname}\At{fraction}</span></tt></a></li>
+<li><a class="reference internal" href="#polfloateval-polname-atexpr-numerical-expression" id="id107"><tt class="docutils literal"><span class="pre">\PolFloatEval{polname}\AtExpr{numerical</span> expression}</tt></a></li>
+<li><a class="reference internal" href="#polfloateval-polname-at-fraction" id="id108"><tt class="docutils literal"><span class="pre">\PolFloatEval{polname}\At{fraction}</span></tt></a></li>
+<li><a class="reference internal" href="#polifcoeffisplusorminusone-a-b" id="id109"><tt class="docutils literal"><span class="pre">\PolIfCoeffIsPlusOrMinusOne{A}{B}</span></tt></a></li>
+<li><a class="reference internal" href="#polleadingcoeff-polname" id="id110"><tt class="docutils literal">\PolLeadingCoeff{polname}</tt></a></li>
+<li><a class="reference internal" href="#polnthcoeff-polname-number" id="id111"><tt class="docutils literal"><span class="pre">\PolNthCoeff{polname}{number}</span></tt></a></li>
+<li><a class="reference internal" href="#poldegree-polname" id="id112"><tt class="docutils literal">\PolDegree{polname}</tt></a></li>
+<li><a class="reference internal" href="#policontent-polname" id="id113"><tt class="docutils literal">\PolIContent{polname}</tt></a></li>
+<li><a class="reference internal" href="#poltoexpr-polname" id="id114"><tt class="docutils literal">\PolToExpr{polname}</tt></a><ul>
+<li><a class="reference internal" href="#poltoexproneterm-raw-coeff-number" id="id115"><tt class="docutils literal"><span class="pre">\PolToExprOneTerm{raw_coeff}{number}</span></tt></a></li>
+<li><a class="reference internal" href="#poltoexpronetermstylea-raw-coeff-number" id="id116"><tt class="docutils literal"><span class="pre">\PolToExprOneTermStyleA{raw_coeff}{number}</span></tt></a></li>
+<li><a class="reference internal" href="#poltoexpronetermstyleb-raw-coeff-number" id="id117"><tt class="docutils literal"><span class="pre">\PolToExprOneTermStyleB{raw_coeff}{number}</span></tt></a></li>
+<li><a class="reference internal" href="#poltoexprcmd-raw-coeff" id="id118"><tt class="docutils literal">\PolToExprCmd{raw_coeff}</tt></a></li>
+<li><a class="reference internal" href="#poltoexprtermprefix-raw-coeff" id="id119"><tt class="docutils literal">\PolToExprTermPrefix{raw_coeff}</tt></a></li>
+<li><a class="reference internal" href="#id28" id="id120"><tt class="docutils literal">\PolToExprVar</tt></a></li>
+<li><a class="reference internal" href="#id29" id="id121"><tt class="docutils literal">\PolToExprTimes</tt></a></li>
+</ul>
+</li>
+<li><a class="reference internal" href="#id31" id="id122"><tt class="docutils literal"><span class="pre">\PolToExpr*{polname}</span></tt></a></li>
+<li><a class="reference internal" href="#poltofloatexpr-polname" id="id123"><tt class="docutils literal">\PolToFloatExpr{polname}</tt></a><ul>
+<li><a class="reference internal" href="#poltofloatexproneterm-raw-coeff-number" id="id124"><tt class="docutils literal"><span class="pre">\PolToFloatExprOneTerm{raw_coeff}{number}</span></tt></a></li>
+<li><a class="reference internal" href="#poltofloatexprcmd-raw-coeff" id="id125"><tt class="docutils literal">\PolToFloatExprCmd{raw_coeff}</tt></a></li>
+</ul>
+</li>
+<li><a class="reference internal" href="#id35" id="id126"><tt class="docutils literal"><span class="pre">\PolToFloatExpr*{polname}</span></tt></a></li>
+<li><a class="reference internal" href="#poltolist-polname" id="id127"><tt class="docutils literal">\PolToList{polname}</tt></a></li>
+<li><a class="reference internal" href="#poltocsv-polname" id="id128"><tt class="docutils literal">\PolToCSV{polname}</tt></a></li>
+<li><a class="reference internal" href="#polsturmchainlength-sturmname" id="id129"><tt class="docutils literal">\PolSturmChainLength{sturmname}</tt></a></li>
+<li><a class="reference internal" href="#polsturmifzeroexactlyknown-sturmname-index-a-b" id="id130"><tt class="docutils literal"><span class="pre">\PolSturmIfZeroExactlyKnown{sturmname}{index}{A}{B}</span></tt></a></li>
+<li><a class="reference internal" href="#polsturmisolatedzeroleft-sturmname-index" id="id131"><tt class="docutils literal"><span class="pre">\PolSturmIsolatedZeroLeft{sturmname}{index}</span></tt></a></li>
+<li><a class="reference internal" href="#polsturmisolatedzeroright-sturmname-index" id="id132"><tt class="docutils literal"><span class="pre">\PolSturmIsolatedZeroRight{sturmname}{index}</span></tt></a></li>
+<li><a class="reference internal" href="#polsturmisolatedzeromultiplicity-sturmname-index" id="id133"><tt class="docutils literal"><span class="pre">\PolSturmIsolatedZeroMultiplicity{sturmname}{index}</span></tt></a></li>
+<li><a class="reference internal" href="#polsturmnbofisolatedzeros-sturmname" id="id134"><tt class="docutils literal">\PolSturmNbOfIsolatedZeros{sturmname}</tt></a><ul>
+<li><a class="reference internal" href="#polsturmnbofrootsof-sturmname-lessthanorequalto-value" id="id135"><tt class="docutils literal"><span class="pre">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualTo{value}</span></tt></a></li>
+<li><a class="reference internal" href="#polsturmnbofrootsof-sturmname-lessthanorequaltoexpr-expression" id="id136"><tt class="docutils literal"><span class="pre">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualToExpr{expression}</span></tt></a></li>
+<li><a class="reference internal" href="#polsturmnbwithmultofrootsof-sturmname-lessthanorequalto-value" id="id137"><tt class="docutils literal"><span class="pre">\PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualTo{value}</span></tt></a></li>
+<li><a class="reference internal" href="#polsturmnbwithmultofrootsof-sturmname-lessthanorequaltoexpr-expression" id="id138"><tt class="docutils literal"><span class="pre">\PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualToExpr{expression}</span></tt></a></li>
+</ul>
+</li>
+<li><a class="reference internal" href="#polsturmnbofrationalroots-sturmname" id="id139"><tt class="docutils literal">\PolSturmNbOfRationalRoots{sturmname}</tt></a></li>
+<li><a class="reference internal" href="#polsturmnbofrationalrootswithmultiplicities-sturmname" id="id140"><tt class="docutils literal">\PolSturmNbOfRationalRootsWithMultiplicities{sturmname}</tt></a></li>
+<li><a class="reference internal" href="#polsturmrationalroot-sturmname-k" id="id141"><tt class="docutils literal"><span class="pre">\PolSturmRationalRoot{sturmname}{k}</span></tt></a></li>
+<li><a class="reference internal" href="#polsturmrationalrootindex-sturmname-k" id="id142"><tt class="docutils literal"><span class="pre">\PolSturmRationalRootIndex{sturmname}{k}</span></tt></a></li>
+<li><a class="reference internal" href="#polsturmrationalrootmultiplicity-sturmname-k" id="id143"><tt class="docutils literal"><span class="pre">\PolSturmRationalRootMultiplicity{sturmname}{k}</span></tt></a></li>
+<li><a class="reference internal" href="#polintervalwidth-sturmname-index" id="id144"><tt class="docutils literal"><span class="pre">\PolIntervalWidth{sturmname}{index}</span></tt></a></li>
+<li><a class="reference internal" href="#expandable-macros-for-use-within-execution-of-polprintintervals" id="id145">Expandable macros for use within execution of <tt class="docutils literal">\PolPrintIntervals</tt></a><ul>
+<li><a class="reference internal" href="#polprintintervalsthevar" id="id146"><tt class="docutils literal">\PolPrintIntervalsTheVar</tt></a></li>
+<li><a class="reference internal" href="#polprintintervalstheindex" id="id147"><tt class="docutils literal">\PolPrintIntervalsTheIndex</tt></a></li>
+<li><a class="reference internal" href="#polprintintervalsthesturmname" id="id148"><tt class="docutils literal">\PolPrintIntervalsTheSturmName</tt></a></li>
+<li><a class="reference internal" href="#polprintintervalstheleftendpoint" id="id149"><tt class="docutils literal">\PolPrintIntervalsTheLeftEndPoint</tt></a></li>
+<li><a class="reference internal" href="#polprintintervalstherightendpoint" id="id150"><tt class="docutils literal">\PolPrintIntervalsTheRightEndPoint</tt></a></li>
+<li><a class="reference internal" href="#polprintintervalsthemultiplicity" id="id151"><tt class="docutils literal">\PolPrintIntervalsTheMultiplicity</tt></a></li>
+</ul>
+</li>
+<li><a class="reference internal" href="#poldectostring-decimal-number" id="id152"><tt class="docutils literal">\PolDecToString{decimal number}</tt></a></li>
+</ul>
+</li>
+<li><a class="reference internal" href="#booleans-with-default-setting-as-indicated" id="id153">Booleans (with default setting as indicated)</a><ul>
+<li><a class="reference internal" href="#xintverbosefalse" id="id154"><tt class="docutils literal">\xintverbosefalse</tt></a></li>
+<li><a class="reference internal" href="#poltypesetallfalse" id="id155"><tt class="docutils literal">\poltypesetallfalse</tt></a></li>
+<li><a class="reference internal" href="#poltoexprallfalse" id="id156"><tt class="docutils literal">\poltoexprallfalse</tt></a></li>
+</ul>
+</li>
+<li><a class="reference internal" href="#polexprsetup" id="id157"><tt class="docutils literal">\polexprsetup</tt></a></li>
+<li><a class="reference internal" href="#technicalities" id="id158">Technicalities</a></li>
+<li><a class="reference internal" href="#change-log" id="id159">CHANGE LOG</a></li>
+<li><a class="reference internal" href="#acknowledgments" id="id160">Acknowledgments</a></li>
+</ul>
+</div>
+<div class="section" id="basic-syntax">
+<h1><a class="toc-backref" href="#id38">Basic syntax</a></h1>
+<p>The syntax is:</p>
+<pre class="literal-block">
+\poldef polname(x):= expression in variable x;
+</pre>
+<p>where:</p>
+<ul class="simple">
+<li>in place of <tt class="docutils literal">x</tt> an arbitrary <em>dummy variable</em> is authorized,
+i.e. per default any of <tt class="docutils literal"><span class="pre">[a-z|A-Z]</span></tt> (more letters can be declared
+under Unicode engines.)</li>
+<li><tt class="docutils literal">polname</tt> consists of letters, digits, and the <tt class="docutils literal">_</tt> and
+<tt class="docutils literal">'</tt> characters. It must start with a letter.</li>
+</ul>
+<div class="admonition attention">
+<p class="first admonition-title">Attention!</p>
+<p class="last">The <tt class="docutils literal">'</tt> is authorized since <tt class="docutils literal">0.5.1</tt>. As a result some constructs
+recognized by the <tt class="docutils literal">\xintexpr</tt> parser, such as <tt class="docutils literal">var1 'and' var2</tt>
+will get misinterpreted and cause errors. However these constructs
+are unlikely to be frequently needed in polynomial expressions, and
+the <tt class="docutils literal">\xintexpr</tt> syntax offers alternatives, so it was deemed a
+small evil. Of course the <tt class="docutils literal">\xintexpr</tt> parser is modified only
+temporarily during execution of <tt class="docutils literal">\poldef</tt>.</p>
+</div>
+<p>One can also issue:</p>
+<pre class="literal-block">
+\PolDef{polname}{expression in variable x}
+</pre>
+<p>which admits an optional first argument to modify the variable letter
+from its default <tt class="docutils literal">x</tt>.</p>
+<dl class="docutils">
+<dt><tt class="docutils literal">\poldef <span class="pre">f(x):=</span> <span class="pre">1-x+x^2;</span></tt></dt>
+<dd>defines polynomial <tt class="docutils literal">f</tt>. Polynomial names must start with a
+letter and may contain letters, digits, underscores and the right
+tick character. The
+variable must be a single letter. The colon character is optional.
+The semi-colon at end of expression is mandatory.</dd>
+<dt><tt class="docutils literal"><span class="pre">\PolDef{f}{1-x+x^2}</span></tt></dt>
+<dd>does the same as <tt class="docutils literal">\poldef <span class="pre">f(x):=</span> <span class="pre">1-x+x^2;</span></tt> To use another letter
+than <tt class="docutils literal">x</tt> in the expression, one must pass it as an extra optional
+argument to <tt class="docutils literal">\PolDef</tt>. Useful if the semi-colon has been assigned
+some non-standard catcode by some package.</dd>
+<dt><tt class="docutils literal"><span class="pre">\PolLet{g}={f}</span></tt></dt>
+<dd>saves a copy of <tt class="docutils literal">f</tt> under name <tt class="docutils literal">g</tt>. Also usable without <tt class="docutils literal">=</tt>.</dd>
+<dt><tt class="docutils literal">\poldef <span class="pre">f(z):=</span> <span class="pre">f(z)^2;</span></tt></dt>
+<dd>redefines <tt class="docutils literal">f</tt> in terms of itself.</dd>
+<dt><tt class="docutils literal">\poldef <span class="pre">f(T):=</span> <span class="pre">f(f(T));</span></tt></dt>
+<dd>again redefines <tt class="docutils literal">f</tt> in terms of its (new) self.</dd>
+<dt><tt class="docutils literal">\poldef <span class="pre">k(z):=</span> <span class="pre">f(z)-g(g(z)^2)^2;</span></tt></dt>
+<dd>should now define the zero polynomial... Let's check:
+<tt class="docutils literal">\[ k(z) = <span class="pre">\PolTypeset[z]{k}</span> \]</tt></dd>
+<dt><tt class="docutils literal"><span class="pre">\PolDiff{f}{f'}</span></tt></dt>
+<dd>sets <tt class="docutils literal">f'</tt> to the derivative of <tt class="docutils literal">f</tt>. The name doesn't have to be
+<tt class="docutils literal">f'</tt> (in fact the <tt class="docutils literal">'</tt> is licit only since <tt class="docutils literal">0.5.1</tt>).</dd>
+</dl>
+<div class="admonition important">
+<p class="first admonition-title">Important</p>
+<p class="last">This is not done automatically. If some new definition needs to use
+the derivative of some available polynomial, that derivative
+polynomial must have been defined via <tt class="docutils literal">\PolDiff</tt>: something like
+<tt class="docutils literal"><span class="pre">T'(x)^2</span></tt> will not work without a prior <tt class="docutils literal"><span class="pre">\PolDiff{T}{T'}</span></tt>.</p>
+</div>
+<dl class="docutils">
+<dt><tt class="docutils literal"><span class="pre">\PolDiff{f'}{f''}</span></tt></dt>
+<dd>obtains second derivative.</dd>
+<dt><tt class="docutils literal"><span class="pre">\PolDiff[3]{f}{f'''}</span></tt></dt>
+<dd>computes the third derivative.</dd>
+</dl>
+<pre class="literal-block">
+$f(z) = \PolTypeset[z]{f} $\newline
+$f'(z) = \PolTypeset[z]{f'} $\newline
+$f''(z) = \PolTypeset[z]{f''} $\newline
+$f'''(z)= \PolTypeset[z]{f'''} $\par
+</pre>
+<div class="admonition important">
+<p class="first admonition-title">Important</p>
+<p>The package does not currently know rational functions: <tt class="docutils literal">/</tt> in
+a parsed polynomial expression does the Euclidean quotient:</p>
+<pre class="literal-block">
+(1-x^2)/(1-x)
+</pre>
+<p>does give <tt class="docutils literal">1+x</tt> but</p>
+<pre class="literal-block">
+(1/(1-x))*(1-x^2)
+</pre>
+<p>evaluates to zero. This will work as expected:</p>
+<pre class="last literal-block">
+\poldef k(x):= (x-1)(x-2)(x-3)(x-4)/(x^2-5x+4);
+</pre>
+</div>
+<div class="admonition attention" id="warningtacit">
+<p class="first admonition-title">Attention!</p>
+<p><tt class="docutils literal">1/2 x^2</tt> skips the space and is treated like <tt class="docutils literal"><span class="pre">1/(2*x^2)</span></tt> because
+of the tacit multiplication rules of xintexpr. But this means it
+gives zero! Thus one must use <tt class="docutils literal">(1/2)x^2</tt> or <tt class="docutils literal">1/2*x^2</tt> or
+<tt class="docutils literal"><span class="pre">(1/2)*x^2</span></tt> for disambiguation: <tt class="docutils literal">x - 1/2*x^2 + <span class="pre">1/3*x^3...</span></tt>. It is
+even simpler to move the denominator to the right: <tt class="docutils literal">x - x^2/2 +
+x^3/3 - ...</tt>.</p>
+<p class="last">It is worth noting that <tt class="docutils literal"><span class="pre">1/2(x-1)(x-2)</span></tt> suffers the same issue:
+<a class="reference external" href="http://www.ctan.org/pkg/xint">xint</a> tacit multiplication always &quot;ties more&quot;, hence this gets
+interpreted as <tt class="docutils literal"><span class="pre">1/(2*(x-1)*(x-2))</span></tt> which gives zero by polynomial
+division. Thus, use one of <tt class="docutils literal"><span class="pre">(1/2)(x-1)(x-2)</span></tt>, <tt class="docutils literal"><span class="pre">1/2*(x-1)(x-2)</span></tt> or
+<tt class="docutils literal"><span class="pre">(x-1)(x-2)/2</span></tt>.</p>
+</div>
+<p>After:</p>
+<pre class="literal-block">
+\poldef f_1(x):= 25(x-1)(x^2-2)(x-3)(x-4)(x-5);%
+\poldef f_2(x):= 37(x-1)(x^2-2)(x-6)(x-7)(x-8);%
+</pre>
+<p>the macro call <tt class="docutils literal"><span class="pre">\PolGCD{f_1}{f_2}{k}</span></tt> sets <tt class="docutils literal">k</tt> to the (unitary) GCD of
+<tt class="docutils literal">f_1</tt> and <tt class="docutils literal">f_2</tt> (hence to the expansion of <tt class="docutils literal"><span class="pre">(x-1)(x^2-2)</span></tt>.)</p>
+<dl class="docutils">
+<dt><tt class="docutils literal">\PolToExpr{k}</tt></dt>
+<dd>will (expandably) give in this case <tt class="docutils literal"><span class="pre">x^3-x^2-2*x+2</span></tt>. This is
+useful for console or file output (the syntax is Maple- and
+PSTricks-compatible; the letter used in output can be
+(non-expandably) changed via a redefinition of <a class="reference internal" href="#poltoexprvar">\PolToExprVar</a>.)</dd>
+<dt><tt class="docutils literal"><span class="pre">\PolToExpr*{k}</span></tt></dt>
+<dd>gives ascending powers: <tt class="docutils literal"><span class="pre">2-2*x-x^2+x^3</span></tt>.</dd>
+</dl>
+</div>
+<div class="section" id="examples-of-localization-of-roots">
+<h1><a class="toc-backref" href="#id39">Examples of localization of roots</a></h1>
+<ul>
+<li><p class="first">To make printed decimal numbers more enjoyable than via
+<tt class="docutils literal">\xintSignedFrac</tt>:</p>
+<pre class="literal-block">
+\renewcommand\PolTypesetOne[1]{\PolDecToString{\xintREZ{#1}}}%
+</pre>
+<p><tt class="docutils literal">\PolDecToString</tt> will use decimal notation to incorporate the power
+of ten part; and the <tt class="docutils literal">\xintREZ</tt> will have the effect to suppress
+trailing zeros if present in raw numerator (if those digits end up
+after decimal mark.) Notice that the above are expandable macros and
+that one can also do:</p>
+<pre class="literal-block">
+\renewcommand\PolToExprCmd[1]{\PolDecToString{\xintREZ{#1}}}%
+</pre>
+<p>to modify output of <a class="reference internal" href="#poltoexpr-polname">\PolToExpr{polname}</a>.</p>
+</li>
+<li><p class="first">For extra info in log file use <tt class="docutils literal">\xintverbosetrue</tt>.</p>
+</li>
+<li><p class="first">Only for some of these examples is the output included here.</p>
+</li>
+</ul>
+<div class="section" id="a-typical-example">
+<h2><a class="toc-backref" href="#id40">A typical example</a></h2>
+<p>In this example the polynomial is square-free.</p>
+<pre class="literal-block">
+\poldef f(x) := x^7 - x^6 - 2x + 1;
+
+\PolToSturm{f}{f}
+\PolSturmIsolateZeros{f}
+The \PolTypeset{f} polynomial has \PolSturmNbOfIsolatedZeros{f} distinct real
+roots which are located in the following intervals:
+\PolPrintIntervals{f}
+Here is the second root with ten more decimal digits:
+\PolRefineInterval[10]{f}{2}
+\[\PolSturmIsolatedZeroLeft{f}{2}&lt;Z_2&lt;\PolSturmIsolatedZeroRight{f}{2}\]
+And here is the first root with twenty digits after decimal mark:
+\PolEnsureIntervalLength{f}{1}{-20}
+\[\PolSturmIsolatedZeroLeft{f}{1}&lt;Z_1&lt;\PolSturmIsolatedZeroRight{f}{1}\]
+The first element of the Sturm chain has degree $\PolDegree{f_0}$. As
+this is the original degreee $\PolDegree{f}$ we know that $f$ is square free.
