diff options
author | Karl Berry <karl@freefriends.org> | 2021-04-10 21:13:16 +0000 |
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committer | Karl Berry <karl@freefriends.org> | 2021-04-10 21:13:16 +0000 |
commit | f83ae24524b685073a4bd225d9a814e1b5850daf (patch) | |
tree | b75236a60f69657356a312a2cb911cce756819ee /Master/texmf-dist/doc/latex/polexpr | |
parent | 89f4f0573f09dfd63100d90c8d80d1da160e3550 (diff) |
polexpr (now generic)
git-svn-id: svn://tug.org/texlive/trunk@58821 c570f23f-e606-0410-a88d-b1316a301751
Diffstat (limited to 'Master/texmf-dist/doc/latex/polexpr')
-rw-r--r-- | Master/texmf-dist/doc/latex/polexpr/README.md | 102 | ||||
-rw-r--r-- | Master/texmf-dist/doc/latex/polexpr/polexpr.html | 2911 | ||||
-rw-r--r-- | Master/texmf-dist/doc/latex/polexpr/polexpr.txt | 2598 |
3 files changed, 0 insertions, 5611 deletions
diff --git a/Master/texmf-dist/doc/latex/polexpr/README.md b/Master/texmf-dist/doc/latex/polexpr/README.md deleted file mode 100644 index a901b56adc7..00000000000 --- a/Master/texmf-dist/doc/latex/polexpr/README.md +++ /dev/null @@ -1,102 +0,0 @@ -Package polexpr README -====================== - -License -------- - -Copyright (C) 2018-2020 Jean-François Burnol - -See documentation of package [xint](http://www.ctan.org/pkg/xint) for -contact information. - -This Work may be distributed and/or modified under the conditions of the -LaTeX Project Public License version 1.3c. This version of this license -is in - -> <http://www.latex-project.org/lppl/lppl-1-3c.txt> - -and version 1.3 or later is part of all distributions of LaTeX version -2005/12/01 or later. - -This Work has the LPPL maintenance status author-maintained. - -The Author of this Work is Jean-François Burnol. - -This Work consists of the package file polexpr.sty, this README.md and -the documentation file polexpr.txt. - -Abstract --------- - -The package provides `\poldef`. This a parser of polynomial expressions -based upon the `\xintdeffunc` mechanism of xintexpr. - -The parsed expressions use the operations of algebra (inclusive of -composition of functions) with standard operators, fractional numbers -(possibly in scientific notation) and previously defined polynomial -functions or other constructs as recognized by the `\xintexpr` numerical -parser. - -The polynomials are then not only genuine `\xintexpr` (and -`\xintfloatexpr`) numerical functions but additionally are known to the -package via their coefficients. This allows dedicated macros to -implement polynomial algorithmics. - -Releases --------- - -- 0.1 (2018/01/11) - Initial release (files README, polexpr.sty). -- 0.2 (2018/01/14) - Documentation moved to polexpr.{txt,html}. -- 0.3 (2018/01/17) - Make polynomials known to `\xintfloatexpr` and improve - documentation. -- 0.3.1 (2018/01/18) - Fix two typos in documentation. -- 0.4 (2018/02/16) - - Revert 0.3 automatic generation of floating point variants. - - Move CHANGE LOG from README.md to HTML documentation. - - A few bug fixes and breaking changes. Please refer to - `polexpr.html`. - - Main new feature: root localization via [Sturm - Theorem](https://en.wikipedia.org/wiki/Sturm%27s_theorem). -- 0.4.1 (2018/03/01) - Synced with xint 1.3. -- 0.4.2 (2018/03/03) - Documentation fix. -- 0.5 (2018/04/08) - - new macros `\PolMakePrimitive` and `\PolIContent`. - - main (breaking) change: `\PolToSturm` creates a chain of primitive - integer coefficients polynomials. -- 0.5.1 (2018/04/22) - The `'` character can be used in polynomial names. -- 0.6 (2018/11/20) - New feature: multiplicity of roots. -- 0.7 (2018/12/08), 0.7.1 (bugfix), 0.7.2 (bugfix) (2018/12/09) - New feature: finding all rational roots. -- 0.7.3 (2019/02/04) - Bugfix: polynomial names ending in digits caused errors. Thanks to - Thomas Söll for report. -- 0.7.4 (2019/02/12) - Bugfix: 20000000000 is too big for \numexpr, shouldn't I know that? - Thanks to Jürgen Gilg for report. -- 0.7.5 (2020/01/31) - Synced with xint 1.4. Requires it. - -Files of 0.7.5 release: - -- README.md, -- polexpr.sty (package file), -- polexpr.txt (documentation), -- polexpr.html (conversion via - [DocUtils](http://docutils.sourceforge.net/docs/index.html) - rst2html.py) - -Acknowledgments ---------------- - -Thanks to Jürgen Gilg whose question about -[xint](http://www.ctan.org/pkg/xint) usage for differentiating -polynomials was the initial trigger leading to this package, and to -Jürgen Gilg and Thomas Söll for testing it on some concrete problems. diff --git a/Master/texmf-dist/doc/latex/polexpr/polexpr.html b/Master/texmf-dist/doc/latex/polexpr/polexpr.html deleted file mode 100644 index 74963329764..00000000000 --- a/Master/texmf-dist/doc/latex/polexpr/polexpr.html +++ /dev/null @@ -1,2911 +0,0 @@ -<?xml version="1.0" encoding="utf-8" ?> -<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"> -<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en" lang="en"> -<head> -<meta http-equiv="Content-Type" content="text/html; charset=utf-8" /> -<meta name="generator" content="Docutils 0.14: http://docutils.sourceforge.net/" /> -<title>Package polexpr documentation</title> -<style type="text/css"> - -/* -:Author: David Goodger (goodger@python.org) -:Id: $Id: html4css1.css 7952 2016-07-26 18:15:59Z milde $ -:Copyright: This stylesheet has been placed in the public domain. - -Default cascading style sheet for the HTML output of Docutils. - -See http://docutils.sf.net/docs/howto/html-stylesheets.html for how to -customize this style sheet. -*/ - -/* used to remove borders from tables and images */ -.borderless, table.borderless td, table.borderless th { - border: 0 } - -table.borderless td, table.borderless th { - /* Override padding for "table.docutils td" with "! important". - The right padding separates the table cells. */ - padding: 0 0.5em 0 0 ! important } - -.first { - /* Override more specific margin styles with "! important". */ - 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.literal.string { color: #0C5404 } -pre.code .name.builtin, code .name.builtin { color: #352B84 } -pre.code .deleted, code .deleted { background-color: #DEB0A1} -pre.code .inserted, code .inserted { background-color: #A3D289} - -span.classifier { - font-family: sans-serif ; - font-style: oblique } - -span.classifier-delimiter { - font-family: sans-serif ; - font-weight: bold } - -span.interpreted { - font-family: sans-serif } - -span.option { - white-space: nowrap } - -span.pre { - white-space: pre } - -span.problematic { - color: red } - -span.section-subtitle { - /* font-size relative to parent (h1..h6 element) */ - font-size: 80% } - -table.citation { - border-left: solid 1px gray; - margin-left: 1px } - -table.docinfo { - margin: 2em 4em } - -table.docutils { - margin-top: 0.5em ; - margin-bottom: 0.5em } - -table.footnote { - border-left: solid 1px black; - margin-left: 1px } - -table.docutils td, table.docutils th, -table.docinfo td, table.docinfo th { - padding-left: 0.5em ; - padding-right: 0.5em ; - vertical-align: top } - -table.docutils th.field-name, table.docinfo th.docinfo-name { - font-weight: bold ; - text-align: left ; - white-space: nowrap ; - padding-left: 0 } - -/* "booktabs" style (no vertical lines) */ -table.docutils.booktabs { - border: 0px; - border-top: 2px solid; - border-bottom: 2px solid; - border-collapse: collapse; -} -table.docutils.booktabs * { - border: 0px; -} -table.docutils.booktabs th { - border-bottom: thin solid; - text-align: left; -} - -h1 tt.docutils, h2 tt.docutils, h3 tt.docutils, -h4 tt.docutils, h5 tt.docutils, h6 tt.docutils { - font-size: 100% } - -ul.auto-toc { - list-style-type: none } - -</style> -</head> -<body> -<div class="document" id="package-polexpr-documentation"> -<h1 class="title">Package polexpr documentation</h1> -<h2 class="subtitle" id="id1">0.7.5 (2020/01/31)</h2> - -<!-- comment: -*- fill-column: 72; mode: rst; -*- --> -<div class="contents topic" id="contents"> -<p class="topic-title first">Contents</p> -<ul class="simple"> -<li><a class="reference internal" href="#basic-syntax" id="id38">Basic syntax</a></li> -<li><a class="reference internal" href="#examples-of-localization-of-roots" id="id39">Examples of localization of roots</a><ul> -<li><a class="reference internal" href="#a-typical-example" id="id40">A typical example</a></li> -<li><a class="reference internal" href="#a-degree-four-polynomial-with-nearby-roots" id="id41">A degree four polynomial with nearby roots</a></li> -<li><a class="reference internal" href="#the-degree-nine-polynomial-with-0-99-0-999-0-9999-as-triple-roots" id="id42">The degree nine polynomial with 0.99, 0.999, 0.9999 as triple roots</a></li> -<li><a class="reference internal" href="#a-degree-five-polynomial-with-three-rational-roots" id="id43">A degree five polynomial with three rational roots</a></li> -<li><a class="reference internal" href="#a-mignotte-type-polynomial" id="id44">A Mignotte type polynomial</a></li> -<li><a class="reference internal" href="#the-wilkinson-polynomial" id="id45">The Wilkinson polynomial</a></li> -<li><a class="reference internal" href="#the-second-wilkinson-polynomial" id="id46">The second Wilkinson polynomial</a></li> -<li><a class="reference internal" href="#the-degree-41-polynomial-with-2-1-9-1-8-0-0-1-1-9-2-as-roots" id="id47">The degree 41 polynomial with -2, -1.9, -1.8, ..., 0, 0.1, ..., 1.9, 2 as roots</a></li> -<li><a class="reference internal" href="#roots-of-chebyshev-polynomials" id="id48">Roots of Chebyshev polynomials</a></li> -</ul> -</li> -<li><a class="reference internal" href="#non-expandable-macros" id="id49">Non-expandable macros</a><ul> -<li><a class="reference internal" href="#poldef-polname-letter-expression-in-letter" id="id50"><tt class="docutils literal">\poldef <span class="pre">polname(letter):=</span> expression in letter;</tt></a></li> -<li><a class="reference internal" href="#poldef-letter-polname-expression-in-letter" id="id51"><tt class="docutils literal"><span class="pre">\PolDef[letter]{polname}{expression</span> in letter}</tt></a></li> -<li><a class="reference internal" href="#polgenfloatvariant-polname" id="id52"><tt class="docutils literal">\PolGenFloatVariant{polname}</tt></a></li> -<li><a class="reference internal" href="#pollet-polname-2-polname-1" id="id53"><tt class="docutils literal"><span class="pre">\PolLet{polname_2}={polname_1}</span></tt></a></li> -<li><a class="reference internal" href="#polgloballet-polname-2-polname-1" id="id54"><tt class="docutils literal"><span class="pre">\PolGlobalLet{polname_2}={polname_1}</span></tt></a></li> -<li><a class="reference internal" href="#polassign-polname-toarray-macro" id="id55"><tt class="docutils literal"><span class="pre">\PolAssign{polname}\toarray\macro</span></tt></a></li> -<li><a class="reference internal" href="#polget-polname-fromarray-macro" id="id56"><tt class="docutils literal"><span class="pre">\PolGet{polname}\fromarray\macro</span></tt></a></li> -<li><a class="reference internal" href="#polfromcsv-polname-csv" id="id57"><tt class="docutils literal"><span class="pre">\PolFromCSV{polname}{<csv>}</span></tt></a></li> -<li><a class="reference internal" href="#poltypeset-polname" id="id58"><tt class="docutils literal">\PolTypeset{polname}</tt></a><ul> -<li><a class="reference internal" href="#poltypesetcmd-raw-coeff" id="id59"><tt class="docutils literal">\PolTypesetCmd{raw_coeff}</tt></a></li> -<li><a class="reference internal" href="#poltypesetone-raw-coeff" id="id60"><tt class="docutils literal">\PolTypesetOne{raw_coeff}</tt></a></li> -<li><a class="reference internal" href="#id6" id="id61"><tt class="docutils literal">\PolTypesetMonomialCmd</tt></a></li> -<li><a class="reference internal" href="#poltypesetcmdprefix-raw-coeff" id="id62"><tt class="docutils literal">\PolTypesetCmdPrefix{raw_coeff}</tt></a></li> -</ul> -</li> -<li><a class="reference internal" href="#id8" id="id63"><tt class="docutils literal"><span class="pre">\PolTypeset*{polname}</span></tt></a></li> -<li><a class="reference internal" href="#poldiff-polname-1-polname-2" id="id64"><tt class="docutils literal"><span class="pre">\PolDiff{polname_1}{polname_2}</span></tt></a></li> -<li><a class="reference internal" href="#poldiff-n-polname-1-polname-2" id="id65"><tt class="docutils literal"><span class="pre">\PolDiff[N]{polname_1}{polname_2}</span></tt></a></li> -<li><a class="reference internal" href="#polantidiff-polname-1-polname-2" id="id66"><tt class="docutils literal"><span class="pre">\PolAntiDiff{polname_1}{polname_2}</span></tt></a></li> -<li><a class="reference internal" href="#polantidiff-n-polname-1-polname-2" id="id67"><tt class="docutils literal"><span class="pre">\PolAntiDiff[N]{polname_1}{polname_2}</span></tt></a></li> -<li><a class="reference internal" href="#poldivide-polname-1-polname-2-polname-q-polname-r" id="id68"><tt class="docutils literal"><span class="pre">\PolDivide{polname_1}{polname_2}{polname_Q}{polname_R}</span></tt></a></li> -<li><a class="reference internal" href="#polquo-polname-1-polname-2-polname-q" id="id69"><tt class="docutils literal"><span class="pre">\PolQuo{polname_1}{polname_2}{polname_Q}</span></tt></a></li> -<li><a class="reference internal" href="#polrem-polname-1-polname-2-polname-r" id="id70"><tt class="docutils literal"><span class="pre">\PolRem{polname_1}{polname_2}{polname_R}</span></tt></a></li> -<li><a class="reference internal" href="#polgcd-polname-1-polname-2-polname-gcd" id="id71"><tt class="docutils literal"><span class="pre">\PolGCD{polname_1}{polname_2}{polname_GCD}</span></tt></a></li> -<li><a class="reference internal" href="#poltosturm-polname-sturmname" id="id72"><tt class="docutils literal"><span class="pre">\PolToSturm{polname}{sturmname}</span></tt></a></li> -<li><a class="reference internal" href="#id10" id="id73"><tt class="docutils literal"><span class="pre">\PolToSturm*{polname}{sturmname}</span></tt></a></li> -<li><a class="reference internal" href="#polsettosturmchainsignchangesat-macro-sturmname-fraction" id="id74"><tt class="docutils literal"><span class="pre">\PolSetToSturmChainSignChangesAt{\macro}{sturmname}{fraction}</span></tt></a></li> -<li><a class="reference internal" href="#polsettonbofzeroswithin-macro-sturmname-value-a-value-b" id="id75"><tt class="docutils literal"><span class="pre">\PolSetToNbOfZerosWithin{\macro}{sturmname}{value_a}{value_b}</span></tt></a></li> -<li><a class="reference internal" href="#polsturmisolatezeros-sturmname" id="id76"><tt class="docutils literal">\PolSturmIsolateZeros{sturmname}</tt></a></li> -<li><a class="reference internal" href="#id12" id="id77"><tt class="docutils literal"><span class="pre">\PolSturmIsolateZeros*{sturmname}</span></tt></a></li> -<li><a class="reference internal" href="#id14" id="id78"><tt class="docutils literal"><span class="pre">\PolSturmIsolateZeros**{sturmname}</span></tt></a></li> -<li><a class="reference internal" href="#polsturmisolatezerosandgetmultiplicities-sturmname" id="id79"><tt class="docutils literal">\PolSturmIsolateZerosAndGetMultiplicities{sturmname}</tt></a></li> -<li><a class="reference internal" href="#polsturmisolatezerosgetmultiplicitiesandrationalroots-sturmname" id="id80"><tt class="docutils literal">\PolSturmIsolateZerosGetMultiplicitiesAndRationalRoots{sturmname}</tt></a></li> -<li><a class="reference internal" href="#polsturmisolatezerosandfindrationalroots-sturmname" id="id81"><tt class="docutils literal">\PolSturmIsolateZerosAndFindRationalRoots{sturmname}</tt></a></li> -<li><a class="reference internal" href="#polrefineinterval-sturmname-index" id="id82"><tt class="docutils literal"><span class="pre">\PolRefineInterval*{sturmname}{index}</span></tt></a></li> -<li><a class="reference internal" href="#polrefineinterval-n-sturmname-index" id="id83"><tt class="docutils literal"><span class="pre">\PolRefineInterval[N]{sturmname}{index}</span></tt></a></li> -<li><a class="reference internal" href="#polensureintervallength-sturmname-index-e" id="id84"><tt class="docutils literal"><span class="pre">\PolEnsureIntervalLength{sturmname}{index}{E}</span></tt></a></li> -<li><a class="reference internal" href="#polensureintervallengths-sturmname-e" id="id85"><tt class="docutils literal"><span class="pre">\PolEnsureIntervalLengths{sturmname}{E}</span></tt></a></li> -<li><a class="reference internal" href="#polprintintervals-varname-sturmname" id="id86"><tt class="docutils literal"><span class="pre">\PolPrintIntervals[varname]{sturmname}</span></tt></a><ul> -<li><a class="reference internal" href="#polprintintervalsnorealroots" id="id87"><tt class="docutils literal">\PolPrintIntervalsNoRealRoots</tt></a></li> -<li><a class="reference internal" href="#polprintintervalsbeginenv" id="id88"><tt class="docutils literal">\PolPrintIntervalsBeginEnv</tt></a></li> -<li><a class="reference internal" href="#polprintintervalsendenv" id="id89"><tt class="docutils literal">\PolPrintIntervalsEndEnv</tt></a></li> -<li><a class="reference internal" href="#polprintintervalsknownroot" id="id90"><tt class="docutils literal">\PolPrintIntervalsKnownRoot</tt></a></li> -<li><a class="reference internal" href="#polprintintervalsunknownroot" id="id91"><tt class="docutils literal">\PolPrintIntervalsUnknownRoot</tt></a></li> -<li><a class="reference internal" href="#id15" id="id92"><tt class="docutils literal">\PolPrintIntervalsPrintExactZero</tt></a></li> -<li><a class="reference internal" href="#id16" id="id93"><tt class="docutils literal">\PolPrintIntervalsPrintLeftEndPoint</tt></a></li> -<li><a class="reference internal" href="#id17" id="id94"><tt class="docutils literal">\PolPrintIntervalsPrintRightEndPoint</tt></a></li> -</ul> -</li> -<li><a class="reference internal" href="#id19" id="id95"><tt class="docutils literal"><span class="pre">\PolPrintIntervals*[varname]{sturmname}</span></tt></a><ul> -<li><a class="reference internal" href="#polprintintervalsprintmultiplicity" id="id96"><tt class="docutils literal">\PolPrintIntervalsPrintMultiplicity</tt></a></li> -</ul> -</li> -<li><a class="reference internal" href="#polmapcoeffs-macro-polname" id="id97"><tt class="docutils literal"><span class="pre">\PolMapCoeffs{\macro}{polname}</span></tt></a></li> -<li><a class="reference internal" href="#polreducecoeffs-polname" id="id98"><tt class="docutils literal">\PolReduceCoeffs{polname}</tt></a></li> -<li><a class="reference internal" href="#id21" id="id99"><tt class="docutils literal"><span class="pre">\PolReduceCoeffs*{polname}</span></tt></a></li> -<li><a class="reference internal" href="#polmakemonic-polname" id="id100"><tt class="docutils literal">\PolMakeMonic{polname}</tt></a></li> -<li><a class="reference internal" href="#polmakeprimitive-polname" id="id101"><tt class="docutils literal">\PolMakePrimitive{polname}</tt></a></li> -</ul> -</li> -<li><a class="reference internal" href="#expandable-macros" id="id102">Expandable macros</a><ul> -<li><a class="reference internal" href="#poleval-polname-atexpr-numerical-expression" id="id103"><tt class="docutils literal"><span class="pre">\PolEval{polname}\AtExpr{numerical</span> expression}</tt></a></li> -<li><a class="reference internal" href="#poleval-polname-at-fraction" id="id104"><tt class="docutils literal"><span class="pre">\PolEval{polname}\At{fraction}</span></tt></a></li> -<li><a class="reference internal" href="#polevalreduced-polname-atexpr-numerical-expression" id="id105"><tt class="docutils literal"><span class="pre">\PolEvalReduced{polname}\AtExpr{numerical</span> expression}</tt></a></li> -<li><a class="reference internal" href="#polevalreduced-polname-at-fraction" id="id106"><tt class="docutils literal"><span class="pre">\PolEvalReduced{polname}\At{fraction}</span></tt></a></li> -<li><a class="reference internal" href="#polfloateval-polname-atexpr-numerical-expression" id="id107"><tt class="docutils literal"><span class="pre">\PolFloatEval{polname}\AtExpr{numerical</span> expression}</tt></a></li> -<li><a class="reference internal" href="#polfloateval-polname-at-fraction" id="id108"><tt class="docutils literal"><span class="pre">\PolFloatEval{polname}\At{fraction}</span></tt></a></li> -<li><a class="reference internal" href="#polifcoeffisplusorminusone-a-b" id="id109"><tt class="docutils literal"><span class="pre">\PolIfCoeffIsPlusOrMinusOne{A}{B}</span></tt></a></li> -<li><a class="reference internal" href="#polleadingcoeff-polname" id="id110"><tt class="docutils literal">\PolLeadingCoeff{polname}</tt></a></li> -<li><a class="reference internal" href="#polnthcoeff-polname-number" id="id111"><tt class="docutils literal"><span class="pre">\PolNthCoeff{polname}{number}</span></tt></a></li> -<li><a class="reference internal" href="#poldegree-polname" id="id112"><tt class="docutils literal">\PolDegree{polname}</tt></a></li> -<li><a class="reference internal" href="#policontent-polname" id="id113"><tt class="docutils literal">\PolIContent{polname}</tt></a></li> -<li><a class="reference internal" href="#poltoexpr-polname" id="id114"><tt class="docutils literal">\PolToExpr{polname}</tt></a><ul> -<li><a class="reference internal" href="#poltoexproneterm-raw-coeff-number" id="id115"><tt class="docutils literal"><span class="pre">\PolToExprOneTerm{raw_coeff}{number}</span></tt></a></li> -<li><a class="reference internal" href="#poltoexpronetermstylea-raw-coeff-number" id="id116"><tt class="docutils literal"><span class="pre">\PolToExprOneTermStyleA{raw_coeff}{number}</span></tt></a></li> -<li><a class="reference internal" href="#poltoexpronetermstyleb-raw-coeff-number" id="id117"><tt class="docutils literal"><span class="pre">\PolToExprOneTermStyleB{raw_coeff}{number}</span></tt></a></li> -<li><a class="reference internal" href="#poltoexprcmd-raw-coeff" id="id118"><tt class="docutils literal">\PolToExprCmd{raw_coeff}</tt></a></li> -<li><a class="reference internal" href="#poltoexprtermprefix-raw-coeff" id="id119"><tt class="docutils literal">\PolToExprTermPrefix{raw_coeff}</tt></a></li> -<li><a class="reference internal" href="#id28" id="id120"><tt class="docutils literal">\PolToExprVar</tt></a></li> -<li><a class="reference internal" href="#id29" id="id121"><tt class="docutils literal">\PolToExprTimes</tt></a></li> -</ul> -</li> -<li><a class="reference internal" href="#id31" id="id122"><tt class="docutils literal"><span class="pre">\PolToExpr*{polname}</span></tt></a></li> -<li><a class="reference internal" href="#poltofloatexpr-polname" id="id123"><tt class="docutils literal">\PolToFloatExpr{polname}</tt></a><ul> -<li><a class="reference internal" href="#poltofloatexproneterm-raw-coeff-number" id="id124"><tt class="docutils literal"><span class="pre">\PolToFloatExprOneTerm{raw_coeff}{number}</span></tt></a></li> -<li><a class="reference internal" href="#poltofloatexprcmd-raw-coeff" id="id125"><tt class="docutils literal">\PolToFloatExprCmd{raw_coeff}</tt></a></li> -</ul> -</li> -<li><a class="reference internal" href="#id35" id="id126"><tt class="docutils literal"><span class="pre">\PolToFloatExpr*{polname}</span></tt></a></li> -<li><a class="reference internal" href="#poltolist-polname" id="id127"><tt class="docutils literal">\PolToList{polname}</tt></a></li> -<li><a class="reference internal" href="#poltocsv-polname" id="id128"><tt class="docutils literal">\PolToCSV{polname}</tt></a></li> -<li><a class="reference internal" href="#polsturmchainlength-sturmname" id="id129"><tt class="docutils literal">\PolSturmChainLength{sturmname}</tt></a></li> -<li><a class="reference internal" href="#polsturmifzeroexactlyknown-sturmname-index-a-b" id="id130"><tt class="docutils literal"><span class="pre">\PolSturmIfZeroExactlyKnown{sturmname}{index}{A}{B}</span></tt></a></li> -<li><a class="reference internal" href="#polsturmisolatedzeroleft-sturmname-index" id="id131"><tt class="docutils literal"><span class="pre">\PolSturmIsolatedZeroLeft{sturmname}{index}</span></tt></a></li> -<li><a class="reference internal" href="#polsturmisolatedzeroright-sturmname-index" id="id132"><tt class="docutils literal"><span class="pre">\PolSturmIsolatedZeroRight{sturmname}{index}</span></tt></a></li> -<li><a class="reference internal" href="#polsturmisolatedzeromultiplicity-sturmname-index" id="id133"><tt class="docutils literal"><span class="pre">\PolSturmIsolatedZeroMultiplicity{sturmname}{index}</span></tt></a></li> -<li><a class="reference internal" href="#polsturmnbofisolatedzeros-sturmname" id="id134"><tt class="docutils literal">\PolSturmNbOfIsolatedZeros{sturmname}</tt></a><ul> -<li><a class="reference internal" href="#polsturmnbofrootsof-sturmname-lessthanorequalto-value" id="id135"><tt class="docutils literal"><span class="pre">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualTo{value}</span></tt></a></li> -<li><a class="reference internal" href="#polsturmnbofrootsof-sturmname-lessthanorequaltoexpr-expression" id="id136"><tt class="docutils literal"><span class="pre">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualToExpr{expression}</span></tt></a></li> -<li><a class="reference internal" href="#polsturmnbwithmultofrootsof-sturmname-lessthanorequalto-value" id="id137"><tt class="docutils literal"><span class="pre">\PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualTo{value}</span></tt></a></li> -<li><a class="reference internal" href="#polsturmnbwithmultofrootsof-sturmname-lessthanorequaltoexpr-expression" id="id138"><tt class="docutils literal"><span class="pre">\PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualToExpr{expression}</span></tt></a></li> -</ul> -</li> -<li><a class="reference internal" href="#polsturmnbofrationalroots-sturmname" id="id139"><tt class="docutils literal">\PolSturmNbOfRationalRoots{sturmname}</tt></a></li> -<li><a class="reference internal" href="#polsturmnbofrationalrootswithmultiplicities-sturmname" id="id140"><tt class="docutils literal">\PolSturmNbOfRationalRootsWithMultiplicities{sturmname}</tt></a></li> -<li><a class="reference internal" href="#polsturmrationalroot-sturmname-k" id="id141"><tt class="docutils literal"><span class="pre">\PolSturmRationalRoot{sturmname}{k}</span></tt></a></li> -<li><a class="reference internal" href="#polsturmrationalrootindex-sturmname-k" id="id142"><tt class="docutils literal"><span class="pre">\PolSturmRationalRootIndex{sturmname}{k}</span></tt></a></li> -<li><a class="reference internal" href="#polsturmrationalrootmultiplicity-sturmname-k" id="id143"><tt class="docutils literal"><span class="pre">\PolSturmRationalRootMultiplicity{sturmname}{k}</span></tt></a></li> -<li><a class="reference internal" href="#polintervalwidth-sturmname-index" id="id144"><tt class="docutils literal"><span class="pre">\PolIntervalWidth{sturmname}{index}</span></tt></a></li> -<li><a class="reference internal" href="#expandable-macros-for-use-within-execution-of-polprintintervals" id="id145">Expandable macros for use within execution of <tt class="docutils literal">\PolPrintIntervals</tt></a><ul> -<li><a class="reference internal" href="#polprintintervalsthevar" id="id146"><tt class="docutils literal">\PolPrintIntervalsTheVar</tt></a></li> -<li><a class="reference internal" href="#polprintintervalstheindex" id="id147"><tt class="docutils literal">\PolPrintIntervalsTheIndex</tt></a></li> -<li><a class="reference internal" href="#polprintintervalsthesturmname" id="id148"><tt class="docutils literal">\PolPrintIntervalsTheSturmName</tt></a></li> -<li><a class="reference internal" href="#polprintintervalstheleftendpoint" id="id149"><tt class="docutils literal">\PolPrintIntervalsTheLeftEndPoint</tt></a></li> -<li><a class="reference internal" href="#polprintintervalstherightendpoint" id="id150"><tt class="docutils literal">\PolPrintIntervalsTheRightEndPoint</tt></a></li> -<li><a class="reference internal" href="#polprintintervalsthemultiplicity" id="id151"><tt class="docutils literal">\PolPrintIntervalsTheMultiplicity</tt></a></li> -</ul> -</li> -<li><a class="reference internal" href="#poldectostring-decimal-number" id="id152"><tt class="docutils literal">\PolDecToString{decimal number}</tt></a></li> -</ul> -</li> -<li><a class="reference internal" href="#booleans-with-default-setting-as-indicated" id="id153">Booleans (with default setting as indicated)</a><ul> -<li><a class="reference internal" href="#xintverbosefalse" id="id154"><tt class="docutils literal">\xintverbosefalse</tt></a></li> -<li><a class="reference internal" href="#poltypesetallfalse" id="id155"><tt class="docutils literal">\poltypesetallfalse</tt></a></li> -<li><a class="reference internal" href="#poltoexprallfalse" id="id156"><tt class="docutils literal">\poltoexprallfalse</tt></a></li> -</ul> -</li> -<li><a class="reference internal" href="#polexprsetup" id="id157"><tt class="docutils literal">\polexprsetup</tt></a></li> -<li><a class="reference internal" href="#technicalities" id="id158">Technicalities</a></li> -<li><a class="reference internal" href="#change-log" id="id159">CHANGE LOG</a></li> -<li><a class="reference internal" href="#acknowledgments" id="id160">Acknowledgments</a></li> -</ul> -</div> -<div class="section" id="basic-syntax"> -<h1><a class="toc-backref" href="#id38">Basic syntax</a></h1> -<p>The syntax is:</p> -<pre class="literal-block"> -\poldef polname(x):= expression in variable x; -</pre> -<p>where:</p> -<ul class="simple"> -<li>in place of <tt class="docutils literal">x</tt> an arbitrary <em>dummy variable</em> is authorized, -i.e. per default any of <tt class="docutils literal"><span class="pre">[a-z|A-Z]</span></tt> (more letters can be declared -under Unicode engines.)</li> -<li><tt class="docutils literal">polname</tt> consists of letters, digits, and the <tt class="docutils literal">_</tt> and -<tt class="docutils literal">'</tt> characters. It must start with a letter.</li> -</ul> -<div class="admonition attention"> -<p class="first admonition-title">Attention!</p> -<p class="last">The <tt class="docutils literal">'</tt> is authorized since <tt class="docutils literal">0.5.1</tt>. As a result some constructs -recognized by the <tt class="docutils literal">\xintexpr</tt> parser, such as <tt class="docutils literal">var1 'and' var2</tt> -will get misinterpreted and cause errors. However these constructs -are unlikely to be frequently needed in polynomial expressions, and -the <tt class="docutils literal">\xintexpr</tt> syntax offers alternatives, so it was deemed a -small evil. Of course the <tt class="docutils literal">\xintexpr</tt> parser is modified only -temporarily during execution of <tt class="docutils literal">\poldef</tt>.</p> -</div> -<p>One can also issue:</p> -<pre class="literal-block"> -\PolDef{polname}{expression in variable x} -</pre> -<p>which admits an optional first argument to modify the variable letter -from its default <tt class="docutils literal">x</tt>.</p> -<dl class="docutils"> -<dt><tt class="docutils literal">\poldef <span class="pre">f(x):=</span> <span class="pre">1-x+x^2;</span></tt></dt> -<dd>defines polynomial <tt class="docutils literal">f</tt>. Polynomial names must start with a -letter and may contain letters, digits, underscores and the right -tick character. The -variable must be a single letter. The colon character is optional. -The semi-colon at end of expression is mandatory.</dd> -<dt><tt class="docutils literal"><span class="pre">\PolDef{f}{1-x+x^2}</span></tt></dt> -<dd>does the same as <tt class="docutils literal">\poldef <span class="pre">f(x):=</span> <span class="pre">1-x+x^2;</span></tt> To use another letter -than <tt class="docutils literal">x</tt> in the expression, one must pass it as an extra optional -argument to <tt class="docutils literal">\PolDef</tt>. Useful if the semi-colon has been assigned -some non-standard catcode by some package.</dd> -<dt><tt class="docutils literal"><span class="pre">\PolLet{g}={f}</span></tt></dt> -<dd>saves a copy of <tt class="docutils literal">f</tt> under name <tt class="docutils literal">g</tt>. Also usable without <tt class="docutils literal">=</tt>.</dd> -<dt><tt class="docutils literal">\poldef <span class="pre">f(z):=</span> <span class="pre">f(z)^2;</span></tt></dt> -<dd>redefines <tt class="docutils literal">f</tt> in terms of itself.</dd> -<dt><tt class="docutils literal">\poldef <span class="pre">f(T):=</span> <span class="pre">f(f(T));</span></tt></dt> -<dd>again redefines <tt class="docutils literal">f</tt> in terms of its (new) self.</dd> -<dt><tt class="docutils literal">\poldef <span class="pre">k(z):=</span> <span class="pre">f(z)-g(g(z)^2)^2;</span></tt></dt> -<dd>should now define the zero polynomial... Let's check: -<tt class="docutils literal">\[ k(z) = <span class="pre">\PolTypeset[z]{k}</span> \]</tt></dd> -<dt><tt class="docutils literal"><span class="pre">\PolDiff{f}{f'}</span></tt></dt> -<dd>sets <tt class="docutils literal">f'</tt> to the derivative of <tt class="docutils literal">f</tt>. The name doesn't have to be -<tt class="docutils literal">f'</tt> (in fact the <tt class="docutils literal">'</tt> is licit only since <tt class="docutils literal">0.5.1</tt>).</dd> -</dl> -<div class="admonition important"> -<p class="first admonition-title">Important</p> -<p class="last">This is not done automatically. If some new definition needs to use -the derivative of some available polynomial, that derivative -polynomial must have been defined via <tt class="docutils literal">\PolDiff</tt>: something like -<tt class="docutils literal"><span class="pre">T'(x)^2</span></tt> will not work without a prior <tt class="docutils literal"><span class="pre">\PolDiff{T}{T'}</span></tt>.