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@@ -362,76 +362,128 @@ ul.auto-toc {
<body>
<div class="document" id="package-polexpr-documentation">
<h1 class="title">Package polexpr documentation</h1>
-<h2 class="subtitle" id="id1">0.3.1 (2018/01/18)</h2>
+<h2 class="subtitle" id="id1">0.4 (2018/02/16)</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="#first-examples" id="id15">First Examples</a></li>
-<li><a class="reference internal" href="#non-expandable-macros" id="id16">Non-expandable macros</a><ul>
-<li><a class="reference internal" href="#poldef-polname-letter-expression-in-letter" id="id17"><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="id18"><tt class="docutils literal"><span class="pre">\PolDef[letter]{polname}{expression</span> in letter}</tt></a></li>
-<li><a class="reference internal" href="#pollet-polname-2-polname-1" id="id19"><tt class="docutils literal"><span class="pre">\PolLet{polname_2}={polname_1}</span></tt></a></li>
-<li><a class="reference internal" href="#polassign-polname-toarray-macro" id="id20"><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="id21"><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="id22"><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="id23"><tt class="docutils literal">\PolTypeset{polname}</tt></a><ul>
-<li><a class="reference internal" href="#poltypesetcmd-raw-coeff" id="id24"><tt class="docutils literal">\PolTypesetCmd{raw_coeff}</tt></a></li>
-<li><a class="reference internal" href="#poltypesetcmdprefix-raw-coeff" id="id25"><tt class="docutils literal">\PolTypesetCmdPrefix{raw_coeff}</tt></a></li>
-<li><a class="reference internal" href="#poltypesetmonomialcmd" id="id26"><tt class="docutils literal">\PolTypesetMonomialCmd</tt></a></li>
+<li><a class="reference internal" href="#basic-examples" id="id33">Basic Examples</a></li>
+<li><a class="reference internal" href="#examples-of-localization-of-roots" id="id34">Examples of localization of roots</a><ul>
+<li><a class="reference internal" href="#a-typical-example" id="id35">A typical example</a></li>
+<li><a class="reference internal" href="#a-degree-four-polynomial-with-nearby-roots" id="id36">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="id37">The degree nine polynomial with 0.99, 0.999, 0.9999 as triple roots</a></li>
+<li><a class="reference internal" href="#a-mignotte-type-polynomial" id="id38">A Mignotte type 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="id39">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="id40">Roots of Chebyshev polynomials</a></li>
</ul>
</li>
-<li><a class="reference internal" href="#id5" id="id27"><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="id28"><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="id29"><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="id30"><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="id31"><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="id32"><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="#polgcd-polname-1-polname-2-polname-gcd" id="id33"><tt class="docutils literal"><span class="pre">\PolGCD{polname_1}{polname_2}{polname_GCD}</span></tt></a></li>
-<li><a class="reference internal" href="#polmapcoeffs-macro-polname" id="id34"><tt class="docutils literal"><span class="pre">\PolMapCoeffs{\macro}{polname}</span></tt></a></li>
-<li><a class="reference internal" href="#polreducecoeffs-polname" id="id35"><tt class="docutils literal">\PolReduceCoeffs{polname}</tt></a></li>
+<li><a class="reference internal" href="#non-expandable-macros" id="id41">Non-expandable macros</a><ul>
+<li><a class="reference internal" href="#poldef-polname-letter-expression-in-letter" id="id42"><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="id43"><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="id44"><tt class="docutils literal">\PolGenFloatVariant{polname}</tt></a></li>
+<li><a class="reference internal" href="#pollet-polname-2-polname-1" id="id45"><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="id46"><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="id47"><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="id48"><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="id49"><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="id50"><tt class="docutils literal">\PolTypeset{polname}</tt></a><ul>
+<li><a class="reference internal" href="#poltypesetcmd-raw-coeff" id="id51"><tt class="docutils literal">\PolTypesetCmd{raw_coeff}</tt></a></li>
+<li><a class="reference internal" href="#poltypesetone-raw-coeff" id="id52"><tt class="docutils literal">\PolTypesetOne{raw_coeff}</tt></a></li>
+<li><a class="reference internal" href="#id6" id="id53"><tt class="docutils literal">\PolTypesetMonomialCmd</tt></a></li>
+<li><a class="reference internal" href="#poltypesetcmdprefix-raw-coeff" id="id54"><tt class="docutils literal">\PolTypesetCmdPrefix{raw_coeff}</tt></a></li>
</ul>
</li>
-<li><a class="reference internal" href="#expandable-macros" id="id36">Expandable macros</a><ul>
-<li><a class="reference internal" href="#poleval-polname-at-numerical-expression" id="id37"><tt class="docutils literal"><span class="pre">\PolEval{polname}\At{numerical</span> expression}</tt></a></li>
-<li><a class="reference internal" href="#polevalreduced-polname-at-numerical-expression" id="id38"><tt class="docutils literal"><span class="pre">\PolEvalReduced{polname}\At{numerical</span> expression}</tt></a></li>
-<li><a class="reference internal" href="#polfloateval-polname-at-numerical-expression" id="id39"><tt class="docutils literal"><span class="pre">\PolFloatEval{polname}\At{numerical</span> expression}</tt></a></li>
-<li><a class="reference internal" href="#polnthcoeff-polname-number" id="id40"><tt class="docutils literal"><span class="pre">\PolNthCoeff{polname}{number}</span></tt></a></li>
-<li><a class="reference internal" href="#poldegree-polname" id="id41"><tt class="docutils literal">\PolDegree{polname}</tt></a></li>
-<li><a class="reference internal" href="#poltoexpr-polname" id="id42"><tt class="docutils literal">\PolToExpr{polname}</tt></a><ul>
-<li><a class="reference internal" href="#poltoexproneterm-raw-coeff-number" id="id43"><tt class="docutils literal"><span class="pre">\PolToExprOneTerm{raw_coeff}{number}</span></tt></a></li>
-<li><a class="reference internal" href="#poltoexpronetermstyleb-raw-coeff-number" id="id44"><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="id45"><tt class="docutils literal">\PolToExprCmd{raw_coeff}</tt></a></li>
-<li><a class="reference internal" href="#poltoexprtermprefix-raw-coeff" id="id46"><tt class="docutils literal">\PolToExprTermPrefix{raw_coeff}</tt></a></li>
-<li><a class="reference internal" href="#poltoexprvar" id="id47"><tt class="docutils literal">\PolToExprVar</tt></a></li>
-<li><a class="reference internal" href="#poltoexprtimes" id="id48"><tt class="docutils literal">\PolToExprTimes</tt></a></li>
+<li><a class="reference internal" href="#id8" id="id55"><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="id56"><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="id57"><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="id58"><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="id59"><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="id60"><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="id61"><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="id62"><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="id63"><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="id64"><tt class="docutils literal"><span class="pre">\PolToSturm{polname}{sturmname}</span></tt></a></li>
+<li><a class="reference internal" href="#id10" id="id65"><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="id66"><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="id67"><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="id68"><tt class="docutils literal">\PolSturmIsolateZeros{sturmname}</tt></a></li>
+<li><a class="reference internal" href="#polrefineinterval-sturmname-index" id="id69"><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="id70"><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="id71"><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="id72"><tt class="docutils literal"><span class="pre">\PolEnsureIntervalLengths{sturmname}{E}</span></tt></a></li>
+<li><a class="reference internal" href="#polprintintervals-varname-sturmname" id="id73"><tt class="docutils literal"><span class="pre">\PolPrintIntervals[varname]{sturmname}</span></tt></a><ul>
+<li><a class="reference internal" href="#id11" id="id74"><tt class="docutils literal">\PolPrintIntervalsPrintExactZero</tt></a></li>
+<li><a class="reference internal" href="#id12" id="id75"><tt class="docutils literal">\PolPrintIntervalsPrintLeftEndPoint</tt></a></li>
+<li><a class="reference internal" href="#id13" id="id76"><tt class="docutils literal">\PolPrintIntervalsPrintRightEndPoint</tt></a></li>
</ul>
</li>
-<li><a class="reference internal" href="#id12" id="id49"><tt class="docutils literal"><span class="pre">\PolToExpr*{polname}</span></tt></a></li>