+Its derivative is up to a constant \PolTypeset{f_1} (in this example
+it is identical with it).
+\PolToSturm{f_1}{f_1}\PolSturmIsolateZeros{f_1}%
+The derivative has \PolSturmNbOfIsolatedZeros{f_1} distinct real
+roots:
+\PolPrintIntervals[W]{f_1}
+\PolEnsureIntervalLengths{f_1}{-10}%
+Here they are with ten digits after decimal mark:
+\PolPrintIntervals[W]{f_1}
+\PolDiff{f_1}{f''}
+\PolToSturm{f''}{f''}
+\PolSturmIsolateZeros{f''}
+The second derivative is \PolTypeset{f''}.
+It has \PolSturmNbOfIsolatedZeros{f''} distinct real
+roots:
+\PolPrintIntervals[X]{f''}
+Here is the positive one with 20 digits after decimal mark:
+\PolEnsureIntervalLength{f''}{2}{-20}%
+\[X_2 = \PolSturmIsolatedZeroLeft{f''}{2}\dots\]
+The more mathematically advanced among our dear readers will be able
+to give the exact value for $X_2$!
+</pre>
+</div>
+<div class="section" id="a-degree-four-polynomial-with-nearby-roots">
+<h2><a class="toc-backref" href="#id41">A degree four polynomial with nearby roots</a></h2>
+<p>Notice that this example is a bit outdated as <tt class="docutils literal">0.7</tt> release has
+added <tt class="docutils literal"><span class="pre">\PolSturmIsolateZeros**{sturmname}</span></tt> which would find exactly
+the roots. The steps here retain their interest when one is interested
+in finding isolating intervals for example to prepare some demonstration
+of dichotomy method.</p>
+<pre class="literal-block">
+\PolDef{Q}{(x-1.050001)(x-1.105001)(x-1.110501)(x-1.111051)}
+\PolTypeset{Q}
+\PolToSturm{Q}{Q} % it is allowed to use same prefix for Sturm chain
+\PolSturmIsolateZeros{Q}
+\PolPrintIntervals{Q}
+% reports 1.0 &lt; Z_1 &lt; 1.1, 1.10 &lt; Z_2 &lt; 1.11, 1.110 &lt; Z_3 &lt; 1.111, and 1.111 &lt; Z_4 &lt; 1.112
+% but the above bounds do not allow minimizing separation between roots
+% so we refine:
+\PolRefineInterval*{Q}{1}
+\PolRefineInterval*{Q}{2}
+\PolRefineInterval*{Q}{3}
+\PolRefineInterval*{Q}{4}
+\PolPrintIntervals{Q}
+% reports 1.05 &lt; Z_1 &lt; 1.06, 1.105 &lt; Z_2 &lt; 1.106, 1.1105 &lt; Z_3 &lt; 1.1106,
+% and 1.11105 &lt; Z_4 &lt; 1.11106.
+\PolEnsureIntervalLengths{Q}{-6}
+\PolPrintIntervals{Q}
+% of course finds here all roots exactly
+</pre>
+</div>
+<div class="section" id="the-degree-nine-polynomial-with-0-99-0-999-0-9999-as-triple-roots">
+<h2><a class="toc-backref" href="#id42">The degree nine polynomial with 0.99, 0.999, 0.9999 as triple roots</a></h2>
+<pre class="literal-block">
+% define a user command (xinttools is loaded automatically by polexpr)
+\newcommand\showmultiplicities[1]{% #1 = &quot;sturmname&quot;
+\xintFor* ##1 in {\xintSeq{1}{\PolSturmNbOfIsolatedZeros{#1}}}\do{%
+ The multiplicity is \PolSturmIsolatedZeroMultiplicity{#1}{##1}
+ \PolSturmIfZeroExactlyKnown{#1}{##1}%
+ {at the root $x=\PolSturmIsolatedZeroLeft{#1}{##1}$}
+ {for the root such that
+ $\PolSturmIsolatedZeroLeft{#1}{##1}&lt;x&lt;\PolSturmIsolatedZeroRight{#1}{##1}$}
+ \par
+}}%
+\PolDef{f}{(x-0.99)^3(x-0.999)^3(x-0.9999)^3}
+\renewcommand\PolTypesetOne[1]{\PolDecToString{\xintREZ{#1}}}
+\PolTypeset{f}\par
+\PolToSturm{f}{f}% it is allowed to use &quot;polname&quot; as &quot;sturmname&quot; too
+\PolSturmIsolateZerosAndGetMultiplicities{f}% use the &quot;sturmname&quot; here
+% or \PolSturmIsolateZeros*{f} which is exactly the same, but shorter..
+
+\showmultiplicities{f}
+</pre>
+<p>In this example, the output will look like this (but using math mode):</p>
+<pre class="literal-block">
+x^9 - 8.9667x^8 + 35.73400293x^7 - 83.070418400109x^6 + 124.143648875193123x^5
+- 123.683070924326075877x^4 + 82.149260397553075617891x^3
+- 35.07602992699900159127007x^2 + 8.7364078733314648368671733x
+- 0.967100824643585986488103299
+
+The multiplicity is 3 at the root x = 0.99
+The multiplicity is 3 at the root x = 0.999
+The multiplicity is 3 at the root x = 0.9999
+</pre>
+<p>On first pass, these rational roots were found (due to their relative
+magnitudes, using <tt class="docutils literal">\PolSturmIsolateZeros**</tt> was not needed here). But
+multiplicity computation works also with (decimal) roots not yet
+identified or with non-decimal or irrational roots.</p>
+<p>It is fun to modify only a tiny bit the polynomial and see if polexpr
+survives:</p>
+<pre class="literal-block">
+\PolDef{g}{f(x)+1e-27}
+\PolTypeset{g}\par
+\PolToSturm{g}{g}
+\PolSturmIsolateZeros*{g}
+
+\showmultiplicities{g}
+</pre>
+<p>This produces:</p>
+<pre class="literal-block">
+x^9 - 8.9667x^8 + 35.73400293x^7 - 83.070418400109x^6 + 124.143648875193123x^5
+- 123.683070924326075877x^4 + 82.149260397553075617891x^3
+- 35.07602992699900159127007x^2 + 8.7364078733314648368671733x
+- 0.967100824643585986488103298
+
+The multiplicity is 1 for the root such that 0.98 &lt; x &lt; 0.99
+The multiplicity is 1 for the root such that 0.9991 &lt; x &lt; 0.9992
+The multiplicity is 1 for the root such that 0.9997 &lt; x &lt; 0.9998
+</pre>
+<p>Which means that the multiplicity-3 roots each became a real and a pair of
+complex ones. Let's see them better:</p>
+<pre class="literal-block">
+\PolEnsureIntervalLengths{g}{-10}
+
+\showmultiplicities{g}
+</pre>
+<p>which produces:</p>
+<pre class="literal-block">
+The multiplicity is 1 for the root such that 0.9899888032 &lt; x &lt; 0.9899888033
+The multiplicity is 1 for the root such that 0.9991447980 &lt; x &lt; 0.9991447981
+The multiplicity is 1 for the root such that 0.9997663986 &lt; x &lt; 0.9997663987
+</pre>
+</div>
+<div class="section" id="a-degree-five-polynomial-with-three-rational-roots">
+<h2><a class="toc-backref" href="#id43">A degree five polynomial with three rational roots</a></h2>
+<pre class="literal-block">
+\poldef Q(x) := 1581755751184441 x^5
+ -14907697165025339 x^4
+ +48415668972339336 x^3
+ -63952057791306264 x^2
+ +46833913221154895 x
+ -49044360626280925;
+
+\PolToSturm{Q}{Q}
+%\begin{flushleft}
+ \renewcommand\PolTypesetCmdPrefix[1]{\allowbreak\xintiiifSgn{#1}{}{+}{+}}%
+ $Q_0(x) = \PolTypeset{Q_0}$
+%\end{flushleft}
+\PolSturmIsolateZeros**{Q}
+\PolPrintIntervals{Q}
+
+$Q_{norr}(x) = \PolTypeset{Q_norr}$
+</pre>
+<p>Here, all real roots are rational:</p>
+<pre class="literal-block">
+Z_1 = 833719/265381
+Z_2 = 165707065/52746197
+Z_3 = 355/113
+
+Q_norr(x) = x^2 + 1
+</pre>
+<p>And let's get their decimal expansion too:</p>
+<pre class="literal-block">
+% print decimal expansion of the found roots
+\renewcommand\PolPrintIntervalsPrintExactZero
+ {\xintTrunc{20}{\PolPrintIntervalsTheLeftEndPoint}\dots}
+\PolPrintIntervals{Q}
+
+Z_1 = 3.14159265358107777120...
+Z_2 = 3.14159265358979340254...
+Z_3 = 3.14159292035398230088...
+</pre>
+</div>
+<div class="section" id="a-mignotte-type-polynomial">
+<h2><a class="toc-backref" href="#id44">A Mignotte type polynomial</a></h2>
+<pre class="literal-block">
+\PolDef{P}{x^10 - (10x-1)^2}%
+\PolTypeset{P} % prints it in expanded form
+\PolToSturm{P}{P} % we can use same prefix for Sturm chain
+\PolSturmIsolateZeros{P} % finds 4 real roots
+This polynomial has \PolSturmNbOfIsolatedZeros{P} distinct real roots:
+\PolPrintIntervals{P}%
+% reports -2 &lt; Z_1 &lt; -1, 0.09 &lt; Z_2 &lt; 0.10, 0.1 &lt; Z_3 &lt; 0.2, 1 &lt; Z_4 &lt; 2
+Let us refine the second and third intervals to separate the corresponding
+roots:
+\PolRefineInterval*{P}{2}% will refine to 0.0999990 &lt; Z_2 &lt; 0.0999991
+\PolRefineInterval*{P}{3}% will refine to 0.100001 &lt; Z_3 &lt; 0.100002
+\PolPrintIntervals{P}%
+Let us now get to know all roots with 10 digits after decimal mark:
+\PolEnsureIntervalLengths{P}{-10}%
+\PolPrintIntervals{P}% now all roots are known 10 decimal digits after mark
+Finally, we display 20 digits of the second root:
+\PolEnsureIntervalLength{P}{2}{-20}% makes Z_2 known with 20 digits after mark
+\[\PolSturmIsolatedZeroLeft{P}{2}&lt;Z_2&lt;\PolSturmIsolatedZeroRight{P}{2}\]
+</pre>
+<p>The last line produces:</p>
+<pre class="literal-block">
+0.09999900004999650028 &lt; Z_2 &lt; 0.09999900004999650029
+</pre>
+</div>
+<div class="section" id="the-wilkinson-polynomial">
+<h2><a class="toc-backref" href="#id45">The Wilkinson polynomial</a></h2>
+<p>See <a class="reference external" href="https://en.wikipedia.org/wiki/Wilkinson%27s_polynomial">Wilkinson polynomial</a>.</p>
+<pre class="literal-block">
+\documentclass{article}
+\usepackage{polexpr}
+\begin{document}
+%\xintverbosetrue % for the curious...
+
+\poldef f(x) := mul((x - i), i = 1..20);
+
+\renewcommand\PolTypesetCmdPrefix[1]{\allowbreak\xintiiifSgn{#1}{}{+}{+}}%
+\renewcommand\PolTypesetOne[1]{\xintDecToString{#1}}%
+
+\noindent\PolTypeset{f}
+
+\PolToSturm{f}{f}
+\PolSturmIsolateZeros{f}
+\PolPrintIntervals{f}
+
+\clearpage
+
+\poldef g(x) := f(x) - 2**{-23} x**19;
+
+% be patient!
+\PolToSturm{g}{g}
+\noindent\PolTypeset{g_0}% integer coefficient primitive polynomial
+
+\PolSturmIsolateZeros{g}
+\PolEnsureIntervalLengths{g}{-10}
+
+\renewcommand\PolPrintIntervalsPrintMultiplicity{}
+\PolPrintIntervals*{g}
+
+\end{document}
+</pre>
+<p>The first polynomial:</p>
+<pre class="literal-block">
+f(x) = x**20
+- 210 x**19
++ 20615 x**18
+- 1256850 x**17
++ 53327946 x**16
+- 1672280820 x**15
++ 40171771630 x**14
+- 756111184500 x**13
++ 11310276995381 x**12
+- 135585182899530 x**11
++ 1307535010540395 x**10
+- 10142299865511450 x**9
++ 63030812099294896 x**8
+- 311333643161390640 x**7
++ 1206647803780373360 x**6
+- 3599979517947607200 x**5
++ 8037811822645051776 x**4
+- 12870931245150988800 x**3
++ 13803759753640704000 x**2
+- 8752948036761600000 x
++ 2432902008176640000
+</pre>
+<p>is handled fast enough (a few seconds), but the modified one <tt class="docutils literal">f(x) -
+<span class="pre">2**-23</span> <span class="pre">x**19</span></tt> takes about 20x longer (the Sturm chain polynomials
+have integer coefficients with up to 321 digits, whereas (surprisingly
+perhaps) those of the Sturm chain polynomials derived from <tt class="docutils literal">f</tt> never
+have more than 21 digits ...).</p>
+<p>Once the Sturm chain is computed and the zeros isolated, obtaining their
+decimal digits is relatively faster. Here is for the ten real roots of
+<tt class="docutils literal">f(x) - <span class="pre">2**-23</span> <span class="pre">x**19</span></tt> as computed by the code above:</p>
+<pre class="literal-block">
+Z_1 = 0.9999999999...
+Z_2 = 2.0000000000...
+Z_3 = 2.9999999999...
+Z_4 = 4.0000000002...
+Z_5 = 4.9999999275...
+Z_6 = 6.0000069439...
+Z_7 = 6.9996972339...
+Z_8 = 8.0072676034...
+Z_9 = 8.9172502485...
+Z_10 = 20.8469081014...
+</pre>
+</div>
+<div class="section" id="the-second-wilkinson-polynomial">
+<h2><a class="toc-backref" href="#id46">The second Wilkinson polynomial</a></h2>
+<pre class="literal-block">
+\documentclass{article}
+\usepackage{polexpr}
+\begin{document}
+\poldef f(x) := mul(x - 2^-i, i = 1..20);
+
+%\PolTypeset{f}
+
+\PolToSturm{f}{f}
+\PolSturmIsolateZeros**{f}
+\PolPrintIntervals{f}
+\end{document}
+</pre>
+<p>This takes more time than the polynomial with 1, 2, .., 20 as roots but
+less than the latter modified by the <tt class="docutils literal"><span class="pre">2**-23</span></tt> change in one
+coefficient.</p>
+<p>Here is the output (with release 0.7.2):</p>
+<pre class="literal-block">
+Z_1 = 0.00000095367431640625
+Z_2 = 0.0000019073486328125
+Z_3 = 0.000003814697265625
+Z_4 = 0.00000762939453125
+Z_5 = 0.0000152587890625
+Z_6 = 0.000030517578125
+Z_7 = 0.00006103515625
+Z_8 = 0.0001220703125
+Z_9 = 1/4096
+Z_10 = 1/2048
+Z_11 = 1/1024
+Z_12 = 1/512
+Z_13 = 1/256
+Z_14 = 1/128
+Z_15 = 0.015625
+Z_16 = 0.03125
+Z_17 = 0.0625
+Z_18 = 0.125
+Z_19 = 0.25
+Z_20 = 0.5
+</pre>
+<p>There is some incoherence in output format which has its source in the
+fact that some roots are found in branches which can only find decimal
+roots, whereas some are found in branches which could find general
+fractions and they use <tt class="docutils literal">\xintIrr</tt> before storage of the found root.
+This may evolve in future.</p>
+</div>
+<div class="section" id="the-degree-41-polynomial-with-2-1-9-1-8-0-0-1-1-9-2-as-roots">
+<h2><a class="toc-backref" href="#id47">The degree 41 polynomial with -2, -1.9, -1.8, ..., 0, 0.1, ..., 1.9, 2 as roots</a></h2>
+<pre class="literal-block">
+\PolDef{P}{mul((x-i*1e-1), i=-20..20)}% i/10 is same but less efficient
+</pre>
+<p>In the defining expression we could have used <tt class="docutils literal">i/10</tt> but this gives
+less efficient internal form for the coefficients (the <tt class="docutils literal">10</tt>'s end up
+in denominators). Using <tt class="docutils literal">\PolToExpr{P}</tt> after having done</p>
+<pre class="literal-block">
+\renewcommand\PolToExprCmd[1]{\PolDecToString{\xintREZ{#1}}}
+</pre>
+<p>we get this expanded form:</p>
+<pre class="literal-block">
+x^41
+-28.7*x^39
++375.7117*x^37
+-2975.11006*x^35
++15935.28150578*x^33
+-61167.527674162*x^31
++173944.259366417394*x^29
+-373686.963560544648*x^27
++613012.0665016658846445*x^25
+-771182.31133138163125495*x^23
++743263.86672885754888959569*x^21
+-545609.076599482896371978698*x^19
++301748.325708943677229642930528*x^17
+-123655.8987669450434698869844544*x^15
++36666.1782054884005855608205864192*x^13
+-7607.85821367459445649518380016128*x^11
++1053.15135918687298508885950223794176*x^9
+-90.6380005918141132650786081964032*x^7
++4.33701563847327366842552218288128*x^5
+-0.0944770968420804735498178265088*x^3
++0.00059190121813899276854174416896*x
+</pre>
+<p>which shows coefficients with up to 36 significant digits...</p>
+<p>Stress test: not a hard challenge to <tt class="docutils literal">xint + polexpr</tt>, but be a bit patient!</p>
+<pre class="literal-block">
+\PolDef{P}{mul((x-i*1e-1), i=-20..20)}%
+\PolToSturm{P}{S} % dutifully computes S_0, ..., S_{41}
+% the [1] optional argument limits the search to interval (-10,10)
+\PolSturmIsolateZeros[1]{S} % finds *exactly* (but a bit slowly) all 41 roots!
+\PolPrintIntervals{S} % nice, isn't it?
+</pre>
+<div class="admonition note">
+<p class="first admonition-title">Note</p>
+<p>Release <tt class="docutils literal">0.5</tt> has <em>experimental</em> addition of optional argument
+<tt class="docutils literal">E</tt> to <tt class="docutils literal">\PolSturmIsolateZeros</tt>. It instructs to search roots only
+in interval <tt class="docutils literal"><span class="pre">(-10^E,</span> 10^E)</tt>. Important: the extremities are
+<em>assumed to not be roots</em>. In this example, the <tt class="docutils literal">[1]</tt> in
+<tt class="docutils literal"><span class="pre">\PolSturmIsolateZeros[1]{S}</span></tt> gives some speed gain; without it, it
+turns out in this case that <tt class="docutils literal">polexpr</tt> would have started with
+<tt class="docutils literal"><span class="pre">(-10^6,</span> 10^6)</tt> interval.</p>
+<p class="last">Please note that this will probably get replaced in future by the
+specification of a general interval. Do not rely on meaning of this
+optional argument keeping the same.</p>
+</div>
+</div>
+<div class="section" id="roots-of-chebyshev-polynomials">
+<h2><a class="toc-backref" href="#id48">Roots of Chebyshev polynomials</a></h2>
+<pre class="literal-block">
+\newcount\mycount
+\poldef T_0(x) := 1;
+\poldef T_1(x) := x;
+\mycount 2
+\xintloop
+ \poldef T_\the\mycount(x) :=
+ 2x*T_\the\numexpr\mycount-1(x)
+ - T_\the\numexpr\mycount-2(x);
+\ifnum\mycount&lt;15
+\advance\mycount 1
+\repeat
+
+\[T_{15} = \PolTypeset[X]{T_15}\]
+\PolToSturm{T_15}{T_15}
+\PolSturmIsolateZeros{T_15}
+\PolEnsureIntervalLengths{T_15}{-10}
+\PolPrintIntervals{T_15}
+</pre>
+</div>
+</div>
+<div class="section" id="non-expandable-macros">
+<h1><a class="toc-backref" href="#id49">Non-expandable macros</a></h1>
+<div class="section" id="poldef-polname-letter-expression-in-letter">
+<span id="poldef"></span><h2><a class="toc-backref" href="#id50"><tt class="docutils literal">\poldef <span class="pre">polname(letter):=</span> expression in letter;</tt></a></h2>
+<blockquote>
+<p>This evaluates the <em>polynomial expression</em> and stores the coefficients
+in a private structure accessible later via other package macros,
+under the user-chosen <tt class="docutils literal">polname</tt>. Of course the <em>expression</em> can
+use other previously defined polynomials. Names must start with a
+letter and are constituted of letters, digits, underscores and
+(since <tt class="docutils literal">0.5.1</tt>) the right tick <tt class="docutils literal">'</tt>.