</p> -</div> -<dl class="docutils"> -<dt><tt class="docutils literal"><span class="pre">\PolDiff{f'}{f''}</span></tt></dt> -<dd>obtains second derivative.</dd> -<dt><tt class="docutils literal"><span class="pre">\PolDiff[3]{f}{f'''}</span></tt></dt> -<dd>computes the third derivative.</dd> -</dl> -<pre class="literal-block"> -$f(z) = \PolTypeset[z]{f} $\newline -$f'(z) = \PolTypeset[z]{f'} $\newline -$f''(z) = \PolTypeset[z]{f''} $\newline -$f'''(z)= \PolTypeset[z]{f'''} $\par -</pre> -<div class="admonition important"> -<p class="first admonition-title">Important</p> -<p>The package does not currently know rational functions: <tt class="docutils literal">/</tt> in -a parsed polynomial expression does the Euclidean quotient:</p> -<pre class="literal-block"> -(1-x^2)/(1-x) -</pre> -<p>does give <tt class="docutils literal">1+x</tt> but</p> -<pre class="literal-block"> -(1/(1-x))*(1-x^2) -</pre> -<p>evaluates to zero. This will work as expected:</p> -<pre class="last literal-block"> -\poldef k(x):= (x-1)(x-2)(x-3)(x-4)/(x^2-5x+4); -</pre> -</div> -<div class="admonition attention" id="warningtacit"> -<p class="first admonition-title">Attention!</p> -<p><tt class="docutils literal">1/2 x^2</tt> skips the space and is treated like <tt class="docutils literal"><span class="pre">1/(2*x^2)</span></tt> because -of the tacit multiplication rules of xintexpr. But this means it -gives zero! Thus one must use <tt class="docutils literal">(1/2)x^2</tt> or <tt class="docutils literal">1/2*x^2</tt> or -<tt class="docutils literal"><span class="pre">(1/2)*x^2</span></tt> for disambiguation: <tt class="docutils literal">x - 1/2*x^2 + <span class="pre">1/3*x^3...</span></tt>. It is -even simpler to move the denominator to the right: <tt class="docutils literal">x - x^2/2 + -x^3/3 - ...</tt>.</p> -<p class="last">It is worth noting that <tt class="docutils literal"><span class="pre">1/2(x-1)(x-2)</span></tt> suffers the same issue: -<a class="reference external" href="http://www.ctan.org/pkg/xint">xint</a> tacit multiplication always "ties more", hence this gets -interpreted as <tt class="docutils literal"><span class="pre">1/(2*(x-1)*(x-2))</span></tt> which gives zero by polynomial -division. Thus, use one of <tt class="docutils literal"><span class="pre">(1/2)(x-1)(x-2)</span></tt>, <tt class="docutils literal"><span class="pre">1/2*(x-1)(x-2)</span></tt> or -<tt class="docutils literal"><span class="pre">(x-1)(x-2)/2</span></tt>.</p> -</div> -<p>After:</p> -<pre class="literal-block"> -\poldef f_1(x):= 25(x-1)(x^2-2)(x-3)(x-4)(x-5);% -\poldef f_2(x):= 37(x-1)(x^2-2)(x-6)(x-7)(x-8);% -</pre> -<p>the macro call <tt class="docutils literal"><span class="pre">\PolGCD{f_1}{f_2}{k}</span></tt> sets <tt class="docutils literal">k</tt> to the (unitary) GCD of -<tt class="docutils literal">f_1</tt> and <tt class="docutils literal">f_2</tt> (hence to the expansion of <tt class="docutils literal"><span class="pre">(x-1)(x^2-2)</span></tt>.)</p> -<dl class="docutils"> -<dt><tt class="docutils literal">\PolToExpr{k}</tt></dt> -<dd>will (expandably) give in this case <tt class="docutils literal"><span class="pre">x^3-x^2-2*x+2</span></tt>. This is -useful for console or file output (the syntax is Maple- and -PSTricks-compatible; the letter used in output can be -(non-expandably) changed via a redefinition of <a class="reference internal" href="#poltoexprvar">\PolToExprVar</a>.)</dd> -<dt><tt class="docutils literal"><span class="pre">\PolToExpr*{k}</span></tt></dt> -<dd>gives ascending powers: <tt class="docutils literal"><span class="pre">2-2*x-x^2+x^3</span></tt>.</dd> -</dl> -</div> -<div class="section" id="examples-of-localization-of-roots"> -<h1><a class="toc-backref" href="#id39">Examples of localization of roots</a></h1> -<ul> -<li><p class="first">To make printed decimal numbers more enjoyable than via -<tt class="docutils literal">\xintSignedFrac</tt>:</p> -<pre class="literal-block"> -\renewcommand\PolTypesetOne[1]{\PolDecToString{\xintREZ{#1}}}% -</pre> -<p><tt class="docutils literal">\PolDecToString</tt> will use decimal notation to incorporate the power -of ten part; and the <tt class="docutils literal">\xintREZ</tt> will have the effect to suppress -trailing zeros if present in raw numerator (if those digits end up -after decimal mark.) Notice that the above are expandable macros and -that one can also do:</p> -<pre class="literal-block"> -\renewcommand\PolToExprCmd[1]{\PolDecToString{\xintREZ{#1}}}% -</pre> -<p>to modify output of <a class="reference internal" href="#poltoexpr-polname">\PolToExpr{polname}</a>.</p> -</li> -<li><p class="first">For extra info in log file use <tt class="docutils literal">\xintverbosetrue</tt>.</p> -</li> -<li><p class="first">Only for some of these examples is the output included here.</p> -</li> -</ul> -<div class="section" id="a-typical-example"> -<h2><a class="toc-backref" href="#id40">A typical example</a></h2> -<p>In this example the polynomial is square-free.</p> -<pre class="literal-block"> -\poldef f(x) := x^7 - x^6 - 2x + 1; - -\PolToSturm{f}{f} -\PolSturmIsolateZeros{f} -The \PolTypeset{f} polynomial has \PolSturmNbOfIsolatedZeros{f} distinct real -roots which are located in the following intervals: -\PolPrintIntervals{f} -Here is the second root with ten more decimal digits: -\PolRefineInterval[10]{f}{2} -\[\PolSturmIsolatedZeroLeft{f}{2}<Z_2<\PolSturmIsolatedZeroRight{f}{2}\] -And here is the first root with twenty digits after decimal mark: -\PolEnsureIntervalLength{f}{1}{-20} -\[\PolSturmIsolatedZeroLeft{f}{1}<Z_1<\PolSturmIsolatedZeroRight{f}{1}\] -The first element of the Sturm chain has degree $\PolDegree{f_0}$. As -this is the original degreee $\PolDegree{f}$ we know that $f$ is square free. -Its derivative is up to a constant \PolTypeset{f_1} (in this example -it is identical with it). -\PolToSturm{f_1}{f_1}\PolSturmIsolateZeros{f_1}% -The derivative has \PolSturmNbOfIsolatedZeros{f_1} distinct real -roots: -\PolPrintIntervals[W]{f_1} -\PolEnsureIntervalLengths{f_1}{-10}% -Here they are with ten digits after decimal mark: -\PolPrintIntervals[W]{f_1} -\PolDiff{f_1}{f''} -\PolToSturm{f''}{f''} -\PolSturmIsolateZeros{f''} -The second derivative is \PolTypeset{f''}. -It has \PolSturmNbOfIsolatedZeros{f''} distinct real -roots: -\PolPrintIntervals[X]{f''} -Here is the positive one with 20 digits after decimal mark: -\PolEnsureIntervalLength{f''}{2}{-20}% -\[X_2 = \PolSturmIsolatedZeroLeft{f''}{2}\dots\] -The more mathematically advanced among our dear readers will be able -to give the exact value for $X_2$! -</pre> -</div> -<div class="section" id="a-degree-four-polynomial-with-nearby-roots"> -<h2><a class="toc-backref" href="#id41">A degree four polynomial with nearby roots</a></h2> -<p>Notice that this example is a bit outdated as <tt class="docutils literal">0.7</tt> release has -added <tt class="docutils literal"><span class="pre">\PolSturmIsolateZeros**{sturmname}</span></tt> which would find exactly -the roots. The steps here retain their interest when one is interested -in finding isolating intervals for example to prepare some demonstration -of dichotomy method.</p> -<pre class="literal-block"> -\PolDef{Q}{(x-1.050001)(x-1.105001)(x-1.110501)(x-1.111051)} -\PolTypeset{Q} -\PolToSturm{Q}{Q} % it is allowed to use same prefix for Sturm chain -\PolSturmIsolateZeros{Q} -\PolPrintIntervals{Q} -% reports 1.0 < Z_1 < 1.1, 1.10 < Z_2 < 1.11, 1.110 < Z_3 < 1.111, and 1.111 < Z_4 < 1.112 -% but the above bounds do not allow minimizing separation between roots -% so we refine: -\PolRefineInterval*{Q}{1} -\PolRefineInterval*{Q}{2} -\PolRefineInterval*{Q}{3} -\PolRefineInterval*{Q}{4} -\PolPrintIntervals{Q} -% reports 1.05 < Z_1 < 1.06, 1.105 < Z_2 < 1.106, 1.1105 < Z_3 < 1.1106, -% and 1.11105 < Z_4 < 1.11106. -\PolEnsureIntervalLengths{Q}{-6} -\PolPrintIntervals{Q} -% of course finds here all roots exactly -</pre> -</div> -<div class="section" id="the-degree-nine-polynomial-with-0-99-0-999-0-9999-as-triple-roots"> -<h2><a class="toc-backref" href="#id42">The degree nine polynomial with 0.99, 0.999, 0.9999 as triple roots</a></h2> -<pre class="literal-block"> -% define a user command (xinttools is loaded automatically by polexpr) -\newcommand\showmultiplicities[1]{% #1 = "sturmname" -\xintFor* ##1 in {\xintSeq{1}{\PolSturmNbOfIsolatedZeros{#1}}}\do{% - The multiplicity is \PolSturmIsolatedZeroMultiplicity{#1}{##1} - \PolSturmIfZeroExactlyKnown{#1}{##1}% - {at the root $x=\PolSturmIsolatedZeroLeft{#1}{##1}$} - {for the root such that - $\PolSturmIsolatedZeroLeft{#1}{##1}<x<\PolSturmIsolatedZeroRight{#1}{##1}$} - \par -}}% -\PolDef{f}{(x-0.99)^3(x-0.999)^3(x-0.9999)^3} -\renewcommand\PolTypesetOne[1]{\PolDecToString{\xintREZ{#1}}} -\PolTypeset{f}\par -\PolToSturm{f}{f}% it is allowed to use "polname" as "sturmname" too -\PolSturmIsolateZerosAndGetMultiplicities{f}% use the "sturmname" here -% or \PolSturmIsolateZeros*{f} which is exactly the same, but shorter.. - -\showmultiplicities{f} -</pre> -<p>In this example, the output will look like this (but using math mode):</p> -<pre class="literal-block"> -x^9 - 8.9667x^8 + 35.73400293x^7 - 83.070418400109x^6 + 124.143648875193123x^5 -- 123.683070924326075877x^4 + 82.149260397553075617891x^3 -- 35.07602992699900159127007x^2 + 8.7364078733314648368671733x -- 0.967100824643585986488103299 - -The multiplicity is 3 at the root x = 0.99 -The multiplicity is 3 at the root x = 0.999 -The multiplicity is 3 at the root x = 0.9999 -</pre> -<p>On first pass, these rational roots were found (due to their relative -magnitudes, using <tt class="docutils literal">\PolSturmIsolateZeros**</tt> was not needed here). But -multiplicity computation works also with (decimal) roots not yet -identified or with non-decimal or irrational roots.</p> -<p>It is fun to modify only a tiny bit the polynomial and see if polexpr -survives:</p> -<pre class="literal-block"> -\PolDef{g}{f(x)+1e-27} -\PolTypeset{g}\par -\PolToSturm{g}{g} -\PolSturmIsolateZeros*{g} - -\showmultiplicities{g} -</pre> -<p>This produces:</p> -<pre class="literal-block"> -x^9 - 8.9667x^8 + 35.73400293x^7 - 83.070418400109x^6 + 124.143648875193123x^5 -- 123.683070924326075877x^4 + 82.149260397553075617891x^3 -- 35.07602992699900159127007x^2 + 8.7364078733314648368671733x -- 0.967100824643585986488103298 - -The multiplicity is 1 for the root such that 0.98 < x < 0.99 -The multiplicity is 1 for the root such that 0.9991 < x < 0.9992 -The multiplicity is 1 for the root such that 0.9997 < x < 0.9998 -</pre> -<p>Which means that the multiplicity-3 roots each became a real and a pair of -complex ones. Let's see them better:</p> -<pre class="literal-block"> -\PolEnsureIntervalLengths{g}{-10} - -\showmultiplicities{g} -</pre> -<p>which produces:</p> -<pre class="literal-block"> -The multiplicity is 1 for the root such that 0.9899888032 < x < 0.9899888033 -The multiplicity is 1 for the root such that 0.9991447980 < x < 0.9991447981 -The multiplicity is 1 for the root such that 0.9997663986 < x < 0.9997663987 -</pre> -</div> -<div class="section" id="a-degree-five-polynomial-with-three-rational-roots"> -<h2><a class="toc-backref" href="#id43">A degree five polynomial with three rational roots</a></h2> -<pre class="literal-block"> -\poldef Q(x) := 1581755751184441 x^5 - -14907697165025339 x^4 - +48415668972339336 x^3 - -63952057791306264 x^2 - +46833913221154895 x - -49044360626280925; - -\PolToSturm{Q}{Q} -%\begin{flushleft} - \renewcommand\PolTypesetCmdPrefix[1]{\allowbreak\xintiiifSgn{#1}{}{+}{+}}% - $Q_0(x) = \PolTypeset{Q_0}$ -%\end{flushleft} -\PolSturmIsolateZeros**{Q} -\PolPrintIntervals{Q} - -$Q_{norr}(x) = \PolTypeset{Q_norr}$ -</pre> -<p>Here, all real roots are rational:</p> -<pre class="literal-block"> -Z_1 = 833719/265381 -Z_2 = 165707065/52746197 -Z_3 = 355/113 - -Q_norr(x) = x^2 + 1 -</pre> -<p>And let's get their decimal expansion too:</p> -<pre class="literal-block"> -% print decimal expansion of the found roots -\renewcommand\PolPrintIntervalsPrintExactZero - {\xintTrunc{20}{\PolPrintIntervalsTheLeftEndPoint}\dots} -\PolPrintIntervals{Q} - -Z_1 = 3.14159265358107777120... -Z_2 = 3.14159265358979340254... -Z_3 = 3.14159292035398230088... -</pre> -</div> -<div class="section" id="a-mignotte-type-polynomial"> -<h2><a class="toc-backref" href="#id44">A Mignotte type polynomial</a></h2> -<pre class="literal-block"> -\PolDef{P}{x^10 - (10x-1)^2}% -\PolTypeset{P} % prints it in expanded form -\PolToSturm{P}{P} % we can use same prefix for Sturm chain -\PolSturmIsolateZeros{P} % finds 4 real roots -This polynomial has \PolSturmNbOfIsolatedZeros{P} distinct real roots: -\PolPrintIntervals{P}% -% reports -2 < Z_1 < -1, 0.09 < Z_2 < 0.10, 0.1 < Z_3 < 0.2, 1 < Z_4 < 2 -Let us refine the second and third intervals to separate the corresponding -roots: -\PolRefineInterval*{P}{2}% will refine to 0.0999990 < Z_2 < 0.0999991 -\PolRefineInterval*{P}{3}% will refine to 0.100001 < Z_3 < 0.100002 -\PolPrintIntervals{P}% -Let us now get to know all roots with 10 digits after decimal mark: -\PolEnsureIntervalLengths{P}{-10}% -\PolPrintIntervals{P}% now all roots are known 10 decimal digits after mark -Finally, we display 20 digits of the second root: -\PolEnsureIntervalLength{P}{2}{-20}% makes Z_2 known with 20 digits after mark -\[\PolSturmIsolatedZeroLeft{P}{2}<Z_2<\PolSturmIsolatedZeroRight{P}{2}\] -</pre> -<p>The last line produces:</p> -<pre class="literal-block"> -0.09999900004999650028 < Z_2 < 0.09999900004999650029 -</pre> -</div> -<div class="section" id="the-wilkinson-polynomial"> -<h2><a class="toc-backref" href="#id45">The Wilkinson polynomial</a></h2> -<p>See <a class="reference external" href="https://en.wikipedia.org/wiki/Wilkinson%27s_polynomial">Wilkinson polynomial</a>.</p> -<pre class="literal-block"> -\documentclass{article} -\usepackage{polexpr} -\begin{document} -%\xintverbosetrue % for the curious... - -\poldef f(x) := mul((x - i), i = 1..20); - -\renewcommand\PolTypesetCmdPrefix[1]{\allowbreak\xintiiifSgn{#1}{}{+}{+}}% -\renewcommand\PolTypesetOne[1]{\xintDecToString{#1}}% - -\noindent\PolTypeset{f} - -\PolToSturm{f}{f} -\PolSturmIsolateZeros{f} -\PolPrintIntervals{f} - -\clearpage - -\poldef g(x) := f(x) - 2**{-23} x**19; - -% be patient! -\PolToSturm{g}{g} -\noindent\PolTypeset{g_0}% integer coefficient primitive polynomial - -\PolSturmIsolateZeros{g} -\PolEnsureIntervalLengths{g}{-10} - -\renewcommand\PolPrintIntervalsPrintMultiplicity{} -\PolPrintIntervals*{g} - -\end{document} -</pre> -<p>The first polynomial:</p> -<pre class="literal-block"> -f(x) = x**20 -- 210 x**19 -+ 20615 x**18 -- 1256850 x**17 -+ 53327946 x**16 -- 1672280820 x**15 -+ 40171771630 x**14 -- 756111184500 x**13 -+ 11310276995381 x**12 -- 135585182899530 x**11 -+ 1307535010540395 x**10 -- 10142299865511450 x**9 -+ 63030812099294896 x**8 -- 311333643161390640 x**7 -+ 1206647803780373360 x**6 -- 3599979517947607200 x**5 -+ 8037811822645051776 x**4 -- 12870931245150988800 x**3 -+ 13803759753640704000 x**2 -- 8752948036761600000 x -+ 2432902008176640000 -</pre> -<p>is handled fast enough (a few seconds), but the modified one <tt class="docutils literal">f(x) - -<span class="pre">2**-23</span> <span class="pre">x**19</span></tt> takes about 20x longer (the Sturm chain polynomials -have integer coefficients with up to 321 digits, whereas (surprisingly -perhaps) those of the Sturm chain polynomials derived from <tt class="docutils literal">f</tt> never -have more than 21 digits ...).</p> -<p>Once the Sturm chain is computed and the zeros isolated, obtaining their -decimal digits is relatively faster. Here is for the ten real roots of -<tt class="docutils literal">f(x) - <span class="pre">2**-23</span> <span class="pre">x**19</span></tt> as computed by the code above:</p> -<pre class="literal-block"> -Z_1 = 0.9999999999... -Z_2 = 2.0000000000... -Z_3 = 2.9999999999... -Z_4 = 4.0000000002... -Z_5 = 4.9999999275... -Z_6 = 6.0000069439... -Z_7 = 6.9996972339... -Z_8 = 8.0072676034... -Z_9 = 8.9172502485... -Z_10 = 20.8469081014... -</pre> -</div> -<div class="section" id="the-second-wilkinson-polynomial"> -<h2><a class="toc-backref" href="#id46">The second Wilkinson polynomial</a></h2> -<pre class="literal-block"> -\documentclass{article} -\usepackage{polexpr} -\begin{document} -\poldef f(x) := mul(x - 2^-i, i = 1..20); - -%\PolTypeset{f} - -\PolToSturm{f}{f} -\PolSturmIsolateZeros**{f} -\PolPrintIntervals{f} -\end{document} -</pre> -<p>This takes more time than the polynomial with 1, 2, .., 20 as roots but -less than the latter modified by the <tt class="docutils literal"><span class="pre">2**-23</span></tt> change in one -coefficient.</p> -<p>Here is the output (with release 0.7.2):</p> -<pre class="literal-block"> -Z_1 = 0.00000095367431640625 -Z_2 = 0.0000019073486328125 -Z_3 = 0.000003814697265625 -Z_4 = 0.00000762939453125 -Z_5 = 0.0000152587890625 -Z_6 = 0.000030517578125 -Z_7 = 0.00006103515625 -Z_8 = 0.0001220703125 -Z_9 = 1/4096 -Z_10 = 1/2048 -Z_11 = 1/1024 -Z_12 = 1/512 -Z_13 = 1/256 -Z_14 = 1/128 -Z_15 = 0.015625 -Z_16 = 0.03125 -Z_17 = 0.0625 -Z_18 = 0.125 -Z_19 = 0.25 -Z_20 = 0.5 -</pre> -<p>There is some incoherence in output format which has its source in the -fact that some roots are found in branches which can only find decimal -roots, whereas some are found in branches which could find general -fractions and they use <tt class="docutils literal">\xintIrr</tt> before storage of the found root. -This may evolve in future.</p> -</div> -<div class="section" id="the-degree-41-polynomial-with-2-1-9-1-8-0-0-1-1-9-2-as-roots"> -<h2><a class="toc-backref" href="#id47">The degree 41 polynomial with -2, -1.9, -1.8, ..., 0, 0.1, ..., 1.9, 2 as roots</a></h2> -<pre class="literal-block"> -\PolDef{P}{mul((x-i*1e-1), i=-20..20)}% i/10 is same but less efficient -</pre> -<p>In the defining expression we could have used <tt class="docutils literal">i/10</tt> but this gives -less efficient internal form for the coefficients (the <tt class="docutils literal">10</tt>'s end up -in denominators). Using <tt class="docutils literal">\PolToExpr{P}</tt> after having done</p> -<pre class="literal-block"> -\renewcommand\PolToExprCmd[1]{\PolDecToString{\xintREZ{#1}}} -</pre> -<p>we get this expanded form:</p> -<pre class="literal-block"> -x^41 --28.7*x^39 -+375.7117*x^37 --2975.11006*x^35 -+15935.28150578*x^33 --61167.527674162*x^31 -+173944.259366417394*x^29 --373686.963560544648*x^27 -+613012.0665016658846445*x^25 --771182.31133138163125495*x^23 -+743263.86672885754888959569*x^21 --545609.076599482896371978698*x^19 -+301748.325708943677229642930528*x^17 --123655.8987669450434698869844544*x^15 -+36666.1782054884005855608205864192*x^13 --7607.85821367459445649518380016128*x^11 -+1053.15135918687298508885950223794176*x^9 --90.6380005918141132650786081964032*x^7 -+4.33701563847327366842552218288128*x^5 --0.0944770968420804735498178265088*x^3 -+0.00059190121813899276854174416896*x -</pre> -<p>which shows coefficients with up to 36 significant digits...</p> -<p>Stress test: not a hard challenge to <tt class="docutils literal">xint + polexpr</tt>, but be a bit patient!</p> -<pre class="literal-block"> -\PolDef{P}{mul((x-i*1e-1), i=-20..20)}% -\PolToSturm{P}{S} % dutifully computes S_0, ..., S_{41} -% the [1] optional argument limits the search to interval (-10,10) -\PolSturmIsolateZeros[1]{S} % finds *exactly* (but a bit slowly) all 41 roots! -\PolPrintIntervals{S} % nice, isn't it? -</pre> -<div class="admonition note"> -<p class="first admonition-title">Note</p> -<p>Release <tt class="docutils literal">0.5</tt> has <em>experimental</em> addition of optional argument -<tt class="docutils literal">E</tt> to <tt class="docutils literal">\PolSturmIsolateZeros</tt>. It instructs to search roots only -in interval <tt class="docutils literal"><span class="pre">(-10^E,</span> 10^E)</tt>. Important: the extremities are -<em>assumed to not be roots</em>. In this example, the <tt class="docutils literal">[1]</tt> in -<tt class="docutils literal"><span class="pre">\PolSturmIsolateZeros[1]{S}</span></tt> gives some speed gain; without it, it -turns out in this case that <tt class="docutils literal">polexpr</tt> would have started with -<tt class="docutils literal"><span class="pre">(-10^6,</span> 10^6)</tt> interval.</p> -<p class="last">Please note that this will probably get replaced in future by the -specification of a general interval. Do not rely on meaning of this -optional argument keeping the same.</p> -</div> -</div> -<div class="section" id="roots-of-chebyshev-polynomials"> -<h2><a class="toc-backref" href="#id48">Roots of Chebyshev polynomials</a></h2> -<pre class="literal-block"> -\newcount\mycount -\poldef T_0(x) := 1; -\poldef T_1(x) := x; -\mycount 2 -\xintloop - \poldef T_\the\mycount(x) := - 2x*T_\the\numexpr\mycount-1(x) - - T_\the\numexpr\mycount-2(x); -\ifnum\mycount<15 -\advance\mycount 1 -\repeat - -\[T_{15} = \PolTypeset[X]{T_15}\] -\PolToSturm{T_15}{T_15} -\PolSturmIsolateZeros{T_15} -\PolEnsureIntervalLengths{T_15}{-10} -\PolPrintIntervals{T_15} -</pre> -</div> -</div> -<div class="section" id="non-expandable-macros"> -<h1><a class="toc-backref" href="#id49">Non-expandable macros</a></h1> -<div class="section" id="poldef-polname-letter-expression-in-letter"> -<span id="poldef"></span><h2><a class="toc-backref" href="#id50"><tt class="docutils literal">\poldef <span class="pre">polname(letter):=</span> expression in letter;</tt></a></h2> -<blockquote> -<p>This evaluates the <em>polynomial expression</em> and stores the coefficients -in a private structure accessible later via other package macros, -under the user-chosen <tt class="docutils literal">polname</tt>. Of course the <em>expression</em> can -use other previously defined polynomials. Names must start with a -letter and are constituted of letters, digits, underscores and -(since <tt class="docutils literal">0.5.1</tt>) the right tick <tt class="docutils literal">'</tt>. -The whole <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> syntax is authorized:</p> -<pre class="literal-block"> -\poldef sin(z) := add((-1)^i z^(2i+1)/(2i+1)!, i = 0..10); -</pre> -<p>With fractional coefficients, beware the <a class="reference internal" href="#warningtacit">tacit multiplication issue</a>.</p> -<p>As a side effect the function <tt class="docutils literal">polname()</tt> is recognized as a -genuine <tt class="docutils literal"><span class="pre">\xintexpr...\relax</span></tt> function for (exact) numerical -evaluation (or within an <tt class="docutils literal">\xintdefvar</tt> assignment.) It computes -values not according to the original expression but via the Horner -scheme corresponding to the polynomial coefficients.</p> -<div class="admonition attention"> -<p class="first admonition-title">Attention!</p> -<p>Release <tt class="docutils literal">0.3</tt> also did the necessary set-up to let the -polynomial be known to the <tt class="docutils literal">\xintfloatexpr</tt> (or -<tt class="docutils literal">\xintdeffloatvar</tt>) parser.</p> -<p>Since <tt class="docutils literal">0.4</tt> this isn't done automatically. Even more, a -previously existing floating point variant of the same name will -be let undefined again, to avoid hard to debug mismatches between -exact and floating point polynomials. This also applies when the -polynomial is produced not via <tt class="docutils literal">\poldef</tt> or <tt class="docutils literal">\PolDef</tt> but as -a product of the other package macros.</p> -<p class="last">See <a class="reference internal" href="#polgenfloatvariant-polname">\PolGenFloatVariant{polname}</a>.</p> -</div> -<p>The original expression is lost after parsing, and in particular -the package provides no way to typeset it. This has to be done -manually, if needed.</p> -</blockquote> -</div> -<div class="section" id="poldef-letter-polname-expression-in-letter"> -<span id="id2"></span><h2><a class="toc-backref" href="#id51"><tt class="docutils literal"><span class="pre">\PolDef[letter]{polname}{expression</span> in letter}</tt></a></h2> -<blockquote> -Does the same as <a class="reference external" href="poldef;">\poldef</a> in an undelimited macro -format (thus avoiding potential problems with the catcode of the -semi-colon in presence of some packages.) In absence of the -<tt class="docutils literal">[letter]</tt> optional argument, the variable is assumed to be <tt class="docutils literal">x</tt>.</blockquote> -</div> -<div class="section" id="polgenfloatvariant-polname"> -<span id="polgenfloatvariant"></span><h2><a class="toc-backref" href="#id52"><tt class="docutils literal">\PolGenFloatVariant{polname}</tt></a></h2> -<blockquote> -<p>Makes the polynomial also usable in the <tt class="docutils literal">\xintfloatexpr</tt> parser. -It will therein evaluates via an Horner scheme with coefficients -already pre-rounded to the float precision.</p> -<p>See also <a class="reference internal" href="#poltofloatexpr-polname">\PolToFloatExpr{polname}</a>.</p> -<div class="admonition attention"> -<p class="first admonition-title">Attention!</p> -<p>Release <tt class="docutils literal">0.3</tt> did this automatically on <tt class="docutils literal">\PolDef</tt> and -<tt class="docutils literal">\poldef</tt> but this was removed at <tt class="docutils literal">0.4</tt> for optimization.</p> -<p class="last">Any operation, for example generating the derivative polynomial, -or dividing two polynomials or using the <tt class="docutils literal">\PolLet</tt>, <strong>must</strong> be -followed by explicit usage of <tt class="docutils literal">\PolGenFloatVariant{polname}</tt> if -the new polynomial is to be used in <tt class="docutils literal">\xintfloatexpr</tt> or alike -context.</p> -</div> -</blockquote> -</div> -<div class="section" id="pollet-polname-2-polname-1"> -<span id="pollet"></span><h2><a class="toc-backref" href="#id53"><tt class="docutils literal"><span class="pre">\PolLet{polname_2}={polname_1}</span></tt></a></h2> -<blockquote> -Makes a copy of the already defined polynomial <tt class="docutils literal">polname_1</tt> to a -new one <tt class="docutils literal">polname_2</tt>. Same effect as -<tt class="docutils literal"><span class="pre">\PolDef{polname_2}{polname_1(x)}</span></tt> but with less overhead. The -<tt class="docutils literal">=</tt> is optional.</blockquote> -</div> -<div class="section" id="polgloballet-polname-2-polname-1"> -<span id="polgloballet"></span><h2><a class="toc-backref" href="#id54"><tt class="docutils literal"><span class="pre">\PolGlobalLet{polname_2}={polname_1}</span></tt></a></h2> -<blockquote> -Acts globally.</blockquote> -</div> -<div class="section" id="polassign-polname-toarray-macro"> -<span id="polassign"></span><h2><a class="toc-backref" href="#id55"><tt class="docutils literal"><span class="pre">\PolAssign{polname}\toarray\macro</span></tt></a></h2> -<blockquote> -<p>Defines a one-argument expandable macro <tt class="docutils literal"><span class="pre">\macro{#1}</span></tt> which expands -to the (raw) #1th polynomial coefficient.</p> -<ul class="simple"> -<li>Attention, coefficients here are indexed starting at 1.</li> -<li>With #1=-1, -2, ..., <tt class="docutils literal"><span class="pre">\macro{#1}</span></tt> returns leading coefficients.</li> -<li>With #1=0, returns the number of coefficients, i.e. <tt class="docutils literal">1 + deg f</tt> -for non-zero polynomials.</li> -<li>Out-of-range #1's return <tt class="docutils literal">0/1[0]</tt>.</li> -</ul> -<p>See also <a class="reference internal" href="#polnthcoeff-polname-number">\PolNthCoeff{polname}{number}</a>. The main difference is that -with <tt class="docutils literal">\PolAssign</tt>, <tt class="docutils literal">\macro</tt> is made a prefix to <tt class="docutils literal">1 + deg f</tt> -already defined (hidden to user) macros holding individually the -coefficients but <a class="reference internal" href="#polnthcoeff-polname-number">\PolNthCoeff{polname}{number}</a> does each time the job -to expandably recover the <tt class="docutils literal">Nth</tt> coefficient, and due to -expandability can not store it in a macro for future usage (of course, -it can be an argument in an <tt class="docutils literal">\edef</tt>.) The other difference -is the shift by one in indexing, mentioned above (negative -indices act the same in both.)</p> -</blockquote> -</div> -<div class="section" id="polget-polname-fromarray-macro"> -<span id="polget"></span><h2><a class="toc-backref" href="#id56"><tt class="docutils literal"><span class="pre">\PolGet{polname}\fromarray\macro</span></tt></a></h2> -<blockquote> -<p>Does the converse operation to -<tt class="docutils literal"><span class="pre">\PolAssign{polname}\toarray\macro</span></tt>. Each individual -<tt class="docutils literal">\macro{number}</tt> gets expanded in an <tt class="docutils literal">\edef</tt> and then normalized -via <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a>'s macro <tt class="docutils literal">\xintRaw</tt>.</p> -<p>The leading zeros are removed from the polynomial.</p> -<p>(contrived) Example:</p> -<pre class="literal-block"> -\xintAssignArray{1}{-2}{5}{-3}\to\foo -\PolGet{f}\fromarray\foo -</pre> -<p>This will define <tt class="docutils literal">f</tt> as would have <tt class="docutils literal">\poldef <span class="pre">f(x):=1-2x+5x^2-3x^3;</span></tt>.</p> -<div class="admonition note"> -<p class="first admonition-title">Note</p> -<p class="last">Prior to <tt class="docutils literal">0.5</tt>, coefficients were not normalized via -<tt class="docutils literal">\xintRaw</tt> for internal storage.</p> -</div> -</blockquote> -</div> -<div class="section" id="polfromcsv-polname-csv"> -<span id="polfromcsv"></span><h2><a class="toc-backref" href="#id57"><tt class="docutils literal"><span class="pre">\PolFromCSV{polname}{<csv>}</span></tt></a></h2> -<blockquote> -<p>Defines a polynomial directly from the comma separated list of values -(or a macro expanding to such a list) of its coefficients, the <em>first -item</em> gives the constant term, the <em>last item</em> gives the leading -coefficient, except if zero, then it is dropped (iteratively). List -items are each expanded in an <tt class="docutils literal">\edef</tt> and then put into normalized -form via <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a>'s macro <tt class="docutils literal">\xintRaw</tt>.</p> -<p>As leading zero coefficients are removed:</p> -<pre class="literal-block"> -\PolFromCSV{f}{0, 0, 0, 0, 0, 0, 0, 0, 0, 0} -</pre> -<p>defines the zero polynomial, which holds only one coefficient.</p> -<p>See also expandable macro <a class="reference internal" href="#poltocsv-polname">\PolToCSV</a>.</p> -<div class="admonition note"> -<p class="first admonition-title">Note</p> -<p class="last">Prior to <tt class="docutils literal">0.5</tt>, coefficients were not normalized via -<tt class="docutils literal">\xintRaw</tt> for internal storage.</p> -</div> -</blockquote> -</div> -<div class="section" id="poltypeset-polname"> -<span id="poltypeset"></span><h2><a class="toc-backref" href="#id58"><tt class="docutils literal">\PolTypeset{polname}</tt></a></h2> -<blockquote> -<p>Typesets in descending powers in math mode. It uses letter <tt class="docutils literal">x</tt> but -this can be changed via an optional argument:</p> -<pre class="literal-block"> -\PolTypeset[z]{polname} -</pre> -<p>By default zero coefficients are skipped (issue <tt class="docutils literal">\poltypesetalltrue</tt> -to get all of them in output).</p> -<p>These commands (whose meanings will be found in the package code) -can be re-defined for customization. Their default definitions are -expandable, but this is not a requirement.</p> -</blockquote> -<div class="section" id="poltypesetcmd-raw-coeff"> -<span id="poltypesetcmd"></span><h3><a class="toc-backref" href="#id59"><tt class="docutils literal">\PolTypesetCmd{raw_coeff}</tt></a></h3> -<blockquote> -<p>Checks if the coefficient is <tt class="docutils literal">1</tt> or <tt class="docutils literal"><span class="pre">-1</span></tt> and then skips printing -the <tt class="docutils literal">1</tt>, except for the constant term. Also it sets conditional -<a class="reference internal" href="#polifcoeffisplusorminusone-a-b">\PolIfCoeffIsPlusOrMinusOne{A}{B}</a>.</p> -<p>The actual printing of the coefficients, when not equal to plus or -minus one is handled by <a class="reference internal" href="#poltypesetone-raw-coeff">\PolTypesetOne{raw_coeff}</a>.</p> -</blockquote> -</div> -<div class="section" id="poltypesetone-raw-coeff"> -<span id="poltypesetone"></span><h3><a class="toc-backref" href="#id60"><tt class="docutils literal">\PolTypesetOne{raw_coeff}</tt></a></h3> -<blockquote> -<p>The default is <tt class="docutils literal">\xintSignedFrac</tt> but this macro is annoying as it -insists to use a power of ten, and not decimal notation.</p> -<p>One can do things such as for example: <a class="footnote-reference" href="#id5" id="id4">[1]</a></p> -<pre class="literal-block"> -\renewcommand\PolTypesetOne[1]{\num{\xintPFloat[5]{#1}}} -\renewcommand\PolTypesetOne[1]{\num{\xintRound{4}{#1}}} -</pre> -<p>where e.g. we used the <tt class="docutils literal">\num</tt> macro of <tt class="docutils literal">siunitx</tt> as it -understands floating point notation.