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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/generic | |
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/generic')
-rw-r--r-- | Master/texmf-dist/doc/generic/polexpr/README.md | 123 | ||||
-rw-r--r-- | Master/texmf-dist/doc/generic/polexpr/polexpr.html | 4086 |
2 files changed, 4209 insertions, 0 deletions
diff --git a/Master/texmf-dist/doc/generic/polexpr/README.md b/Master/texmf-dist/doc/generic/polexpr/README.md new file mode 100644 index 00000000000..b32b8f056df --- /dev/null +++ b/Master/texmf-dist/doc/generic/polexpr/README.md @@ -0,0 +1,123 @@ +Package polexpr README +====================== + +Usage +----- + +The package can be used with TeX based formats incorporating the +e-TeX primitives. The `\expanded` primitive available generally +since TeXLive 2019 is required. + + \input polexpr.sty + +with Plain or other non-LaTeX macro formats, or: + + \usepackage{polexpr} + +with the LaTeX macro format. + +The package currently requires xintexpr.sty `1.4d` or later. + +Abstract +-------- + +The package provides a parser `\poldef` of algebraic polynomial +expressions. As it is based on +[xintexpr](http://www.ctan.org/pkg/xint) +the coefficients are allowed to be arbitrary rational numbers. + +Once defined, a polynomial is usable by its name either as a numerical +function in `\xintexpr/\xinteval`, or for additional polynomial +definitions, or as argument to the package macros. The localization of +real roots to arbitrary precision as well as the determination of all +rational roots is implemented via such macros. + +Since release `0.8`, polexpr extends the +[xintexpr](http://www.ctan.org/pkg/xint) syntax to recognize +polynomials as a new variable type (and not only as functions). +Functionality which previously was implemented via macros such as the +computation of a greatest common divisor is now available directly in +`\xintexpr`, `\xinteval` or `\poldef` via infix or functional syntax. + +Releases +-------- + +- 0.1 (2018/01/11) + Initial release (files README, polexpr.sty). +- 0.2 (2018/01/14) + Documentation moved to polexpr.{txt,html}. +- 0.3 (2018/01/17) + Make polynomials known to `\xintfloatexpr` and improve + documentation. +- 0.3.1 (2018/01/18) + Fix two typos in documentation. +- 0.4 (2018/02/16) + - Revert 0.3 automatic generation of floating point variants. + - Move CHANGE LOG from README.md to HTML documentation. + - A few bug fixes and breaking changes. Please refer to + `polexpr.html`. + - Main new feature: root localization via [Sturm + Theorem](https://en.wikipedia.org/wiki/Sturm%27s_theorem). +- 0.4.1 (2018/03/01) + Synced with xintexpr 1.3. +- 0.4.2 (2018/03/03) + Documentation fix. +- 0.5 (2018/04/08) + - new macros `\PolMakePrimitive` and `\PolIContent`. + - main (breaking) change: `\PolToSturm` creates a chain of primitive + integer coefficients polynomials. +- 0.5.1 (2018/04/22) + The `'` character can be used in polynomial names. +- 0.6 (2018/11/20) + New feature: multiplicity of roots. +- 0.7 (2018/12/08), 0.7.1 (bugfix), 0.7.2 (bugfix) (2018/12/09) + New feature: finding all rational roots. +- 0.7.3 (2019/02/04) + Bugfix: polynomial names ending in digits caused errors. Thanks to + Thomas Söll for report. +- 0.7.4 (2019/02/12) + Bugfix: 20000000000 is too big for \numexpr, shouldn't I know that? + Thanks to Jürgen Gilg for report. +- 0.7.5 (2020/01/31) + Synced with xintexpr 1.4. Requires it. +- 0.8 (2021/03/29) + Complete refactoring of the package core for better integration with + and enhancement of xintexpr 1.4. + +Files of 0.8 release: + +- polexpr.sty, polexprcore.tex, polexprexpr.tex, polexprsturm.tex, +- README.md, +- polexpr.html (documentation) + +Acknowledgments +--------------- + +Thanks to Jürgen Gilg whose question about +[xintexpr](http://www.ctan.org/pkg/xintexpr) usage for differentiating +polynomials was the initial trigger leading to this package, and to +Jürgen Gilg and Thomas Söll for testing it on some concrete problems. + +License +------- + +Copyright (C) 2018-2021 Jean-François Burnol + +See documentation of package [xintexpr](http://www.ctan.org/pkg/xint) for +contact information. + +This Work may be distributed and/or modified under the conditions of the +LaTeX Project Public License version 1.3c. This version of this license +is in + +> <http://www.latex-project.org/lppl/lppl-1-3c.txt> + +and version 1.3 or later is part of all distributions of LaTeX version +2005/12/01 or later. + +This Work has the LPPL maintenance status author-maintained. + +The Author of this Work is Jean-François Burnol. + +This Work consists of the package files polexpr.sty, polexprcore.tex, +polexprexpr.tex, polexprsturm.tex, this README.md and polexpr.html. diff --git a/Master/texmf-dist/doc/generic/polexpr/polexpr.html b/Master/texmf-dist/doc/generic/polexpr/polexpr.html new file mode 100644 index 00000000000..81cd2b4b7c1 --- /dev/null +++ b/Master/texmf-dist/doc/generic/polexpr/polexpr.html @@ -0,0 +1,4086 @@ +<!DOCTYPE html> +<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en" lang="en"> +<head> +<meta charset="utf-8"/> +<meta name="generator" content="Docutils 0.16: http://docutils.sourceforge.net/" /> +<title>Package polexpr documentation</title> +<style type="text/css"> + +/* Minimal style sheet for the HTML output of Docutils. */ +/* */ +/* :Author: Günter Milde, based on html4css1.css by David Goodger */ +/* :Id: $Id: minimal.css 8397 2019-09-20 11:09:34Z milde $ */ +/* :Copyright: © 2015 Günter Milde. */ +/* :License: Released under the terms of the `2-Clause BSD license`_, */ +/* in short: */ +/* */ +/* Copying and distribution of this file, with or without modification, */ +/* are permitted in any medium without royalty provided the copyright */ +/* notice and this notice are preserved. */ +/* */ +/* This file is offered as-is, without any warranty. */ +/* */ +/* .. _2-Clause BSD license: http://www.spdx.org/licenses/BSD-2-Clause */ + +/* This CSS2.1_ stylesheet defines rules for Docutils elements without */ +/* HTML equivalent. It is required to make the document semantic visible. */ +/* */ +/* .. _CSS2.1: http://www.w3.org/TR/CSS2 */ +/* .. _validates: http://jigsaw.w3.org/css-validator/validator$link */ + +/* alignment of text and inline objects inside block objects*/ +.align-left { text-align: left; } +.align-right { text-align: right; } +.align-center { clear: both; text-align: center; } +.align-top { vertical-align: top; } +.align-middle { vertical-align: middle; } +.align-bottom { vertical-align: bottom; } + +/* titles */ +h1.title, p.subtitle { + text-align: center; +} +p.topic-title, +p.sidebar-title, +p.rubric, +p.admonition-title, +p.system-message-title { + font-weight: bold; +} +h1 + p.subtitle, +h1 + p.section-subtitle { + font-size: 1.6em; +} +h2 + p.section-subtitle { font-size: 1.28em; } +p.subtitle, +p.section-subtitle, +p.sidebar-subtitle { + font-weight: bold; + margin-top: -0.5em; +} +p.sidebar-title, +p.rubric { + font-size: larger; +} +p.rubric { color: maroon; } +a.toc-backref { + color: black; + text-decoration: none; } + +/* Warnings, Errors */ +div.caution p.admonition-title, +div.attention p.admonition-title, +div.danger p.admonition-title, +div.error p.admonition-title, +div.warning p.admonition-title, +div.system-messages h1, +div.error, +span.problematic, +p.system-message-title { + color: red; +} + +/* inline literals */ +span.docutils.literal { + font-family: monospace; + white-space: pre-wrap; +} +/* do not wraph at hyphens and similar: */ +.literal > span.pre { white-space: nowrap; } + +/* Lists */ + +/* compact and simple lists: no margin between items */ +.simple li, .compact li, +.simple ul, .compact ul, +.simple ol, .compact ol, +.simple > li p, .compact > li p, +dl.simple > dd, dl.compact > dd { + margin-top: 0; + margin-bottom: 0; +} + +/* Table of Contents */ +div.topic.contents { margin: 0.5em 0; } +div.topic.contents ul { + list-style-type: none; + padding-left: 1.5em; +} + +/* Enumerated Lists */ +ol.arabic { list-style: decimal } +ol.loweralpha { list-style: lower-alpha } +ol.upperalpha { list-style: upper-alpha } +ol.lowerroman { list-style: lower-roman } +ol.upperroman { list-style: upper-roman } + +dt span.classifier { font-style: italic } +dt span.classifier:before { + font-style: normal; + margin: 0.5em; + content: ":"; +} + +/* Field Lists and drivatives */ +/* bold field name, content starts on the same line */ +dl.field-list > dt, +dl.option-list > dt, +dl.docinfo > dt, +dl.footnote > dt, +dl.citation > dt { + font-weight: bold; + clear: left; + float: left; + margin: 0; + padding: 0; + padding-right: 0.5em; +} +/* Offset for field content (corresponds to the --field-name-limit option) */ +dl.field-list > dd, +dl.option-list > dd, +dl.docinfo > dd { + margin-left: 9em; /* ca. 14 chars in the test examples */ +} +/* start field-body on a new line after long field names */ +dl.field-list > dd > *:first-child, +dl.option-list > dd > *:first-child +{ + display: inline-block; + width: 100%; + margin: 0; +} +/* field names followed by a colon */ +dl.field-list > dt:after, +dl.docinfo > dt:after { + content: ":"; +} + +/* Bibliographic Fields (docinfo) */ +pre.address { font: inherit; } +dd.authors > p { margin: 0; } + +/* Option Lists */ +dl.option-list { margin-left: 1.5em; } +dl.option-list > dt { font-weight: normal; } +span.option { white-space: nowrap; } + +/* Footnotes and Citations */ +dl.footnote.superscript > dd {margin-left: 1em; } +dl.footnote.brackets > dd {margin-left: 2em; } +dl > dt.label { font-weight: normal; } +a.footnote-reference.brackets:before, +dt.label > span.brackets:before { content: "["; } +a.footnote-reference.brackets:after, +dt.label > span.brackets:after { content: "]"; } +a.footnote-reference.superscript, +dl.footnote.superscript > dt.label { + vertical-align: super; + font-size: smaller; +} +dt.label > span.fn-backref { margin-left: 0.2em; } +dt.label > span.fn-backref > a { font-style: italic; } + +/* Line Blocks */ +div.line-block { display: block; } +div.line-block div.line-block { + margin-top: 0; + margin-bottom: 0; + margin-left: 40px; +} + +/* Figures, Images, and Tables */ +.figure.align-left, +figure.align-left, +img.align-left, +object.align-left, +table.align-left { + margin-right: auto; +} +.figure.align-center, +figure.align-center, +img.align-center, +object.align-center, +table.align-center { + margin-left: auto; + margin-right: auto; +} +.figure.align-right, +figure.align-right, +img.align-right, +object.align-right, +table.align-right { + margin-left: auto; +} +.figure.align-center, .figure.align-right, +figure.align-center, figure.align-right, +img.align-center, img.align-right, +object.align-center, object.align-right { + display: block; +} +/* reset inner alignment in figures and tables */ +.figure.align-left, .figure.align-right, +figure.align-left, figure.align-right, +table.align-left, table.align-center, table.align-right { + text-align: inherit; +} + +/* Admonitions and System Messages */ +div.admonition, +div.system-message, +div.sidebar, +aside.sidebar { + margin: 1em 1.5em; + border: medium outset; + padding-top: 0.5em; + padding-bottom: 0.5em; + padding-right: 1em; + padding-left: 1em; +} + +/* Sidebar */ +div.sidebar, +aside.sidebar { + width: 30%; + max-width: 26em; + float: right; + clear: right; +} + +/* Text Blocks */ +blockquote, +div.topic, +pre.literal-block, +pre.doctest-block, +pre.math, +pre.code { + margin-left: 1.5em; + margin-right: 1.5em; +} +pre.code .ln { color: gray; } /* line numbers */ + +/* Tables */ +table { border-collapse: collapse; } +td, th { + border-style: solid; + border-color: silver; + padding: 0 1ex; + border-width: thin; +} +td > p:first-child, th > p:first-child { margin-top: 0; } +td > p, th > p { margin-bottom: 0; } + +table > caption { + text-align: left; + margin-bottom: 0.25em +} + +table.borderless td, table.borderless th { + border: 0; + padding: 0; + padding-right: 0.5em /* separate table cells */ +} + +/* Document Header and Footer */ +/* div.header, */ +/* header { border-bottom: 1px solid black; } */ +/* div.footer, */ +/* footer { border-top: 1px solid black; } */ + +/* new HTML5 block elements: set display for older browsers */ +header, section, footer, aside, nav, main, article, figure { + display: block; +} + +</style> +<style type="text/css"> + +/* CSS31_ style sheet for the output of Docutils HTML writers. */ +/* Rules for easy reading and pre-defined style variants. */ +/* */ +/* :Author: Günter Milde, based on html4css1.css by David Goodger */ +/* :Id: $Id: plain.css 8397 2019-09-20 11:09:34Z milde $ */ +/* :Copyright: © 2015 Günter Milde. */ +/* :License: Released under the terms of the `2-Clause BSD license`_, */ +/* in short: */ +/* */ +/* Copying and distribution of this file, with or without modification, */ +/* are permitted in any medium without royalty provided the copyright */ +/* notice and this notice are preserved. */ +/* */ +/* This file is offered as-is, without any warranty. */ +/* */ +/* .. _2-Clause BSD license: http://www.spdx.org/licenses/BSD-2-Clause */ +/* .. _CSS3: http://www.w3.org/TR/CSS3 */ + + +/* Document Structure */ +/* ****************** */ + +/* "page layout" */ +body { + margin: 0; + background-color: #dbdbdb; +} +div.document, +main { + line-height:1.3; + counter-reset: table; + /* counter-reset: figure; */ + /* avoid long lines --> better reading */ + /* OTOH: lines should not be too short because of missing hyphenation, */ + max-width: 50em; + padding: 1px 2%; /* 1px on top avoids grey bar above title (mozilla) */ + margin: auto; + background-color: white; +} + +/* Sections */ + +/* Transitions */ + +hr.docutils { + width: 80%; + margin-top: 1em; + margin-bottom: 1em; + clear: both; +} + +/* Paragraphs */ +/* ========== */ + +/* vertical space (parskip) */ +p, ol, ul, dl, +div.line-block, +div.topic, +table { + margin-top: 0.5em; + margin-bottom: 0.5em; +} +p:first-child { margin-top: 0; } +/* (:last-child is new in CSS 3) */ +p:last-child { margin-bottom: 0; } + +h1, h2, h3, h4, h5, h6, +dl > dd { + margin-bottom: 0.5em; +} + +/* Lists */ +/* ===== */ + +/* Definition Lists */ + +/* lists nested in definition lists */ +/* (:only-child is new in CSS 3) */ +dd > ul:only-child, dd > ol:only-child { padding-left: 1em; } + +/* Description Lists */ +/* styled like in most dictionaries, encyclopedias etc. */ +dl.description > dt { + font-weight: bold; + clear: left; + float: left; + margin: 0; + padding: 0; + padding-right: 0.5em; +} + +/* Field Lists */ + +/* example for custom field-name width */ +dl.field-list.narrow > dd { + margin-left: 5em; +} +/* run-in: start field-body on same line after long field names */ +dl.field-list.run-in > dd p { + display: block; +} + +/* Bibliographic Fields */ + +/* generally, bibliographic fields use special definition list dl.docinfo */ +/* but dedication and abstract are placed into "topic" divs */ +div.abstract p.topic-title { + text-align: center; +} +div.dedication { + margin: 2em 5em; + text-align: center; + font-style: italic; +} +div.dedication p.topic-title { + font-style: normal; +} + +/* Citations */ +dl.citation dt.label { + font-weight: bold; +} +span.fn-backref { + font-weight: normal; +} + +/* Text Blocks */ +/* =========== */ + +/* Literal Blocks */ + +pre.literal-block, +pre.doctest-block, +pre.math, +pre.code { + font-family: monospace; +} + +/* Block Quotes */ + +blockquote > table, +div.topic > table { + margin-top: 0; + margin-bottom: 0; +} +blockquote p.attribution, +div.topic p.attribution { + text-align: right; + margin-left: 20%; +} + +/* Tables */ +/* ====== */ + +/* th { vertical-align: bottom; } */ + +table tr { text-align: left; } + +/* "booktabs" style (no vertical lines) */ +table.booktabs { + border: 0; + border-top: 2px solid; + border-bottom: 2px solid; + border-collapse: collapse; +} +table.booktabs * { + border: 0; +} +table.booktabs th { + border-bottom: thin solid; +} + +/* numbered tables (counter defined in div.document) */ +table.numbered > caption:before { + counter-increment: table; + content: "Table " counter(table) ": "; + font-weight: bold; +} + +/* Explicit Markup Blocks */ +/* ====================== */ + +/* Footnotes and Citations */ +/* ----------------------- */ + +/* line on the left */ +dl.footnote { + padding-left: 1ex; + border-left: solid; + border-left-width: thin; +} + +/* Directives */ +/* ---------- */ + +/* Body Elements */ +/* ~~~~~~~~~~~~~ */ + +/* Images and Figures */ + +/* let content flow to the side of aligned images and figures */ +.figure.align-left, +figure.align-left, +img.align-left, +object.align-left { + display: block; + clear: left; + float: left; + margin-right: 1em; +} +.figure.align-right, +figure.align-right, +img.align-right, +object.align-right { + display: block; + clear: right; + float: right; + margin-left: 1em; +} +/* Stop floating sidebars, images and figures at section level 1,2,3 */ +h1, h2, h3 { clear: both; } + +/* Sidebar */ + +/* Move right. In a layout with fixed margins, */ +/* it can be moved into the margin. */ +div.sidebar, +aside.sidebar { + width: 30%; + max-width: 26em; + margin-left: 1em; + margin-right: -2%; + background-color: #ffffee; +} + +/* Code */ + +pre.code { padding: 0.7ex } +pre.code, code { background-color: #eeeeee } +pre.code .ln { color: gray; } /* line numbers */ +/* basic highlighting: for a complete scheme, see */ +/* http://docutils.sourceforge.net/sandbox/stylesheets/ */ +pre.code .comment, code .comment { color: #5C6576 } +pre.code .keyword, code .keyword { color: #3B0D06; font-weight: bold } +pre.code .literal.string, code .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} + +/* Math */ +/* styled separately (see math.css for math-output=HTML) */ + +/* Epigraph */ +/* Highlights */ +/* Pull-Quote */ +/* Compound Paragraph */ +/* Container */ + +/* can be styled in a custom stylesheet */ + +/* Document Header and Footer */ + +footer, header, +div.footer, div.header { + font-size: smaller; + clear: both; + padding: 0.5em 2%; + background-color: #ebebee; + border: none; +} + +/* Inline Markup */ +/* ============= */ + +/* Emphasis */ +/* em */ +/* Strong Emphasis */ +/* strong */ +/* Interpreted Text */ +/* span.interpreted */ +/* Title Reference */ +/* cite */ + +/* Inline Literals */ +/* possible values: normal, nowrap, pre, pre-wrap, pre-line */ +/* span.docutils.literal { white-space: pre-wrap; } */ + +/* Hyperlink References */ +a { text-decoration: none; } + +/* External Targets */ +/* span.target.external */ +/* Internal Targets */ +/* span.target.internal */ +/* Footnote References */ +/* a.footnote-reference */ +/* Citation References */ +/* a.citation-reference */ + +</style> +</head> +<body> +<div class="document" id="package-polexpr-documentation"> +<h1 class="title">Package polexpr documentation</h1> +<p class="subtitle" id="id1">0.8 (2021/03/29)</p> + +<div class="contents topic" id="contents"> +<p class="topic-title">Contents</p> +<ul class="simple"> +<li><p><a class="reference internal" href="#usage" id="id41">Usage</a></p></li> +<li><p><a class="reference internal" href="#abstract" id="id42">Abstract</a></p></li> +<li><p><a class="reference internal" href="#prerequisites" id="id43">Prerequisites</a></p></li> +<li><p><a class="reference internal" href="#quick-syntax-overview" id="id44">Quick syntax overview</a></p></li> +<li><p><a class="reference internal" href="#the-polexpr-0-8-extensions-to-the-xintexpr-syntax" id="id45">The polexpr <span class="docutils literal">0.8</span> extensions to the <span class="docutils literal">\xintexpr</span> syntax</a></p> +<ul> +<li><p><a class="reference internal" href="#warning-about-unstability-of-the-new-syntax" id="id46">Warning about unstability of the new syntax</a></p></li> +<li><p><a class="reference internal" href="#infix-operators" id="id47">Infix operators <span class="docutils literal">+, <span class="pre">-,</span> *, /, **, ^</span></a></p></li> +<li><p><a class="reference internal" href="#experimental-infix-operators" id="id48">Experimental infix operators <span class="docutils literal">//, /:</span></a></p></li> +<li><p><a class="reference internal" href="#comparison-operators" id="id49">Comparison operators <span class="docutils literal"><, >, <=, >=, ==, !=</span></a></p></li> +<li><p><a class="reference internal" href="#pol-nutple-expression" id="id50"><span class="docutils literal"><span class="pre">pol(<nutple</span> expression>)</span></a></p></li> +<li><p><a class="reference internal" href="#xinteval-pol-expr" id="id51"><span class="docutils literal"><span class="pre">\xinteval{<pol.</span> <span class="pre">expr.>}</span></span></a></p></li> +<li><p><a class="reference internal" href="#evalp-pol-expr-pol-expr" id="id52"><span class="docutils literal"><span class="pre">evalp(<pol.</span> <span class="pre">expr.>,</span> <pol. expr>)</span></a></p></li> +<li><p><a class="reference internal" href="#deg-pol-expr" id="id53"><span class="docutils literal"><span class="pre">deg(<pol.</span> <span class="pre">expr.>)</span></span></a></p></li> +<li><p><a class="reference internal" href="#coeffs-pol-expr" id="id54"><span class="docutils literal"><span class="pre">coeffs(<pol.</span> <span class="pre">expr.>)</span></span></a></p></li> +<li><p><a class="reference internal" href="#coeff-pol-expr-num-expr" id="id55"><span class="docutils literal"><span class="pre">coeff(<pol.</span> <span class="pre">expr.>,</span> <num. <span class="pre">expr.>)</span></span></a></p></li> +<li><p><a class="reference internal" href="#lcoeff-pol-expr" id="id56"><span class="docutils literal"><span class="pre">lcoeff(<pol.</span> <span class="pre">expr.>)</span></span></a></p></li> +<li><p><a class="reference internal" href="#monicpart-pol-expr" id="id57"><span class="docutils literal"><span class="pre">monicpart(<pol.</span> <span class="pre">expr.>)</span></span></a></p></li> +<li><p><a class="reference internal" href="#icontent-pol-expr" id="id58"><span class="docutils literal"><span class="pre">icontent(<pol.</span> <span class="pre">expr.>)</span></span></a></p></li> +<li><p><a class="reference internal" href="#primpart-pol-expr" id="id59"><span class="docutils literal"><span class="pre">primpart(<pol.</span> <span class="pre">expr.>)</span></span></a></p></li> +<li><p><a class="reference internal" href="#quorem-pol-expr-pol-expr" id="id60"><span class="docutils literal"><span class="pre">quorem(<pol.</span> <span class="pre">expr.>,</span> <pol. <span class="pre">expr.>)</span></span></a></p></li> +<li><p><a class="reference internal" href="#quo-pol-expr-pol-expr" id="id61"><span class="docutils literal"><span class="pre">quo(<pol.</span> <span class="pre">expr.>,</span> <pol. <span class="pre">expr.>)</span></span></a></p></li> +<li><p><a class="reference internal" href="#rem-pol-expr-pol-expr" id="id62"><span class="docutils literal"><span class="pre">rem(<pol.</span> <span class="pre">expr.>,</span> <pol. <span class="pre">expr.>)</span></span></a></p></li> +<li><p><a class="reference internal" href="#prem-pol-expr-1-pol-expr-2" id="id63"><span class="docutils literal"><span class="pre">prem(<pol.</span> expr. 1>, <pol. expr. 2>)</span></a></p></li> +<li><p><a class="reference internal" href="#divmod-pol-expr-1-pol-expr-2" id="id64"><span class="docutils literal"><span class="pre">divmod(<pol.</span> expr. 1>, <pol. expr. 2>)</span></a></p></li> +<li><p><a class="reference internal" href="#mod-pol-expr-1-pol-expr-2" id="id65"><span class="docutils literal"><span class="pre">mod(<pol.</span> expr. 1>, <pol. expr. 2>)</span></a></p></li> +<li><p><a class="reference internal" href="#polgcd-pol-expr-1-pol-expr-2" id="id66"><span class="docutils literal"><span class="pre">polgcd(<pol.</span> expr. 1>, <pol. expr. 2>, <span class="pre">...)</span></span></a></p></li> +<li><p><a class="reference internal" href="#resultant-pol-expr-1-pol-expr-2" id="id67"><span class="docutils literal"><span class="pre">resultant(<pol.</span> expr. 1>, <pol. expr. 2>)</span></a></p></li> +<li><p><a class="reference internal" href="#disc-pol-expr" id="id68"><span class="docutils literal"><span class="pre">disc(<pol.</span> <span class="pre">expr.>)</span></span></a></p></li> +<li><p><a class="reference internal" href="#polpowmod-pol-expr-1-num-expr-pol-expr-2" id="id69"><span class="docutils literal"><span class="pre">polpowmod(<pol.</span> expr. 1>, <num. <span class="pre">expr.>,</span> <pol. expr. 2>)</span></a></p></li> +<li><p><a class="reference internal" href="#rdcoeffs-pol-expr" id="id70"><span class="docutils literal"><span class="pre">rdcoeffs(<pol.</span> <span class="pre">expr.>)</span></span></a></p></li> +<li><p><a class="reference internal" href="#rdzcoeffs-pol-expr" id="id71"><span class="docutils literal"><span class="pre">rdzcoeffs(<pol.</span> <span class="pre">expr.>)</span></span></a></p></li> +<li><p><a class="reference internal" href="#diff1-pol-expr" id="id72"><span class="docutils literal"><span class="pre">diff1(<pol.</span> <span class="pre">expr.>)</span></span></a></p></li> +<li><p><a class="reference internal" href="#diff2-pol-expr" id="id73"><span class="docutils literal"><span class="pre">diff2(<pol.</span> <span class="pre">expr.>)</span></span></a></p></li> +<li><p><a class="reference internal" href="#diffn-pol-expr-p-num-expr-n" id="id74"><span class="docutils literal"><span class="pre">diffn(<pol.</span> expr. P>, <num. expr. n>)</span></a></p></li> +<li><p><a class="reference internal" href="#antider-pol-expr-p" id="id75"><span class="docutils literal"><span class="pre">antider(<pol.</span> expr. P>)</span></a></p></li> +<li><p><a class="reference internal" href="#intfrom-pol-expr-p-pol-expr-c" id="id76"><span class="docutils literal"><span class="pre">intfrom(<pol.</span> expr. P>, <pol. expr. c>)</span></a></p></li> +<li><p><a class="reference internal" href="#integral-pol-expr-p-pol-expr-a-pol-expr-b" id="id77"><span class="docutils literal"><span class="pre">integral(<pol.