+The whole <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> syntax is authorized:</p>
+<pre class="literal-block">
+\poldef sin(z) := add((-1)^i z^(2i+1)/(2i+1)!, i = 0..10);
+</pre>
+<p>With fractional coefficients, beware the <a class="reference internal" href="#warningtacit">tacit multiplication issue</a>.</p>
+<p>As a side effect the function <tt class="docutils literal">polname()</tt> is recognized as a
+genuine <tt class="docutils literal"><span class="pre">\xintexpr...\relax</span></tt> function for (exact) numerical
+evaluation (or within an <tt class="docutils literal">\xintdefvar</tt> assignment.) It computes
+values not according to the original expression but via the Horner
+scheme corresponding to the polynomial coefficients.</p>
+<div class="admonition attention">
+<p class="first admonition-title">Attention!</p>
+<p>Release <tt class="docutils literal">0.3</tt> also did the necessary set-up to let the
+polynomial be known to the <tt class="docutils literal">\xintfloatexpr</tt> (or
+<tt class="docutils literal">\xintdeffloatvar</tt>) parser.</p>
+<p>Since <tt class="docutils literal">0.4</tt> this isn't done automatically. Even more, a
+previously existing floating point variant of the same name will
+be let undefined again, to avoid hard to debug mismatches between
+exact and floating point polynomials. This also applies when the
+polynomial is produced not via <tt class="docutils literal">\poldef</tt> or <tt class="docutils literal">\PolDef</tt> but as
+a product of the other package macros.</p>
+<p class="last">See <a class="reference internal" href="#polgenfloatvariant-polname">\PolGenFloatVariant{polname}</a>.</p>
+</div>
+<p>The original expression is lost after parsing, and in particular
+the package provides no way to typeset it. This has to be done
+manually, if needed.</p>
+</blockquote>
+</div>
+<div class="section" id="poldef-letter-polname-expression-in-letter">
+<span id="id2"></span><h2><a class="toc-backref" href="#id51"><tt class="docutils literal"><span class="pre">\PolDef[letter]{polname}{expression</span> in letter}</tt></a></h2>
+<blockquote>
+Does the same as <a class="reference external" href="poldef;">\poldef</a> in an undelimited macro
+format (thus avoiding potential problems with the catcode of the
+semi-colon in presence of some packages.) In absence of the
+<tt class="docutils literal">[letter]</tt> optional argument, the variable is assumed to be <tt class="docutils literal">x</tt>.</blockquote>
+</div>
+<div class="section" id="polgenfloatvariant-polname">
+<span id="polgenfloatvariant"></span><h2><a class="toc-backref" href="#id52"><tt class="docutils literal">\PolGenFloatVariant{polname}</tt></a></h2>
+<blockquote>
+<p>Makes the polynomial also usable in the <tt class="docutils literal">\xintfloatexpr</tt> parser.
+It will therein evaluates via an Horner scheme with coefficients
+already pre-rounded to the float precision.</p>
+<p>See also <a class="reference internal" href="#poltofloatexpr-polname">\PolToFloatExpr{polname}</a>.</p>
+<div class="admonition attention">
+<p class="first admonition-title">Attention!</p>
+<p>Release <tt class="docutils literal">0.3</tt> did this automatically on <tt class="docutils literal">\PolDef</tt> and
+<tt class="docutils literal">\poldef</tt> but this was removed at <tt class="docutils literal">0.4</tt> for optimization.</p>
+<p class="last">Any operation, for example generating the derivative polynomial,
+or dividing two polynomials or using the <tt class="docutils literal">\PolLet</tt>, <strong>must</strong> be
+followed by explicit usage of <tt class="docutils literal">\PolGenFloatVariant{polname}</tt> if
+the new polynomial is to be used in <tt class="docutils literal">\xintfloatexpr</tt> or alike
+context.</p>
+</div>
+</blockquote>
+</div>
+<div class="section" id="pollet-polname-2-polname-1">
+<span id="pollet"></span><h2><a class="toc-backref" href="#id53"><tt class="docutils literal"><span class="pre">\PolLet{polname_2}={polname_1}</span></tt></a></h2>
+<blockquote>
+Makes a copy of the already defined polynomial <tt class="docutils literal">polname_1</tt> to a
+new one <tt class="docutils literal">polname_2</tt>. Same effect as
+<tt class="docutils literal"><span class="pre">\PolDef{polname_2}{polname_1(x)}</span></tt> but with less overhead. The
+<tt class="docutils literal">=</tt> is optional.</blockquote>
+</div>
+<div class="section" id="polgloballet-polname-2-polname-1">
+<span id="polgloballet"></span><h2><a class="toc-backref" href="#id54"><tt class="docutils literal"><span class="pre">\PolGlobalLet{polname_2}={polname_1}</span></tt></a></h2>
+<blockquote>
+Acts globally.</blockquote>
+</div>
+<div class="section" id="polassign-polname-toarray-macro">
+<span id="polassign"></span><h2><a class="toc-backref" href="#id55"><tt class="docutils literal"><span class="pre">\PolAssign{polname}\toarray\macro</span></tt></a></h2>
+<blockquote>
+<p>Defines a one-argument expandable macro <tt class="docutils literal"><span class="pre">\macro{#1}</span></tt> which expands
+to the (raw) #1th polynomial coefficient.</p>
+<ul class="simple">
+<li>Attention, coefficients here are indexed starting at 1.</li>
+<li>With #1=-1, -2, ..., <tt class="docutils literal"><span class="pre">\macro{#1}</span></tt> returns leading coefficients.</li>
+<li>With #1=0, returns the number of coefficients, i.e. <tt class="docutils literal">1 + deg f</tt>
+for non-zero polynomials.</li>
+<li>Out-of-range #1's return <tt class="docutils literal">0/1[0]</tt>.</li>
+</ul>
+<p>See also <a class="reference internal" href="#polnthcoeff-polname-number">\PolNthCoeff{polname}{number}</a>. The main difference is that
+with <tt class="docutils literal">\PolAssign</tt>, <tt class="docutils literal">\macro</tt> is made a prefix to <tt class="docutils literal">1 + deg f</tt>
+already defined (hidden to user) macros holding individually the
+coefficients but <a class="reference internal" href="#polnthcoeff-polname-number">\PolNthCoeff{polname}{number}</a> does each time the job
+to expandably recover the <tt class="docutils literal">Nth</tt> coefficient, and due to
+expandability can not store it in a macro for future usage (of course,
+it can be an argument in an <tt class="docutils literal">\edef</tt>.) The other difference
+is the shift by one in indexing, mentioned above (negative
+indices act the same in both.)</p>
+</blockquote>
+</div>
+<div class="section" id="polget-polname-fromarray-macro">
+<span id="polget"></span><h2><a class="toc-backref" href="#id56"><tt class="docutils literal"><span class="pre">\PolGet{polname}\fromarray\macro</span></tt></a></h2>
+<blockquote>
+<p>Does the converse operation to
+<tt class="docutils literal"><span class="pre">\PolAssign{polname}\toarray\macro</span></tt>. Each individual
+<tt class="docutils literal">\macro{number}</tt> gets expanded in an <tt class="docutils literal">\edef</tt> and then normalized
+via <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a>'s macro <tt class="docutils literal">\xintRaw</tt>.</p>
+<p>The leading zeros are removed from the polynomial.</p>
+<p>(contrived) Example:</p>
+<pre class="literal-block">
+\xintAssignArray{1}{-2}{5}{-3}\to\foo
+\PolGet{f}\fromarray\foo
+</pre>
+<p>This will define <tt class="docutils literal">f</tt> as would have <tt class="docutils literal">\poldef <span class="pre">f(x):=1-2x+5x^2-3x^3;</span></tt>.</p>
+<div class="admonition note">
+<p class="first admonition-title">Note</p>
+<p class="last">Prior to <tt class="docutils literal">0.5</tt>, coefficients were not normalized via
+<tt class="docutils literal">\xintRaw</tt> for internal storage.</p>
+</div>
+</blockquote>
+</div>
+<div class="section" id="polfromcsv-polname-csv">
+<span id="polfromcsv"></span><h2><a class="toc-backref" href="#id57"><tt class="docutils literal"><span class="pre">\PolFromCSV{polname}{&lt;csv&gt;}</span></tt></a></h2>
+<blockquote>
+<p>Defines a polynomial directly from the comma separated list of values
+(or a macro expanding to such a list) of its coefficients, the <em>first
+item</em> gives the constant term, the <em>last item</em> gives the leading
+coefficient, except if zero, then it is dropped (iteratively). List
+items are each expanded in an <tt class="docutils literal">\edef</tt> and then put into normalized
+form via <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a>'s macro <tt class="docutils literal">\xintRaw</tt>.</p>
+<p>As leading zero coefficients are removed:</p>
+<pre class="literal-block">
+\PolFromCSV{f}{0, 0, 0, 0, 0, 0, 0, 0, 0, 0}
+</pre>
+<p>defines the zero polynomial, which holds only one coefficient.</p>
+<p>See also expandable macro <a class="reference internal" href="#poltocsv-polname">\PolToCSV</a>.</p>
+<div class="admonition note">
+<p class="first admonition-title">Note</p>
+<p class="last">Prior to <tt class="docutils literal">0.5</tt>, coefficients were not normalized via
+<tt class="docutils literal">\xintRaw</tt> for internal storage.</p>
+</div>
+</blockquote>
+</div>
+<div class="section" id="poltypeset-polname">
+<span id="poltypeset"></span><h2><a class="toc-backref" href="#id58"><tt class="docutils literal">\PolTypeset{polname}</tt></a></h2>
+<blockquote>
+<p>Typesets in descending powers in math mode. It uses letter <tt class="docutils literal">x</tt> but
+this can be changed via an optional argument:</p>
+<pre class="literal-block">
+\PolTypeset[z]{polname}
+</pre>
+<p>By default zero coefficients are skipped (issue <tt class="docutils literal">\poltypesetalltrue</tt>
+to get all of them in output).</p>
+<p>These commands (whose meanings will be found in the package code)
+can be re-defined for customization. Their default definitions are
+expandable, but this is not a requirement.</p>
+</blockquote>
+<div class="section" id="poltypesetcmd-raw-coeff">
+<span id="poltypesetcmd"></span><h3><a class="toc-backref" href="#id59"><tt class="docutils literal">\PolTypesetCmd{raw_coeff}</tt></a></h3>
+<blockquote>
+<p>Checks if the coefficient is <tt class="docutils literal">1</tt> or <tt class="docutils literal"><span class="pre">-1</span></tt> and then skips printing
+the <tt class="docutils literal">1</tt>, except for the constant term. Also it sets conditional
+<a class="reference internal" href="#polifcoeffisplusorminusone-a-b">\PolIfCoeffIsPlusOrMinusOne{A}{B}</a>.</p>
+<p>The actual printing of the coefficients, when not equal to plus or
+minus one is handled by <a class="reference internal" href="#poltypesetone-raw-coeff">\PolTypesetOne{raw_coeff}</a>.</p>
+</blockquote>
+</div>
+<div class="section" id="poltypesetone-raw-coeff">
+<span id="poltypesetone"></span><h3><a class="toc-backref" href="#id60"><tt class="docutils literal">\PolTypesetOne{raw_coeff}</tt></a></h3>
+<blockquote>
+<p>The default is <tt class="docutils literal">\xintSignedFrac</tt> but this macro is annoying as it
+insists to use a power of ten, and not decimal notation.</p>
+<p>One can do things such as for example: <a class="footnote-reference" href="#id5" id="id4">[1]</a></p>
+<pre class="literal-block">
+\renewcommand\PolTypesetOne[1]{\num{\xintPFloat[5]{#1}}}
+\renewcommand\PolTypesetOne[1]{\num{\xintRound{4}{#1}}}
+</pre>
+<p>where e.g. we used the <tt class="docutils literal">\num</tt> macro of <tt class="docutils literal">siunitx</tt> as it
+understands floating point notation.</p>
+<table class="docutils footnote" frame="void" id="id5" rules="none">
+<colgroup><col class="label" /><col /></colgroup>
+<tbody valign="top">
+<tr><td class="label"><a class="fn-backref" href="#id4">[1]</a></td><td>the difference in the syntaxes of <tt class="docutils literal">\xintPFloat</tt> and
+<tt class="docutils literal">\xintRound</tt> is explained from the fact that
+<tt class="docutils literal">\xintPFloat</tt> by default uses the prevailing precision
+hence the extra argument like here <tt class="docutils literal">5</tt> is an optional one.</td></tr>
+</tbody>
+</table>
+<p>One can also give a try to using <a class="reference internal" href="#poldectostring-decimal-number">\PolDecToString{decimal number}</a>
+which uses decimal notation (at least for the numerator part).</p>
+</blockquote>
+</div>
+<div class="section" id="id6">
+<span id="poltypesetmonomialcmd"></span><h3><a class="toc-backref" href="#id61"><tt class="docutils literal">\PolTypesetMonomialCmd</tt></a></h3>
+<blockquote>
+This decides how a monomial (in variable <tt class="docutils literal">\PolVar</tt> and with
+exponent <tt class="docutils literal">\PolIndex</tt>) is to be printed. The default does nothing
+for the constant term, <tt class="docutils literal">\PolVar</tt> for the first degree and
+<tt class="docutils literal"><span class="pre">\PolVar^{\PolIndex}</span></tt> for higher degrees monomials. Beware that
+<tt class="docutils literal">\PolIndex</tt> expands to digit tokens and needs termination in
+<tt class="docutils literal">\ifnum</tt> tests.</blockquote>
+</div>
+<div class="section" id="poltypesetcmdprefix-raw-coeff">
+<span id="poltypesetcmdprefix"></span><h3><a class="toc-backref" href="#id62"><tt class="docutils literal">\PolTypesetCmdPrefix{raw_coeff}</tt></a></h3>
+<blockquote>
+Expands to a <tt class="docutils literal">+</tt> if the <tt class="docutils literal">raw_coeff</tt> is zero or positive, and to
+nothing if <tt class="docutils literal">raw_coeff</tt> is negative, as in latter case the
+<tt class="docutils literal">\xintSignedFrac</tt> used by <a class="reference internal" href="#poltypesetcmd-raw-coeff">\PolTypesetCmd{raw_coeff}</a> will put
+the <tt class="docutils literal">-</tt> sign in front of the fraction (if it is a fraction) and
+this will thus serve as separator in the typeset formula. Not used
+for the first term.</blockquote>
+</div>
+</div>
+<div class="section" id="id8">
+<span id="id7"></span><h2><a class="toc-backref" href="#id63"><tt class="docutils literal"><span class="pre">\PolTypeset*{polname}</span></tt></a></h2>
+<blockquote>
+Typesets in ascending powers. Use e.g. <tt class="docutils literal">[h]</tt> optional argument
+(after the <tt class="docutils literal">*</tt>) to use letter <tt class="docutils literal">h</tt> rather than <tt class="docutils literal">x</tt>.</blockquote>
+</div>
+<div class="section" id="poldiff-polname-1-polname-2">
+<span id="poldiff"></span><h2><a class="toc-backref" href="#id64"><tt class="docutils literal"><span class="pre">\PolDiff{polname_1}{polname_2}</span></tt></a></h2>
+<blockquote>
+<p>This sets <tt class="docutils literal">polname_2</tt> to the first derivative of <tt class="docutils literal">polname_1</tt>. It
+is allowed to issue <tt class="docutils literal"><span class="pre">\PolDiff{f}{f}</span></tt>, effectively replacing <tt class="docutils literal">f</tt>
+by <tt class="docutils literal">f'</tt>.</p>
+<p>Coefficients of the result <tt class="docutils literal">polname_2</tt> are irreducible fractions
+(see <a class="reference internal" href="#technicalities">Technicalities</a> for the whole story.)</p>
+</blockquote>
+</div>
+<div class="section" id="poldiff-n-polname-1-polname-2">
+<span id="poldiff-n"></span><h2><a class="toc-backref" href="#id65"><tt class="docutils literal"><span class="pre">\PolDiff[N]{polname_1}{polname_2}</span></tt></a></h2>
+<blockquote>
+This sets <tt class="docutils literal">polname_2</tt> to the <tt class="docutils literal">N</tt>-th derivative of <tt class="docutils literal">polname_1</tt>.
+Identical arguments is allowed. With <tt class="docutils literal">N=0</tt>, same effect as
+<tt class="docutils literal"><span class="pre">\PolLet{polname_2}={polname_1}</span></tt>. With negative <tt class="docutils literal">N</tt>, switches to
+using <tt class="docutils literal">\PolAntiDiff</tt>.</blockquote>
+</div>
+<div class="section" id="polantidiff-polname-1-polname-2">
+<span id="polantidiff"></span><h2><a class="toc-backref" href="#id66"><tt class="docutils literal"><span class="pre">\PolAntiDiff{polname_1}{polname_2}</span></tt></a></h2>
+<blockquote>
+<p>This sets <tt class="docutils literal">polname_2</tt> to the primitive of <tt class="docutils literal">polname_1</tt> vanishing
+at zero.</p>
+<p>Coefficients of the result <tt class="docutils literal">polname_2</tt> are irreducible fractions
+(see <a class="reference internal" href="#technicalities">Technicalities</a> for the whole story.)</p>
+</blockquote>
+</div>
+<div class="section" id="polantidiff-n-polname-1-polname-2">
+<span id="polantidiff-n"></span><h2><a class="toc-backref" href="#id67"><tt class="docutils literal"><span class="pre">\PolAntiDiff[N]{polname_1}{polname_2}</span></tt></a></h2>
+<blockquote>
+This sets <tt class="docutils literal">polname_2</tt> to the result of <tt class="docutils literal">N</tt> successive integrations on
+<tt class="docutils literal">polname_1</tt>. With negative <tt class="docutils literal">N</tt>, it switches to using <tt class="docutils literal">\PolDiff</tt>.</blockquote>
+</div>
+<div class="section" id="poldivide-polname-1-polname-2-polname-q-polname-r">
+<span id="poldivide"></span><h2><a class="toc-backref" href="#id68"><tt class="docutils literal"><span class="pre">\PolDivide{polname_1}{polname_2}{polname_Q}{polname_R}</span></tt></a></h2>
+<blockquote>
+This sets <tt class="docutils literal">polname_Q</tt> and <tt class="docutils literal">polname_R</tt> to be the quotient and
+remainder in the Euclidean division of <tt class="docutils literal">polname_1</tt> by
+<tt class="docutils literal">polname_2</tt>.</blockquote>
+</div>
+<div class="section" id="polquo-polname-1-polname-2-polname-q">
+<span id="polquo"></span><h2><a class="toc-backref" href="#id69"><tt class="docutils literal"><span class="pre">\PolQuo{polname_1}{polname_2}{polname_Q}</span></tt></a></h2>
+<blockquote>
+This sets <tt class="docutils literal">polname_Q</tt> to be the quotient in the Euclidean division
+of <tt class="docutils literal">polname_1</tt> by <tt class="docutils literal">polname_2</tt>.</blockquote>
+</div>
+<div class="section" id="polrem-polname-1-polname-2-polname-r">
+<span id="polrem"></span><h2><a class="toc-backref" href="#id70"><tt class="docutils literal"><span class="pre">\PolRem{polname_1}{polname_2}{polname_R}</span></tt></a></h2>
+<blockquote>
+This sets <tt class="docutils literal">polname_R</tt> to be the remainder in the Euclidean division
+of <tt class="docutils literal">polname_1</tt> by <tt class="docutils literal">polname_2</tt>.</blockquote>
+</div>
+<div class="section" id="polgcd-polname-1-polname-2-polname-gcd">
+<span id="polgcd"></span><h2><a class="toc-backref" href="#id71"><tt class="docutils literal"><span class="pre">\PolGCD{polname_1}{polname_2}{polname_GCD}</span></tt></a></h2>
+<blockquote>
+This sets <tt class="docutils literal">polname_GCD</tt> to be the (monic) GCD of the two first
+polynomials. It is a unitary polynomial except if both <tt class="docutils literal">polname_1</tt>
+and <tt class="docutils literal">polname_2</tt> vanish, then <tt class="docutils literal">polname_GCD</tt> is the zero
+polynomial.</blockquote>
+<!-- ``\PolIGCD{polname_1}{polname_2}{polname_iGCD}``
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ **NOT YET**
+
+ This **assumes** that the two polynomials have integer coefficients.
+ It then computes the greatest common divisor in the integer
+ polynomial ring, normalized to have a positive leading coefficient
+ (if the inputs are not both zero).