</p> -<table class="docutils footnote" frame="void" id="id5" rules="none"> -<colgroup><col class="label" /><col /></colgroup> -<tbody valign="top"> -<tr><td class="label"><a class="fn-backref" href="#id4">[1]</a></td><td>the difference in the syntaxes of <tt class="docutils literal">\xintPFloat</tt> and -<tt class="docutils literal">\xintRound</tt> is explained from the fact that -<tt class="docutils literal">\xintPFloat</tt> by default uses the prevailing precision -hence the extra argument like here <tt class="docutils literal">5</tt> is an optional one.</td></tr> -</tbody> -</table> -<p>One can also give a try to using <a class="reference internal" href="#poldectostring-decimal-number">\PolDecToString{decimal number}</a> -which uses decimal notation (at least for the numerator part).</p> -</blockquote> -</div> -<div class="section" id="id6"> -<span id="poltypesetmonomialcmd"></span><h3><a class="toc-backref" href="#id61"><tt class="docutils literal">\PolTypesetMonomialCmd</tt></a></h3> -<blockquote> -This decides how a monomial (in variable <tt class="docutils literal">\PolVar</tt> and with -exponent <tt class="docutils literal">\PolIndex</tt>) is to be printed. The default does nothing -for the constant term, <tt class="docutils literal">\PolVar</tt> for the first degree and -<tt class="docutils literal"><span class="pre">\PolVar^{\PolIndex}</span></tt> for higher degrees monomials. Beware that -<tt class="docutils literal">\PolIndex</tt> expands to digit tokens and needs termination in -<tt class="docutils literal">\ifnum</tt> tests.</blockquote> -</div> -<div class="section" id="poltypesetcmdprefix-raw-coeff"> -<span id="poltypesetcmdprefix"></span><h3><a class="toc-backref" href="#id62"><tt class="docutils literal">\PolTypesetCmdPrefix{raw_coeff}</tt></a></h3> -<blockquote> -Expands to a <tt class="docutils literal">+</tt> if the <tt class="docutils literal">raw_coeff</tt> is zero or positive, and to -nothing if <tt class="docutils literal">raw_coeff</tt> is negative, as in latter case the -<tt class="docutils literal">\xintSignedFrac</tt> used by <a class="reference internal" href="#poltypesetcmd-raw-coeff">\PolTypesetCmd{raw_coeff}</a> will put -the <tt class="docutils literal">-</tt> sign in front of the fraction (if it is a fraction) and -this will thus serve as separator in the typeset formula. Not used -for the first term.</blockquote> -</div> -</div> -<div class="section" id="id8"> -<span id="id7"></span><h2><a class="toc-backref" href="#id63"><tt class="docutils literal"><span class="pre">\PolTypeset*{polname}</span></tt></a></h2> -<blockquote> -Typesets in ascending powers. Use e.g. <tt class="docutils literal">[h]</tt> optional argument -(after the <tt class="docutils literal">*</tt>) to use letter <tt class="docutils literal">h</tt> rather than <tt class="docutils literal">x</tt>.</blockquote> -</div> -<div class="section" id="poldiff-polname-1-polname-2"> -<span id="poldiff"></span><h2><a class="toc-backref" href="#id64"><tt class="docutils literal"><span class="pre">\PolDiff{polname_1}{polname_2}</span></tt></a></h2> -<blockquote> -<p>This sets <tt class="docutils literal">polname_2</tt> to the first derivative of <tt class="docutils literal">polname_1</tt>. It -is allowed to issue <tt class="docutils literal"><span class="pre">\PolDiff{f}{f}</span></tt>, effectively replacing <tt class="docutils literal">f</tt> -by <tt class="docutils literal">f'</tt>.</p> -<p>Coefficients of the result <tt class="docutils literal">polname_2</tt> are irreducible fractions -(see <a class="reference internal" href="#technicalities">Technicalities</a> for the whole story.)</p> -</blockquote> -</div> -<div class="section" id="poldiff-n-polname-1-polname-2"> -<span id="poldiff-n"></span><h2><a class="toc-backref" href="#id65"><tt class="docutils literal"><span class="pre">\PolDiff[N]{polname_1}{polname_2}</span></tt></a></h2> -<blockquote> -This sets <tt class="docutils literal">polname_2</tt> to the <tt class="docutils literal">N</tt>-th derivative of <tt class="docutils literal">polname_1</tt>. -Identical arguments is allowed. With <tt class="docutils literal">N=0</tt>, same effect as -<tt class="docutils literal"><span class="pre">\PolLet{polname_2}={polname_1}</span></tt>. With negative <tt class="docutils literal">N</tt>, switches to -using <tt class="docutils literal">\PolAntiDiff</tt>.</blockquote> -</div> -<div class="section" id="polantidiff-polname-1-polname-2"> -<span id="polantidiff"></span><h2><a class="toc-backref" href="#id66"><tt class="docutils literal"><span class="pre">\PolAntiDiff{polname_1}{polname_2}</span></tt></a></h2> -<blockquote> -<p>This sets <tt class="docutils literal">polname_2</tt> to the primitive of <tt class="docutils literal">polname_1</tt> vanishing -at zero.</p> -<p>Coefficients of the result <tt class="docutils literal">polname_2</tt> are irreducible fractions -(see <a class="reference internal" href="#technicalities">Technicalities</a> for the whole story.)</p> -</blockquote> -</div> -<div class="section" id="polantidiff-n-polname-1-polname-2"> -<span id="polantidiff-n"></span><h2><a class="toc-backref" href="#id67"><tt class="docutils literal"><span class="pre">\PolAntiDiff[N]{polname_1}{polname_2}</span></tt></a></h2> -<blockquote> -This sets <tt class="docutils literal">polname_2</tt> to the result of <tt class="docutils literal">N</tt> successive integrations on -<tt class="docutils literal">polname_1</tt>. With negative <tt class="docutils literal">N</tt>, it switches to using <tt class="docutils literal">\PolDiff</tt>.</blockquote> -</div> -<div class="section" id="poldivide-polname-1-polname-2-polname-q-polname-r"> -<span id="poldivide"></span><h2><a class="toc-backref" href="#id68"><tt class="docutils literal"><span class="pre">\PolDivide{polname_1}{polname_2}{polname_Q}{polname_R}</span></tt></a></h2> -<blockquote> -This sets <tt class="docutils literal">polname_Q</tt> and <tt class="docutils literal">polname_R</tt> to be the quotient and -remainder in the Euclidean division of <tt class="docutils literal">polname_1</tt> by -<tt class="docutils literal">polname_2</tt>.</blockquote> -</div> -<div class="section" id="polquo-polname-1-polname-2-polname-q"> -<span id="polquo"></span><h2><a class="toc-backref" href="#id69"><tt class="docutils literal"><span class="pre">\PolQuo{polname_1}{polname_2}{polname_Q}</span></tt></a></h2> -<blockquote> -This sets <tt class="docutils literal">polname_Q</tt> to be the quotient in the Euclidean division -of <tt class="docutils literal">polname_1</tt> by <tt class="docutils literal">polname_2</tt>.</blockquote> -</div> -<div class="section" id="polrem-polname-1-polname-2-polname-r"> -<span id="polrem"></span><h2><a class="toc-backref" href="#id70"><tt class="docutils literal"><span class="pre">\PolRem{polname_1}{polname_2}{polname_R}</span></tt></a></h2> -<blockquote> -This sets <tt class="docutils literal">polname_R</tt> to be the remainder in the Euclidean division -of <tt class="docutils literal">polname_1</tt> by <tt class="docutils literal">polname_2</tt>.</blockquote> -</div> -<div class="section" id="polgcd-polname-1-polname-2-polname-gcd"> -<span id="polgcd"></span><h2><a class="toc-backref" href="#id71"><tt class="docutils literal"><span class="pre">\PolGCD{polname_1}{polname_2}{polname_GCD}</span></tt></a></h2> -<blockquote> -This sets <tt class="docutils literal">polname_GCD</tt> to be the (monic) GCD of the two first -polynomials. It is a unitary polynomial except if both <tt class="docutils literal">polname_1</tt> -and <tt class="docutils literal">polname_2</tt> vanish, then <tt class="docutils literal">polname_GCD</tt> is the zero -polynomial.</blockquote> -<!-- ``\PolIGCD{polname_1}{polname_2}{polname_iGCD}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - **NOT YET** - - This **assumes** that the two polynomials have integer coefficients. - It then computes the greatest common divisor in the integer - polynomial ring, normalized to have a positive leading coefficient - (if the inputs are not both zero). - -``\PolIContent{polname}`` -~~~~~~~~~~~~~~~~~~~~~~~~~ - - **NOT YET** - - This computes a positive rational number such that dividing the - polynomial with it returns an integer coefficients polynomial with - no common factor among the coefficients. --> -</div> -<div class="section" id="poltosturm-polname-sturmname"> -<span id="poltosturm"></span><h2><a class="toc-backref" href="#id72"><tt class="docutils literal"><span class="pre">\PolToSturm{polname}{sturmname}</span></tt></a></h2> -<blockquote> -<p>With <tt class="docutils literal">polname</tt> being for example <tt class="docutils literal">P</tt>, the macro starts by -computing polynomials <tt class="docutils literal">P</tt> and <tt class="docutils literal">P'</tt>, then computes the (opposite -of the) remainder in euclidean division, iteratively.</p> -<p>The last non-zero remainder <tt class="docutils literal">P_N_</tt> (where <tt class="docutils literal">N</tt> is obtainable as -<a class="reference internal" href="#polsturmchainlength-sturmname">\PolSturmChainLength{sturmname}</a>) is up to a factor -the GCD of <tt class="docutils literal">P</tt> and <tt class="docutils literal">P'</tt> hence it is a constant if and only if -<tt class="docutils literal">P</tt> is square-free.</p> -<div class="admonition note"> -<p class="first admonition-title">Note</p> -<ul class="last simple"> -<li>Since <tt class="docutils literal">0.5</tt> all these polynomials are divided by their rational -content, so they have integer coefficients with no common factor, -and the last one if a constant is either <tt class="docutils literal">1</tt> or <tt class="docutils literal"><span class="pre">-1</span></tt>.</li> -<li>After this normalization to primitive polynomials, they are -stored internally as <tt class="docutils literal">sturmname_k_</tt>, <tt class="docutils literal">k=0,1, ...</tt>.</li> -<li>These polynomials are used internally only. To keep them as -genuine declared polynomials also after the macro call, use the -starred variant <a class="reference internal" href="#id9">PolToSturm*</a>.</li> -</ul> -</div> -<div class="admonition note"> -<p class="first admonition-title">Note</p> -<p class="last">It is perfectly allowed to use the polynomial name as Sturm chain name: -<tt class="docutils literal"><span class="pre">\PolToSturm{f}(f}</span></tt>.</p> -</div> -<p>The macro then declares <tt class="docutils literal">sturmname_0</tt>, <tt class="docutils literal">sturmname_1</tt>, ..., which are -the (non-declared) <tt class="docutils literal">sturmname_k_</tt> divided by the last one. Division is -not done if this last one is the constant <tt class="docutils literal">1</tt> or <tt class="docutils literal"><span class="pre">-1</span></tt>, i.e. if the -original polynomial was square-free. These polynomials are primitive -polynomials too, i.e. with integer coefficients having no common factor.</p> -<p>Thus <tt class="docutils literal">sturmname_0</tt> has exactly the same real and complex roots as -polynomial <tt class="docutils literal">polname</tt>, but with each root now of multiplicity one: -i.e. it is the "square-free part" of original polynomial <tt class="docutils literal">polname</tt>.</p> -<p>Notice that <tt class="docutils literal">sturmname_1</tt> isn't necessarily the derivative of -<tt class="docutils literal">sturmname_0</tt> due to the various normalizations.</p> -<p>The polynomials <tt class="docutils literal">sturmname_k</tt> main utility is for the execution of -<a class="reference internal" href="#polsturmisolatezeros-sturmname">\PolSturmIsolateZeros{sturmname}</a>. Be careful not to use these -names <tt class="docutils literal">sturmname_0</tt>, <tt class="docutils literal">sturmname_1</tt>, etc... for defining other -polynomials after having done <tt class="docutils literal"><span class="pre">\PolToSturm{polname}{sturmname}</span></tt> and -before executing <tt class="docutils literal">\PolSturmIsolateZeros{sturmname}</tt> else the -latter will behave erroneously.</p> -<p><a class="reference internal" href="#polsturmchainlength-sturmname">\PolSturmChainLength{sturmname}</a> gives the index of the last -element of the Sturm chain.</p> -</blockquote> -</div> -<div class="section" id="id10"> -<span id="id9"></span><h2><a class="toc-backref" href="#id73"><tt class="docutils literal"><span class="pre">\PolToSturm*{polname}{sturmname}</span></tt></a></h2> -<blockquote> -<p>Does the same as <a class="reference internal" href="#poltosturm">un-starred version</a> and additionally it -keeps for user usage the memory of the <em>un-normalized</em> Sturm chain -polynomials <tt class="docutils literal">sturmname_k_</tt>, <tt class="docutils literal">k=0,1, <span class="pre">...,</span> N</tt>, with -<tt class="docutils literal">N</tt> being <a class="reference internal" href="#polsturmchainlength-sturmname">\PolSturmChainLength{sturmname}</a>.</p> -<div class="admonition note"> -<p class="first admonition-title">Note</p> -<p class="last">This behaviour was modified at <tt class="docutils literal">0.6</tt>, anyhow the macro was -broken at <tt class="docutils literal">0.5</tt>.</p> -</div> -<div class="admonition hint"> -<p class="first admonition-title">Hint</p> -<p class="last">The square-free part of <tt class="docutils literal">polname</tt> is <tt class="docutils literal">sturmname_0</tt>, and their -quotient is the polynomial with name -<tt class="docutils literal">sturname_\PolSturmChainLength{sturmname}_</tt>. It thus easy to -set-up a loop iteratively computing the latter until the last one -is a constant, thus obtaining the decomposition of an <tt class="docutils literal">f</tt> as -a product <tt class="docutils literal">c f_1 f_2 f_3 ...</tt> of a constant and square-free (primitive) -polynomials, where each <tt class="docutils literal">f_i</tt> divides its predecessor.</p> -</div> -</blockquote> -</div> -<div class="section" id="polsettosturmchainsignchangesat-macro-sturmname-fraction"> -<span id="polsettosturmchainsignchangesat"></span><h2><a class="toc-backref" href="#id74"><tt class="docutils literal"><span class="pre">\PolSetToSturmChainSignChangesAt{\macro}{sturmname}{fraction}</span></tt></a></h2> -<blockquote> -<p>Sets macro <tt class="docutils literal">\macro</tt> to the number of sign changes in the Sturm -chain with name prefix <tt class="docutils literal">sturmname</tt>, at location <tt class="docutils literal">fraction</tt> -(which must be in format as acceptable by the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> macros.)</p> -<div class="admonition note"> -<p class="first admonition-title">Note</p> -<p>The author was lazy and did not provide rather an expandable -variant, where one would do <tt class="docutils literal"><span class="pre">\edef\macro{\PolNbOf...}</span></tt>.</p> -<p>This will presumably get added in a future release.</p> -<p class="last">After some hesitation it was decided the macro would by default -act globally. To make the scope of its macro definition local, -use <tt class="docutils literal">[\empty]</tt> as extra optional argument.</p> -</div> -</blockquote> -</div> -<div class="section" id="polsettonbofzeroswithin-macro-sturmname-value-a-value-b"> -<span id="polsettonbofzeroswithin"></span><h2><a class="toc-backref" href="#id75"><tt class="docutils literal"><span class="pre">\PolSetToNbOfZerosWithin{\macro}{sturmname}{value_a}{value_b}</span></tt></a></h2> -<blockquote> -<p>Applies the <a class="reference external" href="https://en.wikipedia.org/wiki/Sturm%27s_theorem">Sturm Theorem</a> to set <tt class="docutils literal">\macro</tt> to the exact number -of <strong>distinct</strong> roots of <tt class="docutils literal">sturmname_0</tt> in the interval <tt class="docutils literal">(value_a, -value_b]</tt> (the macro first re-orders the value for <tt class="docutils literal">value_a <= -value_b</tt> to hold).</p> -<div class="admonition note"> -<p class="first admonition-title">Note</p> -<p>The author was lazy and did not provide rather an expandable -variant, where one would do <tt class="docutils literal"><span class="pre">\edef\macro{\PolNbOf...}</span></tt>.</p> -<p>This will presumably get added in future.</p> -<p class="last">After some hesitation it was decided the macro would by default -act globally. To make the scope of its macro definition local, -use <tt class="docutils literal">[\empty]</tt> as extra optional argument.</p> -</div> -<p>See also the expandable -<a class="reference internal" href="#polsturmnbofrootsof-sturmname-lessthanorequalto-value">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualTo{value}</a>, from -which it is immediate (with <tt class="docutils literal">\numexpr</tt>) to create an expandable -variant of this macro. However the difference is that this macro -requires only <a class="reference internal" href="#poltosturm">\PolToSturm</a> to have been executed, -whereas the expandable variant requires prior execution of -<a class="reference internal" href="#polsturmisolatezeros">\PolSturmIsolateZeros</a>.</p> -<p>See also the expandable -<a class="reference internal" href="#polsturmnbwithmultofrootsof-sturmname-lessthanorequalto-value">\PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualTo{value}</a> -which requires prior execution of -<a class="reference internal" href="#id11">\PolSturmIsolateZeros*</a>.</p> -</blockquote> -</div> -<div class="section" id="polsturmisolatezeros-sturmname"> -<span id="polsturmisolatezeros"></span><h2><a class="toc-backref" href="#id76"><tt class="docutils literal">\PolSturmIsolateZeros{sturmname}</tt></a></h2> -<blockquote> -<p>The macros locates, using <a class="reference external" href="https://en.wikipedia.org/wiki/Sturm%27s_theorem">Sturm theorem</a>, as many disjoint -intervals as there are (real) roots.</p> -<div class="admonition important"> -<p class="first admonition-title">Important</p> -<p>The Sturm chain must have been produced by an earlier -<a class="reference internal" href="#poltosturm-polname-sturmname">\PolToSturm{polname}{sturmname}</a>.</p> -<p>Why does this macro ask for argument the name of Sturm chain, -rather than the name of a polynomial? well this is mainly for -legacy reason, and because it is accompanied by other macros for -which it is simpler to assume the argument will be the name of an -already computed Sturm chain.</p> -<p>Notice that <tt class="docutils literal"><span class="pre">\PolToSturm{f}{f}</span></tt> is perfectly legal (the -<tt class="docutils literal">sturmname</tt> can be same as the <tt class="docutils literal">polname</tt>): it defines -polynomials <tt class="docutils literal">f_0</tt>, <tt class="docutils literal">f_1</tt>, ... having <tt class="docutils literal">f</tt> has name prefix.</p> -<p class="last">Such a prior call -to <tt class="docutils literal">\PolToSturm</tt> must have been made at any rate for -<tt class="docutils literal">\PolSturmIsolateZeros</tt> to be usable.</p> -</div> -<p>After its execution they are two types of such intervals (stored in -memory and accessible via macros or <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> variables, see below):</p> -<ul class="simple"> -<li>singleton <tt class="docutils literal">{a}</tt>: then <tt class="docutils literal">a</tt> is a root, (necessarily a decimal -number, but not all such decimal numbers are exactly identified yet).</li> -<li>open intervals <tt class="docutils literal">(a,b)</tt>: then there is exactly one root <tt class="docutils literal">z</tt> -such that <tt class="docutils literal">a < z < b</tt>, and the end points are guaranteed to not -be roots.</li> -</ul> -<p>The interval boundaries are decimal numbers, originating -in iterated decimal subdivision from initial intervals -<tt class="docutils literal"><span class="pre">(-10^E,</span> 0)</tt> and <tt class="docutils literal">(0, 10^E)</tt> with <tt class="docutils literal">E</tt> chosen initially large -enough so that all roots are enclosed; if zero is a root it is always -identified as such. The non-singleton intervals are of the -type <tt class="docutils literal">(a/10^f, <span class="pre">(a+1)/10^f)</span></tt> with <tt class="docutils literal">a</tt> an integer, which is -neither <tt class="docutils literal">0</tt> nor <tt class="docutils literal"><span class="pre">-1</span></tt>. Hence either <tt class="docutils literal">a</tt> and <tt class="docutils literal">a+1</tt> are both positive -or they are both negative.</p> -<p>One does not <em>a priori</em> know what will be the lengths of these -intervals (except that they are always powers of ten), they -vary depending on how many digits two successive roots have in -common in their respective decimal expansions.</p> -<div class="admonition important"> -<p class="first admonition-title">Important</p> -<p>If some two consecutive intervals share an end-point, no -information is yet gained about the separation between the two -roots which could at this stage be arbitrarily small.</p> -<p class="last">See <a class="reference internal" href="#polrefineinterval-sturmname-index">\PolRefineInterval*{sturmname}{index}</a> which addresses -this issue.</p> -</div> -<!-- This procedure is covariant -with the independent variable ``x`` becoming ``-x``. -Hmm, pas sûr et trop fatigué --> -<p>The interval boundaries (and exactly found roots) are made available -for future computations in <tt class="docutils literal">\xintexpr</tt>-essions or polynomial -definitions as variables <tt class="docutils literal"><sturmname>L_1</tt>, -<tt class="docutils literal"><sturmname>L_2</tt>, etc..., for the left end-points and -<tt class="docutils literal"><sturmname>R_1</tt>, <tt class="docutils literal"><sturmname>R_2</tt>, ..., for the right -end-points.</p> -<p>Thus for example, if <tt class="docutils literal">sturmname</tt> is <tt class="docutils literal">f</tt>, one can use the -<a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> variables <tt class="docutils literal">fL_1</tt>, <tt class="docutils literal">fL_2</tt>, ... to refer in expressions -to the left end-points (or to the exact root, if left and right end -points coincide). Additionally, <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> variable <tt class="docutils literal">fZ_1_isknown</tt> -will have value <tt class="docutils literal">1</tt> if the root in the first interval is known, -and <tt class="docutils literal">0</tt> otherwise. And similarly for the other intervals.</p> -<p>Also, macros <a class="reference internal" href="#polsturmisolatedzeroleft-sturmname-index">\PolSturmIsolatedZeroLeft{sturmname}{index}</a> and -<a class="reference internal" href="#polsturmisolatedzeroright-sturmname-index">\PolSturmIsolatedZeroRight{sturmname}{index}</a> are provided which -expand to these same values, written in decimal notation (i.e. -pre-processed by <a class="reference internal" href="#poldectostring">\PolDecToString</a>.) And there -is also <a class="reference internal" href="#polsturmifzeroexactlyknown-sturmname-index-a-b">\PolSturmIfZeroExactlyKnown{sturmname}{index}{A}{B}</a>.</p> -<div class="admonition important"> -<p class="first admonition-title">Important</p> -<p class="last">Trailing zeroes in the stored decimal numbers accessible via the -macros are significant: they are also present in the decimal -expansion of the exact root.</p> -</div> -<p>These variables and macros are automatically updated when one next -uses macros such as <a class="reference internal" href="#polrefineinterval-sturmname-index">\PolRefineInterval*{sturmname}{index}</a>.</p> -<p>The start of decimal expansion of a positive <tt class="docutils literal">k</tt>-th root is given -by <a class="reference internal" href="#polsturmisolatedzeroleft">\PolSturmIsolatedZeroLeft{sturmname}{k}</a>, and for a negative root it is given -by <a class="reference internal" href="#polsturmisolatedzeroright">PolSturmIsolatedZeroRight{sturmname}{k}</a>. These two decimal -numbers are either both zero or both of the same sign.</p> -<p>The number of distinct roots is obtainable expandably as -<a class="reference internal" href="#polsturmnbofisolatedzeros-sturmname">\PolSturmNbOfIsolatedZeros{sturmname}</a>.</p> -<p>Furthermore -<a class="reference internal" href="#polsturmnbofrootsof-sturmname-lessthanorequalto-value">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualTo{value}</a> and -<a class="reference internal" href="#polsturmnbofrootsof-sturmname-lessthanorequaltoexpr-expression">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualToExpr{expression}</a>. -will expandably compute respectively the number of real roots at -most equal to <tt class="docutils literal">value</tt> or <tt class="docutils literal">expression</tt>, and the same but with -multiplicities.</p> -<div class="admonition note"> -<p class="first admonition-title">Note</p> -<p class="last">In the current implementation the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> variables -and <a class="reference external" href="http://www.ctan.org/pkg/xint">xinttools</a> arrays are globally defined. On the -other hand the Sturm sequence polynomials obey the current scope.</p> -</div> -<div class="admonition note"> -<p class="first admonition-title">Note</p> -<p>As all computations are done <em>exactly</em> there can be no errors... -apart those due to bad coding by author. The results are exact -bounds for the mathematically exact real roots.</p> -<p class="last">Future releases will perhaps also provide macros based on Newton -or Regula Falsi methods. Exact computations with such methods -lead however quickly to very big fractions, and this forces usage -of some rounding scheme for the abscissas if computation times -are to remain reasonable. This raises issues of its own, which -are studied in numerical mathematics.</p> -</div> -</blockquote> -</div> -<div class="section" id="id12"> -<span id="id11"></span><h2><a class="toc-backref" href="#id77"><tt class="docutils literal"><span class="pre">\PolSturmIsolateZeros*{sturmname}</span></tt></a></h2> -<blockquote> -<p>The macro does the same as <a class="reference internal" href="#polsturmisolatezeros-sturmname">\PolSturmIsolateZeros{sturmname}</a> and -then in addition it does the extra work to determine all -multiplicities (of the real roots): -after executing this macro, -<a class="reference internal" href="#polsturmisolatedzeromultiplicity-sturmname-index">\PolSturmIsolatedZeroMultiplicity{sturmname}{index}</a> will expand -to the multiplicity of the root located in the <tt class="docutils literal">index</tt>-th -interval (intervals are enumerated from left to right, with index -starting at <tt class="docutils literal">1</tt>).</p> -<p>Furthermore, if for example the <tt class="docutils literal">sturmname</tt> is <tt class="docutils literal">f</tt>, <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> -variables <tt class="docutils literal">fM_1</tt>, <tt class="docutils literal">fM_2</tt>... hold the multiplicities thus -computed.</p> -<div class="admonition note"> -<p class="first admonition-title">Note</p> -<p class="last">It is <strong>not</strong> necessary to have executed the <a class="reference internal" href="#id9">PolToSturm*</a> starred -variant, as the non-starred variant keeps internally the memory of the -original GCD (and even of the full non-normalized original Sturm -chain), even though it does not make the declarations as <em>user-level</em> -genuine polynomials.</p> -</div> -<p>See <a class="reference internal" href="#the-degree-nine-polynomial-with-0-99-0-999-0-9999-as-triple-roots">The degree nine polynomial with 0.99, 0.999, 0.9999 as triple -roots</a> for an example.</p> -</blockquote> -</div> -<div class="section" id="id14"> -<span id="id13"></span><h2><a class="toc-backref" href="#id78"><tt class="docutils literal"><span class="pre">\PolSturmIsolateZeros**{sturmname}</span></tt></a></h2> -<blockquote> -<p>The macro does the same as <a class="reference internal" href="#id12">\PolSturmIsolateZeros*{sturmname}</a> and -in addition it does the extra work to determine all the <em>rational</em> -roots.</p> -<div class="admonition note"> -<p class="first admonition-title">Note</p> -<p class="last">After execution of this macro, a root is "known" if and only if -it is rational.</p> -</div> -<p>Furthermore, primitive polynomial <tt class="docutils literal">sturmname_sqf_norr</tt> is created -to match the (square-free) <tt class="docutils literal">sturmname_0</tt> from which all rational -roots have been removed (see <a class="reference internal" href="#polexprsetup">\polexprsetup</a> for customizing this -name). The number of distinct rational roots is thus the difference -between the degrees of these two polynomials (see also -<a class="reference internal" href="#polsturmnbofrationalroots-sturmname">\PolSturmNbOfRationalRoots{sturmname}</a>).</p> -<p>And <tt class="docutils literal">sturmname_norr</tt> is <tt class="docutils literal">sturmname_0_</tt> from which all rational -roots have been removed (see <a class="reference internal" href="#polexprsetup">\polexprsetup</a>), i.e. it contains -the irrational roots of the original polynomial, with the same -multiplicities.</p> -<p>See <a class="reference internal" href="#a-degree-five-polynomial-with-three-rational-roots">A degree five polynomial with three rational -roots</a> for an example.</p> -</blockquote> -</div> -<div class="section" id="polsturmisolatezerosandgetmultiplicities-sturmname"> -<span id="polsturmisolatezerosandgetmultiplicities"></span><h2><a class="toc-backref" href="#id79"><tt class="docutils literal">\PolSturmIsolateZerosAndGetMultiplicities{sturmname}</tt></a></h2> -<blockquote> -This is another name for <a class="reference internal" href="#id12">\PolSturmIsolateZeros*{sturmname}</a>.</blockquote> -</div> -<div class="section" id="polsturmisolatezerosgetmultiplicitiesandrationalroots-sturmname"> -<span id="polsturmisolatezerosgetmultiplicitiesandrationalroots"></span><h2><a class="toc-backref" href="#id80"><tt class="docutils literal">\PolSturmIsolateZerosGetMultiplicitiesAndRationalRoots{sturmname}</tt></a></h2> -<blockquote> -This is another name for <a class="reference internal" href="#id14">\PolSturmIsolateZeros**{sturmname}</a>.</blockquote> -</div> -<div class="section" id="polsturmisolatezerosandfindrationalroots-sturmname"> -<h2><a class="toc-backref" href="#id81"><tt class="docutils literal">\PolSturmIsolateZerosAndFindRationalRoots{sturmname}</tt></a></h2> -<blockquote> -<p>This works exactly like <a class="reference internal" href="#id14">\PolSturmIsolateZeros**{sturmname}</a> -(inclusive of declaring the polynomials <tt class="docutils literal">sturmname_sqf_norr</tt> and -<tt class="docutils literal">sturmname_norr</tt> with no rational roots) except that it does <em>not</em> -compute the multiplicities of the <em>non-rational</em> roots.</p> -<div class="admonition note"> -<p class="first admonition-title">Note</p> -<p class="last">There is no macro to find the rational roots but not compute -their multiplicities at the same time.</p> -</div> -<div class="admonition attention"> -<p class="first admonition-title">Attention!</p> -<p>This macro does <em>not</em> define <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> variables -<tt class="docutils literal">sturmnameM_1</tt>, <tt class="docutils literal">sturmnameM_2</tt>, ... holding the -multiplicities and it leaves the multiplicity array (whose accessor -is <a class="reference internal" href="#polsturmisolatedzeromultiplicity-sturmname-index">\PolSturmIsolatedZeroMultiplicity{sturmname}{index}</a>) into -a broken state, as all non-rational roots will supposedly have -multiplicity one. This means that the output of -<a class="reference internal" href="#id18">\PolPrintIntervals*</a> for example will be -erroneous for the intervals with irrational roots.</p> -<p class="last">I decided to document it because finding multiplicities of the -non rational roots is somewhat costly, and one may be interested -only into finding the rational roots (of course random -polynomials with integer coefficients will not have <em>any</em> -rational root anyhow).</p> -</div> -</blockquote> -</div> -<div class="section" id="polrefineinterval-sturmname-index"> -<span id="polrefineinterval"></span><h2><a class="toc-backref" href="#id82"><tt class="docutils literal"><span class="pre">\PolRefineInterval*{sturmname}{index}</span></tt></a></h2> -<blockquote> -The <tt class="docutils literal">index</tt>-th interval (starting indexing at one) is further -subdivided as many times as is necessary in order for the newer -interval to have both its end-points distinct from the end-points of -the original interval. This means that the <tt class="docutils literal">k</tt>th root is then -strictly separated from the other roots.</blockquote> -</div> -<div class="section" id="polrefineinterval-n-sturmname-index"> -<span id="polrefineinterval-n"></span><h2><a class="toc-backref" href="#id83"><tt class="docutils literal"><span class="pre">\PolRefineInterval[N]{sturmname}{index}</span></tt></a></h2> -<blockquote> -The <tt class="docutils literal">index</tt>-th interval (starting count at one) is further -subdivided once, reducing its length by a factor of 10. This is done -<tt class="docutils literal">N</tt> times if the optional argument <tt class="docutils literal">[N]</tt> is present.