</span> expr. P>, [<pol. expr. a>, <pol. expr. <span class="pre">b>])</span></span></a></p></li> +</ul> +</li> +<li><p><a class="reference internal" href="#examples-of-localization-of-roots" id="id78">Examples of localization of roots</a></p> +<ul> +<li><p><a class="reference internal" href="#a-typical-example" id="id79">A typical example</a></p></li> +<li><p><a class="reference internal" href="#a-degree-four-polynomial-with-nearby-roots" id="id80">A degree four polynomial with nearby roots</a></p></li> +<li><p><a class="reference internal" href="#the-degree-nine-polynomial-with-0-99-0-999-0-9999-as-triple-roots" id="id81">The degree nine polynomial with 0.99, 0.999, 0.9999 as triple roots</a></p></li> +<li><p><a class="reference internal" href="#a-degree-five-polynomial-with-three-rational-roots" id="id82">A degree five polynomial with three rational roots</a></p></li> +<li><p><a class="reference internal" href="#a-mignotte-type-polynomial" id="id83">A Mignotte type polynomial</a></p></li> +<li><p><a class="reference internal" href="#the-wilkinson-polynomial" id="id84">The Wilkinson polynomial</a></p></li> +<li><p><a class="reference internal" href="#the-second-wilkinson-polynomial" id="id85">The second Wilkinson polynomial</a></p></li> +<li><p><a class="reference internal" href="#the-degree-41-polynomial-with-2-1-9-1-8-0-0-1-1-9-2-as-roots" id="id86">The degree 41 polynomial with -2, -1.9, -1.8, ..., 0, 0.1, ..., 1.9, 2 as roots</a></p></li> +<li><p><a class="reference internal" href="#roots-of-chebyshev-polynomials" id="id87">Roots of Chebyshev polynomials</a></p></li> +</ul> +</li> +<li><p><a class="reference internal" href="#non-expandable-macros" id="id88">Non-expandable macros</a></p> +<ul> +<li><p><a class="reference internal" href="#poldef-polname-letter-expression-using-the-letter-as-indeterminate" id="id89"><span class="docutils literal">\poldef <span class="pre">polname(letter):=</span> expression using the letter as indeterminate;</span></a></p></li> +<li><p><a class="reference internal" href="#poldef-letter-polname-expression-using-the-letter-as-indeterminate" id="id90"><span class="docutils literal"><span class="pre">\PolDef[letter]{polname}{expression</span> using the letter as indeterminate}</span></a></p></li> +<li><p><a class="reference internal" href="#polgenfloatvariant-polname" id="id91"><span class="docutils literal">\PolGenFloatVariant{polname}</span></a></p></li> +<li><p><a class="reference internal" href="#pollet-polname-2-polname-1" id="id92"><span class="docutils literal"><span class="pre">\PolLet{polname_2}={polname_1}</span></span></a></p></li> +<li><p><a class="reference internal" href="#polgloballet-polname-2-polname-1" id="id93"><span class="docutils literal"><span class="pre">\PolGlobalLet{polname_2}={polname_1}</span></span></a></p></li> +<li><p><a class="reference internal" href="#polassign-polname-toarray-macro" id="id94"><span class="docutils literal"><span class="pre">\PolAssign{polname}\toarray\macro</span></span></a></p></li> +<li><p><a class="reference internal" href="#polget-polname-fromarray-macro" id="id95"><span class="docutils literal"><span class="pre">\PolGet{polname}\fromarray\macro</span></span></a></p></li> +<li><p><a class="reference internal" href="#polfromcsv-polname-csv" id="id96"><span class="docutils literal"><span class="pre">\PolFromCSV{polname}{<csv>}</span></span></a></p></li> +<li><p><a class="reference internal" href="#poltypeset-pol-expr" id="id97"><span class="docutils literal"><span class="pre">\PolTypeset{<pol.</span> <span class="pre">expr.>}</span></span></a></p> +<ul> +<li><p><a class="reference internal" href="#poltypesetcmd-raw-coeff" id="id98"><span class="docutils literal">\PolTypesetCmd{raw_coeff}</span></a></p></li> +<li><p><a class="reference internal" href="#poltypesetone-raw-coeff" id="id99"><span class="docutils literal">\PolTypesetOne{raw_coeff}</span></a></p></li> +<li><p><a class="reference internal" href="#id9" id="id100"><span class="docutils literal">\PolTypesetMonomialCmd</span></a></p></li> +<li><p><a class="reference internal" href="#poltypesetcmdprefix-raw-coeff" id="id101"><span class="docutils literal">\PolTypesetCmdPrefix{raw_coeff}</span></a></p></li> +</ul> +</li> +<li><p><a class="reference internal" href="#id11" id="id102"><span class="docutils literal"><span class="pre">\PolTypeset*{<pol.</span> <span class="pre">expr.>}</span></span></a></p></li> +<li><p><a class="reference internal" href="#poldiff-polname-1-polname-2" id="id103"><span class="docutils literal"><span class="pre">\PolDiff{polname_1}{polname_2}</span></span></a></p></li> +<li><p><a class="reference internal" href="#poldiff-n-polname-1-polname-2" id="id104"><span class="docutils literal"><span class="pre">\PolDiff[N]{polname_1}{polname_2}</span></span></a></p></li> +<li><p><a class="reference internal" href="#polantidiff-polname-1-polname-2" id="id105"><span class="docutils literal"><span class="pre">\PolAntiDiff{polname_1}{polname_2}</span></span></a></p></li> +<li><p><a class="reference internal" href="#polantidiff-n-polname-1-polname-2" id="id106"><span class="docutils literal"><span class="pre">\PolAntiDiff[N]{polname_1}{polname_2}</span></span></a></p></li> +<li><p><a class="reference internal" href="#poldivide-polname-1-polname-2-polname-q-polname-r" id="id107"><span class="docutils literal"><span class="pre">\PolDivide{polname_1}{polname_2}{polname_Q}{polname_R}</span></span></a></p></li> +<li><p><a class="reference internal" href="#polquo-polname-1-polname-2-polname-q" id="id108"><span class="docutils literal"><span class="pre">\PolQuo{polname_1}{polname_2}{polname_Q}</span></span></a></p></li> +<li><p><a class="reference internal" href="#polrem-polname-1-polname-2-polname-r" id="id109"><span class="docutils literal"><span class="pre">\PolRem{polname_1}{polname_2}{polname_R}</span></span></a></p></li> +<li><p><a class="reference internal" href="#polgcd-polname-1-polname-2-polname-gcd" id="id110"><span class="docutils literal"><span class="pre">\PolGCD{polname_1}{polname_2}{polname_GCD}</span></span></a></p></li> +<li><p><a class="reference internal" href="#non-expandable-macros-related-to-the-root-localization-routines" id="id111">Non-expandable macros related to the root localization routines</a></p> +<ul> +<li><p><a class="reference internal" href="#poltosturm-polname-sturmname" id="id112"><span class="docutils literal"><span class="pre">\PolToSturm{polname}{sturmname}</span></span></a></p></li> +<li><p><a class="reference internal" href="#id13" id="id113"><span class="docutils literal"><span class="pre">\PolToSturm*{polname}{sturmname}</span></span></a></p></li> +<li><p><a class="reference internal" href="#polsettosturmchainsignchangesat-macro-sturmname-fraction" id="id114"><span class="docutils literal"><span class="pre">\PolSetToSturmChainSignChangesAt{\macro}{sturmname}{fraction}</span></span></a></p></li> +<li><p><a class="reference internal" href="#polsettonbofzeroswithin-macro-sturmname-value-a-value-b" id="id115"><span class="docutils literal"><span class="pre">\PolSetToNbOfZerosWithin{\macro}{sturmname}{value_a}{value_b}</span></span></a></p></li> +<li><p><a class="reference internal" href="#polsturmisolatezeros-sturmname" id="id116"><span class="docutils literal">\PolSturmIsolateZeros{sturmname}</span></a></p></li> +<li><p><a class="reference internal" href="#id15" id="id117"><span class="docutils literal"><span class="pre">\PolSturmIsolateZeros*{sturmname}</span></span></a></p></li> +<li><p><a class="reference internal" href="#id17" id="id118"><span class="docutils literal"><span class="pre">\PolSturmIsolateZeros**{sturmname}</span></span></a></p></li> +<li><p><a class="reference internal" href="#polsturmisolatezerosandgetmultiplicities-sturmname" id="id119"><span class="docutils literal">\PolSturmIsolateZerosAndGetMultiplicities{sturmname}</span></a></p></li> +<li><p><a class="reference internal" href="#polsturmisolatezerosgetmultiplicitiesandrationalroots-sturmname" id="id120"><span class="docutils literal">\PolSturmIsolateZerosGetMultiplicitiesAndRationalRoots{sturmname}</span></a></p></li> +<li><p><a class="reference internal" href="#polsturmisolatezerosandfindrationalroots-sturmname" id="id121"><span class="docutils literal">\PolSturmIsolateZerosAndFindRationalRoots{sturmname}</span></a></p></li> +<li><p><a class="reference internal" href="#polrefineinterval-sturmname-index" id="id122"><span class="docutils literal"><span class="pre">\PolRefineInterval*{sturmname}{index}</span></span></a></p></li> +<li><p><a class="reference internal" href="#polrefineinterval-n-sturmname-index" id="id123"><span class="docutils literal"><span class="pre">\PolRefineInterval[N]{sturmname}{index}</span></span></a></p></li> +<li><p><a class="reference internal" href="#polensureintervallength-sturmname-index-e" id="id124"><span class="docutils literal"><span class="pre">\PolEnsureIntervalLength{sturmname}{index}{E}</span></span></a></p></li> +<li><p><a class="reference internal" href="#polensureintervallengths-sturmname-e" id="id125"><span class="docutils literal"><span class="pre">\PolEnsureIntervalLengths{sturmname}{E}</span></span></a></p></li> +<li><p><a class="reference internal" href="#polprintintervals-varname-sturmname" id="id126"><span class="docutils literal"><span class="pre">\PolPrintIntervals[varname]{sturmname}</span></span></a></p> +<ul> +<li><p><a class="reference internal" href="#polprintintervalsnorealroots" id="id127"><span class="docutils literal">\PolPrintIntervalsNoRealRoots</span></a></p></li> +<li><p><a class="reference internal" href="#polprintintervalsbeginenv" id="id128"><span class="docutils literal">\PolPrintIntervalsBeginEnv</span></a></p></li> +<li><p><a class="reference internal" href="#polprintintervalsendenv" id="id129"><span class="docutils literal">\PolPrintIntervalsEndEnv</span></a></p></li> +<li><p><a class="reference internal" href="#polprintintervalsknownroot" id="id130"><span class="docutils literal">\PolPrintIntervalsKnownRoot</span></a></p></li> +<li><p><a class="reference internal" href="#polprintintervalsunknownroot" id="id131"><span class="docutils literal">\PolPrintIntervalsUnknownRoot</span></a></p></li> +<li><p><a class="reference internal" href="#id18" id="id132"><span class="docutils literal">\PolPrintIntervalsPrintExactZero</span></a></p></li> +<li><p><a class="reference internal" href="#id19" id="id133"><span class="docutils literal">\PolPrintIntervalsPrintLeftEndPoint</span></a></p></li> +<li><p><a class="reference internal" href="#id20" id="id134"><span class="docutils literal">\PolPrintIntervalsPrintRightEndPoint</span></a></p></li> +</ul> +</li> +<li><p><a class="reference internal" href="#id22" id="id135"><span class="docutils literal"><span class="pre">\PolPrintIntervals*[varname]{sturmname}</span></span></a></p> +<ul> +<li><p><a class="reference internal" href="#polprintintervalsprintmultiplicity" id="id136"><span class="docutils literal">\PolPrintIntervalsPrintMultiplicity</span></a></p></li> +</ul> +</li> +</ul> +</li> +<li><p><a class="reference internal" href="#polmapcoeffs-macro-polname" id="id137"><span class="docutils literal"><span class="pre">\PolMapCoeffs{\macro}{polname}</span></span></a></p></li> +<li><p><a class="reference internal" href="#polreducecoeffs-polname" id="id138"><span class="docutils literal">\PolReduceCoeffs{polname}</span></a></p></li> +<li><p><a class="reference internal" href="#id24" id="id139"><span class="docutils literal"><span class="pre">\PolReduceCoeffs*{polname}</span></span></a></p></li> +<li><p><a class="reference internal" href="#polmakemonic-polname" id="id140"><span class="docutils literal">\PolMakeMonic{polname}</span></a></p></li> +<li><p><a class="reference internal" href="#polmakeprimitive-polname" id="id141"><span class="docutils literal">\PolMakePrimitive{polname}</span></a></p></li> +</ul> +</li> +<li><p><a class="reference internal" href="#expandable-macros" id="id142">Expandable macros</a></p> +<ul> +<li><p><a class="reference internal" href="#poleval-polname-atexpr-numerical-expression" id="id143"><span class="docutils literal"><span class="pre">\PolEval{polname}\AtExpr{numerical</span> expression}</span></a></p></li> +<li><p><a class="reference internal" href="#poleval-polname-at-fraction" id="id144"><span class="docutils literal"><span class="pre">\PolEval{polname}\At{fraction}</span></span></a></p></li> +<li><p><a class="reference internal" href="#polevalreduced-polname-atexpr-numerical-expression" id="id145"><span class="docutils literal"><span class="pre">\PolEvalReduced{polname}\AtExpr{numerical</span> expression}</span></a></p></li> +<li><p><a class="reference internal" href="#polevalreduced-polname-at-fraction" id="id146"><span class="docutils literal"><span class="pre">\PolEvalReduced{polname}\At{fraction}</span></span></a></p></li> +<li><p><a class="reference internal" href="#polfloateval-polname-atexpr-numerical-expression" id="id147"><span class="docutils literal"><span class="pre">\PolFloatEval{polname}\AtExpr{numerical</span> expression}</span></a></p></li> +<li><p><a class="reference internal" href="#polfloateval-polname-at-fraction" id="id148"><span class="docutils literal"><span class="pre">\PolFloatEval{polname}\At{fraction}</span></span></a></p></li> +<li><p><a class="reference internal" href="#polifcoeffisplusorminusone-a-b" id="id149"><span class="docutils literal"><span class="pre">\PolIfCoeffIsPlusOrMinusOne{A}{B}</span></span></a></p></li> +<li><p><a class="reference internal" href="#polleadingcoeff-polname" id="id150"><span class="docutils literal">\PolLeadingCoeff{polname}</span></a></p></li> +<li><p><a class="reference internal" href="#polnthcoeff-polname-number" id="id151"><span class="docutils literal"><span class="pre">\PolNthCoeff{polname}{number}</span></span></a></p></li> +<li><p><a class="reference internal" href="#poldegree-polname" id="id152"><span class="docutils literal">\PolDegree{polname}</span></a></p></li> +<li><p><a class="reference internal" href="#policontent-polname" id="id153"><span class="docutils literal">\PolIContent{polname}</span></a></p></li> +<li><p><a class="reference internal" href="#poltoexpr-pol-expr" id="id154"><span class="docutils literal"><span class="pre">\PolToExpr{<pol.</span> <span class="pre">expr.>}</span></span></a></p> +<ul> +<li><p><a class="reference internal" href="#id31" id="id155"><span class="docutils literal">\PolToExprVar</span></a></p></li> +<li><p><a class="reference internal" href="#poltoexprinvar" id="id156"><span class="docutils literal">\PolToExprInVar</span></a></p></li> +<li><p><a class="reference internal" href="#id32" id="id157"><span class="docutils literal">\PolToExprTimes</span></a></p></li> +<li><p><a class="reference internal" href="#poltoexprcaret" id="id158"><span class="docutils literal">\PolToExprCaret</span></a></p></li> +<li><p><a class="reference internal" href="#poltoexprcmd-raw-coeff" id="id159"><span class="docutils literal">\PolToExprCmd{raw_coeff}</span></a></p></li> +<li><p><a class="reference internal" href="#poltoexproneterm-raw-coeff-number" id="id160"><span class="docutils literal"><span class="pre">\PolToExprOneTerm{raw_coeff}{number}</span></span></a></p></li> +<li><p><a class="reference internal" href="#poltoexpronetermstylea-raw-coeff-number" id="id161"><span class="docutils literal"><span class="pre">\PolToExprOneTermStyleA{raw_coeff}{number}</span></span></a></p></li> +<li><p><a class="reference internal" href="#poltoexpronetermstyleb-raw-coeff-number" id="id162"><span class="docutils literal"><span class="pre">\PolToExprOneTermStyleB{raw_coeff}{number}</span></span></a></p></li> +<li><p><a class="reference internal" href="#poltoexprtermprefix-raw-coeff" id="id163"><span class="docutils literal">\PolToExprTermPrefix{raw_coeff}</span></a></p></li> +</ul> +</li> +<li><p><a class="reference internal" href="#id34" id="id164"><span class="docutils literal"><span class="pre">\PolToExpr*{<pol.</span> <span class="pre">expr.>}</span></span></a></p></li> +<li><p><a class="reference internal" href="#poltofloatexpr-pol-expr" id="id165"><span class="docutils literal"><span class="pre">\PolToFloatExpr{<pol.</span> <span class="pre">expr.>}</span></span></a></p> +<ul> +<li><p><a class="reference internal" href="#poltofloatexproneterm-raw-coeff-number" id="id166"><span class="docutils literal"><span class="pre">\PolToFloatExprOneTerm{raw_coeff}{number}</span></span></a></p></li> +<li><p><a class="reference internal" href="#poltofloatexprcmd-raw-coeff" id="id167"><span class="docutils literal">\PolToFloatExprCmd{raw_coeff}</span></a></p></li> +</ul> +</li> +<li><p><a class="reference internal" href="#id38" id="id168"><span class="docutils literal"><span class="pre">\PolToFloatExpr*{<pol.</span> <span class="pre">expr.>}</span></span></a></p></li> +<li><p><a class="reference internal" href="#poltolist-polname" id="id169"><span class="docutils literal">\PolToList{polname}</span></a></p></li> +<li><p><a class="reference internal" href="#poltocsv-polname" id="id170"><span class="docutils literal">\PolToCSV{polname}</span></a></p></li> +<li><p><a class="reference internal" href="#expandable-macros-related-to-the-root-localization-routines" id="id171">Expandable macros related to the root localization routines</a></p> +<ul> +<li><p><a class="reference internal" href="#polsturmchainlength-sturmname" id="id172"><span class="docutils literal">\PolSturmChainLength{sturmname}</span></a></p></li> +<li><p><a class="reference internal" href="#polsturmifzeroexactlyknown-sturmname-index-a-b" id="id173"><span class="docutils literal"><span class="pre">\PolSturmIfZeroExactlyKnown{sturmname}{index}{A}{B}</span></span></a></p></li> +<li><p><a class="reference internal" href="#polsturmisolatedzeroleft-sturmname-index" id="id174"><span class="docutils literal"><span class="pre">\PolSturmIsolatedZeroLeft{sturmname}{index}</span></span></a></p></li> +<li><p><a class="reference internal" href="#polsturmisolatedzeroright-sturmname-index" id="id175"><span class="docutils literal"><span class="pre">\PolSturmIsolatedZeroRight{sturmname}{index}</span></span></a></p></li> +<li><p><a class="reference internal" href="#polsturmisolatedzeromultiplicity-sturmname-index" id="id176"><span class="docutils literal"><span class="pre">\PolSturmIsolatedZeroMultiplicity{sturmname}{index}</span></span></a></p></li> +<li><p><a class="reference internal" href="#polsturmnbofisolatedzeros-sturmname" id="id177"><span class="docutils literal">\PolSturmNbOfIsolatedZeros{sturmname}</span></a></p></li> +<li><p><a class="reference internal" href="#polsturmnbofrootsof-sturmname-lessthanorequalto-value" id="id178"><span class="docutils literal"><span class="pre">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualTo{value}</span></span></a></p></li> +<li><p><a class="reference internal" href="#polsturmnbofrootsof-sturmname-lessthanorequaltoexpr-expression" id="id179"><span class="docutils literal"><span class="pre">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualToExpr{expression}</span></span></a></p></li> +<li><p><a class="reference internal" href="#polsturmnbwithmultofrootsof-sturmname-lessthanorequalto-value" id="id180"><span class="docutils literal"><span class="pre">\PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualTo{value}</span></span></a></p></li> +<li><p><a class="reference internal" href="#polsturmnbwithmultofrootsof-sturmname-lessthanorequaltoexpr-expression" id="id181"><span class="docutils literal"><span class="pre">\PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualToExpr{expression}</span></span></a></p></li> +<li><p><a class="reference internal" href="#polsturmnbofrationalroots-sturmname" id="id182"><span class="docutils literal">\PolSturmNbOfRationalRoots{sturmname}</span></a></p></li> +<li><p><a class="reference internal" href="#polsturmnbofrationalrootswithmultiplicities-sturmname" id="id183"><span class="docutils literal">\PolSturmNbOfRationalRootsWithMultiplicities{sturmname}</span></a></p></li> +<li><p><a class="reference internal" href="#polsturmrationalroot-sturmname-k" id="id184"><span class="docutils literal"><span class="pre">\PolSturmRationalRoot{sturmname}{k}</span></span></a></p></li> +<li><p><a class="reference internal" href="#polsturmrationalrootindex-sturmname-k" id="id185"><span class="docutils literal"><span class="pre">\PolSturmRationalRootIndex{sturmname}{k}</span></span></a></p></li> +<li><p><a class="reference internal" href="#polsturmrationalrootmultiplicity-sturmname-k" id="id186"><span class="docutils literal"><span class="pre">\PolSturmRationalRootMultiplicity{sturmname}{k}</span></span></a></p></li> +<li><p><a class="reference internal" href="#polintervalwidth-sturmname-index" id="id187"><span class="docutils literal"><span class="pre">\PolIntervalWidth{sturmname}{index}</span></span></a></p></li> +</ul> +</li> +<li><p><a class="reference internal" href="#expandable-macros-for-use-within-execution-of-polprintintervals" id="id188">Expandable macros for use within execution of <span class="docutils literal">\PolPrintIntervals</span></a></p> +<ul> +<li><p><a class="reference internal" href="#polprintintervalsthevar" id="id189"><span class="docutils literal">\PolPrintIntervalsTheVar</span></a></p></li> +<li><p><a class="reference internal" href="#polprintintervalstheindex" id="id190"><span class="docutils literal">\PolPrintIntervalsTheIndex</span></a></p></li> +<li><p><a class="reference internal" href="#polprintintervalsthesturmname" id="id191"><span class="docutils literal">\PolPrintIntervalsTheSturmName</span></a></p></li> +<li><p><a class="reference internal" href="#polprintintervalstheleftendpoint" id="id192"><span class="docutils literal">\PolPrintIntervalsTheLeftEndPoint</span></a></p></li> +<li><p><a class="reference internal" href="#polprintintervalstherightendpoint" id="id193"><span class="docutils literal">\PolPrintIntervalsTheRightEndPoint</span></a></p></li> +<li><p><a class="reference internal" href="#polprintintervalsthemultiplicity" id="id194"><span class="docutils literal">\PolPrintIntervalsTheMultiplicity</span></a></p></li> +</ul> +</li> +</ul> +</li> +<li><p><a class="reference internal" href="#booleans-with-default-setting-as-indicated" id="id195">Booleans (with default setting as indicated)</a></p> +<ul> +<li><p><a class="reference internal" href="#xintverbosefalse" id="id196"><span class="docutils literal">\xintverbosefalse</span></a></p></li> +<li><p><a class="reference internal" href="#polnewpolverbosefalse" id="id197"><span class="docutils literal">\polnewpolverbosefalse</span></a></p></li> +<li><p><a class="reference internal" href="#poltypesetallfalse" id="id198"><span class="docutils literal">\poltypesetallfalse</span></a></p></li> +<li><p><a class="reference internal" href="#poltoexprallfalse" id="id199"><span class="docutils literal">\poltoexprallfalse</span></a></p></li> +</ul> +</li> +<li><p><a class="reference internal" href="#utilies" id="id200">Utilies</a></p> +<ul> +<li><p><a class="reference internal" href="#poldectostring-decimal-number" id="id201"><span class="docutils literal">\PolDecToString{decimal number}</span></a></p></li> +<li><p><a class="reference internal" href="#polexprsetup" id="id202"><span class="docutils literal">\polexprsetup</span></a></p></li> +</ul> +</li> +<li><p><a class="reference internal" href="#technicalities" id="id203">Technicalities</a></p></li> +<li><p><a class="reference internal" href="#change-log" id="id204">CHANGE LOG</a></p></li> +<li><p><a class="reference internal" href="#acknowledgments" id="id205">Acknowledgments</a></p></li> +</ul> +</div> +<div class="section" id="usage"> +<h1><a class="toc-backref" href="#id41">Usage</a></h1> +<p>The package can be used with TeX based formats incorporating the e-TeX +primitives. The <span class="docutils literal">\expanded</span> primitive available generally since +TeXLive 2019 is required.</p> +<pre class="literal-block">\input polexpr.sty</pre> +<p>with Plain or other non-LaTeX macro formats, or:</p> +<pre class="literal-block">\usepackage{polexpr}</pre> +<p>with the LaTeX macro format.</p> +<p>The package requires <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> <span class="docutils literal">1.4d</span> or later.</p> +<div class="admonition note"> +<p class="admonition-title">Note</p> +<p>Until <span class="docutils literal">0.8</span> the package only had a LaTeX interface. As a result, +parts of this documentation may still give examples using LaTeX syntax such +as <span class="docutils literal">\newcommand</span>. Please convert to the syntax appropriate to the +TeX macro format used if needed.</p> +</div> +</div> +<div class="section" id="abstract"> +<h1><a class="toc-backref" href="#id42">Abstract</a></h1> +<p>The package provides a parser <span class="docutils literal">\poldef</span> of algebraic polynomial +expressions. As it is based on <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> +the coefficients are allowed to be arbitrary rational numbers.</p> +<p>Once defined, a polynomial is usable by its name either as a numerical +function in <span class="docutils literal"><span class="pre">\xintexpr/\xinteval</span></span>, or for additional polynomial +definitions, or as argument to the package macros. The localization of +real roots to arbitrary precision as well as the determination of all +rational roots is implemented via such macros.</p> +<p>Since release <span class="docutils literal">0.8</span>, polexpr extends the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> +syntax to recognize +polynomials as a new variable type (and not only as functions). +Functionality which previously was implemented via macros such as the +computation of a greatest common divisor is now available directly in +<span class="docutils literal">\xintexpr</span>, <span class="docutils literal">\xinteval</span> or <span class="docutils literal">\poldef</span> via infix or functional +syntax.</p> +</div> +<div class="section" id="prerequisites"> +<h1><a class="toc-backref" href="#id43">Prerequisites</a></h1> +<ul> +<li><p>The user must have some understanding of TeX as a macro-expansion +based programming interface, and in particular of how <span class="docutils literal">\edef</span> +differs from <span class="docutils literal">\def</span>: functionalities of the package as described in +the <a class="reference internal" href="#expandable-macros">Expandable macros</a> section are suitable for usage in <span class="docutils literal">\edef</span>, +<span class="docutils literal">\write</span> or <span class="docutils literal">\xinteval</span> context. At <span class="docutils literal">0.8</span> some of these +macros have an even more convenient functional interface inside +<span class="docutils literal">\xinteval</span>, as is described in a <a class="reference internal" href="#polexpr08">dedicated section</a>.</p> +<p>Despite its name <span class="docutils literal">\poldef</span> is more to be seen as an <span class="docutils literal">\edef</span> +although it does not define a TeX macro (at user level); and of course +<span class="docutils literal">\edef</span> would do usually nothing on the typical input parsed by +<span class="docutils literal">\poldef</span> which generally has no backslash in it: but if this input +does contain macros, they will then be expanded fully and are supposed to +produce recognizable syntax elements in this expansion only context.</p> +<p>Note that the <span class="docutils literal">def</span> in <span class="docutils literal">\poldef</span> reminds us that the macro does +some assignments hence is not usable in expandable only context. Its +whole point is rather to define entities which, them, can then be used +in the expandable only <span class="docutils literal">\xinteval</span> (or <span class="docutils literal">\poldef</span>) context.