+
+``\PolIContent{polname}``
+~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ **NOT YET**
+
+ This computes a positive rational number such that dividing the
+ polynomial with it returns an integer coefficients polynomial with
+ no common factor among the coefficients. -->
+</div>
+<div class="section" id="poltosturm-polname-sturmname">
+<span id="poltosturm"></span><h2><a class="toc-backref" href="#id72"><tt class="docutils literal"><span class="pre">\PolToSturm{polname}{sturmname}</span></tt></a></h2>
+<blockquote>
+<p>With <tt class="docutils literal">polname</tt> being for example <tt class="docutils literal">P</tt>, the macro starts by
+computing polynomials <tt class="docutils literal">P</tt> and <tt class="docutils literal">P'</tt>, then computes the (opposite
+of the) remainder in euclidean division, iteratively.</p>
+<p>The last non-zero remainder <tt class="docutils literal">P_N_</tt> (where <tt class="docutils literal">N</tt> is obtainable as
+<a class="reference internal" href="#polsturmchainlength-sturmname">\PolSturmChainLength{sturmname}</a>) is up to a factor
+the GCD of <tt class="docutils literal">P</tt> and <tt class="docutils literal">P'</tt> hence it is a constant if and only if
+<tt class="docutils literal">P</tt> is square-free.</p>
+<div class="admonition note">
+<p class="first admonition-title">Note</p>
+<ul class="last simple">
+<li>Since <tt class="docutils literal">0.5</tt> all these polynomials are divided by their rational
+content, so they have integer coefficients with no common factor,
+and the last one if a constant is either <tt class="docutils literal">1</tt> or <tt class="docutils literal"><span class="pre">-1</span></tt>.</li>
+<li>After this normalization to primitive polynomials, they are
+stored internally as <tt class="docutils literal">sturmname_k_</tt>, <tt class="docutils literal">k=0,1, ...</tt>.</li>
+<li>These polynomials are used internally only. To keep them as
+genuine declared polynomials also after the macro call, use the
+starred variant <a class="reference internal" href="#id9">PolToSturm*</a>.</li>
+</ul>
+</div>
+<div class="admonition note">
+<p class="first admonition-title">Note</p>
+<p class="last">It is perfectly allowed to use the polynomial name as Sturm chain name:
+<tt class="docutils literal"><span class="pre">\PolToSturm{f}(f}</span></tt>.</p>
+</div>
+<p>The macro then declares <tt class="docutils literal">sturmname_0</tt>, <tt class="docutils literal">sturmname_1</tt>, ..., which are
+the (non-declared) <tt class="docutils literal">sturmname_k_</tt> divided by the last one. Division is
+not done if this last one is the constant <tt class="docutils literal">1</tt> or <tt class="docutils literal"><span class="pre">-1</span></tt>, i.e. if the
+original polynomial was square-free. These polynomials are primitive
+polynomials too, i.e. with integer coefficients having no common factor.</p>
+<p>Thus <tt class="docutils literal">sturmname_0</tt> has exactly the same real and complex roots as
+polynomial <tt class="docutils literal">polname</tt>, but with each root now of multiplicity one:
+i.e. it is the &quot;square-free part&quot; of original polynomial <tt class="docutils literal">polname</tt>.</p>
+<p>Notice that <tt class="docutils literal">sturmname_1</tt> isn't necessarily the derivative of
+<tt class="docutils literal">sturmname_0</tt> due to the various normalizations.</p>
+<p>The polynomials <tt class="docutils literal">sturmname_k</tt> main utility is for the execution of
+<a class="reference internal" href="#polsturmisolatezeros-sturmname">\PolSturmIsolateZeros{sturmname}</a>. Be careful not to use these
+names <tt class="docutils literal">sturmname_0</tt>, <tt class="docutils literal">sturmname_1</tt>, etc... for defining other
+polynomials after having done <tt class="docutils literal"><span class="pre">\PolToSturm{polname}{sturmname}</span></tt> and
+before executing <tt class="docutils literal">\PolSturmIsolateZeros{sturmname}</tt> else the
+latter will behave erroneously.</p>
+<p><a class="reference internal" href="#polsturmchainlength-sturmname">\PolSturmChainLength{sturmname}</a> gives the index of the last
+element of the Sturm chain.</p>
+</blockquote>
+</div>
+<div class="section" id="id10">
+<span id="id9"></span><h2><a class="toc-backref" href="#id73"><tt class="docutils literal"><span class="pre">\PolToSturm*{polname}{sturmname}</span></tt></a></h2>
+<blockquote>
+<p>Does the same as <a class="reference internal" href="#poltosturm">un-starred version</a> and additionally it
+keeps for user usage the memory of the <em>un-normalized</em> Sturm chain
+polynomials <tt class="docutils literal">sturmname_k_</tt>, <tt class="docutils literal">k=0,1, <span class="pre">...,</span> N</tt>, with
+<tt class="docutils literal">N</tt> being <a class="reference internal" href="#polsturmchainlength-sturmname">\PolSturmChainLength{sturmname}</a>.</p>
+<div class="admonition note">
+<p class="first admonition-title">Note</p>
+<p class="last">This behaviour was modified at <tt class="docutils literal">0.6</tt>, anyhow the macro was
+broken at <tt class="docutils literal">0.5</tt>.</p>
+</div>
+<div class="admonition hint">
+<p class="first admonition-title">Hint</p>
+<p class="last">The square-free part of <tt class="docutils literal">polname</tt> is <tt class="docutils literal">sturmname_0</tt>, and their
+quotient is the polynomial with name
+<tt class="docutils literal">sturname_\PolSturmChainLength{sturmname}_</tt>. It thus easy to
+set-up a loop iteratively computing the latter until the last one
+is a constant, thus obtaining the decomposition of an <tt class="docutils literal">f</tt> as
+a product <tt class="docutils literal">c f_1 f_2 f_3 ...</tt> of a constant and square-free (primitive)
+polynomials, where each <tt class="docutils literal">f_i</tt> divides its predecessor.</p>
+</div>
+</blockquote>
+</div>
+<div class="section" id="polsettosturmchainsignchangesat-macro-sturmname-fraction">
+<span id="polsettosturmchainsignchangesat"></span><h2><a class="toc-backref" href="#id74"><tt class="docutils literal"><span class="pre">\PolSetToSturmChainSignChangesAt{\macro}{sturmname}{fraction}</span></tt></a></h2>
+<blockquote>
+<p>Sets macro <tt class="docutils literal">\macro</tt> to the number of sign changes in the Sturm
+chain with name prefix <tt class="docutils literal">sturmname</tt>, at location <tt class="docutils literal">fraction</tt>
+(which must be in format as acceptable by the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> macros.)</p>
+<div class="admonition note">
+<p class="first admonition-title">Note</p>
+<p>The author was lazy and did not provide rather an expandable
+variant, where one would do <tt class="docutils literal"><span class="pre">\edef\macro{\PolNbOf...}</span></tt>.</p>
+<p>This will presumably get added in a future release.</p>
+<p class="last">After some hesitation it was decided the macro would by default
+act globally. To make the scope of its macro definition local,
+use <tt class="docutils literal">[\empty]</tt> as extra optional argument.</p>
+</div>
+</blockquote>
+</div>
+<div class="section" id="polsettonbofzeroswithin-macro-sturmname-value-a-value-b">
+<span id="polsettonbofzeroswithin"></span><h2><a class="toc-backref" href="#id75"><tt class="docutils literal"><span class="pre">\PolSetToNbOfZerosWithin{\macro}{sturmname}{value_a}{value_b}</span></tt></a></h2>
+<blockquote>
+<p>Applies the <a class="reference external" href="https://en.wikipedia.org/wiki/Sturm%27s_theorem">Sturm Theorem</a> to set <tt class="docutils literal">\macro</tt> to the exact number
+of <strong>distinct</strong> roots of <tt class="docutils literal">sturmname_0</tt> in the interval <tt class="docutils literal">(value_a,
+value_b]</tt> (the macro first re-orders the value for <tt class="docutils literal">value_a &lt;=
+value_b</tt> to hold).</p>
+<div class="admonition note">
+<p class="first admonition-title">Note</p>
+<p>The author was lazy and did not provide rather an expandable
+variant, where one would do <tt class="docutils literal"><span class="pre">\edef\macro{\PolNbOf...}</span></tt>.</p>
+<p>This will presumably get added in future.</p>
+<p class="last">After some hesitation it was decided the macro would by default
+act globally. To make the scope of its macro definition local,
+use <tt class="docutils literal">[\empty]</tt> as extra optional argument.</p>
+</div>
+<p>See also the expandable
+<a class="reference internal" href="#polsturmnbofrootsof-sturmname-lessthanorequalto-value">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualTo{value}</a>, from
+which it is immediate (with <tt class="docutils literal">\numexpr</tt>) to create an expandable
+variant of this macro. However the difference is that this macro
+requires only <a class="reference internal" href="#poltosturm">\PolToSturm</a> to have been executed,
+whereas the expandable variant requires prior execution of
+<a class="reference internal" href="#polsturmisolatezeros">\PolSturmIsolateZeros</a>.</p>
+<p>See also the expandable
+<a class="reference internal" href="#polsturmnbwithmultofrootsof-sturmname-lessthanorequalto-value">\PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualTo{value}</a>
+which requires prior execution of
+<a class="reference internal" href="#id11">\PolSturmIsolateZeros*</a>.</p>
+</blockquote>
+</div>
+<div class="section" id="polsturmisolatezeros-sturmname">
+<span id="polsturmisolatezeros"></span><h2><a class="toc-backref" href="#id76"><tt class="docutils literal">\PolSturmIsolateZeros{sturmname}</tt></a></h2>
+<blockquote>
+<p>The macros locates, using <a class="reference external" href="https://en.wikipedia.org/wiki/Sturm%27s_theorem">Sturm theorem</a>, as many disjoint
+intervals as there are (real) roots.</p>
+<div class="admonition important">
+<p class="first admonition-title">Important</p>
+<p>The Sturm chain must have been produced by an earlier
+<a class="reference internal" href="#poltosturm-polname-sturmname">\PolToSturm{polname}{sturmname}</a>.</p>
+<p>Why does this macro ask for argument the name of Sturm chain,
+rather than the name of a polynomial? well this is mainly for
+legacy reason, and because it is accompanied by other macros for
+which it is simpler to assume the argument will be the name of an
+already computed Sturm chain.</p>
+<p>Notice that <tt class="docutils literal"><span class="pre">\PolToSturm{f}{f}</span></tt> is perfectly legal (the
+<tt class="docutils literal">sturmname</tt> can be same as the <tt class="docutils literal">polname</tt>): it defines
+polynomials <tt class="docutils literal">f_0</tt>, <tt class="docutils literal">f_1</tt>, ... having <tt class="docutils literal">f</tt> has name prefix.</p>
+<p class="last">Such a prior call
+to <tt class="docutils literal">\PolToSturm</tt> must have been made at any rate for
+<tt class="docutils literal">\PolSturmIsolateZeros</tt> to be usable.</p>
+</div>
+<p>After its execution they are two types of such intervals (stored in
+memory and accessible via macros or <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> variables, see below):</p>
+<ul class="simple">
+<li>singleton <tt class="docutils literal">{a}</tt>: then <tt class="docutils literal">a</tt> is a root, (necessarily a decimal
+number, but not all such decimal numbers are exactly identified yet).</li>
+<li>open intervals <tt class="docutils literal">(a,b)</tt>: then there is exactly one root <tt class="docutils literal">z</tt>
+such that <tt class="docutils literal">a &lt; z &lt; b</tt>, and the end points are guaranteed to not
+be roots.</li>
+</ul>
+<p>The interval boundaries are decimal numbers, originating
+in iterated decimal subdivision from initial intervals
+<tt class="docutils literal"><span class="pre">(-10^E,</span> 0)</tt> and <tt class="docutils literal">(0, 10^E)</tt> with <tt class="docutils literal">E</tt> chosen initially large
+enough so that all roots are enclosed; if zero is a root it is always
+identified as such. The non-singleton intervals are of the
+type <tt class="docutils literal">(a/10^f, <span class="pre">(a+1)/10^f)</span></tt> with <tt class="docutils literal">a</tt> an integer, which is
+neither <tt class="docutils literal">0</tt> nor <tt class="docutils literal"><span class="pre">-1</span></tt>. Hence either <tt class="docutils literal">a</tt> and <tt class="docutils literal">a+1</tt> are both positive
+or they are both negative.</p>
+<p>One does not <em>a priori</em> know what will be the lengths of these
+intervals (except that they are always powers of ten), they
+vary depending on how many digits two successive roots have in
+common in their respective decimal expansions.</p>
+<div class="admonition important">
+<p class="first admonition-title">Important</p>
+<p>If some two consecutive intervals share an end-point, no
+information is yet gained about the separation between the two
+roots which could at this stage be arbitrarily small.</p>
+<p class="last">See <a class="reference internal" href="#polrefineinterval-sturmname-index">\PolRefineInterval*{sturmname}{index}</a> which addresses
+this issue.</p>
+</div>
+<!-- This procedure is covariant
+with the independent variable ``x`` becoming ``-x``.
+Hmm, pas sûr et trop fatigué -->
+<p>The interval boundaries (and exactly found roots) are made available
+for future computations in <tt class="docutils literal">\xintexpr</tt>-essions or polynomial
+definitions as variables <tt class="docutils literal">&lt;sturmname&gt;L_1</tt>,
+<tt class="docutils literal">&lt;sturmname&gt;L_2</tt>, etc..., for the left end-points and
+<tt class="docutils literal">&lt;sturmname&gt;R_1</tt>, <tt class="docutils literal">&lt;sturmname&gt;R_2</tt>, ..., for the right
+end-points.</p>
+<p>Thus for example, if <tt class="docutils literal">sturmname</tt> is <tt class="docutils literal">f</tt>, one can use the
+<a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> variables <tt class="docutils literal">fL_1</tt>, <tt class="docutils literal">fL_2</tt>, ... to refer in expressions
+to the left end-points (or to the exact root, if left and right end
+points coincide). Additionally, <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> variable <tt class="docutils literal">fZ_1_isknown</tt>
+will have value <tt class="docutils literal">1</tt> if the root in the first interval is known,
+and <tt class="docutils literal">0</tt> otherwise. And similarly for the other intervals.</p>
+<p>Also, macros <a class="reference internal" href="#polsturmisolatedzeroleft-sturmname-index">\PolSturmIsolatedZeroLeft{sturmname}{index}</a> and
+<a class="reference internal" href="#polsturmisolatedzeroright-sturmname-index">\PolSturmIsolatedZeroRight{sturmname}{index}</a> are provided which
+expand to these same values, written in decimal notation (i.e.
+pre-processed by <a class="reference internal" href="#poldectostring">\PolDecToString</a>.) And there
+is also <a class="reference internal" href="#polsturmifzeroexactlyknown-sturmname-index-a-b">\PolSturmIfZeroExactlyKnown{sturmname}{index}{A}{B}</a>.</p>
+<div class="admonition important">
+<p class="first admonition-title">Important</p>
+<p class="last">Trailing zeroes in the stored decimal numbers accessible via the
+macros are significant: they are also present in the decimal
+expansion of the exact root.</p>
+</div>
+<p>These variables and macros are automatically updated when one next
+uses macros such as <a class="reference internal" href="#polrefineinterval-sturmname-index">\PolRefineInterval*{sturmname}{index}</a>.</p>
+<p>The start of decimal expansion of a positive <tt class="docutils literal">k</tt>-th root is given
+by <a class="reference internal" href="#polsturmisolatedzeroleft">\PolSturmIsolatedZeroLeft{sturmname}{k}</a>, and for a negative root it is given
+by <a class="reference internal" href="#polsturmisolatedzeroright">PolSturmIsolatedZeroRight{sturmname}{k}</a>. These two decimal
+numbers are either both zero or both of the same sign.</p>
+<p>The number of distinct roots is obtainable expandably as
+<a class="reference internal" href="#polsturmnbofisolatedzeros-sturmname">\PolSturmNbOfIsolatedZeros{sturmname}</a>.</p>
+<p>Furthermore
+<a class="reference internal" href="#polsturmnbofrootsof-sturmname-lessthanorequalto-value">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualTo{value}</a> and
+<a class="reference internal" href="#polsturmnbofrootsof-sturmname-lessthanorequaltoexpr-expression">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualToExpr{expression}</a>.
+will expandably compute respectively the number of real roots at
+most equal to <tt class="docutils literal">value</tt> or <tt class="docutils literal">expression</tt>, and the same but with
+multiplicities.</p>
+<div class="admonition note">
+<p class="first admonition-title">Note</p>
+<p class="last">In the current implementation the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> variables
+and <a class="reference external" href="http://www.ctan.org/pkg/xint">xinttools</a> arrays are globally defined. On the
+other hand the Sturm sequence polynomials obey the current scope.</p>
+</div>
+<div class="admonition note">
+<p class="first admonition-title">Note</p>
+<p>As all computations are done <em>exactly</em> there can be no errors...