-<li><a class="reference internal" href="#poltofloatexpr-polname" id="id50"><tt class="docutils literal">\PolToFloatExpr{polname}</tt></a><ul>
-<li><a class="reference internal" href="#poltofloatexproneterm-raw-coeff-number" id="id51"><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="id52"><tt class="docutils literal">\PolToFloatExprCmd{raw_coeff}</tt></a></li>
+<li><a class="reference internal" href="#polmapcoeffs-macro-polname" id="id77"><tt class="docutils literal"><span class="pre">\PolMapCoeffs{\macro}{polname}</span></tt></a></li>
+<li><a class="reference internal" href="#polreducecoeffs-polname" id="id78"><tt class="docutils literal">\PolReduceCoeffs{polname}</tt></a></li>
+<li><a class="reference internal" href="#id15" id="id79"><tt class="docutils literal"><span class="pre">\PolReduceCoeffs*{polname}</span></tt></a></li>
+<li><a class="reference internal" href="#polmakemonic-polname" id="id80"><tt class="docutils literal">\PolMakeMonic{polname}</tt></a></li>
</ul>
</li>
-<li><a class="reference internal" href="#id13" id="id53"><tt class="docutils literal"><span class="pre">\PolToFloatExpr*{polname}</span></tt></a></li>
-<li><a class="reference internal" href="#poltolist-polname" id="id54"><tt class="docutils literal">\PolToList{polname}</tt></a></li>
-<li><a class="reference internal" href="#poltocsv-polname" id="id55"><tt class="docutils literal">\PolToCSV{polname}</tt></a></li>
+<li><a class="reference internal" href="#expandable-macros" id="id81">Expandable macros</a><ul>
+<li><a class="reference internal" href="#poleval-polname-atexpr-numerical-expression" id="id82"><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="id83"><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="id84"><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="id85"><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="id86"><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="id87"><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="id88"><tt class="docutils literal"><span class="pre">\PolIfCoeffIsPlusOrMinusOne{A}{B}</span></tt></a></li>
+<li><a class="reference internal" href="#polleadingcoeff-polname" id="id89"><tt class="docutils literal">\PolLeadingCoeff{polname}</tt></a></li>
+<li><a class="reference internal" href="#polnthcoeff-polname-number" id="id90"><tt class="docutils literal"><span class="pre">\PolNthCoeff{polname}{number}</span></tt></a></li>
+<li><a class="reference internal" href="#poldegree-polname" id="id91"><tt class="docutils literal">\PolDegree{polname}</tt></a></li>
+<li><a class="reference internal" href="#poltoexpr-polname" id="id92"><tt class="docutils literal">\PolToExpr{polname}</tt></a><ul>
+<li><a class="reference internal" href="#poltoexproneterm-raw-coeff-number" id="id93"><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="id94"><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="id95"><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="id96"><tt class="docutils literal">\PolToExprCmd{raw_coeff}</tt></a></li>
+<li><a class="reference internal" href="#poltoexprtermprefix-raw-coeff" id="id97"><tt class="docutils literal">\PolToExprTermPrefix{raw_coeff}</tt></a></li>
+<li><a class="reference internal" href="#id23" id="id98"><tt class="docutils literal">\PolToExprVar</tt></a></li>
+<li><a class="reference internal" href="#id24" id="id99"><tt class="docutils literal">\PolToExprTimes</tt></a></li>
</ul>
</li>
-<li><a class="reference internal" href="#booleans-with-default-setting-as-indicated" id="id56">Booleans (with default setting as indicated)</a><ul>
-<li><a class="reference internal" href="#xintverbosefalse" id="id57"><tt class="docutils literal">\xintverbosefalse</tt></a></li>
-<li><a class="reference internal" href="#poltypesetallfalse" id="id58"><tt class="docutils literal">\poltypesetallfalse</tt></a></li>
-<li><a class="reference internal" href="#poltoexprallfalse" id="id59"><tt class="docutils literal">\poltoexprallfalse</tt></a></li>
+<li><a class="reference internal" href="#id26" id="id100"><tt class="docutils literal"><span class="pre">\PolToExpr*{polname}</span></tt></a></li>
+<li><a class="reference internal" href="#poltofloatexpr-polname" id="id101"><tt class="docutils literal">\PolToFloatExpr{polname}</tt></a><ul>
+<li><a class="reference internal" href="#poltofloatexproneterm-raw-coeff-number" id="id102"><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="id103"><tt class="docutils literal">\PolToFloatExprCmd{raw_coeff}</tt></a></li>
</ul>
</li>
-<li><a class="reference internal" href="#technicalities" id="id60">Technicalities</a></li>
-<li><a class="reference internal" href="#releases" id="id61">RELEASES</a></li>
-<li><a class="reference internal" href="#acknowledgments" id="id62">Acknowledgments</a></li>
+<li><a class="reference internal" href="#id30" id="id104"><tt class="docutils literal"><span class="pre">\PolToFloatExpr*{polname}</span></tt></a></li>
+<li><a class="reference internal" href="#poltolist-polname" id="id105"><tt class="docutils literal">\PolToList{polname}</tt></a></li>
+<li><a class="reference internal" href="#poltocsv-polname" id="id106"><tt class="docutils literal">\PolToCSV{polname}</tt></a></li>
+<li><a class="reference internal" href="#polsturmchainlength-sturmname" id="id107"><tt class="docutils literal">\PolSturmChainLength{sturmname}</tt></a></li>
+<li><a class="reference internal" href="#polsturmifzeroexactlyknown-sturmname-index-a-b" id="id108"><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="id109"><tt class="docutils literal"><span class="pre">\PolSturmIsolatedZeroLeft{sturmname}{index}</span></tt></a></li>
+<li><a class="reference internal" href="#polsturmisolatedzeroright-sturmname-index" id="id110"><tt class="docutils literal"><span class="pre">\PolSturmIsolatedZeroRight{sturmname}{index}</span></tt></a></li>
+<li><a class="reference internal" href="#polsturmnbofisolatedzeros-sturmname" id="id111"><tt class="docutils literal">\PolSturmNbOfIsolatedZeros{sturmname}</tt></a></li>
+<li><a class="reference internal" href="#polintervalwidth-sturmname-index" id="id112"><tt class="docutils literal"><span class="pre">\PolIntervalWidth{sturmname}{index}</span></tt></a></li>
+<li><a class="reference internal" href="#macros-for-use-within-execution-of-polprintintervals" id="id113">Macros for use within execution of <tt class="docutils literal">\PolPrintIntervals</tt></a><ul>
+<li><a class="reference internal" href="#id31" id="id114"><tt class="docutils literal">\PolPrintIntervalsTheEndPoint</tt></a></li>
+<li><a class="reference internal" href="#id32" id="id115"><tt class="docutils literal">\PolPrintIntervalsTheIndex</tt></a></li>
+<li><a class="reference internal" href="#polifendpointispositive-a-b" id="id116"><tt class="docutils literal"><span class="pre">\PolIfEndPointIsPositive{A}{B}</span></tt></a></li>
+<li><a class="reference internal" href="#polifendpointisnegative-a-b" id="id117"><tt class="docutils literal"><span class="pre">\PolIfEndPointIsNegative{A}{B}</span></tt></a></li>
+<li><a class="reference internal" href="#polifendpointiszero-a-b" id="id118"><tt class="docutils literal"><span class="pre">\PolIfEndPointIsZero{A}{B}</span></tt></a></li>
+</ul>
+</li>
+<li><a class="reference internal" href="#poldectostring-decimal-number" id="id119"><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="id120">Booleans (with default setting as indicated)</a><ul>
+<li><a class="reference internal" href="#xintverbosefalse" id="id121"><tt class="docutils literal">\xintverbosefalse</tt></a></li>
+<li><a class="reference internal" href="#poltypesetallfalse" id="id122"><tt class="docutils literal">\poltypesetallfalse</tt></a></li>
+<li><a class="reference internal" href="#poltoexprallfalse" id="id123"><tt class="docutils literal">\poltoexprallfalse</tt></a></li>
+</ul>
+</li>
+<li><a class="reference internal" href="#technicalities" id="id124">Technicalities</a></li>
+<li><a class="reference internal" href="#change-log" id="id125">CHANGE LOG</a></li>
+<li><a class="reference internal" href="#acknowledgments" id="id126">Acknowledgments</a></li>
</ul>
</div>
-<div class="section" id="first-examples">
-<h1><a class="toc-backref" href="#id15">First Examples</a></h1>
+<div class="section" id="basic-examples">
+<h1><a class="toc-backref" href="#id33">Basic Examples</a></h1>
<p>The syntax is:</p>
<pre class="literal-block">
\poldef polname(x):= expression in variable x;
@@ -526,10 +578,200 @@ PSTricks-compatible; the letter used in output can be
<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="#id34">Examples of localization of roots</a></h1>
+<p>First some remarks about auxiliaries.</p>
+<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 <tt class="docutils literal">\xintverbosetrue</tt>. In fact one can also
+work from command line then (I recommend <tt class="docutils literal">rlwrap</tt> for encapsulating
+<tt class="docutils literal">latex</tt>).</p>
+</li>
+</ul>
+<div class="section" id="a-typical-example">
+<h2><a class="toc-backref" href="#id35">A typical example</a></h2>
+<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 derivative polynomial is \PolTypeset{f_1}.