</p> +</li> +<li><p>The user must have some familiarity with <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> and in +particular must know what <span class="docutils literal">\xintexpr</span>, <span class="docutils literal">\xinttheexpr</span>, +<span class="docutils literal">\xinteval</span> and <span class="docutils literal">\xintfloatexpr</span>, <span class="docutils literal">\xintthefloatexpr</span>, +<span class="docutils literal">\xintfloateval</span> mean and what are the good practices with them.</p></li> +<li><p>The user will become quickly aware that exact computations with +fractions easily lead to very big ones in very few steps; see +<a class="reference internal" href="#polreducecoeffs-polname">\PolReduceCoeffs{polname}</a> in this context.</p></li> +<li><p>Finally, it is mandatory to read the entire documentation before +starting to use the package.</p></li> +</ul> +</div> +<div class="section" id="quick-syntax-overview"> +<h1><a class="toc-backref" href="#id44">Quick syntax overview</a></h1> +<p>The syntax is:</p> +<pre class="literal-block">\poldef polname(x):= expression in variable x;</pre> +<ul> +<li><p>In place of <span class="docutils literal">x</span> an arbitrary <em>dummy variable</em> is authorized, +i.e. per default one <span class="docutils literal">a, .., z, A, .., Z</span> (more letters can be declared +under Unicode engines).</p></li> +<li><p><span class="docutils literal">polname</span> consists of letters, digits, and the <span class="docutils literal">_</span> and <span class="docutils literal">'</span> +characters. It <strong>must</strong> start with a letter: do not use the +underscore <span class="docutils literal">_</span> as <em>first character</em> of a polynomial name (even +if of catcode letter). No warning is emitted but dire consequences +will result.</p> +<div class="admonition hint"> +<p class="admonition-title">Hint</p> +<p>The <span class="docutils literal">@</span> is usable too, independently of whether it is of catcode +letter or other. This has always been the case, but was not +documented by polexpr prior to <span class="docutils literal">0.8</span>, as the author has never +found the time to provide some official guidelines on how to name +temporary variables and the <span class="docutils literal">@</span> is used already as such internally +to package; time has still not yet been found for <span class="docutils literal">0.8</span> to review +the situation but it seems reasonable to recommend at any rate to +restrict usage of <span class="docutils literal">@</span> to scratch variables of defined macros and +to avoid using it to name document variable.</p> +</div> +</li> +<li><p>The colon before the equality sign is optional and its catcode does +not matter.</p></li> +<li><p>The semi-colon at the end of the expression is mandatory. Its catcode +does not matter if <span class="docutils literal">\poldef</span> is not used inside the argument of +another macro.</p></li> +</ul> +<p>There is an alternative syntax</p> +<pre class="literal-block">\PolDef[optional letter]{polname}{expression in the letter}</pre> +<p>Its optional first argument defaults to <span class="docutils literal">x</span>.</p> +<dl> +<dt><span class="docutils literal">\poldef <span class="pre">f(x):=</span> 1 - x + quo(x^5,1 - x + x^2);</span></dt> +<dd><p>defines polynomial <span class="docutils literal">f</span>. The indeterminate <span class="docutils literal">x</span> must be +only submitted to algebraic operations.</p> +<p>The <span class="docutils literal">quo()</span> function (new at <span class="docutils literal">0.8</span>) computes the euclidean +division quotient.</p> +</dd> +</dl> +<div class="admonition important"> +<p class="admonition-title">Important</p> +<p>For backwards compatibility one can currently also use:</p> +<pre class="literal-block">\poldef f(x):= 1 - x + x^5/(1 - x + x^2);</pre> +<p>Due to precedence rules the first operand is <span class="docutils literal">x^5</span>, not of course +<span class="docutils literal"><span class="pre">1-x+x^5</span></span>.</p> +<p>Note that <span class="docutils literal"><span class="pre">(1-x^2)/(1-x)</span></span> produces <span class="docutils literal">1+x</span> +but <span class="docutils literal"><span class="pre">(1/(1-x))*(1-x^2)</span></span> produces zero! One also has to be aware +of some precedence rules, for example:</p> +<pre class="literal-block">\poldef k(x):= (x-1)(x-2)(x-3)(x-4)/(x^2-5x+4);</pre> +<p>does compute a degree 2 polynomial because the tacit multiplication +ties more than the division operator.</p> +<p>In short, it is safer to use the <span class="docutils literal">quo()</span> function which avoids +surprises.</p> +</div> +<div class="admonition attention" id="warningtacit"> +<p class="admonition-title">Attention!</p> +<p>Tacit multiplication means that +<span class="docutils literal">1/2 x^2</span> skips the space and is treated like <span class="docutils literal"><span class="pre">1/(2*x^2)</span></span>. +But then it gives zero!</p> +<p>Thus one must use <span class="docutils literal">(1/2)x^2</span> or <span class="docutils literal">1/2*x^2</span> or +<span class="docutils literal"><span class="pre">(1/2)*x^2</span></span> for disambiguation: <span class="docutils literal">x - 1/2*x^2 + <span class="pre">1/3*x^3...</span></span>. It is +simpler to move the denominator to the right: <span class="docutils literal">x - x^2/2 + x^3/3 - ...</span>.</p> +<p>It is worth noting that <span class="docutils literal"><span class="pre">1/2(x-1)(x-2)</span></span> suffers the same issue: +<a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a>'s tacit multiplication always "ties more", hence this +gets interpreted as <span class="docutils literal"><span class="pre">1/(2*(x-1)*(x-2))</span></span> which gives zero by +polynomial division. Thus, use in such cases one of +<span class="docutils literal"><span class="pre">(1/2)(x-1)(x-2)</span></span>, <span class="docutils literal"><span class="pre">1/2*(x-1)(x-2)</span></span> or <span class="docutils literal"><span class="pre">(x-1)(x-2)/2</span></span>.</p> +</div> +<div class="admonition warning"> +<p class="admonition-title">Warning</p> +<p>The package does not currently know rational functions, but in order +to leave open this as a future possibility, the usage of <span class="docutils literal">/</span> to stand +for the +euclidean quotient is <strong>deprecated</strong>.</p> +<p>Please start using rather the <span class="docutils literal">quo()</span> function. It is possible +that in a future major relase <span class="docutils literal">A/B</span> with <span class="docutils literal">B</span> a non-scalar will +raise an error. Or, who knows, rational functions will be +implemented sometime during the next decades, and then <span class="docutils literal">A/B</span> will +naturally be the rational function.</p> +</div> +<div class="admonition important"> +<p class="admonition-title">Important</p> +<p><span class="docutils literal">\poldef <span class="pre">P(x):=...;</span></span> defines <span class="docutils literal">P</span> both as a <em>function</em>, +to be used as:</p> +<pre class="literal-block">P(..numeric or even polynomial expression..)</pre> +<p>and as a <em>variable</em> which can used inside polynomial expressions or +as argument to some polynomial specific functions such as <span class="docutils literal">deg()</span> +or <span class="docutils literal">polgcd()</span> <a class="footnote-reference brackets" href="#id3" id="id2">1</a>.</p> +<dl class="footnote brackets"> +<dt class="label" id="id3"><span class="brackets"><a class="fn-backref" href="#id2">1</a></span></dt> +<dd><p>Functional syntax accepts expressions as arguments; but the +TeX <strong>macros</strong> described in the documentation, even the +expandable ones, work only (there are a few exceptions to the +general rule) with arguments being <em>names of declared +polynomials</em>.</p> +</dd> +</dl> +<p>One needs to have a clear understanding of the difference between +<span class="docutils literal">P</span> used a function and <span class="docutils literal">P</span> used as a variable: if <span class="docutils literal">P</span> and +<span class="docutils literal">Q</span> are both declared polynomials then:</p> +<pre class="literal-block">(P+Q)(3)% <--- attention!</pre> +<p>is currently evaluated as <span class="docutils literal"><span class="pre">(P+Q)*3</span></span>, because <span class="docutils literal">P+Q</span> is not known +as a <em>function</em>, but <em>only as a variable of polynomial type</em>. +Even worse:</p> +<pre class="literal-block">(P)(3)% <--- attention!</pre> +<p>will compute <span class="docutils literal">P*3</span>, because one can not in current <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> syntax +enclose a function name in parentheses: consequently it is the variable +which is used here. There is a <em>meager possibility</em> that in future +some internal changes to <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> would let <span class="docutils literal"><span class="pre">(P)(3)</span></span> actually +compute <span class="docutils literal">P(3)</span> and <span class="docutils literal"><span class="pre">(P+Q)(3)</span></span> compute <span class="docutils literal">P(3) + Q(3)</span>, but note +that <span class="docutils literal"><span class="pre">(P)(P)</span></span> will then do <span class="docutils literal">P(P)</span> and not <span class="docutils literal">P*P</span>, +the latter, current interpretation, looking more +intuitive. Anyway, do not rely too extensively on tacit <span class="docutils literal">*</span> and use +explicit <span class="docutils literal"><span class="pre">(P+Q)*(1+2)</span></span> if this is what is intended.</p> +<p>As an alternative to explicit <span class="docutils literal"><span class="pre">P(3)+Q(3)</span></span> there is <span class="docutils literal">evalp(P+Q,3)</span>.</p> +</div> +<dl> +<dt><span class="docutils literal"><span class="pre">\PolLet{g}={f}</span></span></dt> +<dd><p>saves a copy of <span class="docutils literal">f</span> under name <span class="docutils literal">g</span>. Also usable without <span class="docutils literal">=</span>.</p> +<p>Has exactly the same effect as <span class="docutils literal">\poldef <span class="pre">g(x):=f;</span></span> or <span class="docutils literal">\poldef <span class="pre">g(w):=f(w);</span></span>.</p> +</dd> +<dt><span class="docutils literal">\poldef <span class="pre">f(z):=</span> f^2;</span></dt> +<dd><p>redefines <span class="docutils literal">f</span> in terms of itself. Prior to <span class="docutils literal">0.8</span> one needed +the right hand side to be <span class="docutils literal"><span class="pre">f(z)^2</span></span>. Also, now <span class="docutils literal">sqr(f)</span> is +possible (also <span class="docutils literal">sqr(f(x))</span> but not <span class="docutils literal"><span class="pre">sqr(f)(x)</span></span>).</p> +</dd> +</dl> +<div class="admonition important"> +<p class="admonition-title">Important</p> +<p>Note that <span class="docutils literal">f^2(z)</span> or <span class="docutils literal"><span class="pre">sqr(f)(z)</span></span> will give a logical but +perhaps unexpected result: first <span class="docutils literal">f^2</span> is computed, then the +opening parenthesis is seen which inserts a tacit multiplication +<span class="docutils literal">*</span>, so in the end it is as if the input had been <span class="docutils literal">f^2 * z</span>. +Although <span class="docutils literal">f</span> is both a variable and a function, <span class="docutils literal">f^2</span> is +computed as a polynomial <em>variable</em> and ceases being a function.</p> +</div> +<dl> +<dt><span class="docutils literal">\poldef <span class="pre">f(T):=</span> f(f);</span></dt> +<dd><p>again modifies <span class="docutils literal">f</span>. Here it is used both as variable and as +a function. Prior to <span class="docutils literal">0.8</span> it needed to be <span class="docutils literal">f(f(T))</span>.</p> +</dd> +<dt><span class="docutils literal">\poldef <span class="pre">k(z):=</span> <span class="pre">f-g(g^2)^2;</span></span></dt> +<dd><p>if everybody followed, this should now define the zero polynomial... +And <span class="docutils literal"><span class="pre">f-sqr(g(sqr(g)))</span></span> computes the same thing.</p> +<p>We can check this in a typeset document like this:</p> +<pre class="literal-block">\poldef f(x):= 1 - x + quo(x^5,1 - x + x^2);% +\PolLet{g}={f}% +\poldef f(z):= f^2;% +\poldef f(T):= f(f);% +\poldef k(w):= f-sqr(g(sqr(g)));% +$$f(x) = \vcenter{\hsize10cm \PolTypeset{f}} $$ +$$g(z) = \PolTypeset{g} $$ +$$k(z) = \PolTypeset{k} $$ +\immediate\write128{f(x)=\PolToExpr{f}}% ah, here we see it also</pre> +</dd> +<dt><span class="docutils literal">\poldef <span class="pre">f'(x):=</span> diff1(f);</span></dt> +<dd><p>(new at <span class="docutils literal">0.8</span>)</p> +</dd> +<dt><span class="docutils literal"><span class="pre">\PolDiff{f}{f'}</span></span></dt> +<dd><p>Both set <span class="docutils literal">f'</span> (or any other chosen name) to the derivative +of <span class="docutils literal">f</span>.</p> +</dd> +</dl> +<div class="admonition important"> +<p class="admonition-title">Important</p> +<p>This is not done automatically. If some new definition needs to use +the derivative of some available polynomial, that derivative +polynomial must have been previously defined: something such as +<span class="docutils literal"><span class="pre">f'(3)^2</span></span> will not work without a prior definition of <span class="docutils literal">f'</span>.</p> +<p>But one can now use <span class="docutils literal">diff1(f)</span> for on-the-spot construction with no +permanent declaration, so here <span class="docutils literal"><span class="pre">evalp(diff1(f),3)^2</span></span>. And +<span class="docutils literal"><span class="pre">diff1(f)^2</span></span> is same as <span class="docutils literal"><span class="pre">f'^2</span></span>, assuming here <span class="docutils literal">f'</span> was declared +to be the derived polynomial.</p> +<p>Notice that the name <span class="docutils literal">diff1()</span> is experimental and may change. Use +<span class="docutils literal"><span class="pre">\PolDiff{f}{f'}</span></span> as the stable interface.</p> +</div> +<dl> +<dt><span class="docutils literal">\PolTypeset{P}</span></dt> +<dd><p>Typesets (switching to math mode if in text mode):</p> +<pre class="literal-block">\poldef f(x):=(3+x)^5;% +\PolDiff{f}{f'}\PolDiff{f'}{f''}\PolDiff{f''}{f'''}% +$$f(z) = \PolTypeset[z]{f} $$ +$$f'(z) = \PolTypeset[z]{f'} $$ +$$f''(z) = \PolTypeset[z]{f''} $$ +$$f'''(z)= \PolTypeset[z]{f'''} $$</pre> +<p>See <a class="reference internal" href="#poltypeset">the documentation</a> for the configurability +via macros.</p> +<p>Since <span class="docutils literal">0.8</span> <a class="reference internal" href="#poltypeset">\PolTypeset</a> accepts directly an +expression, it does not have to be a pre-declared polynomial name:</p> +<pre class="literal-block">\PolTypeset{mul(x-i,i=1..5)}</pre> +</dd> +<dt><span class="docutils literal">\PolToExpr{P}</span></dt> +<dd><p>Expandably (contrarily to <a class="reference internal" href="#poltypeset">\PolTypeset</a>) +produces <span class="docutils literal">c_n*x^n + ... + c_0</span> starting from the leading +coefficient. The <span class="docutils literal">+</span> signs are omitted if followed by negative +coefficients.</p> +<p>This is useful for console or file output. This syntax is Maple and +PSTricks <span class="docutils literal">\psplot[algebraic]</span> compatible; and also it is +compatible with <span class="docutils literal">\poldef</span> input syntax, of course. See +<a class="reference internal" href="#poltoexprcaret">\PolToExprCaret</a> for configuration of the <span class="docutils literal">^</span>, for example to +use rather <span class="docutils literal">**</span> for Python syntax compliance.</p> +<p>Changed at <span class="docutils literal">0.8</span>: the <span class="docutils literal">^</span> in output is by default of catcode 12 +so in a draft document one can use <span class="docutils literal">\PolToExpr{P}</span> inside the +typesetting flow (without requiring math mode, where the <span class="docutils literal">*</span> would +be funny and <span class="docutils literal">^12</span> would only put the <span class="docutils literal">1</span> as exponent anyhow; +but arguably in text mode the <span class="docutils literal">+</span> and <span class="docutils literal">-</span> are not satisfactory +for math, except sometimes in monospace typeface, and anyhow TeX is +unable to break the expression across lines, barring special help).</p> +<p>See <a class="reference internal" href="#poltoexpr-pol-expr">\PolToExpr{<pol. expr.>}</a> and related macros for customization.</p> +<p>Extended at <span class="docutils literal">0.8</span> to accept as argument not only the name of a +polynomial variable but more generally any polynomial expression.</p> +</dd> +</dl> +</div> +<div class="section" id="the-polexpr-0-8-extensions-to-the-xintexpr-syntax"> +<span id="polexpr08"></span><h1><a class="toc-backref" href="#id45">The polexpr <span class="docutils literal">0.8</span> extensions to the <span class="docutils literal">\xintexpr</span> syntax</a></h1> +<p>All the syntax elements described in this section can be used in the +<span class="docutils literal"><span class="pre">\xintexpr/\xinteval</span></span> context (where polynomials can be obtained from +the <span class="docutils literal"><span class="pre">pol([])</span></span> constructor, once polexpr is loaded): their usage is +not limited to only <span class="docutils literal">\poldef</span> context.</p> +<div class="admonition note"> +<p class="admonition-title">Note</p> +<p>If a variable <span class="docutils literal">myPol</span> defined via <span class="docutils literal">\xintdefvar</span> turns out +to be a polynomial, the difference with those declared via <span class="docutils literal">\poldef</span> +will be:</p> +<ol class="arabic"> +<li><p><span class="docutils literal">myPol</span> is not usable as <em>function</em>, but only as a variable. +Attention that <span class="docutils literal">f(x)</span> if <span class="docutils literal">f</span> is only a variable (even a +polynomial one) will actually compute <span class="docutils literal">f * x</span>.</p></li> +<li><p><span class="docutils literal">myPol</span> is not known to the polexpr package, hence for example the +macros to achieve localization of its roots are unavailable.</p> +<p>In a parallel universe I perhaps have implemented this expandably +which means it could then be accessible with syntax such as +<span class="docutils literal"><span class="pre">rightmostroot(pol([42,1,34,2,-8,1]))</span></span> but...</p> +</li> +</ol> +</div> +<div class="section" id="warning-about-unstability-of-the-new-syntax"> +<h2><a class="toc-backref" href="#id46">Warning about unstability of the new syntax</a></h2> +<div class="admonition warning"> +<p class="admonition-title">Warning</p> +<p>Consider the entirety of this section as <strong>UNSTABLE</strong> and +<strong>EXPERIMENTAL</strong> (except perhaps regarding <span class="docutils literal">+</span>, <span class="docutils literal">-</span> and <span class="docutils literal">*</span>).</p> +<p>And this applies even to items not explicitly flagged with one of +<strong>unstable</strong>, <strong>Unstable</strong>, or <strong>UNSTABLE</strong> which only reflect that +documentation was written over a period of time exceeding one minute, +enough for the author mood changes to kick in.</p> +<p>It is hard to find good names at the start of a life-long extension +program of functionalities, and perhaps in future it will be +preferred to rename everything or give to some functions other +meanings. Such quasi-complete renamings happened already a few times +during the week devoted to development.</p> +</div> +</div> +<div class="section" id="infix-operators"> +<h2><a class="toc-backref" href="#id47">Infix operators <span class="docutils literal">+, <span class="pre">-,</span> *, /, **, ^</span></a></h2> +<blockquote> +<p>As has been explained in the <a class="reference internal" href="#quick-syntax-overview">Quick syntax overview</a> these infix +operators have been made polynomial aware, not only in the +<span class="docutils literal">\poldef</span> context, but generally in any <span class="docutils literal"><span class="pre">\xintexpr/\xinteval</span></span> +context, inclusive of <span class="docutils literal">\xintdeffunc</span>.</p> +<p>Conversely functions declared via <span class="docutils literal">\xintdeffunc</span> and making use of +these operators will automatically be able to accept polynomials +declared from <span class="docutils literal">\poldef</span> as variables.</p> +<p>Usage of <span class="docutils literal">/</span> for euclidean division of polynomials is <strong>deprecated</strong>. +Only in case of a scalar denominator is it to be considered stable. +Please use rather <span class="docutils literal">quo()</span>.</p> +</blockquote> +<div class="admonition warning"> +<p class="admonition-title">Warning</p> +<p>The <span class="docutils literal">pow(x,a)</span> function of <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> for <span class="docutils literal">x^a</span> with fractional +<span class="docutils literal">a</span> will not (with current <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> <span class="docutils literal">1.4d</span>) accept a polynomial +as first variable even if the second argument is an integer.</p> +<p>It is possible (via <span class="docutils literal">\poormanloghack</span>) to instruct <span class="docutils literal">\xintexpr</span> to +let <span class="docutils literal"><span class="pre">x**a</span></span> or <span class="docutils literal">x^a</span> be as <span class="docutils literal">pow(x,a)</span>. If this is done <span class="docutils literal">**</span> +(resp. <span class="docutils literal">^</span>) will become unusable with polynomials (i.e. will create +a low-level TeX error).</p> +<p>And vice versa if polexpr gets loaded after the <span class="docutils literal">\poormanloghack</span> +was used, <span class="docutils literal">**</span> and <span class="docutils literal">^</span> in <span class="docutils literal"><span class="pre">\xintexpr/\xinteval</span></span> will again only +accept integer powers.</p> +<p>Thus employ <span class="docutils literal">\poormanloghack</span> for at most one of <span class="docutils literal">**</span> or <span class="docutils literal">^</span> +in order to keep one of them available for polynomials and integer +powers.</p> +</div> +</div> +<div class="section" id="experimental-infix-operators"> +<h2><a class="toc-backref" href="#id48">Experimental infix operators <span class="docutils literal">//, /:</span></a></h2> +<blockquote> +<p>Here is the tentative behaviour of <span class="docutils literal"><span class="pre">A//B</span></span> according to types:</p> +<ul class="simple"> +<li><p><span class="docutils literal">A</span> non scalar and <span class="docutils literal">B</span> non scalar: euclidean quotient,</p></li> +<li><p><span class="docutils literal">A</span> scalar and <span class="docutils literal">B</span> scalar: floored division,</p></li> +<li><p><span class="docutils literal">A</span> scalar and <span class="docutils literal">B</span> non scalar: produces zero,</p></li> +<li><p><span class="docutils literal">A</span> non scalar and <span class="docutils literal">B</span> scalar: coefficient per +coefficient floored division.</p></li> +</ul> +<p>This is an <strong>experimental</strong> overloading of the <span class="docutils literal">//</span> and <span class="docutils literal">/:</span> +from <span class="docutils literal">\xintexpr</span>.</p> +<p>The behaviour in the last case, but not only, is to be considerd +<strong>unstable</strong>. The alternative would be for <span class="docutils literal"><span class="pre">A//B</span></span> with <span class="docutils literal">B</span> +scalar to act as <span class="docutils literal">quo(A,B)</span>. But, we have currently chosen to let +<span class="docutils literal">//B</span> for a scalar <span class="docutils literal">B</span> act coefficient-wise on the numerator. +Beware that it thus means it can be employed with the idea of doing +euclidean division only by checking that <span class="docutils literal">B</span> is non-scalar.</p> +<p>The <span class="docutils literal">/:</span> operator provides the associated remainder so always +<span class="docutils literal">A</span> is reconstructed from <span class="docutils literal"><span class="pre">(A//B)*B</span> + <span class="pre">A/:B</span></span>.</p> +<p>If <span class="docutils literal">:</span> is active character use <span class="docutils literal">/\string:</span> (it is safer to use +<span class="docutils literal">/\string :</span> if it is not known if <span class="docutils literal">:</span> has catcode other, letter, +or is active, but note that <span class="docutils literal">/:</span> is fine and needs no precaution if +<span class="docutils literal">:</span> has catcode letter, it is only an active <span class="docutils literal">:</span> which is +problematic, like for all other characters possibly used in an +expression).</p> +<blockquote> +<p><strong>UNSTABLE</strong></p> +<p>As explained above, there are (among other things) hesitations +about behaviour with <span class="docutils literal">pol2</span> a scalar.</p> +</blockquote> +</blockquote> +</div> +<div class="section" id="comparison-operators"> +<h2><a class="toc-backref" href="#id49">Comparison operators <span class="docutils literal"><, >, <=, >=, ==, !=</span></a></h2> +<blockquote> +<p><strong>NOT YET IMPLEMENTED</strong></p> +<p>As the internal representation by <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> and <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> of +fractions does not currently require them to be in reduced terms, +such operations would be a bit costly as they could not benefit from +the <span class="docutils literal">\pdfstrcmp</span> engine primitive. In fact <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> does not use +it yet anywhere, even for normalized pure integers, although it could +speed up signifcantly certain aspects of core arithmetic.</p> +<p>Equality of polynomials can currently be tested by computing the +difference, which is a bit costly. And of course the <span class="docutils literal">deg()</span> +function allows comparing degrees. In this context note the +following syntax:</p> +<pre class="literal-block">(deg(Q)) ?? { zero } { non-zero scalar } { non-scalar }</pre> +<p>for branching.