+apart those due to bad coding by author. The results are exact
+bounds for the mathematically exact real roots.</p>
+<p class="last">Future releases will perhaps also provide macros based on Newton
+or Regula Falsi methods. Exact computations with such methods
+lead however quickly to very big fractions, and this forces usage
+of some rounding scheme for the abscissas if computation times
+are to remain reasonable. This raises issues of its own, which
+are studied in numerical mathematics.</p>
+</div>
+</blockquote>
+</div>
+<div class="section" id="id12">
+<span id="id11"></span><h2><a class="toc-backref" href="#id77"><tt class="docutils literal"><span class="pre">\PolSturmIsolateZeros*{sturmname}</span></tt></a></h2>
+<blockquote>
+<p>The macro does the same as <a class="reference internal" href="#polsturmisolatezeros-sturmname">\PolSturmIsolateZeros{sturmname}</a> and
+then in addition it does the extra work to determine all
+multiplicities (of the real roots):
+after executing this macro,
+<a class="reference internal" href="#polsturmisolatedzeromultiplicity-sturmname-index">\PolSturmIsolatedZeroMultiplicity{sturmname}{index}</a> will expand
+to the multiplicity of the root located in the <tt class="docutils literal">index</tt>-th
+interval (intervals are enumerated from left to right, with index
+starting at <tt class="docutils literal">1</tt>).</p>
+<p>Furthermore, if for example the <tt class="docutils literal">sturmname</tt> is <tt class="docutils literal">f</tt>, <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a>
+variables <tt class="docutils literal">fM_1</tt>, <tt class="docutils literal">fM_2</tt>... hold the multiplicities thus
+computed.</p>
+<div class="admonition note">
+<p class="first admonition-title">Note</p>
+<p class="last">It is <strong>not</strong> necessary to have executed the <a class="reference internal" href="#id9">PolToSturm*</a> starred
+variant, as the non-starred variant keeps internally the memory of the
+original GCD (and even of the full non-normalized original Sturm
+chain), even though it does not make the declarations as <em>user-level</em>
+genuine polynomials.</p>
+</div>
+<p>See <a class="reference internal" href="#the-degree-nine-polynomial-with-0-99-0-999-0-9999-as-triple-roots">The degree nine polynomial with 0.99, 0.999, 0.9999 as triple
+roots</a> for an example.</p>
+</blockquote>
+</div>
+<div class="section" id="id14">
+<span id="id13"></span><h2><a class="toc-backref" href="#id78"><tt class="docutils literal"><span class="pre">\PolSturmIsolateZeros**{sturmname}</span></tt></a></h2>
+<blockquote>
+<p>The macro does the same as <a class="reference internal" href="#id12">\PolSturmIsolateZeros*{sturmname}</a> and
+in addition it does the extra work to determine all the <em>rational</em>
+roots.</p>
+<div class="admonition note">
+<p class="first admonition-title">Note</p>
+<p class="last">After execution of this macro, a root is &quot;known&quot; if and only if
+it is rational.</p>
+</div>
+<p>Furthermore, primitive polynomial <tt class="docutils literal">sturmname_sqf_norr</tt> is created
+to match the (square-free) <tt class="docutils literal">sturmname_0</tt> from which all rational
+roots have been removed (see <a class="reference internal" href="#polexprsetup">\polexprsetup</a> for customizing this
+name). The number of distinct rational roots is thus the difference
+between the degrees of these two polynomials (see also
+<a class="reference internal" href="#polsturmnbofrationalroots-sturmname">\PolSturmNbOfRationalRoots{sturmname}</a>).</p>
+<p>And <tt class="docutils literal">sturmname_norr</tt> is <tt class="docutils literal">sturmname_0_</tt> from which all rational
+roots have been removed (see <a class="reference internal" href="#polexprsetup">\polexprsetup</a>), i.e. it contains
+the irrational roots of the original polynomial, with the same
+multiplicities.</p>
+<p>See <a class="reference internal" href="#a-degree-five-polynomial-with-three-rational-roots">A degree five polynomial with three rational
+roots</a> for an example.</p>
+</blockquote>
+</div>
+<div class="section" id="polsturmisolatezerosandgetmultiplicities-sturmname">
+<span id="polsturmisolatezerosandgetmultiplicities"></span><h2><a class="toc-backref" href="#id79"><tt class="docutils literal">\PolSturmIsolateZerosAndGetMultiplicities{sturmname}</tt></a></h2>
+<blockquote>
+This is another name for <a class="reference internal" href="#id12">\PolSturmIsolateZeros*{sturmname}</a>.</blockquote>
+</div>
+<div class="section" id="polsturmisolatezerosgetmultiplicitiesandrationalroots-sturmname">
+<span id="polsturmisolatezerosgetmultiplicitiesandrationalroots"></span><h2><a class="toc-backref" href="#id80"><tt class="docutils literal">\PolSturmIsolateZerosGetMultiplicitiesAndRationalRoots{sturmname}</tt></a></h2>
+<blockquote>
+This is another name for <a class="reference internal" href="#id14">\PolSturmIsolateZeros**{sturmname}</a>.</blockquote>
+</div>
+<div class="section" id="polsturmisolatezerosandfindrationalroots-sturmname">
+<h2><a class="toc-backref" href="#id81"><tt class="docutils literal">\PolSturmIsolateZerosAndFindRationalRoots{sturmname}</tt></a></h2>
+<blockquote>
+<p>This works exactly like <a class="reference internal" href="#id14">\PolSturmIsolateZeros**{sturmname}</a>
+(inclusive of declaring the polynomials <tt class="docutils literal">sturmname_sqf_norr</tt> and
+<tt class="docutils literal">sturmname_norr</tt> with no rational roots) except that it does <em>not</em>
+compute the multiplicities of the <em>non-rational</em> roots.</p>
+<div class="admonition note">
+<p class="first admonition-title">Note</p>
+<p class="last">There is no macro to find the rational roots but not compute
+their multiplicities at the same time.</p>
+</div>
+<div class="admonition attention">
+<p class="first admonition-title">Attention!</p>
+<p>This macro does <em>not</em> define <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> variables
+<tt class="docutils literal">sturmnameM_1</tt>, <tt class="docutils literal">sturmnameM_2</tt>, ... holding the
+multiplicities and it leaves the multiplicity array (whose accessor
+is <a class="reference internal" href="#polsturmisolatedzeromultiplicity-sturmname-index">\PolSturmIsolatedZeroMultiplicity{sturmname}{index}</a>) into
+a broken state, as all non-rational roots will supposedly have
+multiplicity one. This means that the output of
+<a class="reference internal" href="#id18">\PolPrintIntervals*</a> for example will be
+erroneous for the intervals with irrational roots.</p>
+<p class="last">I decided to document it because finding multiplicities of the
+non rational roots is somewhat costly, and one may be interested
+only into finding the rational roots (of course random
+polynomials with integer coefficients will not have <em>any</em>
+rational root anyhow).</p>
+</div>
+</blockquote>
+</div>
+<div class="section" id="polrefineinterval-sturmname-index">
+<span id="polrefineinterval"></span><h2><a class="toc-backref" href="#id82"><tt class="docutils literal"><span class="pre">\PolRefineInterval*{sturmname}{index}</span></tt></a></h2>
+<blockquote>
+The <tt class="docutils literal">index</tt>-th interval (starting indexing at one) is further
+subdivided as many times as is necessary in order for the newer
+interval to have both its end-points distinct from the end-points of
+the original interval. This means that the <tt class="docutils literal">k</tt>th root is then
+strictly separated from the other roots.</blockquote>
+</div>
+<div class="section" id="polrefineinterval-n-sturmname-index">
+<span id="polrefineinterval-n"></span><h2><a class="toc-backref" href="#id83"><tt class="docutils literal"><span class="pre">\PolRefineInterval[N]{sturmname}{index}</span></tt></a></h2>
+<blockquote>
+The <tt class="docutils literal">index</tt>-th interval (starting count at one) is further
+subdivided once, reducing its length by a factor of 10. This is done
+<tt class="docutils literal">N</tt> times if the optional argument <tt class="docutils literal">[N]</tt> is present.</blockquote>
+</div>
+<div class="section" id="polensureintervallength-sturmname-index-e">
+<span id="polensureintervallength"></span><h2><a class="toc-backref" href="#id84"><tt class="docutils literal"><span class="pre">\PolEnsureIntervalLength{sturmname}{index}{E}</span></tt></a></h2>
+<blockquote>
+The <tt class="docutils literal">index</tt>-th interval is subdivided until its length becomes at
+most <tt class="docutils literal">10^E</tt>. This means (for <tt class="docutils literal">E&lt;0</tt>) that the first <tt class="docutils literal"><span class="pre">-E</span></tt> digits
+after decimal mark of the <tt class="docutils literal">k</tt>th root will then be known exactly.</blockquote>
+</div>
+<div class="section" id="polensureintervallengths-sturmname-e">
+<span id="polensureintervallengths"></span><h2><a class="toc-backref" href="#id85"><tt class="docutils literal"><span class="pre">\PolEnsureIntervalLengths{sturmname}{E}</span></tt></a></h2>
+<blockquote>
+<p>The intervals as obtained from <tt class="docutils literal">\PolSturmIsolateZeros</tt> are (if
+necessary) subdivided further by (base 10) dichotomy in order for
+each of them to have length at most <tt class="docutils literal">10^E</tt> (length will be shorter
+than <tt class="docutils literal">10^E</tt> in output only if it did not change or became zero.)</p>
+<p>This means that decimal expansions of all roots will be known with
+<tt class="docutils literal"><span class="pre">-E</span></tt> digits (for <tt class="docutils literal">E&lt;0</tt>) after decimal mark.</p>
+</blockquote>
+</div>
+<div class="section" id="polprintintervals-varname-sturmname">
+<span id="polprintintervals"></span><h2><a class="toc-backref" href="#id86"><tt class="docutils literal"><span class="pre">\PolPrintIntervals[varname]{sturmname}</span></tt></a></h2>
+<blockquote>
+<p>This is a convenience macro which prints the bounds for the roots
+<tt class="docutils literal">Z_1</tt>, <tt class="docutils literal">Z_2</tt>, ... (the optional argument <tt class="docutils literal">varname</tt> allows to
+specify a replacement for the default <tt class="docutils literal">Z</tt>). This will be done (by
+default) in a
+math mode <tt class="docutils literal">array</tt>, one interval per row, and pattern <tt class="docutils literal">rcccl</tt>,
+where the second and fourth column hold the <tt class="docutils literal">&lt;</tt> sign, except when
+the interval reduces to a singleton, which means the root is known
+exactly.</p>
+<div class="admonition attention">
+<p class="first admonition-title">Attention!</p>
+<p class="last">This macro was refactored at 0.7, its default output remained
+identical but the ways to customize it got completely
+modified.</p>
+</div>
+<p>See next macros which govern its output.</p>
+</blockquote>
+<div class="section" id="polprintintervalsnorealroots">
+<h3><a class="toc-backref" href="#id87"><tt class="docutils literal">\PolPrintIntervalsNoRealRoots</tt></a></h3>
+<blockquote>
+<p>Executed in place of an <tt class="docutils literal">array</tt> environment, when there are no
+real roots. Default definition:</p>
+<pre class="literal-block">
+\newcommand\PolPrintIntervalsNoRealRoots{}
+</pre>
+</blockquote>
+</div>
+<div class="section" id="polprintintervalsbeginenv">
+<h3><a class="toc-backref" href="#id88"><tt class="docutils literal">\PolPrintIntervalsBeginEnv</tt></a></h3>
+<blockquote>
+<p>Default definition:</p>
+<pre class="literal-block">
+\newcommand\PolPrintIntervalsBeginEnv{\[\begin{array}{rcccl}}
+</pre>
+</blockquote>
+</div>
+<div class="section" id="polprintintervalsendenv">
+<h3><a class="toc-backref" href="#id89"><tt class="docutils literal">\PolPrintIntervalsEndEnv</tt></a></h3>
+<blockquote>
+<p>Default definition:</p>
+<pre class="literal-block">
+\newcommand\PolPrintIntervalsEndEnv{\end{array}\]}
+</pre>
+</blockquote>
+</div>
+<div class="section" id="polprintintervalsknownroot">
+<h3><a class="toc-backref" href="#id90"><tt class="docutils literal">\PolPrintIntervalsKnownRoot</tt></a></h3>
+<blockquote>
+<p>Default definition:</p>
+<pre class="literal-block">
+\newcommand\PolPrintIntervalsKnownRoot{%
+ &amp;&amp;\PolPrintIntervalsTheVar_{\PolPrintIntervalsTheIndex}%
+ &amp;=&amp;\PolPrintIntervalsPrintExactZero
+}
+</pre>
+</blockquote>
+</div>
+<div class="section" id="polprintintervalsunknownroot">
+<h3><a class="toc-backref" href="#id91"><tt class="docutils literal">\PolPrintIntervalsUnknownRoot</tt></a></h3>
+<blockquote>
+<p>Default definition:</p>
+<pre class="literal-block">
+\newcommand\PolPrintIntervalsUnknownRoot{%
+ \PolPrintIntervalsPrintLeftEndPoint&amp;&lt;&amp;%
+ \PolPrintIntervalsTheVar_{\PolPrintIntervalsTheIndex}&amp;&lt;&amp;%
+ \PolPrintIntervalsPrintRightEndPoint
+}
+</pre>
+</blockquote>
+</div>
+<div class="section" id="id15">
+<span id="polprintintervalsprintexactzero"></span><h3><a class="toc-backref" href="#id92"><tt class="docutils literal">\PolPrintIntervalsPrintExactZero</tt></a></h3>
+<blockquote>
+<p>Default definition:</p>
+<pre class="literal-block">
+\newcommand\PolPrintIntervalsPrintExactZero{\PolPrintIntervalsTheLeftEndPoint}
+</pre>
+</blockquote>
+</div>
+<div class="section" id="id16">
+<span id="polprintintervalsprintleftendpoint"></span><h3><a class="toc-backref" href="#id93"><tt class="docutils literal">\PolPrintIntervalsPrintLeftEndPoint</tt></a></h3>
+<blockquote>
+<p>Default definition:</p>
+<pre class="literal-block">
+\newcommand\PolPrintIntervalsPrintLeftEndPoint{\PolPrintIntervalsTheLeftEndPoint}
+</pre>
+</blockquote>
+</div>
+<div class="section" id="id17">
+<span id="polprintintervalsprintrightendpoint"></span><h3><a class="toc-backref" href="#id94"><tt class="docutils literal">\PolPrintIntervalsPrintRightEndPoint</tt></a></h3>
+<blockquote>
+<p>Default definition is:</p>
+<pre class="literal-block">
+\newcommand\PolPrintIntervalsPrintRightEndPoint{\PolPrintIntervalsTheRightEndPoint}
+</pre>
+</blockquote>
+</div>
+</div>
+<div class="section" id="id19">
+<span id="id18"></span><h2><a class="toc-backref" href="#id95"><tt class="docutils literal"><span class="pre">\PolPrintIntervals*[varname]{sturmname}</span></tt></a></h2>
+<blockquote>
+<p>This starred variant produces an alternative output (which
+displays the root multiplicity), and is provided as an
+example of customization.</p>
+<p>As replacement for <a class="reference internal" href="#polprintintervalsknownroot">\PolPrintIntervalsKnownRoot</a>,
+<a class="reference internal" href="#polprintintervalsprintexactzero">\PolPrintIntervalsPrintExactZero</a>,
+<a class="reference internal" href="#polprintintervalsunknownroot">\PolPrintIntervalsUnknownRoot</a> it uses its own
+<tt class="docutils literal"><span class="pre">\POL&#64;&#64;PrintIntervals...</span></tt> macros. We only reproduce here one
+definition:</p>
+<pre class="literal-block">
+\newcommand\POL&#64;&#64;PrintIntervalsPrintExactZero{%
+ \displaystyle
+ \xintSignedFrac{\PolPrintIntervalsTheLeftEndPoint}%
+}%
+</pre>
+<p>Multiplicities are printed using this auxiliary macro:</p>
+</blockquote>
+<div class="section" id="polprintintervalsprintmultiplicity">
+<h3><a class="toc-backref" href="#id96"><tt class="docutils literal">\PolPrintIntervalsPrintMultiplicity</tt></a></h3>
+<blockquote>
+<p>whose default definition is:</p>
+<pre class="literal-block">
+\newcommand\PolPrintIntervalsPrintMultiplicity{(\mbox{mult. }\PolPrintIntervalsTheMultiplicity)}
+</pre>
+</blockquote>
+</div>
+</div>
+<div class="section" id="polmapcoeffs-macro-polname">
+<span id="polmapcoeffs"></span><h2><a class="toc-backref" href="#id97"><tt class="docutils literal"><span class="pre">\PolMapCoeffs{\macro}{polname}</span></tt></a></h2>
+<blockquote>
+<p>It modifies ('in-place': original coefficients get lost) each
+coefficient of the defined polynomial via the <em>expandable</em> macro
+<tt class="docutils literal">\macro</tt>. The degree is adjusted as necessary if some leading
+coefficients vanish after the operation. In replacement text of
+<tt class="docutils literal">\macro</tt>, <tt class="docutils literal">\index</tt> expands to the coefficient index (which is
+defined to be zero for the constant term).</p>
+<p>Notice that <tt class="docutils literal">\macro</tt> will have to handle inputs of the shape
+<tt class="docutils literal">A/B[N]</tt> (<a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> internal notation). This means that it probably
+will have to be expressed in terms of macros from <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> package.</p>
+<p>Example:</p>
+<pre class="literal-block">
+\def\foo#1{\xintMul{#1}{\the\numexpr\index*\index\relax}}
+</pre>
+<p>(or with <tt class="docutils literal"><span class="pre">\xintSqr{\index}</span></tt>) to replace <tt class="docutils literal">n</tt>-th coefficient
+<tt class="docutils literal">f_n</tt> by <tt class="docutils literal">f_n*n^2</tt>.</p>
+</blockquote>
+</div>
+<div class="section" id="polreducecoeffs-polname">
+<span id="polreducecoeffs"></span><h2><a class="toc-backref" href="#id98"><tt class="docutils literal">\PolReduceCoeffs{polname}</tt></a></h2>
+<blockquote>
+About the same as <tt class="docutils literal"><span class="pre">\PolMapCoeffs{\xintIrr}{polname}</span></tt> (but
+maintaining a <tt class="docutils literal">[0]</tt> postfix for speedier <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> parsing when
+polynomial function is used for computations.) This is a
+one-argument macro, working 'in-place'.</blockquote>
+</div>
+<div class="section" id="id21">
+<span id="id20"></span><h2><a class="toc-backref" href="#id99"><tt class="docutils literal"><span class="pre">\PolReduceCoeffs*{polname}</span></tt></a></h2>
+<blockquote>
+<p>This starred variant leaves un-touched the decimal exponent in the
+internal representation of the fractional coefficients, i.e. if a
+coefficient is internally <tt class="docutils literal">A/B[N]</tt>, then <tt class="docutils literal">A/B</tt> is reduced to
+smallest terms, but the <tt class="docutils literal">10^N</tt> part is kept as is. Note: if the
+polynomial is freshly defined directly via <a class="reference internal" href="#polfromcsv">\PolFromCSV</a> its coefficients might still be internally in some
+format like <tt class="docutils literal">1.5e7</tt>; the macro will anyhow always first do the
+needed conversion to strict format <tt class="docutils literal">A/B[N]</tt>.</p>
+<p>Evaluations with polynomials treated by this can be much faster than
+with those handled by the non-starred variant
+<a class="reference internal" href="#polreducecoeffs-polname">\PolReduceCoeffs{polname}</a>: as the numerators and denominators
+remain smaller, this proves very beneficial in favorable cases
+(especially when the coefficients are decimal numbers) to the
+expansion speed of the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> macros used internally by
+<a class="reference internal" href="#polevalat">\PolEval</a>.</p>
+</blockquote>
+</div>
+<div class="section" id="polmakemonic-polname">
+<span id="polmakemonic"></span><h2><a class="toc-backref" href="#id100"><tt class="docutils literal">\PolMakeMonic{polname}</tt></a></h2>
+<blockquote>
+Divides by the leading coefficient. It is recommended to execute
+<a class="reference internal" href="#id21">\PolReduceCoeffs*{polname}</a> immediately afterwards. This is not
+done automatically, due to the case the original polynomial had integer
+coefficients and we want to keep the leading one as common
+denominator.</blockquote>
+</div>
+<div class="section" id="polmakeprimitive-polname">
+<span id="polmakeprimitive"></span><h2><a class="toc-backref" href="#id101"><tt class="docutils literal">\PolMakePrimitive{polname}</tt></a></h2>
+<blockquote>
+Divides by the integer content see (<a class="reference internal" href="#policontent">\PolIContent</a>). This thus produces a polynomial with integer
+coefficients having no common factor. The sign of the leading
+coefficient is not modified.</blockquote>
+</div>
+</div>
+<div class="section" id="expandable-macros">
+<h1><a class="toc-backref" href="#id102">Expandable macros</a></h1>
+<p>All these macros expand completely in two steps except <tt class="docutils literal">\PolToExpr</tt>
+and <tt class="docutils literal">\PolToFloatExpr</tt> (and their auxiliaries) which need a
+<tt class="docutils literal">\write</tt>, <tt class="docutils literal">\edef</tt> or a <tt class="docutils literal"><span class="pre">\csname...\endcsname</span></tt> context.</p>
+<div class="section" id="poleval-polname-atexpr-numerical-expression">
+<span id="polevalatexpr"></span><h2><a class="toc-backref" href="#id103"><tt class="docutils literal"><span class="pre">\PolEval{polname}\AtExpr{numerical</span> expression}</tt></a></h2>
+<blockquote>
+It boils down to
+<tt class="docutils literal">\xinttheexpr polname(numerical <span class="pre">expression)\relax</span></tt>.</blockquote>
+</div>
+<div class="section" id="poleval-polname-at-fraction">
+<span id="polevalat"></span><h2><a class="toc-backref" href="#id104"><tt class="docutils literal"><span class="pre">\PolEval{polname}\At{fraction}</span></tt></a></h2>
+<blockquote>
+Evaluates the polynomial at value <tt class="docutils literal">fraction</tt> which must be in (or
+expand to) a format acceptable to the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> macros.</blockquote>
+</div>
+<div class="section" id="polevalreduced-polname-atexpr-numerical-expression">
+<span id="polevalreducedatexpr"></span><h2><a class="toc-backref" href="#id105"><tt class="docutils literal"><span class="pre">\PolEvalReduced{polname}\AtExpr{numerical</span> expression}</tt></a></h2>
+<blockquote>
+Boils down to <tt class="docutils literal">\xinttheexpr reduce(polname(numerical <span class="pre">expression))\relax</span></tt>.</blockquote>
+</div>
+<div class="section" id="polevalreduced-polname-at-fraction">
+<span id="polevalreducedat"></span><h2><a class="toc-backref" href="#id106"><tt class="docutils literal"><span class="pre">\PolEvalReduced{polname}\At{fraction}</span></tt></a></h2>
+<blockquote>