+\PolToSturm{f_1}{f_1}\PolSturmIsolateZeros{f_1}%
+It 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_xx}
+\PolToSturm{f_xx}{f_xx}
+\PolSturmIsolateZeros{f_xx}
+The second derivative is \PolTypeset{f_xx}.
+It has \PolSturmNbOfIsolatedZeros{f_xx} distinct real
+roots:
+\PolPrintIntervals[X]{f_xx}
+Here is the positive one with 20 digits after decimal mark:
+\PolEnsureIntervalLength{f_xx}{2}{-20}%
+\[X_2 = \PolSturmIsolatedZeroLeft{f_xx}{2}\dots\]
+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="#id36">A degree four polynomial with nearby roots</a></h2>
+<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="#id37">The degree nine polynomial with 0.99, 0.999, 0.9999 as triple roots</a></h2>
+<pre class="literal-block">
+\PolDef{P}{(x-0.99)^3(x-0.999)^3(x-0.9999)^3}
+\PolTypeset{P}\par
+\PolToSturm{P}{P}%
+\PolLet{Psqfree}{P_0}\PolMakeMonic{Psqfree}\PolReduceCoeffs*{Psqfree}
+\par
+The monic square-free radical is \PolTypeset{Psqfree}.
+\PolSturmIsolateZeros{P}
+\par
+It has \PolSturmNbOfIsolatedZeros{P} real roots.
+\PolPrintIntervals{P}% all three roots found exactly
+</pre>
+</div>
+<div class="section" id="a-mignotte-type-polynomial">
+<h2><a class="toc-backref" href="#id38">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
+\PolPrintIntervals{P}%
+% reports -2 &lt; Z_1 &lt; -1, 0 &lt; Z_2 &lt; 0.1, 0.1 &lt; Z_3 &lt; 0.2, 1 &lt; Z_4 &lt; 2
+\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}%
+\PolEnsureIntervalLengths{P}{-10}%
+\PolPrintIntervals{P}% now all roots are known 10 decimal digits after mark
+\PolEnsureIntervalLength{P}{2}{-20}% makes Z_2 known with 20 digits after mark
+There are $\PolSturmNbOfIsolatedZeros{P}$ distinct real roots and there holds
+$\PolSturmIsolatedZeroLeft{P}{2}&lt;Z_2&lt;\PolSturmIsolatedZeroRight{P}{2}$.
+</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-degree-41-polynomial-with-2-1-9-1-8-0-0-1-1-9-2-as-roots">
+<h2><a class="toc-backref" href="#id39">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}
+\PolSturmIsolateZeros{S} % finds *exactly* (but a bit slowly) all 41 roots!
+\PolPrintIntervals{S} % nice, isn't it?
+</pre>
+</div>
+<div class="section" id="roots-of-chebyshev-polynomials">
+<h2><a class="toc-backref" href="#id40">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="#id16">Non-expandable macros</a></h1>
+<h1><a class="toc-backref" href="#id41">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="#id17"><tt class="docutils literal">\poldef <span class="pre">polname(letter):=</span> expression in letter;</tt></a></h2>
+<span id="poldef"></span><h2><a class="toc-backref" href="#id42"><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,
@@ -546,25 +788,18 @@ genuine <tt class="docutils literal"><span class="pre">\xintexpr...\relax</span>
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>
-<p>Also, a function with the same name is created for use within
-<tt class="docutils literal">\xintfloatexpr</tt> (or <tt class="docutils literal">\xintdeffloatvar</tt>.) This is indispensible
-for numerical algorithms as exact computations very quickly lead to
-very big fractions. Addition and multiplication steps of the Horner
-scheme will be executed as floating-point operations. The
-coefficients have already been rounded at time of definition,
-according to the then prevailing <tt class="docutils literal">\xinttheDigits</tt> value.</p>
-<div class="admonition important">
-<p class="first admonition-title">Important</p>
-<p>Package macros (such as derivatives or Euclidean division)
-operate with the &quot;exact&quot; polynomials; &quot;floating point&quot;
-polynomials are always obtained in a second step.</p>
-<p>To modifiy &quot;in-place&quot; the original coefficients of a polynomial
-and round them to float precision:</p>
-<pre class="literal-block">
-\PolMapCoeffs{\xintFloat}{polname}
-% or \xintFloat[P] for precision P digits
-</pre>
-<p class="last">See <a class="reference internal" href="#polmapcoeffs-macro-polname">\PolMapCoeffs{\macro}{polname}</a>.</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
@@ -572,23 +807,47 @@ 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="#id18"><tt class="docutils literal"><span class="pre">\PolDef[letter]{polname}{expression</span> in letter}</tt></a></h2>
+<span id="id2"></span><h2><a class="toc-backref" href="#id43"><tt class="docutils literal"><span class="pre">\PolDef[letter]{polname}{expression</span> in letter}</tt></a></h2>
<blockquote>
-Does the same 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>
+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="#id44"><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">
-<h2><a class="toc-backref" href="#id19"><tt class="docutils literal"><span class="pre">\PolLet{polname_2}={polname_1}</span></tt></a></h2>
+<span id="pollet"></span><h2><a class="toc-backref" href="#id45"><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="#id46"><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">
-<h2><a class="toc-backref" href="#id20"><tt class="docutils literal"><span class="pre">\PolAssign{polname}\toarray\macro</span></tt></a></h2>
+<span id="polassign"></span><h2><a class="toc-backref" href="#id47"><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>
@@ -611,7 +870,7 @@ indices act the same in both.)</p>
</blockquote>
</div>
<div class="section" id="polget-polname-fromarray-macro">
-<h2><a class="toc-backref" href="#id21"><tt class="docutils literal"><span class="pre">\PolGet{polname}\fromarray\macro</span></tt></a></h2>
+<span id="polget"></span><h2><a class="toc-backref" href="#id48"><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>. No
error checks on validity of coefficients as numbers. Each
@@ -625,18 +884,18 @@ polynomial.</p>
</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>.
However the coefficients are still in their original form (i.e.
-they were not subjected to <tt class="docutils literal">\xintRaw</tt> or similar xintfrac macro.)</p>
+they were not subjected to <tt class="docutils literal">\xintRaw</tt> or similar <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> macro.)</p>
</blockquote>
</div>
<div class="section" id="polfromcsv-polname-csv">
-<h2><a class="toc-backref" href="#id22"><tt class="docutils literal"><span class="pre">\PolFromCSV{polname}{&lt;csv&gt;}</span></tt></a></h2>
+<span id="polfromcsv"></span><h2><a class="toc-backref" href="#id49"><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 constant term being the first item. No validity checks. Spaces
from the list argument are trimmed. List items are each expanded in
an <tt class="docutils literal">\edef</tt>, but currently left in their original form like e.g.