</p> +</blockquote> +</div> +<div class="section" id="pol-nutple-expression"> +<h2><a class="toc-backref" href="#id50"><span class="docutils literal"><span class="pre">pol(<nutple</span> expression>)</span></a></h2> +<blockquote> +<p>This converts a nutple <span class="docutils literal"><span class="pre">[c0,c1,...,cN]</span></span> into the polynomial +variable having these coefficients. Attention that the square +brackets are <strong>mandatory</strong>, except of course if the argument is +actually an expression producing such a "nutple".</p> +<blockquote> +<p>Currently, this process will not normalize the coefficients (such +as reducing to lowest terms), it only trims out the leading zero +coefficients.</p> +</blockquote> +<p>Inside <span class="docutils literal">\xintexpr</span>, this is the only (allowed) way to create ex +nihilo a polynomial variable; inside <span class="docutils literal">\poldef</span> it is an alternative +input syntax which is more efficient than typing <span class="docutils literal">c0 + c1 * x + c2 * x^2 + ...</span>.</p> +</blockquote> +<div class="admonition important"> +<p class="admonition-title">Important</p> +<p>Whenever an expression with polynomials collapses to a constant, it +becomes a scalar. There is currently no distinction during the +parsing of expressions by <span class="docutils literal">\poldef</span> +or <span class="docutils literal">\xintexpr</span> between constant polynomial variables and scalar +variables.</p> +<p>Naturally, <span class="docutils literal">\poldef</span> can be used to declare a constant polynomial +<span class="docutils literal">P</span>, then <span class="docutils literal">P</span> can also be used as function having a value +independent of argument, but as a variable, it is non-distinguishable +from a scalar (of course functions such as <span class="docutils literal">deg()</span> tacitly +consider scalars to be constant polynomials).</p> +<p>Notice that we tend to use the vocable "variable" to refer to +arbitrary expressions used as function arguments, without implying +that we are actually referring to pre-declared variables in the sense +of <span class="docutils literal">\xintdefvar</span>.</p> +</div> +</div> +<div class="section" id="xinteval-pol-expr"> +<h2><a class="toc-backref" href="#id51"><span class="docutils literal"><span class="pre">\xinteval{<pol.</span> <span class="pre">expr.>}</span></span></a></h2> +<blockquote> +<p>This is documented here for lack of a better place: it evaluates the +polynomial expression then outputs the "string" <span class="docutils literal"><span class="pre">pol([c0,</span> c1, <span class="pre">...,</span> cN])</span> +if the degree <span class="docutils literal">N</span> is at least one (and the usual scalar output else).</p> +<p>The "pol" word uses letter catcodes, which is actually mandatory for +this output to be usable as input, but it does not make sense to use +this inside <span class="docutils literal">\poldef</span> or <span class="docutils literal">\xintexpr</span> at it means basically +executing <span class="docutils literal"><span class="pre">pol(coeffs(..expression..))</span></span> which is but a convoluted +way to obtain the same result as <span class="docutils literal"><span class="pre">(..expression..)</span></span> (the +parentheses delimiting the polynomial expression).</p> +<p>For example, <span class="docutils literal"><span class="pre">\xinteval{(1+pol([0,1]))^10}</span></span> expands (in two steps) +to:</p> +<pre class="literal-block">pol([1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1])</pre> +<p>You do need loading polexpr for this, else of course <span class="docutils literal"><span class="pre">pol([])</span></span> +remains unknown to <span class="docutils literal">\xinteval{}</span> as well as the polynomial algebra ! +This example can also be done as +<span class="docutils literal"><span class="pre">\xinteval{subs((1+x)^10,x=pol([0,1]))}</span></span>.</p> +<p>I hesitated using as output the polynomial notation as produced by +<a class="reference internal" href="#poltoexpr">\PolToExpr{}</a>, but finally opted for this.</p> +</blockquote> +</div> +<div class="section" id="evalp-pol-expr-pol-expr"> +<h2><a class="toc-backref" href="#id52"><span class="docutils literal"><span class="pre">evalp(<pol.</span> <span class="pre">expr.>,</span> <pol. expr>)</span></a></h2> +<blockquote> +<p>Evaluates the first argument as a polynomial function of the +second. Usually the second argument will be scalar, but this is not +required:</p> +<pre class="literal-block">\poldef K(x):= evalp(-3x^3-5x+1,-27x^4+5x-2);</pre> +<p>If the first argument is an already declared polynomial <span class="docutils literal">P</span>, use +rather the functional form <span class="docutils literal">P()</span> (which can accept a numerical as +well as polynomial argument) as it is more efficient.</p> +<p>One can also use <span class="docutils literal">subs()</span> syntax <a class="footnote-reference brackets" href="#id5" id="id4">2</a> (see <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> documentation):</p> +<pre class="literal-block">\poldef K(x):= subs(-3y^3-5y+1, y = -27x^4+5x-2);</pre> +<p>but the <span class="docutils literal">evalp()</span> will use a Horner evaluation scheme which is +usually more efficient.</p> +<dl class="footnote brackets"> +<dt class="label" id="id5"><span class="brackets"><a class="fn-backref" href="#id4">2</a></span></dt> +<dd><p>by the way Maple uses the opposite, hence wrong, order +<span class="docutils literal"><span class="pre">subs(x=...,</span> P)</span> but was written before computer science +reached the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> heights. However it makes validating +Maple results by polexpr sometimes cumbersome, but perhaps +they will update it at some point.</p> +</dd> +</dl> +<blockquote> +<p><strong>name unstable</strong></p> +<p><span class="docutils literal">poleval</span>? <span class="docutils literal">evalpol</span>? <span class="docutils literal">peval</span>? <span class="docutils literal">evalp</span>? <span class="docutils literal">value</span>? +<span class="docutils literal">eval</span>? <span class="docutils literal">evalat</span>? <span class="docutils literal">eval1at2</span>? <span class="docutils literal">evalat2nd</span>?</p> +<p>Life is so complicated when one asks questions. Not everybody does, +though, as is amply demonstrated these days.</p> +<p><strong>syntax unstable</strong></p> +<p>I am hesitating about permuting the order of the arguments.</p> +</blockquote> +</blockquote> +</div> +<div class="section" id="deg-pol-expr"> +<h2><a class="toc-backref" href="#id53"><span class="docutils literal"><span class="pre">deg(<pol.</span> <span class="pre">expr.>)</span></span></a></h2> +<blockquote> +<p>Computes the degree.</p> +</blockquote> +<div class="admonition important"> +<p class="admonition-title">Important</p> +<p>As <span class="docutils literal">\xintexpr</span> does not yet support infinities, the degree of +the zero polynomial is <span class="docutils literal"><span class="pre">-1</span></span>. Beware that this breaks additivity +of degrees, but <span class="docutils literal"><span class="pre">deg(P)<0</span></span> correctly detects the zero polynomial, +and <span class="docutils literal"><span class="pre">deg(P)<=0</span></span> detects scalars.</p> +</div> +</div> +<div class="section" id="coeffs-pol-expr"> +<h2><a class="toc-backref" href="#id54"><span class="docutils literal"><span class="pre">coeffs(<pol.</span> <span class="pre">expr.>)</span></span></a></h2> +<blockquote> +<p>Produces the nutple <span class="docutils literal"><span class="pre">[c0,c1,...,cN]</span></span> of coefficients. The highest +degree coefficient is always non zero (except for the zero +polynomial...).</p> +<blockquote> +<p><strong>name unstable</strong></p> +<p>I am considering in particular using <span class="docutils literal">polcoeffs()</span> to avoid +having to overload <span class="docutils literal">coeffs()</span> in future when matrix type +will be added to <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a>.</p> +</blockquote> +</blockquote> +</div> +<div class="section" id="coeff-pol-expr-num-expr"> +<h2><a class="toc-backref" href="#id55"><span class="docutils literal"><span class="pre">coeff(<pol.</span> <span class="pre">expr.>,</span> <num. <span class="pre">expr.>)</span></span></a></h2> +<blockquote> +<p>As expected. Produces zero if <span class="docutils literal">n</span> is negative or higher than the +degree.</p> +<blockquote> +<p><strong>name and syntax unstable</strong></p> +<p>I am hesitating with <span class="docutils literal">coeff(n,pol)</span> syntax and also perhaps +using <span class="docutils literal">polcoeff()</span> in order to avoid having to overload +<span class="docutils literal">coeff()</span> when matrix type will be added to <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a>.</p> +</blockquote> +</blockquote> +</div> +<div class="section" id="lcoeff-pol-expr"> +<h2><a class="toc-backref" href="#id56"><span class="docutils literal"><span class="pre">lcoeff(<pol.</span> <span class="pre">expr.>)</span></span></a></h2> +<blockquote> +<p>The leading coefficient.</p> +</blockquote> +</div> +<div class="section" id="monicpart-pol-expr"> +<h2><a class="toc-backref" href="#id57"><span class="docutils literal"><span class="pre">monicpart(<pol.</span> <span class="pre">expr.>)</span></span></a></h2> +<blockquote> +<p>Divides by the leading coefficient, except that <span class="docutils literal"><span class="pre">monicpart(0)==0</span></span>.</p> +<blockquote> +<p><strong>unstable</strong></p> +<p>Currently the coefficients are reduced to lowest terms (contrarily +to legacy behaviour of <a class="reference internal" href="#polmakemonic">\PolMakeMonic</a>), and +additionally the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> <span class="docutils literal">\xintREZ</span> macro is applied which +extracts powers of ten from numerator or denominator and stores +them internally separately. This is generally beneficial to +efficiency of multiplication.</p> +</blockquote> +</blockquote> +</div> +<div class="section" id="icontent-pol-expr"> +<h2><a class="toc-backref" href="#id58"><span class="docutils literal"><span class="pre">icontent(<pol.</span> <span class="pre">expr.>)</span></span></a></h2> +<blockquote> +<p>The gcd of the (possibly fractional) polynomial coefficients. It is +always produced as an irreducible (non-negative) fraction. According +to Gauss theorem the content of a product is the product of the +contents.</p> +<blockquote> +<p><strong>name unstable</strong></p> +<p>Some hesitation with using <span class="docutils literal">content()</span> rather.</p> +</blockquote> +</blockquote> +</div> +<div class="section" id="primpart-pol-expr"> +<h2><a class="toc-backref" href="#id59"><span class="docutils literal"><span class="pre">primpart(<pol.</span> <span class="pre">expr.>)</span></span></a></h2> +<blockquote> +<p>The quotient (except for the zero polynomial) by +<span class="docutils literal"><span class="pre">icontent(<pol.</span> <span class="pre">expr.>)</span></span>. This is thus a polynomial with +integer coefficients having <span class="docutils literal">1</span> as greatest common divisor. The +sign of the leading coefficient is the same as in the original.</p> +<p>And <span class="docutils literal"><span class="pre">primpart(0)==0</span></span>.</p> +<p>The trailing zeros of the integer coefficients are extracted +into a power of ten exponent part, in the internal representation.</p> +</blockquote> +</div> +<div class="section" id="quorem-pol-expr-pol-expr"> +<h2><a class="toc-backref" href="#id60"><span class="docutils literal"><span class="pre">quorem(<pol.</span> <span class="pre">expr.>,</span> <pol. <span class="pre">expr.>)</span></span></a></h2> +<blockquote> +<p>Produces a nutple <span class="docutils literal">[Q,R]</span> with <span class="docutils literal">Q</span> the euclidean quotient and +<span class="docutils literal">R</span> the remainder.</p> +<blockquote> +<p><strong>name unstable</strong></p> +<p><span class="docutils literal">poldiv()</span>?</p> +</blockquote> +</blockquote> +</div> +<div class="section" id="quo-pol-expr-pol-expr"> +<h2><a class="toc-backref" href="#id61"><span class="docutils literal"><span class="pre">quo(<pol.</span> <span class="pre">expr.>,</span> <pol. <span class="pre">expr.>)</span></span></a></h2> +<blockquote> +<p>The euclidean quotient.</p> +<p>The deprecated <span class="docutils literal">pol1/pol2</span> syntax computes the same polynomial.</p> +</blockquote> +</div> +<div class="section" id="rem-pol-expr-pol-expr"> +<h2><a class="toc-backref" href="#id62"><span class="docutils literal"><span class="pre">rem(<pol.</span> <span class="pre">expr.>,</span> <pol. <span class="pre">expr.>)</span></span></a></h2> +<blockquote> +<p>The euclidean remainder. If <span class="docutils literal">pol2</span> is a (non-zero) scalar, this is +zero.</p> +<p>There is no infix operator associated to this, for lack of evident +notation. Please advise.</p> +<p><span class="docutils literal">/:</span> can be used if one is certain that <span class="docutils literal">pol2</span> is of +degree at least one. But read the warning about it being unstable +even in that case.</p> +</blockquote> +</div> +<div class="section" id="prem-pol-expr-1-pol-expr-2"> +<span id="prem"></span><h2><a class="toc-backref" href="#id63"><span class="docutils literal"><span class="pre">prem(<pol.</span> expr. 1>, <pol. expr. 2>)</span></a></h2> +<blockquote> +<p>Produces a nutple <span class="docutils literal">[m, spR]</span> where <span class="docutils literal">spR</span> is the (special) pseudo +Euclidean remainder. Its description is:</p> +<ul> +<li><p>the standard euclidean remainder <span class="docutils literal">R</span> is <span class="docutils literal">spR/m</span></p></li> +<li><p><span class="docutils literal">m = b^f</span> with <span class="docutils literal">b</span> equal to the <strong>absolute value</strong> of the +leading coefficient of <span class="docutils literal">pol2</span>,</p></li> +<li><p><span class="docutils literal">f</span> is the number of non-zero coefficients in the euclidean +quotient, if <span class="docutils literal"><span class="pre">deg(pol2)>0</span></span> (even if the remainder vanishes).</p> +<p>If <span class="docutils literal">pol2</span> is a scalar however, the function outputs <span class="docutils literal">[1,0]</span>.</p> +</li> +</ul> +<p>With these definitions one can show that if both <span class="docutils literal">pol1</span> and +<span class="docutils literal">pol2</span> have integer coefficients, then this is also the case of +<span class="docutils literal">spR</span>, which makes its interest (and also <span class="docutils literal">m*Q</span> has integer +coefficients, with <span class="docutils literal">Q</span> the euclidean quotient, if <span class="docutils literal"><span class="pre">deg(pol2)>0</span></span>). +Also, <span class="docutils literal">prem()</span> is computed faster than <span class="docutils literal">rem()</span> for such integer +coefficients polynomials.</p> +<div class="admonition hint"> +<p class="admonition-title">Hint</p> +<p>If you want the euclidean quotient <span class="docutils literal">R</span> evaluated via <span class="docutils literal">spR/m</span> +(which may be faster, even with non integer coefficients) use +<span class="docutils literal"><span class="pre">subs(last(x)/first(x),x=prem(P,Q))</span></span> syntax as it avoids +computing <span class="docutils literal">prem(P,Q)</span> twice. This does the trick both in +<span class="docutils literal">\poldef</span> or in <span class="docutils literal">\xintdefvar</span>.</p> +<p>However, as is explained in the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> documentation, using +such syntax in an <span class="docutils literal">\xintdeffunc</span> is (a.t.t.o.w) illusory, due to +technicalities of how <span class="docutils literal">subs()</span> gets converted into nested +expandable macros. One needs an auxiliary function like this:</p> +<pre class="literal-block">\xintdeffunc lastoverfirst(x):=last(x)/first(x); +\xintdeffunc myR(x)=lastoverfirst(prem(x));</pre> +<p>Then, <span class="docutils literal">myR(pol1,pol2)</span> will evaluate <span class="docutils literal">prem(pol1,pol2)</span> only +once and compute a polynomial identical to the euclidean +remainder (internal representations of coefficients may differ).</p> +</div> +<p>In this case of integer coefficients polynomials, the polexpr +internal representation of the integer coefficients in the pseudo +remainder will be with unit denominators only if that was already the +case for those of <span class="docutils literal">pol1</span> and <span class="docutils literal">pol2</span> (no automatic reduction to +lowest terms is made prior or after computation).</p> +<p>Pay attention here that <span class="docutils literal">b</span> is the <strong>absolute value</strong> of the +leading coefficient of <span class="docutils literal">pol2</span>. Thus the coefficients of the +pseudo-remainder have the same signs as those of the standard +remainder. This diverges from Maple's function with the same name.</p> +</blockquote> +</div> +<div class="section" id="divmod-pol-expr-1-pol-expr-2"> +<h2><a class="toc-backref" href="#id64"><span class="docutils literal"><span class="pre">divmod(<pol.</span> expr. 1>, <pol. expr. 2>)</span></a></h2> +<blockquote> +<p>Overloads the scalar <span class="docutils literal">divmod()</span> and associates it with the +experimental <span class="docutils literal">//</span> and <span class="docutils literal">/:</span> as extended to the polynomial type.</p> +<p>In particular when both <span class="docutils literal">pol1</span> and <span class="docutils literal">pol2</span> are scalars, this is +the usual <span class="docutils literal">divmod()</span> (as in Python) and for <span class="docutils literal">pol1</span> and <span class="docutils literal">pol2</span> +non constant polynomials, this is the same as <span class="docutils literal">quorem()</span>.</p> +<blockquote> +<p><strong>Highly unstable</strong> overloading of <span class="docutils literal">\xinteval</span>'s <span class="docutils literal">divmod()</span>.</p> +</blockquote> +</blockquote> +</div> +<div class="section" id="mod-pol-expr-1-pol-expr-2"> +<h2><a class="toc-backref" href="#id65"><span class="docutils literal"><span class="pre">mod(<pol.</span> expr. 1>, <pol. expr. 2>)</span></a></h2> +<blockquote> +<p>The <span class="docutils literal">R</span> of the <span class="docutils literal">divmod()</span> output. Same as <span class="docutils literal">R</span> of <span class="docutils literal">quorem()</span> +when the second argument <span class="docutils literal">pol2</span> is of degree at least one.</p> +<blockquote> +<p><strong>Highly unstable</strong> overloading of <span class="docutils literal">\xinteval</span>'s <span class="docutils literal">mod()</span>.</p> +</blockquote> +</blockquote> +</div> +<div class="section" id="polgcd-pol-expr-1-pol-expr-2"> +<h2><a class="toc-backref" href="#id66"><span class="docutils literal"><span class="pre">polgcd(<pol.</span> expr. 1>, <pol. expr. 2>, <span class="pre">...)</span></span></a></h2> +<blockquote> +<p>Evaluates to the greatest common polynomial divisor of all the +polynomial inputs. The output is a <strong>primitive</strong> (in particular, +with integer coefficients) polynomial. It is zero if and only if all +inputs vanish.</p> +<p>Attention, there must be either at least two polynomial variables, or +alternatively, only one argument which then must be a bracketed list +or some expression or variable evaluating to such a "nutple" whose +items are polynomials (see the documentation of the scalar <span class="docutils literal">gcd()</span> +in <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a>).</p> +<blockquote> +<p>The two variable case could (and was, during development) have been +defined at user level like this:</p> +<pre class="literal-block">\xintdeffunc polgcd_(P,Q):= + (deg(Q))??{P}{1}{polgcd_(Q,primpart(last(prem(P,Q))))}; +\xintdeffunc polgcd(P,Q):=polgcd_(primpart(P),primpart(Q));%</pre> +<p>This is basically what is done internally for two polynomials, up +to some internal optimizations.</p> +</blockquote> +<p><strong>UNSTABLE</strong></p> +<p>I hesitate between returning a <em>primitive</em> or a <em>monic</em> polynomial. +Maple returns a primitive polynomial if all inputs <a class="footnote-reference brackets" href="#id7" id="id6">3</a> have integer +coefficients, else it returns a monic polynomial, but this is +complicated technically for us to add such a check and would add +serious overhead.</p> +<p>Internally, computations are done using primitive +integer-coefficients polynomials (as can be seen in the function +template above). So I decided finally to output a primitive +polynomial, as one can always apply <span class="docutils literal">monicpart()</span> to it.</p> +<p>Attention that this is at odds with behaviour of the legacy +<a class="reference internal" href="#polgcd">\PolGCD</a> (non expandable) macro.</p> +<dl class="footnote brackets"> +<dt class="label" id="id7"><span class="brackets"><a class="fn-backref" href="#id6">3</a></span></dt> +<dd><p>actually, only two polynomial arguments are allowed by Maple's +<span class="docutils literal">gcd()</span> as far as I know.</p> +</dd> +</dl> +</blockquote> +</div> +<div class="section" id="resultant-pol-expr-1-pol-expr-2"> +<h2><a class="toc-backref" href="#id67"><span class="docutils literal"><span class="pre">resultant(<pol.</span> expr. 1>, <pol. expr. 2>)</span></a></h2> +<blockquote> +<p>The resultant.</p> +<blockquote> +<p><strong>NOT YET IMPLEMENTED</strong></p> +</blockquote> +</blockquote> +</div> +<div class="section" id="disc-pol-expr"> +<h2><a class="toc-backref" href="#id68"><span class="docutils literal"><span class="pre">disc(<pol.</span> <span class="pre">expr.>)</span></span></a></h2> +<blockquote> +<p>The discriminant.</p> +<blockquote> +<p><strong>NOT YET IMPLEMENTED</strong></p> +</blockquote> +</blockquote> +</div> +<div class="section" id="polpowmod-pol-expr-1-num-expr-pol-expr-2"> +<h2><a class="toc-backref" href="#id69"><span class="docutils literal"><span class="pre">polpowmod(<pol.</span> expr. 1>, <num. <span class="pre">expr.>,</span> <pol. expr. 2>)</span></a></h2> +<blockquote> +<p>Modular exponentiation: <span class="docutils literal">mod(pol1^N, pol2)</span> in a more efficient +manner than first computing <span class="docutils literal">pol1^N</span> then reducing modulo <span class="docutils literal">pol2</span>.</p> +<p>Attention that this is using the <span class="docutils literal">mod()</span> operation, whose current +experimental status is as follows:</p> +<ul class="simple"> +<li><p>if <span class="docutils literal"><span class="pre">deg(pol2)>0</span></span>, the euclidean remainder operation,</p></li> +<li><p>if <span class="docutils literal">pol2</span> is a scalar, coefficient-wise reduction modulo <span class="docutils literal">pol2</span>.</p></li> +</ul> +<p><strong>UNSTABLE</strong></p> +<blockquote> +<p>This is currently implemented at high level via <span class="docutils literal">\xintdeffunc</span> and +recursive definitions, which were copied over from a scalar example +in the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> manual:</p> +<pre class="literal-block">\xintdeffunc polpowmod_(P, m, Q) := + isone(m)? + % m=1: return P modulo Q + { mod(P,Q) } + % m > 1: test if odd or even and do recursive call + { odd(m)? { mod(P*sqr(polpowmod_(P, m//2, Q)), Q) } + { mod( sqr(polpowmod_(P, m//2, Q)), Q) } + } + ;% +\xintdeffunc polpowmod(P, m, Q) := (m)?{polpowmod_(P, m, Q)}{1};%</pre> +<p>Negative exponents are not currently implemented.</p> +<p>For example:</p> +<pre class="literal-block">\xinteval{subs(polpowmod(1+x,100,x^7),x=pol([0,1]))} +\xinteval{subs(polpowmod(1+x,20,10), x=pol([0,1]))}</pre> +<p>produce respectively:</p> +<pre class="literal-block">pol([1, 100, 4950, 161700, 3921225, 75287520, 1192052400]) +pol([1, 0, 0, 0, 5, 4, 0, 0, 0, 0, 6, 0, 0, 0, 0, 4, 5, 0, 0, 0, 1])</pre> +</blockquote> +</blockquote> +</div> +<div class="section" id="rdcoeffs-pol-expr"> +<h2><a class="toc-backref" href="#id70"><span class="docutils literal"><span class="pre">rdcoeffs(<pol.</span> <span class="pre">expr.>)</span></span></a></h2> +<blockquote> +<p>This operates on the internal representation of the coefficients, +reducing them to lowest terms.</p> +<blockquote> +<p><strong>name HIGHLY undecided</strong></p> +</blockquote> +</blockquote> +</div> +<div class="section" id="rdzcoeffs-pol-expr"> +<h2><a class="toc-backref" href="#id71"><span class="docutils literal"><span class="pre">rdzcoeffs(<pol.</span> <span class="pre">expr.>)</span></span></a></h2> +<blockquote> +<p>This operates on the internal representation of the coefficients, +reducing them to lowest terms then extracting from numerator +or denominator the maximal power of ten to store as a decimal +exponent.</p> +<p>This is sometimes favourable to more efficient polynomial algebra +computations.</p> +<blockquote> +<p><strong>name HIGHLY undecided</strong></p> +</blockquote> +</blockquote> +</div> +<div class="section" id="diff1-pol-expr"> +<h2><a class="toc-backref" href="#id72"><span class="docutils literal"><span class="pre">diff1(<pol.</span> <span class="pre">expr.>)</span></span></a></h2> +<blockquote> +<p>The first derivative.</p> +<blockquote> +<p><strong>name UNSTABLE</strong></p> +<p>This name may be used in future to be the partial derivative with +respect to a first variable.</p> +</blockquote> +</blockquote> +</div> +<div class="section" id="diff2-pol-expr"> +<h2><a class="toc-backref" href="#id73"><span class="docutils literal"><span class="pre">diff2(<pol.</span> <span class="pre">expr.>)</span></span></a></h2> +<blockquote> +<p>The second derivative.</p> +<blockquote> +<p><strong>name UNSTABLE</strong></p> +<p>This name may be used in future to be the partial derivative with +respect to a second variable.</p> +</blockquote> +</blockquote> +</div> +<div class="section" id="diffn-pol-expr-p-num-expr-n"> +<h2><a class="toc-backref" href="#id74"><span class="docutils literal"><span class="pre">diffn(<pol.</span> expr. P>, <num. expr. n>)</span></a></h2> +<blockquote> +<p>The <span class="docutils literal">n</span>th derivative of <span class="docutils literal">P</span>. For <span class="docutils literal">n<0</span> computes iterated primitives +vanishing at the origin.</p> +<p>The coefficients are not reduced to lowest terms.</p> +<blockquote> +<p><strong>name and syntax UNSTABLE</strong></p> +<p>I am also considering reversing the order of the arguments.