+Evaluates the polynomial at value <tt class="docutils literal">fraction</tt> which must be in (or
+expand to) a format acceptable to the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> macros, and produce
+an irreducible fraction.</blockquote>
+</div>
+<div class="section" id="polfloateval-polname-atexpr-numerical-expression">
+<span id="polfloatevalatexpr"></span><h2><a class="toc-backref" href="#id107"><tt class="docutils literal"><span class="pre">\PolFloatEval{polname}\AtExpr{numerical</span> expression}</tt></a></h2>
+<blockquote>
+<p>Boils down to <tt class="docutils literal">\xintthefloatexpr polname(numerical <span class="pre">expression)\relax</span></tt>.</p>
+<p>This is done via a Horner Scheme (see <a class="reference internal" href="#poldef">\poldef</a> and
+<a class="reference internal" href="#polgenfloatvariant-polname">\PolGenFloatVariant{polname}</a>), with already rounded
+coefficients. <a class="footnote-reference" href="#id24" id="id22">[2]</a> To use the <em>exact coefficients</em> with <em>exactly
+executed</em> additions and multiplications, just insert it in the float
+expression as in this example: <a class="footnote-reference" href="#id25" id="id23">[3]</a></p>
+<pre class="literal-block">
+\xintthefloatexpr 3.27*\xintexpr f(2.53)\relax^2\relax
+</pre>
+<p>The <tt class="docutils literal">f(2.53)</tt> is exactly computed then rounded at the time of
+getting raised to the power <tt class="docutils literal">2</tt>. Moving the <tt class="docutils literal">^2</tt> inside, that
+operation would also be treated exactly.</p>
+<table class="docutils footnote" frame="void" id="id24" rules="none">
+<colgroup><col class="label" /><col /></colgroup>
+<tbody valign="top">
+<tr><td class="label"><a class="fn-backref" href="#id22">[2]</a></td><td>Anyway each floating point operation starts by rounding its
+operands to the floating point precision.</td></tr>
+</tbody>
+</table>
+<table class="docutils footnote" frame="void" id="id25" rules="none">
+<colgroup><col class="label" /><col /></colgroup>
+<tbody valign="top">
+<tr><td class="label"><a class="fn-backref" href="#id23">[3]</a></td><td>The <tt class="docutils literal">\xintexpr</tt> here could be <tt class="docutils literal">\xinttheexpr</tt> but that
+would be less efficient. Cf. <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> documentation about
+nested expressions.</td></tr>
+</tbody>
+</table>
+</blockquote>
+</div>
+<div class="section" id="polfloateval-polname-at-fraction">
+<span id="polfloatevalat"></span><h2><a class="toc-backref" href="#id108"><tt class="docutils literal"><span class="pre">\PolFloatEval{polname}\At{fraction}</span></tt></a></h2>
+<blockquote>
+Evaluates the polynomial at value <tt class="docutils literal">fraction</tt> which must be in (or
+expand to) a format acceptable to the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> macros, and produces
+a floating point number.</blockquote>
+</div>
+<div class="section" id="polifcoeffisplusorminusone-a-b">
+<span id="polifcoeffisplusorminusone"></span><h2><a class="toc-backref" href="#id109"><tt class="docutils literal"><span class="pre">\PolIfCoeffIsPlusOrMinusOne{A}{B}</span></tt></a></h2>
+<blockquote>
+<p>This macro is a priori undefined.</p>
+<p>It is defined via the default <a class="reference internal" href="#poltypesetcmd-raw-coeff">\PolTypesetCmd{raw_coeff}</a> to be
+used if needed in the execution of <a class="reference internal" href="#poltypesetmonomialcmd">\PolTypesetMonomialCmd</a>,
+e.g. to insert a <tt class="docutils literal">\cdot</tt> in front of <tt class="docutils literal"><span class="pre">\PolVar^{\PolIndex}</span></tt> if
+the coefficient is not plus or minus one.</p>
+<p>The macro will execute <tt class="docutils literal">A</tt> if the coefficient has been found to be
+plus or minus one, and <tt class="docutils literal">B</tt> if not.</p>
+</blockquote>
+</div>
+<div class="section" id="polleadingcoeff-polname">
+<span id="polleadingcoeff"></span><h2><a class="toc-backref" href="#id110"><tt class="docutils literal">\PolLeadingCoeff{polname}</tt></a></h2>
+<blockquote>
+Expands to the leading coefficient.</blockquote>
+</div>
+<div class="section" id="polnthcoeff-polname-number">
+<span id="polnthcoeff"></span><h2><a class="toc-backref" href="#id111"><tt class="docutils literal"><span class="pre">\PolNthCoeff{polname}{number}</span></tt></a></h2>
+<blockquote>
+It expands to the raw <tt class="docutils literal">N</tt>-th coefficient (<tt class="docutils literal">0/1[0]</tt> if the index
+number is out of range). With <tt class="docutils literal"><span class="pre">N=-1</span></tt>, <tt class="docutils literal"><span class="pre">-2</span></tt>, ... expands to the
+leading coefficients.</blockquote>
+</div>
+<div class="section" id="poldegree-polname">
+<span id="poldegree"></span><h2><a class="toc-backref" href="#id112"><tt class="docutils literal">\PolDegree{polname}</tt></a></h2>
+<blockquote>
+It expands to the degree. This is <tt class="docutils literal"><span class="pre">-1</span></tt> if zero polynomial but this
+may change in future. Should it then expand to <tt class="docutils literal"><span class="pre">-\infty</span></tt> ?</blockquote>
+</div>
+<div class="section" id="policontent-polname">
+<span id="policontent"></span><h2><a class="toc-backref" href="#id113"><tt class="docutils literal">\PolIContent{polname}</tt></a></h2>
+<blockquote>
+<p>It expands to the contents of the polynomial, i.e. to the positive
+fraction such that dividing by this fraction produces a polynomial
+with integer coefficients having no common prime divisor.</p>
+<p>See <a class="reference internal" href="#polmakeprimitive">\PolMakePrimitive</a>.</p>
+</blockquote>
+</div>
+<div class="section" id="poltoexpr-polname">
+<span id="poltoexpr"></span><h2><a class="toc-backref" href="#id114"><tt class="docutils literal">\PolToExpr{polname}</tt></a></h2>
+<blockquote>
+<p>Expands <a class="footnote-reference" href="#id27" id="id26">[4]</a> to <tt class="docutils literal"><span class="pre">coeff_N*x^N+...</span></tt> (descending powers.)</p>
+<table class="docutils footnote" frame="void" id="id27" rules="none">
+<colgroup><col class="label" /><col /></colgroup>
+<tbody valign="top">
+<tr><td class="label"><a class="fn-backref" href="#id26">[4]</a></td><td>in a <tt class="docutils literal">\write</tt>, <tt class="docutils literal">\edef</tt>, or <tt class="docutils literal"><span class="pre">\csname...\endcsname</span></tt>, but
+not under <tt class="docutils literal"><span class="pre">\romannumeral-`0</span></tt>.</td></tr>
+</tbody>
+</table>
+<p>By default zero coefficients are skipped (issue <tt class="docutils literal">\poltoexpralltrue</tt> to
+get all of them in output).</p>
+<p>By default, no <tt class="docutils literal">+</tt> sign before negative coefficients, for
+compliance with Maple input format (but see
+<a class="reference internal" href="#poltoexprtermprefix-raw-coeff">\PolToExprTermPrefix{raw_coeff}</a>.) Also, like the default
+behaviour of <a class="reference internal" href="#poltypeset-polname">\PolTypeset{polname}</a>, does not print (for the non
+constant terms) coefficients equal to plus or minus one. The degree
+one monomial is output as <tt class="docutils literal">x</tt>, not <tt class="docutils literal">x^1</tt>. Complete customization is
+possible, see next macros.</p>
+<p>Of course <tt class="docutils literal">\PolToExpr{f}</tt> can be inserted in a <tt class="docutils literal">\poldef</tt>, as the
+latter expands token by token, hence will force complete expansion
+of <tt class="docutils literal">\PolToExpr{f}</tt>, but a simple <tt class="docutils literal">f(x)</tt> is more efficient for
+the identical result.</p>
+</blockquote>
+<div class="section" id="poltoexproneterm-raw-coeff-number">
+<span id="poltoexproneterm"></span><h3><a class="toc-backref" href="#id115"><tt class="docutils literal"><span class="pre">\PolToExprOneTerm{raw_coeff}{number}</span></tt></a></h3>
+<blockquote>
+<p>This two argument expandable command takes care of the monomial and
+its coefficient. The default definition is done in order for
+coefficients of absolute value <tt class="docutils literal">1</tt> not be printed explicitely
+(except of course for the constant term). Also by default, the
+monomial of degree one is <tt class="docutils literal">x</tt> not <tt class="docutils literal">x^1</tt>, and <tt class="docutils literal">x^0</tt> is skipped.</p>
+<p>For compatibility with Maple input requirements, by default a <tt class="docutils literal">*</tt>
+always precedes the <tt class="docutils literal">x^number</tt>, except if the coefficient is a one
+or a minus one. See <a class="reference internal" href="#poltoexprtimes">\PolToExprTimes</a>.</p>
+</blockquote>
+</div>
+<div class="section" id="poltoexpronetermstylea-raw-coeff-number">
+<span id="poltoexpronetermstylea"></span><h3><a class="toc-backref" href="#id116"><tt class="docutils literal"><span class="pre">\PolToExprOneTermStyleA{raw_coeff}{number}</span></tt></a></h3>
+<blockquote>
+Holds the default package meaning of
+<a class="reference internal" href="#poltoexproneterm-raw-coeff-number">\PolToExprOneTerm{raw_coeff}{number}</a>.</blockquote>
+</div>
+<div class="section" id="poltoexpronetermstyleb-raw-coeff-number">
+<span id="poltoexpronetermstyleb"></span><h3><a class="toc-backref" href="#id117"><tt class="docutils literal"><span class="pre">\PolToExprOneTermStyleB{raw_coeff}{number}</span></tt></a></h3>
+<blockquote>
+<p>For output in this style:</p>
+<pre class="literal-block">
+2*x^11/3+3*x^8/7-x^5-x^4/4-x^3-x^2/2-2*x+1
+</pre>
+<p>issue <tt class="docutils literal">\let\PolToExprOneTerm\PolToExprOneTermStyleB</tt> before usage of
+<tt class="docutils literal">\PolToExpr</tt>. Note that then <tt class="docutils literal">\PolToExprCmd</tt> isn't used at all.
+To revert to package default, issue
+<tt class="docutils literal">\let\PolToExprOneTerm\PolToExprOneTermStyleA</tt>.</p>
+<p>To suppress the <tt class="docutils literal">*</tt>'s, cf. <a class="reference internal" href="#poltoexprtimes">\PolToExprTimes</a>.</p>
+</blockquote>
+</div>
+<div class="section" id="poltoexprcmd-raw-coeff">
+<span id="poltoexprcmd"></span><h3><a class="toc-backref" href="#id118"><tt class="docutils literal">\PolToExprCmd{raw_coeff}</tt></a></h3>
+<blockquote>
+It is the one-argument macro used by the package definition of
+<tt class="docutils literal">\PolToExprOneTerm</tt> for the coefficients themselves (when not
+equal to plus or minus one), and it defaults to
+<tt class="docutils literal"><span class="pre">\xintPRaw{\xintRawWithZeros{#1}}</span></tt>. One will have to redefine it
+to <tt class="docutils literal"><span class="pre">\xintIrr{#1}</span></tt> or to <tt class="docutils literal"><span class="pre">\xintPRaw{\xintIrr{#1}}</span></tt> to obtain in the
+output forcefully reduced coefficients.</blockquote>
+</div>
+<div class="section" id="poltoexprtermprefix-raw-coeff">
+<span id="poltoexprtermprefix"></span><h3><a class="toc-backref" href="#id119"><tt class="docutils literal">\PolToExprTermPrefix{raw_coeff}</tt></a></h3>
+<blockquote>
+Defined identically as <a class="reference internal" href="#poltypesetcmdprefix-raw-coeff">\PolTypesetCmdPrefix{raw_coeff}</a>. It
+prefixes with a plus sign for non-negative coefficients, because
+they don't carry one by themselves.</blockquote>
+</div>
+<div class="section" id="id28">
+<span id="poltoexprvar"></span><h3><a class="toc-backref" href="#id120"><tt class="docutils literal">\PolToExprVar</tt></a></h3>
+<blockquote>
+This expands to the variable to use in output (it does not have to
+be a single letter, may be an expandable macro.) Initial definition
+is <tt class="docutils literal">x</tt>.</blockquote>
+</div>
+<div class="section" id="id29">
+<span id="poltoexprtimes"></span><h3><a class="toc-backref" href="#id121"><tt class="docutils literal">\PolToExprTimes</tt></a></h3>
+<blockquote>
+This expands to the symbol used for multiplication of an
+<tt class="docutils literal"><span class="pre">x^{number}</span></tt> by the corresponding coefficient. The default is
+<tt class="docutils literal">*</tt>. Redefine the macro to expand to nothing to get rid of it (but
+this will give output incompatible with some professional computer
+algebra software).</blockquote>
+</div>
+</div>
+<div class="section" id="id31">
+<span id="id30"></span><h2><a class="toc-backref" href="#id122"><tt class="docutils literal"><span class="pre">\PolToExpr*{polname}</span></tt></a></h2>
+<blockquote>
+Expands to <tt class="docutils literal"><span class="pre">coeff_0+coeff_1*x+coeff_2*x^2+...</span></tt> (ascending powers).
+Customizable like <a class="reference internal" href="#poltoexpr-polname">\PolToExpr{polname}</a> via the same macros.</blockquote>
+</div>
+<div class="section" id="poltofloatexpr-polname">
+<span id="poltofloatexpr"></span><h2><a class="toc-backref" href="#id123"><tt class="docutils literal">\PolToFloatExpr{polname}</tt></a></h2>
+<blockquote>
+<p>Similar to <a class="reference internal" href="#poltoexpr-polname">\PolToExpr{polname}</a> but uses <a class="reference external" href="\PolToFloatExprCmd{raw_coeff}">\PolToFloatExprCmd</a>
+which by default rounds and converts the coefficients to floating
+point format.</p>
+<div class="admonition note">
+<p class="first admonition-title">Note</p>
+<p>It is not necessary to have issued
+<a class="reference internal" href="#polgenfloatvariant-polname">\PolGenFloatVariant{polname}</a>. The rounded coefficients are
+not easily recoverable from the <tt class="docutils literal">\xintfloatexpr</tt> polynomial
+function hence <tt class="docutils literal">\PolToFloatExprCmd</tt> operates from the <em>exact</em>
+coefficients anew.</p>
+<p class="last">Attention that both macros obey the prevailing float precision.
+If it is changed between those macro calls, then a mismatch
+exists between the coefficients as used in <tt class="docutils literal">\xintfloatexpr</tt> and
+those output by <tt class="docutils literal">\PolToFloatExpr{polname}</tt>.</p>
+</div>
+</blockquote>
+<div class="section" id="poltofloatexproneterm-raw-coeff-number">
+<span id="poltofloatexproneterm"></span><h3><a class="toc-backref" href="#id124"><tt class="docutils literal"><span class="pre">\PolToFloatExprOneTerm{raw_coeff}{number}</span></tt></a></h3>
+<blockquote>
+Similar to <a class="reference external" href="\PolToExprOneTerm{raw_coeff}{number}">\PolToExprOneTerm</a>. But does not treat
+especially coefficients equal to plus or minus one.</blockquote>
+</div>
+<div class="section" id="poltofloatexprcmd-raw-coeff">
+<span id="id33"></span><h3><a class="toc-backref" href="#id125"><tt class="docutils literal">\PolToFloatExprCmd{raw_coeff}</tt></a></h3>
+<blockquote>
+<p>It is the one-argument macro used by <tt class="docutils literal">\PolToFloatExprOneTerm</tt>.
+Its package definition is <tt class="docutils literal"><span class="pre">\xintFloat{#1}</span></tt>.</p>
+<div class="admonition caution">
+<p class="first admonition-title">Caution!</p>
+<p>Currently (<a class="reference external" href="http://www.ctan.org/pkg/xint">xint</a> <tt class="docutils literal">1.3c</tt>) <tt class="docutils literal">\xintFloat{0}</tt> outputs <tt class="docutils literal">0.e0</tt>
+which is perfectly acceptable input for Python, but not for
+Maple. Thus, one should better leave the <a class="reference internal" href="#poltoexprallfalse">\poltoexprallfalse</a>
+toggle to its default <tt class="docutils literal">\iffalse</tt> state, if one intends to use
+the output in a Maple worksheet.</p>
+<p>But even then the zero polynomial will cause a problem. Workaround:</p>
+<pre class="literal-block">
+\renewcommand\PolToFloatExprCmd[1]{\xintiiifZero{#1}{0.0}{\xintFloat{#1}}}
+</pre>
+<p class="last">Usage of <tt class="docutils literal">\xintiiifZero</tt> and not <tt class="docutils literal">\xintifZero</tt> is only for
+optimization (I can't help it) because <tt class="docutils literal">#1</tt> is known to be
+in <tt class="docutils literal">xintfrac</tt> raw format.</p>
+</div>
+</blockquote>
+</div>
+</div>
+<div class="section" id="id35">
+<span id="id34"></span><h2><a class="toc-backref" href="#id126"><tt class="docutils literal"><span class="pre">\PolToFloatExpr*{polname}</span></tt></a></h2>
+<blockquote>
+Typesets in ascending powers.</blockquote>
+</div>
+<div class="section" id="poltolist-polname">
+<span id="poltolist"></span><h2><a class="toc-backref" href="#id127"><tt class="docutils literal">\PolToList{polname}</tt></a></h2>
+<blockquote>
+Expands to <tt class="docutils literal"><span class="pre">{coeff_0}{coeff_1}...{coeff_N}</span></tt> with <tt class="docutils literal">N</tt> = degree, and
+<tt class="docutils literal">coeff_N</tt> the leading coefficient
+(the zero polynomial does give <tt class="docutils literal">{0/1[0]}</tt> and not an
+empty output.)</blockquote>
+</div>
+<div class="section" id="poltocsv-polname">
+<span id="poltocsv"></span><h2><a class="toc-backref" href="#id128"><tt class="docutils literal">\PolToCSV{polname}</tt></a></h2>
+<blockquote>
+Expands to <tt class="docutils literal">coeff_0, coeff_1, coeff_2, <span class="pre">.....,</span> coeff_N</tt>, starting
+with constant term and ending with leading coefficient. Converse
+to <a class="reference internal" href="#polfromcsv-polname-csv">\PolFromCSV</a>.</blockquote>
+</div>
+<div class="section" id="polsturmchainlength-sturmname">
+<span id="polsturmchainlength"></span><h2><a class="toc-backref" href="#id129"><tt class="docutils literal">\PolSturmChainLength{sturmname}</tt></a></h2>
+<blockquote>
+<p>Returns the integer <tt class="docutils literal">N</tt> such that <tt class="docutils literal">sturmname_N</tt> is the last one
+in the Sturm chain <tt class="docutils literal">sturmname_0</tt>, <tt class="docutils literal">sturmname_1</tt>, ...</p>
+<p>See <a class="reference internal" href="#poltosturm-polname-sturmname">\PolToSturm{polname}{sturmname}</a>.</p>
+</blockquote>
+</div>
+<div class="section" id="polsturmifzeroexactlyknown-sturmname-index-a-b">
+<span id="polsturmifzeroexactlyknown"></span><h2><a class="toc-backref" href="#id130"><tt class="docutils literal"><span class="pre">\PolSturmIfZeroExactlyKnown{sturmname}{index}{A}{B}</span></tt></a></h2>
+<blockquote>
+<p>Executes <tt class="docutils literal">A</tt> if the <tt class="docutils literal">index</tt>-th interval reduces to a singleton,
+i.e. the root is known exactly, else <tt class="docutils literal">B</tt>.</p>
+<div class="admonition note">
+<p class="first admonition-title">Note</p>
+<p class="last"><tt class="docutils literal">index</tt> is allowed to be something like <tt class="docutils literal">1+2*3</tt> as it is fed
+to <tt class="docutils literal"><span class="pre">\the\numexpr...\relax</span></tt>.</p>
+</div>
+</blockquote>
+</div>
+<div class="section" id="polsturmisolatedzeroleft-sturmname-index">
+<span id="polsturmisolatedzeroleft"></span><h2><a class="toc-backref" href="#id131"><tt class="docutils literal"><span class="pre">\PolSturmIsolatedZeroLeft{sturmname}{index}</span></tt></a></h2>
+<blockquote>
+<p>Expands to the left end-point for the <tt class="docutils literal">index</tt>-th interval, as
+computed by some earlier <a class="reference internal" href="#polsturmisolatezeros-sturmname">\PolSturmIsolateZeros{sturmname}</a>.</p>
+<div class="admonition note">
+<p class="first admonition-title">Note</p>
+<p class="last">Of course, this is kept updated by macros such as
+<a class="reference internal" href="#polrefineinterval-n">\PolRefineInterval{sturmname}{index}</a>.</p>
+</div>
+<p>The value is pre-formatted using <a class="reference internal" href="#poldectostring">\PolDecTostring</a>.</p>
+</blockquote>
+</div>
+<div class="section" id="polsturmisolatedzeroright-sturmname-index">
+<span id="polsturmisolatedzeroright"></span><h2><a class="toc-backref" href="#id132"><tt class="docutils literal"><span class="pre">\PolSturmIsolatedZeroRight{sturmname}{index}</span></tt></a></h2>
+<blockquote>
+<p>Expands to the right end-point for the <tt class="docutils literal">index</tt>-th interval as
+computed by some earlier <a class="reference internal" href="#polsturmisolatezeros-sturmname">\PolSturmIsolateZeros{sturmname}</a> and
+possibly refined afterwards.</p>
+<p>The value is pre-formatted using <a class="reference internal" href="#poldectostring">\PolDecTostring</a>.</p>
+</blockquote>
+</div>
+<div class="section" id="polsturmisolatedzeromultiplicity-sturmname-index">
+<span id="polsturmisolatedzeromultiplicity"></span><h2><a class="toc-backref" href="#id133"><tt class="docutils literal"><span class="pre">\PolSturmIsolatedZeroMultiplicity{sturmname}{index}</span></tt></a></h2>
+<blockquote>
+<p>Expands to the multiplicity of the unique root contained in the
+<tt class="docutils literal">index</tt>-th interval.</p>
+<div class="admonition attention">
+<p class="first admonition-title">Attention!</p>
+<p class="last">A prior execution of <a class="reference internal" href="#id12">\PolSturmIsolateZeros*{sturmname}</a> is mandatory.</p>
+</div>
+<p>See <a class="reference internal" href="#the-degree-nine-polynomial-with-0-99-0-999-0-9999-as-triple-roots">The degree nine polynomial with 0.99, 0.999, 0.9999 as triple
+roots</a> for an example of use.</p>
+</blockquote>
+</div>
+<div class="section" id="polsturmnbofisolatedzeros-sturmname">
+<span id="polsturmnbofisolatedzeros"></span><h2><a class="toc-backref" href="#id134"><tt class="docutils literal">\PolSturmNbOfIsolatedZeros{sturmname}</tt></a></h2>
+<blockquote>
+Expands to the number of real roots of the polynomial
+<tt class="docutils literal">&lt;sturmname&gt;_0</tt>, i.e. the number of distinct real roots of the
+polynomial originally used to create the Sturm chain via
+<a class="reference internal" href="#poltosturm-polname-sturmname">\PolToSturm{polname}{sturmname}</a>.</blockquote>
+<div class="admonition warning">
+<p class="first admonition-title">Warning</p>
+<p class="last">The next few macros counting roots, with or without multiplicities,
+less than or equal to some value, are under evaluation and may be
+removed from the package if their utility is judged to be not high