-<tt class="docutils literal">1.5e3</tt> which is not converted to <tt class="docutils literal">15/1[2]</tt> <em>raw</em> xintfrac
+<tt class="docutils literal">1.5e3</tt> which is not converted to <tt class="docutils literal">15/1[2]</tt> <em>raw</em> <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a>
format (this may change).</p>
<p>Leading zero coefficients are removed:</p>
<pre class="literal-block">
@@ -647,7 +906,7 @@ format (this may change).</p>
</blockquote>
</div>
<div class="section" id="poltypeset-polname">
-<h2><a class="toc-backref" href="#id23"><tt class="docutils literal">\PolTypeset{polname}</tt></a></h2>
+<span id="poltypeset"></span><h2><a class="toc-backref" href="#id50"><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>
@@ -661,31 +920,52 @@ 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">
-<h3><a class="toc-backref" href="#id24"><tt class="docutils literal">\PolTypesetCmd{raw_coeff}</tt></a></h3>
+<span id="poltypesetcmd"></span><h3><a class="toc-backref" href="#id51"><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="#id52"><tt class="docutils literal">\PolTypesetOne{raw_coeff}</tt></a></h3>
<blockquote>
-<p>Basically will use <tt class="docutils literal">\xintSignedFrac</tt> from <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a>, but 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...</p>
-<p>One can do things such as for example: <a class="footnote-reference" href="#id4" id="id3">[1]</a></p>
+<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\PolTypesetCmd[1]{\num{\xintPFloat[5]{#1}}}
-\renewcommand\PolTypesetCmd[1]{\num{\xintRound{4}{#1}}}
+\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="id4" rules="none">
+<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="#id3">[1]</a></td><td>the difference in the syntaxes of <tt class="docutils literal">\xintPFloat</tt> and
+<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="#id53"><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">
-<h3><a class="toc-backref" href="#id25"><tt class="docutils literal">\PolTypesetCmdPrefix{raw_coeff}</tt></a></h3>
+<span id="poltypesetcmdprefix"></span><h3><a class="toc-backref" href="#id54"><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
@@ -694,25 +974,15 @@ the <tt class="docutils literal">-</tt> sign in front of the fraction (if it is
this will thus serve as separator in the typeset formula. Not used
for the first term.</blockquote>
</div>
-<div class="section" id="poltypesetmonomialcmd">
-<h3><a class="toc-backref" href="#id26"><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>
-<div class="section" id="id5">
-<h2><a class="toc-backref" href="#id27"><tt class="docutils literal"><span class="pre">\PolTypeset*{polname}</span></tt></a></h2>
+<div class="section" id="id8">
+<span id="id7"></span><h2><a class="toc-backref" href="#id55"><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">
-<h2><a class="toc-backref" href="#id28"><tt class="docutils literal"><span class="pre">\PolDiff{polname_1}{polname_2}</span></tt></a></h2>
+<span id="poldiff"></span><h2><a class="toc-backref" href="#id56"><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>
@@ -722,7 +992,7 @@ by <tt class="docutils literal">f'</tt>.</p>
</blockquote>
</div>
<div class="section" id="poldiff-n-polname-1-polname-2">
-<h2><a class="toc-backref" href="#id29"><tt class="docutils literal"><span class="pre">\PolDiff[N]{polname_1}{polname_2}</span></tt></a></h2>
+<span id="poldiff-n"></span><h2><a class="toc-backref" href="#id57"><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
@@ -730,7 +1000,7 @@ Identical arguments is allowed. With <tt class="docutils literal">N=0</tt>, same
using <tt class="docutils literal">\PolAntiDiff</tt>.</blockquote>
</div>
<div class="section" id="polantidiff-polname-1-polname-2">
-<h2><a class="toc-backref" href="#id30"><tt class="docutils literal"><span class="pre">\PolAntiDiff{polname_1}{polname_2}</span></tt></a></h2>
+<span id="polantidiff"></span><h2><a class="toc-backref" href="#id58"><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>
@@ -739,27 +1009,287 @@ at zero.</p>
</blockquote>
</div>
<div class="section" id="polantidiff-n-polname-1-polname-2">
-<h2><a class="toc-backref" href="#id31"><tt class="docutils literal"><span class="pre">\PolAntiDiff[N]{polname_1}{polname_2}</span></tt></a></h2>
+<span id="polantidiff-n"></span><h2><a class="toc-backref" href="#id59"><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">
-<h2><a class="toc-backref" href="#id32"><tt class="docutils literal"><span class="pre">\PolDivide{polname_1}{polname_2}{polname_Q}{polname_R}</span></tt></a></h2>
+<span id="poldivide"></span><h2><a class="toc-backref" href="#id60"><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="#id61"><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="#id62"><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">
-<h2><a class="toc-backref" href="#id33"><tt class="docutils literal"><span class="pre">\PolGCD{polname_1}{polname_2}{polname_GCD}</span></tt></a></h2>
+<span id="polgcd"></span><h2><a class="toc-backref" href="#id63"><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="#id64"><tt class="docutils literal"><span class="pre">\PolToSturm{polname}{sturmname}</span></tt></a></h2>
+<blockquote>
+<p>With, for example, <tt class="docutils literal">polname</tt> being <tt class="docutils literal">P</tt> and <tt class="docutils literal">sturmname</tt> being
+<tt class="docutils literal">S</tt>, the macro starts by computing polynomials <tt class="docutils literal">S_0 = P</tt>, <tt class="docutils literal">S_1
+= P'</tt>, ..., with <tt class="docutils literal">S_{n+1}</tt> the opposite of the remainder of
+euclidean division of <tt class="docutils literal"><span class="pre">S_{n-1}</span></tt> by <tt class="docutils literal">S_{n}</tt>. The last non-zero
+remainder <tt class="docutils literal">S_N</tt> is up to a factor the GCD of <tt class="docutils literal">P</tt> and <tt class="docutils literal">P'</tt>
+hence a constant if and only if <tt class="docutils literal">P</tt> is square-free.</p>
+<p>In case <tt class="docutils literal">S_N</tt> is not a constant, the macro then goes on with
+dividing all <tt class="docutils literal">S_k</tt>'s with <tt class="docutils literal">S_N</tt> (which becomes <tt class="docutils literal">1</tt>).</p>
+<p>Thus <tt class="docutils literal">S_0</tt> now has exactly the same real and complex
+roots as polynomial <tt class="docutils literal">polname</tt>, but each with multiplicity one.</p>
+</blockquote>
+</div>
+<div class="section" id="id10">
+<span id="id9"></span><h2><a class="toc-backref" href="#id65"><tt class="docutils literal"><span class="pre">\PolToSturm*{polname}{sturmname}</span></tt></a></h2>
+<blockquote>
+Does not divide the Sturm chain by its last element.</blockquote>
+</div>
+<div class="section" id="polsettosturmchainsignchangesat-macro-sturmname-fraction">
+<span id="polsettosturmchainsignchangesat"></span><h2><a class="toc-backref" href="#id66"><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="#id67"><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 distinct 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>
+</blockquote>
+</div>
+<div class="section" id="polsturmisolatezeros-sturmname">
+<span id="polsturmisolatezeros"></span><h2><a class="toc-backref" href="#id68"><tt class="docutils literal">\PolSturmIsolateZeros{sturmname}</tt></a></h2>
+<blockquote>
+<p>First, it evaluates using <a class="reference external" href="https://en.wikipedia.org/wiki/Sturm%27s_theorem">Sturm theorem</a> the number of distinct
+real roots of <tt class="docutils literal">sturmname_0</tt>.</p>
+<div class="admonition important">
+<p class="first admonition-title">Important</p>
+<p class="last">The Sturm chain <strong>must</strong> be of the reduced type, i.e.
+as constructed via <a class="reference internal" href="#poltosturm-polname-sturmname">\PolToSturm{polname}{sturmname}</a>.</p>
+</div>
+<p>Then it locates, again using <a class="reference external" href="https://en.wikipedia.org/wiki/Sturm%27s_theorem">Sturm theorem</a>, as many disjoint
+intervals as there are roots. Some intervals reduce to singleton
+which are roots. Non-singleton intervals get refined to make sure
+none of their two limit points is a root: they contain each a single
+root, in their respective interiors.</p>
+<!-- This procedure is covariant
+with the independent variable ``x`` becoming ``-x``.
+Hmm, pas sûr et trop fatigué -->
+<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>; if zero is a root it is always
+identified individually. 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 <tt class="docutils literal">a</tt> and <tt class="docutils literal">a+1</tt> are both positive
+or both negative.</p>
+<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>Also two macro arrays (in the sense of
+<a class="reference external" href="http://www.ctan.org/pkg/xint">xinttools</a>'s <tt class="docutils literal">\xintAssignArray</tt>) are created for holding the
+interval end-points written out in standard decimal notation
+(see <a class="reference internal" href="#poldectostring-decimal-number">\PolDecToString{decimal number}</a>).