</blockquote> -</div> -<div class="section" id="polensureintervallength-sturmname-index-e"> -<span id="polensureintervallength"></span><h2><a class="toc-backref" href="#id84"><tt class="docutils literal"><span class="pre">\PolEnsureIntervalLength{sturmname}{index}{E}</span></tt></a></h2> -<blockquote> -The <tt class="docutils literal">index</tt>-th interval is subdivided until its length becomes at -most <tt class="docutils literal">10^E</tt>. This means (for <tt class="docutils literal">E<0</tt>) that the first <tt class="docutils literal"><span class="pre">-E</span></tt> digits -after decimal mark of the <tt class="docutils literal">k</tt>th root will then be known exactly.</blockquote> -</div> -<div class="section" id="polensureintervallengths-sturmname-e"> -<span id="polensureintervallengths"></span><h2><a class="toc-backref" href="#id85"><tt class="docutils literal"><span class="pre">\PolEnsureIntervalLengths{sturmname}{E}</span></tt></a></h2> -<blockquote> -<p>The intervals as obtained from <tt class="docutils literal">\PolSturmIsolateZeros</tt> are (if -necessary) subdivided further by (base 10) dichotomy in order for -each of them to have length at most <tt class="docutils literal">10^E</tt> (length will be shorter -than <tt class="docutils literal">10^E</tt> in output only if it did not change or became zero.)</p> -<p>This means that decimal expansions of all roots will be known with -<tt class="docutils literal"><span class="pre">-E</span></tt> digits (for <tt class="docutils literal">E<0</tt>) after decimal mark.</p> -</blockquote> -</div> -<div class="section" id="polprintintervals-varname-sturmname"> -<span id="polprintintervals"></span><h2><a class="toc-backref" href="#id86"><tt class="docutils literal"><span class="pre">\PolPrintIntervals[varname]{sturmname}</span></tt></a></h2> -<blockquote> -<p>This is a convenience macro which prints the bounds for the roots -<tt class="docutils literal">Z_1</tt>, <tt class="docutils literal">Z_2</tt>, ... (the optional argument <tt class="docutils literal">varname</tt> allows to -specify a replacement for the default <tt class="docutils literal">Z</tt>). This will be done (by -default) in a -math mode <tt class="docutils literal">array</tt>, one interval per row, and pattern <tt class="docutils literal">rcccl</tt>, -where the second and fourth column hold the <tt class="docutils literal"><</tt> sign, except when -the interval reduces to a singleton, which means the root is known -exactly.</p> -<div class="admonition attention"> -<p class="first admonition-title">Attention!</p> -<p class="last">This macro was refactored at 0.7, its default output remained -identical but the ways to customize it got completely -modified.</p> -</div> -<p>See next macros which govern its output.</p> -</blockquote> -<div class="section" id="polprintintervalsnorealroots"> -<h3><a class="toc-backref" href="#id87"><tt class="docutils literal">\PolPrintIntervalsNoRealRoots</tt></a></h3> -<blockquote> -<p>Executed in place of an <tt class="docutils literal">array</tt> environment, when there are no -real roots. Default definition:</p> -<pre class="literal-block"> -\newcommand\PolPrintIntervalsNoRealRoots{} -</pre> -</blockquote> -</div> -<div class="section" id="polprintintervalsbeginenv"> -<h3><a class="toc-backref" href="#id88"><tt class="docutils literal">\PolPrintIntervalsBeginEnv</tt></a></h3> -<blockquote> -<p>Default definition:</p> -<pre class="literal-block"> -\newcommand\PolPrintIntervalsBeginEnv{\[\begin{array}{rcccl}} -</pre> -</blockquote> -</div> -<div class="section" id="polprintintervalsendenv"> -<h3><a class="toc-backref" href="#id89"><tt class="docutils literal">\PolPrintIntervalsEndEnv</tt></a></h3> -<blockquote> -<p>Default definition:</p> -<pre class="literal-block"> -\newcommand\PolPrintIntervalsEndEnv{\end{array}\]} -</pre> -</blockquote> -</div> -<div class="section" id="polprintintervalsknownroot"> -<h3><a class="toc-backref" href="#id90"><tt class="docutils literal">\PolPrintIntervalsKnownRoot</tt></a></h3> -<blockquote> -<p>Default definition:</p> -<pre class="literal-block"> -\newcommand\PolPrintIntervalsKnownRoot{% - &&\PolPrintIntervalsTheVar_{\PolPrintIntervalsTheIndex}% - &=&\PolPrintIntervalsPrintExactZero -} -</pre> -</blockquote> -</div> -<div class="section" id="polprintintervalsunknownroot"> -<h3><a class="toc-backref" href="#id91"><tt class="docutils literal">\PolPrintIntervalsUnknownRoot</tt></a></h3> -<blockquote> -<p>Default definition:</p> -<pre class="literal-block"> -\newcommand\PolPrintIntervalsUnknownRoot{% - \PolPrintIntervalsPrintLeftEndPoint&<&% - \PolPrintIntervalsTheVar_{\PolPrintIntervalsTheIndex}&<&% - \PolPrintIntervalsPrintRightEndPoint -} -</pre> -</blockquote> -</div> -<div class="section" id="id15"> -<span id="polprintintervalsprintexactzero"></span><h3><a class="toc-backref" href="#id92"><tt class="docutils literal">\PolPrintIntervalsPrintExactZero</tt></a></h3> -<blockquote> -<p>Default definition:</p> -<pre class="literal-block"> -\newcommand\PolPrintIntervalsPrintExactZero{\PolPrintIntervalsTheLeftEndPoint} -</pre> -</blockquote> -</div> -<div class="section" id="id16"> -<span id="polprintintervalsprintleftendpoint"></span><h3><a class="toc-backref" href="#id93"><tt class="docutils literal">\PolPrintIntervalsPrintLeftEndPoint</tt></a></h3> -<blockquote> -<p>Default definition:</p> -<pre class="literal-block"> -\newcommand\PolPrintIntervalsPrintLeftEndPoint{\PolPrintIntervalsTheLeftEndPoint} -</pre> -</blockquote> -</div> -<div class="section" id="id17"> -<span id="polprintintervalsprintrightendpoint"></span><h3><a class="toc-backref" href="#id94"><tt class="docutils literal">\PolPrintIntervalsPrintRightEndPoint</tt></a></h3> -<blockquote> -<p>Default definition is:</p> -<pre class="literal-block"> -\newcommand\PolPrintIntervalsPrintRightEndPoint{\PolPrintIntervalsTheRightEndPoint} -</pre> -</blockquote> -</div> -</div> -<div class="section" id="id19"> -<span id="id18"></span><h2><a class="toc-backref" href="#id95"><tt class="docutils literal"><span class="pre">\PolPrintIntervals*[varname]{sturmname}</span></tt></a></h2> -<blockquote> -<p>This starred variant produces an alternative output (which -displays the root multiplicity), and is provided as an -example of customization.</p> -<p>As replacement for <a class="reference internal" href="#polprintintervalsknownroot">\PolPrintIntervalsKnownRoot</a>, -<a class="reference internal" href="#polprintintervalsprintexactzero">\PolPrintIntervalsPrintExactZero</a>, -<a class="reference internal" href="#polprintintervalsunknownroot">\PolPrintIntervalsUnknownRoot</a> it uses its own -<tt class="docutils literal"><span class="pre">\POL@@PrintIntervals...</span></tt> macros. We only reproduce here one -definition:</p> -<pre class="literal-block"> -\newcommand\POL@@PrintIntervalsPrintExactZero{% - \displaystyle - \xintSignedFrac{\PolPrintIntervalsTheLeftEndPoint}% -}% -</pre> -<p>Multiplicities are printed using this auxiliary macro:</p> -</blockquote> -<div class="section" id="polprintintervalsprintmultiplicity"> -<h3><a class="toc-backref" href="#id96"><tt class="docutils literal">\PolPrintIntervalsPrintMultiplicity</tt></a></h3> -<blockquote> -<p>whose default definition is:</p> -<pre class="literal-block"> -\newcommand\PolPrintIntervalsPrintMultiplicity{(\mbox{mult. }\PolPrintIntervalsTheMultiplicity)} -</pre> -</blockquote> -</div> -</div> -<div class="section" id="polmapcoeffs-macro-polname"> -<span id="polmapcoeffs"></span><h2><a class="toc-backref" href="#id97"><tt class="docutils literal"><span class="pre">\PolMapCoeffs{\macro}{polname}</span></tt></a></h2> -<blockquote> -<p>It modifies ('in-place': original coefficients get lost) each -coefficient of the defined polynomial via the <em>expandable</em> macro -<tt class="docutils literal">\macro</tt>. The degree is adjusted as necessary if some leading -coefficients vanish after the operation. In replacement text of -<tt class="docutils literal">\macro</tt>, <tt class="docutils literal">\index</tt> expands to the coefficient index (which is -defined to be zero for the constant term).</p> -<p>Notice that <tt class="docutils literal">\macro</tt> will have to handle inputs of the shape -<tt class="docutils literal">A/B[N]</tt> (<a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> internal notation). This means that it probably -will have to be expressed in terms of macros from <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> package.</p> -<p>Example:</p> -<pre class="literal-block"> -\def\foo#1{\xintMul{#1}{\the\numexpr\index*\index\relax}} -</pre> -<p>(or with <tt class="docutils literal"><span class="pre">\xintSqr{\index}</span></tt>) to replace <tt class="docutils literal">n</tt>-th coefficient -<tt class="docutils literal">f_n</tt> by <tt class="docutils literal">f_n*n^2</tt>.</p> -</blockquote> -</div> -<div class="section" id="polreducecoeffs-polname"> -<span id="polreducecoeffs"></span><h2><a class="toc-backref" href="#id98"><tt class="docutils literal">\PolReduceCoeffs{polname}</tt></a></h2> -<blockquote> -About the same as <tt class="docutils literal"><span class="pre">\PolMapCoeffs{\xintIrr}{polname}</span></tt> (but -maintaining a <tt class="docutils literal">[0]</tt> postfix for speedier <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> parsing when -polynomial function is used for computations.) This is a -one-argument macro, working 'in-place'.</blockquote> -</div> -<div class="section" id="id21"> -<span id="id20"></span><h2><a class="toc-backref" href="#id99"><tt class="docutils literal"><span class="pre">\PolReduceCoeffs*{polname}</span></tt></a></h2> -<blockquote> -<p>This starred variant leaves un-touched the decimal exponent in the -internal representation of the fractional coefficients, i.e. if a -coefficient is internally <tt class="docutils literal">A/B[N]</tt>, then <tt class="docutils literal">A/B</tt> is reduced to -smallest terms, but the <tt class="docutils literal">10^N</tt> part is kept as is. Note: if the -polynomial is freshly defined directly via <a class="reference internal" href="#polfromcsv">\PolFromCSV</a> its coefficients might still be internally in some -format like <tt class="docutils literal">1.5e7</tt>; the macro will anyhow always first do the -needed conversion to strict format <tt class="docutils literal">A/B[N]</tt>.</p> -<p>Evaluations with polynomials treated by this can be much faster than -with those handled by the non-starred variant -<a class="reference internal" href="#polreducecoeffs-polname">\PolReduceCoeffs{polname}</a>: as the numerators and denominators -remain smaller, this proves very beneficial in favorable cases -(especially when the coefficients are decimal numbers) to the -expansion speed of the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> macros used internally by -<a class="reference internal" href="#polevalat">\PolEval</a>.</p> -</blockquote> -</div> -<div class="section" id="polmakemonic-polname"> -<span id="polmakemonic"></span><h2><a class="toc-backref" href="#id100"><tt class="docutils literal">\PolMakeMonic{polname}</tt></a></h2> -<blockquote> -Divides by the leading coefficient. It is recommended to execute -<a class="reference internal" href="#id21">\PolReduceCoeffs*{polname}</a> immediately afterwards. This is not -done automatically, due to the case the original polynomial had integer -coefficients and we want to keep the leading one as common -denominator.</blockquote> -</div> -<div class="section" id="polmakeprimitive-polname"> -<span id="polmakeprimitive"></span><h2><a class="toc-backref" href="#id101"><tt class="docutils literal">\PolMakePrimitive{polname}</tt></a></h2> -<blockquote> -Divides by the integer content see (<a class="reference internal" href="#policontent">\PolIContent</a>). This thus produces a polynomial with integer -coefficients having no common factor. The sign of the leading -coefficient is not modified.</blockquote> -</div> -</div> -<div class="section" id="expandable-macros"> -<h1><a class="toc-backref" href="#id102">Expandable macros</a></h1> -<p>All these macros expand completely in two steps except <tt class="docutils literal">\PolToExpr</tt> -and <tt class="docutils literal">\PolToFloatExpr</tt> (and their auxiliaries) which need a -<tt class="docutils literal">\write</tt>, <tt class="docutils literal">\edef</tt> or a <tt class="docutils literal"><span class="pre">\csname...\endcsname</span></tt> context.</p> -<div class="section" id="poleval-polname-atexpr-numerical-expression"> -<span id="polevalatexpr"></span><h2><a class="toc-backref" href="#id103"><tt class="docutils literal"><span class="pre">\PolEval{polname}\AtExpr{numerical</span> expression}</tt></a></h2> -<blockquote> -It boils down to -<tt class="docutils literal">\xinttheexpr polname(numerical <span class="pre">expression)\relax</span></tt>.</blockquote> -</div> -<div class="section" id="poleval-polname-at-fraction"> -<span id="polevalat"></span><h2><a class="toc-backref" href="#id104"><tt class="docutils literal"><span class="pre">\PolEval{polname}\At{fraction}</span></tt></a></h2> -<blockquote> -Evaluates the polynomial at value <tt class="docutils literal">fraction</tt> which must be in (or -expand to) a format acceptable to the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> macros.</blockquote> -</div> -<div class="section" id="polevalreduced-polname-atexpr-numerical-expression"> -<span id="polevalreducedatexpr"></span><h2><a class="toc-backref" href="#id105"><tt class="docutils literal"><span class="pre">\PolEvalReduced{polname}\AtExpr{numerical</span> expression}</tt></a></h2> -<blockquote> -Boils down to <tt class="docutils literal">\xinttheexpr reduce(polname(numerical <span class="pre">expression))\relax</span></tt>.</blockquote> -</div> -<div class="section" id="polevalreduced-polname-at-fraction"> -<span id="polevalreducedat"></span><h2><a class="toc-backref" href="#id106"><tt class="docutils literal"><span class="pre">\PolEvalReduced{polname}\At{fraction}</span></tt></a></h2> -<blockquote> -Evaluates the polynomial at value <tt class="docutils literal">fraction</tt> which must be in (or -expand to) a format acceptable to the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> macros, and produce -an irreducible fraction.</blockquote> -</div> -<div class="section" id="polfloateval-polname-atexpr-numerical-expression"> -<span id="polfloatevalatexpr"></span><h2><a class="toc-backref" href="#id107"><tt class="docutils literal"><span class="pre">\PolFloatEval{polname}\AtExpr{numerical</span> expression}</tt></a></h2> -<blockquote> -<p>Boils down to <tt class="docutils literal">\xintthefloatexpr polname(numerical <span class="pre">expression)\relax</span></tt>.</p> -<p>This is done via a Horner Scheme (see <a class="reference internal" href="#poldef">\poldef</a> and -<a class="reference internal" href="#polgenfloatvariant-polname">\PolGenFloatVariant{polname}</a>), with already rounded -coefficients. <a class="footnote-reference" href="#id24" id="id22">[2]</a> To use the <em>exact coefficients</em> with <em>exactly -executed</em> additions and multiplications, just insert it in the float -expression as in this example: <a class="footnote-reference" href="#id25" id="id23">[3]</a></p> -<pre class="literal-block"> -\xintthefloatexpr 3.27*\xintexpr f(2.53)\relax^2\relax -</pre> -<p>The <tt class="docutils literal">f(2.53)</tt> is exactly computed then rounded at the time of -getting raised to the power <tt class="docutils literal">2</tt>. Moving the <tt class="docutils literal">^2</tt> inside, that -operation would also be treated exactly.</p> -<table class="docutils footnote" frame="void" id="id24" rules="none"> -<colgroup><col class="label" /><col /></colgroup> -<tbody valign="top"> -<tr><td class="label"><a class="fn-backref" href="#id22">[2]</a></td><td>Anyway each floating point operation starts by rounding its -operands to the floating point precision.</td></tr> -</tbody> -</table> -<table class="docutils footnote" frame="void" id="id25" rules="none"> -<colgroup><col class="label" /><col /></colgroup> -<tbody valign="top"> -<tr><td class="label"><a class="fn-backref" href="#id23">[3]</a></td><td>The <tt class="docutils literal">\xintexpr</tt> here could be <tt class="docutils literal">\xinttheexpr</tt> but that -would be less efficient. Cf. <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> documentation about -nested expressions.</td></tr> -</tbody> -</table> -</blockquote> -</div> -<div class="section" id="polfloateval-polname-at-fraction"> -<span id="polfloatevalat"></span><h2><a class="toc-backref" href="#id108"><tt class="docutils literal"><span class="pre">\PolFloatEval{polname}\At{fraction}</span></tt></a></h2> -<blockquote> -Evaluates the polynomial at value <tt class="docutils literal">fraction</tt> which must be in (or -expand to) a format acceptable to the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> macros, and produces -a floating point number.</blockquote> -</div> -<div class="section" id="polifcoeffisplusorminusone-a-b"> -<span id="polifcoeffisplusorminusone"></span><h2><a class="toc-backref" href="#id109"><tt class="docutils literal"><span class="pre">\PolIfCoeffIsPlusOrMinusOne{A}{B}</span></tt></a></h2> -<blockquote> -<p>This macro is a priori undefined.</p> -<p>It is defined via the default <a class="reference internal" href="#poltypesetcmd-raw-coeff">\PolTypesetCmd{raw_coeff}</a> to be -used if needed in the execution of <a class="reference internal" href="#poltypesetmonomialcmd">\PolTypesetMonomialCmd</a>, -e.g. to insert a <tt class="docutils literal">\cdot</tt> in front of <tt class="docutils literal"><span class="pre">\PolVar^{\PolIndex}</span></tt> if -the coefficient is not plus or minus one.</p> -<p>The macro will execute <tt class="docutils literal">A</tt> if the coefficient has been found to be -plus or minus one, and <tt class="docutils literal">B</tt> if not.</p> -</blockquote> -</div> -<div class="section" id="polleadingcoeff-polname"> -<span id="polleadingcoeff"></span><h2><a class="toc-backref" href="#id110"><tt class="docutils literal">\PolLeadingCoeff{polname}</tt></a></h2> -<blockquote> -Expands to the leading coefficient.</blockquote> -</div> -<div class="section" id="polnthcoeff-polname-number"> -<span id="polnthcoeff"></span><h2><a class="toc-backref" href="#id111"><tt class="docutils literal"><span class="pre">\PolNthCoeff{polname}{number}</span></tt></a></h2> -<blockquote> -It expands to the raw <tt class="docutils literal">N</tt>-th coefficient (<tt class="docutils literal">0/1[0]</tt> if the index -number is out of range). With <tt class="docutils literal"><span class="pre">N=-1</span></tt>, <tt class="docutils literal"><span class="pre">-2</span></tt>, ... expands to the -leading coefficients.</blockquote> -</div> -<div class="section" id="poldegree-polname"> -<span id="poldegree"></span><h2><a class="toc-backref" href="#id112"><tt class="docutils literal">\PolDegree{polname}</tt></a></h2> -<blockquote> -It expands to the degree. This is <tt class="docutils literal"><span class="pre">-1</span></tt> if zero polynomial but this -may change in future. Should it then expand to <tt class="docutils literal"><span class="pre">-\infty</span></tt> ?</blockquote> -</div> -<div class="section" id="policontent-polname"> -<span id="policontent"></span><h2><a class="toc-backref" href="#id113"><tt class="docutils literal">\PolIContent{polname}</tt></a></h2> -<blockquote> -<p>It expands to the contents of the polynomial, i.e. to the positive -fraction such that dividing by this fraction produces a polynomial -with integer coefficients having no common prime divisor.</p> -<p>See <a class="reference internal" href="#polmakeprimitive">\PolMakePrimitive</a>.</p> -</blockquote> -</div> -<div class="section" id="poltoexpr-polname"> -<span id="poltoexpr"></span><h2><a class="toc-backref" href="#id114"><tt class="docutils literal">\PolToExpr{polname}</tt></a></h2> -<blockquote> -<p>Expands <a class="footnote-reference" href="#id27" id="id26">[4]</a> to <tt class="docutils literal"><span class="pre">coeff_N*x^N+...</span></tt> (descending powers.)</p> -<table class="docutils footnote" frame="void" id="id27" rules="none"> -<colgroup><col class="label" /><col /></colgroup> -<tbody valign="top"> -<tr><td class="label"><a class="fn-backref" href="#id26">[4]</a></td><td>in a <tt class="docutils literal">\write</tt>, <tt class="docutils literal">\edef</tt>, or <tt class="docutils literal"><span class="pre">\csname...\endcsname</span></tt>, but -not under <tt class="docutils literal"><span class="pre">\romannumeral-`0</span></tt>.</td></tr> -</tbody> -</table> -<p>By default zero coefficients are skipped (issue <tt class="docutils literal">\poltoexpralltrue</tt> to -get all of them in output).</p> -<p>By default, no <tt class="docutils literal">+</tt> sign before negative coefficients, for -compliance with Maple input format (but see -<a class="reference internal" href="#poltoexprtermprefix-raw-coeff">\PolToExprTermPrefix{raw_coeff}</a>.) Also, like the default -behaviour of <a class="reference internal" href="#poltypeset-polname">\PolTypeset{polname}</a>, does not print (for the non -constant terms) coefficients equal to plus or minus one. The degree -one monomial is output as <tt class="docutils literal">x</tt>, not <tt class="docutils literal">x^1</tt>. Complete customization is -possible, see next macros.</p> -<p>Of course <tt class="docutils literal">\PolToExpr{f}</tt> can be inserted in a <tt class="docutils literal">\poldef</tt>, as the -latter expands token by token, hence will force complete expansion -of <tt class="docutils literal">\PolToExpr{f}</tt>, but a simple <tt class="docutils literal">f(x)</tt> is more efficient for -the identical result.</p> -</blockquote> -<div class="section" id="poltoexproneterm-raw-coeff-number"> -<span id="poltoexproneterm"></span><h3><a class="toc-backref" href="#id115"><tt class="docutils literal"><span class="pre">\PolToExprOneTerm{raw_coeff}{number}</span></tt></a></h3> -<blockquote> -<p>This two argument expandable command takes care of the monomial and -its coefficient. The default definition is done in order for -coefficients of absolute value <tt class="docutils literal">1</tt> not be printed explicitely -(except of course for the constant term). Also by default, the -monomial of degree one is <tt class="docutils literal">x</tt> not <tt class="docutils literal">x^1</tt>, and <tt class="docutils literal">x^0</tt> is skipped.</p> -<p>For compatibility with Maple input requirements, by default a <tt class="docutils literal">*</tt> -always precedes the <tt class="docutils literal">x^number</tt>, except if the coefficient is a one -or a minus one. See <a class="reference internal" href="#poltoexprtimes">\PolToExprTimes</a>.</p> -</blockquote> -</div> -<div class="section" id="poltoexpronetermstylea-raw-coeff-number"> -<span id="poltoexpronetermstylea"></span><h3><a class="toc-backref" href="#id116"><tt class="docutils literal"><span class="pre">\PolToExprOneTermStyleA{raw_coeff}{number}</span></tt></a></h3> -<blockquote> -Holds the default package meaning of -<a class="reference internal" href="#poltoexproneterm-raw-coeff-number">\PolToExprOneTerm{raw_coeff}{number}</a>.</blockquote> -</div> -<div class="section" id="poltoexpronetermstyleb-raw-coeff-number"> -<span id="poltoexpronetermstyleb"></span><h3><a class="toc-backref" href="#id117"><tt class="docutils literal"><span class="pre">\PolToExprOneTermStyleB{raw_coeff}{number}</span></tt></a></h3> -<blockquote> -<p>For output in this style:</p> -<pre class="literal-block"> -2*x^11/3+3*x^8/7-x^5-x^4/4-x^3-x^2/2-2*x+1 -</pre> -<p>issue <tt class="docutils literal">\let\PolToExprOneTerm\PolToExprOneTermStyleB</tt> before usage of -<tt class="docutils literal">\PolToExpr</tt>. Note that then <tt class="docutils literal">\PolToExprCmd</tt> isn't used at all. -To revert to package default, issue -<tt class="docutils literal">\let\PolToExprOneTerm\PolToExprOneTermStyleA</tt>.</p> -<p>To suppress the <tt class="docutils literal">*</tt>'s, cf. <a class="reference internal" href="#poltoexprtimes">\PolToExprTimes</a>.</p> -</blockquote> -</div> -<div class="section" id="poltoexprcmd-raw-coeff"> -<span id="poltoexprcmd"></span><h3><a class="toc-backref" href="#id118"><tt class="docutils literal">\PolToExprCmd{raw_coeff}</tt></a></h3> -<blockquote> -It is the one-argument macro used by the package definition of -<tt class="docutils literal">\PolToExprOneTerm</tt> for the coefficients themselves (when not -equal to plus or minus one), and it defaults to -<tt class="docutils literal"><span class="pre">\xintPRaw{\xintRawWithZeros{#1}}</span></tt>. One will have to redefine it -to <tt class="docutils literal"><span class="pre">\xintIrr{#1}</span></tt> or to <tt class="docutils literal"><span class="pre">\xintPRaw{\xintIrr{#1}}</span></tt> to obtain in the -output forcefully reduced coefficients.</blockquote> -</div> -<div class="section" id="poltoexprtermprefix-raw-coeff"> -<span id="poltoexprtermprefix"></span><h3><a class="toc-backref" href="#id119"><tt class="docutils literal">\PolToExprTermPrefix{raw_coeff}</tt></a></h3> -<blockquote> -Defined identically as <a class="reference internal" href="#poltypesetcmdprefix-raw-coeff">\PolTypesetCmdPrefix{raw_coeff}</a>. It -prefixes with a plus sign for non-negative coefficients, because -they don't carry one by themselves.</blockquote> -</div> -<div class="section" id="id28"> -<span id="poltoexprvar"></span><h3><a class="toc-backref" href="#id120"><tt class="docutils literal">\PolToExprVar</tt></a></h3> -<blockquote> -This expands to the variable to use in output (it does not have to -be a single letter, may be an expandable macro.) Initial definition -is <tt class="docutils literal">x</tt>.</blockquote> -</div> -<div class="section" id="id29"> -<span id="poltoexprtimes"></span><h3><a class="toc-backref" href="#id121"><tt class="docutils literal">\PolToExprTimes</tt></a></h3> -<blockquote> -This expands to the symbol used for multiplication of an -<tt class="docutils literal"><span class="pre">x^{number}</span></tt> by the corresponding coefficient. The default is -<tt class="docutils literal">*</tt>. Redefine the macro to expand to nothing to get rid of it (but -this will give output incompatible with some professional computer -algebra software).</blockquote> -</div> -</div> -<div class="section" id="id31"> -<span id="id30"></span><h2><a class="toc-backref" href="#id122"><tt class="docutils literal"><span class="pre">\PolToExpr*{polname}</span></tt></a></h2> -<blockquote> -Expands to <tt class="docutils literal"><span class="pre">coeff_0+coeff_1*x+coeff_2*x^2+...</span></tt> (ascending powers). -Customizable like <a class="reference internal" href="#poltoexpr-polname">\PolToExpr{polname}</a> via the same macros.</blockquote> -</div> -<div class="section" id="poltofloatexpr-polname"> -<span id="poltofloatexpr"></span><h2><a class="toc-backref" href="#id123"><tt class="docutils literal">\PolToFloatExpr{polname}</tt></a></h2> -<blockquote> -<p>Similar to <a class="reference internal" href="#poltoexpr-polname">\PolToExpr{polname}</a> but uses <a class="reference external" href="\PolToFloatExprCmd{raw_coeff}">\PolToFloatExprCmd</a> -which by default rounds and converts the coefficients to floating -point format.</p> -<div class="admonition note"> -<p class="first admonition-title">Note</p> -<p>It is not necessary to have issued -<a class="reference internal" href="#polgenfloatvariant-polname">\PolGenFloatVariant{polname}</a>. The rounded coefficients are -not easily recoverable from the <tt class="docutils literal">\xintfloatexpr</tt> polynomial -function hence <tt class="docutils literal">\PolToFloatExprCmd</tt> operates from the <em>exact</em> -coefficients anew.</p> -<p class="last">Attention that both macros obey the prevailing float precision. -If it is changed between those macro calls, then a mismatch -exists between the coefficients as used in <tt class="docutils literal">\xintfloatexpr</tt> and -those output by <tt class="docutils literal">\PolToFloatExpr{polname}</tt>.</p> -</div> -</blockquote> -<div class="section" id="poltofloatexproneterm-raw-coeff-number"> -<span id="poltofloatexproneterm"></span><h3><a class="toc-backref" href="#id124"><tt class="docutils literal"><span class="pre">\PolToFloatExprOneTerm{raw_coeff}{number}</span></tt></a></h3> -<blockquote> -Similar to <a class="reference external" href="\PolToExprOneTerm{raw_coeff}{number}">\PolToExprOneTerm</a>. But does not treat -especially coefficients equal to plus or minus one.</blockquote> -</div> -<div class="section" id="poltofloatexprcmd-raw-coeff"> -<span id="id33"></span><h3><a class="toc-backref" href="#id125"><tt class="docutils literal">\PolToFloatExprCmd{raw_coeff}</tt></a></h3> -<blockquote> -<p>It is the one-argument macro used by <tt class="docutils literal">\PolToFloatExprOneTerm</tt>. -Its package definition is <tt class="docutils literal"><span class="pre">\xintFloat{#1}</span></tt>.</p> -<div class="admonition caution"> -<p class="first admonition-title">Caution!</p> -<p>Currently (<a class="reference external" href="http://www.ctan.org/pkg/xint">xint</a> <tt class="docutils literal">1.3c</tt>) <tt class="docutils literal">\xintFloat{0}</tt> outputs <tt class="docutils literal">0.e0</tt> -which is perfectly acceptable input for Python, but not for -Maple. Thus, one should better leave the <a class="reference internal" href="#poltoexprallfalse">\poltoexprallfalse</a> -toggle to its default <tt class="docutils literal">\iffalse</tt> state, if one intends to use -the output in a Maple worksheet.</p> -<p>But even then the zero polynomial will cause a problem. Workaround:</p> -<pre class="literal-block"> -\renewcommand\PolToFloatExprCmd[1]{\xintiiifZero{#1}{0.0}{\xintFloat{#1}}} -</pre> -<p class="last">Usage of <tt class="docutils literal">\xintiiifZero</tt> and not <tt class="docutils literal">\xintifZero</tt> is only for -optimization (I can't help it) because <tt class="docutils literal">#1</tt> is known to be -in <tt class="docutils literal">xintfrac</tt> raw format.</p> -</div> -</blockquote> -</div> -</div> -<div class="section" id="id35"> -<span id="id34"></span><h2><a class="toc-backref" href="#id126"><tt class="docutils literal"><span class="pre">\PolToFloatExpr*{polname}</span></tt></a></h2> -<blockquote> -Typesets in ascending powers.</blockquote> -</div> -<div class="section" id="poltolist-polname"> -<span id="poltolist"></span><h2><a class="toc-backref" href="#id127"><tt class="docutils literal">\PolToList{polname}</tt></a></h2> -<blockquote> -Expands to <tt class="docutils literal"><span class="pre">{coeff_0}{coeff_1}...{coeff_N}</span></tt> with <tt class="docutils literal">N</tt> = degree, and -<tt class="docutils literal">coeff_N</tt> the leading coefficient -(the zero polynomial does give <tt class="docutils literal">{0/1[0]}</tt> and not an -empty output.)</blockquote> -</div> -<div class="section" id="poltocsv-polname"> -<span id="poltocsv"></span><h2><a class="toc-backref" href="#id128"><tt class="docutils literal">\PolToCSV{polname}</tt></a></h2> -<blockquote> -Expands to <tt class="docutils literal">coeff_0, coeff_1, coeff_2, <span class="pre">.....,</span> coeff_N</tt>, starting -with constant term and ending with leading coefficient. Converse -to <a class="reference internal" href="#polfromcsv-polname-csv">\PolFromCSV</a>.</blockquote> -</div> -<div class="section" id="polsturmchainlength-sturmname"> -<span id="polsturmchainlength"></span><h2><a class="toc-backref" href="#id129"><tt class="docutils literal">\PolSturmChainLength{sturmname}</tt></a></h2> -<blockquote> -<p>Returns the integer <tt class="docutils literal">N</tt> such that <tt class="docutils literal">sturmname_N</tt> is the last one -in the Sturm chain <tt class="docutils literal">sturmname_0</tt>, <tt class="docutils literal">sturmname_1</tt>, ...</p> -<p>See <a class="reference internal" href="#poltosturm-polname-sturmname">\PolToSturm{polname}{sturmname}</a>.</p> -</blockquote> -</div> -<div class="section" id="polsturmifzeroexactlyknown-sturmname-index-a-b"> -<span id="polsturmifzeroexactlyknown"></span><h2><a class="toc-backref" href="#id130"><tt class="docutils literal"><span class="pre">\PolSturmIfZeroExactlyKnown{sturmname}{index}{A}{B}</span></tt></a></h2> -<blockquote> -<p>Executes <tt class="docutils literal">A</tt> if the <tt class="docutils literal">index</tt>-th interval reduces to a singleton, -i.e. the root is known exactly, else <tt class="docutils literal">B</tt>.</p> -<div class="admonition note"> -<p class="first admonition-title">Note</p> -<p class="last"><tt class="docutils literal">index</tt> is allowed to be something like <tt class="docutils literal">1+2*3</tt> as it is fed -to <tt class="docutils literal"><span class="pre">\the\numexpr...\relax</span></tt>.</p> -</div> -</blockquote> -</div> -<div class="section" id="polsturmisolatedzeroleft-sturmname-index"> -<span id="polsturmisolatedzeroleft"></span><h2><a class="toc-backref" href="#id131"><tt class="docutils literal"><span class="pre">\PolSturmIsolatedZeroLeft{sturmname}{index}</span></tt></a></h2> -<blockquote> -<p>Expands to the left end-point for the <tt class="docutils literal">index</tt>-th interval, as -computed by some earlier <a class="reference internal" href="#polsturmisolatezeros-sturmname">\PolSturmIsolateZeros{sturmname}</a>.