+enough. They can be re-coded at user level on the basis of the other
+documented package macros anyway.</p>
+</div>
+<div class="section" id="polsturmnbofrootsof-sturmname-lessthanorequalto-value">
+<h3><a class="toc-backref" href="#id135"><tt class="docutils literal"><span class="pre">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualTo{value}</span></tt></a></h3>
+<blockquote>
+<p>Expands to the number of distinct roots (of the polynomial used to
+create the Sturm chain) less than or equal to the <tt class="docutils literal">value</tt> (i.e. a
+number of fraction recognizable by the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> macros).</p>
+<div class="admonition attention">
+<p class="first admonition-title">Attention!</p>
+<p><a class="reference internal" href="#polsturmisolatezeros-sturmname">\PolSturmIsolateZeros{sturmname}</a> must have been executed
+beforehand.</p>
+<p class="last">And the argument is a <tt class="docutils literal">sturmname</tt>, not a <tt class="docutils literal">polname</tt> (this is
+why the macro contains Sturm in its name), simply to be reminded
+of the above constraint.</p>
+</div>
+</blockquote>
+</div>
+<div class="section" id="polsturmnbofrootsof-sturmname-lessthanorequaltoexpr-expression">
+<h3><a class="toc-backref" href="#id136"><tt class="docutils literal"><span class="pre">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualToExpr{expression}</span></tt></a></h3>
+<blockquote>
+<p>Expands to the number of distinct roots (of the polynomial
+used to create the Sturm chain) which are less than or equal to the
+given <tt class="docutils literal">expression</tt>.</p>
+<div class="admonition attention">
+<p class="first admonition-title">Attention!</p>
+<p class="last"><a class="reference internal" href="#polsturmisolatezeros-sturmname">\PolSturmIsolateZeros{sturmname}</a> must have been executed
+beforehand.</p>
+</div>
+</blockquote>
+</div>
+<div class="section" id="polsturmnbwithmultofrootsof-sturmname-lessthanorequalto-value">
+<h3><a class="toc-backref" href="#id137"><tt class="docutils literal"><span class="pre">\PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualTo{value}</span></tt></a></h3>
+<blockquote>
+<p>Expands to the number counted with multiplicities of the roots (of
+the polynomial used to create the Sturm chain) which are less than
+or equal to the given <tt class="docutils literal">value</tt>.</p>
+<div class="admonition attention">
+<p class="first admonition-title">Attention!</p>
+<p class="last"><a class="reference internal" href="#id12">\PolSturmIsolateZeros*{sturmname}</a> (or the double starred
+variant) must have been executed beforehand.</p>
+</div>
+</blockquote>
+</div>
+<div class="section" id="polsturmnbwithmultofrootsof-sturmname-lessthanorequaltoexpr-expression">
+<h3><a class="toc-backref" href="#id138"><tt class="docutils literal"><span class="pre">\PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualToExpr{expression}</span></tt></a></h3>
+<blockquote>
+<p>Expands to the total number of roots (counted with multiplicities)
+which are less than or equal to the given <tt class="docutils literal">expression</tt>.</p>
+<div class="admonition attention">
+<p class="first admonition-title">Attention!</p>
+<p class="last"><a class="reference internal" href="#id12">\PolSturmIsolateZeros*{sturmname}</a> (or the double starred
+variant) must have been executed beforehand.</p>
+</div>
+</blockquote>
+</div>
+</div>
+<div class="section" id="polsturmnbofrationalroots-sturmname">
+<h2><a class="toc-backref" href="#id139"><tt class="docutils literal">\PolSturmNbOfRationalRoots{sturmname}</tt></a></h2>
+<blockquote>
+<p>Expands to the number of rational roots (without multiplicities).</p>
+<div class="admonition attention">
+<p class="first admonition-title">Attention!</p>
+<p class="last"><a class="reference internal" href="#id14">\PolSturmIsolateZeros**{sturmname}</a> must have been executed
+beforehand.</p>
+</div>
+</blockquote>
+</div>
+<div class="section" id="polsturmnbofrationalrootswithmultiplicities-sturmname">
+<h2><a class="toc-backref" href="#id140"><tt class="docutils literal">\PolSturmNbOfRationalRootsWithMultiplicities{sturmname}</tt></a></h2>
+<blockquote>
+<p>Expands to the number of rational roots (counted with multiplicities).</p>
+<div class="admonition attention">
+<p class="first admonition-title">Attention!</p>
+<p class="last"><a class="reference internal" href="#id14">\PolSturmIsolateZeros**{sturmname}</a> must have been executed
+beforehand.</p>
+</div>
+</blockquote>
+</div>
+<div class="section" id="polsturmrationalroot-sturmname-k">
+<h2><a class="toc-backref" href="#id141"><tt class="docutils literal"><span class="pre">\PolSturmRationalRoot{sturmname}{k}</span></tt></a></h2>
+<blockquote>
+<p>Expands to the <tt class="docutils literal">k</tt>th rational root (they are ordered and indexed
+starting at 1 for the most negative).</p>
+<div class="admonition attention">
+<p class="first admonition-title">Attention!</p>
+<p class="last"><a class="reference internal" href="#id14">\PolSturmIsolateZeros**{sturmname}</a> must have been executed
+beforehand.</p>
+</div>
+</blockquote>
+</div>
+<div class="section" id="polsturmrationalrootindex-sturmname-k">
+<h2><a class="toc-backref" href="#id142"><tt class="docutils literal"><span class="pre">\PolSturmRationalRootIndex{sturmname}{k}</span></tt></a></h2>
+<blockquote>
+<p>Expands to <tt class="docutils literal">index</tt> of the <tt class="docutils literal">k</tt>th rational root as part of the
+ordered real roots (without multiplicities). I.e., above macro
+<a class="reference internal" href="#polsturmrationalroot-sturmname-k">\PolSturmRationalRoot{sturmname}{k}</a> is equivalent to this
+nested call:</p>
+<pre class="literal-block">
+\PolSturmIsolatedZeroLeft{sturmname}{\PolSturmRationalRootIndex{sturmname}{k}}
+</pre>
+<div class="admonition attention">
+<p class="first admonition-title">Attention!</p>
+<p class="last"><a class="reference internal" href="#id14">\PolSturmIsolateZeros**{sturmname}</a> must have been executed
+beforehand.</p>
+</div>
+</blockquote>
+</div>
+<div class="section" id="polsturmrationalrootmultiplicity-sturmname-k">
+<h2><a class="toc-backref" href="#id143"><tt class="docutils literal"><span class="pre">\PolSturmRationalRootMultiplicity{sturmname}{k}</span></tt></a></h2>
+<blockquote>
+<p>Expands to the multiplicity of the <tt class="docutils literal">k</tt>th rational root.</p>
+<div class="admonition attention">
+<p class="first admonition-title">Attention!</p>
+<p class="last"><a class="reference internal" href="#id14">\PolSturmIsolateZeros**{sturmname}</a> must have been executed
+beforehand.</p>
+</div>
+</blockquote>
+</div>
+<div class="section" id="polintervalwidth-sturmname-index">
+<span id="polintervalwidth"></span><h2><a class="toc-backref" href="#id144"><tt class="docutils literal"><span class="pre">\PolIntervalWidth{sturmname}{index}</span></tt></a></h2>
+<blockquote>
+The <tt class="docutils literal">10^E</tt> width of the current <tt class="docutils literal">index</tt>-th root localization
+interval. Output is in <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> raw <tt class="docutils literal">1/1[E]</tt> format (if not zero).</blockquote>
+</div>
+<div class="section" id="expandable-macros-for-use-within-execution-of-polprintintervals">
+<h2><a class="toc-backref" href="#id145">Expandable macros for use within execution of <tt class="docutils literal">\PolPrintIntervals</tt></a></h2>
+<p>These macros are for usage within custom user redefinitions of
+<a class="reference internal" href="#polprintintervalsknownroot">\PolPrintIntervalsKnownRoot</a>, <a class="reference internal" href="#polprintintervalsunknownroot">\PolPrintIntervalsUnknownRoot</a>, or
+in redefinitions of <a class="reference internal" href="#polprintintervalsprintexactzero">PolPrintIntervalsPrintExactZero</a> (used in the
+default for the former) and of <a class="reference internal" href="#polprintintervalsprintleftendpoint">\PolPrintIntervalsPrintLeftEndPoint</a>,
+<a class="reference internal" href="#polprintintervalsprintrightendpoint">\PolPrintIntervalsPrintRightEndPoint</a> (used in the default for the
+latter).</p>
+<div class="admonition attention">
+<p class="first admonition-title">Attention!</p>
+<p class="last">Some macros formerly mentioned here got removed at 0.7:
+<tt class="docutils literal">\PolPrintIntervalsTheEndPoint</tt>,
+<tt class="docutils literal"><span class="pre">\PolIfEndPointIsPositive{A}{B}</span></tt>,
+<tt class="docutils literal"><span class="pre">\PolIfEndPointIsNegative{A}{B}</span></tt>,
+<tt class="docutils literal"><span class="pre">\PolIfEndPointIsZero{A}{B}</span></tt>.</p>
+</div>
+<div class="section" id="polprintintervalsthevar">
+<h3><a class="toc-backref" href="#id146"><tt class="docutils literal">\PolPrintIntervalsTheVar</tt></a></h3>
+<blockquote>
+Expands to the name (default <tt class="docutils literal">Z</tt>) used for representing the roots,
+which was passed as optional argument <tt class="docutils literal">varname</tt> to
+<a class="reference internal" href="#polprintintervals-varname-sturmname">\PolPrintIntervals[varname]{sturmname}</a>.</blockquote>
+</div>
+<div class="section" id="polprintintervalstheindex">
+<h3><a class="toc-backref" href="#id147"><tt class="docutils literal">\PolPrintIntervalsTheIndex</tt></a></h3>
+<blockquote>
+Expands to the index of the considered interval (indexing starting
+at 1 for the leftmost interval).</blockquote>
+</div>
+<div class="section" id="polprintintervalsthesturmname">
+<h3><a class="toc-backref" href="#id148"><tt class="docutils literal">\PolPrintIntervalsTheSturmName</tt></a></h3>
+<blockquote>
+Expands to the argument which was passed as <tt class="docutils literal">sturmname</tt> to
+<a class="reference internal" href="#polprintintervals-varname-sturmname">\PolPrintIntervals[varname]{sturmname}</a>.</blockquote>
+</div>
+<div class="section" id="polprintintervalstheleftendpoint">
+<h3><a class="toc-backref" href="#id149"><tt class="docutils literal">\PolPrintIntervalsTheLeftEndPoint</tt></a></h3>
+<blockquote>
+The left end point of the interval, as would be produced by
+<a class="reference internal" href="#polsturmisolatedzeroleft">\PolSturmIsolatedZeroLeft</a> if it was
+used with arguments the Sturm chain name and interval index returned
+by <a class="reference internal" href="#polprintintervalsthesturmname">\PolPrintIntervalsTheSturmName</a> and
+<a class="reference internal" href="#polprintintervalstheindex">\PolPrintIntervalsTheIndex</a>.</blockquote>
+</div>
+<div class="section" id="polprintintervalstherightendpoint">
+<h3><a class="toc-backref" href="#id150"><tt class="docutils literal">\PolPrintIntervalsTheRightEndPoint</tt></a></h3>
+<blockquote>
+The right end point of the interval, as would be produced by
+<a class="reference internal" href="#polsturmisolatedzeroright">\PolSturmIsolatedZeroRight</a> for
+this Sturm chain name and index.</blockquote>
+</div>
+<div class="section" id="polprintintervalsthemultiplicity">
+<h3><a class="toc-backref" href="#id151"><tt class="docutils literal">\PolPrintIntervalsTheMultiplicity</tt></a></h3>
+<blockquote>
+The multiplicity of the unique root within the interval of index
+<a class="reference internal" href="#polprintintervalstheindex">\PolPrintIntervalsTheIndex</a>. Makes sense only if the starred (or
+double-starred) variant of <a class="reference internal" href="#polsturmisolatezeros">\PolSturmIsolateZeros</a> was used earlier.</blockquote>
+</div>
+</div>
+<div class="section" id="poldectostring-decimal-number">
+<span id="poldectostring"></span><h2><a class="toc-backref" href="#id152"><tt class="docutils literal">\PolDecToString{decimal number}</tt></a></h2>
+<blockquote>
+<p>This is a utility macro to print decimal numbers. It has been
+backported to <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> (release <tt class="docutils literal">1.3</tt> of <tt class="docutils literal">2018/03/01</tt>) under
+the name <tt class="docutils literal">\xintDecToString</tt>, and the <tt class="docutils literal">polexpr</tt> macro is simply
+now an alias to it.</p>
+<p>For example
+<tt class="docutils literal"><span class="pre">\PolDecToString{123.456e-8}</span></tt> will expand to <tt class="docutils literal">0.00000123456</tt>
+and <tt class="docutils literal"><span class="pre">\PolDecToString{123.450e-8}</span></tt> to <tt class="docutils literal">0.00000123450</tt> which
+illustrates that trailing zeros are not trimmed. To trim trailing
+zeroes, one can use <tt class="docutils literal"><span class="pre">\PolDecToString{\xintREZ{#1}}</span></tt>.</p>
+<p>The precise behaviour of this macro may evolve in future releases of
+<a class="reference external" href="http://www.ctan.org/pkg/xint">xint</a>.</p>
+</blockquote>
+</div>
+</div>
+<div class="section" id="booleans-with-default-setting-as-indicated">
+<h1><a class="toc-backref" href="#id153">Booleans (with default setting as indicated)</a></h1>
+<div class="section" id="xintverbosefalse">
+<h2><a class="toc-backref" href="#id154"><tt class="docutils literal">\xintverbosefalse</tt></a></h2>
+<blockquote>
+<p>This is actually an <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> configuration. Setting it to
+<tt class="docutils literal">true</tt> triggers the writing of information to the log when new
+polynomials are defined.</p>
+<div class="admonition caution">
+<p class="first admonition-title">Caution!</p>
+<p class="last">The macro meanings as written to the log are to be considered
+unstable and undocumented internal structures.</p>
+</div>
+</blockquote>
+</div>
+<div class="section" id="poltypesetallfalse">
+<h2><a class="toc-backref" href="#id155"><tt class="docutils literal">\poltypesetallfalse</tt></a></h2>
+<blockquote>
+If <tt class="docutils literal">true</tt>, <a class="reference internal" href="#poltypeset-polname">\PolTypeset{polname}</a> will also typeset the vanishing
+coefficients.</blockquote>
+</div>
+<div class="section" id="poltoexprallfalse">
+<h2><a class="toc-backref" href="#id156"><tt class="docutils literal">\poltoexprallfalse</tt></a></h2>
+<blockquote>
+If <tt class="docutils literal">true</tt>, <a class="reference internal" href="#poltoexpr-polname">\PolToExpr{polname}</a> and <a class="reference internal" href="#poltofloatexpr-polname">\PolToFloatExpr{polname}</a> will
+also include the vanishing coefficients in their outputs.</blockquote>
+</div>
+</div>
+<div class="section" id="polexprsetup">
+<h1><a class="toc-backref" href="#id157"><tt class="docutils literal">\polexprsetup</tt></a></h1>
+<blockquote>
+<p>Serves to customize the package. Currently only two keys are
+recognized:</p>
+<ul class="simple">
+<li><tt class="docutils literal">norr</tt>: the postfix that <a class="reference internal" href="#id14">\PolSturmIsolateZeros**{sturmname}</a>
+should append to <tt class="docutils literal">sturmname</tt> to declare the primitive polynomial
+obtained from original one after removal of all rational roots.
+The default value is <tt class="docutils literal">_norr</tt> (standing for “no rational roots”).</li>
+<li><tt class="docutils literal">sqfnorr</tt>: the postfix that <a class="reference internal" href="#id14">\PolSturmIsolateZeros**{sturmname}</a>
+should append to <tt class="docutils literal">sturmname</tt> to declare the primitive polynomial
+obtained from original one after removal of all rational roots and
+suppression of all multiplicities.
+The default value is <tt class="docutils literal">_sqf_norr</tt> (standing for “square-free with
+no rational roots”).</li>
+</ul>
+<p>The package executes <tt class="docutils literal">\polexprsetup{norr=_norr,
+sqfnorr=_sqf_norr}</tt> as default.</p>
+</blockquote>
+</div>
+<div class="section" id="technicalities">
+<h1><a class="toc-backref" href="#id158">Technicalities</a></h1>
+<ul>
+<li><p class="first">The catcode of the semi-colon is reset temporarily by <a class="reference internal" href="#poldef">\poldef</a> macro in case some other package (for example the French
+babel module) may have made it active. This will fail though if the
+whole thing was already part of a macro argument, in such cases one
+can use <a class="reference internal" href="#id2">\PolDef{f}{P(x)}</a>
+rather. The colon in <tt class="docutils literal">:=</tt> may be active with no consequences.</p>
+</li>
+<li><p class="first">As a consequence of <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> addition and subtraction always using
+least common multiples for the denominators <a class="footnote-reference" href="#id37" id="id36">[5]</a>, user-chosen common
+denominators survive additions and multiplications. For example, this:</p>
+<pre class="literal-block">
+\poldef P(x):= 1/2 + 2/2*x + 3/2*x^3 + 4/2*x^4;
+\poldef Q(x):= 1/3 + (2/3)x + (3/3)x^3 + (4/3)x^4;
+\poldef PQ(x):= P(x)Q(x);
+</pre>
+<p>gives internally the polynomial:</p>
+<pre class="literal-block">
+1/6+4/6*x^1+4/6*x^2+6/6*x^3+20/6*x^4+16/6*x^5+9/6*x^6+24/6*x^7+16/6*x^8
+</pre>
+<p>where all coefficients have the same denominator 6. Notice though that
+<tt class="docutils literal">\PolToExpr{PQ}</tt> outputs the <tt class="docutils literal">6/6*x^3</tt> as <tt class="docutils literal">x^3</tt> because (by
+default) it recognizes and filters out coefficients equal to one or
+minus one (since release <tt class="docutils literal">0.3</tt>). One can use for example
+<tt class="docutils literal">\PolToCSV{PQ}</tt> to see the internally stored coefficients.</p>
+<table class="docutils footnote" frame="void" id="id37" rules="none">
+<colgroup><col class="label" /><col /></colgroup>
+<tbody valign="top">
+<tr><td class="label"><a class="fn-backref" href="#id36">[5]</a></td><td><p class="first last">prior to <tt class="docutils literal">0.4.1</tt>, <tt class="docutils literal">polexpr</tt> used to temporarily patch
+during the parsing of polynomials the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> macros. This
+patch was backported to <a class="reference external" href="http://www.ctan.org/pkg/xint">xint</a> at release <tt class="docutils literal">1.3</tt>.</p>
+</td></tr>
+</tbody>
+</table>
+</li>
+<li><p class="first"><a class="reference internal" href="#poldiff-polname-1-polname-2">\PolDiff{polname_1}{polname_2}</a> always applies <tt class="docutils literal">\xintIrr</tt> to the
+resulting coefficients, except that the <em>power of ten</em> part <tt class="docutils literal">[N]</tt>
+(for example an input in scientific notation such as <tt class="docutils literal">1.23e5</tt> gives
+<tt class="docutils literal">123/1[3]</tt> internally in xintfrac) is not taken into account in the
+reduction of the fraction. This is tentative and may change.</p>
+<p>Same remark for <a class="reference internal" href="#polantidiff-polname-1-polname-2">\PolAntiDiff{polname_1}{polname_2}</a>.</p>
+</li>
+<li><p class="first">Currently, the package stores all coefficients from index <tt class="docutils literal">0</tt> to
+index equal to the polynomial degree inside a single macro, as a list.
+This data structure is obviously very inefficient for polynomials of
+high degree and few coefficients (as an example with <tt class="docutils literal">\poldef
+<span class="pre">f(x):=x^1000</span> + x^500;</tt> the subsequent definition <tt class="docutils literal">\poldef <span class="pre">g(x):=</span>
+<span class="pre">f(x)^2;</span></tt> will do of the order of 1,000,000 multiplications and
+additions involvings only zeroes... which does take time). This
+may change in the future.</p>
+</li>
+<li><p class="first">As is to be expected internal structures of the package are barely
+documented and unstable. Don't use them.</p>
+</li>
+</ul>
+</div>
+<div class="section" id="change-log">
+<h1><a class="toc-backref" href="#id159">CHANGE LOG</a></h1>
+<ul>
+<li><p class="first">v0.1 (2018/01/11): initial release. Features:</p>
+<ul class="simple">
+<li>The <a class="reference internal" href="#poldef">\poldef</a> parser itself,</li>
+<li>Differentiation and anti-differentiation,</li>
+<li>Euclidean division and GCDs,</li>
+<li>Various utilities such as <a class="reference internal" href="#polfromcsv">\PolFromCSV</a>,
+<a class="reference internal" href="#polmapcoeffs">\PolMapCoeffs</a>,
+<a class="reference internal" href="#poltocsv">\PolToCSV</a>, <a class="reference internal" href="#poltoexpr">\PolToExpr</a>, ...</li>
+</ul>
+<p>Only one-variable polynomials so far.</p>
+</li>
+<li><p class="first">v0.2 (2018/01/14)</p>
+<ul class="simple">
+<li>Fix: <tt class="docutils literal">&quot;README thinks \numexpr recognizes ^ operator&quot;</tt>.</li>
+<li>Convert README to reStructuredText markup.</li>
+<li>Move main documentation from README to separate <tt class="docutils literal">polexpr.txt</tt> file.</li>
+<li>Provide <tt class="docutils literal">polexpr.html</tt> as obtained via <a class="reference external" href="http://docutils.sourceforge.net/docs/index.html">DocUtils</a> <tt class="docutils literal">rst2html.py</tt>.</li>
+<li>Convert README to (CTAN compatible) Markdown markup.</li>
+</ul>
+<p>Due to lack of available time the test suite might not be extensive
+enough. Bug reports are very welcome!</p>
+</li>
+<li><p class="first">v0.3 (2018/01/17)</p>
+<ul>
+<li><p class="first">bug fixes:</p>
+<ul>
+<li><p class="first">the <tt class="docutils literal">0.1</tt> <a class="reference internal" href="#polevalat">\PolEval</a> accepted expressions for its second
+argument, but this was removed by mistake at <tt class="docutils literal">0.2</tt>. Restored.</p>
+<p><strong>Attention</strong>: at <tt class="docutils literal">0.4</tt> this has been reverted again, and
+<a class="reference internal" href="#polevalatexpr">\PolEval{P}\AtExpr{foo}</a> syntax is needed for
+using expressions in the second argument.</p>
+</li>
+</ul>
+</li>
+<li><p class="first">incompatible or breaking changes:</p>
+<ul class="simple">
+<li><a class="reference internal" href="#poltoexpr">\PolToExpr</a> now by default uses <em>descending</em>
+powers (it also treats differently coefficients equal to 1 or -1.)