</p> +</blockquote> +</blockquote> +</div> +<div class="section" id="antider-pol-expr-p"> +<h2><a class="toc-backref" href="#id75"><span class="docutils literal"><span class="pre">antider(<pol.</span> expr. P>)</span></a></h2> +<blockquote> +<p>The primitive of <span class="docutils literal">P</span> with no constant term. Same as <span class="docutils literal"><span class="pre">diffn(P,-1)</span></span>.</p> +</blockquote> +</div> +<div class="section" id="intfrom-pol-expr-p-pol-expr-c"> +<h2><a class="toc-backref" href="#id76"><span class="docutils literal"><span class="pre">intfrom(<pol.</span> expr. P>, <pol. expr. c>)</span></a></h2> +<blockquote> +<p>The primitive of <span class="docutils literal">P</span> vanishing at <span class="docutils literal">c</span>, i.e. <span class="docutils literal">\int_c^x P(t)dt</span>.</p> +<p>Also <span class="docutils literal">c</span> can be a polynomial... so if <span class="docutils literal">c</span> is monomial <span class="docutils literal">x</span> +this will give zero!</p> +<blockquote> +<p><strong>UNSTABLE</strong></p> +<p>Allowing general polynomial variable for <span class="docutils literal">c</span> adds a bit of +overhead to the case of a pure scalar. So I am hesitating +maintaining this feature whose interest appears dubious.</p> +</blockquote> +</blockquote> +</div> +<div class="section" id="integral-pol-expr-p-pol-expr-a-pol-expr-b"> +<h2><a class="toc-backref" href="#id77"><span class="docutils literal"><span class="pre">integral(<pol.</span> expr. P>, [<pol. expr. a>, <pol. expr. <span class="pre">b>])</span></span></a></h2> +<blockquote> +<p><span class="docutils literal">\int_a^b P(t)dt</span>.</p> +<p>The brackets here are not denoting an optional argument +but a <em>mandatory</em> nutple argument <span class="docutils literal">[a, b]</span> with <em>two items</em>.</p> +<p><span class="docutils literal">a</span> and <span class="docutils literal">b</span> are not restricted to be scalars, they can be +polynomials.</p> +<blockquote> +<p>To compute <span class="docutils literal"><span class="pre">\int_{x-1}^x</span> P(t)dt</span> it is more efficient to use +<span class="docutils literal"><span class="pre">intfrom(x-1)</span></span>.</p> +<p>Similary to compute <span class="docutils literal"><span class="pre">\int_x^{x+1}</span> P(t)dt</span>, use <span class="docutils literal"><span class="pre">-intfrom(x+1)</span></span>.</p> +<p><strong>UNSTABLE</strong></p> +<p>Am I right to allow general polynomials <span class="docutils literal">a</span> and <span class="docutils literal">b</span> hence add +overhead to the pure scalar case ?</p> +</blockquote> +</blockquote> +</div> +</div> +<div class="section" id="examples-of-localization-of-roots"> +<h1><a class="toc-backref" href="#id78">Examples of localization of roots</a></h1> +<div class="admonition note"> +<p class="admonition-title">Note</p> +<p>As of <span class="docutils literal">0.8</span>, <span class="docutils literal">polexpr</span> is usable with Plain TeX and not only with +LaTeX, the examples of this section have been converted to use a +syntax which (at least at time of writing, March 2021) works in both.</p> +<p>This is done in order for the examples to be easy to copy-paste to +documents using either macro format.</p> +</div> +<ul> +<li><p>To make printed decimal numbers more enjoyable than via +<span class="docutils literal">\xintSignedFrac</span> (or <span class="docutils literal">\xintSignedFwOver</span> with Plain):</p> +<pre class="literal-block">\def\PolTypesetOne#1{\PolDecToString{\xintREZ{#1}}}%</pre> +<p><span class="docutils literal">\PolDecToString</span> will use decimal notation to incorporate the power +of ten part; and the <span class="docutils literal">\xintREZ</span> will have the effect to suppress +trailing zeros if present in raw numerator (if those digits end up +after decimal mark.) Notice that the above are expandable macros and +that one can also do:</p> +<pre class="literal-block">\def\PolToExprCmd#1{\PolDecToString{\xintREZ{#1}}}%</pre> +<p>to modify output of <a class="reference internal" href="#poltoexpr-pol-expr">\PolToExpr{<pol. expr.>}</a>.</p> +</li> +<li><p>For extra info in log file use <span class="docutils literal">\xintverbosetrue</span>.</p></li> +</ul> +<div class="section" id="a-typical-example"> +<h2><a class="toc-backref" href="#id79">A typical example</a></h2> +<p>In this example the polynomial is square-free.</p> +<pre class="literal-block">\poldef f(x) := x^7 - x^6 - 2x + 1; + +\PolToSturm{f}{f} +\PolSturmIsolateZeros{f} +The \PolTypeset{f} polynomial has \PolSturmNbOfIsolatedZeros{f} distinct real +roots which are located in the following intervals: +\PolPrintIntervals{f} +Here is the second root with ten more decimal digits: +\PolRefineInterval[10]{f}{2} +$$\PolSturmIsolatedZeroLeft{f}{2}<Z_2<\PolSturmIsolatedZeroRight{f}{2}$$ +And here is the first root with twenty digits after decimal mark: +\PolEnsureIntervalLength{f}{1}{-20} +$$\PolSturmIsolatedZeroLeft{f}{1}<Z_1<\PolSturmIsolatedZeroRight{f}{1}$$ +The first element of the Sturm chain has degree $\PolDegree{f_0}$. As +this is the original degreee $\PolDegree{f}$ we know that $f$ is square free. +Its derivative is up to a constant \PolTypeset{f_1} (in this example +it is identical with it). +\PolToSturm{f_1}{f_1}\PolSturmIsolateZeros{f_1}% +The derivative has \PolSturmNbOfIsolatedZeros{f_1} distinct real +roots: +\PolPrintIntervals[W]{f_1} +\PolEnsureIntervalLengths{f_1}{-10}% +Here they are with ten digits after decimal mark: +\PolPrintIntervals[W]{f_1} +\PolDiff{f_1}{f''} +\PolToSturm{f''}{f''} +\PolSturmIsolateZeros{f''} +The second derivative is \PolTypeset{f''}. +It has \PolSturmNbOfIsolatedZeros{f''} distinct real +roots: +\PolPrintIntervals[X]{f''} +Here is the positive one with 20 digits after decimal mark: +\PolEnsureIntervalLength{f''}{2}{-20}% +$$X_2 = \PolSturmIsolatedZeroLeft{f''}{2}\dots$$ +The more mathematically advanced among our dear readers will be able +to give the exact value for $X_2$!</pre> +</div> +<div class="section" id="a-degree-four-polynomial-with-nearby-roots"> +<h2><a class="toc-backref" href="#id80">A degree four polynomial with nearby roots</a></h2> +<p>Notice that this example is a bit outdated as <span class="docutils literal">0.7</span> release has +added <span class="docutils literal"><span class="pre">\PolSturmIsolateZeros**{sturmname}</span></span> which would find exactly +the roots. The steps here retain their interest when one is interested +in finding isolating intervals for example to prepare some demonstration +of dichotomy method.</p> +<pre class="literal-block">\PolDef{Q}{(x-1.050001)(x-1.105001)(x-1.110501)(x-1.111051)} +\PolTypeset{Q} +\PolToSturm{Q}{Q} % it is allowed to use same prefix for Sturm chain +\PolSturmIsolateZeros{Q} +\PolPrintIntervals{Q} +% reports 1.0 < Z_1 < 1.1, 1.10 < Z_2 < 1.11, 1.110 < Z_3 < 1.111, and 1.111 < Z_4 < 1.112 +% but the above bounds do not allow minimizing separation between roots +% so we refine: +\PolRefineInterval*{Q}{1} +\PolRefineInterval*{Q}{2} +\PolRefineInterval*{Q}{3} +\PolRefineInterval*{Q}{4} +\PolPrintIntervals{Q} +% reports 1.05 < Z_1 < 1.06, 1.105 < Z_2 < 1.106, 1.1105 < Z_3 < 1.1106, +% and 1.11105 < Z_4 < 1.11106. +\PolEnsureIntervalLengths{Q}{-6} +\PolPrintIntervals{Q} +% of course finds here all roots exactly</pre> +</div> +<div class="section" id="the-degree-nine-polynomial-with-0-99-0-999-0-9999-as-triple-roots"> +<h2><a class="toc-backref" href="#id81">The degree nine polynomial with 0.99, 0.999, 0.9999 as triple roots</a></h2> +<pre class="literal-block">% define a user command (xinttools is loaded automatically by polexpr) +\def\showmultiplicities#1{% #1 = "sturmname" +\xintFor* ##1 in {\xintSeq{1}{\PolSturmNbOfIsolatedZeros{#1}}}\do{% + The multiplicity is \PolSturmIsolatedZeroMultiplicity{#1}{##1} + \PolSturmIfZeroExactlyKnown{#1}{##1}% + {at the root $x=\PolSturmIsolatedZeroLeft{#1}{##1}$} + {for the root such that + $\PolSturmIsolatedZeroLeft{#1}{##1}<x<\PolSturmIsolatedZeroRight{#1}{##1}$} + \par +}}% +\PolDef{f}{(x-0.99)^3(x-0.999)^3(x-0.9999)^3} +\def\PolTypesetOne#1{\PolDecToString{\xintREZ{#1}}} +\PolTypeset{f}\par +\PolToSturm{f}{f}% it is allowed to use "polname" as "sturmname" too +\PolSturmIsolateZerosAndGetMultiplicities{f}% use the "sturmname" here +% or \PolSturmIsolateZeros*{f} which is exactly the same, but shorter.. + +\showmultiplicities{f}</pre> +<p>In this example, the output will look like this (but using math mode):</p> +<pre class="literal-block">x^9 - 8.9667x^8 + 35.73400293x^7 - 83.070418400109x^6 + 124.143648875193123x^5 +- 123.683070924326075877x^4 + 82.149260397553075617891x^3 +- 35.07602992699900159127007x^2 + 8.7364078733314648368671733x +- 0.967100824643585986488103299 + +The multiplicity is 3 at the root x = 0.99 +The multiplicity is 3 at the root x = 0.999 +The multiplicity is 3 at the root x = 0.9999</pre> +<p>On first pass, these rational roots were found (due to their relative +magnitudes, using <span class="docutils literal">\PolSturmIsolateZeros**</span> was not needed here). But +multiplicity computation works also with (decimal) roots not yet +identified or with non-decimal or irrational roots.</p> +<p>It is fun to modify only a tiny bit the polynomial and see if polexpr +survives:</p> +<pre class="literal-block">\PolDef{g}{f(x)+1e-27} +\PolTypeset{g}\par +\PolToSturm{g}{g} +\PolSturmIsolateZeros*{g} + +\showmultiplicities{g}</pre> +<p>This produces:</p> +<pre class="literal-block">x^9 - 8.9667x^8 + 35.73400293x^7 - 83.070418400109x^6 + 124.143648875193123x^5 +- 123.683070924326075877x^4 + 82.149260397553075617891x^3 +- 35.07602992699900159127007x^2 + 8.7364078733314648368671733x +- 0.967100824643585986488103298 + +The multiplicity is 1 for the root such that 0.98 < x < 0.99 +The multiplicity is 1 for the root such that 0.9991 < x < 0.9992 +The multiplicity is 1 for the root such that 0.9997 < x < 0.9998</pre> +<p>Which means that the multiplicity-3 roots each became a real and a pair of +complex ones. Let's see them better:</p> +<pre class="literal-block">\PolEnsureIntervalLengths{g}{-10} + +\showmultiplicities{g}</pre> +<p>which produces:</p> +<pre class="literal-block">The multiplicity is 1 for the root such that 0.9899888032 < x < 0.9899888033 +The multiplicity is 1 for the root such that 0.9991447980 < x < 0.9991447981 +The multiplicity is 1 for the root such that 0.9997663986 < x < 0.9997663987</pre> +</div> +<div class="section" id="a-degree-five-polynomial-with-three-rational-roots"> +<h2><a class="toc-backref" href="#id82">A degree five polynomial with three rational roots</a></h2> +<pre class="literal-block">\poldef Q(x) := 1581755751184441 x^5 + -14907697165025339 x^4 + +48415668972339336 x^3 + -63952057791306264 x^2 + +46833913221154895 x + -49044360626280925; + +\PolToSturm{Q}{Q} + \def\PolTypesetCmdPrefix#1{\allowbreak\xintiiifSgn{#1}{}{+}{+}}% + $Q_0(x) = \PolTypeset{Q_0}$ +\PolSturmIsolateZeros**{Q} +\PolPrintIntervals{Q} + +$Q_{norr}(x) = \PolTypeset{Q_norr}$</pre> +<p>Here, all real roots are rational:</p> +<pre class="literal-block">Z_1 = 833719/265381 +Z_2 = 165707065/52746197 +Z_3 = 355/113 + +Q_norr(x) = x^2 + 1</pre> +<p>And let's get their decimal expansion too:</p> +<pre class="literal-block">% print decimal expansion of the found roots +\def\PolPrintIntervalsPrintExactZero + {\xintTrunc{20}{\PolPrintIntervalsTheLeftEndPoint}\dots} +\PolPrintIntervals{Q} + +Z_1 = 3.14159265358107777120... +Z_2 = 3.14159265358979340254... +Z_3 = 3.14159292035398230088...</pre> +</div> +<div class="section" id="a-mignotte-type-polynomial"> +<h2><a class="toc-backref" href="#id83">A Mignotte type polynomial</a></h2> +<pre class="literal-block">\PolDef{P}{x^10 - (10x-1)^2}% +\PolTypeset{P} % prints it in expanded form +\PolToSturm{P}{P} % we can use same prefix for Sturm chain +\PolSturmIsolateZeros{P} % finds 4 real roots +This polynomial has \PolSturmNbOfIsolatedZeros{P} distinct real roots: +\PolPrintIntervals{P}% +% reports -2 < Z_1 < -1, 0.09 < Z_2 < 0.10, 0.1 < Z_3 < 0.2, 1 < Z_4 < 2 +Let us refine the second and third intervals to separate the corresponding +roots: +\PolRefineInterval*{P}{2}% will refine to 0.0999990 < Z_2 < 0.0999991 +\PolRefineInterval*{P}{3}% will refine to 0.100001 < Z_3 < 0.100002 +\PolPrintIntervals{P}% +Let us now get to know all roots with 10 digits after decimal mark: +\PolEnsureIntervalLengths{P}{-10}% +\PolPrintIntervals{P}% now all roots are known 10 decimal digits after mark +Finally, we display 20 digits of the second root: +\PolEnsureIntervalLength{P}{2}{-20}% makes Z_2 known with 20 digits after mark +$$\PolSturmIsolatedZeroLeft{P}{2}<Z_2<\PolSturmIsolatedZeroRight{P}{2}$$</pre> +<p>The last line produces:</p> +<pre class="literal-block">0.09999900004999650028 < Z_2 < 0.09999900004999650029</pre> +</div> +<div class="section" id="the-wilkinson-polynomial"> +<h2><a class="toc-backref" href="#id84">The Wilkinson polynomial</a></h2> +<p>See <a class="reference external" href="https://en.wikipedia.org/wiki/Wilkinson%27s_polynomial">Wilkinson polynomial</a>.</p> +<pre class="literal-block">%\xintverbosetrue % for the curious... + +\poldef f(x) := mul((x - i), i = 1..20); + +\def\PolTypesetCmdPrefix#1{\allowbreak\xintiiifSgn{#1}{}{+}{+}}% +\def\PolTypesetOne#1{\xintDecToString{#1}}% + +\noindent\PolTypeset{f} + +\PolToSturm{f}{f} +\PolSturmIsolateZeros{f} +\PolPrintIntervals{f} + +% \vfill\eject + +% This page is commented out because it takes about 30s on a 2GHz CPU +% \poldef g(x) := f(x) - 2**{-23} x**19; + +% \PolToSturm{g}{g} +% \noindent\PolTypeset{g_0}% integer coefficient primitive polynomial + +% \PolSturmIsolateZeros{g} +% \PolEnsureIntervalLengths{g}{-10} + +% \let\PolPrintIntervalsPrintMultiplicity\empty +% \PolPrintIntervals*{g}</pre> +<p>The first polynomial:</p> +<pre class="literal-block">f(x) = x**20 +- 210 x**19 ++ 20615 x**18 +- 1256850 x**17 ++ 53327946 x**16 +- 1672280820 x**15 ++ 40171771630 x**14 +- 756111184500 x**13 ++ 11310276995381 x**12 +- 135585182899530 x**11 ++ 1307535010540395 x**10 +- 10142299865511450 x**9 ++ 63030812099294896 x**8 +- 311333643161390640 x**7 ++ 1206647803780373360 x**6 +- 3599979517947607200 x**5 ++ 8037811822645051776 x**4 +- 12870931245150988800 x**3 ++ 13803759753640704000 x**2 +- 8752948036761600000 x ++ 2432902008176640000</pre> +<p>is handled fast enough, but the modified one <span class="docutils literal">f(x) - <span class="pre">2**-23</span> <span class="pre">x**19</span></span> takes about 20x longer.</p> +<p>The Sturm chain polynomials +have integer coefficients with up to 321 digits, whereas (surprisingly +perhaps) those of the Sturm chain polynomials derived from <span class="docutils literal">f</span> never +have more than 21 digits ...</p> +<p>Once the Sturm chain is computed and the zeros isolated, obtaining their +decimal digits is relatively faster. Here is for the ten real roots of +<span class="docutils literal">f(x) - <span class="pre">2**-23</span> <span class="pre">x**19</span></span> as computed by the code above:</p> +<pre class="literal-block">Z_1 = 0.9999999999... +Z_2 = 2.0000000000... +Z_3 = 2.9999999999... +Z_4 = 4.0000000002... +Z_5 = 4.9999999275... +Z_6 = 6.0000069439... +Z_7 = 6.9996972339... +Z_8 = 8.0072676034... +Z_9 = 8.9172502485... +Z_10 = 20.8469081014...</pre> +</div> +<div class="section" id="the-second-wilkinson-polynomial"> +<h2><a class="toc-backref" href="#id85">The second Wilkinson polynomial</a></h2> +<pre class="literal-block">\poldef f(x) := mul(x - 2^-i, i = 1..20); + +%\PolTypeset{f} + +\PolToSturm{f}{f} +\PolSturmIsolateZeros**{f} +\PolPrintIntervals{f}</pre> +<p>This takes more time than the polynomial with 1, 2, .., 20 as roots but +less than the latter modified by the <span class="docutils literal"><span class="pre">2**-23</span></span> tiny change to one of its +coefficient.</p> +<p>Here is the output (with release 0.7.2):</p> +<pre class="literal-block">Z_1 = 0.00000095367431640625 +Z_2 = 0.0000019073486328125 +Z_3 = 0.000003814697265625 +Z_4 = 0.00000762939453125 +Z_5 = 0.0000152587890625 +Z_6 = 0.000030517578125 +Z_7 = 0.00006103515625 +Z_8 = 0.0001220703125 +Z_9 = 1/4096 +Z_10 = 1/2048 +Z_11 = 1/1024 +Z_12 = 1/512 +Z_13 = 1/256 +Z_14 = 1/128 +Z_15 = 0.015625 +Z_16 = 0.03125 +Z_17 = 0.0625 +Z_18 = 0.125 +Z_19 = 0.25 +Z_20 = 0.5</pre> +<p>There is some incoherence in output format which has its source in the +fact that some roots are found in branches which can only find decimal +roots, whereas some are found in branches which could find general +fractions and they use <span class="docutils literal">\xintIrr</span> before storage of the found root. +This may evolve in future.</p> +</div> +<div class="section" id="the-degree-41-polynomial-with-2-1-9-1-8-0-0-1-1-9-2-as-roots"> +<h2><a class="toc-backref" href="#id86">The degree 41 polynomial with -2, -1.9, -1.8, ..., 0, 0.1, ..., 1.9, 2 as roots</a></h2> +<pre class="literal-block">\PolDef{P}{mul((x-i*1e-1), i=-20..20)}% i/10 is same but less efficient</pre> +<p>In the defining expression we could have used <span class="docutils literal">i/10</span> but this gives +less efficient internal form for the coefficients (the <span class="docutils literal">10</span>'s end up +in denominators).</p> +<p>Using <span class="docutils literal">\PolToExpr{P}</span> after having done</p> +<pre class="literal-block">\def\PolToExprCmd#1{\PolDecToString{\xintREZ{#1}}}</pre> +<p>we get this expanded form:</p> +<pre class="literal-block">x^41 +-28.7*x^39 ++375.7117*x^37 +-2975.11006*x^35 ++15935.28150578*x^33 +-61167.527674162*x^31 ++173944.259366417394*x^29 +-373686.963560544648*x^27 ++613012.0665016658846445*x^25 +-771182.31133138163125495*x^23 ++743263.86672885754888959569*x^21 +-545609.076599482896371978698*x^19 ++301748.325708943677229642930528*x^17 +-123655.8987669450434698869844544*x^15 ++36666.1782054884005855608205864192*x^13 +-7607.85821367459445649518380016128*x^11 ++1053.15135918687298508885950223794176*x^9 +-90.6380005918141132650786081964032*x^7 ++4.33701563847327366842552218288128*x^5 +-0.0944770968420804735498178265088*x^3 ++0.00059190121813899276854174416896*x</pre> +<p>which shows coefficients with up to 36 significant digits...</p> +<p>Stress test: not a hard challenge to <span class="docutils literal">xint + polexpr</span>, but be a bit +patient!</p> +<pre class="literal-block">\PolDef{P}{mul((x-i*1e-1), i=-20..20)}% +\PolToSturm{P}{S} % dutifully computes S_0, ..., S_{41} +% the [1] optional argument limits the search to interval (-10,10) +\PolSturmIsolateZeros[1]{S} % finds *exactly* (but a bit slowly) all 41 roots! +\PolPrintIntervals{S} % nice, isn't it?</pre> +<div class="admonition note"> +<p class="admonition-title">Note</p> +<p>Release <span class="docutils literal">0.5</span> has <em>experimental</em> addition of optional argument +<span class="docutils literal">E</span> to <span class="docutils literal">\PolSturmIsolateZeros</span>. It instructs to search roots only +in interval <span class="docutils literal"><span class="pre">(-10^E,</span> 10^E)</span>. Important: the extremities are +<em>assumed to not be roots</em>. In this example, the <span class="docutils literal">[1]</span> in +<span class="docutils literal"><span class="pre">\PolSturmIsolateZeros[1]{S}</span></span> gives some speed gain; without it, it +turns out in this case that <span class="docutils literal">polexpr</span> would have started with +<span class="docutils literal"><span class="pre">(-10^6,</span> 10^6)</span> interval.</p> +<p>Please note that this will probably get replaced in future by the +specification of a general interval. Do not rely on meaning of this +optional argument keeping the same.</p> +</div> +</div> +<div class="section" id="roots-of-chebyshev-polynomials"> +<h2><a class="toc-backref" href="#id87">Roots of Chebyshev polynomials</a></h2> +<pre class="literal-block">\newcount\mycount +\poldef T_0(x) := 1; +\poldef T_1(x) := x; +\mycount 2 +\xintloop + \poldef T_\the\mycount(x) := + 2x*T_\the\numexpr\mycount-1(x) + - T_\the\numexpr\mycount-2(x); +\ifnum\mycount<15 +\advance\mycount 1 +\repeat + +$$T_{15} = \PolTypeset[X]{T_15}$$ +\PolToSturm{T_15}{T_15} +\PolSturmIsolateZeros{T_15} +\PolEnsureIntervalLengths{T_15}{-10} +\PolPrintIntervals{T_15}</pre> +</div> +</div> +<div class="section" id="non-expandable-macros"> +<h1><a class="toc-backref" href="#id88">Non-expandable macros</a></h1> +<div class="admonition note"> +<p class="admonition-title">Note</p> +<p>At <span class="docutils literal">0.8</span> <span class="docutils literal">polexpr</span> is usable with Plain TeX and not only with +LaTeX. Some examples given in this section may be using LaTeX syntax +such as <span class="docutils literal">\renewcommand</span>. Convert to TeX primitives as appropriate +if testing with a non LaTeX macro format.</p> +</div> +<div class="section" id="poldef-polname-letter-expression-using-the-letter-as-indeterminate"> +<span id="poldef"></span><h2><a class="toc-backref" href="#id89"><span class="docutils literal">\poldef <span class="pre">polname(letter):=</span> expression using the letter as indeterminate;</span></a></h2> +<blockquote> +<p>This evaluates the <em>polynomial expression</em> and stores the +coefficients in a private structure accessible later via other +package macros, when used with argument the chosen <span class="docutils literal">polname</span>. Of +course the <em>expression</em> can use other previously defined +polynomials.</p> +<p>Polynomial names must start with a letter and are constituted of +letters, digits, underscores and the right tick <span class="docutils literal">'</span>.</p> +<p>The whole <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> syntax is authorized:</p> +<pre class="literal-block">\poldef mypol(z) := add((-1)^i z^(2i+1)/(2i+1)!, i = 0..10);</pre> +<p>With fractional coefficients, beware the <a class="reference internal" href="#warningtacit">tacit multiplication issue</a>.</p> +<p>Furthermore:</p> +<ul class="simple"> +<li><p>a variable <span class="docutils literal">mypol</span> is defined which can be used in <span class="docutils literal">\poldef</span> +as well as in <span class="docutils literal">\xinteval</span> for algebraic computations or as +argument to polynomial aware functions,</p></li> +<li><p>a function <span class="docutils literal">mypol()</span> is defined which can be used in <span class="docutils literal">\poldef</span> +as well as in <span class="docutils literal">\xinteval</span>. It accepts there as argument scalars +and also other polynomials (via their names, thanks to previous +item).</p></li> +</ul> +<p>Notice that any function defined via <span class="docutils literal">\xintdeffunc</span> and using +only algebraic operations (and ople indexing or slicing operations) +should work fine in <span class="docutils literal"><span class="pre">\xintexpr/\xinteval</span></span> with such polynomial +names as argument.</p> +<p>In the case of a constant polynomial, the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> variable (not the +internal data structure on which the package macros operate) +associated to it is indistinguishable from a scalar, it is actually +a scalar and has lost all traces from its origins as a polynomial +(so for example can be used as argument to the <span class="docutils literal">cos()</span> function). +<strong>THIS MAY CHANGE</strong></p> +<p>The <em>function</em> on the other hand remains a one-argument function, +which simply has a constant value.</p> +<div class="admonition attention"> +<p class="admonition-title">Attention!</p> +<p>The function <span class="docutils literal">mypol()</span> is defined <strong>only</strong> for +<span class="docutils literal"><span class="pre">\xintexpr/\xinteval</span></span> +context. It will be unknown to <span class="docutils literal">\xintfloateval</span>.</p> +<p>Worse, a +previously existing floating point function of the same name will +be let undefined again, to avoid hard to debug mismatches between +exact and floating point polynomials. This also applies when the +polynomial is produced not via <span class="docutils literal">\poldef</span> or <span class="docutils literal">\PolDef</span> but +as result of usage of the other package macros.</p> +<p>See <a class="reference internal" href="#polgenfloatvariant-polname">\PolGenFloatVariant{polname}</a> to generate a <strong>function</strong> +usable in <span class="docutils literal">\xintfloateval</span>. Such a function can only be +used with scalar input, see next warning.</p> +</div> +<div class="admonition attention"> +<p class="admonition-title">Attention!</p> +<p>Using the <strong>variable</strong> <span class="docutils literal">mypol</span> inside <span class="docutils literal">\xintfloateval</span> will +generate low-level errors because the infix operators there are +not polynomial-aware, and the polynomial specific functions such +as <span class="docutils literal">deg()</span> are only defined for usage inside <span class="docutils literal">\xintexpr</span>.</p> +<p>In short, currently polynomials defined via <span class="docutils literal">polexpr</span> can +be used in floating point context only for numerical evaluations, +via <strong>functions</strong> obtained from <a class="reference internal" href="#polgenfloatvariant-polname">\PolGenFloatVariant{polname}</a> +usage.</p> +<p>Changes to the original polynomial via package macros are not +automatically mapped to the numerical floating point evaluator +which must be manually updated as necessary when the original +rational coefficient polynomial is modified.</p> +<p><strong>THIS MAY CHANGE</strong></p> +</div> +<p>The original expression is lost after parsing, and in particular the +package provides no way to typeset it (of course the package +provides macros to typeset the computed polynomial). Typesetting +the original expression has to be done manually, if needed.</p> +</blockquote> +</div> +<div class="section" id="poldef-letter-polname-expression-using-the-letter-as-indeterminate"> +<span id="id8"></span><h2><a class="toc-backref" href="#id90"><span class="docutils literal"><span class="pre">\PolDef[letter]{polname}{expression</span> using the letter as indeterminate}</span></a></h2> +<blockquote> +<p>Does the same as <a class="reference internal" href="#poldef">\poldef</a> in an undelimited macro +format (thus avoiding potential problems with the catcode of the +semi-colon in presence of some packages.) In absence of the +<span class="docutils literal">[letter]</span> optional argument, the variable is assumed to be <span class="docutils literal">x</span>.</p> +</blockquote> +</div> +<div class="section" id="polgenfloatvariant-polname"> +<span id="polgenfloatvariant"></span><h2><a class="toc-backref" href="#id91"><span class="docutils literal">\PolGenFloatVariant{polname}</span></a></h2> +<blockquote> +<p>Makes the polynomial also usable in the <span class="docutils literal">\xintfloatexpr</span> parser. +It will therein evaluates via an Horner scheme with coefficients +already pre-rounded to the float precision.