</p> -<div class="admonition note"> -<p class="first admonition-title">Note</p> -<p class="last">Of course, this is kept updated by macros such as -<a class="reference internal" href="#polrefineinterval-n">\PolRefineInterval{sturmname}{index}</a>.</p> -</div> -<p>The value is pre-formatted using <a class="reference internal" href="#poldectostring">\PolDecTostring</a>.</p> -</blockquote> -</div> -<div class="section" id="polsturmisolatedzeroright-sturmname-index"> -<span id="polsturmisolatedzeroright"></span><h2><a class="toc-backref" href="#id132"><tt class="docutils literal"><span class="pre">\PolSturmIsolatedZeroRight{sturmname}{index}</span></tt></a></h2> -<blockquote> -<p>Expands to the right end-point for the <tt class="docutils literal">index</tt>-th interval as -computed by some earlier <a class="reference internal" href="#polsturmisolatezeros-sturmname">\PolSturmIsolateZeros{sturmname}</a> and -possibly refined afterwards.</p> -<p>The value is pre-formatted using <a class="reference internal" href="#poldectostring">\PolDecTostring</a>.</p> -</blockquote> -</div> -<div class="section" id="polsturmisolatedzeromultiplicity-sturmname-index"> -<span id="polsturmisolatedzeromultiplicity"></span><h2><a class="toc-backref" href="#id133"><tt class="docutils literal"><span class="pre">\PolSturmIsolatedZeroMultiplicity{sturmname}{index}</span></tt></a></h2> -<blockquote> -<p>Expands to the multiplicity of the unique root contained in the -<tt class="docutils literal">index</tt>-th interval.</p> -<div class="admonition attention"> -<p class="first admonition-title">Attention!</p> -<p class="last">A prior execution of <a class="reference internal" href="#id12">\PolSturmIsolateZeros*{sturmname}</a> is mandatory.</p> -</div> -<p>See <a class="reference internal" href="#the-degree-nine-polynomial-with-0-99-0-999-0-9999-as-triple-roots">The degree nine polynomial with 0.99, 0.999, 0.9999 as triple -roots</a> for an example of use.</p> -</blockquote> -</div> -<div class="section" id="polsturmnbofisolatedzeros-sturmname"> -<span id="polsturmnbofisolatedzeros"></span><h2><a class="toc-backref" href="#id134"><tt class="docutils literal">\PolSturmNbOfIsolatedZeros{sturmname}</tt></a></h2> -<blockquote> -Expands to the number of real roots of the polynomial -<tt class="docutils literal"><sturmname>_0</tt>, i.e. the number of distinct real roots of the -polynomial originally used to create the Sturm chain via -<a class="reference internal" href="#poltosturm-polname-sturmname">\PolToSturm{polname}{sturmname}</a>.</blockquote> -<div class="admonition warning"> -<p class="first admonition-title">Warning</p> -<p class="last">The next few macros counting roots, with or without multiplicities, -less than or equal to some value, are under evaluation and may be -removed from the package if their utility is judged to be not high -enough. They can be re-coded at user level on the basis of the other -documented package macros anyway.</p> -</div> -<div class="section" id="polsturmnbofrootsof-sturmname-lessthanorequalto-value"> -<h3><a class="toc-backref" href="#id135"><tt class="docutils literal"><span class="pre">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualTo{value}</span></tt></a></h3> -<blockquote> -<p>Expands to the number of distinct roots (of the polynomial used to -create the Sturm chain) less than or equal to the <tt class="docutils literal">value</tt> (i.e. a -number of fraction recognizable by the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> macros).</p> -<div class="admonition attention"> -<p class="first admonition-title">Attention!</p> -<p><a class="reference internal" href="#polsturmisolatezeros-sturmname">\PolSturmIsolateZeros{sturmname}</a> must have been executed -beforehand.</p> -<p class="last">And the argument is a <tt class="docutils literal">sturmname</tt>, not a <tt class="docutils literal">polname</tt> (this is -why the macro contains Sturm in its name), simply to be reminded -of the above constraint.</p> -</div> -</blockquote> -</div> -<div class="section" id="polsturmnbofrootsof-sturmname-lessthanorequaltoexpr-expression"> -<h3><a class="toc-backref" href="#id136"><tt class="docutils literal"><span class="pre">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualToExpr{expression}</span></tt></a></h3> -<blockquote> -<p>Expands to the number of distinct roots (of the polynomial -used to create the Sturm chain) which are less than or equal to the -given <tt class="docutils literal">expression</tt>.</p> -<div class="admonition attention"> -<p class="first admonition-title">Attention!</p> -<p class="last"><a class="reference internal" href="#polsturmisolatezeros-sturmname">\PolSturmIsolateZeros{sturmname}</a> must have been executed -beforehand.</p> -</div> -</blockquote> -</div> -<div class="section" id="polsturmnbwithmultofrootsof-sturmname-lessthanorequalto-value"> -<h3><a class="toc-backref" href="#id137"><tt class="docutils literal"><span class="pre">\PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualTo{value}</span></tt></a></h3> -<blockquote> -<p>Expands to the number counted with multiplicities of the roots (of -the polynomial used to create the Sturm chain) which are less than -or equal to the given <tt class="docutils literal">value</tt>.</p> -<div class="admonition attention"> -<p class="first admonition-title">Attention!</p> -<p class="last"><a class="reference internal" href="#id12">\PolSturmIsolateZeros*{sturmname}</a> (or the double starred -variant) must have been executed beforehand.</p> -</div> -</blockquote> -</div> -<div class="section" id="polsturmnbwithmultofrootsof-sturmname-lessthanorequaltoexpr-expression"> -<h3><a class="toc-backref" href="#id138"><tt class="docutils literal"><span class="pre">\PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualToExpr{expression}</span></tt></a></h3> -<blockquote> -<p>Expands to the total number of roots (counted with multiplicities) -which are less than or equal to the given <tt class="docutils literal">expression</tt>.</p> -<div class="admonition attention"> -<p class="first admonition-title">Attention!</p> -<p class="last"><a class="reference internal" href="#id12">\PolSturmIsolateZeros*{sturmname}</a> (or the double starred -variant) must have been executed beforehand.</p> -</div> -</blockquote> -</div> -</div> -<div class="section" id="polsturmnbofrationalroots-sturmname"> -<h2><a class="toc-backref" href="#id139"><tt class="docutils literal">\PolSturmNbOfRationalRoots{sturmname}</tt></a></h2> -<blockquote> -<p>Expands to the number of rational roots (without multiplicities).</p> -<div class="admonition attention"> -<p class="first admonition-title">Attention!</p> -<p class="last"><a class="reference internal" href="#id14">\PolSturmIsolateZeros**{sturmname}</a> must have been executed -beforehand.</p> -</div> -</blockquote> -</div> -<div class="section" id="polsturmnbofrationalrootswithmultiplicities-sturmname"> -<h2><a class="toc-backref" href="#id140"><tt class="docutils literal">\PolSturmNbOfRationalRootsWithMultiplicities{sturmname}</tt></a></h2> -<blockquote> -<p>Expands to the number of rational roots (counted with multiplicities).</p> -<div class="admonition attention"> -<p class="first admonition-title">Attention!</p> -<p class="last"><a class="reference internal" href="#id14">\PolSturmIsolateZeros**{sturmname}</a> must have been executed -beforehand.</p> -</div> -</blockquote> -</div> -<div class="section" id="polsturmrationalroot-sturmname-k"> -<h2><a class="toc-backref" href="#id141"><tt class="docutils literal"><span class="pre">\PolSturmRationalRoot{sturmname}{k}</span></tt></a></h2> -<blockquote> -<p>Expands to the <tt class="docutils literal">k</tt>th rational root (they are ordered and indexed -starting at 1 for the most negative).</p> -<div class="admonition attention"> -<p class="first admonition-title">Attention!</p> -<p class="last"><a class="reference internal" href="#id14">\PolSturmIsolateZeros**{sturmname}</a> must have been executed -beforehand.</p> -</div> -</blockquote> -</div> -<div class="section" id="polsturmrationalrootindex-sturmname-k"> -<h2><a class="toc-backref" href="#id142"><tt class="docutils literal"><span class="pre">\PolSturmRationalRootIndex{sturmname}{k}</span></tt></a></h2> -<blockquote> -<p>Expands to <tt class="docutils literal">index</tt> of the <tt class="docutils literal">k</tt>th rational root as part of the -ordered real roots (without multiplicities). I.e., above macro -<a class="reference internal" href="#polsturmrationalroot-sturmname-k">\PolSturmRationalRoot{sturmname}{k}</a> is equivalent to this -nested call:</p> -<pre class="literal-block"> -\PolSturmIsolatedZeroLeft{sturmname}{\PolSturmRationalRootIndex{sturmname}{k}} -</pre> -<div class="admonition attention"> -<p class="first admonition-title">Attention!</p> -<p class="last"><a class="reference internal" href="#id14">\PolSturmIsolateZeros**{sturmname}</a> must have been executed -beforehand.</p> -</div> -</blockquote> -</div> -<div class="section" id="polsturmrationalrootmultiplicity-sturmname-k"> -<h2><a class="toc-backref" href="#id143"><tt class="docutils literal"><span class="pre">\PolSturmRationalRootMultiplicity{sturmname}{k}</span></tt></a></h2> -<blockquote> -<p>Expands to the multiplicity of the <tt class="docutils literal">k</tt>th rational root.</p> -<div class="admonition attention"> -<p class="first admonition-title">Attention!</p> -<p class="last"><a class="reference internal" href="#id14">\PolSturmIsolateZeros**{sturmname}</a> must have been executed -beforehand.</p> -</div> -</blockquote> -</div> -<div class="section" id="polintervalwidth-sturmname-index"> -<span id="polintervalwidth"></span><h2><a class="toc-backref" href="#id144"><tt class="docutils literal"><span class="pre">\PolIntervalWidth{sturmname}{index}</span></tt></a></h2> -<blockquote> -The <tt class="docutils literal">10^E</tt> width of the current <tt class="docutils literal">index</tt>-th root localization -interval. Output is in <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> raw <tt class="docutils literal">1/1[E]</tt> format (if not zero).</blockquote> -</div> -<div class="section" id="expandable-macros-for-use-within-execution-of-polprintintervals"> -<h2><a class="toc-backref" href="#id145">Expandable macros for use within execution of <tt class="docutils literal">\PolPrintIntervals</tt></a></h2> -<p>These macros are for usage within custom user redefinitions of -<a class="reference internal" href="#polprintintervalsknownroot">\PolPrintIntervalsKnownRoot</a>, <a class="reference internal" href="#polprintintervalsunknownroot">\PolPrintIntervalsUnknownRoot</a>, or -in redefinitions of <a class="reference internal" href="#polprintintervalsprintexactzero">PolPrintIntervalsPrintExactZero</a> (used in the -default for the former) and of <a class="reference internal" href="#polprintintervalsprintleftendpoint">\PolPrintIntervalsPrintLeftEndPoint</a>, -<a class="reference internal" href="#polprintintervalsprintrightendpoint">\PolPrintIntervalsPrintRightEndPoint</a> (used in the default for the -latter).</p> -<div class="admonition attention"> -<p class="first admonition-title">Attention!</p> -<p class="last">Some macros formerly mentioned here got removed at 0.7: -<tt class="docutils literal">\PolPrintIntervalsTheEndPoint</tt>, -<tt class="docutils literal"><span class="pre">\PolIfEndPointIsPositive{A}{B}</span></tt>, -<tt class="docutils literal"><span class="pre">\PolIfEndPointIsNegative{A}{B}</span></tt>, -<tt class="docutils literal"><span class="pre">\PolIfEndPointIsZero{A}{B}</span></tt>.</p> -</div> -<div class="section" id="polprintintervalsthevar"> -<h3><a class="toc-backref" href="#id146"><tt class="docutils literal">\PolPrintIntervalsTheVar</tt></a></h3> -<blockquote> -Expands to the name (default <tt class="docutils literal">Z</tt>) used for representing the roots, -which was passed as optional argument <tt class="docutils literal">varname</tt> to -<a class="reference internal" href="#polprintintervals-varname-sturmname">\PolPrintIntervals[varname]{sturmname}</a>.</blockquote> -</div> -<div class="section" id="polprintintervalstheindex"> -<h3><a class="toc-backref" href="#id147"><tt class="docutils literal">\PolPrintIntervalsTheIndex</tt></a></h3> -<blockquote> -Expands to the index of the considered interval (indexing starting -at 1 for the leftmost interval).</blockquote> -</div> -<div class="section" id="polprintintervalsthesturmname"> -<h3><a class="toc-backref" href="#id148"><tt class="docutils literal">\PolPrintIntervalsTheSturmName</tt></a></h3> -<blockquote> -Expands to the argument which was passed as <tt class="docutils literal">sturmname</tt> to -<a class="reference internal" href="#polprintintervals-varname-sturmname">\PolPrintIntervals[varname]{sturmname}</a>.</blockquote> -</div> -<div class="section" id="polprintintervalstheleftendpoint"> -<h3><a class="toc-backref" href="#id149"><tt class="docutils literal">\PolPrintIntervalsTheLeftEndPoint</tt></a></h3> -<blockquote> -The left end point of the interval, as would be produced by -<a class="reference internal" href="#polsturmisolatedzeroleft">\PolSturmIsolatedZeroLeft</a> if it was -used with arguments the Sturm chain name and interval index returned -by <a class="reference internal" href="#polprintintervalsthesturmname">\PolPrintIntervalsTheSturmName</a> and -<a class="reference internal" href="#polprintintervalstheindex">\PolPrintIntervalsTheIndex</a>.</blockquote> -</div> -<div class="section" id="polprintintervalstherightendpoint"> -<h3><a class="toc-backref" href="#id150"><tt class="docutils literal">\PolPrintIntervalsTheRightEndPoint</tt></a></h3> -<blockquote> -The right end point of the interval, as would be produced by -<a class="reference internal" href="#polsturmisolatedzeroright">\PolSturmIsolatedZeroRight</a> for -this Sturm chain name and index.</blockquote> -</div> -<div class="section" id="polprintintervalsthemultiplicity"> -<h3><a class="toc-backref" href="#id151"><tt class="docutils literal">\PolPrintIntervalsTheMultiplicity</tt></a></h3> -<blockquote> -The multiplicity of the unique root within the interval of index -<a class="reference internal" href="#polprintintervalstheindex">\PolPrintIntervalsTheIndex</a>. Makes sense only if the starred (or -double-starred) variant of <a class="reference internal" href="#polsturmisolatezeros">\PolSturmIsolateZeros</a> was used earlier.</blockquote> -</div> -</div> -<div class="section" id="poldectostring-decimal-number"> -<span id="poldectostring"></span><h2><a class="toc-backref" href="#id152"><tt class="docutils literal">\PolDecToString{decimal number}</tt></a></h2> -<blockquote> -<p>This is a utility macro to print decimal numbers. It has been -backported to <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> (release <tt class="docutils literal">1.3</tt> of <tt class="docutils literal">2018/03/01</tt>) under -the name <tt class="docutils literal">\xintDecToString</tt>, and the <tt class="docutils literal">polexpr</tt> macro is simply -now an alias to it.</p> -<p>For example -<tt class="docutils literal"><span class="pre">\PolDecToString{123.456e-8}</span></tt> will expand to <tt class="docutils literal">0.00000123456</tt> -and <tt class="docutils literal"><span class="pre">\PolDecToString{123.450e-8}</span></tt> to <tt class="docutils literal">0.00000123450</tt> which -illustrates that trailing zeros are not trimmed. To trim trailing -zeroes, one can use <tt class="docutils literal"><span class="pre">\PolDecToString{\xintREZ{#1}}</span></tt>.</p> -<p>The precise behaviour of this macro may evolve in future releases of -<a class="reference external" href="http://www.ctan.org/pkg/xint">xint</a>.</p> -</blockquote> -</div> -</div> -<div class="section" id="booleans-with-default-setting-as-indicated"> -<h1><a class="toc-backref" href="#id153">Booleans (with default setting as indicated)</a></h1> -<div class="section" id="xintverbosefalse"> -<h2><a class="toc-backref" href="#id154"><tt class="docutils literal">\xintverbosefalse</tt></a></h2> -<blockquote> -<p>This is actually an <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> configuration. Setting it to -<tt class="docutils literal">true</tt> triggers the writing of information to the log when new -polynomials are defined.</p> -<div class="admonition caution"> -<p class="first admonition-title">Caution!</p> -<p class="last">The macro meanings as written to the log are to be considered -unstable and undocumented internal structures.</p> -</div> -</blockquote> -</div> -<div class="section" id="poltypesetallfalse"> -<h2><a class="toc-backref" href="#id155"><tt class="docutils literal">\poltypesetallfalse</tt></a></h2> -<blockquote> -If <tt class="docutils literal">true</tt>, <a class="reference internal" href="#poltypeset-polname">\PolTypeset{polname}</a> will also typeset the vanishing -coefficients.</blockquote> -</div> -<div class="section" id="poltoexprallfalse"> -<h2><a class="toc-backref" href="#id156"><tt class="docutils literal">\poltoexprallfalse</tt></a></h2> -<blockquote> -If <tt class="docutils literal">true</tt>, <a class="reference internal" href="#poltoexpr-polname">\PolToExpr{polname}</a> and <a class="reference internal" href="#poltofloatexpr-polname">\PolToFloatExpr{polname}</a> will -also include the vanishing coefficients in their outputs.</blockquote> -</div> -</div> -<div class="section" id="polexprsetup"> -<h1><a class="toc-backref" href="#id157"><tt class="docutils literal">\polexprsetup</tt></a></h1> -<blockquote> -<p>Serves to customize the package. Currently only two keys are -recognized:</p> -<ul class="simple"> -<li><tt class="docutils literal">norr</tt>: the postfix that <a class="reference internal" href="#id14">\PolSturmIsolateZeros**{sturmname}</a> -should append to <tt class="docutils literal">sturmname</tt> to declare the primitive polynomial -obtained from original one after removal of all rational roots. -The default value is <tt class="docutils literal">_norr</tt> (standing for “no rational roots”).</li> -<li><tt class="docutils literal">sqfnorr</tt>: the postfix that <a class="reference internal" href="#id14">\PolSturmIsolateZeros**{sturmname}</a> -should append to <tt class="docutils literal">sturmname</tt> to declare the primitive polynomial -obtained from original one after removal of all rational roots and -suppression of all multiplicities. -The default value is <tt class="docutils literal">_sqf_norr</tt> (standing for “square-free with -no rational roots”).</li> -</ul> -<p>The package executes <tt class="docutils literal">\polexprsetup{norr=_norr, -sqfnorr=_sqf_norr}</tt> as default.</p> -</blockquote> -</div> -<div class="section" id="technicalities"> -<h1><a class="toc-backref" href="#id158">Technicalities</a></h1> -<ul> -<li><p class="first">The catcode of the semi-colon is reset temporarily by <a class="reference internal" href="#poldef">\poldef</a> macro in case some other package (for example the French -babel module) may have made it active. This will fail though if the -whole thing was already part of a macro argument, in such cases one -can use <a class="reference internal" href="#id2">\PolDef{f}{P(x)}</a> -rather. The colon in <tt class="docutils literal">:=</tt> may be active with no consequences.</p> -</li> -<li><p class="first">As a consequence of <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> addition and subtraction always using -least common multiples for the denominators <a class="footnote-reference" href="#id37" id="id36">[5]</a>, user-chosen common -denominators survive additions and multiplications. For example, this:</p> -<pre class="literal-block"> -\poldef P(x):= 1/2 + 2/2*x + 3/2*x^3 + 4/2*x^4; -\poldef Q(x):= 1/3 + (2/3)x + (3/3)x^3 + (4/3)x^4; -\poldef PQ(x):= P(x)Q(x); -</pre> -<p>gives internally the polynomial:</p> -<pre class="literal-block"> -1/6+4/6*x^1+4/6*x^2+6/6*x^3+20/6*x^4+16/6*x^5+9/6*x^6+24/6*x^7+16/6*x^8 -</pre> -<p>where all coefficients have the same denominator 6. Notice though that -<tt class="docutils literal">\PolToExpr{PQ}</tt> outputs the <tt class="docutils literal">6/6*x^3</tt> as <tt class="docutils literal">x^3</tt> because (by -default) it recognizes and filters out coefficients equal to one or -minus one (since release <tt class="docutils literal">0.3</tt>). One can use for example -<tt class="docutils literal">\PolToCSV{PQ}</tt> to see the internally stored coefficients.</p> -<table class="docutils footnote" frame="void" id="id37" rules="none"> -<colgroup><col class="label" /><col /></colgroup> -<tbody valign="top"> -<tr><td class="label"><a class="fn-backref" href="#id36">[5]</a></td><td><p class="first last">prior to <tt class="docutils literal">0.4.1</tt>, <tt class="docutils literal">polexpr</tt> used to temporarily patch -during the parsing of polynomials the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> macros. This -patch was backported to <a class="reference external" href="http://www.ctan.org/pkg/xint">xint</a> at release <tt class="docutils literal">1.3</tt>.</p> -</td></tr> -</tbody> -</table> -</li> -<li><p class="first"><a class="reference internal" href="#poldiff-polname-1-polname-2">\PolDiff{polname_1}{polname_2}</a> always applies <tt class="docutils literal">\xintIrr</tt> to the -resulting coefficients, except that the <em>power of ten</em> part <tt class="docutils literal">[N]</tt> -(for example an input in scientific notation such as <tt class="docutils literal">1.23e5</tt> gives -<tt class="docutils literal">123/1[3]</tt> internally in xintfrac) is not taken into account in the -reduction of the fraction. This is tentative and may change.</p> -<p>Same remark for <a class="reference internal" href="#polantidiff-polname-1-polname-2">\PolAntiDiff{polname_1}{polname_2}</a>.</p> -</li> -<li><p class="first">Currently, the package stores all coefficients from index <tt class="docutils literal">0</tt> to -index equal to the polynomial degree inside a single macro, as a list. -This data structure is obviously very inefficient for polynomials of -high degree and few coefficients (as an example with <tt class="docutils literal">\poldef -<span class="pre">f(x):=x^1000</span> + x^500;</tt> the subsequent definition <tt class="docutils literal">\poldef <span class="pre">g(x):=</span> -<span class="pre">f(x)^2;</span></tt> will do of the order of 1,000,000 multiplications and -additions involvings only zeroes... which does take time). This -may change in the future.</p> -</li> -<li><p class="first">As is to be expected internal structures of the package are barely -documented and unstable. Don't use them.</p> -</li> -</ul> -</div> -<div class="section" id="change-log"> -<h1><a class="toc-backref" href="#id159">CHANGE LOG</a></h1> -<ul> -<li><p class="first">v0.1 (2018/01/11): initial release. Features:</p> -<ul class="simple"> -<li>The <a class="reference internal" href="#poldef">\poldef</a> parser itself,</li> -<li>Differentiation and anti-differentiation,</li> -<li>Euclidean division and GCDs,</li> -<li>Various utilities such as <a class="reference internal" href="#polfromcsv">\PolFromCSV</a>, -<a class="reference internal" href="#polmapcoeffs">\PolMapCoeffs</a>, -<a class="reference internal" href="#poltocsv">\PolToCSV</a>, <a class="reference internal" href="#poltoexpr">\PolToExpr</a>, ...</li> -</ul> -<p>Only one-variable polynomials so far.</p> -</li> -<li><p class="first">v0.2 (2018/01/14)</p> -<ul class="simple"> -<li>Fix: <tt class="docutils literal">"README thinks \numexpr recognizes ^ operator"</tt>.</li> -<li>Convert README to reStructuredText markup.</li> -<li>Move main documentation from README to separate <tt class="docutils literal">polexpr.txt</tt> file.</li> -<li>Provide <tt class="docutils literal">polexpr.html</tt> as obtained via <a class="reference external" href="http://docutils.sourceforge.net/docs/index.html">DocUtils</a> <tt class="docutils literal">rst2html.py</tt>.</li> -<li>Convert README to (CTAN compatible) Markdown markup.</li> -</ul> -<p>Due to lack of available time the test suite might not be extensive -enough. Bug reports are very welcome!</p> -</li> -<li><p class="first">v0.3 (2018/01/17)</p> -<ul> -<li><p class="first">bug fixes:</p> -<ul> -<li><p class="first">the <tt class="docutils literal">0.1</tt> <a class="reference internal" href="#polevalat">\PolEval</a> accepted expressions for its second -argument, but this was removed by mistake at <tt class="docutils literal">0.2</tt>. Restored.</p> -<p><strong>Attention</strong>: at <tt class="docutils literal">0.4</tt> this has been reverted again, and -<a class="reference internal" href="#polevalatexpr">\PolEval{P}\AtExpr{foo}</a> syntax is needed for -using expressions in the second argument.</p> -</li> -</ul> -</li> -<li><p class="first">incompatible or breaking changes:</p> -<ul class="simple"> -<li><a class="reference internal" href="#poltoexpr">\PolToExpr</a> now by default uses <em>descending</em> -powers (it also treats differently coefficients equal to 1 or -1.) -Use <a class="reference internal" href="#id30">\PolToExpr*</a> for <em>ascending</em> powers.</li> -<li><a class="reference internal" href="#polevalat">\PolEval</a> reduced the output to smallest terms, -but as this is costly with big fractions and not needed if e.g. -wrapped in an <tt class="docutils literal">\xintRound</tt> or <tt class="docutils literal">\xintFloat</tt>, this step has been -removed; the former meaning is available as <a class="reference internal" href="#polevalreducedat">\PolEvalReduced</a>.</li> -</ul> -</li> -<li><p class="first">new (or newly documented) macros:</p> -<ul class="simple"> -<li><a class="reference internal" href="#poltypesetcmd">\PolTypesetCmd</a></li> -<li><a class="reference internal" href="#poltypesetcmdprefix">\PolTypesetCmdPrefix</a></li> -<li><a class="reference internal" href="#poltypesetmonomialcmd">\PolTypesetMonomialCmd</a></li> -<li><a class="reference internal" href="#polevalreducedat">\PolEvalReducedAt</a></li> -<li><a class="reference internal" href="#poltofloatexpr">\PolToFloatExpr</a></li> -<li><a class="reference internal" href="#poltoexproneterm">\PolToExprOneTerm</a></li> -<li><a class="reference internal" href="#poltofloatexproneterm">\PolToFloatExprOneTerm</a></li> -<li><a class="reference internal" href="#poltoexprcmd">\PolToExprCmd</a></li> -<li><a class="reference internal" href="#id33">\PolToFloatExprCmd</a></li> -<li><a class="reference internal" href="#poltoexprtermprefix">\PolToExprTermPrefix</a></li> -<li><a class="reference internal" href="#poltoexprvar">\PolToExprVar</a></li> -<li><a class="reference internal" href="#poltoexprtimes">\PolToExprTimes</a></li> -</ul> -</li> -<li><p class="first">improvements:</p> -<ul> -<li><p class="first">documentation has a table of contents, internal hyperlinks, -standardized signature notations and added explanations.</p> -</li> -<li><p class="first">one can do <tt class="docutils literal"><span class="pre">\PolLet{g}={f}</span></tt> or <tt class="docutils literal"><span class="pre">\PolLet{g}{f}</span></tt>.</p> -</li> -<li><p class="first"><tt class="docutils literal">\PolToExpr{f}</tt> is highly customizable.</p> -</li> -<li><p class="first"><a class="reference internal" href="#poldef">\poldef</a> and other defining macros prepare the polynomial -functions for usage within <tt class="docutils literal">\xintthefloatexpr</tt> (or -<tt class="docutils literal">\xintdeffloatvar</tt>). Coefficients are pre-rounded to the -floating point precision. Indispensible for numerical algorithms, -as exact fractions, even reduced, quickly become very big. See the -documentation about how to use the exact polynomials also in -floating point context.</p> -<p><strong>Attention</strong>: this has been reverted at <tt class="docutils literal">0.4</tt>. The macro -<a class="reference internal" href="#polgenfloatvariant">\PolGenFloatVariant</a> must be used for -generation floating point polynomial functions.</p> -</li> -</ul> -</li> -</ul> -</li> -<li><p class="first">v0.3.1 (2018/01/18)</p> -<p>Fixes two typos in example code included in the documentation.</p> -</li> -<li><p class="first">v0.4 (2018/02/16)</p> -<ul> -<li><p class="first">bug fixes:</p> -<ul class="simple"> -<li>when Euclidean division gave a zero remainder, the internal -representation of this zero polynomial could be faulty; this -could cause mysterious bugs in conjunction with other package -macros such as <a class="reference internal" href="#polmapcoeffs">\PolMapCoeffs</a>.</li> -<li><a class="reference internal" href="#polgcd">\PolGCD</a> was buggy in case of first polynomial being -of lesser degree than the second one.</li> -</ul> -</li> -<li><p class="first">breaking changes:</p> -<ul> -<li><p class="first">formerly <a class="reference internal" href="#polevalat">\PolEval{P}\At{foo}</a> allowed <tt class="docutils literal">foo</tt> to -be an expression, which was transparently handled via -<tt class="docutils literal">\xinttheexpr</tt>. Now, <tt class="docutils literal">foo</tt> must be a fraction (or a macro -expanding to such) in the format acceptable by <tt class="docutils literal">xintfrac.sty</tt> -macros. Use <a class="reference internal" href="#polevalatexpr">\PolEval{P}\AtExpr{foo}</a> for more -general arguments using expression syntax. E.g., if <tt class="docutils literal">foo</tt> is the -name of a variable known to <tt class="docutils literal">\xintexpr</tt>.</p> -<p>The same holds for <a class="reference internal" href="#polevalreducedat">\PolEvalReduced</a> -and <a class="reference internal" href="#polfloatevalat">\PolFloatEval</a>.</p> -</li> -<li><p class="first">the <tt class="docutils literal">3.0</tt> automatic generation of floating point variants has -been reverted. Not only do <em>not</em> the package macros automatically -generate floating point variants of newly created polynomials, -they actually make pre-existing such variant undefined.</p> -<p>See <a class="reference internal" href="#polgenfloatvariant">\PolGenFloatVariant</a>.</p> -</li> -</ul> -</li> -<li><p class="first">new non-expandable macros:</p> -<ul class="simple"> -<li><a class="reference internal" href="#polgenfloatvariant">\PolGenFloatVariant</a></li> -<li><a class="reference internal" href="#polgloballet">\PolGlobalLet</a></li> -<li><a class="reference internal" href="#poltypesetone">\PolTypesetOne</a></li> -<li><a class="reference internal" href="#polquo">\PolQuo</a></li> -<li><a class="reference internal" href="#polrem">\PolRem</a></li> -<li><a class="reference internal" href="#poltosturm">\PolToSturm</a></li> -<li><a class="reference internal" href="#id9">\PolToSturm*</a></li> -<li><a class="reference internal" href="#polsettosturmchainsignchangesat">\PolSetToSturmChainSignChangesAt</a></li> -<li><a class="reference internal" href="#polsettonbofzeroswithin">\PolSetToNbOfZerosWithin</a></li> -<li><a class="reference internal" href="#polsturmisolatezeros">\PolSturmIsolateZeros</a></li> -<li><a class="reference internal" href="#polrefineinterval">\PolRefineInterval*</a></li> -<li><a class="reference internal" href="#polrefineinterval-n">\PolRefineInterval[N]</a></li> -<li><a class="reference internal" href="#polensureintervallength">\PolEnsureIntervalLength</a></li> -<li><a class="reference internal" href="#polensureintervallengths">\PolEnsureIntervalLengths</a></li> -<li><a class="reference internal" href="#polprintintervals">\PolPrintIntervals</a></li> -<li><a class="reference internal" href="#polprintintervalsprintexactzero">\PolPrintIntervalsPrintExactZero</a></li> -<li><a class="reference internal" href="#polprintintervalsprintleftendpoint">\PolPrintIntervalsPrintLeftEndPoint</a></li> -<li><a class="reference internal" href="#polprintintervalsprintrightendpoint">\PolPrintIntervalsPrintRightEndPoint</a></li> -<li><a class="reference internal" href="#id20">\PolReduceCoeffs*</a></li> -<li><a class="reference internal" href="#polmakemonic">\PolMakeMonic</a></li> -</ul> -</li> -<li><p class="first">new expandable macros:</p> -<ul class="simple"> -<li><a class="reference internal" href="#poltoexpronetermstylea">\PolToExprOneTermStyleA</a></li> -<li><a class="reference internal" href="#polifcoeffisplusorminusone">\PolIfCoeffIsPlusOrMinusOne</a></li> -<li><a class="reference internal" href="#polleadingcoeff">\PolLeadingCoeff</a></li> -<li><a class="reference internal" href="#polsturmchainlength">\PolSturmChainLength</a></li> -<li><a class="reference internal" href="#polsturmnbofisolatedzeros">\PolSturmNbOfIsolatedZeros</a></li> -<li><a class="reference internal" href="#polsturmifzeroexactlyknown">\PolSturmIfZeroExactlyKnown</a></li> -<li><a class="reference internal" href="#polsturmisolatedzeroleft">\PolSturmIsolatedZeroLeft</a></li> -<li><a class="reference internal" href="#polsturmisolatedzeroright">\PolSturmIsolatedZeroRight</a></li> -<li><tt class="docutils literal">\PolPrintIntervalsTheEndPoint</tt> (removed at 0.7)</li> -<li><a class="reference internal" href="#polprintintervalstheindex">\PolPrintIntervalsTheIndex</a></li> -<li><tt class="docutils literal">\PolIfEndPointIsPositive</tt> (removed at 0.7)</li> -<li><tt class="docutils literal">\PolIfEndPointIsNegative</tt> (removed at 0.7)</li> -<li><tt class="docutils literal">\PolIfEndPointIsZero</tt> (removed at 0.7)</li> -<li><a class="reference internal" href="#polintervalwidth">\PolIntervalWidth</a></li> -<li><a class="reference internal" href="#poldectostring">\PolDecToString</a></li> -</ul> -</li> -<li><p class="first">improvements:</p> -<p>The main new feature is implementation of the <a class="reference external" href="https://en.wikipedia.org/wiki/Sturm%27s_theorem">Sturm algorithm</a> -for localization of the real roots of polynomials.