+To access these values, 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.</p>
+<div class="admonition important">
+<p class="first admonition-title">Important</p>
+<p class="last">Trailing zeroes in these stored decimal numbers are significant:
+they are also present in the decimal expansion of the exact root.</p>
+</div>
+<div class="admonition note">
+<p class="first admonition-title">Note</p>
+<p class="last">The actual array macros are <tt class="docutils literal">\POL_ZeroInt&lt;sturmname&gt;L</tt> and
+<tt class="docutils literal">\POL_ZeroInt&lt;sturmname&gt;R</tt> but as these names use the
+non-letter character <tt class="docutils literal">_</tt> and possibly also digits from
+<tt class="docutils literal">sturmname</tt>, the accessor macros above have been made part of
+the package.</p>
+</div>
+<p>The start of decimal expansion of a positive <tt class="docutils literal">k</tt>-th root is given
+by <tt class="docutils literal"><span class="pre">\PolSturmIsolatedZeroLeft{sturmname}{k}</span></tt>, and for a negative
+root it is given by <tt class="docutils literal"><span class="pre">\PolSturmIsolatedZeroRight{sturmname}{k}</span></tt>.
+These two decimal numbers are either both zero or both of the same
+sign.</p>
+<p>The number of distinct roots is obtainable as
+<tt class="docutils literal">\PolSturmNbOfIsolatedZeros{sturmname}</tt>.</p>
+<div class="admonition note">
+<p class="first admonition-title">Note</p>
+<p class="last">In the current implementation the <tt class="docutils literal"><span class="pre">&lt;sturmname&gt;...</span></tt> variables
+and the <tt class="docutils literal"><span class="pre">\POL_ZeroInt...</span></tt> 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 class="last">When two successive roots are located in adjacent intervals, the
+separation between them is not lower bounded. See
+<a class="reference internal" href="#polrefineinterval-sturmname-index">\PolRefineInterval*{sturmname}{index}</a>.</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="polrefineinterval-sturmname-index">
+<span id="polrefineinterval"></span><h2><a class="toc-backref" href="#id69"><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="#id70"><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="#id71"><tt class="docutils literal"><span class="pre">\PolEnsureIntervalLength{sturmname}{index}{E}</span></tt></a></h2>
<blockquote>
-This sets <tt class="docutils literal">polname_GCD</tt> to be the G.C.D. 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>
+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="#id72"><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="#id73"><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 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. The user is invited to renewcommand the macro if some other
+type of tabular environment for example is wanted.</p>
+<p>In each array cell the corresponding interval end-point (which may
+be an exactly known root) is available as macro
+<a class="reference internal" href="#polprintintervalstheendpoint">\PolPrintIntervalsTheEndPoint</a> (in decimal notation). And the
+corresponding interval index is available as
+<a class="reference internal" href="#polprintintervalstheindex">\PolPrintIntervalsTheIndex</a>.</p>
+<p>These values may be tested to decide some on-the-fly customization
+(color for example), via the following auxiliaries which can be
+modified by user. Furthermore these auxiliaries can also use the
+following conditionals: <a class="reference internal" href="#polifendpointispositive-a-b">\PolIfEndPointIsPositive{A}{B}</a>,
+<a class="reference internal" href="#polifendpointisnegative-a-b">\PolIfEndPointIsNegative{A}{B}</a>, <a class="reference internal" href="#polifendpointiszero-a-b">\PolIfEndPointIsZero{A}{B}</a>.</p>
+</blockquote>
+<div class="section" id="id11">
+<span id="polprintintervalsprintexactzero"></span><h3><a class="toc-backref" href="#id74"><tt class="docutils literal">\PolPrintIntervalsPrintExactZero</tt></a></h3>
+<blockquote>
+<p>This is provided to help customize how an exactly known root is
+printed in the right most column of the array. The package
+definition is:</p>
+<pre class="literal-block">
+\newcommand\PolPrintIntervalsPrintExactZero{\PolPrintIntervalsTheEndPoint}%
+</pre>
+<p>Recall that this is expanded in an array cell.</p>
+<p>If for example you want to print in red the third root, known
+exactly, the macro could make a test for the value of
+<a class="reference internal" href="#polprintintervalstheindex">\PolPrintIntervalsTheIndex</a> and act accordingly.</p>
+</blockquote>
+</div>
+<div class="section" id="id12">
+<span id="polprintintervalsprintleftendpoint"></span><h3><a class="toc-backref" href="#id75"><tt class="docutils literal">\PolPrintIntervalsPrintLeftEndPoint</tt></a></h3>
+<blockquote>
+<p>Package definition is:</p>
+<pre class="literal-block">
+\newcommand\PolPrintIntervalsPrintLeftEndPoint{\PolPrintIntervalsTheEndPoint}%
+</pre>
+</blockquote>
+</div>
+<div class="section" id="id13">
+<span id="polprintintervalsprintrightendpoint"></span><h3><a class="toc-backref" href="#id76"><tt class="docutils literal">\PolPrintIntervalsPrintRightEndPoint</tt></a></h3>
+<blockquote>
+<p>Package definition is:</p>
+<pre class="literal-block">
+\newcommand\PolPrintIntervalsPrintRightEndPoint{\PolPrintIntervalsTheEndPoint}%
+</pre>
+</blockquote>
+</div>
</div>
<div class="section" id="polmapcoeffs-macro-polname">
-<h2><a class="toc-backref" href="#id34"><tt class="docutils literal"><span class="pre">\PolMapCoeffs{\macro}{polname}</span></tt></a></h2>
+<span id="polmapcoeffs"></span><h2><a class="toc-backref" href="#id77"><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
@@ -769,7 +1299,7 @@ coefficients vanish after the operation. In replacement text of
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 xintfrac package.</p>
+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}}
@@ -779,94 +1309,176 @@ will have to be expressed in terms of macros from xintfrac package.</p>
</blockquote>
</div>
<div class="section" id="polreducecoeffs-polname">
-<h2><a class="toc-backref" href="#id35"><tt class="docutils literal">\PolReduceCoeffs{polname}</tt></a></h2>
+<span id="polreducecoeffs"></span><h2><a class="toc-backref" href="#id78"><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="id15">
+<span id="id14"></span><h2><a class="toc-backref" href="#id79"><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="#id80"><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="#id15">\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>
<div class="section" id="expandable-macros">
-<h1><a class="toc-backref" href="#id36">Expandable macros</a></h1>
+<h1><a class="toc-backref" href="#id81">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-at-numerical-expression">
-<h2><a class="toc-backref" href="#id37"><tt class="docutils literal"><span class="pre">\PolEval{polname}\At{numerical</span> expression}</tt></a></h2>
+<div class="section" id="poleval-polname-atexpr-numerical-expression">
+<span id="polevalatexpr"></span><h2><a class="toc-backref" href="#id82"><tt class="docutils literal"><span class="pre">\PolEval{polname}\AtExpr{numerical</span> expression}</tt></a></h2>
<blockquote>
-<p>It boils down to <tt class="docutils literal">\xinttheexpr polname(numerical <span class="pre">expression)\relax</span></tt>.</p>
-<div class="admonition note">
-<p class="first admonition-title">Note</p>
-<p>The <tt class="docutils literal">0.2</tt> version stupidly tried to be clever and as a result
-of a misguided optimization choked if <tt class="docutils literal">value</tt> was not a number
-but a numerical expression (a sum e.g.), but the more powerful
-behaviour has been reinstored at <tt class="docutils literal">0.3</tt> release.</p>
-<p class="last">The <tt class="docutils literal">0.1</tt> and <tt class="docutils literal">0.2</tt> version did a <tt class="docutils literal">reduce</tt> which however is
-costly on big fractions and irrelevant if the output is served as
-argument of <tt class="docutils literal">\xintRound</tt> or <tt class="docutils literal">\xintFloat</tt>. Thus <tt class="docutils literal">reduce</tt> was
-removed, and former meaning is now available as
-<a class="reference internal" href="#polevalreduced-polname-at-numerical-expression">\PolEvalReduced{polname}\At{numerical expression}</a></p>
+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="#id83"><tt class="docutils literal"><span class="pre">\PolEval{polname}\At{fraction}</span></tt></a></h2>
+<blockquote>
+<p>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.</p>
+<div class="admonition attention">