</p> +<p>See also <a class="reference internal" href="#poltofloatexpr-pol-expr">\PolToFloatExpr{<pol. expr.>}</a>.</p> +<div class="admonition attention"> +<p class="admonition-title">Attention!</p> +<p>Any operation, for example generating the derivative polynomial, +or dividing two polynomials or using the <span class="docutils literal">\PolLet</span>, <strong>must</strong> be +followed by explicit usage of <span class="docutils literal">\PolGenFloatVariant{polname}</span> if +the new polynomial is to be used in <span class="docutils literal">\xintfloateval</span> <strong>as a +function</strong>.</p> +</div> +</blockquote> +</div> +<div class="section" id="pollet-polname-2-polname-1"> +<span id="pollet"></span><h2><a class="toc-backref" href="#id92"><span class="docutils literal"><span class="pre">\PolLet{polname_2}={polname_1}</span></span></a></h2> +<blockquote> +<p>Makes a copy of the already defined polynomial <span class="docutils literal">polname_1</span> to a +new one <span class="docutils literal">polname_2</span>. Same effect as +<span class="docutils literal"><span class="pre">\PolDef{polname_2}{polname_1(x)}</span></span> but with less overhead. The +<span class="docutils literal">=</span> is optional.</p> +</blockquote> +</div> +<div class="section" id="polgloballet-polname-2-polname-1"> +<span id="polgloballet"></span><h2><a class="toc-backref" href="#id93"><span class="docutils literal"><span class="pre">\PolGlobalLet{polname_2}={polname_1}</span></span></a></h2> +<blockquote> +<p>Acts globally.</p> +</blockquote> +</div> +<div class="section" id="polassign-polname-toarray-macro"> +<span id="polassign"></span><h2><a class="toc-backref" href="#id94"><span class="docutils literal"><span class="pre">\PolAssign{polname}\toarray\macro</span></span></a></h2> +<blockquote> +<p>Defines a one-argument expandable macro <span class="docutils literal"><span class="pre">\macro{#1}</span></span> which expands +to the (raw) #1th polynomial coefficient.</p> +<ul class="simple"> +<li><p>Attention, coefficients here are indexed starting at 1.</p></li> +<li><p>With #1=-1, -2, ..., <span class="docutils literal"><span class="pre">\macro{#1}</span></span> returns leading coefficients.</p></li> +<li><p>With #1=0, returns the number of coefficients, i.e. <span class="docutils literal">1 + deg f</span> +for non-zero polynomials.</p></li> +<li><p>Out-of-range #1's return <span class="docutils literal">0/1[0]</span>.</p></li> +</ul> +<p>See also <a class="reference internal" href="#polnthcoeff-polname-number">\PolNthCoeff{polname}{number}</a>. The main difference is that +with <span class="docutils literal">\PolAssign</span>, <span class="docutils literal">\macro</span> is made a prefix to <span class="docutils literal">1 + deg f</span> +already defined (hidden to user) macros holding individually the +coefficients but <a class="reference internal" href="#polnthcoeff-polname-number">\PolNthCoeff{polname}{number}</a> does each time the job +to expandably recover the <span class="docutils literal">Nth</span> coefficient, and due to +expandability can not store it in a macro for future usage (of course, +it can be an argument in an <span class="docutils literal">\edef</span>.) The other difference +is the shift by one in indexing, mentioned above (negative +indices act the same in both.)</p> +</blockquote> +</div> +<div class="section" id="polget-polname-fromarray-macro"> +<span id="polget"></span><h2><a class="toc-backref" href="#id95"><span class="docutils literal"><span class="pre">\PolGet{polname}\fromarray\macro</span></span></a></h2> +<blockquote> +<p>Does the converse operation to +<span class="docutils literal"><span class="pre">\PolAssign{polname}\toarray\macro</span></span>. Each individual +<span class="docutils literal">\macro{number}</span> gets expanded in an <span class="docutils literal">\edef</span> and then normalized +via <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a>'s macro <span class="docutils literal">\xintRaw</span>.</p> +<p>The leading zeros are removed from the polynomial.</p> +<p>(contrived) Example:</p> +<pre class="literal-block">\xintAssignArray{1}{-2}{5}{-3}\to\foo +\PolGet{f}\fromarray\foo</pre> +<p>This will define <span class="docutils literal">f</span> as would have <span class="docutils literal">\poldef <span class="pre">f(x):=1-2x+5x^2-3x^3;</span></span>.</p> +<div class="admonition note"> +<p class="admonition-title">Note</p> +<p>Prior to <span class="docutils literal">0.5</span>, coefficients were not normalized via +<span class="docutils literal">\xintRaw</span> for internal storage.</p> +</div> +</blockquote> +</div> +<div class="section" id="polfromcsv-polname-csv"> +<span id="polfromcsv"></span><h2><a class="toc-backref" href="#id96"><span class="docutils literal"><span class="pre">\PolFromCSV{polname}{<csv>}</span></span></a></h2> +<blockquote> +<p>Defines a polynomial directly from the comma separated list of values +(or a macro expanding to such a list) of its coefficients, the <em>first +item</em> gives the constant term, the <em>last item</em> gives the leading +coefficient, except if zero, then it is dropped (iteratively). List +items are each expanded in an <span class="docutils literal">\edef</span> and then put into normalized +form via <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a>'s macro <span class="docutils literal">\xintRaw</span>.</p> +<p>As leading zero coefficients are removed:</p> +<pre class="literal-block">\PolFromCSV{f}{0, 0, 0, 0, 0, 0, 0, 0, 0, 0}</pre> +<p>defines the zero polynomial, which holds only one coefficient.</p> +<p>See also expandable macro <a class="reference internal" href="#poltocsv-polname">\PolToCSV</a>.</p> +<div class="admonition note"> +<p class="admonition-title">Note</p> +<p>Prior to <span class="docutils literal">0.5</span>, coefficients were not normalized via +<span class="docutils literal">\xintRaw</span> for internal storage.</p> +</div> +</blockquote> +</div> +<div class="section" id="poltypeset-pol-expr"> +<span id="poltypeset"></span><h2><a class="toc-backref" href="#id97"><span class="docutils literal"><span class="pre">\PolTypeset{<pol.</span> <span class="pre">expr.>}</span></span></a></h2> +<blockquote> +<p>Typesets in descending powers, switching to math mode if in text +mode, after evaluating the polynomial expression:</p> +<pre class="literal-block">\PolTypeset{mul(x-i,i=1..5)}% possible since polexpr 0.8</pre> +<p>The letter used in the input expression is by default <span class="docutils literal">x</span>, +but can be modified by a redefinition of <a class="reference internal" href="#poltoexprinvar">\PolToExprInVar</a>.</p> +<p>It uses also by default the letter <span class="docutils literal">x</span> on output but this one can +be changed via an optional argument:</p> +<pre class="literal-block">\PolTypeset[z]{polname or polynomial expression}</pre> +<p>By default zero coefficients are skipped (use <span class="docutils literal">\poltypesetalltrue</span> +to get all of them in output).</p> +<p>The following macros (whose meanings will be found in the package code) +can be re-defined for customization. Their default definitions are +expandable, but this is not a requirement.</p> +</blockquote> +<div class="section" id="poltypesetcmd-raw-coeff"> +<span id="poltypesetcmd"></span><h3><a class="toc-backref" href="#id98"><span class="docutils literal">\PolTypesetCmd{raw_coeff}</span></a></h3> +<blockquote> +<p>Checks if the coefficient is <span class="docutils literal">1</span> or <span class="docutils literal"><span class="pre">-1</span></span> and then skips printing +the <span class="docutils literal">1</span>, except for the constant term. Also it sets conditional +<a class="reference internal" href="#polifcoeffisplusorminusone-a-b">\PolIfCoeffIsPlusOrMinusOne{A}{B}</a>.</p> +<p>The actual printing of the coefficients, when not equal to plus or +minus one is handled by <a class="reference internal" href="#poltypesetone-raw-coeff">\PolTypesetOne{raw_coeff}</a>.</p> +</blockquote> +</div> +<div class="section" id="poltypesetone-raw-coeff"> +<span id="poltypesetone"></span><h3><a class="toc-backref" href="#id99"><span class="docutils literal">\PolTypesetOne{raw_coeff}</span></a></h3> +<blockquote> +<p>Defaults to <span class="docutils literal">\xintSignedFrac</span> (LaTeX) or <span class="docutils literal">\xintSignedFwOver</span> +(else). But these <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> very old legacy macros are a bit +annoyin as they insist in exhibiting a power of ten rather than +using simpler decimal notation.</p> +<p>As alternative one can do things such as:</p> +<pre class="literal-block">\def\PolTypesetOne#1{\xintDecToString{\xintREZ{#1}}} +% or with LaTeX+siunitx for example +\renewcommand\PolTypesetOne[1]{\num{\xintPFloat[5]{#1}}} +% (as \num of siunitx understands floating point notation) +\renewcommand\PolTypesetOne[1]{\num{\xintRound{4}{#1}}}</pre> +</blockquote> +</div> +<div class="section" id="id9"> +<span id="poltypesetmonomialcmd"></span><h3><a class="toc-backref" href="#id100"><span class="docutils literal">\PolTypesetMonomialCmd</span></a></h3> +<blockquote> +<p>This decides how a monomial (in variable <span class="docutils literal">\PolVar</span> and with +exponent <span class="docutils literal">\PolIndex</span>) is to be printed. The default does nothing +for the constant term, <span class="docutils literal">\PolVar</span> for the first degree and +<span class="docutils literal"><span class="pre">\PolVar^{\PolIndex}</span></span> for higher degrees monomials. Beware that +<span class="docutils literal">\PolIndex</span> expands to digit tokens and needs termination in +<span class="docutils literal">\ifnum</span> tests.</p> +</blockquote> +</div> +<div class="section" id="poltypesetcmdprefix-raw-coeff"> +<span id="poltypesetcmdprefix"></span><h3><a class="toc-backref" href="#id101"><span class="docutils literal">\PolTypesetCmdPrefix{raw_coeff}</span></a></h3> +<blockquote> +<p>Expands to a <span class="docutils literal">+</span> if the <span class="docutils literal">raw_coeff</span> is zero or positive, and to +nothing if <span class="docutils literal">raw_coeff</span> is negative, as in latter case the +<span class="docutils literal">\xintSignedFrac</span> (or <span class="docutils literal">\xintSignedFwOver</span>) used by +<a class="reference internal" href="#poltypesetcmd-raw-coeff">\PolTypesetCmd{raw_coeff}</a> will put the <span class="docutils literal">-</span> sign in front of +the fraction (if it is a fraction) and this will thus serve as +separator in the typeset formula. Not used for the first term.</p> +</blockquote> +</div> +</div> +<div class="section" id="id11"> +<span id="id10"></span><h2><a class="toc-backref" href="#id102"><span class="docutils literal"><span class="pre">\PolTypeset*{<pol.</span> <span class="pre">expr.>}</span></span></a></h2> +<blockquote> +<p>Typesets in ascending powers. Use e.g. <span class="docutils literal">[h]</span> optional argument +(after the <span class="docutils literal">*</span>) to use letter <span class="docutils literal">h</span> rather than <span class="docutils literal">x</span>.</p> +<p>Extended at <span class="docutils literal">0.8</span> to accept general expressions and not only +polynomial names. Redefine <a class="reference internal" href="#poltoexprinvar">\PolToExprInVar</a> to use in the +expression another letter than default <span class="docutils literal">x</span>.</p> +</blockquote> +</div> +<div class="section" id="poldiff-polname-1-polname-2"> +<span id="poldiff"></span><h2><a class="toc-backref" href="#id103"><span class="docutils literal"><span class="pre">\PolDiff{polname_1}{polname_2}</span></span></a></h2> +<blockquote> +<p>This sets <span class="docutils literal">polname_2</span> to the first derivative of <span class="docutils literal">polname_1</span>. It +is allowed to issue <span class="docutils literal"><span class="pre">\PolDiff{f}{f}</span></span>, effectively replacing <span class="docutils literal">f</span> +by <span class="docutils literal">f'</span>.</p> +<p>Coefficients of the result <span class="docutils literal">polname_2</span> are irreducible fractions +(see <a class="reference internal" href="#technicalities">Technicalities</a> for the whole story.)</p> +</blockquote> +</div> +<div class="section" id="poldiff-n-polname-1-polname-2"> +<span id="poldiff-n"></span><h2><a class="toc-backref" href="#id104"><span class="docutils literal"><span class="pre">\PolDiff[N]{polname_1}{polname_2}</span></span></a></h2> +<blockquote> +<p>This sets <span class="docutils literal">polname_2</span> to the <span class="docutils literal">N</span>-th derivative of <span class="docutils literal">polname_1</span>. +Identical arguments is allowed. With <span class="docutils literal">N=0</span>, same effect as +<span class="docutils literal"><span class="pre">\PolLet{polname_2}={polname_1}</span></span>. With negative <span class="docutils literal">N</span>, switches to +using <span class="docutils literal">\PolAntiDiff</span>.</p> +</blockquote> +</div> +<div class="section" id="polantidiff-polname-1-polname-2"> +<span id="polantidiff"></span><h2><a class="toc-backref" href="#id105"><span class="docutils literal"><span class="pre">\PolAntiDiff{polname_1}{polname_2}</span></span></a></h2> +<blockquote> +<p>This sets <span class="docutils literal">polname_2</span> to the primitive of <span class="docutils literal">polname_1</span> vanishing +at zero.</p> +<p>Coefficients of the result <span class="docutils literal">polname_2</span> are irreducible fractions +(see <a class="reference internal" href="#technicalities">Technicalities</a> for the whole story.)</p> +</blockquote> +</div> +<div class="section" id="polantidiff-n-polname-1-polname-2"> +<span id="polantidiff-n"></span><h2><a class="toc-backref" href="#id106"><span class="docutils literal"><span class="pre">\PolAntiDiff[N]{polname_1}{polname_2}</span></span></a></h2> +<blockquote> +<p>This sets <span class="docutils literal">polname_2</span> to the result of <span class="docutils literal">N</span> successive integrations on +<span class="docutils literal">polname_1</span>. With negative <span class="docutils literal">N</span>, it switches to using <span class="docutils literal">\PolDiff</span>.</p> +</blockquote> +</div> +<div class="section" id="poldivide-polname-1-polname-2-polname-q-polname-r"> +<span id="poldivide"></span><h2><a class="toc-backref" href="#id107"><span class="docutils literal"><span class="pre">\PolDivide{polname_1}{polname_2}{polname_Q}{polname_R}</span></span></a></h2> +<blockquote> +<p>This sets <span class="docutils literal">polname_Q</span> and <span class="docutils literal">polname_R</span> to be the quotient and +remainder in the Euclidean division of <span class="docutils literal">polname_1</span> by +<span class="docutils literal">polname_2</span>.</p> +</blockquote> +</div> +<div class="section" id="polquo-polname-1-polname-2-polname-q"> +<span id="polquo"></span><h2><a class="toc-backref" href="#id108"><span class="docutils literal"><span class="pre">\PolQuo{polname_1}{polname_2}{polname_Q}</span></span></a></h2> +<blockquote> +<p>This sets <span class="docutils literal">polname_Q</span> to be the quotient in the Euclidean division +of <span class="docutils literal">polname_1</span> by <span class="docutils literal">polname_2</span>.</p> +</blockquote> +</div> +<div class="section" id="polrem-polname-1-polname-2-polname-r"> +<span id="polrem"></span><h2><a class="toc-backref" href="#id109"><span class="docutils literal"><span class="pre">\PolRem{polname_1}{polname_2}{polname_R}</span></span></a></h2> +<blockquote> +<p>This sets <span class="docutils literal">polname_R</span> to be the remainder in the Euclidean division +of <span class="docutils literal">polname_1</span> by <span class="docutils literal">polname_2</span>.</p> +</blockquote> +</div> +<div class="section" id="polgcd-polname-1-polname-2-polname-gcd"> +<span id="polgcd"></span><h2><a class="toc-backref" href="#id110"><span class="docutils literal"><span class="pre">\PolGCD{polname_1}{polname_2}{polname_GCD}</span></span></a></h2> +<blockquote> +<p>This sets <span class="docutils literal">polname_GCD</span> to be the (monic) GCD of the two first +polynomials. It is a unitary polynomial except if both <span class="docutils literal">polname_1</span> +and <span class="docutils literal">polname_2</span> vanish, then <span class="docutils literal">polname_GCD</span> is the zero +polynomial.</p> +</blockquote> +</div> +<div class="section" id="non-expandable-macros-related-to-the-root-localization-routines"> +<h2><a class="toc-backref" href="#id111">Non-expandable macros related to the root localization routines</a></h2> +<div class="section" id="poltosturm-polname-sturmname"> +<span id="poltosturm"></span><h3><a class="toc-backref" href="#id112"><span class="docutils literal"><span class="pre">\PolToSturm{polname}{sturmname}</span></span></a></h3> +<blockquote> +<p>With <span class="docutils literal">polname</span> being for example <span class="docutils literal">P</span>, the macro starts by +computing polynomials <span class="docutils literal">P</span> and <span class="docutils literal">P'</span>, then computes the (opposite +of the) remainder in euclidean division, iteratively.</p> +<p>The last non-zero remainder <span class="docutils literal">P_N_</span> (where <span class="docutils literal">N</span> is obtainable as +<a class="reference internal" href="#polsturmchainlength-sturmname">\PolSturmChainLength{sturmname}</a>) is up to a factor +the GCD of <span class="docutils literal">P</span> and <span class="docutils literal">P'</span> hence it is a constant if and only if +<span class="docutils literal">P</span> is square-free.</p> +<div class="admonition note"> +<p class="admonition-title">Note</p> +<ul class="simple"> +<li><p>Since <span class="docutils literal">0.5</span> all these polynomials are divided by their rational +content, so they have integer coefficients with no common factor, +and the last one if a constant is either <span class="docutils literal">1</span> or <span class="docutils literal"><span class="pre">-1</span></span>.</p></li> +<li><p>After this normalization to primitive polynomials, they are +stored internally as <span class="docutils literal">sturmname_k_</span>, <span class="docutils literal">k=0,1, ...</span>.</p></li> +<li><p>These polynomials are used internally only. To keep them as +genuine declared polynomials also after the macro call, use the +starred variant <a class="reference internal" href="#id12">PolToSturm*</a>.</p></li> +</ul> +</div> +<div class="admonition note"> +<p class="admonition-title">Note</p> +<p>It is perfectly allowed to use the polynomial name as Sturm chain name: +<span class="docutils literal"><span class="pre">\PolToSturm{f}(f}</span></span>.</p> +</div> +<p>The macro then declares <span class="docutils literal">sturmname_0</span>, <span class="docutils literal">sturmname_1</span>, ..., which are +the (non-declared) <span class="docutils literal">sturmname_k_</span> divided by the last one. Division is +not done if this last one is the constant <span class="docutils literal">1</span> or <span class="docutils literal"><span class="pre">-1</span></span>, i.e. if the +original polynomial was square-free. These polynomials are primitive +polynomials too, i.e. with integer coefficients having no common factor.</p> +<p>Thus <span class="docutils literal">sturmname_0</span> has exactly the same real and complex roots as +polynomial <span class="docutils literal">polname</span>, but with each root now of multiplicity one: +i.e. it is the "square-free part" of original polynomial <span class="docutils literal">polname</span>.</p> +<p>Notice that <span class="docutils literal">sturmname_1</span> isn't necessarily the derivative of +<span class="docutils literal">sturmname_0</span> due to the various normalizations.</p> +<p>The polynomials <span class="docutils literal">sturmname_k</span> main utility is for the execution of +<a class="reference internal" href="#polsturmisolatezeros-sturmname">\PolSturmIsolateZeros{sturmname}</a>. Be careful not to use these +names <span class="docutils literal">sturmname_0</span>, <span class="docutils literal">sturmname_1</span>, etc... for defining other +polynomials after having done <span class="docutils literal"><span class="pre">\PolToSturm{polname}{sturmname}</span></span> and +before executing <span class="docutils literal">\PolSturmIsolateZeros{sturmname}</span> else the +latter will behave erroneously.</p> +<p><a class="reference internal" href="#polsturmchainlength-sturmname">\PolSturmChainLength{sturmname}</a> gives the index of the last +element of the Sturm chain.</p> +</blockquote> +</div> +<div class="section" id="id13"> +<span id="id12"></span><h3><a class="toc-backref" href="#id113"><span class="docutils literal"><span class="pre">\PolToSturm*{polname}{sturmname}</span></span></a></h3> +<blockquote> +<p>Does the same as <a class="reference internal" href="#poltosturm">un-starred version</a> and additionally it +keeps for user usage the memory of the <em>un-normalized</em> Sturm chain +polynomials <span class="docutils literal">sturmname_k_</span>, <span class="docutils literal">k=0,1, <span class="pre">...,</span> N</span>, with +<span class="docutils literal">N</span> being <a class="reference internal" href="#polsturmchainlength-sturmname">\PolSturmChainLength{sturmname}</a>.</p> +<div class="admonition note"> +<p class="admonition-title">Note</p> +<p>This behaviour was modified at <span class="docutils literal">0.6</span>, anyhow the macro was +broken at <span class="docutils literal">0.5</span>.</p> +</div> +<div class="admonition hint"> +<p class="admonition-title">Hint</p> +<p>The square-free part of <span class="docutils literal">polname</span> is <span class="docutils literal">sturmname_0</span>, and their +quotient is the polynomial with name +<span class="docutils literal">sturname_\PolSturmChainLength{sturmname}_</span>. It thus easy to +set-up a loop iteratively computing the latter until the last one +is a constant, thus obtaining the decomposition of an <span class="docutils literal">f</span> as +a product <span class="docutils literal">c f_1 f_2 f_3 ...</span> of a constant and square-free (primitive) +polynomials, where each <span class="docutils literal">f_i</span> divides its predecessor.</p> +</div> +</blockquote> +</div> +<div class="section" id="polsettosturmchainsignchangesat-macro-sturmname-fraction"> +<span id="polsettosturmchainsignchangesat"></span><h3><a class="toc-backref" href="#id114"><span class="docutils literal"><span class="pre">\PolSetToSturmChainSignChangesAt{\macro}{sturmname}{fraction}</span></span></a></h3> +<blockquote> +<p>Sets macro <span class="docutils literal">\macro</span> to the number of sign changes in the Sturm +chain with name prefix <span class="docutils literal">sturmname</span>, at location <span class="docutils literal">fraction</span> +(which must be in format as acceptable by the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> macros.)</p> +<div class="admonition note"> +<p class="admonition-title">Note</p> +<p>The author was lazy and did not provide rather an expandable +variant, where one would do <span class="docutils literal"><span class="pre">\edef\macro{\PolNbOf...}</span></span>.</p> +<p>This will presumably get added in a future release.</p> +<p>After some hesitation it was decided the macro would by default +act globally. To make the scope of its macro definition local, +use <span class="docutils literal">[\empty]</span> as extra optional argument.</p> +</div> +</blockquote> +</div> +<div class="section" id="polsettonbofzeroswithin-macro-sturmname-value-a-value-b"> +<span id="polsettonbofzeroswithin"></span><h3><a class="toc-backref" href="#id115"><span class="docutils literal"><span class="pre">\PolSetToNbOfZerosWithin{\macro}{sturmname}{value_a}{value_b}</span></span></a></h3> +<blockquote> +<p>Applies the <a class="reference external" href="https://en.wikipedia.org/wiki/Sturm%27s_theorem">Sturm Theorem</a> to set <span class="docutils literal">\macro</span> to the exact number +of <strong>distinct</strong> roots of <span class="docutils literal">sturmname_0</span> in the interval <span class="docutils literal">(value_a, value_b]</span> (the macro first re-orders the value for <span class="docutils literal">value_a <= value_b</span> to hold).</p> +<div class="admonition note"> +<p class="admonition-title">Note</p> +<p>The author was lazy and did not provide rather an expandable +variant, where one would do <span class="docutils literal"><span class="pre">\edef\macro{\PolNbOf...}</span></span>.</p> +<p>This will presumably get added in future.</p> +<p>After some hesitation it was decided the macro would by default +act globally. To make the scope of its macro definition local, +use <span class="docutils literal">[\empty]</span> as extra optional argument.</p> +</div> +<p>See also the expandable +<a class="reference internal" href="#polsturmnbofrootsof-sturmname-lessthanorequalto-value">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualTo{value}</a>, from +which it is immediate (with <span class="docutils literal">\numexpr</span>) to create an expandable +variant of this macro. However the difference is that this macro +requires only <a class="reference internal" href="#poltosturm">\PolToSturm</a> to have been executed, +whereas the expandable variant requires prior execution of +<a class="reference internal" href="#polsturmisolatezeros">\PolSturmIsolateZeros</a>.</p> +<p>See also the expandable +<a class="reference internal" href="#polsturmnbwithmultofrootsof-sturmname-lessthanorequalto-value">\PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualTo{value}</a> +which requires prior execution of +<a class="reference internal" href="#id14">\PolSturmIsolateZeros*</a>.</p> +</blockquote> +</div> +<div class="section" id="polsturmisolatezeros-sturmname"> +<span id="polsturmisolatezeros"></span><h3><a class="toc-backref" href="#id116"><span class="docutils literal">\PolSturmIsolateZeros{sturmname}</span></a></h3> +<blockquote> +<p>The macros locates, using <a class="reference external" href="https://en.wikipedia.org/wiki/Sturm%27s_theorem">Sturm theorem</a>, as many disjoint +intervals as there are (real) roots.</p> +<div class="admonition important"> +<p class="admonition-title">Important</p> +<p>The Sturm chain must have been produced by an earlier +<a class="reference internal" href="#poltosturm-polname-sturmname">\PolToSturm{polname}{sturmname}</a>.