</p> -</li> -</ul> -</li> -<li><p class="first">v0.4.1 (2018/03/01)</p> -<p>Synced with xint 1.3.</p> -</li> -<li><p class="first">v0.4.2 (2018/03/03)</p> -<p>Documentation fix.</p> -</li> -<li><p class="first">v0.5 (2018/04/08)</p> -<ul class="simple"> -<li>bug fixes:<ul> -<li><a class="reference internal" href="#polget-polname-fromarray-macro">\PolGet{polname}\fromarray\macro</a> crashed when <tt class="docutils literal">\macro</tt> was -an <a class="reference external" href="http://www.ctan.org/pkg/xint">xinttools</a> array macro with no items. It now produces the zero -polynomial.</li> -</ul> -</li> -<li>breaking changes:<ul> -<li><a class="reference internal" href="#poltosturm">\PolToSturm</a> creates primitive integer coefficients polynomials. -This speeds up localization of roots via -<a class="reference internal" href="#polsturmisolatezeros">\PolSturmIsolateZeros</a>. In case of user protests the author -will make available again the code producing the bona fide Sturm -polynomials as used formerly.</li> -<li>polynomials created from <a class="reference internal" href="#polfromcsv">\PolFromCSV</a> or <a class="reference internal" href="#polget">\PolGet</a> -get their coefficients normalized via <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a>'s <tt class="docutils literal">\xintRaw</tt>.</li> -</ul> -</li> -<li>experimental change:<ul> -<li>optional argument to <a class="reference internal" href="#polsturmisolatezeros">\PolSturmIsolateZeros</a> (see <a class="reference internal" href="#the-degree-41-polynomial-with-2-1-9-1-8-0-0-1-1-9-2-as-roots">The -degree 41 polynomial with -2, -1.9, -1.8, ..., 0, 0.1, ..., 1.9, 2 -as roots</a> for usage). It will presumably be replaced in future by -an interval specification.</li> -</ul> -</li> -<li>new non-expandable macro:<ul> -<li><a class="reference internal" href="#polmakeprimitive">\PolMakePrimitive</a></li> -</ul> -</li> -<li>new expandable macro:<ul> -<li><a class="reference internal" href="#policontent">\PolIContent</a></li> -</ul> -</li> -</ul> -</li> -<li><p class="first">v0.5.1 (2018/04/22)</p> -<ul class="simple"> -<li>new feature:<ul> -<li>the character <tt class="docutils literal">'</tt> can be used in polynomial names.</li> -</ul> -</li> -</ul> -</li> -<li><p class="first">v0.6 (2018/11/20)</p> -<ul class="simple"> -<li>bugfix:<ul> -<li>the starred variant <a class="reference internal" href="#id10">\PolToSturm*{polname}{sturmname}</a> was -broken. On the occasion of the fix, its meaning has been modified, -see its documentation.</li> -<li>using <a class="reference internal" href="#poltosturm">\PolToSturm</a> with a constant polynomial -caused a division by zero error.</li> -</ul> -</li> -<li>new macro:<ul> -<li><a class="reference internal" href="#id11">\PolSturmIsolateZeros*</a> -acts like the <a class="reference internal" href="#polsturmisolatezeros">non-starred variant</a> then computes all the multiplicities.</li> -</ul> -</li> -<li>new expandable macros:<ul> -<li><a class="reference internal" href="#polsturmisolatedzeromultiplicity-sturmname-index">\PolSturmIsolatedZeroMultiplicity{sturmname}{index}</a></li> -<li><a class="reference internal" href="#polsturmnbofrootsof-sturmname-lessthanorequalto-value">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualTo{value}</a></li> -<li><a class="reference internal" href="#polsturmnbofrootsof-sturmname-lessthanorequaltoexpr-expression">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualToExpr{expression}</a></li> -<li><a class="reference internal" href="#polsturmnbwithmultofrootsof-sturmname-lessthanorequalto-value">\PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualTo{value}</a></li> -<li><a class="reference internal" href="#polsturmnbwithmultofrootsof-sturmname-lessthanorequaltoexpr-expression">\PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualToExpr{expression}</a></li> -</ul> -</li> -</ul> -</li> -<li><p class="first">v0.7 (2018/12/08), v0.7.1 (bugfix), v0.7.2 (2nd bugfix) (2018/12/09)</p> -<ul class="simple"> -<li>breaking changes:<ul> -<li>although <a class="reference internal" href="#polprintintervals-varname-sturmname">\PolPrintIntervals[varname]{sturmname}</a> default output -remains the same, some auxiliary macros for user-customization -have been removed: <tt class="docutils literal">\PolPrintIntervalsTheEndPoint</tt>, -<tt class="docutils literal"><span class="pre">\PolIfEndPointIsPositive{A}{B}</span></tt>, -<tt class="docutils literal"><span class="pre">\PolIfEndPointIsNegative{A}{B}</span></tt>, and -<tt class="docutils literal"><span class="pre">\PolIfEndPointIsZero{A}{B}</span></tt>.</li> -</ul> -</li> -<li>bugfix:<ul> -<li>it could happen that, contrarily to documentation, an interval -computed by <a class="reference internal" href="#polsturmisolatezeros-sturmname">\PolSturmIsolateZeros{sturmname}</a> had zero as an -endpoint,</li> -<li><a class="reference internal" href="#polensureintervallength-sturmname-index-e">\PolEnsureIntervalLength{sturmname}{index}{E}</a> could under -certain circumstances erroneously replace a non-zero root by -zero,</li> -<li><a class="reference internal" href="#polensureintervallengths-sturmname-e">\PolEnsureIntervalLengths{sturmname}{E}</a> crashed when used with -a polynomial with no real roots, hence for which no isolation intervals -existed (thanks to Thomas Söll for report).</li> -</ul> -</li> -<li>new macros:<ul> -<li><a class="reference internal" href="#id14">\PolSturmIsolateZeros**{sturmname}</a></li> -<li><a class="reference internal" href="#polsturmisolatezerosgetmultiplicitiesandrationalroots-sturmname">\PolSturmIsolateZerosGetMultiplicitiesAndRationalRoots{sturmname}</a></li> -<li><a class="reference internal" href="#polsturmisolatezerosandfindrationalroots-sturmname">\PolSturmIsolateZerosAndFindRationalRoots{sturmname}</a></li> -<li><a class="reference internal" href="#polexprsetup">\polexprsetup</a></li> -<li><a class="reference internal" href="#id18">\PolPrintIntervals*</a></li> -<li><a class="reference internal" href="#polprintintervalsnorealroots">\PolPrintIntervalsNoRealRoots</a></li> -<li><a class="reference internal" href="#polprintintervalsbeginenv">\PolPrintIntervalsBeginEnv</a></li> -<li><a class="reference internal" href="#polprintintervalsendenv">\PolPrintIntervalsEndEnv</a></li> -<li><a class="reference internal" href="#polprintintervalsknownroot">\PolPrintIntervalsKnownRoot</a></li> -<li><a class="reference internal" href="#polprintintervalsunknownroot">\PolPrintIntervalsUnknownRoot</a></li> -<li><a class="reference internal" href="#polprintintervalsprintmultiplicity">\PolPrintIntervalsPrintMultiplicity</a></li> -</ul> -</li> -<li>new expandable macros:<ul> -<li><a class="reference internal" href="#polsturmnbofrationalroots-sturmname">\PolSturmNbOfRationalRoots{sturmname}</a></li> -<li><a class="reference internal" href="#polsturmnbofrationalrootswithmultiplicities-sturmname">\PolSturmNbOfRationalRootsWithMultiplicities{sturmname}</a></li> -<li><a class="reference internal" href="#polsturmrationalroot-sturmname-k">\PolSturmRationalRoot{sturmname}{k}</a></li> -<li><a class="reference internal" href="#polsturmrationalrootindex-sturmname-k">\PolSturmRationalRootIndex{sturmname}{k}</a></li> -<li><a class="reference internal" href="#polsturmrationalrootmultiplicity-sturmname-k">\PolSturmRationalRootMultiplicity{sturmname}{k}</a></li> -<li><a class="reference internal" href="#polprintintervalsthevar">\PolPrintIntervalsTheVar</a></li> -<li><a class="reference internal" href="#polprintintervalsthesturmname">\PolPrintIntervalsTheSturmName</a></li> -<li><a class="reference internal" href="#polprintintervalsthemultiplicity">\PolPrintIntervalsTheMultiplicity</a></li> -</ul> -</li> -</ul> -</li> -<li><p class="first">v0.7.3 (2019/02/04)</p> -<ul class="simple"> -<li>bugfix:<ul> -<li>Debugging information not destined to user showed in log if root -finding was done under <tt class="docutils literal">\xintverbosetrue</tt> regime.</li> -<li><a class="reference internal" href="#polprintintervalsthevar">\PolPrintIntervalsTheVar</a> remained defined after -<a class="reference internal" href="#polprintintervals">\PolPrintIntervals</a> but was left undefined after -<a class="reference internal" href="#id18">\PolPrintIntervals*</a> (reported by Jürgen Gilg). Now remains -defined in both cases, and <a class="reference internal" href="#polprintintervalsthesturmname">\PolPrintIntervalsTheSturmName</a> -also.</li> -<li>Polynomial names ending in digits caused errors (reported by Thomas -Söll).</li> -</ul> -</li> -</ul> -</li> -<li><p class="first">v0.7.4 (2019/02/12)</p> -<ul class="simple"> -<li>bugfix:<ul> -<li>20000000000 is too big for <tt class="docutils literal">\numexpr</tt>, shouldn't I know that? -Thanks to Jürgen Gilg for report.</li> -</ul> -</li> -</ul> -</li> -<li><p class="first">v0.7.5 (2020/01/31)</p> -<p>Synced with xint 1.4. Requires it.</p> -</li> -</ul> -</div> -<div class="section" id="acknowledgments"> -<h1><a class="toc-backref" href="#id160">Acknowledgments</a></h1> -<p>Thanks to Jürgen Gilg whose question about <a class="reference external" href="http://www.ctan.org/pkg/xint">xint</a> usage for -differentiating polynomials was the initial trigger leading to this -package, and to Jürgen Gilg and Thomas Söll for testing it on some -concrete problems.</p> -<p>Renewed thanks to them on occasion of the <tt class="docutils literal">0.6</tt> and <tt class="docutils literal">0.7</tt> releases for their -continued interest.</p> -<p>See README.md for the License.</p> -</div> -</div> -</body> -</html> diff --git a/Master/texmf-dist/doc/latex/polexpr/polexpr.txt b/Master/texmf-dist/doc/latex/polexpr/polexpr.txt deleted file mode 100644 index 898375926b0..00000000000 --- a/Master/texmf-dist/doc/latex/polexpr/polexpr.txt +++ /dev/null @@ -1,2598 +0,0 @@ -.. comment: -*- fill-column: 72; mode: rst; -*- - -=============================== - Package polexpr documentation -=============================== - -0.7.5 (2020/01/31) -================== - -.. contents:: - -Basic syntax ------------- - -The syntax is:: - - \poldef polname(x):= expression in variable x; - -where: - -- in place of ``x`` an arbitrary *dummy variable* is authorized, - i.e. per default any of ``[a-z|A-Z]`` (more letters can be declared - under Unicode engines.) - -- ``polname`` consists of letters, digits, and the ``_`` and - ``'`` characters. It must start with a letter. - -.. attention:: - - The ``'`` is authorized since ``0.5.1``. As a result some constructs - recognized by the ``\xintexpr`` parser, such as ``var1 'and' var2`` - will get misinterpreted and cause errors. However these constructs - are unlikely to be frequently needed in polynomial expressions, and - the ``\xintexpr`` syntax offers alternatives, so it was deemed a - small evil. Of course the ``\xintexpr`` parser is modified only - temporarily during execution of ``\poldef``. - -One can also issue:: - - \PolDef{polname}{expression in variable x} - -which admits an optional first argument to modify the variable letter -from its default ``x``. - -``\poldef f(x):= 1-x+x^2;`` - defines polynomial ``f``. Polynomial names must start with a - letter and may contain letters, digits, underscores and the right - tick character. The - variable must be a single letter. The colon character is optional. - The semi-colon at end of expression is mandatory. - -``\PolDef{f}{1-x+x^2}`` - does the same as ``\poldef f(x):= 1-x+x^2;`` To use another letter - than ``x`` in the expression, one must pass it as an extra optional - argument to ``\PolDef``. Useful if the semi-colon has been assigned - some non-standard catcode by some package. - -``\PolLet{g}={f}`` - saves a copy of ``f`` under name ``g``. Also usable without ``=``. - -``\poldef f(z):= f(z)^2;`` - redefines ``f`` in terms of itself. - -``\poldef f(T):= f(f(T));`` - again redefines ``f`` in terms of its (new) self. - -``\poldef k(z):= f(z)-g(g(z)^2)^2;`` - should now define the zero polynomial... Let's check: - ``\[ k(z) = \PolTypeset[z]{k} \]`` - -``\PolDiff{f}{f'}`` - sets ``f'`` to the derivative of ``f``. The name doesn't have to be - ``f'`` (in fact the ``'`` is licit only since ``0.5.1``). - -.. important:: - - This is not done automatically. If some new definition needs to use - the derivative of some available polynomial, that derivative - polynomial must have been defined via ``\PolDiff``: something like - ``T'(x)^2`` will not work without a prior ``\PolDiff{T}{T'}``. - -``\PolDiff{f'}{f''}`` - obtains second derivative. - -``\PolDiff[3]{f}{f'''}`` - computes the third derivative. - -:: - - $f(z) = \PolTypeset[z]{f} $\newline - $f'(z) = \PolTypeset[z]{f'} $\newline - $f''(z) = \PolTypeset[z]{f''} $\newline - $f'''(z)= \PolTypeset[z]{f'''} $\par - -.. important:: - - The package does not currently know rational functions: ``/`` in - a parsed polynomial expression does the Euclidean quotient:: - - (1-x^2)/(1-x) - - does give ``1+x`` but :: - - (1/(1-x))*(1-x^2) - - evaluates to zero. This will work as expected:: - - \poldef k(x):= (x-1)(x-2)(x-3)(x-4)/(x^2-5x+4); - -.. _warningtacit: - -.. attention:: - - ``1/2 x^2`` skips the space and is treated like ``1/(2*x^2)`` because - of the tacit multiplication rules of \xintexpr. But this means it - gives zero! Thus one must use ``(1/2)x^2`` or ``1/2*x^2`` or - ``(1/2)*x^2`` for disambiguation: ``x - 1/2*x^2 + 1/3*x^3...``. It is - even simpler to move the denominator to the right: ``x - x^2/2 + - x^3/3 - ...``. - - It is worth noting that ``1/2(x-1)(x-2)`` suffers the same issue: - xint_ tacit multiplication always "ties more", hence this gets - interpreted as ``1/(2*(x-1)*(x-2))`` which gives zero by polynomial - division. Thus, use one of ``(1/2)(x-1)(x-2)``, ``1/2*(x-1)(x-2)`` or - ``(x-1)(x-2)/2``. - -After:: - - \poldef f_1(x):= 25(x-1)(x^2-2)(x-3)(x-4)(x-5);% - \poldef f_2(x):= 37(x-1)(x^2-2)(x-6)(x-7)(x-8);% - -the macro call ``\PolGCD{f_1}{f_2}{k}`` sets ``k`` to the (unitary) GCD of -``f_1`` and ``f_2`` (hence to the expansion of ``(x-1)(x^2-2)``.) - -``\PolToExpr{k}`` - will (expandably) give in this case ``x^3-x^2-2*x+2``. This is - useful for console or file output (the syntax is Maple- and - PSTricks-compatible; the letter used in output can be - (non-expandably) changed via a redefinition of `\\PolToExprVar`_.) - -``\PolToExpr*{k}`` - gives ascending powers: ``2-2*x-x^2+x^3``. - -Examples of localization of roots ---------------------------------- - -- To make printed decimal numbers more enjoyable than via - ``\xintSignedFrac``:: - - \renewcommand\PolTypesetOne[1]{\PolDecToString{\xintREZ{#1}}}% - - ``\PolDecToString`` will use decimal notation to incorporate the power - of ten part; and the ``\xintREZ`` will have the effect to suppress - trailing zeros if present in raw numerator (if those digits end up - after decimal mark.) Notice that the above are expandable macros and - that one can also do:: - - \renewcommand\PolToExprCmd[1]{\PolDecToString{\xintREZ{#1}}}% - - to modify output of `\\PolToExpr{polname}`_. - -- For extra info in log file use ``\xintverbosetrue``. - -- Only for some of these examples is the output included here. - - -A typical example -~~~~~~~~~~~~~~~~~ - -In this example the polynomial is square-free. - -:: - - \poldef f(x) := x^7 - x^6 - 2x + 1; - - \PolToSturm{f}{f} - \PolSturmIsolateZeros{f} - The \PolTypeset{f} polynomial has \PolSturmNbOfIsolatedZeros{f} distinct real - roots which are located in the following intervals: - \PolPrintIntervals{f} - Here is the second root with ten more decimal digits: - \PolRefineInterval[10]{f}{2} - \[\PolSturmIsolatedZeroLeft{f}{2}<Z_2<\PolSturmIsolatedZeroRight{f}{2}\] - And here is the first root with twenty digits after decimal mark: - \PolEnsureIntervalLength{f}{1}{-20} - \[\PolSturmIsolatedZeroLeft{f}{1}<Z_1<\PolSturmIsolatedZeroRight{f}{1}\] - The first element of the Sturm chain has degree $\PolDegree{f_0}$. As - this is the original degreee $\PolDegree{f}$ we know that $f$ is square free. - Its derivative is up to a constant \PolTypeset{f_1} (in this example - it is identical with it). - \PolToSturm{f_1}{f_1}\PolSturmIsolateZeros{f_1}% - The derivative has \PolSturmNbOfIsolatedZeros{f_1} distinct real - roots: - \PolPrintIntervals[W]{f_1} - \PolEnsureIntervalLengths{f_1}{-10}% - Here they are with ten digits after decimal mark: - \PolPrintIntervals[W]{f_1} - \PolDiff{f_1}{f''} - \PolToSturm{f''}{f''} - \PolSturmIsolateZeros{f''} - The second derivative is \PolTypeset{f''}. - It has \PolSturmNbOfIsolatedZeros{f''} distinct real - roots: - \PolPrintIntervals[X]{f''} - Here is the positive one with 20 digits after decimal mark: - \PolEnsureIntervalLength{f''}{2}{-20}% - \[X_2 = \PolSturmIsolatedZeroLeft{f''}{2}\dots\] - The more mathematically advanced among our dear readers will be able - to give the exact value for $X_2$! - -A degree four polynomial with nearby roots -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - -Notice that this example is a bit outdated as ``0.7`` release has -added ``\PolSturmIsolateZeros**{sturmname}`` which would find exactly -the roots. The steps here retain their interest when one is interested -in finding isolating intervals for example to prepare some demonstration -of dichotomy method. - - -:: - - \PolDef{Q}{(x-1.050001)(x-1.105001)(x-1.110501)(x-1.111051)} - \PolTypeset{Q} - \PolToSturm{Q}{Q} % it is allowed to use same prefix for Sturm chain - \PolSturmIsolateZeros{Q} - \PolPrintIntervals{Q} - % reports 1.0 < Z_1 < 1.1, 1.10 < Z_2 < 1.11, 1.110 < Z_3 < 1.111, and 1.111 < Z_4 < 1.112 - % but the above bounds do not allow minimizing separation between roots - % so we refine: - \PolRefineInterval*{Q}{1} - \PolRefineInterval*{Q}{2} - \PolRefineInterval*{Q}{3} - \PolRefineInterval*{Q}{4} - \PolPrintIntervals{Q} - % reports 1.05 < Z_1 < 1.06, 1.105 < Z_2 < 1.106, 1.1105 < Z_3 < 1.1106, - % and 1.11105 < Z_4 < 1.11106. - \PolEnsureIntervalLengths{Q}{-6} - \PolPrintIntervals{Q} - % of course finds here all roots exactly - - -The degree nine polynomial with 0.99, 0.999, 0.9999 as triple roots -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - -:: - - % define a user command (xinttools is loaded automatically by polexpr) - \newcommand\showmultiplicities[1]{% #1 = "sturmname" - \xintFor* ##1 in {\xintSeq{1}{\PolSturmNbOfIsolatedZeros{#1}}}\do{% - The multiplicity is \PolSturmIsolatedZeroMultiplicity{#1}{##1} - \PolSturmIfZeroExactlyKnown{#1}{##1}% - {at the root $x=\PolSturmIsolatedZeroLeft{#1}{##1}$} - {for the root such that - $\PolSturmIsolatedZeroLeft{#1}{##1}<x<\PolSturmIsolatedZeroRight{#1}{##1}$} - \par - }}% - \PolDef{f}{(x-0.99)^3(x-0.999)^3(x-0.9999)^3} - \renewcommand\PolTypesetOne[1]{\PolDecToString{\xintREZ{#1}}} - \PolTypeset{f}\par - \PolToSturm{f}{f}% it is allowed to use "polname" as "sturmname" too - \PolSturmIsolateZerosAndGetMultiplicities{f}% use the "sturmname" here - % or \PolSturmIsolateZeros*{f} which is exactly the same, but shorter.. - - \showmultiplicities{f} - -In this example, the output will look like this (but using math mode):: - - x^9 - 8.9667x^8 + 35.73400293x^7 - 83.070418400109x^6 + 124.143648875193123x^5 - - 123.683070924326075877x^4 + 82.149260397553075617891x^3 - - 35.07602992699900159127007x^2 + 8.7364078733314648368671733x - - 0.967100824643585986488103299 - - The multiplicity is 3 at the root x = 0.99 - The multiplicity is 3 at the root x = 0.999 - The multiplicity is 3 at the root x = 0.9999 - -On first pass, these rational roots were found (due to their relative -magnitudes, using ``\PolSturmIsolateZeros**`` was not needed here). But -multiplicity computation works also with (decimal) roots not yet -identified or with non-decimal or irrational roots. - -It is fun to modify only a tiny bit the polynomial and see if polexpr -survives:: - - \PolDef{g}{f(x)+1e-27} - \PolTypeset{g}\par - \PolToSturm{g}{g} - \PolSturmIsolateZeros*{g} - - \showmultiplicities{g} - -This produces:: - - x^9 - 8.9667x^8 + 35.73400293x^7 - 83.070418400109x^6 + 124.143648875193123x^5 - - 123.683070924326075877x^4 + 82.149260397553075617891x^3 - - 35.07602992699900159127007x^2 + 8.7364078733314648368671733x - - 0.967100824643585986488103298 - - The multiplicity is 1 for the root such that 0.98 < x < 0.99 - The multiplicity is 1 for the root such that 0.9991 < x < 0.9992 - The multiplicity is 1 for the root such that 0.9997 < x < 0.9998 - -Which means that the multiplicity-3 roots each became a real and a pair of -complex ones. Let's see them better:: - - \PolEnsureIntervalLengths{g}{-10} - - \showmultiplicities{g} - -which produces:: - - The multiplicity is 1 for the root such that 0.9899888032 < x < 0.9899888033 - The multiplicity is 1 for the root such that 0.9991447980 < x < 0.9991447981 - The multiplicity is 1 for the root such that 0.9997663986 < x < 0.9997663987 - -A degree five polynomial with three rational roots -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - -:: - - \poldef Q(x) := 1581755751184441 x^5 - -14907697165025339 x^4 - +48415668972339336 x^3 - -63952057791306264 x^2 - +46833913221154895 x - -49044360626280925; - - \PolToSturm{Q}{Q} - %\begin{flushleft} - \renewcommand\PolTypesetCmdPrefix[1]{\allowbreak\xintiiifSgn{#1}{}{+}{+}}% - $Q_0(x) = \PolTypeset{Q_0}$ - %\end{flushleft} - \PolSturmIsolateZeros**{Q} - \PolPrintIntervals{Q} - - $Q_{norr}(x) = \PolTypeset{Q_norr}$ - -Here, all real roots are rational:: - - Z_1 = 833719/265381 - Z_2 = 165707065/52746197 - Z_3 = 355/113 - - Q_norr(x) = x^2 + 1 - -And let's get their decimal expansion too:: - - % print decimal expansion of the found roots - \renewcommand\PolPrintIntervalsPrintExactZero - {\xintTrunc{20}{\PolPrintIntervalsTheLeftEndPoint}\dots} - \PolPrintIntervals{Q} - - Z_1 = 3.14159265358107777120... - Z_2 = 3.14159265358979340254... - Z_3 = 3.14159292035398230088... - - -A Mignotte type polynomial -~~~~~~~~~~~~~~~~~~~~~~~~~~ - -:: - - \PolDef{P}{x^10 - (10x-1)^2}% - \PolTypeset{P} % prints it in expanded form - \PolToSturm{P}{P} % we can use same prefix for Sturm chain - \PolSturmIsolateZeros{P} % finds 4 real roots - This polynomial has \PolSturmNbOfIsolatedZeros{P} distinct real roots: - \PolPrintIntervals{P}% - % reports -2 < Z_1 < -1, 0.09 < Z_2 < 0.10, 0.1 < Z_3 < 0.2, 1 < Z_4 < 2 - Let us refine the second and third intervals to separate the corresponding - roots: - \PolRefineInterval*{P}{2}% will refine to 0.0999990 < Z_2 < 0.0999991 - \PolRefineInterval*{P}{3}% will refine to 0.100001 < Z_3 < 0.100002 - \PolPrintIntervals{P}% - Let us now get to know all roots with 10 digits after decimal mark: - \PolEnsureIntervalLengths{P}{-10}% - \PolPrintIntervals{P}% now all roots are known 10 decimal digits after mark - Finally, we display 20 digits of the second root: - \PolEnsureIntervalLength{P}{2}{-20}% makes Z_2 known with 20 digits after mark - \[\PolSturmIsolatedZeroLeft{P}{2}<Z_2<\PolSturmIsolatedZeroRight{P}{2}\] - -The last line produces:: - - 0.09999900004999650028 < Z_2 < 0.09999900004999650029 - - -The Wilkinson polynomial -~~~~~~~~~~~~~~~~~~~~~~~~ - -See `Wilkinson polynomial`_. - -:: - - \documentclass{article} - \usepackage{polexpr} - \begin{document} - %\xintverbosetrue % for the curious... - - \poldef f(x) := mul((x - i), i = 1..20); - - \renewcommand\PolTypesetCmdPrefix[1]{\allowbreak\xintiiifSgn{#1}{}{+}{+}}% - \renewcommand\PolTypesetOne[1]{\xintDecToString{#1}}% - - \noindent\PolTypeset{f} - - \PolToSturm{f}{f} - \PolSturmIsolateZeros{f} - \PolPrintIntervals{f} - - \clearpage - - \poldef g(x) := f(x) - 2**{-23} x**19; - - % be patient! - \PolToSturm{g}{g} - \noindent\PolTypeset{g_0}% integer coefficient primitive polynomial - - \PolSturmIsolateZeros{g} - \PolEnsureIntervalLengths{g}{-10} - - \renewcommand\PolPrintIntervalsPrintMultiplicity{} - \PolPrintIntervals*{g} - - \end{document} - - -The first polynomial:: - - f(x) = x**20 - - 210 x**19 - + 20615 x**18 - - 1256850 x**17 - + 53327946 x**16 - - 1672280820 x**15 - + 40171771630 x**14 - - 756111184500 x**13 - + 11310276995381 x**12 - - 135585182899530 x**11 - + 1307535010540395 x**10 - - 10142299865511450 x**9 - + 63030812099294896 x**8 - - 311333643161390640 x**7 - + 1206647803780373360 x**6 - - 3599979517947607200 x**5 - + 8037811822645051776 x**4 - - 12870931245150988800 x**3 - + 13803759753640704000 x**2 - - 8752948036761600000 x - + 2432902008176640000 - -is handled fast enough (a few seconds), but the modified one ``f(x) - -2**-23 x**19`` takes about 20x longer (the Sturm chain polynomials -have integer coefficients with up to 321 digits, whereas (surprisingly -perhaps) those of the Sturm chain polynomials derived from ``f`` never -have more than 21 digits ...). - -Once the Sturm chain is computed and the zeros isolated, obtaining their -decimal digits is relatively faster. Here is for the ten real roots of -``f(x) - 2**-23 x**19`` as computed by the code above:: - - Z_1 = 0.9999999999... - Z_2 = 2.0000000000... - Z_3 = 2.9999999999... - Z_4 = 4.0000000002... - Z_5 = 4.9999999275... - Z_6 = 6.0000069439... - Z_7 = 6.9996972339... - Z_8 = 8.0072676034... - Z_9 = 8.9172502485... - Z_10 = 20.8469081014... - -The second Wilkinson polynomial -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - -:: - - \documentclass{article} - \usepackage{polexpr} - \begin{document} - \poldef f(x) := mul(x - 2^-i, i = 1..20); - - %\PolTypeset{f} - - \PolToSturm{f}{f} - \PolSturmIsolateZeros**{f} - \PolPrintIntervals{f} - \end{document} - -This takes more time than the polynomial with 1, 2, .., 20 as roots but -less than the latter modified by the ``2**-23`` change in one -coefficient. - -Here is the output (with release 0.7.2):: - - Z_1 = 0.00000095367431640625 - Z_2 = 0.0000019073486328125 - Z_3 = 0.000003814697265625 - Z_4 = 0.00000762939453125 - Z_5 = 0.0000152587890625 - Z_6 = 0.000030517578125 - Z_7 = 0.00006103515625 - Z_8 = 0.0001220703125 - Z_9 = 1/4096 - Z_10 = 1/2048 - Z_11 = 1/1024 - Z_12 = 1/512 - Z_13 = 1/256 - Z_14 = 1/128 - Z_15 = 0.015625 - Z_16 = 0.03125 - Z_17 = 0.0625 - Z_18 = 0.125 - Z_19 = 0.25 - Z_20 = 0.5 - -There is some incoherence in output format which has its source in the -fact that some roots are found in branches which can only find decimal -roots, whereas some are found in branches which could find general -fractions and they use ``\xintIrr`` before storage of the found root. -This may evolve in future. - - -The degree 41 polynomial with -2, -1.9, -1.8, ..., 0, 0.1, ..., 1.9, 2 as roots -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - -:: - - \PolDef{P}{mul((x-i*1e-1), i=-20..20)}% i/10 is same but less efficient - -In the defining expression we could have used ``i/10`` but this gives -less efficient internal form for the coefficients (the ``10``'s end up -in denominators). Using ``\PolToExpr{P}`` after having done - -:: - - \renewcommand\PolToExprCmd[1]{\PolDecToString{\xintREZ{#1}}} - -we get this expanded form:: - - x^41 - -28.7*x^39 - +375.7117*x^37 - -2975.11006*x^35 - +15935.28150578*x^33 - -61167.527674162*x^31 - +173944.259366417394*x^29 - -373686.963560544648*x^27 - +613012.0665016658846445*x^25 - -771182.31133138163125495*x^23 - +743263.86672885754888959569*x^21 - -545609.076599482896371978698*x^19 - +301748.325708943677229642930528*x^17 - -123655.8987669450434698869844544*x^15 - +36666.1782054884005855608205864192*x^13 - -7607.85821367459445649518380016128*x^11 - +1053.15135918687298508885950223794176*x^9 - -90.6380005918141132650786081964032*x^7 - +4.33701563847327366842552218288128*x^5 - -0.0944770968420804735498178265088*x^3 - +0.00059190121813899276854174416896*x - -which shows coefficients with up to 36 significant digits... - -Stress test: not a hard challenge to ``xint + polexpr``, but be a bit patient! - -:: - - \PolDef{P}{mul((x-i*1e-1), i=-20..20)}% - \PolToSturm{P}{S} % dutifully computes S_0, ..., S_{41} - % the [1] optional argument limits the search to interval (-10,10) - \PolSturmIsolateZeros[1]{S} % finds *exactly* (but a bit slowly) all 41 roots! - \PolPrintIntervals{S} % nice, isn't it? - -.. note:: - - Release ``0.5`` has *experimental* addition of optional argument - ``E`` to ``\PolSturmIsolateZeros``. It instructs to search roots only - in interval ``(-10^E, 10^E)``. Important: the extremities are - *assumed to not be roots*. In this example, the ``[1]`` in - ``\PolSturmIsolateZeros[1]{S}`` gives some speed gain; without it, it - turns out in this case that ``polexpr`` would have started with - ``(-10^6, 10^6)`` interval. - - Please note that this will probably get replaced in future by the - specification of a general interval. Do not rely on meaning of this - optional argument keeping the same. - -Roots of Chebyshev polynomials -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - -:: - - \newcount\mycount - \poldef T_0(x) := 1; - \poldef T_1(x) := x; - \mycount 2 - \xintloop - \poldef T_\the\mycount(x) := - 2x*T_\the\numexpr\mycount-1(x) - - T_\the\numexpr\mycount-2(x); - \ifnum\mycount<15 - \advance\mycount 1 - \repeat - - \[T_{15} = \PolTypeset[X]{T_15}\] - \PolToSturm{T_15}{T_15} - \PolSturmIsolateZeros{T_15} - \PolEnsureIntervalLengths{T_15}{-10} - \PolPrintIntervals{T_15} - - -Non-expandable macros ---------------------- - -.. _poldef;: - -``\poldef polname(letter):= expression in letter;`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - This evaluates the *polynomial expression* and stores the coefficients - in a private structure accessible later via other package macros, - under the user-chosen ``polname``. Of course the *expression* can - use other previously defined polynomials. Names must start with a - letter and are constituted of letters, digits, underscores and - (since ``0.5.1``) the right tick ``'``. - The whole xintexpr_ syntax is authorized:: - - \poldef sin(z) := add((-1)^i z^(2i+1)/(2i+1)!, i = 0..10); - - With fractional coefficients, beware the `tacit multiplication issue - <warningtacit_>`_. - - As a side effect the function ``polname()`` is recognized as a - genuine ``\xintexpr...