+<p class="first admonition-title">Attention!</p>
+<p>Meaning was changed at <tt class="docutils literal">0.4</tt>. Formerly <tt class="docutils literal"><span class="pre">\PolEval{P}\At{foo}</span></tt>
+accepted for <tt class="docutils literal">foo</tt> an expression which was handled by
+<tt class="docutils literal">\xintexpr</tt>. See <a class="reference internal" href="#poleval-polname-atexpr-numerical-expression">\PolEval{polname}\AtExpr{numerical
+expression}</a>.</p>
+<p class="last">In particular, to use an <tt class="docutils literal">\xintexpr</tt> user-declared variable (or
+e.g. the variables as defined by <a class="reference external" href="PolSturmIsolateZeros">\PolSturmIsolateZeros</a>) one <strong>must</strong> use the <tt class="docutils literal">\AtExpr</tt> syntax.</p>
</div>
</blockquote>
</div>
-<div class="section" id="polevalreduced-polname-at-numerical-expression">
-<h2><a class="toc-backref" href="#id38"><tt class="docutils literal"><span class="pre">\PolEvalReduced{polname}\At{numerical</span> expression}</tt></a></h2>
+<div class="section" id="polevalreduced-polname-atexpr-numerical-expression">
+<span id="polevalreducedatexpr"></span><h2><a class="toc-backref" href="#id84"><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="polfloateval-polname-at-numerical-expression">
-<h2><a class="toc-backref" href="#id39"><tt class="docutils literal"><span class="pre">\PolFloatEval{polname}\At{numerical</span> expression}</tt></a></h2>
+<div class="section" id="polevalreduced-polname-at-fraction">
+<span id="polevalreducedat"></span><h2><a class="toc-backref" href="#id85"><tt class="docutils literal"><span class="pre">\PolEvalReduced{polname}\At{fraction}</span></tt></a></h2>
+<blockquote>
+<p>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.</p>
+<div class="admonition attention">
+<p class="first admonition-title">Attention!</p>
+<p class="last">Meaning was changed at <tt class="docutils literal">0.4</tt>. Formerly the evaluation point
+could be given as an expression.</p>
+</div>
+</blockquote>
+</div>
+<div class="section" id="polfloateval-polname-atexpr-numerical-expression">
+<span id="polfloatevalatexpr"></span><h2><a class="toc-backref" href="#id86"><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>), with
-already rounded coefficients. <a class="footnote-reference" href="#id8" id="id6">[2]</a> To use the <em>exact coefficients</em>
-(and <em>exact</em> additions and multiplications), just insert it in the
-float expression as in this example: <a class="footnote-reference" href="#id9" id="id7">[3]</a></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="#id19" id="id17">[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="#id20" id="id18">[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="id8" rules="none">
+<div class="admonition attention">
+<p class="first admonition-title">Attention!</p>
+<p class="last">At <tt class="docutils literal">polexpr 0.3</tt>, polynoms were automatically also prepared for
+use in floating point contexts. This got dropped at <tt class="docutils literal">0.4</tt> for
+optimization purposes. See <a class="reference internal" href="#polgenfloatvariant-polname">\PolGenFloatVariant{polname}</a>.</p>
+</div>
+<table class="docutils footnote" frame="void" id="id19" rules="none">
<colgroup><col class="label" /><col /></colgroup>
<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id6">[2]</a></td><td>Anyway each floating point operation starts by rounding its
+<tr><td class="label"><a class="fn-backref" href="#id17">[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="id9" rules="none">
+<table class="docutils footnote" frame="void" id="id20" rules="none">
<colgroup><col class="label" /><col /></colgroup>
<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id7">[3]</a></td><td>The <tt class="docutils literal">\xintexpr</tt> 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>
+<tr><td class="label"><a class="fn-backref" href="#id18">[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="#id87"><tt class="docutils literal"><span class="pre">\PolFloatEval{polname}\At{fraction}</span></tt></a></h2>
+<blockquote>
+<p>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.</p>
+<div class="admonition attention">
+<p class="first admonition-title">Attention!</p>
+<p class="last">Meaning was changed at <tt class="docutils literal">0.4</tt>. Formerly the evaluation point
+could be given as an expression.</p>
+</div>
+</blockquote>
+</div>
+<div class="section" id="polifcoeffisplusorminusone-a-b">
+<span id="polifcoeffisplusorminusone"></span><h2><a class="toc-backref" href="#id88"><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="#id89"><tt class="docutils literal">\PolLeadingCoeff{polname}</tt></a></h2>
+<blockquote>
+Expands to the leading coefficient.</blockquote>
+</div>
<div class="section" id="polnthcoeff-polname-number">
-<h2><a class="toc-backref" href="#id40"><tt class="docutils literal"><span class="pre">\PolNthCoeff{polname}{number}</span></tt></a></h2>
+<span id="polnthcoeff"></span><h2><a class="toc-backref" href="#id90"><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">
-<h2><a class="toc-backref" href="#id41"><tt class="docutils literal">\PolDegree{polname}</tt></a></h2>
+<span id="poldegree"></span><h2><a class="toc-backref" href="#id91"><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="poltoexpr-polname">
-<h2><a class="toc-backref" href="#id42"><tt class="docutils literal">\PolToExpr{polname}</tt></a></h2>
+<span id="poltoexpr"></span><h2><a class="toc-backref" href="#id92"><tt class="docutils literal">\PolToExpr{polname}</tt></a></h2>
<blockquote>
-<p>Expands <a class="footnote-reference" href="#id11" id="id10">[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="id11" rules="none">
+<p>Expands <a class="footnote-reference" href="#id22" id="id21">[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="id22" rules="none">
<colgroup><col class="label" /><col /></colgroup>
<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id10">[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
+<tr><td class="label"><a class="fn-backref" href="#id21">[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>
@@ -885,7 +1497,7 @@ of <tt class="docutils literal">\PolToExpr{f}</tt>, but a simple <tt class="docu
the identical result.</p>
</blockquote>
<div class="section" id="poltoexproneterm-raw-coeff-number">
-<h3><a class="toc-backref" href="#id43"><tt class="docutils literal"><span class="pre">\PolToExprOneTerm{raw_coeff}{number}</span></tt></a></h3>
+<span id="poltoexproneterm"></span><h3><a class="toc-backref" href="#id93"><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
@@ -897,20 +1509,28 @@ always precedes the <tt class="docutils literal">x^number</tt>, except if the co
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="#id94"><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">
-<h3><a class="toc-backref" href="#id44"><tt class="docutils literal"><span class="pre">\PolToExprOneTermStyleB{raw_coeff}{number}</span></tt></a></h3>
+<span id="poltoexpronetermstyleb"></span><h3><a class="toc-backref" href="#id95"><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 using
-<tt class="docutils literal">\PolToExpr</tt>. Note that then <tt class="docutils literal">\PolToExprCmd</tt> isn't used at all.</p>
+<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">
-<h3><a class="toc-backref" href="#id45"><tt class="docutils literal">\PolToExprCmd{raw_coeff}</tt></a></h3>
+<span id="poltoexprcmd"></span><h3><a class="toc-backref" href="#id96"><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
@@ -920,21 +1540,21 @@ to <tt class="docutils literal"><span class="pre">\xintIrr{#1}</span></tt> or to
output forcefully reduced coefficients.</blockquote>
</div>
<div class="section" id="poltoexprtermprefix-raw-coeff">
-<h3><a class="toc-backref" href="#id46"><tt class="docutils literal">\PolToExprTermPrefix{raw_coeff}</tt></a></h3>
+<span id="poltoexprtermprefix"></span><h3><a class="toc-backref" href="#id97"><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="poltoexprvar">
-<h3><a class="toc-backref" href="#id47"><tt class="docutils literal">\PolToExprVar</tt></a></h3>
+<div class="section" id="id23">
+<span id="poltoexprvar"></span><h3><a class="toc-backref" href="#id98"><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="poltoexprtimes">
-<h3><a class="toc-backref" href="#id48"><tt class="docutils literal">\PolToExprTimes</tt></a></h3>
+<div class="section" id="id24">
+<span id="poltoexprtimes"></span><h3><a class="toc-backref" href="#id99"><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
@@ -943,41 +1563,39 @@ this will give output incompatible with some professional computer