+Use <a class="reference internal" href="#id30">\PolToExpr*</a> for <em>ascending</em> powers.</li>
+<li><a class="reference internal" href="#polevalat">\PolEval</a> reduced the output to smallest terms,
+but as this is costly with big fractions and not needed if e.g.
+wrapped in an <tt class="docutils literal">\xintRound</tt> or <tt class="docutils literal">\xintFloat</tt>, this step has been
+removed; the former meaning is available as <a class="reference internal" href="#polevalreducedat">\PolEvalReduced</a>.</li>
+</ul>
+</li>
+<li><p class="first">new (or newly documented) macros:</p>
+<ul class="simple">
+<li><a class="reference internal" href="#poltypesetcmd">\PolTypesetCmd</a></li>
+<li><a class="reference internal" href="#poltypesetcmdprefix">\PolTypesetCmdPrefix</a></li>
+<li><a class="reference internal" href="#poltypesetmonomialcmd">\PolTypesetMonomialCmd</a></li>
+<li><a class="reference internal" href="#polevalreducedat">\PolEvalReducedAt</a></li>
+<li><a class="reference internal" href="#poltofloatexpr">\PolToFloatExpr</a></li>
+<li><a class="reference internal" href="#poltoexproneterm">\PolToExprOneTerm</a></li>
+<li><a class="reference internal" href="#poltofloatexproneterm">\PolToFloatExprOneTerm</a></li>
+<li><a class="reference internal" href="#poltoexprcmd">\PolToExprCmd</a></li>
+<li><a class="reference internal" href="#id33">\PolToFloatExprCmd</a></li>
+<li><a class="reference internal" href="#poltoexprtermprefix">\PolToExprTermPrefix</a></li>
+<li><a class="reference internal" href="#poltoexprvar">\PolToExprVar</a></li>
+<li><a class="reference internal" href="#poltoexprtimes">\PolToExprTimes</a></li>
+</ul>
+</li>
+<li><p class="first">improvements:</p>
+<ul>
+<li><p class="first">documentation has a table of contents, internal hyperlinks,
+standardized signature notations and added explanations.</p>
+</li>
+<li><p class="first">one can do <tt class="docutils literal"><span class="pre">\PolLet{g}={f}</span></tt> or <tt class="docutils literal"><span class="pre">\PolLet{g}{f}</span></tt>.</p>
+</li>
+<li><p class="first"><tt class="docutils literal">\PolToExpr{f}</tt> is highly customizable.</p>
+</li>
+<li><p class="first"><a class="reference internal" href="#poldef">\poldef</a> and other defining macros prepare the polynomial
+functions for usage within <tt class="docutils literal">\xintthefloatexpr</tt> (or
+<tt class="docutils literal">\xintdeffloatvar</tt>). Coefficients are pre-rounded to the
+floating point precision. Indispensible for numerical algorithms,
+as exact fractions, even reduced, quickly become very big. See the
+documentation about how to use the exact polynomials also in
+floating point context.</p>
+<p><strong>Attention</strong>: this has been reverted at <tt class="docutils literal">0.4</tt>. The macro
+<a class="reference internal" href="#polgenfloatvariant">\PolGenFloatVariant</a> must be used for
+generation floating point polynomial functions.</p>
+</li>
+</ul>
+</li>
+</ul>
+</li>
+<li><p class="first">v0.3.1 (2018/01/18)</p>
+<p>Fixes two typos in example code included in the documentation.</p>
+</li>
+<li><p class="first">v0.4 (2018/02/16)</p>
+<ul>
+<li><p class="first">bug fixes:</p>
+<ul class="simple">
+<li>when Euclidean division gave a zero remainder, the internal
+representation of this zero polynomial could be faulty; this
+could cause mysterious bugs in conjunction with other package
+macros such as <a class="reference internal" href="#polmapcoeffs">\PolMapCoeffs</a>.</li>
+<li><a class="reference internal" href="#polgcd">\PolGCD</a> was buggy in case of first polynomial being
+of lesser degree than the second one.</li>
+</ul>
+</li>
+<li><p class="first">breaking changes:</p>
+<ul>
+<li><p class="first">formerly <a class="reference internal" href="#polevalat">\PolEval{P}\At{foo}</a> allowed <tt class="docutils literal">foo</tt> to
+be an expression, which was transparently handled via
+<tt class="docutils literal">\xinttheexpr</tt>. Now, <tt class="docutils literal">foo</tt> must be a fraction (or a macro
+expanding to such) in the format acceptable by <tt class="docutils literal">xintfrac.sty</tt>
+macros. Use <a class="reference internal" href="#polevalatexpr">\PolEval{P}\AtExpr{foo}</a> for more
+general arguments using expression syntax. E.g., if <tt class="docutils literal">foo</tt> is the
+name of a variable known to <tt class="docutils literal">\xintexpr</tt>.</p>
+<p>The same holds for <a class="reference internal" href="#polevalreducedat">\PolEvalReduced</a>
+and <a class="reference internal" href="#polfloatevalat">\PolFloatEval</a>.</p>
+</li>
+<li><p class="first">the <tt class="docutils literal">3.0</tt> automatic generation of floating point variants has
+been reverted. Not only do <em>not</em> the package macros automatically
+generate floating point variants of newly created polynomials,
+they actually make pre-existing such variant undefined.</p>
+<p>See <a class="reference internal" href="#polgenfloatvariant">\PolGenFloatVariant</a>.</p>
+</li>
+</ul>
+</li>
+<li><p class="first">new non-expandable macros:</p>
+<ul class="simple">
+<li><a class="reference internal" href="#polgenfloatvariant">\PolGenFloatVariant</a></li>
+<li><a class="reference internal" href="#polgloballet">\PolGlobalLet</a></li>
+<li><a class="reference internal" href="#poltypesetone">\PolTypesetOne</a></li>
+<li><a class="reference internal" href="#polquo">\PolQuo</a></li>
+<li><a class="reference internal" href="#polrem">\PolRem</a></li>
+<li><a class="reference internal" href="#poltosturm">\PolToSturm</a></li>
+<li><a class="reference internal" href="#id9">\PolToSturm*</a></li>
+<li><a class="reference internal" href="#polsettosturmchainsignchangesat">\PolSetToSturmChainSignChangesAt</a></li>
+<li><a class="reference internal" href="#polsettonbofzeroswithin">\PolSetToNbOfZerosWithin</a></li>
+<li><a class="reference internal" href="#polsturmisolatezeros">\PolSturmIsolateZeros</a></li>
+<li><a class="reference internal" href="#polrefineinterval">\PolRefineInterval*</a></li>
+<li><a class="reference internal" href="#polrefineinterval-n">\PolRefineInterval[N]</a></li>
+<li><a class="reference internal" href="#polensureintervallength">\PolEnsureIntervalLength</a></li>
+<li><a class="reference internal" href="#polensureintervallengths">\PolEnsureIntervalLengths</a></li>
+<li><a class="reference internal" href="#polprintintervals">\PolPrintIntervals</a></li>
+<li><a class="reference internal" href="#polprintintervalsprintexactzero">\PolPrintIntervalsPrintExactZero</a></li>
+<li><a class="reference internal" href="#polprintintervalsprintleftendpoint">\PolPrintIntervalsPrintLeftEndPoint</a></li>
+<li><a class="reference internal" href="#polprintintervalsprintrightendpoint">\PolPrintIntervalsPrintRightEndPoint</a></li>
+<li><a class="reference internal" href="#id20">\PolReduceCoeffs*</a></li>
+<li><a class="reference internal" href="#polmakemonic">\PolMakeMonic</a></li>
+</ul>
+</li>
+<li><p class="first">new expandable macros:</p>
+<ul class="simple">
+<li><a class="reference internal" href="#poltoexpronetermstylea">\PolToExprOneTermStyleA</a></li>
+<li><a class="reference internal" href="#polifcoeffisplusorminusone">\PolIfCoeffIsPlusOrMinusOne</a></li>
+<li><a class="reference internal" href="#polleadingcoeff">\PolLeadingCoeff</a></li>
+<li><a class="reference internal" href="#polsturmchainlength">\PolSturmChainLength</a></li>
+<li><a class="reference internal" href="#polsturmnbofisolatedzeros">\PolSturmNbOfIsolatedZeros</a></li>
+<li><a class="reference internal" href="#polsturmifzeroexactlyknown">\PolSturmIfZeroExactlyKnown</a></li>
+<li><a class="reference internal" href="#polsturmisolatedzeroleft">\PolSturmIsolatedZeroLeft</a></li>
+<li><a class="reference internal" href="#polsturmisolatedzeroright">\PolSturmIsolatedZeroRight</a></li>
+<li><tt class="docutils literal">\PolPrintIntervalsTheEndPoint</tt> (removed at 0.7)</li>
+<li><a class="reference internal" href="#polprintintervalstheindex">\PolPrintIntervalsTheIndex</a></li>
+<li><tt class="docutils literal">\PolIfEndPointIsPositive</tt> (removed at 0.7)</li>
+<li><tt class="docutils literal">\PolIfEndPointIsNegative</tt> (removed at 0.7)</li>
+<li><tt class="docutils literal">\PolIfEndPointIsZero</tt> (removed at 0.7)</li>
+<li><a class="reference internal" href="#polintervalwidth">\PolIntervalWidth</a></li>
+<li><a class="reference internal" href="#poldectostring">\PolDecToString</a></li>
+</ul>
+</li>
+<li><p class="first">improvements:</p>
+<p>The main new feature is implementation of the <a class="reference external" href="https://en.wikipedia.org/wiki/Sturm%27s_theorem">Sturm algorithm</a>
+for localization of the real roots of polynomials.</p>
+</li>
+</ul>
+</li>
+<li><p class="first">v0.4.1 (2018/03/01)</p>
+<p>Synced with xint 1.3.</p>
+</li>
+<li><p class="first">v0.4.2 (2018/03/03)</p>
+<p>Documentation fix.</p>
+</li>
+<li><p class="first">v0.5 (2018/04/08)</p>
+<ul class="simple">
+<li>bug fixes:<ul>
+<li><a class="reference internal" href="#polget-polname-fromarray-macro">\PolGet{polname}\fromarray\macro</a> crashed when <tt class="docutils literal">\macro</tt> was
+an <a class="reference external" href="http://www.ctan.org/pkg/xint">xinttools</a> array macro with no items. It now produces the zero
+polynomial.</li>
+</ul>
+</li>
+<li>breaking changes:<ul>
+<li><a class="reference internal" href="#poltosturm">\PolToSturm</a> creates primitive integer coefficients polynomials.
+This speeds up localization of roots via
+<a class="reference internal" href="#polsturmisolatezeros">\PolSturmIsolateZeros</a>. In case of user protests the author
+will make available again the code producing the bona fide Sturm
+polynomials as used formerly.</li>
+<li>polynomials created from <a class="reference internal" href="#polfromcsv">\PolFromCSV</a> or <a class="reference internal" href="#polget">\PolGet</a>
+get their coefficients normalized via <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a>'s <tt class="docutils literal">\xintRaw</tt>.</li>
+</ul>
+</li>
+<li>experimental change:<ul>
+<li>optional argument to <a class="reference internal" href="#polsturmisolatezeros">\PolSturmIsolateZeros</a> (see <a class="reference internal" href="#the-degree-41-polynomial-with-2-1-9-1-8-0-0-1-1-9-2-as-roots">The
+degree 41 polynomial with -2, -1.9, -1.8, ..., 0, 0.1, ..., 1.9, 2
+as roots</a> for usage). It will presumably be replaced in future by
+an interval specification.</li>
+</ul>
+</li>
+<li>new non-expandable macro:<ul>
+<li><a class="reference internal" href="#polmakeprimitive">\PolMakePrimitive</a></li>
+</ul>
+</li>
+<li>new expandable macro:<ul>
+<li><a class="reference internal" href="#policontent">\PolIContent</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li><p class="first">v0.5.1 (2018/04/22)</p>
+<ul class="simple">
+<li>new feature:<ul>
+<li>the character <tt class="docutils literal">'</tt> can be used in polynomial names.</li>
+</ul>
+</li>
+</ul>
+</li>
+<li><p class="first">v0.6 (2018/11/20)</p>
+<ul class="simple">
+<li>bugfix:<ul>
+<li>the starred variant <a class="reference internal" href="#id10">\PolToSturm*{polname}{sturmname}</a> was
+broken. On the occasion of the fix, its meaning has been modified,
+see its documentation.</li>
+<li>using <a class="reference internal" href="#poltosturm">\PolToSturm</a> with a constant polynomial
+caused a division by zero error.</li>
+</ul>
+</li>
+<li>new macro:<ul>
+<li><a class="reference internal" href="#id11">\PolSturmIsolateZeros*</a>
+acts like the <a class="reference internal" href="#polsturmisolatezeros">non-starred variant</a> then computes all the multiplicities.</li>
+</ul>
+</li>
+<li>new expandable macros:<ul>
+<li><a class="reference internal" href="#polsturmisolatedzeromultiplicity-sturmname-index">\PolSturmIsolatedZeroMultiplicity{sturmname}{index}</a></li>
+<li><a class="reference internal" href="#polsturmnbofrootsof-sturmname-lessthanorequalto-value">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualTo{value}</a></li>
+<li><a class="reference internal" href="#polsturmnbofrootsof-sturmname-lessthanorequaltoexpr-expression">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualToExpr{expression}</a></li>
+<li><a class="reference internal" href="#polsturmnbwithmultofrootsof-sturmname-lessthanorequalto-value">\PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualTo{value}</a></li>
+<li><a class="reference internal" href="#polsturmnbwithmultofrootsof-sturmname-lessthanorequaltoexpr-expression">\PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualToExpr{expression}</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li><p class="first">v0.7 (2018/12/08), v0.7.1 (bugfix), v0.7.2 (2nd bugfix) (2018/12/09)</p>
+<ul class="simple">
+<li>breaking changes:<ul>
+<li>although <a class="reference internal" href="#polprintintervals-varname-sturmname">\PolPrintIntervals[varname]{sturmname}</a> default output
+remains the same, some auxiliary macros for user-customization
+have been removed: <tt class="docutils literal">\PolPrintIntervalsTheEndPoint</tt>,
+<tt class="docutils literal"><span class="pre">\PolIfEndPointIsPositive{A}{B}</span></tt>,
+<tt class="docutils literal"><span class="pre">\PolIfEndPointIsNegative{A}{B}</span></tt>, and
+<tt class="docutils literal"><span class="pre">\PolIfEndPointIsZero{A}{B}</span></tt>.</li>
+</ul>
+</li>
+<li>bugfix:<ul>
+<li>it could happen that, contrarily to documentation, an interval
+computed by <a class="reference internal" href="#polsturmisolatezeros-sturmname">\PolSturmIsolateZeros{sturmname}</a> had zero as an
+endpoint,</li>
+<li><a class="reference internal" href="#polensureintervallength-sturmname-index-e">\PolEnsureIntervalLength{sturmname}{index}{E}</a> could under
+certain circumstances erroneously replace a non-zero root by
+zero,</li>
+<li><a class="reference internal" href="#polensureintervallengths-sturmname-e">\PolEnsureIntervalLengths{sturmname}{E}</a> crashed when used with
+a polynomial with no real roots, hence for which no isolation intervals
+existed (thanks to Thomas Söll for report).</li>
+</ul>
+</li>
+<li>new macros:<ul>
+<li><a class="reference internal" href="#id14">\PolSturmIsolateZeros**{sturmname}</a></li>
+<li><a class="reference internal" href="#polsturmisolatezerosgetmultiplicitiesandrationalroots-sturmname">\PolSturmIsolateZerosGetMultiplicitiesAndRationalRoots{sturmname}</a></li>
+<li><a class="reference internal" href="#polsturmisolatezerosandfindrationalroots-sturmname">\PolSturmIsolateZerosAndFindRationalRoots{sturmname}</a></li>
+<li><a class="reference internal" href="#polexprsetup">\polexprsetup</a></li>
+<li><a class="reference internal" href="#id18">\PolPrintIntervals*</a></li>
+<li><a class="reference internal" href="#polprintintervalsnorealroots">\PolPrintIntervalsNoRealRoots</a></li>
+<li><a class="reference internal" href="#polprintintervalsbeginenv">\PolPrintIntervalsBeginEnv</a></li>
+<li><a class="reference internal" href="#polprintintervalsendenv">\PolPrintIntervalsEndEnv</a></li>
+<li><a class="reference internal" href="#polprintintervalsknownroot">\PolPrintIntervalsKnownRoot</a></li>
+<li><a class="reference internal" href="#polprintintervalsunknownroot">\PolPrintIntervalsUnknownRoot</a></li>
+<li><a class="reference internal" href="#polprintintervalsprintmultiplicity">\PolPrintIntervalsPrintMultiplicity</a></li>
+</ul>
+</li>
+<li>new expandable macros:<ul>
+<li><a class="reference internal" href="#polsturmnbofrationalroots-sturmname">\PolSturmNbOfRationalRoots{sturmname}</a></li>
+<li><a class="reference internal" href="#polsturmnbofrationalrootswithmultiplicities-sturmname">\PolSturmNbOfRationalRootsWithMultiplicities{sturmname}</a></li>
+<li><a class="reference internal" href="#polsturmrationalroot-sturmname-k">\PolSturmRationalRoot{sturmname}{k}</a></li>
+<li><a class="reference internal" href="#polsturmrationalrootindex-sturmname-k">\PolSturmRationalRootIndex{sturmname}{k}</a></li>
+<li><a class="reference internal" href="#polsturmrationalrootmultiplicity-sturmname-k">\PolSturmRationalRootMultiplicity{sturmname}{k}</a></li>
+<li><a class="reference internal" href="#polprintintervalsthevar">\PolPrintIntervalsTheVar</a></li>
+<li><a class="reference internal" href="#polprintintervalsthesturmname">\PolPrintIntervalsTheSturmName</a></li>
+<li><a class="reference internal" href="#polprintintervalsthemultiplicity">\PolPrintIntervalsTheMultiplicity</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li><p class="first">v0.7.3 (2019/02/04)</p>
+<ul class="simple">
+<li>bugfix:<ul>
+<li>Debugging information not destined to user showed in log if root
+finding was done under <tt class="docutils literal">\xintverbosetrue</tt> regime.</li>
+<li><a class="reference internal" href="#polprintintervalsthevar">\PolPrintIntervalsTheVar</a> remained defined after
+<a class="reference internal" href="#polprintintervals">\PolPrintIntervals</a> but was left undefined after
+<a class="reference internal" href="#id18">\PolPrintIntervals*</a> (reported by Jürgen Gilg). Now remains
+defined in both cases, and <a class="reference internal" href="#polprintintervalsthesturmname">\PolPrintIntervalsTheSturmName</a>
+also.</li>
+<li>Polynomial names ending in digits caused errors (reported by Thomas
+Söll).</li>
+</ul>
+</li>
+</ul>
+</li>
+<li><p class="first">v0.7.4 (2019/02/12)</p>
+<ul class="simple">
+<li>bugfix:<ul>
+<li>20000000000 is too big for <tt class="docutils literal">\numexpr</tt>, shouldn't I know that?
+Thanks to Jürgen Gilg for report.</li>
+</ul>
+</li>
+</ul>
+</li>
+</ul>
+</div>
+<div class="section" id="acknowledgments">
+<h1><a class="toc-backref" href="#id160">Acknowledgments</a></h1>
+<p>Thanks to Jürgen Gilg whose question about <a class="reference external" href="http://www.ctan.org/pkg/xint">xint</a> usage for
+differentiating polynomials was the initial trigger leading to this
+package, and to Jürgen Gilg and Thomas Söll for testing it on some
+concrete problems.</p>
+<p>Renewed thanks to them on occasion of the <tt class="docutils literal">0.6</tt> and <tt class="docutils literal">0.7</tt> releases for their
+continued interest.</p>
+<p>See README.md for the License.</p>
+</div>
+</div>
+</body>
+</html>