\relax`` function for (exact) numerical - evaluation (or within an ``\xintdefvar`` assignment.) It computes - values not according to the original expression but via the Horner - scheme corresponding to the polynomial coefficients. - - .. attention:: - - Release ``0.3`` also did the necessary set-up to let the - polynomial be known to the ``\xintfloatexpr`` (or - ``\xintdeffloatvar``) parser. - - Since ``0.4`` this isn't done automatically. Even more, a - previously existing floating point variant of the same name will - be let undefined again, to avoid hard to debug mismatches between - exact and floating point polynomials. This also applies when the - polynomial is produced not via ``\poldef`` or ``\PolDef`` but as - a product of the other package macros. - - See `\\PolGenFloatVariant{polname}`_. - - The original expression is lost after parsing, and in particular - the package provides no way to typeset it. This has to be done - manually, if needed. - -.. _PolDef: - -``\PolDef[letter]{polname}{expression in letter}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - Does the same as `\\poldef <poldef;>`_ in an undelimited macro - format (thus avoiding potential problems with the catcode of the - semi-colon in presence of some packages.) In absence of the - ``[letter]`` optional argument, the variable is assumed to be ``x``. - -.. _PolGenFloatVariant: - -``\PolGenFloatVariant{polname}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - Makes the polynomial also usable in the ``\xintfloatexpr`` parser. - It will therein evaluates via an Horner scheme with coefficients - already pre-rounded to the float precision. - - See also `\\PolToFloatExpr{polname}`_. - - .. attention:: - - Release ``0.3`` did this automatically on ``\PolDef`` and - ``\poldef`` but this was removed at ``0.4`` for optimization. - - Any operation, for example generating the derivative polynomial, - or dividing two polynomials or using the ``\PolLet``, **must** be - followed by explicit usage of ``\PolGenFloatVariant{polname}`` if - the new polynomial is to be used in ``\xintfloatexpr`` or alike - context. - -.. _PolLet: - -``\PolLet{polname_2}={polname_1}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - Makes a copy of the already defined polynomial ``polname_1`` to a - new one ``polname_2``. Same effect as - ``\PolDef{polname_2}{polname_1(x)}`` but with less overhead. The - ``=`` is optional. - -.. _PolGlobalLet: - -``\PolGlobalLet{polname_2}={polname_1}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - Acts globally. - -.. _PolAssign: - -``\PolAssign{polname}\toarray\macro`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - Defines a one-argument expandable macro ``\macro{#1}`` which expands - to the (raw) #1th polynomial coefficient. - - - Attention, coefficients here are indexed starting at 1. - - - With #1=-1, -2, ..., ``\macro{#1}`` returns leading coefficients. - - - With #1=0, returns the number of coefficients, i.e. ``1 + deg f`` - for non-zero polynomials. - - - Out-of-range #1's return ``0/1[0]``. - - See also `\\PolNthCoeff{polname}{number}`_. The main difference is that - with ``\PolAssign``, ``\macro`` is made a prefix to ``1 + deg f`` - already defined (hidden to user) macros holding individually the - coefficients but `\\PolNthCoeff{polname}{number}`_ does each time the job - to expandably recover the ``Nth`` coefficient, and due to - expandability can not store it in a macro for future usage (of course, - it can be an argument in an ``\edef``.) The other difference - is the shift by one in indexing, mentioned above (negative - indices act the same in both.) - -.. _PolGet: - -``\PolGet{polname}\fromarray\macro`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - Does the converse operation to - ``\PolAssign{polname}\toarray\macro``. Each individual - ``\macro{number}`` gets expanded in an ``\edef`` and then normalized - via xintfrac_\ 's macro ``\xintRaw``. - - The leading zeros are removed from the polynomial. - - (contrived) Example:: - - \xintAssignArray{1}{-2}{5}{-3}\to\foo - \PolGet{f}\fromarray\foo - - This will define ``f`` as would have ``\poldef f(x):=1-2x+5x^2-3x^3;``. - - .. note:: - - Prior to ``0.5``, coefficients were not normalized via - ``\xintRaw`` for internal storage. - -.. _PolFromCSV: - -``\PolFromCSV{polname}{<csv>}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - Defines a polynomial directly from the comma separated list of values - (or a macro expanding to such a list) of its coefficients, the *first - item* gives the constant term, the *last item* gives the leading - coefficient, except if zero, then it is dropped (iteratively). List - items are each expanded in an ``\edef`` and then put into normalized - form via xintfrac_\ 's macro ``\xintRaw``. - - As leading zero coefficients are removed:: - - \PolFromCSV{f}{0, 0, 0, 0, 0, 0, 0, 0, 0, 0} - - defines the zero polynomial, which holds only one coefficient. - - See also expandable macro `\\PolToCSV <\\PolToCSV{polname}_>`_. - - .. note:: - - Prior to ``0.5``, coefficients were not normalized via - ``\xintRaw`` for internal storage. - -.. _PolTypeset: - -``\PolTypeset{polname}`` -~~~~~~~~~~~~~~~~~~~~~~~~ - - Typesets in descending powers in math mode. It uses letter ``x`` but - this can be changed via an optional argument:: - - \PolTypeset[z]{polname} - - By default zero coefficients are skipped (issue ``\poltypesetalltrue`` - to get all of them in output). - - These commands (whose meanings will be found in the package code) - can be re-defined for customization. Their default definitions are - expandable, but this is not a requirement. - -.. _PolTypesetCmd: - -``\PolTypesetCmd{raw_coeff}`` -^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ - - Checks if the coefficient is ``1`` or ``-1`` and then skips printing - the ``1``, except for the constant term. Also it sets conditional - `\\PolIfCoeffIsPlusOrMinusOne{A}{B}`_. - - The actual printing of the coefficients, when not equal to plus or - minus one is handled by `\\PolTypesetOne{raw_coeff}`_. - -.. _PolTypesetOne: - -``\PolTypesetOne{raw_coeff}`` -^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ - - The default is ``\xintSignedFrac`` but this macro is annoying as it - insists to use a power of ten, and not decimal notation. - - One can do things such as for example: [#]_ - - :: - - \renewcommand\PolTypesetOne[1]{\num{\xintPFloat[5]{#1}}} - \renewcommand\PolTypesetOne[1]{\num{\xintRound{4}{#1}}} - - where e.g. we used the ``\num`` macro of ``siunitx`` as it - understands floating point notation. - - .. [#] the difference in the syntaxes of ``\xintPFloat`` and - ``\xintRound`` is explained from the fact that - ``\xintPFloat`` by default uses the prevailing precision - hence the extra argument like here ``5`` is an optional one. - - One can also give a try to using `\\PolDecToString{decimal number}`_ - which uses decimal notation (at least for the numerator part). - -.. _PolTypesetMonomialCmd: - -``\PolTypesetMonomialCmd`` -^^^^^^^^^^^^^^^^^^^^^^^^^^ - - This decides how a monomial (in variable ``\PolVar`` and with - exponent ``\PolIndex``) is to be printed. The default does nothing - for the constant term, ``\PolVar`` for the first degree and - ``\PolVar^{\PolIndex}`` for higher degrees monomials. Beware that - ``\PolIndex`` expands to digit tokens and needs termination in - ``\ifnum`` tests. - -.. _PolTypesetCmdPrefix: - -``\PolTypesetCmdPrefix{raw_coeff}`` -^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ - - Expands to a ``+`` if the ``raw_coeff`` is zero or positive, and to - nothing if ``raw_coeff`` is negative, as in latter case the - ``\xintSignedFrac`` used by `\\PolTypesetCmd{raw_coeff}`_ will put - the ``-`` sign in front of the fraction (if it is a fraction) and - this will thus serve as separator in the typeset formula. Not used - for the first term. - -.. _PolTypeset*: - -``\PolTypeset*{polname}`` -~~~~~~~~~~~~~~~~~~~~~~~~~ - - Typesets in ascending powers. Use e.g. ``[h]`` optional argument - (after the ``*``) to use letter ``h`` rather than ``x``. - -.. _PolDiff: - -``\PolDiff{polname_1}{polname_2}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - This sets ``polname_2`` to the first derivative of ``polname_1``. It - is allowed to issue ``\PolDiff{f}{f}``, effectively replacing ``f`` - by ``f'``. - - Coefficients of the result ``polname_2`` are irreducible fractions - (see `Technicalities`_ for the whole story.) - -.. _PolDiff[N]: - -``\PolDiff[N]{polname_1}{polname_2}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - This sets ``polname_2`` to the ``N``-th derivative of ``polname_1``. - Identical arguments is allowed. With ``N=0``, same effect as - ``\PolLet{polname_2}={polname_1}``. With negative ``N``, switches to - using ``\PolAntiDiff``. - -.. _PolAntiDiff: - -``\PolAntiDiff{polname_1}{polname_2}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - This sets ``polname_2`` to the primitive of ``polname_1`` vanishing - at zero. - - Coefficients of the result ``polname_2`` are irreducible fractions - (see `Technicalities`_ for the whole story.) - -.. _PolAntiDiff[N]: - -``\PolAntiDiff[N]{polname_1}{polname_2}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - This sets ``polname_2`` to the result of ``N`` successive integrations on - ``polname_1``. With negative ``N``, it switches to using ``\PolDiff``. - -.. _PolDivide: - -``\PolDivide{polname_1}{polname_2}{polname_Q}{polname_R}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - This sets ``polname_Q`` and ``polname_R`` to be the quotient and - remainder in the Euclidean division of ``polname_1`` by - ``polname_2``. - -.. _PolQuo: - -``\PolQuo{polname_1}{polname_2}{polname_Q}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - This sets ``polname_Q`` to be the quotient in the Euclidean division - of ``polname_1`` by ``polname_2``. - -.. _PolRem: - -``\PolRem{polname_1}{polname_2}{polname_R}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - This sets ``polname_R`` to be the remainder in the Euclidean division - of ``polname_1`` by ``polname_2``. - -.. _PolGCD: - -``\PolGCD{polname_1}{polname_2}{polname_GCD}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - This sets ``polname_GCD`` to be the (monic) GCD of the two first - polynomials. It is a unitary polynomial except if both ``polname_1`` - and ``polname_2`` vanish, then ``polname_GCD`` is the zero - polynomial. - -.. ``\PolIGCD{polname_1}{polname_2}{polname_iGCD}`` - ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - **NOT YET** - - This **assumes** that the two polynomials have integer coefficients. - It then computes the greatest common divisor in the integer - polynomial ring, normalized to have a positive leading coefficient - (if the inputs are not both zero). - - ``\PolIContent{polname}`` - ~~~~~~~~~~~~~~~~~~~~~~~~~ - - **NOT YET** - - This computes a positive rational number such that dividing the - polynomial with it returns an integer coefficients polynomial with - no common factor among the coefficients. - -.. _PolToSturm: - -``\PolToSturm{polname}{sturmname}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - With ``polname`` being for example ``P``, the macro starts by - computing polynomials ``P`` and ``P'``, then computes the (opposite - of the) remainder in euclidean division, iteratively. - - The last non-zero remainder ``P_N_`` (where ``N`` is obtainable as - `\\PolSturmChainLength{sturmname}`_) is up to a factor - the GCD of ``P`` and ``P'`` hence it is a constant if and only if - ``P`` is square-free. - - .. note:: - - - Since ``0.5`` all these polynomials are divided by their rational - content, so they have integer coefficients with no common factor, - and the last one if a constant is either ``1`` or ``-1``. - - - After this normalization to primitive polynomials, they are - stored internally as ``sturmname_k_``, ``k=0,1, ...``. - - - These polynomials are used internally only. To keep them as - genuine declared polynomials also after the macro call, use the - starred variant `PolToSturm*`_. - - .. note:: - - It is perfectly allowed to use the polynomial name as Sturm chain name: - ``\PolToSturm{f}(f}``. - - The macro then declares ``sturmname_0``, ``sturmname_1``, ..., which are - the (non-declared) ``sturmname_k_`` divided by the last one. Division is - not done if this last one is the constant ``1`` or ``-1``, i.e. if the - original polynomial was square-free. These polynomials are primitive - polynomials too, i.e. with integer coefficients having no common factor. - - Thus ``sturmname_0`` has exactly the same real and complex roots as - polynomial ``polname``, but with each root now of multiplicity one: - i.e. it is the "square-free part" of original polynomial ``polname``. - - Notice that ``sturmname_1`` isn't necessarily the derivative of - ``sturmname_0`` due to the various normalizations. - - The polynomials ``sturmname_k`` main utility is for the execution of - `\\PolSturmIsolateZeros{sturmname}`_. Be careful not to use these - names ``sturmname_0``, ``sturmname_1``, etc... for defining other - polynomials after having done ``\PolToSturm{polname}{sturmname}`` and - before executing ``\PolSturmIsolateZeros{sturmname}`` else the - latter will behave erroneously. - - `\\PolSturmChainLength{sturmname}`_ gives the index of the last - element of the Sturm chain. - -.. _PolToSturm*: - -``\PolToSturm*{polname}{sturmname}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - Does the same as `un-starred version <PolToSturm_>`_ and additionally it - keeps for user usage the memory of the *un-normalized* Sturm chain - polynomials ``sturmname_k_``, ``k=0,1, ..., N``, with - ``N`` being `\\PolSturmChainLength{sturmname}`_. - - .. note:: - - This behaviour was modified at ``0.6``, anyhow the macro was - broken at ``0.5``. - - .. hint:: - - The square-free part of ``polname`` is ``sturmname_0``, and their - quotient is the polynomial with name - ``sturname_\PolSturmChainLength{sturmname}_``. It thus easy to - set-up a loop iteratively computing the latter until the last one - is a constant, thus obtaining the decomposition of an ``f`` as - a product ``c f_1 f_2 f_3 ...`` of a constant and square-free (primitive) - polynomials, where each ``f_i`` divides its predecessor. - -.. _PolSetToSturmChainSignChangesAt: - -``\PolSetToSturmChainSignChangesAt{\macro}{sturmname}{fraction}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - Sets macro ``\macro`` to the number of sign changes in the Sturm - chain with name prefix ``sturmname``, at location ``fraction`` - (which must be in format as acceptable by the xintfrac_ macros.) - - .. note:: - - The author was lazy and did not provide rather an expandable - variant, where one would do ``\edef\macro{\PolNbOf...}``. - - This will presumably get added in a future release. - - After some hesitation it was decided the macro would by default - act globally. To make the scope of its macro definition local, - use ``[\empty]`` as extra optional argument. - -.. _PolSetToNbOfZerosWithin: - -``\PolSetToNbOfZerosWithin{\macro}{sturmname}{value_a}{value_b}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - Applies the `Sturm Theorem`_ to set ``\macro`` to the exact number - of **distinct** roots of ``sturmname_0`` in the interval ``(value_a, - value_b]`` (the macro first re-orders the value for ``value_a <= - value_b`` to hold). - - .. note:: - - The author was lazy and did not provide rather an expandable - variant, where one would do ``\edef\macro{\PolNbOf...}``. - - This will presumably get added in future. - - After some hesitation it was decided the macro would by default - act globally. To make the scope of its macro definition local, - use ``[\empty]`` as extra optional argument. - - See also the expandable - `\\PolSturmNbOfRootsOf{sturmname}\\LessThanOrEqualTo{value}`_, from - which it is immediate (with ``\numexpr``) to create an expandable - variant of this macro. However the difference is that this macro - requires only `\\PolToSturm <PolToSturm_>`_ to have been executed, - whereas the expandable variant requires prior execution of - `\\PolSturmIsolateZeros <PolSturmIsolateZeros_>`_. - - See also the expandable - `\\PolSturmNbWithMultOfRootsOf{sturmname}\\LessThanOrEqualTo{value}`_ - which requires prior execution of - `\\PolSturmIsolateZeros* <PolSturmIsolateZeros*_>`_. - - -.. _PolSturmIsolateZeros: - -``\PolSturmIsolateZeros{sturmname}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - The macros locates, using `Sturm theorem`_, as many disjoint - intervals as there are (real) roots. - - .. important:: - - The Sturm chain must have been produced by an earlier - `\\PolToSturm{polname}{sturmname}`_. - - Why does this macro ask for argument the name of Sturm chain, - rather than the name of a polynomial? well this is mainly for - legacy reason, and because it is accompanied by other macros for - which it is simpler to assume the argument will be the name of an - already computed Sturm chain. - - Notice that ``\PolToSturm{f}{f}`` is perfectly legal (the - ``sturmname`` can be same as the ``polname``): it defines - polynomials ``f_0``, ``f_1``, ... having ``f`` has name prefix. - - Such a prior call - to ``\PolToSturm`` must have been made at any rate for - ``\PolSturmIsolateZeros`` to be usable. - - After its execution they are two types of such intervals (stored in - memory and accessible via macros or xintexpr_ variables, see below): - - - singleton ``{a}``: then ``a`` is a root, (necessarily a decimal - number, but not all such decimal numbers are exactly identified yet). - - - open intervals ``(a,b)``: then there is exactly one root ``z`` - such that ``a < z < b``, and the end points are guaranteed to not - be roots. - - The interval boundaries are decimal numbers, originating - in iterated decimal subdivision from initial intervals - ``(-10^E, 0)`` and ``(0, 10^E)`` with ``E`` chosen initially large - enough so that all roots are enclosed; if zero is a root it is always - identified as such. The non-singleton intervals are of the - type ``(a/10^f, (a+1)/10^f)`` with ``a`` an integer, which is - neither ``0`` nor ``-1``. Hence either ``a`` and ``a+1`` are both positive - or they are both negative. - - One does not *a priori* know what will be the lengths of these - intervals (except that they are always powers of ten), they - vary depending on how many digits two successive roots have in - common in their respective decimal expansions. - - .. important:: - - If some two consecutive intervals share an end-point, no - information is yet gained about the separation between the two - roots which could at this stage be arbitrarily small. - - See `\\PolRefineInterval*{sturmname}{index}`_ which addresses - this issue. - - .. This procedure is covariant - with the independent variable ``x`` becoming ``-x``. - Hmm, pas sûr et trop fatigué - - The interval boundaries (and exactly found roots) are made available - for future computations in ``\xintexpr``-essions or polynomial - definitions as variables ``<sturmname>L_1``, - ``<sturmname>L_2``, etc..., for the left end-points and - ``<sturmname>R_1``, ``<sturmname>R_2``, ..., for the right - end-points. - - Thus for example, if ``sturmname`` is ``f``, one can use the - xintexpr_ variables ``fL_1``, ``fL_2``, ... to refer in expressions - to the left end-points (or to the exact root, if left and right end - points coincide). Additionally, xintexpr_ variable ``fZ_1_isknown`` - will have value ``1`` if the root in the first interval is known, - and ``0`` otherwise. And similarly for the other intervals. - - Also, macros `\\PolSturmIsolatedZeroLeft{sturmname}{index}`_ and - `\\PolSturmIsolatedZeroRight{sturmname}{index}`_ are provided which - expand to these same values, written in decimal notation (i.e. - pre-processed by `\\PolDecToString <PolDecToString_>`_.) And there - is also `\\PolSturmIfZeroExactlyKnown{sturmname}{index}{A}{B}`_. - - .. important:: - - Trailing zeroes in the stored decimal numbers accessible via the - macros are significant: they are also present in the decimal - expansion of the exact root. - - These variables and macros are automatically updated when one next - uses macros such as `\\PolRefineInterval*{sturmname}{index}`_. - - The start of decimal expansion of a positive ``k``-th root is given - by `\\PolSturmIsolatedZeroLeft{sturmname}{k} - <PolSturmIsolatedZeroLeft_>`_, and for a negative root it is given - by `\PolSturmIsolatedZeroRight{sturmname}{k} - <PolSturmIsolatedZeroRight_>`_. These two decimal - numbers are either both zero or both of the same sign. - - The number of distinct roots is obtainable expandably as - `\\PolSturmNbOfIsolatedZeros{sturmname}`_. - - Furthermore - `\\PolSturmNbOfRootsOf{sturmname}\\LessThanOrEqualTo{value}`_ and - `\\PolSturmNbOfRootsOf{sturmname}\\LessThanOrEqualToExpr{expression}`_. - will expandably compute respectively the number of real roots at - most equal to ``value`` or ``expression``, and the same but with - multiplicities. - - .. note:: - - In the current implementation the xintexpr_ variables - and xinttools_ arrays are globally defined. On the - other hand the Sturm sequence polynomials obey the current scope. - - .. note:: - - As all computations are done *exactly* there can be no errors... - apart those due to bad coding by author. The results are exact - bounds for the mathematically exact real roots. - - Future releases will perhaps also provide macros based on Newton - or Regula Falsi methods. Exact computations with such methods - lead however quickly to very big fractions, and this forces usage - of some rounding scheme for the abscissas if computation times - are to remain reasonable. This raises issues of its own, which - are studied in numerical mathematics. - -.. _PolSturmIsolateZeros*: - -``\PolSturmIsolateZeros*{sturmname}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - The macro does the same as `\\PolSturmIsolateZeros{sturmname}`_ and - then in addition it does the extra work to determine all - multiplicities (of the real roots): - after executing this macro, - `\\PolSturmIsolatedZeroMultiplicity{sturmname}{index}`_ will expand - to the multiplicity of the root located in the ``index``\ -th - interval (intervals are enumerated from left to right, with index - starting at ``1``). - - Furthermore, if for example the ``sturmname`` is ``f``, xintexpr_ - variables ``fM_1``, ``fM_2``... hold the multiplicities thus - computed. - - .. note:: - - It is **not** necessary to have executed the `PolToSturm*`_ starred - variant, as the non-starred variant keeps internally the memory of the - original GCD (and even of the full non-normalized original Sturm - chain), even though it does not make the declarations as *user-level* - genuine polynomials. - - See `The degree nine polynomial with 0.99, 0.999, 0.9999 as triple - roots`_ for an example. - -.. _PolSturmIsolateZeros**: - -``\PolSturmIsolateZeros**{sturmname}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - The macro does the same as `\\PolSturmIsolateZeros*{sturmname}`_ and - in addition it does the extra work to determine all the *rational* - roots. - - .. note:: - - After execution of this macro, a root is "known" if and only if - it is rational. - - Furthermore, primitive polynomial ``sturmname_sqf_norr`` is created - to match the (square-free) ``sturmname_0`` from which all rational - roots have been removed (see `\\polexprsetup`_ for customizing this - name). The number of distinct rational roots is thus the difference - between the degrees of these two polynomials (see also - `\\PolSturmNbOfRationalRoots{sturmname}`_). - - And ``sturmname_norr`` is ``sturmname_0_`` from which all rational - roots have been removed (see `\\polexprsetup`_), i.e. it contains - the irrational roots of the original polynomial, with the same - multiplicities. - - See `A degree five polynomial with three rational - roots`_ for an example. - -.. _PolSturmIsolateZerosAndGetMultiplicities: - -``\PolSturmIsolateZerosAndGetMultiplicities{sturmname}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - This is another name for `\\PolSturmIsolateZeros*{sturmname}`_. - -.. _PolSturmIsolateZerosGetMultiplicitiesAndRationalRoots: - -``\PolSturmIsolateZerosGetMultiplicitiesAndRationalRoots{sturmname}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - This is another name for `\\PolSturmIsolateZeros**{sturmname}`_. - - -``\PolSturmIsolateZerosAndFindRationalRoots{sturmname}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - This works exactly like `\\PolSturmIsolateZeros**{sturmname}`_ - (inclusive of declaring the polynomials ``sturmname_sqf_norr`` and - ``sturmname_norr`` with no rational roots) except that it does *not* - compute the multiplicities of the *non-rational* roots. - - .. note:: - - There is no macro to find the rational roots but not compute - their multiplicities at the same time. - - .. attention:: - - This macro does *not* define xintexpr_ variables - ``sturmnameM_1``, ``sturmnameM_2``, ... holding the - multiplicities and it leaves the multiplicity array (whose accessor - is `\\PolSturmIsolatedZeroMultiplicity{sturmname}{index}`_) into - a broken state, as all non-rational roots will supposedly have - multiplicity one. This means that the output of - `\\PolPrintIntervals* <PolPrintIntervals*_>`_ for example will be - erroneous for the intervals with irrational roots. - - I decided to document it because finding multiplicities of the - non rational roots is somewhat costly, and one may be interested - only into finding the rational roots (of course random - polynomials with integer coefficients will not have *any* - rational root anyhow). - - -.. _PolRefineInterval*: - -``\PolRefineInterval*{sturmname}{index}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - The ``index``\ -th interval (starting indexing at one) is further - subdivided as many times as is necessary in order for the newer - interval to have both its end-points distinct from the end-points of - the original interval. This means that the ``k``\ th root is then - strictly separated from the other roots. - -.. _PolRefineInterval[N]: - -``\PolRefineInterval[N]{sturmname}{index}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - The ``index``\ -th interval (starting count at one) is further - subdivided once, reducing its length by a factor of 10. This is done - ``N`` times if the optional argument ``[N]`` is present. - -.. _PolEnsureIntervalLength: - -``\PolEnsureIntervalLength{sturmname}{index}{E}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - The ``index``\ -th interval is subdivided until its length becomes at - most ``10^E``. This means (for ``E<0``) that the first ``-E`` digits - after decimal mark of the ``k``\ th root will then be known exactly. - -.. _PolEnsureIntervalLengths: - -``\PolEnsureIntervalLengths{sturmname}{E}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - The intervals as obtained from ``\PolSturmIsolateZeros`` are (if - necessary) subdivided further by (base 10) dichotomy in order for - each of them to have length at most ``10^E`` (length will be shorter - than ``10^E`` in output only if it did not change or became zero.) - - This means that decimal expansions of all roots will be known with - ``-E`` digits (for ``E<0``) after decimal mark. - -.. _PolPrintIntervals: - -``\PolPrintIntervals[varname]{sturmname}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - This is a convenience macro which prints the bounds for the roots - ``Z_1``, ``Z_2``, ... (the optional argument ``varname`` allows to - specify a replacement for the default ``Z``). This will be done (by - default) in a - math mode ``array``, one interval per row, and pattern ``rcccl``, - where the second and fourth column hold the ``<`` sign, except when - the interval reduces to a singleton, which means the root is known - exactly. - - .. attention:: - - This macro was refactored at 0.7, its default output remained - identical but the ways to customize it got completely - modified. - - See next macros which govern its output. - -``\PolPrintIntervalsNoRealRoots`` -^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ - - Executed in place of an ``array`` environment, when there are no - real roots. Default definition:: - - \newcommand\PolPrintIntervalsNoRealRoots{} - -``\PolPrintIntervalsBeginEnv`` -^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ - - Default definition:: - - \newcommand\PolPrintIntervalsBeginEnv{\[\begin{array}{rcccl}} - -``\PolPrintIntervalsEndEnv`` -^^^^^^^^^^^^^^^^^^^^^^^^^^^^ - - Default definition:: - - \newcommand\PolPrintIntervalsEndEnv{\end{array}\]} - -``\PolPrintIntervalsKnownRoot`` -^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ - - Default definition:: - - \newcommand\PolPrintIntervalsKnownRoot{% - &&\PolPrintIntervalsTheVar_{\PolPrintIntervalsTheIndex}% - &=&\PolPrintIntervalsPrintExactZero - } - -``\PolPrintIntervalsUnknownRoot`` -^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ - - Default definition:: - - \newcommand\PolPrintIntervalsUnknownRoot{% - \PolPrintIntervalsPrintLeftEndPoint&<&% - \PolPrintIntervalsTheVar_{\PolPrintIntervalsTheIndex}&<&% - \PolPrintIntervalsPrintRightEndPoint - } - - -.. _PolPrintIntervalsPrintExactZero: - -``\PolPrintIntervalsPrintExactZero`` -^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ - - Default definition:: - - \newcommand\PolPrintIntervalsPrintExactZero{\PolPrintIntervalsTheLeftEndPoint} - - -.. _PolPrintIntervalsPrintLeftEndPoint: - -``\PolPrintIntervalsPrintLeftEndPoint`` -^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ - - Default definition:: - - \newcommand\PolPrintIntervalsPrintLeftEndPoint{\PolPrintIntervalsTheLeftEndPoint} - -.. _PolPrintIntervalsPrintRightEndPoint: - -``\PolPrintIntervalsPrintRightEndPoint`` -^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ - - Default definition is:: - - \newcommand\PolPrintIntervalsPrintRightEndPoint{\PolPrintIntervalsTheRightEndPoint} - -.. _PolPrintIntervals*: - -``\PolPrintIntervals*[varname]{sturmname}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - This starred variant produces an alternative output (which - displays the root multiplicity), and is provided as an - example of customization. - - As replacement for `\\PolPrintIntervalsKnownRoot`_, - `\\PolPrintIntervalsPrintExactZero`_, - `\\PolPrintIntervalsUnknownRoot`_ it uses its own - ``\POL@@PrintIntervals...`` macros. We only reproduce here one - definition:: - - \newcommand\POL@@PrintIntervalsPrintExactZero{% - \displaystyle - \xintSignedFrac{\PolPrintIntervalsTheLeftEndPoint}% - }% - - Multiplicities are printed using this auxiliary macro: - -``\PolPrintIntervalsPrintMultiplicity`` -^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ - - whose default definition is:: - - \newcommand\PolPrintIntervalsPrintMultiplicity{(\mbox{mult. }\PolPrintIntervalsTheMultiplicity)} - - -.. _PolMapCoeffs: - -``\PolMapCoeffs{\macro}{polname}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - It modifies ('in-place': original coefficients get lost) each - coefficient of the defined polynomial via the *expandable* macro - ``\macro``. The degree is adjusted as necessary if some leading - coefficients vanish after the operation. In replacement text of - ``\macro``, ``\index`` expands to the coefficient index (which is - defined to be zero for the constant term). - - Notice that ``\macro`` will have to handle inputs of the shape - ``A/B[N]`` (xintfrac_ internal notation). This means that it probably - will have to be expressed in terms of macros from xintfrac_ package. - - Example:: - - \def\foo#1{\xintMul{#1}{\the\numexpr\index*\index\relax}} - - (or with ``\xintSqr{\index}``) to replace ``n``-th coefficient - ``f_n`` by ``f_n*n^2``. - -.. _PolReduceCoeffs: - -``\PolReduceCoeffs{polname}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - About the same as ``\PolMapCoeffs{\xintIrr}{polname}`` (but - maintaining a ``[0]`` postfix for speedier xintfrac_ parsing when - polynomial function is used for computations.) This is a - one-argument macro, working 'in-place'. - -.. _PolReduceCoeffs*: - -``\PolReduceCoeffs*{polname}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - This starred variant leaves un-touched the decimal exponent in the - internal representation of the fractional coefficients, i.e. if a - coefficient is internally ``A/B[N]``, then ``A/B`` is reduced to - smallest terms, but the ``10^N`` part is kept as is. Note: if the - polynomial is freshly defined directly via `\\PolFromCSV - <PolFromCSV_>`_ its coefficients might still be internally in some - format like ``1.5e7``; the macro will anyhow always first do the - needed conversion to strict format ``A/B[N]``. - - Evaluations with polynomials treated by this can be much faster than - with those handled by the non-starred variant - `\\PolReduceCoeffs{polname}`_: as the numerators and denominators - remain smaller, this proves very beneficial in favorable cases - (especially when the coefficients are decimal numbers) to the - expansion speed of the xintfrac_ macros used internally by - `\\PolEval <PolEvalAt_>`_. - -.. _PolMakeMonic: - -``\PolMakeMonic{polname}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~ - - Divides by the leading coefficient. It is recommended to execute - `\\PolReduceCoeffs*{polname}`_ immediately afterwards. This is not - done automatically, due to the case the original polynomial had integer - coefficients and we want to keep the leading one as common - denominator. - -.. _PolMakePrimitive: - -``\PolMakePrimitive{polname}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - Divides by the integer content see (`\\PolIContent - <PolIContent_>`_). This thus produces a polynomial with integer - coefficients having no common factor. The sign of the leading - coefficient is not modified. - -Expandable macros ------------------ - -All these macros expand completely in two steps except ``\PolToExpr`` -and ``\PolToFloatExpr`` (and their auxiliaries) which need a -``\write``, ``\edef`` or a ``\csname...\endcsname`` context. - -.. _PolEvalAtExpr: - -``\PolEval{polname}\AtExpr{numerical expression}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - It boils down to - ``\xinttheexpr polname(numerical expression)\relax``. - -.. _PolEvalAt: - -``\PolEval{polname}\At{fraction}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - Evaluates the polynomial at value ``fraction`` which must be in (or - expand to) a format acceptable to the xintfrac_ macros. - -.. _PolEvalReducedAtExpr: - -``\PolEvalReduced{polname}\AtExpr{numerical expression}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - Boils down to ``\xinttheexpr reduce(polname(numerical expression))\relax``. - -.. _PolEvalReducedAt: - -``\PolEvalReduced{polname}\At{fraction}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - Evaluates the polynomial at value ``fraction`` which must be in (or - expand to) a format acceptable to the xintfrac_ macros, and produce - an irreducible fraction. - -.. _PolFloatEvalAtExpr: - -``\PolFloatEval{polname}\AtExpr{numerical expression}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - Boils down to ``\xintthefloatexpr polname(numerical expression)\relax``. - - This is done via a Horner Scheme (see `\\poldef <poldef;_>`_ and - `\\PolGenFloatVariant{polname}`_), with already rounded - coefficients. [#]_ To use the *exact coefficients* with *exactly - executed* additions and multiplications, just insert it in the float - expression as in this example: [#]_ - - :: - - \xintthefloatexpr 3.27*\xintexpr f(2.53)\relax^2\relax - - The ``f(2.53)`` is exactly computed then rounded at the time of - getting raised to the power ``2``. Moving the ``^2`` inside, that - operation would also be treated exactly. - - - .. [#] Anyway each floating point operation starts by rounding its - operands to the floating point precision. - - .. [#] The ``\xintexpr`` here could be ``\xinttheexpr`` but that - would be less efficient. Cf. xintexpr_ documentation about - nested expressions. - -.. _PolFloatEvalAt: - -``\PolFloatEval{polname}\At{fraction}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - Evaluates the polynomial at value ``fraction`` which must be in (or - expand to) a format acceptable to the xintfrac_ macros, and produces - a floating point number. - -.. _PolIfCoeffIsPlusOrMinusOne: - -``\PolIfCoeffIsPlusOrMinusOne{A}{B}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - This macro is a priori undefined. - - It is defined via the default `\\PolTypesetCmd{raw_coeff}`_ to be - used if needed in the execution of `\\PolTypesetMonomialCmd`_, - e.g. to insert a ``\cdot`` in front of ``\PolVar^{\PolIndex}`` if - the coefficient is not plus or minus one. - - The macro will execute ``A`` if the coefficient has been found to be - plus or minus one, and ``B`` if not. - -.. _PolLeadingCoeff: - -``\PolLeadingCoeff{polname}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - Expands to the leading coefficient. - -.. _PolNthCoeff: - -``\PolNthCoeff{polname}{number}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - It expands to the raw ``N``-th coefficient (``0/1[0]`` if the index - number is out of range). With ``N=-1``, ``-2``, ... expands to the - leading coefficients. - -.. _PolDegree: - -``\PolDegree{polname}`` -~~~~~~~~~~~~~~~~~~~~~~~ - - It expands to the degree. This is ``-1`` if zero polynomial but this - may change in future. Should it then expand to ``-\infty`` ? - -.. _PolIContent: - -``\PolIContent{polname}`` -~~~~~~~~~~~~~~~~~~~~~~~~~ - - It expands to the contents of the polynomial, i.e. to the positive - fraction such that dividing by this fraction produces a polynomial - with integer coefficients having no common prime divisor. - - See `\\PolMakePrimitive <PolMakePrimitive_>`_. - -.. _PolToExpr: - -``\PolToExpr{polname}`` -~~~~~~~~~~~~~~~~~~~~~~~ - - Expands [#]_ to ``coeff_N*x^N+...