</p> +<p>Why does this macro ask for argument the name of Sturm chain, +rather than the name of a polynomial? well this is mainly for +legacy reason, and because it is accompanied by other macros for +which it is simpler to assume the argument will be the name of an +already computed Sturm chain.</p> +<p>Notice that <span class="docutils literal"><span class="pre">\PolToSturm{f}{f}</span></span> is perfectly legal (the +<span class="docutils literal">sturmname</span> can be same as the <span class="docutils literal">polname</span>): it defines +polynomials <span class="docutils literal">f_0</span>, <span class="docutils literal">f_1</span>, ... having <span class="docutils literal">f</span> has name prefix.</p> +<p>Such a prior call +to <span class="docutils literal">\PolToSturm</span> must have been made at any rate for +<span class="docutils literal">\PolSturmIsolateZeros</span> to be usable.</p> +</div> +<p>After its execution they are two types of such intervals (stored in +memory and accessible via macros or <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> variables, see below):</p> +<ul class="simple"> +<li><p>singleton <span class="docutils literal">{a}</span>: then <span class="docutils literal">a</span> is a root, (necessarily a decimal +number, but not all such decimal numbers are exactly identified yet).</p></li> +<li><p>open intervals <span class="docutils literal">(a,b)</span>: then there is exactly one root <span class="docutils literal">z</span> +such that <span class="docutils literal">a < z < b</span>, and the end points are guaranteed to not +be roots.</p></li> +</ul> +<p>The interval boundaries are decimal numbers, originating +in iterated decimal subdivision from initial intervals +<span class="docutils literal"><span class="pre">(-10^E,</span> 0)</span> and <span class="docutils literal">(0, 10^E)</span> with <span class="docutils literal">E</span> chosen initially large +enough so that all roots are enclosed; if zero is a root it is always +identified as such. The non-singleton intervals are of the +type <span class="docutils literal">(a/10^f, <span class="pre">(a+1)/10^f)</span></span> with <span class="docutils literal">a</span> an integer, which is +neither <span class="docutils literal">0</span> nor <span class="docutils literal"><span class="pre">-1</span></span>. Hence either <span class="docutils literal">a</span> and <span class="docutils literal">a+1</span> are both positive +or they are both negative.</p> +<p>One does not <em>a priori</em> know what will be the lengths of these +intervals (except that they are always powers of ten), they +vary depending on how many digits two successive roots have in +common in their respective decimal expansions.</p> +<div class="admonition important"> +<p class="admonition-title">Important</p> +<p>If some two consecutive intervals share an end-point, no +information is yet gained about the separation between the two +roots which could at this stage be arbitrarily small.</p> +<p>See <a class="reference internal" href="#polrefineinterval-sturmname-index">\PolRefineInterval*{sturmname}{index}</a> which addresses +this issue.</p> +</div> +<p>The interval boundaries (and exactly found roots) are made available +for future computations in <span class="docutils literal">\xintexpr</span>-essions or polynomial +definitions as variables <span class="docutils literal"><sturmname>L_1</span>, +<span class="docutils literal"><sturmname>L_2</span>, etc..., for the left end-points and +<span class="docutils literal"><sturmname>R_1</span>, <span class="docutils literal"><sturmname>R_2</span>, ..., for the right +end-points.</p> +<p>Thus for example, if <span class="docutils literal">sturmname</span> is <span class="docutils literal">f</span>, one can use the +<a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> variables <span class="docutils literal">fL_1</span>, <span class="docutils literal">fL_2</span>, ... to refer in expressions +to the left end-points (or to the exact root, if left and right end +points coincide). Additionally, <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> variable <span class="docutils literal">fZ_1_isknown</span> +will have value <span class="docutils literal">1</span> if the root in the first interval is known, +and <span class="docutils literal">0</span> otherwise. And similarly for the other intervals.</p> +<p>Also, macros <a class="reference internal" href="#polsturmisolatedzeroleft-sturmname-index">\PolSturmIsolatedZeroLeft{sturmname}{index}</a> and +<a class="reference internal" href="#polsturmisolatedzeroright-sturmname-index">\PolSturmIsolatedZeroRight{sturmname}{index}</a> are provided which +expand to these same values, written in decimal notation (i.e. +pre-processed by <a class="reference internal" href="#poldectostring">\PolDecToString</a>.) And there +is also <a class="reference internal" href="#polsturmifzeroexactlyknown-sturmname-index-a-b">\PolSturmIfZeroExactlyKnown{sturmname}{index}{A}{B}</a>.</p> +<div class="admonition important"> +<p class="admonition-title">Important</p> +<p>Trailing zeroes in the stored decimal numbers accessible via the +macros are significant: they are also present in the decimal +expansion of the exact root.</p> +</div> +<p>These variables and macros are automatically updated when one next +uses macros such as <a class="reference internal" href="#polrefineinterval-sturmname-index">\PolRefineInterval*{sturmname}{index}</a>.</p> +<p>The start of decimal expansion of a positive <span class="docutils literal">k</span>-th root is given +by <a class="reference internal" href="#polsturmisolatedzeroleft">\PolSturmIsolatedZeroLeft{sturmname}{k}</a>, and for a negative root it is given +by <a class="reference internal" href="#polsturmisolatedzeroright">PolSturmIsolatedZeroRight{sturmname}{k}</a>. These two decimal +numbers are either both zero or both of the same sign.</p> +<p>The number of distinct roots is obtainable expandably as +<a class="reference internal" href="#polsturmnbofisolatedzeros-sturmname">\PolSturmNbOfIsolatedZeros{sturmname}</a>.</p> +<p>Furthermore +<a class="reference internal" href="#polsturmnbofrootsof-sturmname-lessthanorequalto-value">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualTo{value}</a> and +<a class="reference internal" href="#polsturmnbofrootsof-sturmname-lessthanorequaltoexpr-expression">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualToExpr{expression}</a>. +will expandably compute respectively the number of real roots at +most equal to <span class="docutils literal">value</span> or <span class="docutils literal">expression</span>, and the same but with +multiplicities.</p> +<div class="admonition note"> +<p class="admonition-title">Note</p> +<p>The current polexpr implementation defines the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> variables +and <a class="reference external" href="http://www.ctan.org/pkg/xint">xinttools</a> arrays described above with <strong>global scpe</strong>. On the +other hand the Sturm sequence polynomials do obey the current scope.</p> +</div> +<div class="admonition note"> +<p class="admonition-title">Note</p> +<p>As all computations are done <em>exactly</em> there can be no errors... +apart those due to bad coding by author. The results are exact +bounds for the mathematically exact real roots.</p> +<p>Future releases will perhaps also provide macros based on Newton +or Regula Falsi methods. Exact computations with such methods +lead however quickly to very big fractions, and this forces usage +of some rounding scheme for the abscissas if computation times +are to remain reasonable. This raises issues of its own, which +are studied in numerical mathematics.</p> +</div> +</blockquote> +</div> +<div class="section" id="id15"> +<span id="id14"></span><h3><a class="toc-backref" href="#id117"><span class="docutils literal"><span class="pre">\PolSturmIsolateZeros*{sturmname}</span></span></a></h3> +<blockquote> +<p>The macro does the same as <a class="reference internal" href="#polsturmisolatezeros-sturmname">\PolSturmIsolateZeros{sturmname}</a> and +then in addition it does the extra work to determine all +multiplicities (of the real roots): +after executing this macro, +<a class="reference internal" href="#polsturmisolatedzeromultiplicity-sturmname-index">\PolSturmIsolatedZeroMultiplicity{sturmname}{index}</a> will expand +to the multiplicity of the root located in the <span class="docutils literal">index</span>-th +interval (intervals are enumerated from left to right, with index +starting at <span class="docutils literal">1</span>).</p> +<p>Furthermore, if for example the <span class="docutils literal">sturmname</span> is <span class="docutils literal">f</span>, <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> +variables <span class="docutils literal">fM_1</span>, <span class="docutils literal">fM_2</span>... hold the multiplicities thus +computed.</p> +<div class="admonition note"> +<p class="admonition-title">Note</p> +<p>It is <strong>not</strong> necessary to have executed the <a class="reference internal" href="#id12">PolToSturm*</a> starred +variant, as the non-starred variant keeps internally the memory of the +original GCD (and even of the full non-normalized original Sturm +chain), even though it does not make the declarations as <em>user-level</em> +genuine polynomials.</p> +</div> +<p>See <a class="reference internal" href="#the-degree-nine-polynomial-with-0-99-0-999-0-9999-as-triple-roots">The degree nine polynomial with 0.99, 0.999, 0.9999 as triple +roots</a> for an example.</p> +</blockquote> +</div> +<div class="section" id="id17"> +<span id="id16"></span><h3><a class="toc-backref" href="#id118"><span class="docutils literal"><span class="pre">\PolSturmIsolateZeros**{sturmname}</span></span></a></h3> +<blockquote> +<p>The macro does the same as <a class="reference internal" href="#id15">\PolSturmIsolateZeros*{sturmname}</a> and +in addition it does the extra work to determine all the <em>rational</em> +roots.</p> +<div class="admonition note"> +<p class="admonition-title">Note</p> +<p>After execution of this macro, a root is "known" if and only if +it is rational.</p> +</div> +<p>Furthermore, primitive polynomial <span class="docutils literal">sturmname_sqf_norr</span> is created +to match the (square-free) <span class="docutils literal">sturmname_0</span> from which all rational +roots have been removed (see <a class="reference internal" href="#polexprsetup">\polexprsetup</a> for customizing this +name). The number of distinct rational roots is thus the difference +between the degrees of these two polynomials (see also +<a class="reference internal" href="#polsturmnbofrationalroots-sturmname">\PolSturmNbOfRationalRoots{sturmname}</a>).</p> +<p>And <span class="docutils literal">sturmname_norr</span> is <span class="docutils literal">sturmname_0_</span> from which all rational +roots have been removed (see <a class="reference internal" href="#polexprsetup">\polexprsetup</a>), i.e. it contains +the irrational roots of the original polynomial, with the same +multiplicities.</p> +<p>See <a class="reference internal" href="#a-degree-five-polynomial-with-three-rational-roots">A degree five polynomial with three rational +roots</a> for an example.</p> +</blockquote> +</div> +<div class="section" id="polsturmisolatezerosandgetmultiplicities-sturmname"> +<span id="polsturmisolatezerosandgetmultiplicities"></span><h3><a class="toc-backref" href="#id119"><span class="docutils literal">\PolSturmIsolateZerosAndGetMultiplicities{sturmname}</span></a></h3> +<blockquote> +<p>This is another name for <a class="reference internal" href="#id15">\PolSturmIsolateZeros*{sturmname}</a>.</p> +</blockquote> +</div> +<div class="section" id="polsturmisolatezerosgetmultiplicitiesandrationalroots-sturmname"> +<span id="polsturmisolatezerosgetmultiplicitiesandrationalroots"></span><h3><a class="toc-backref" href="#id120"><span class="docutils literal">\PolSturmIsolateZerosGetMultiplicitiesAndRationalRoots{sturmname}</span></a></h3> +<blockquote> +<p>This is another name for <a class="reference internal" href="#id17">\PolSturmIsolateZeros**{sturmname}</a>.</p> +</blockquote> +</div> +<div class="section" id="polsturmisolatezerosandfindrationalroots-sturmname"> +<h3><a class="toc-backref" href="#id121"><span class="docutils literal">\PolSturmIsolateZerosAndFindRationalRoots{sturmname}</span></a></h3> +<blockquote> +<p>This works exactly like <a class="reference internal" href="#id17">\PolSturmIsolateZeros**{sturmname}</a> +(inclusive of declaring the polynomials <span class="docutils literal">sturmname_sqf_norr</span> and +<span class="docutils literal">sturmname_norr</span> with no rational roots) except that it does <em>not</em> +compute the multiplicities of the <em>non-rational</em> roots.</p> +<div class="admonition note"> +<p class="admonition-title">Note</p> +<p>There is no macro to find the rational roots but not compute +their multiplicities at the same time.</p> +</div> +<div class="admonition attention"> +<p class="admonition-title">Attention!</p> +<p>This macro does <em>not</em> define <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> variables +<span class="docutils literal">sturmnameM_1</span>, <span class="docutils literal">sturmnameM_2</span>, ... holding the +multiplicities and it leaves the multiplicity array (whose accessor +is <a class="reference internal" href="#polsturmisolatedzeromultiplicity-sturmname-index">\PolSturmIsolatedZeroMultiplicity{sturmname}{index}</a>) into +a broken state, as all non-rational roots will supposedly have +multiplicity one. This means that the output of +<a class="reference internal" href="#id21">\PolPrintIntervals*</a> for example will be +erroneous for the intervals with irrational roots.</p> +<p>I decided to document it because finding multiplicities of the +non rational roots is somewhat costly, and one may be interested +only into finding the rational roots (of course random +polynomials with integer coefficients will not have <em>any</em> +rational root anyhow).</p> +</div> +</blockquote> +</div> +<div class="section" id="polrefineinterval-sturmname-index"> +<span id="polrefineinterval"></span><h3><a class="toc-backref" href="#id122"><span class="docutils literal"><span class="pre">\PolRefineInterval*{sturmname}{index}</span></span></a></h3> +<blockquote> +<p>The <span class="docutils literal">index</span>-th interval (starting indexing at one) is further +subdivided as many times as is necessary in order for the newer +interval to have both its end-points distinct from the end-points of +the original interval. This means that the <span class="docutils literal">k</span>th root is then +strictly separated from the other roots.</p> +</blockquote> +</div> +<div class="section" id="polrefineinterval-n-sturmname-index"> +<span id="polrefineinterval-n"></span><h3><a class="toc-backref" href="#id123"><span class="docutils literal"><span class="pre">\PolRefineInterval[N]{sturmname}{index}</span></span></a></h3> +<blockquote> +<p>The <span class="docutils literal">index</span>-th interval (starting count at one) is further +subdivided once, reducing its length by a factor of 10. This is done +<span class="docutils literal">N</span> times if the optional argument <span class="docutils literal">[N]</span> is present.</p> +</blockquote> +</div> +<div class="section" id="polensureintervallength-sturmname-index-e"> +<span id="polensureintervallength"></span><h3><a class="toc-backref" href="#id124"><span class="docutils literal"><span class="pre">\PolEnsureIntervalLength{sturmname}{index}{E}</span></span></a></h3> +<blockquote> +<p>The <span class="docutils literal">index</span>-th interval is subdivided until its length becomes at +most <span class="docutils literal">10^E</span>. This means (for <span class="docutils literal">E<0</span>) that the first <span class="docutils literal"><span class="pre">-E</span></span> digits +after decimal mark of the <span class="docutils literal">k</span>th root will then be known exactly.</p> +</blockquote> +</div> +<div class="section" id="polensureintervallengths-sturmname-e"> +<span id="polensureintervallengths"></span><h3><a class="toc-backref" href="#id125"><span class="docutils literal"><span class="pre">\PolEnsureIntervalLengths{sturmname}{E}</span></span></a></h3> +<blockquote> +<p>The intervals as obtained from <span class="docutils literal">\PolSturmIsolateZeros</span> are (if +necessary) subdivided further by (base 10) dichotomy in order for +each of them to have length at most <span class="docutils literal">10^E</span> (length will be shorter +than <span class="docutils literal">10^E</span> in output only if it did not change or became zero.)</p> +<p>This means that decimal expansions of all roots will be known with +<span class="docutils literal"><span class="pre">-E</span></span> digits (for <span class="docutils literal">E<0</span>) after decimal mark.</p> +</blockquote> +</div> +<div class="section" id="polprintintervals-varname-sturmname"> +<span id="polprintintervals"></span><h3><a class="toc-backref" href="#id126"><span class="docutils literal"><span class="pre">\PolPrintIntervals[varname]{sturmname}</span></span></a></h3> +<blockquote> +<p>This is a convenience macro which prints the bounds for the roots +<span class="docutils literal">Z_1</span>, <span class="docutils literal">Z_2</span>, ... (the optional argument <span class="docutils literal">varname</span> allows to +specify a replacement for the default <span class="docutils literal">Z</span>). This will be done (by +default) in a +math mode <span class="docutils literal">array</span>, one interval per row, and pattern <span class="docutils literal">rcccl</span>, +where the second and fourth column hold the <span class="docutils literal"><</span> sign, except when +the interval reduces to a singleton, which means the root is known +exactly.</p> +<div class="admonition note"> +<p class="admonition-title">Note</p> +<p>The explanations here and in this section are for LaTeX. With +other TeX macro formats, the LaTeX syntax such as for example +<span class="docutils literal"><span class="pre">\begin{array}{rcccl}</span></span> which appears in the documentation here +is actually replaced with quasi-equivalent direct use of TeX +primitives.</p> +</div> +<p>See next macros which govern its output.</p> +</blockquote> +<div class="section" id="polprintintervalsnorealroots"> +<h4><a class="toc-backref" href="#id127"><span class="docutils literal">\PolPrintIntervalsNoRealRoots</span></a></h4> +<blockquote> +<p>Executed in place of an <span class="docutils literal">array</span> environment, when there are no +real roots. Default definition:</p> +<pre class="literal-block">\newcommand\PolPrintIntervalsNoRealRoots{}</pre> +</blockquote> +</div> +<div class="section" id="polprintintervalsbeginenv"> +<h4><a class="toc-backref" href="#id128"><span class="docutils literal">\PolPrintIntervalsBeginEnv</span></a></h4> +<blockquote> +<p>Default definition:</p> +<pre class="literal-block">\newcommand\PolPrintIntervalsBeginEnv{\[\begin{array}{rcccl}}</pre> +</blockquote> +</div> +<div class="section" id="polprintintervalsendenv"> +<h4><a class="toc-backref" href="#id129"><span class="docutils literal">\PolPrintIntervalsEndEnv</span></a></h4> +<blockquote> +<p>Default definition:</p> +<pre class="literal-block">\newcommand\PolPrintIntervalsEndEnv{\end{array}\]}</pre> +</blockquote> +</div> +<div class="section" id="polprintintervalsknownroot"> +<h4><a class="toc-backref" href="#id130"><span class="docutils literal">\PolPrintIntervalsKnownRoot</span></a></h4> +<blockquote> +<p>Default definition:</p> +<pre class="literal-block">\newcommand\PolPrintIntervalsKnownRoot{% + &&\PolPrintIntervalsTheVar_{\PolPrintIntervalsTheIndex}% + &=&\PolPrintIntervalsPrintExactZero +}</pre> +</blockquote> +</div> +<div class="section" id="polprintintervalsunknownroot"> +<h4><a class="toc-backref" href="#id131"><span class="docutils literal">\PolPrintIntervalsUnknownRoot</span></a></h4> +<blockquote> +<p>Default definition:</p> +<pre class="literal-block">\newcommand\PolPrintIntervalsUnknownRoot{% + \PolPrintIntervalsPrintLeftEndPoint&<&% + \PolPrintIntervalsTheVar_{\PolPrintIntervalsTheIndex}&<&% + \PolPrintIntervalsPrintRightEndPoint +}</pre> +</blockquote> +</div> +<div class="section" id="id18"> +<span id="polprintintervalsprintexactzero"></span><h4><a class="toc-backref" href="#id132"><span class="docutils literal">\PolPrintIntervalsPrintExactZero</span></a></h4> +<blockquote> +<p>Default definition:</p> +<pre class="literal-block">\newcommand\PolPrintIntervalsPrintExactZero{\PolPrintIntervalsTheLeftEndPoint}</pre> +</blockquote> +</div> +<div class="section" id="id19"> +<span id="polprintintervalsprintleftendpoint"></span><h4><a class="toc-backref" href="#id133"><span class="docutils literal">\PolPrintIntervalsPrintLeftEndPoint</span></a></h4> +<blockquote> +<p>Default definition:</p> +<pre class="literal-block">\newcommand\PolPrintIntervalsPrintLeftEndPoint{\PolPrintIntervalsTheLeftEndPoint}</pre> +</blockquote> +</div> +<div class="section" id="id20"> +<span id="polprintintervalsprintrightendpoint"></span><h4><a class="toc-backref" href="#id134"><span class="docutils literal">\PolPrintIntervalsPrintRightEndPoint</span></a></h4> +<blockquote> +<p>Default definition is:</p> +<pre class="literal-block">\newcommand\PolPrintIntervalsPrintRightEndPoint{\PolPrintIntervalsTheRightEndPoint}</pre> +</blockquote> +</div> +</div> +<div class="section" id="id22"> +<span id="id21"></span><h3><a class="toc-backref" href="#id135"><span class="docutils literal"><span class="pre">\PolPrintIntervals*[varname]{sturmname}</span></span></a></h3> +<blockquote> +<p>This starred variant produces an alternative output (which +displays the root multiplicity), and is provided as an +example of customization.</p> +<p>As replacement for <a class="reference internal" href="#polprintintervalsknownroot">\PolPrintIntervalsKnownRoot</a>, +<a class="reference internal" href="#polprintintervalsprintexactzero">\PolPrintIntervalsPrintExactZero</a>, +<a class="reference internal" href="#polprintintervalsunknownroot">\PolPrintIntervalsUnknownRoot</a> it uses its own +<span class="docutils literal"><span class="pre">\POL@@PrintIntervals...</span></span> macros. We only reproduce here one +definition:</p> +<pre class="literal-block">\newcommand\POL@@PrintIntervalsPrintExactZero{% + \displaystyle + \xintSignedFrac{\PolPrintIntervalsTheLeftEndPoint}% +}%</pre> +<p>Multiplicities are printed using this auxiliary macro:</p> +</blockquote> +<div class="section" id="polprintintervalsprintmultiplicity"> +<h4><a class="toc-backref" href="#id136"><span class="docutils literal">\PolPrintIntervalsPrintMultiplicity</span></a></h4> +<blockquote> +<p>whose default definition is:</p> +<pre class="literal-block">\newcommand\PolPrintIntervalsPrintMultiplicity{(\mbox{mult. }\PolPrintIntervalsTheMultiplicity)}</pre> +</blockquote> +</div> +</div> +</div> +<div class="section" id="polmapcoeffs-macro-polname"> +<span id="polmapcoeffs"></span><h2><a class="toc-backref" href="#id137"><span class="docutils literal"><span class="pre">\PolMapCoeffs{\macro}{polname}</span></span></a></h2> +<blockquote> +<p>It modifies ('in-place': original coefficients get lost) each +coefficient of the defined polynomial via the <em>expandable</em> macro +<span class="docutils literal">\macro</span>. The degree is adjusted as necessary if some leading +coefficients vanish after the operation. In replacement text of +<span class="docutils literal">\macro</span>, <span class="docutils literal">\index</span> expands to the coefficient index (which is +defined to be zero for the constant term).</p> +<p>Notice that <span class="docutils literal">\macro</span> will have to handle inputs of the shape +<span class="docutils literal">A/B[N]</span> (<a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> internal notation). This means that it probably +will have to be expressed in terms of macros from <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> package.</p> +<p>Example:</p> +<pre class="literal-block">\def\foo#1{\xintMul{#1}{\the\numexpr\index*\index\relax}}</pre> +<p>(or with <span class="docutils literal"><span class="pre">\xintSqr{\index}</span></span>) to replace <span class="docutils literal">n</span>-th coefficient +<span class="docutils literal">f_n</span> by <span class="docutils literal">f_n*n^2</span>.</p> +</blockquote> +</div> +<div class="section" id="polreducecoeffs-polname"> +<span id="polreducecoeffs"></span><h2><a class="toc-backref" href="#id138"><span class="docutils literal">\PolReduceCoeffs{polname}</span></a></h2> +<blockquote> +<p>About the same as <span class="docutils literal"><span class="pre">\PolMapCoeffs{\xintIrr}{polname}</span></span> (but +maintaining a <span class="docutils literal">[0]</span> postfix for speedier <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> parsing when +polynomial function is used for computations.) This is a +one-argument macro, working 'in-place'.</p> +</blockquote> +</div> +<div class="section" id="id24"> +<span id="id23"></span><h2><a class="toc-backref" href="#id139"><span class="docutils literal"><span class="pre">\PolReduceCoeffs*{polname}</span></span></a></h2> +<blockquote> +<p>This starred variant leaves un-touched the decimal exponent in the +internal representation of the fractional coefficients, i.e. if a +coefficient is internally <span class="docutils literal">A/B[N]</span>, then <span class="docutils literal">A/B</span> is reduced to +smallest terms, but the <span class="docutils literal">10^N</span> part is kept as is. Note: if the +polynomial is freshly defined directly via <a class="reference internal" href="#polfromcsv">\PolFromCSV</a> its coefficients might still be internally in some +format like <span class="docutils literal">1.5e7</span>; the macro will anyhow always first do the +needed conversion to strict format <span class="docutils literal">A/B[N]</span>.</p> +<p>Evaluations with polynomials treated by this can be much faster than +with those handled by the non-starred variant +<a class="reference internal" href="#polreducecoeffs-polname">\PolReduceCoeffs{polname}</a>: as the numerators and denominators +remain smaller, this proves very beneficial in favorable cases +(especially when the coefficients are decimal numbers) to the +expansion speed of the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> macros used internally by +<a class="reference internal" href="#polevalat">\PolEval</a>.</p> +</blockquote> +</div> +<div class="section" id="polmakemonic-polname"> +<span id="polmakemonic"></span><h2><a class="toc-backref" href="#id140"><span class="docutils literal">\PolMakeMonic{polname}</span></a></h2> +<blockquote> +<p>Divides by the leading coefficient. It is recommended to execute +<a class="reference internal" href="#id24">\PolReduceCoeffs*{polname}</a> immediately afterwards. This is not +done automatically, due to the case the original polynomial had integer +coefficients and we want to keep the leading one as common +denominator.