algebra software).</blockquote>
</div>
</div>
-<div class="section" id="id12">
-<h2><a class="toc-backref" href="#id49"><tt class="docutils literal"><span class="pre">\PolToExpr*{polname}</span></tt></a></h2>
+<div class="section" id="id26">
+<span id="id25"></span><h2><a class="toc-backref" href="#id100"><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">
-<h2><a class="toc-backref" href="#id50"><tt class="docutils literal">\PolToFloatExpr{polname}</tt></a></h2>
+<span id="poltofloatexpr"></span><h2><a class="toc-backref" href="#id101"><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>The polynomial function for usage in <tt class="docutils literal">\xintfloatexpr</tt> is
-already prepared with the rounded coefficients, but the latter
-are not easily recoverable (and especially not expandably) from
-this. Thus <tt class="docutils literal">\PolToFloatExprCmd</tt> operates from the <em>exact</em>
-coefficients anew. This means though that if the prevailing float
-precision was changed with <tt class="docutils literal"><span class="pre">\xintDigits:=P;</span></tt> syntax, the output
-will obey this precision <tt class="docutils literal">P</tt>, but the polynomial function was
-defined earlier and operates on floating point numbers with
-coefficients which were rounded at time of definition.</p>
-<p class="last">This may change in future, if the pre-rounded coefficients are
-stored in a more easily accessible data structure.</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">
-<h3><a class="toc-backref" href="#id51"><tt class="docutils literal"><span class="pre">\PolToFloatExprOneTerm{raw_coeff}{number}</span></tt></a></h3>
+<span id="poltofloatexproneterm"></span><h3><a class="toc-backref" href="#id102"><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">
-<h3><a class="toc-backref" href="#id52"><tt class="docutils literal">\PolToFloatExprCmd{raw_coeff}</tt></a></h3>
+<span id="id28"></span><h3><a class="toc-backref" href="#id103"><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>
@@ -999,29 +1617,151 @@ in <tt class="docutils literal">xintfrac</tt> raw format.</p>
</blockquote>
</div>
</div>
-<div class="section" id="id13">
-<h2><a class="toc-backref" href="#id53"><tt class="docutils literal"><span class="pre">\PolToFloatExpr*{polname}</span></tt></a></h2>
+<div class="section" id="id30">
+<span id="id29"></span><h2><a class="toc-backref" href="#id104"><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">
-<h2><a class="toc-backref" href="#id54"><tt class="docutils literal">\PolToList{polname}</tt></a></h2>
+<span id="poltolist"></span><h2><a class="toc-backref" href="#id105"><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
(except zero polynomial which does give <tt class="docutils literal">{0/1[0]}</tt> and not an
empty output.)</blockquote>
</div>
<div class="section" id="poltocsv-polname">
-<h2><a class="toc-backref" href="#id55"><tt class="docutils literal">\PolToCSV{polname}</tt></a></h2>
+<span id="poltocsv"></span><h2><a class="toc-backref" href="#id106"><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>. 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="#id107"><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="#id108"><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><tt class="docutils literal">index</tt> may be a TeX count, or a <tt class="docutils literal">\value{latexcounter}</tt>, or a
+numerical expression as parsable by <tt class="docutils literal">\numexpr</tt>: it does not
+have to be given via explicit digits.</p>
+<p class="last">This remark applies also to the other package macros with
+<tt class="docutils literal">index</tt> being the name of the argument in this documentation.
+There is also an out-of-range check done for some reasonable
+error message (right before everything goes haywire).</p>
+</div>
+</blockquote>
+</div>
+<div class="section" id="polsturmisolatedzeroleft-sturmname-index">
+<span id="polsturmisolatedzeroleft"></span><h2><a class="toc-backref" href="#id109"><tt class="docutils literal"><span class="pre">\PolSturmIsolatedZeroLeft{sturmname}{index}</span></tt></a></h2>
+<blockquote>
+Expands to the left end-point for the <tt class="docutils literal">index</tt>th interval
+obtained via <a class="reference internal" href="#polsturmisolatezeros-sturmname">\PolSturmIsolateZeros{sturmname}</a> and possibly
+refined afterwards.</blockquote>
+</div>
+<div class="section" id="polsturmisolatedzeroright-sturmname-index">
+<span id="polsturmisolatedzeroright"></span><h2><a class="toc-backref" href="#id110"><tt class="docutils literal"><span class="pre">\PolSturmIsolatedZeroRight{sturmname}{index}</span></tt></a></h2>
+<blockquote>
+Expands to the right end-point for the <tt class="docutils literal">index</tt>th interval
+obtained via <a class="reference internal" href="#polsturmisolatezeros-sturmname">\PolSturmIsolateZeros{sturmname}</a> and possibly
+refined afterwards.</blockquote>
+</div>
+<div class="section" id="polsturmnbofisolatedzeros-sturmname">
+<span id="polsturmnbofisolatedzeros"></span><h2><a class="toc-backref" href="#id111"><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> (which is the number of distinct real roots of the
+polynomial used to create the Sturm chain via
+<a class="reference internal" href="#poltosturm-polname-sturmname">\PolToSturm{polname}{sturmname}</a>.</blockquote>
+</div>
+<div class="section" id="polintervalwidth-sturmname-index">
+<span id="polintervalwidth"></span><h2><a class="toc-backref" href="#id112"><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="macros-for-use-within-execution-of-polprintintervals">
+<h2><a class="toc-backref" href="#id113">Macros for use within execution of <tt class="docutils literal">\PolPrintIntervals</tt></a></h2>
+<p>More precisely, they can be used within the replacement texts of the
+<a class="reference internal" href="#polprintintervalsprintleftendpoint">\PolPrintIntervalsPrintLeftEndPoint</a>, etc, macros.</p>
+<div class="section" id="id31">
+<span id="polprintintervalstheendpoint"></span><h3><a class="toc-backref" href="#id114"><tt class="docutils literal">\PolPrintIntervalsTheEndPoint</tt></a></h3>
+<blockquote>
+Within a custom <a class="reference internal" href="#polprintintervalsprintleftendpoint">\PolPrintIntervalsPrintLeftEndPoint</a>, custom
+<a class="reference internal" href="#polprintintervalsprintrightendpoint">\PolPrintIntervalsPrintRightEndPoint</a>, or custom
+<a class="reference internal" href="#polprintintervalsprintexactzero">\PolPrintIntervalsPrintExactZero</a> this macro expands to the left
+or right end point of the considered interval. Serves as default
+replacement for <a class="reference internal" href="#polprintintervalsprintleftendpoint">\PolPrintIntervalsPrintLeftEndPoint</a> , etc...</blockquote>
+</div>
+<div class="section" id="id32">
+<span id="polprintintervalstheindex"></span><h3><a class="toc-backref" href="#id115"><tt class="docutils literal">\PolPrintIntervalsTheIndex</tt></a></h3>
+<blockquote>
+Within a custom <a class="reference internal" href="#polprintintervalsprintleftendpoint">\PolPrintIntervalsPrintLeftEndPoint</a>, custom
+<a class="reference internal" href="#polprintintervalsprintrightendpoint">\PolPrintIntervalsPrintRightEndPoint</a>, or custom
+<a class="reference internal" href="#polprintintervalsprintexactzero">\PolPrintIntervalsPrintExactZero</a> this macro expands to the index
+of the considered interval. For example if user wants to print the
+corresponding end points in red, the index value can thus be tested
+in the replacement text of <a class="reference internal" href="#polprintintervalsprintleftendpoint">\PolPrintIntervalsPrintLeftEndPoint</a> and
+the other two similar macros.</blockquote>
+</div>
+<div class="section" id="polifendpointispositive-a-b">
+<span id="polifendpointispositive"></span><h3><a class="toc-backref" href="#id116"><tt class="docutils literal"><span class="pre">\PolIfEndPointIsPositive{A}{B}</span></tt></a></h3>
+<blockquote>
+Within a custom <a class="reference internal" href="#polprintintervalsprintleftendpoint">\PolPrintIntervalsPrintLeftEndPoint</a>, custom
+<a class="reference internal" href="#polprintintervalsprintrightendpoint">\PolPrintIntervalsPrintRightEndPoint</a>, or custom
+<a class="reference internal" href="#polprintintervalsprintexactzero">\PolPrintIntervalsPrintExactZero</a> this macro executes <tt class="docutils literal">A</tt> if
+the considered interval end-point is positive, else <tt class="docutils literal">B</tt>.</blockquote>
+</div>
+<div class="section" id="polifendpointisnegative-a-b">