`` (descending powers.) - - .. [#] in a ``\write``, ``\edef``, or ``\csname...\endcsname``, but - not under ``\romannumeral-`0``. - - By default zero coefficients are skipped (issue ``\poltoexpralltrue`` to - get all of them in output). - - By default, no ``+`` sign before negative coefficients, for - compliance with Maple input format (but see - `\\PolToExprTermPrefix{raw_coeff}`_.) Also, like the default - behaviour of `\\PolTypeset{polname}`_, does not print (for the non - constant terms) coefficients equal to plus or minus one. The degree - one monomial is output as ``x``, not ``x^1``. Complete customization is - possible, see next macros. - - Of course ``\PolToExpr{f}`` can be inserted in a ``\poldef``, as the - latter expands token by token, hence will force complete expansion - of ``\PolToExpr{f}``, but a simple ``f(x)`` is more efficient for - the identical result. - -.. _PolToExprOneTerm: - -``\PolToExprOneTerm{raw_coeff}{number}`` -^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ - - This two argument expandable command takes care of the monomial and - its coefficient. The default definition is done in order for - coefficients of absolute value ``1`` not be printed explicitely - (except of course for the constant term). Also by default, the - monomial of degree one is ``x`` not ``x^1``, and ``x^0`` is skipped. - - For compatibility with Maple input requirements, by default a ``*`` - always precedes the ``x^number``, except if the coefficient is a one - or a minus one. See `\\PolToExprTimes`_. - -.. _PolToExprOneTermStyleA: - -``\PolToExprOneTermStyleA{raw_coeff}{number}`` -^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ - - Holds the default package meaning of - `\\PolToExprOneTerm{raw_coeff}{number}`_. - -.. _PolToExprOneTermStyleB: - -``\PolToExprOneTermStyleB{raw_coeff}{number}`` -^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ - - For output in this style:: - - 2*x^11/3+3*x^8/7-x^5-x^4/4-x^3-x^2/2-2*x+1 - - issue ``\let\PolToExprOneTerm\PolToExprOneTermStyleB`` before usage of - ``\PolToExpr``. Note that then ``\PolToExprCmd`` isn't used at all. - To revert to package default, issue - ``\let\PolToExprOneTerm\PolToExprOneTermStyleA``. - - To suppress the ``*``'s, cf. `\\PolToExprTimes`_. - -.. _PolToExprCmd: - -``\PolToExprCmd{raw_coeff}`` -^^^^^^^^^^^^^^^^^^^^^^^^^^^^ - - It is the one-argument macro used by the package definition of - ``\PolToExprOneTerm`` for the coefficients themselves (when not - equal to plus or minus one), and it defaults to - ``\xintPRaw{\xintRawWithZeros{#1}}``. One will have to redefine it - to ``\xintIrr{#1}`` or to ``\xintPRaw{\xintIrr{#1}}`` to obtain in the - output forcefully reduced coefficients. - -.. _PolToExprTermPrefix: - -``\PolToExprTermPrefix{raw_coeff}`` -^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ - - Defined identically as `\\PolTypesetCmdPrefix{raw_coeff}`_. It - prefixes with a plus sign for non-negative coefficients, because - they don't carry one by themselves. - -.. _PolToExprVar: - -``\PolToExprVar`` -^^^^^^^^^^^^^^^^^ - - This expands to the variable to use in output (it does not have to - be a single letter, may be an expandable macro.) Initial definition - is ``x``. - -.. _PolToExprTimes: - -``\PolToExprTimes`` -^^^^^^^^^^^^^^^^^^^ - - This expands to the symbol used for multiplication of an - ``x^{number}`` by the corresponding coefficient. The default is - ``*``. Redefine the macro to expand to nothing to get rid of it (but - this will give output incompatible with some professional computer - algebra software). - -.. _PolToExpr*: - -``\PolToExpr*{polname}`` -~~~~~~~~~~~~~~~~~~~~~~~~ - - Expands to ``coeff_0+coeff_1*x+coeff_2*x^2+...`` (ascending powers). - Customizable like `\\PolToExpr{polname}`_ via the same macros. - -.. _PolToFloatExpr: - -``\PolToFloatExpr{polname}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - Similar to `\\PolToExpr{polname}`_ but uses `\\PolToFloatExprCmd - <\\PolToFloatExprCmd{raw_coeff}>`_ - which by default rounds and converts the coefficients to floating - point format. - - .. note:: - - It is not necessary to have issued - `\\PolGenFloatVariant{polname}`_. The rounded coefficients are - not easily recoverable from the ``\xintfloatexpr`` polynomial - function hence ``\PolToFloatExprCmd`` operates from the *exact* - coefficients anew. - - Attention that both macros obey the prevailing float precision. - If it is changed between those macro calls, then a mismatch - exists between the coefficients as used in ``\xintfloatexpr`` and - those output by ``\PolToFloatExpr{polname}``. - -.. _PolToFloatExprOneTerm: - -``\PolToFloatExprOneTerm{raw_coeff}{number}`` -^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ - - Similar to `\\PolToExprOneTerm - <\\PolToExprOneTerm{raw_coeff}{number}>`_. But does not treat - especially coefficients equal to plus or minus one. - -.. _PolToFloatExprCmd: - -``\PolToFloatExprCmd{raw_coeff}`` -^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ - - It is the one-argument macro used by ``\PolToFloatExprOneTerm``. - Its package definition is ``\xintFloat{#1}``. - - .. caution:: - - Currently (xint_ ``1.3c``) ``\xintFloat{0}`` outputs ``0.e0`` - which is perfectly acceptable input for Python, but not for - Maple. Thus, one should better leave the `\\poltoexprallfalse`_ - toggle to its default ``\iffalse`` state, if one intends to use - the output in a Maple worksheet. - - But even then the zero polynomial will cause a problem. Workaround:: - - \renewcommand\PolToFloatExprCmd[1]{\xintiiifZero{#1}{0.0}{\xintFloat{#1}}} - - Usage of ``\xintiiifZero`` and not ``\xintifZero`` is only for - optimization (I can't help it) because ``#1`` is known to be - in ``xintfrac`` raw format. - -.. _PolToFloatExpr*: - -``\PolToFloatExpr*{polname}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - Typesets in ascending powers. - -.. _PolToList: - -``\PolToList{polname}`` -~~~~~~~~~~~~~~~~~~~~~~~ - - Expands to ``{coeff_0}{coeff_1}...{coeff_N}`` with ``N`` = degree, and - ``coeff_N`` the leading coefficient - (the zero polynomial does give ``{0/1[0]}`` and not an - empty output.) - -.. _PolToCSV: - -``\PolToCSV{polname}`` -~~~~~~~~~~~~~~~~~~~~~~ - - Expands to ``coeff_0, coeff_1, coeff_2, ....., coeff_N``, starting - with constant term and ending with leading coefficient. Converse - to `\\PolFromCSV <\\PolFromCSV{polname}{\<csv\>}_>`_. - -.. _PolSturmChainLength: - -``\PolSturmChainLength{sturmname}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - Returns the integer ``N`` such that ``sturmname_N`` is the last one - in the Sturm chain ``sturmname_0``, ``sturmname_1``, ... - - See `\\PolToSturm{polname}{sturmname}`_. - -.. _PolSturmIfZeroExactlyKnown: - -``\PolSturmIfZeroExactlyKnown{sturmname}{index}{A}{B}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - Executes ``A`` if the ``index``\ -th interval reduces to a singleton, - i.e. the root is known exactly, else ``B``. - - .. note:: - - ``index`` is allowed to be something like ``1+2*3`` as it is fed - to ``\the\numexpr...\relax``. - -.. _PolSturmIsolatedZeroLeft: - -``\PolSturmIsolatedZeroLeft{sturmname}{index}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - Expands to the left end-point for the ``index``\ -th interval, as - computed by some earlier `\\PolSturmIsolateZeros{sturmname}`_. - - .. note:: - - Of course, this is kept updated by macros such as - `\\PolRefineInterval{sturmname}{index} <PolRefineInterval[N]_>`_. - - The value is pre-formatted using `\\PolDecTostring - <PolDecToString_>`_. - -.. _PolSturmIsolatedZeroRight: - -``\PolSturmIsolatedZeroRight{sturmname}{index}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - Expands to the right end-point for the ``index``\ -th interval as - computed by some earlier `\\PolSturmIsolateZeros{sturmname}`_ and - possibly refined afterwards. - - The value is pre-formatted using `\\PolDecTostring - <PolDecToString_>`_. - -.. _PolSturmIsolatedZeroMultiplicity: - -``\PolSturmIsolatedZeroMultiplicity{sturmname}{index}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - Expands to the multiplicity of the unique root contained in the - ``index``\ -th interval. - - .. attention:: - - A prior execution of `\\PolSturmIsolateZeros*{sturmname}`_ is mandatory. - - See `The degree nine polynomial with 0.99, 0.999, 0.9999 as triple - roots`_ for an example of use. - -.. _PolSturmNbOfIsolatedZeros: - -``\PolSturmNbOfIsolatedZeros{sturmname}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - Expands to the number of real roots of the polynomial - ``<sturmname>_0``, i.e. the number of distinct real roots of the - polynomial originally used to create the Sturm chain via - `\\PolToSturm{polname}{sturmname}`_. - -.. warning:: - - The next few macros counting roots, with or without multiplicities, - less than or equal to some value, are under evaluation and may be - removed from the package if their utility is judged to be not high - enough. They can be re-coded at user level on the basis of the other - documented package macros anyway. - -``\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualTo{value}`` -^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ - - Expands to the number of distinct roots (of the polynomial used to - create the Sturm chain) less than or equal to the ``value`` (i.e. a - number of fraction recognizable by the xintfrac_ macros). - - .. attention:: - - `\\PolSturmIsolateZeros{sturmname}`_ must have been executed - beforehand. - - And the argument is a ``sturmname``, not a ``polname`` (this is - why the macro contains Sturm in its name), simply to be reminded - of the above constraint. - -``\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualToExpr{expression}`` -^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ - - Expands to the number of distinct roots (of the polynomial - used to create the Sturm chain) which are less than or equal to the - given ``expression``. - - .. attention:: - - `\\PolSturmIsolateZeros{sturmname}`_ must have been executed - beforehand. - -``\PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualTo{value}`` -^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ - - Expands to the number counted with multiplicities of the roots (of - the polynomial used to create the Sturm chain) which are less than - or equal to the given ``value``. - - .. attention:: - - `\\PolSturmIsolateZeros*{sturmname}`_ (or the double starred - variant) must have been executed beforehand. - -``\PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualToExpr{expression}`` -^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ - - Expands to the total number of roots (counted with multiplicities) - which are less than or equal to the given ``expression``. - - .. attention:: - - `\\PolSturmIsolateZeros*{sturmname}`_ (or the double starred - variant) must have been executed beforehand. - -``\PolSturmNbOfRationalRoots{sturmname}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - Expands to the number of rational roots (without multiplicities). - - .. attention:: - - `\\PolSturmIsolateZeros**{sturmname}`_ must have been executed - beforehand. - -``\PolSturmNbOfRationalRootsWithMultiplicities{sturmname}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - Expands to the number of rational roots (counted with multiplicities). - - .. attention:: - - `\\PolSturmIsolateZeros**{sturmname}`_ must have been executed - beforehand. - -``\PolSturmRationalRoot{sturmname}{k}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - Expands to the ``k``\ th rational root (they are ordered and indexed - starting at 1 for the most negative). - - .. attention:: - - `\\PolSturmIsolateZeros**{sturmname}`_ must have been executed - beforehand. - -``\PolSturmRationalRootIndex{sturmname}{k}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - Expands to ``index`` of the ``k``\ th rational root as part of the - ordered real roots (without multiplicities). I.e., above macro - `\\PolSturmRationalRoot{sturmname}{k}`_ is equivalent to this - nested call:: - - \PolSturmIsolatedZeroLeft{sturmname}{\PolSturmRationalRootIndex{sturmname}{k}} - - .. attention:: - - `\\PolSturmIsolateZeros**{sturmname}`_ must have been executed - beforehand. - -``\PolSturmRationalRootMultiplicity{sturmname}{k}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - Expands to the multiplicity of the ``k``\ th rational root. - - .. attention:: - - `\\PolSturmIsolateZeros**{sturmname}`_ must have been executed - beforehand. - -.. _PolIntervalWidth: - -``\PolIntervalWidth{sturmname}{index}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - The ``10^E`` width of the current ``index``\ -th root localization - interval. Output is in xintfrac_ raw ``1/1[E]`` format (if not zero). - -Expandable macros for use within execution of ``\PolPrintIntervals`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - -These macros are for usage within custom user redefinitions of -`\\PolPrintIntervalsKnownRoot`_, `\\PolPrintIntervalsUnknownRoot`_, or -in redefinitions of `\PolPrintIntervalsPrintExactZero`_ (used in the -default for the former) and of `\\PolPrintIntervalsPrintLeftEndPoint`_, -`\\PolPrintIntervalsPrintRightEndPoint`_ (used in the default for the -latter). - -.. attention:: - - Some macros formerly mentioned here got removed at 0.7: - ``\PolPrintIntervalsTheEndPoint``, - ``\PolIfEndPointIsPositive{A}{B}``, - ``\PolIfEndPointIsNegative{A}{B}``, - ``\PolIfEndPointIsZero{A}{B}``. - -``\PolPrintIntervalsTheVar`` -^^^^^^^^^^^^^^^^^^^^^^^^^^^^ - - Expands to the name (default ``Z``) used for representing the roots, - which was passed as optional argument ``varname`` to - `\\PolPrintIntervals[varname]{sturmname}`_. - -``\PolPrintIntervalsTheIndex`` -^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ - - Expands to the index of the considered interval (indexing starting - at 1 for the leftmost interval). - -``\PolPrintIntervalsTheSturmName`` -^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ - - Expands to the argument which was passed as ``sturmname`` to - `\\PolPrintIntervals[varname]{sturmname}`_. - -``\PolPrintIntervalsTheLeftEndPoint`` -^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ - - The left end point of the interval, as would be produced by - `\\PolSturmIsolatedZeroLeft <PolSturmIsolatedZeroLeft_>`_ if it was - used with arguments the Sturm chain name and interval index returned - by `\\PolPrintIntervalsTheSturmName`_ and - `\\PolPrintIntervalsTheIndex`_. - -``\PolPrintIntervalsTheRightEndPoint`` -^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ - - The right end point of the interval, as would be produced by - `\\\PolSturmIsolatedZeroRight <PolSturmIsolatedZeroRight_>`_ for - this Sturm chain name and index. - -``\PolPrintIntervalsTheMultiplicity`` -^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ - - The multiplicity of the unique root within the interval of index - `\\PolPrintIntervalsTheIndex`_. Makes sense only if the starred (or - double-starred) variant of `\\PolSturmIsolateZeros - <PolSturmIsolateZeros_>`_ was used earlier. - -.. _PolDecToString: - -``\PolDecToString{decimal number}`` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - This is a utility macro to print decimal numbers. It has been - backported to xintfrac_ (release ``1.3`` of ``2018/03/01``) under - the name ``\xintDecToString``, and the ``polexpr`` macro is simply - now an alias to it. - - For example - ``\PolDecToString{123.456e-8}`` will expand to ``0.00000123456`` - and ``\PolDecToString{123.450e-8}`` to ``0.00000123450`` which - illustrates that trailing zeros are not trimmed. To trim trailing - zeroes, one can use ``\PolDecToString{\xintREZ{#1}}``. - - The precise behaviour of this macro may evolve in future releases of - xint_. - -Booleans (with default setting as indicated) --------------------------------------------- - -``\xintverbosefalse`` -~~~~~~~~~~~~~~~~~~~~~ - - This is actually an xintexpr_ configuration. Setting it to - ``true`` triggers the writing of information to the log when new - polynomials are defined. - - .. caution:: - - The macro meanings as written to the log are to be considered - unstable and undocumented internal structures. - -``\poltypesetallfalse`` -~~~~~~~~~~~~~~~~~~~~~~~ - - If ``true``, `\\PolTypeset{polname}`_ will also typeset the vanishing - coefficients. - - -``\poltoexprallfalse`` -~~~~~~~~~~~~~~~~~~~~~~ - - If ``true``, `\\PolToExpr{polname}`_ and `\\PolToFloatExpr{polname}`_ will - also include the vanishing coefficients in their outputs. - -``\polexprsetup`` ------------------ - - Serves to customize the package. Currently only two keys are - recognized: - - - ``norr``: the postfix that `\\PolSturmIsolateZeros**{sturmname}`_ - should append to ``sturmname`` to declare the primitive polynomial - obtained from original one after removal of all rational roots. - The default value is ``_norr`` (standing for “no rational roots”). - - - ``sqfnorr``: the postfix that `\\PolSturmIsolateZeros**{sturmname}`_ - should append to ``sturmname`` to declare the primitive polynomial - obtained from original one after removal of all rational roots and - suppression of all multiplicities. - The default value is ``_sqf_norr`` (standing for “square-free with - no rational roots”). - - The package executes ``\polexprsetup{norr=_norr, - sqfnorr=_sqf_norr}`` as default. - -Technicalities --------------- - -- The catcode of the semi-colon is reset temporarily by `\\poldef - <poldef;_>`_ macro in case some other package (for example the French - babel module) may have made it active. This will fail though if the - whole thing was already part of a macro argument, in such cases one - can use `\\PolDef{f}{P(x)} <PolDef_>`_ - rather. The colon in ``:=`` may be active with no consequences. - -- As a consequence of xintfrac_ addition and subtraction always using - least common multiples for the denominators [#]_, user-chosen common - denominators survive additions and multiplications. For example, this:: - - \poldef P(x):= 1/2 + 2/2*x + 3/2*x^3 + 4/2*x^4; - \poldef Q(x):= 1/3 + (2/3)x + (3/3)x^3 + (4/3)x^4; - \poldef PQ(x):= P(x)Q(x); - - gives internally the polynomial:: - - 1/6+4/6*x^1+4/6*x^2+6/6*x^3+20/6*x^4+16/6*x^5+9/6*x^6+24/6*x^7+16/6*x^8 - - where all coefficients have the same denominator 6. Notice though that - ``\PolToExpr{PQ}`` outputs the ``6/6*x^3`` as ``x^3`` because (by - default) it recognizes and filters out coefficients equal to one or - minus one (since release ``0.3``). One can use for example - ``\PolToCSV{PQ}`` to see the internally stored coefficients. - - .. [#] prior to ``0.4.1``, ``polexpr`` used to temporarily patch - during the parsing of polynomials the xintfrac_ macros. This - patch was backported to xint_ at release ``1.3``. - -- `\\PolDiff{polname_1}{polname_2}`_ always applies ``\xintIrr`` to the - resulting coefficients, except that the *power of ten* part ``[N]`` - (for example an input in scientific notation such as ``1.23e5`` gives - ``123/1[3]`` internally in xintfrac) is not taken into account in the - reduction of the fraction. This is tentative and may change. - - Same remark for `\\PolAntiDiff{polname_1}{polname_2}`_. - -- Currently, the package stores all coefficients from index ``0`` to - index equal to the polynomial degree inside a single macro, as a list. - This data structure is obviously very inefficient for polynomials of - high degree and few coefficients (as an example with ``\poldef - f(x):=x^1000 + x^500;`` the subsequent definition ``\poldef g(x):= - f(x)^2;`` will do of the order of 1,000,000 multiplications and - additions involvings only zeroes... which does take time). This - may change in the future. - -- As is to be expected internal structures of the package are barely - documented and unstable. Don't use them. - - -CHANGE LOG ----------- - -- v0.1 (2018/01/11): initial release. Features: - - * The `\\poldef <poldef;_>`_ parser itself, - * Differentiation and anti-differentiation, - * Euclidean division and GCDs, - * Various utilities such as `\\PolFromCSV <PolFromCSV_>`_, - `\\PolMapCoeffs <PolMapCoeffs_>`_, - `\\PolToCSV <PolToCSV_>`_, `\\PolToExpr <PolToExpr_>`_, ... - - Only one-variable polynomials so far. - -- v0.2 (2018/01/14) - - * Fix: ``"README thinks \numexpr recognizes ^ operator"``. - * Convert README to reStructuredText markup. - * Move main documentation from README to separate ``polexpr.txt`` file. - * Provide ``polexpr.html`` as obtained via DocUtils_ ``rst2html.py``. - * Convert README to (CTAN compatible) Markdown markup. - - Due to lack of available time the test suite might not be extensive - enough. Bug reports are very welcome! - -- v0.3 (2018/01/17) - - * bug fixes: - - - the ``0.1`` `\\PolEval <PolEvalAt_>`_ accepted expressions for its second - argument, but this was removed by mistake at ``0.2``. Restored. - - **Attention**: at ``0.4`` this has been reverted again, and - `\\PolEval{P}\\AtExpr{foo} <PolEvalAtExpr_>`_ syntax is needed for - using expressions in the second argument. - * incompatible or breaking changes: - - - `\\PolToExpr <PolToExpr_>`_ now by default uses *descending* - powers (it also treats differently coefficients equal to 1 or -1.) - Use `\\PolToExpr* <PolToExpr*_>`_ for *ascending* powers. - - `\\PolEval <PolEvalAt_>`_ reduced the output to smallest terms, - but as this is costly with big fractions and not needed if e.g. - wrapped in an ``\xintRound`` or ``\xintFloat``, this step has been - removed; the former meaning is available as `\\PolEvalReduced - <PolEvalReducedAt_>`_. - * new (or newly documented) macros: - - - `\\PolTypesetCmd <PolTypesetCmd_>`_ - - `\\PolTypesetCmdPrefix <PolTypesetCmdPrefix_>`_ - - `\\PolTypesetMonomialCmd <PolTypesetMonomialCmd_>`_ - - `\\PolEvalReducedAt <PolEvalReducedAt_>`_ - - `\\PolToFloatExpr <PolToFloatExpr_>`_ - - `\\PolToExprOneTerm <PolToExprOneTerm_>`_ - - `\\PolToFloatExprOneTerm <PolToFloatExprOneTerm_>`_ - - `\\PolToExprCmd <PolToExprCmd_>`_ - - `\\PolToFloatExprCmd <PolToFloatExprCmd_>`_ - - `\\PolToExprTermPrefix <PolToExprTermPrefix_>`_ - - `\\PolToExprVar <PolToExprVar_>`_ - - `\\PolToExprTimes <PolToExprTimes_>`_ - * improvements: - - - documentation has a table of contents, internal hyperlinks, - standardized signature notations and added explanations. - - one can do ``\PolLet{g}={f}`` or ``\PolLet{g}{f}``. - - ``\PolToExpr{f}`` is highly customizable. - - `\\poldef <poldef;_>`_ and other defining macros prepare the polynomial - functions for usage within ``\xintthefloatexpr`` (or - ``\xintdeffloatvar``). Coefficients are pre-rounded to the - floating point precision. Indispensible for numerical algorithms, - as exact fractions, even reduced, quickly become very big. See the - documentation about how to use the exact polynomials also in - floating point context. - - **Attention**: this has been reverted at ``0.4``. The macro - `\\PolGenFloatVariant <PolGenFloatVariant_>`_ must be used for - generation floating point polynomial functions. - -- v0.3.1 (2018/01/18) - - Fixes two typos in example code included in the documentation. - -- v0.4 (2018/02/16) - - * bug fixes: - - - when Euclidean division gave a zero remainder, the internal - representation of this zero polynomial could be faulty; this - could cause mysterious bugs in conjunction with other package - macros such as `\\PolMapCoeffs <PolMapCoeffs_>`_. - - `\\PolGCD <PolGCD_>`_ was buggy in case of first polynomial being - of lesser degree than the second one. - * breaking changes: - - - formerly `\\PolEval{P}\\At{foo} <PolEvalAt_>`_ allowed ``foo`` to - be an expression, which was transparently handled via - ``\xinttheexpr``. Now, ``foo`` must be a fraction (or a macro - expanding to such) in the format acceptable by ``xintfrac.sty`` - macros. Use `\\PolEval{P}\\AtExpr{foo} <PolEvalAtExpr_>`_ for more - general arguments using expression syntax. E.g., if ``foo`` is the - name of a variable known to ``\xintexpr``. - - The same holds for `\\PolEvalReduced <PolEvalReducedAt_>`_ - and `\\PolFloatEval <PolFloatEvalAt_>`_. - - the ``3.0`` automatic generation of floating point variants has - been reverted. Not only do *not* the package macros automatically - generate floating point variants of newly created polynomials, - they actually make pre-existing such variant undefined. - - See `\\PolGenFloatVariant <PolGenFloatVariant_>`_. - * new non-expandable macros: - - - `\\PolGenFloatVariant <PolGenFloatVariant_>`_ - - `\\PolGlobalLet <PolGlobalLet_>`_ - - `\\PolTypesetOne <PolTypesetOne_>`_ - - `\\PolQuo <PolQuo_>`_ - - `\\PolRem <PolRem_>`_ - - `\\PolToSturm <PolToSturm_>`_ - - `\\PolToSturm\* <PolToSturm*_>`_ - - `\\PolSetToSturmChainSignChangesAt <PolSetToSturmChainSignChangesAt_>`_ - - `\\PolSetToNbOfZerosWithin <PolSetToNbOfZerosWithin_>`_ - - `\\PolSturmIsolateZeros <PolSturmIsolateZeros_>`_ - - `\\PolRefineInterval* <PolRefineInterval*_>`_ - - `\\PolRefineInterval[N] <PolRefineInterval[N]_>`_ - - `\\PolEnsureIntervalLength <PolEnsureIntervalLength_>`_ - - `\\PolEnsureIntervalLengths <PolEnsureIntervalLengths_>`_ - - `\\PolPrintIntervals <PolPrintIntervals_>`_ - - `\\PolPrintIntervalsPrintExactZero <PolPrintIntervalsPrintExactZero_>`_ - - `\\PolPrintIntervalsPrintLeftEndPoint <PolPrintIntervalsPrintLeftEndPoint_>`_ - - `\\PolPrintIntervalsPrintRightEndPoint <PolPrintIntervalsPrintRightEndPoint_>`_ - - `\\PolReduceCoeffs* <PolReduceCoeffs*_>`_ - - `\\PolMakeMonic <PolMakeMonic_>`_ - * new expandable macros: - - - `\\PolToExprOneTermStyleA <PolToExprOneTermStyleA_>`_ - - `\\PolIfCoeffIsPlusOrMinusOne <PolIfCoeffIsPlusOrMinusOne_>`_ - - `\\PolLeadingCoeff <PolLeadingCoeff_>`_ - - `\\PolSturmChainLength <PolSturmChainLength_>`_ - - `\\PolSturmNbOfIsolatedZeros <PolSturmNbOfIsolatedZeros_>`_ - - `\\PolSturmIfZeroExactlyKnown <PolSturmIfZeroExactlyKnown_>`_ - - `\\PolSturmIsolatedZeroLeft <PolSturmIsolatedZeroLeft_>`_ - - `\\PolSturmIsolatedZeroRight <PolSturmIsolatedZeroRight_>`_ - - ``\PolPrintIntervalsTheEndPoint`` (removed at 0.7) - - `\\PolPrintIntervalsTheIndex`_ - - ``\PolIfEndPointIsPositive`` (removed at 0.7) - - ``\PolIfEndPointIsNegative`` (removed at 0.7) - - ``\PolIfEndPointIsZero`` (removed at 0.7) - - `\\PolIntervalWidth <PolIntervalWidth_>`_ - - `\\PolDecToString <PolDecToString_>`_ - * improvements: - - The main new feature is implementation of the `Sturm algorithm`_ - for localization of the real roots of polynomials. - -- v0.4.1 (2018/03/01) - - Synced with xint 1.3. - -- v0.4.2 (2018/03/03) - - Documentation fix. - -- v0.5 (2018/04/08) - - * bug fixes: - - - `\\PolGet{polname}\\fromarray\\macro`_ crashed when ``\macro`` was - an xinttools_ array macro with no items. It now produces the zero - polynomial. - * breaking changes: - - - `\\PolToSturm`_ creates primitive integer coefficients polynomials. - This speeds up localization of roots via - `\\PolSturmIsolateZeros`_. In case of user protests the author - will make available again the code producing the bona fide Sturm - polynomials as used formerly. - - polynomials created from `\\PolFromCSV`_ or `\\PolGet <PolGet_>`_ - get their coefficients normalized via xintfrac_\ 's ``\xintRaw``. - * experimental change: - - - optional argument to `\\PolSturmIsolateZeros`_ (see `The - degree 41 polynomial with -2, -1.9, -1.8, ..., 0, 0.1, ..., 1.9, 2 - as roots`_ for usage). It will presumably be replaced in future by - an interval specification. - * new non-expandable macro: - - - `\\PolMakePrimitive`_ - * new expandable macro: - - - `\\PolIContent`_ - -- v0.5.1 (2018/04/22) - - * new feature: - - - the character ``'`` can be used in polynomial names. - -- v0.6 (2018/11/20) - - * bugfix: - - - the starred variant `\\PolToSturm*{polname}{sturmname}`_ was - broken. On the occasion of the fix, its meaning has been modified, - see its documentation. - - - using `\\PolToSturm <PolToSturm_>`_ with a constant polynomial - caused a division by zero error. - - * new macro: - - - `\\PolSturmIsolateZeros* <PolSturmIsolateZeros*_>`_ - acts like the `non-starred variant - <PolSturmIsolateZeros_>`_ then computes all the multiplicities. - - * new expandable macros: - - - `\\PolSturmIsolatedZeroMultiplicity{sturmname}{index}`_ - - `\\PolSturmNbOfRootsOf{sturmname}\\LessThanOrEqualTo{value}`_ - - `\\PolSturmNbOfRootsOf{sturmname}\\LessThanOrEqualToExpr{expression}`_ - - `\\PolSturmNbWithMultOfRootsOf{sturmname}\\LessThanOrEqualTo{value}`_ - - `\\PolSturmNbWithMultOfRootsOf{sturmname}\\LessThanOrEqualToExpr{expression}`_ - -- v0.7 (2018/12/08), v0.7.1 (bugfix), v0.7.2 (2nd bugfix) (2018/12/09) - - * breaking changes: - - - although `\\PolPrintIntervals[varname]{sturmname}`_ default output - remains the same, some auxiliary macros for user-customization - have been removed: ``\PolPrintIntervalsTheEndPoint``, - ``\PolIfEndPointIsPositive{A}{B}``, - ``\PolIfEndPointIsNegative{A}{B}``, and - ``\PolIfEndPointIsZero{A}{B}``. - - * bugfix: - - - it could happen that, contrarily to documentation, an interval - computed by `\\PolSturmIsolateZeros{sturmname}`_ had zero as an - endpoint, - - `\\PolEnsureIntervalLength{sturmname}{index}{E}`_ could under - certain circumstances erroneously replace a non-zero root by - zero, - - `\\PolEnsureIntervalLengths{sturmname}{E}`_ crashed when used with - a polynomial with no real roots, hence for which no isolation intervals - existed (thanks to Thomas Söll for report). - - * new macros: - - - `\\PolSturmIsolateZeros**{sturmname}`_ - - `\\PolSturmIsolateZerosGetMultiplicitiesAndRationalRoots{sturmname}`_ - - `\\PolSturmIsolateZerosAndFindRationalRoots{sturmname}`_ - - `\\polexprsetup`_ - - `\\PolPrintIntervals* <PolPrintIntervals*_>`_ - - `\\PolPrintIntervalsNoRealRoots`_ - - `\\PolPrintIntervalsBeginEnv`_ - - `\\PolPrintIntervalsEndEnv`_ - - `\\PolPrintIntervalsKnownRoot`_ - - `\\PolPrintIntervalsUnknownRoot`_ - - `\\PolPrintIntervalsPrintMultiplicity`_ - - * new expandable macros: - - - `\\PolSturmNbOfRationalRoots{sturmname}`_ - - `\\PolSturmNbOfRationalRootsWithMultiplicities{sturmname}`_ - - `\\PolSturmRationalRoot{sturmname}{k}`_ - - `\\PolSturmRationalRootIndex{sturmname}{k}`_ - - `\\PolSturmRationalRootMultiplicity{sturmname}{k}`_ - - `\\PolPrintIntervalsTheVar`_ - - `\\PolPrintIntervalsTheSturmName`_ - - `\\PolPrintIntervalsTheMultiplicity`_ - -- v0.7.3 (2019/02/04) - - * bugfix: - - - Debugging information not destined to user showed in log if root - finding was done under ``\xintverbosetrue`` regime. - - `\\PolPrintIntervalsTheVar`_ remained defined after - `\\PolPrintIntervals`_ but was left undefined after - `\\PolPrintIntervals*`_ (reported by Jürgen Gilg). Now remains - defined in both cases, and `\\PolPrintIntervalsTheSturmName`_ - also. - - Polynomial names ending in digits caused errors (reported by Thomas - Söll). - -- v0.7.4 (2019/02/12) - - * bugfix: - - - 20000000000 is too big for ``\numexpr``, shouldn't I know that? - Thanks to Jürgen Gilg for report. - -- v0.7.5 (2020/01/31) - - Synced with xint 1.4. Requires it. - - -Acknowledgments ---------------- - -Thanks to Jürgen Gilg whose question about xint_ usage for -differentiating polynomials was the initial trigger leading to this -package, and to Jürgen Gilg and Thomas Söll for testing it on some -concrete problems. - -Renewed thanks to them on occasion of the ``0.6`` and ``0.7`` releases for their -continued interest. - -See README.md for the License. - -.. _xinttools: -.. _xintfrac: -.. _xintexpr: -.. _xint: http://www.ctan.org/pkg/xint - -.. _Wilkinson polynomial: https://en.wikipedia.org/wiki/Wilkinson%27s_polynomial - -.. _Sturm algorithm: -.. _Sturm Theorem: https://en.wikipedia.org/wiki/Sturm%27s_theorem - -.. _DocUtils: http://docutils.sourceforge.net/docs/index.html |