+<span id="polifendpointisnegative"></span><h3><a class="toc-backref" href="#id117"><tt class="docutils literal"><span class="pre">\PolIfEndPointIsNegative{A}{B}</span></tt></a></h3>
+<blockquote>
+Within a custom <a class="reference internal" href="#polprintintervalsprintleftendpoint">\PolPrintIntervalsPrintLeftEndPoint</a>, custom
+<a class="reference internal" href="#polprintintervalsprintrightendpoint">\PolPrintIntervalsPrintRightEndPoint</a>, or custom
+<a class="reference internal" href="#polprintintervalsprintexactzero">\PolPrintIntervalsPrintExactZero</a> this macro executes <tt class="docutils literal">A</tt> if
+the considered interval end-point is negative, else <tt class="docutils literal">B</tt>.</blockquote>
+</div>
+<div class="section" id="polifendpointiszero-a-b">
+<span id="polifendpointiszero"></span><h3><a class="toc-backref" href="#id118"><tt class="docutils literal"><span class="pre">\PolIfEndPointIsZero{A}{B}</span></tt></a></h3>
+<blockquote>
+Within a custom <a class="reference internal" href="#polprintintervalsprintleftendpoint">\PolPrintIntervalsPrintLeftEndPoint</a>, custom
+<a class="reference internal" href="#polprintintervalsprintrightendpoint">\PolPrintIntervalsPrintRightEndPoint</a>, or custom
+<a class="reference internal" href="#polprintintervalsprintexactzero">\PolPrintIntervalsPrintExactZero</a> this macro executes <tt class="docutils literal">A</tt> if
+the considered interval end-point is zero, else <tt class="docutils literal">B</tt>.</blockquote>
+</div>
+</div>
+<div class="section" id="poldectostring-decimal-number">
+<span id="poldectostring"></span><h2><a class="toc-backref" href="#id119"><tt class="docutils literal">\PolDecToString{decimal number}</tt></a></h2>
+<blockquote>
+<p>This is a utility macro to print decimal numbers. Indeed for legacy
+reasons, <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> does not yet have user-level ready-to-use macros
+handling specifically the printing of decimal numbers from their
+internal representations such as <tt class="docutils literal">A/1[N]</tt>.</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>. This
+illustrates that trailing zeros are not trimmed (to achieve that one
+can use <tt class="docutils literal"><span class="pre">\PolDecToString{\xintREZ{#1}}</span></tt>.)</p>
+<p>The macro does not try to identify if the fraction has a denominator
+consisting only of two's and five's; such a denominator will be left
+at right-end of output.</p>
+<p>This utility macro will presumably be incorporated (possibly in a
+more powerful form) to <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> (or rather to a decimal module) in
+a future <a class="reference external" href="http://www.ctan.org/pkg/xint">xint</a> release.</p>
+</blockquote>
+</div>
</div>
<div class="section" id="booleans-with-default-setting-as-indicated">
-<h1><a class="toc-backref" href="#id56">Booleans (with default setting as indicated)</a></h1>
+<h1><a class="toc-backref" href="#id120">Booleans (with default setting as indicated)</a></h1>
<div class="section" id="xintverbosefalse">
-<h2><a class="toc-backref" href="#id57"><tt class="docutils literal">\xintverbosefalse</tt></a></h2>
+<h2><a class="toc-backref" href="#id121"><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
@@ -1034,20 +1774,20 @@ unstable and undocumented internal structures.</p>
</blockquote>
</div>
<div class="section" id="poltypesetallfalse">
-<h2><a class="toc-backref" href="#id58"><tt class="docutils literal">\poltypesetallfalse</tt></a></h2>
+<h2><a class="toc-backref" href="#id122"><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="#id59"><tt class="docutils literal">\poltoexprallfalse</tt></a></h2>
+<h2><a class="toc-backref" href="#id123"><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="technicalities">
-<h1><a class="toc-backref" href="#id60">Technicalities</a></h1>
+<h1><a class="toc-backref" href="#id124">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
@@ -1058,7 +1798,7 @@ rather. The colon in <tt class="docutils literal">:=</tt> may be active with no
<li><p class="first">During execution of polynomial operations by <tt class="docutils literal">\poldef</tt> (but not
during the initial purely numerical parsing of the expression), the
<a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> macro <tt class="docutils literal">\xintAdd</tt> is temporarily patched to always express
-<tt class="docutils literal">a/b + c/d</tt> with <tt class="docutils literal"><span class="pre">l.c.m.(b,d)</span></tt> as denominator. Indeed the current
+<tt class="docutils literal">a/b + c/d</tt> with <tt class="docutils literal">lcm(b,d)</tt> as denominator. Indeed the current
(xint 1.2p) <tt class="docutils literal">\xintAdd</tt> uses <tt class="docutils literal"><span class="pre">(ad+bc)/bd</span></tt> formula except if <tt class="docutils literal">b</tt>
divides <tt class="docutils literal">d</tt> or <tt class="docutils literal">d</tt> divides <tt class="docutils literal">b</tt>, which quickly leads in real life
to big denominators.</p>
@@ -1082,7 +1822,8 @@ survive addition and multiplications:</p>
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 (which in this
-example is the <tt class="docutils literal">l.c.m</tt> of the denominators of the reduced coefficients.)</p>
+example is the least common multiple of the denominators of the
+reduced coefficients.)</p>
</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>
@@ -1111,53 +1852,193 @@ high degree and few coefficients (as an example with <tt class="docutils literal
additions involvings only zeroes... which does take time). This
may change in the future.</p>
</li>
-<li><p class="first">Tests have been made with Newton's iteration (for which computing
-exactly the derivative is precisely what this package is made for) or
-Regula Falsi method for locating roots: using exact computations leads
-quickly to gigantic fractions (but dichotomy method much less so). It
-is thus recommended to use <tt class="docutils literal">\xintdeffloatvar</tt> or
-<tt class="docutils literal">\xintthefloatexpr</tt> contexts for any kind of numerical mathematics.
-Of course, exact computations are invaluable for number theory or
-combinatorics...</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="releases">
-<h1><a class="toc-backref" href="#id61">RELEASES</a></h1>
+<div class="section" id="change-log">
+<h1><a class="toc-backref" href="#id125">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">0.1 (2018/01/11)</p>
-<p>Initial release (files README, polexpr.sty).</p>
+<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>
-<li><p class="first">0.2 (2018/01/14)</p>
-<p>Documentation moved to polexpr.{txt,html}.</p>
+</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="#id25">\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="#id28">\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">0.3 (2018/01/17)</p>
-<p>Make polynomials known to <tt class="docutils literal">\xintfloatexpr</tt> and improve
-documentation.</p>
+<li><p class="first"><tt class="docutils literal">\PolToExpr{f}</tt> is highly customizable.</p>
</li>
-<li><p class="first">0.3.1 (2018/01/18)</p>
-<p>Fix two typos in documentation.</p>
+<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>
-<p>Files of 0.3.1 release:</p>
+</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>README.md,</li>
-<li>polexpr.sty (package file),</li>
-<li>polexpr.txt (documentation),</li>
-<li>polexpr.html (conversion via <a class="reference external" href="http://docutils.sourceforge.net/docs/index.html">DocUtils</a> rst2html.py)</li>
+<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="#id14">\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><a class="reference internal" href="#polprintintervalstheendpoint">\PolPrintIntervalsTheEndPoint</a></li>
+<li><a class="reference internal" href="#polprintintervalstheindex">\PolPrintIntervalsTheIndex</a></li>
+<li><a class="reference internal" href="#polifendpointispositive">\PolIfEndPointIsPositive</a></li>
+<li><a class="reference internal" href="#polifendpointisnegative">\PolIfEndPointIsNegative</a></li>
+<li><a class="reference internal" href="#polifendpointiszero">\PolIfEndPointIsZero</a></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>
</ul>
-<p>See README.md for the License and the change log (there were
-some breaking changes from 0.2 to 0.3).</p>
</div>
<div class="section" id="acknowledgments">
-<h1><a class="toc-backref" href="#id62">Acknowledgments</a></h1>
+<h1><a class="toc-backref" href="#id126">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 on occasion of <tt class="docutils literal">0.4</tt> release!</p>
+<p>See README.md for the License.</p>
</div>
</div>
</body>