</p> +</blockquote> +</div> +<div class="section" id="polmakeprimitive-polname"> +<span id="polmakeprimitive"></span><h2><a class="toc-backref" href="#id141"><span class="docutils literal">\PolMakePrimitive{polname}</span></a></h2> +<blockquote> +<p>Divides by the integer content see (<a class="reference internal" href="#policontent">\PolIContent</a>). This thus produces a polynomial with integer +coefficients having no common factor. The sign of the leading +coefficient is not modified.</p> +</blockquote> +</div> +</div> +<div class="section" id="expandable-macros"> +<h1><a class="toc-backref" href="#id142">Expandable macros</a></h1> +<div class="admonition note"> +<p class="admonition-title">Note</p> +<p>At <span class="docutils literal">0.8</span> <span class="docutils literal">polexpr</span> is usable with Plain TeX and not only with +LaTeX. Some examples given in this section may be using LaTeX syntax +such as <span class="docutils literal">\renewcommand</span>. Convert to TeX primitives as appropriate +if testing with a non LaTeX macro format.</p> +</div> +<p>All these macros expand completely in two steps except <span class="docutils literal">\PolToExpr</span> +and <span class="docutils literal">\PolToFloatExpr</span> (and their auxiliaries) which need a +<span class="docutils literal">\write</span>, <span class="docutils literal">\edef</span> or a <span class="docutils literal"><span class="pre">\csname...\endcsname</span></span> context.</p> +<div class="section" id="poleval-polname-atexpr-numerical-expression"> +<span id="polevalatexpr"></span><h2><a class="toc-backref" href="#id143"><span class="docutils literal"><span class="pre">\PolEval{polname}\AtExpr{numerical</span> expression}</span></a></h2> +<blockquote> +<p>It boils down to +<span class="docutils literal">\xinttheexpr polname(numerical <span class="pre">expression)\relax</span></span>.</p> +</blockquote> +</div> +<div class="section" id="poleval-polname-at-fraction"> +<span id="polevalat"></span><h2><a class="toc-backref" href="#id144"><span class="docutils literal"><span class="pre">\PolEval{polname}\At{fraction}</span></span></a></h2> +<blockquote> +<p>Evaluates the polynomial at value <span class="docutils literal">fraction</span> which must be in (or +expand to) a format acceptable to the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> macros.</p> +</blockquote> +</div> +<div class="section" id="polevalreduced-polname-atexpr-numerical-expression"> +<span id="polevalreducedatexpr"></span><h2><a class="toc-backref" href="#id145"><span class="docutils literal"><span class="pre">\PolEvalReduced{polname}\AtExpr{numerical</span> expression}</span></a></h2> +<blockquote> +<p>Boils down to <span class="docutils literal">\xinttheexpr reduce(polname(numerical <span class="pre">expression))\relax</span></span>.</p> +</blockquote> +</div> +<div class="section" id="polevalreduced-polname-at-fraction"> +<span id="polevalreducedat"></span><h2><a class="toc-backref" href="#id146"><span class="docutils literal"><span class="pre">\PolEvalReduced{polname}\At{fraction}</span></span></a></h2> +<blockquote> +<p>Evaluates the polynomial at value <span class="docutils literal">fraction</span> which must be in (or +expand to) a format acceptable to the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> macros, and produce +an irreducible fraction.</p> +</blockquote> +</div> +<div class="section" id="polfloateval-polname-atexpr-numerical-expression"> +<span id="polfloatevalatexpr"></span><h2><a class="toc-backref" href="#id147"><span class="docutils literal"><span class="pre">\PolFloatEval{polname}\AtExpr{numerical</span> expression}</span></a></h2> +<blockquote> +<p>Boils down to <span class="docutils literal">\xintthefloatexpr polname(numerical <span class="pre">expression)\relax</span></span>.</p> +<p>This is done via a Horner Scheme (see <a class="reference internal" href="#poldef">\poldef</a> and +<a class="reference internal" href="#polgenfloatvariant-polname">\PolGenFloatVariant{polname}</a>), with already rounded +coefficients. <a class="footnote-reference brackets" href="#id27" id="id25">4</a> To use the <em>exact coefficients</em> with <em>exactly +executed</em> additions and multiplications, just insert it in the float +expression as in this example: <a class="footnote-reference brackets" href="#id28" id="id26">5</a></p> +<pre class="literal-block">\xintthefloatexpr 3.27*\xintexpr f(2.53)\relax^2\relax</pre> +<p>The <span class="docutils literal">f(2.53)</span> is exactly computed then rounded at the time of +getting raised to the power <span class="docutils literal">2</span>. Moving the <span class="docutils literal">^2</span> inside, that +operation would also be treated exactly.</p> +<dl class="footnote brackets"> +<dt class="label" id="id27"><span class="brackets"><a class="fn-backref" href="#id25">4</a></span></dt> +<dd><p>Anyway each floating point operation starts by rounding its +operands to the floating point precision.</p> +</dd> +<dt class="label" id="id28"><span class="brackets"><a class="fn-backref" href="#id26">5</a></span></dt> +<dd><p>The <span class="docutils literal">\xintexpr</span> here could be <span class="docutils literal">\xinttheexpr</span> but that +would be less efficient. Cf. <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> documentation about +nested expressions.</p> +</dd> +</dl> +</blockquote> +</div> +<div class="section" id="polfloateval-polname-at-fraction"> +<span id="polfloatevalat"></span><h2><a class="toc-backref" href="#id148"><span class="docutils literal"><span class="pre">\PolFloatEval{polname}\At{fraction}</span></span></a></h2> +<blockquote> +<p>Evaluates the polynomial at value <span class="docutils literal">fraction</span> which must be in (or +expand to) a format acceptable to the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> macros, and produces +a floating point number.</p> +</blockquote> +</div> +<div class="section" id="polifcoeffisplusorminusone-a-b"> +<span id="polifcoeffisplusorminusone"></span><h2><a class="toc-backref" href="#id149"><span class="docutils literal"><span class="pre">\PolIfCoeffIsPlusOrMinusOne{A}{B}</span></span></a></h2> +<blockquote> +<p>This macro is a priori undefined.</p> +<p>It is defined via the default <a class="reference internal" href="#poltypesetcmd-raw-coeff">\PolTypesetCmd{raw_coeff}</a> to be +used if needed in the execution of <a class="reference internal" href="#poltypesetmonomialcmd">\PolTypesetMonomialCmd</a>, +e.g. to insert a <span class="docutils literal">\cdot</span> in front of <span class="docutils literal"><span class="pre">\PolVar^{\PolIndex}</span></span> if +the coefficient is not plus or minus one.</p> +<p>The macro will execute <span class="docutils literal">A</span> if the coefficient has been found to be +plus or minus one, and <span class="docutils literal">B</span> if not.</p> +</blockquote> +</div> +<div class="section" id="polleadingcoeff-polname"> +<span id="polleadingcoeff"></span><h2><a class="toc-backref" href="#id150"><span class="docutils literal">\PolLeadingCoeff{polname}</span></a></h2> +<blockquote> +<p>Expands to the leading coefficient.</p> +</blockquote> +</div> +<div class="section" id="polnthcoeff-polname-number"> +<span id="polnthcoeff"></span><h2><a class="toc-backref" href="#id151"><span class="docutils literal"><span class="pre">\PolNthCoeff{polname}{number}</span></span></a></h2> +<blockquote> +<p>It expands to the raw <span class="docutils literal">N</span>-th coefficient (<span class="docutils literal">0/1[0]</span> if the index +number is out of range). With <span class="docutils literal"><span class="pre">N=-1</span></span>, <span class="docutils literal"><span class="pre">-2</span></span>, ... expands to the +leading coefficients.</p> +</blockquote> +</div> +<div class="section" id="poldegree-polname"> +<span id="poldegree"></span><h2><a class="toc-backref" href="#id152"><span class="docutils literal">\PolDegree{polname}</span></a></h2> +<blockquote> +<p>It expands to the degree. This is <span class="docutils literal"><span class="pre">-1</span></span> if zero polynomial but this +may change in future. Should it then expand to <span class="docutils literal"><span class="pre">-\infty</span></span> ?</p> +</blockquote> +</div> +<div class="section" id="policontent-polname"> +<span id="policontent"></span><h2><a class="toc-backref" href="#id153"><span class="docutils literal">\PolIContent{polname}</span></a></h2> +<blockquote> +<p>It expands to the contents of the polynomial, i.e. to the positive +fraction such that dividing by this fraction produces a polynomial +with integer coefficients having no common prime divisor.</p> +<p>See <a class="reference internal" href="#polmakeprimitive">\PolMakePrimitive</a>.</p> +</blockquote> +</div> +<div class="section" id="poltoexpr-pol-expr"> +<span id="poltoexpr"></span><h2><a class="toc-backref" href="#id154"><span class="docutils literal"><span class="pre">\PolToExpr{<pol.</span> <span class="pre">expr.>}</span></span></a></h2> +<blockquote> +<p>Produces expandably <a class="footnote-reference brackets" href="#id30" id="id29">6</a> the string <span class="docutils literal"><span class="pre">coeff_N*x^N+...</span></span>, i.e. the +polynomial is using descending powers.</p> +<dl class="footnote brackets"> +<dt class="label" id="id30"><span class="brackets"><a class="fn-backref" href="#id29">6</a></span></dt> +<dd><p>requires exhaustive expansion, for example as triggered by +<span class="docutils literal">\write</span> or <span class="docutils literal">\edef</span>.</p> +</dd> +</dl> +<p>Since <span class="docutils literal">0.8</span> the input is not restricted to be a polynomial name but +is allowed to be an arbitrary expression (where by default the +letter <span class="docutils literal">x</span> is recognized as the indeterminate; see +<a class="reference internal" href="#poltoexprinvar">\PolToExprInVar</a>).</p> +<p>The default output (which also by default uses the letter <span class="docutils literal">x</span> and is +completely configurable, see in particular <a class="reference internal" href="#poltoexprvar">\PolToExprVar</a>) is +compatible with both</p> +<ul class="simple"> +<li><p>the Maple's input format,</p></li> +<li><p>and the PSTricks <span class="docutils literal">\psplot[algebraic]</span> input format.</p></li> +</ul> +<p>Attention that it is not compatible with Python, but see +<a class="reference internal" href="#poltoexprcaret">\PolToExprCaret</a> in this regard.</p> +<p>It has the following characteristics:</p> +<ul class="simple"> +<li><p>vanishing coefficients are skipped (issue <span class="docutils literal">\poltoexpralltrue</span> to +override this and produce output such as <span class="docutils literal">x^3+0*x^2+0*x^1+0</span>),</p></li> +<li><p>negative coefficients are not prefixed by a <span class="docutils literal">+</span> sign (else, +Maple would not be happy),</p></li> +<li><p>coefficients numerically equal to <span class="docutils literal">1</span> (or <span class="docutils literal"><span class="pre">-1</span></span>) are present +only via their sign,</p></li> +<li><p>the letter <span class="docutils literal">x</span> is used and the degree one monomial is output as +<span class="docutils literal">x</span>, not as <span class="docutils literal">x^1</span>.</p></li> +<li><p>(<span class="docutils literal">0.8</span>) the caret <span class="docutils literal">^</span> is of catcode 12. This means that one +can for convenience typeset in regular text mode, for example +using <span class="docutils literal">\texttt</span> (in LaTeX). But TeX will not know how to break +the expression across end-of-lines anyhow. Formerly <span class="docutils literal">^</span> was +suitable for math mode but as the exponent is not braced this +worked only for polynomials of degrees at most 9. Anyhow this +is not supposed to be a typesetting macro.</p></li> +</ul> +<p>Complete customization is possible, see the next macros. Any user +redefinition must maintain the expandability property.</p> +</blockquote> +<div class="section" id="id31"> +<span id="poltoexprvar"></span><h3><a class="toc-backref" href="#id155"><span class="docutils literal">\PolToExprVar</span></a></h3> +<blockquote> +<p>Defaults to <span class="docutils literal">x</span>. The letter used in input.</p> +</blockquote> +</div> +<div class="section" id="poltoexprinvar"> +<h3><a class="toc-backref" href="#id156"><span class="docutils literal">\PolToExprInVar</span></a></h3> +<blockquote> +<p>Defaults to <span class="docutils literal">x</span>: the letter used as the polynomial indeterminate.</p> +<p>Recall that declared polynomials are more efficiently used in +algebraic expressions without the <span class="docutils literal">(x)</span>, i.e. <span class="docutils literal">P*Q</span> is better +than <span class="docutils literal"><span class="pre">P(x)*Q(x)</span></span>. Thus the input, even if an expression, does not +have to contain any <span class="docutils literal">x</span>.</p> +<p>(new with <span class="docutils literal">0.8</span>)</p> +</blockquote> +</div> +<div class="section" id="id32"> +<span id="poltoexprtimes"></span><h3><a class="toc-backref" href="#id157"><span class="docutils literal">\PolToExprTimes</span></a></h3> +<blockquote> +<p>Defaults to <span class="docutils literal">*</span>.</p> +</blockquote> +</div> +<div class="section" id="poltoexprcaret"> +<h3><a class="toc-backref" href="#id158"><span class="docutils literal">\PolToExprCaret</span></a></h3> +<blockquote> +<p>Defaults to <span class="docutils literal">^</span> of catcode 12. Set it to +expand to <span class="docutils literal">**</span> for Python compatible output.</p> +<p>(new with <span class="docutils literal">0.8</span>)</p> +</blockquote> +</div> +<div class="section" id="poltoexprcmd-raw-coeff"> +<span id="poltoexprcmd"></span><h3><a class="toc-backref" href="#id159"><span class="docutils literal">\PolToExprCmd{raw_coeff}</span></a></h3> +<blockquote> +<p>Defaults to <span class="docutils literal"><span class="pre">\xintPRaw{\xintRawWithZeros{#1}}</span></span>.</p> +<p>This means that the coefficient value is printed-out as a fraction +<span class="docutils literal">a/b</span>, skipping the <span class="docutils literal">/b</span> part if <span class="docutils literal">b</span> turns out to be one.</p> +<p>Configure it to be <span class="docutils literal"><span class="pre">\xintPRaw{\xintIrr{#1}}</span></span> if the fractions +must be in irreducible terms.</p> +<p>An alternative is <span class="docutils literal"><span class="pre">\xintDecToString{\xintREZ{#1}}</span></span> which uses +integer or decimal fixed point format such as <span class="docutils literal">23.0071</span> if the +internal representation of the number only has a power of ten as +denominator (the effect of <span class="docutils literal">\xintREZ</span> here is to remove trailing +decimal zeros). The behaviour of <span class="docutils literal">\xintDecToString</span> is not yet +stable for other cases, and for example at time of writing no +attempt is made to identify inputs having a finite decimal expansion +so for example <span class="docutils literal">23.007/2</span> or <span class="docutils literal">23.007/25</span> can appear in output +and not their finite decimal expansion with no denominator.</p> +</blockquote> +</div> +<div class="section" id="poltoexproneterm-raw-coeff-number"> +<span id="poltoexproneterm"></span><h3><a class="toc-backref" href="#id160"><span class="docutils literal"><span class="pre">\PolToExprOneTerm{raw_coeff}{number}</span></span></a></h3> +<blockquote> +<p>This is the macro which from the coefficient and the exponent +produces the corresponding term in output, such as <span class="docutils literal">2/3*x^7</span>.</p> +<p>For its default definition, see the source code. It uses +<a class="reference internal" href="#poltoexprcmd">\PolToExprCmd</a>, <a class="reference internal" href="#poltoexprtimes">\PolToExprTimes</a>, <a class="reference internal" href="#poltoexprvar">\PolToExprVar</a> and +<a class="reference internal" href="#poltoexprcaret">\PolToExprCaret</a>.</p> +</blockquote> +</div> +<div class="section" id="poltoexpronetermstylea-raw-coeff-number"> +<span id="poltoexpronetermstylea"></span><h3><a class="toc-backref" href="#id161"><span class="docutils literal"><span class="pre">\PolToExprOneTermStyleA{raw_coeff}{number}</span></span></a></h3> +<blockquote> +<p>This holds the default package meaning of <span class="docutils literal">\PolToExprOneTerm</span>.</p> +</blockquote> +</div> +<div class="section" id="poltoexpronetermstyleb-raw-coeff-number"> +<span id="poltoexpronetermstyleb"></span><h3><a class="toc-backref" href="#id162"><span class="docutils literal"><span class="pre">\PolToExprOneTermStyleB{raw_coeff}{number}</span></span></a></h3> +<blockquote> +<p>This holds an alternative meaning, which puts the fractional part of +a coefficient after the monomial, i.e. like this:</p> +<pre class="literal-block">2*x^11/3+3*x^8/7-x^5-x^4/4-x^3-x^2/2-2*x+1</pre> +<p><a class="reference internal" href="#poltoexprcmd">\PolToExprCmd</a> isn't used at all in this style. But +<a class="reference internal" href="#poltoexprtimes">\PolToExprTimes</a>, <a class="reference internal" href="#poltoexprvar">\PolToExprVar</a> and <a class="reference internal" href="#poltoexprcaret">\PolToExprCaret</a> are obeyed.</p> +<p>To activate it use <span class="docutils literal">\let\PolToExprOneTerm\PolToExprOneTermStyleB</span>. +To revert to the package default behaviour, issue +<span class="docutils literal">\let\PolToExprOneTerm\PolToExprOneTermStyleA</span>.</p> +</blockquote> +</div> +<div class="section" id="poltoexprtermprefix-raw-coeff"> +<span id="poltoexprtermprefix"></span><h3><a class="toc-backref" href="#id163"><span class="docutils literal">\PolToExprTermPrefix{raw_coeff}</span></a></h3> +<blockquote> +<p>It receives as argument the coefficient. Its default behaviour is +to produce a <span class="docutils literal">+</span> if the coefficient is positive, which will thus +serve to separate the monomials in the output. This is to match +the default for <a class="reference internal" href="#poltoexprcmd-raw-coeff">\PolToExprCmd{raw_coeff}</a> which in case of a +positive coefficient does not output an explicit <span class="docutils literal">+</span> prefix.</p> +</blockquote> +</div> +</div> +<div class="section" id="id34"> +<span id="id33"></span><h2><a class="toc-backref" href="#id164"><span class="docutils literal"><span class="pre">\PolToExpr*{<pol.</span> <span class="pre">expr.>}</span></span></a></h2> +<blockquote> +<p>Ascending powers: <span class="docutils literal"><span class="pre">coeff_0+coeff_1*x+coeff_2*x^2+...</span></span>.</p> +<p>Extended at <span class="docutils literal">0.8</span> to accept general expressions as input.</p> +<p>Customizable with the same macros as for +<a class="reference internal" href="#poltoexpr-pol-expr">\PolToExpr{<pol. expr.>}</a>.</p> +</blockquote> +</div> +<div class="section" id="poltofloatexpr-pol-expr"> +<span id="poltofloatexpr"></span><h2><a class="toc-backref" href="#id165"><span class="docutils literal"><span class="pre">\PolToFloatExpr{<pol.</span> <span class="pre">expr.>}</span></span></a></h2> +<blockquote> +<p>Similar to <a class="reference internal" href="#poltoexpr-pol-expr">\PolToExpr{<pol. expr.>}</a> but using <a class="reference external" href="\PolToFloatExprCmd{raw_coeff}">\PolToFloatExprCmd</a> which by default rounds and +converts the coefficients to floating point format.</p> +<div class="admonition note"> +<p class="admonition-title">Note</p> +<p>This is unrelated to <a class="reference internal" href="#polgenfloatvariant-polname">\PolGenFloatVariant{polname}</a>: +<a class="reference internal" href="#poltofloatexprcmd-raw-coeff">\PolToFloatExprCmd{raw_coeff}</a> operates on the <em>exact</em> +coefficients anew (and may thus produce something else than +the coefficients of the polynomial function acting +in <span class="docutils literal">\xintfloateval</span> if the floating point precision was changed +in between).</p> +</div> +<p>Extended at <span class="docutils literal">0.8</span> to accept general expressions as input.</p> +</blockquote> +<div class="section" id="poltofloatexproneterm-raw-coeff-number"> +<span id="poltofloatexproneterm"></span><h3><a class="toc-backref" href="#id166"><span class="docutils literal"><span class="pre">\PolToFloatExprOneTerm{raw_coeff}{number}</span></span></a></h3> +<blockquote> +<p>Similar to <a class="reference external" href="\PolToExprOneTerm{raw_coeff}{number}">\PolToExprOneTerm</a>. But does not treat +especially coefficients equal to plus or minus one.</p> +</blockquote> +</div> +<div class="section" id="poltofloatexprcmd-raw-coeff"> +<span id="id36"></span><h3><a class="toc-backref" href="#id167"><span class="docutils literal">\PolToFloatExprCmd{raw_coeff}</span></a></h3> +<blockquote> +<p>The one-argument macro used by <span class="docutils literal">\PolToFloatExprOneTerm</span>. +It defaults to <span class="docutils literal"><span class="pre">\xintFloat{#1}</span></span>.</p> +<div class="admonition caution"> +<p class="admonition-title">Caution!</p> +<p>Currently <span class="docutils literal">\xintFloat{0}</span> outputs <span class="docutils literal">0.e0</span> +which is perfectly acceptable input for Python, but not for +Maple. Thus, one should better leave the <span class="docutils literal">\\ifpoltoexprall</span> TeX +Boolean to its default <a class="reference internal" href="#poltoexprallfalse">\poltoexprallfalse</a>, if one intends to use +the output in a Maple worksheet.</p> +<p>But even then the zero polynomial will cause a problem. Workaround:</p> +<pre class="literal-block">\renewcommand\PolToFloatExprCmd[1]{\xintiiifZero{#1}{0.0}{\xintFloat{#1}}}</pre> +<p>Usage of <span class="docutils literal">\xintiiifZero</span> and not <span class="docutils literal">\xintifZero</span> is only for +optimization (I can't help it) because <span class="docutils literal">#1</span> is known to be +in <span class="docutils literal">xintfrac</span> raw format.</p> +</div> +</blockquote> +</div> +</div> +<div class="section" id="id38"> +<span id="id37"></span><h2><a class="toc-backref" href="#id168"><span class="docutils literal"><span class="pre">\PolToFloatExpr*{<pol.</span> <span class="pre">expr.>}</span></span></a></h2> +<blockquote> +<p>Ascending powers.</p> +<p>Extended at <span class="docutils literal">0.8</span> to accept general expressions as input.</p> +</blockquote> +</div> +<div class="section" id="poltolist-polname"> +<span id="poltolist"></span><h2><a class="toc-backref" href="#id169"><span class="docutils literal">\PolToList{polname}</span></a></h2> +<blockquote> +<p>Expands to <span class="docutils literal"><span class="pre">{coeff_0}{coeff_1}...{coeff_N}</span></span> with <span class="docutils literal">N</span> = degree, and +<span class="docutils literal">coeff_N</span> the leading coefficient +(the zero polynomial does give <span class="docutils literal">{0/1[0]}</span> and not an +empty output.)</p> +</blockquote> +</div> +<div class="section" id="poltocsv-polname"> +<span id="poltocsv"></span><h2><a class="toc-backref" href="#id170"><span class="docutils literal">\PolToCSV{polname}</span></a></h2> +<blockquote> +<p>Expands to <span class="docutils literal">coeff_0, coeff_1, coeff_2, <span class="pre">.....,</span> coeff_N</span>, starting +with constant term and ending with leading coefficient. Converse +to <a class="reference internal" href="#polfromcsv-polname-csv">\PolFromCSV</a>.</p> +</blockquote> +</div> +<div class="section" id="expandable-macros-related-to-the-root-localization-routines"> +<h2><a class="toc-backref" href="#id171">Expandable macros related to the root localization routines</a></h2> +<div class="section" id="polsturmchainlength-sturmname"> +<span id="polsturmchainlength"></span><h3><a class="toc-backref" href="#id172"><span class="docutils literal">\PolSturmChainLength{sturmname}</span></a></h3> +<blockquote> +<p>Returns the integer <span class="docutils literal">N</span> such that <span class="docutils literal">sturmname_N</span> is the last one +in the Sturm chain <span class="docutils literal">sturmname_0</span>, <span class="docutils literal">sturmname_1</span>, ...</p> +<p>See <a class="reference internal" href="#poltosturm-polname-sturmname">\PolToSturm{polname}{sturmname}</a>.</p> +</blockquote> +</div> +<div class="section" id="polsturmifzeroexactlyknown-sturmname-index-a-b"> +<span id="polsturmifzeroexactlyknown"></span><h3><a class="toc-backref" href="#id173"><span class="docutils literal"><span class="pre">\PolSturmIfZeroExactlyKnown{sturmname}{index}{A}{B}</span></span></a></h3> +<blockquote> +<p>Executes <span class="docutils literal">A</span> if the <span class="docutils literal">index</span>-th interval reduces to a singleton, +i.e. the root is known exactly, else <span class="docutils literal">B</span>.</p> +<div class="admonition note"> +<p class="admonition-title">Note</p> +<p><span class="docutils literal">index</span> is allowed to be something like <span class="docutils literal">1+2*3</span> as it is fed +to <span class="docutils literal"><span class="pre">\the\numexpr...\relax</span></span>.</p> +</div> +</blockquote> +</div> +<div class="section" id="polsturmisolatedzeroleft-sturmname-index"> +<span id="polsturmisolatedzeroleft"></span><h3><a class="toc-backref" href="#id174"><span class="docutils literal"><span class="pre">\PolSturmIsolatedZeroLeft{sturmname}{index}</span></span></a></h3> +<blockquote> +<p>Expands to the left end-point for the <span class="docutils literal">index</span>-th interval, as +computed by some earlier <a class="reference internal" href="#polsturmisolatezeros-sturmname">\PolSturmIsolateZeros{sturmname}</a>.</p> +<div class="admonition note"> +<p class="admonition-title">Note</p> +<p>Of course, this is kept updated by macros such as +<a class="reference internal" href="#polrefineinterval-n">\PolRefineInterval{sturmname}{index}</a>.</p> +</div> +<p>The value is pre-formatted using <a class="reference internal" href="#poldectostring">\PolDecTostring</a>.</p> +</blockquote> +</div> +<div class="section" id="polsturmisolatedzeroright-sturmname-index"> +<span id="polsturmisolatedzeroright"></span><h3><a class="toc-backref" href="#id175"><span class="docutils literal"><span class="pre">\PolSturmIsolatedZeroRight{sturmname}{index}</span></span></a></h3> +<blockquote> +<p>Expands to the right end-point for the <span class="docutils literal">index</span>-th interval as +computed by some earlier <a class="reference internal" href="#polsturmisolatezeros-sturmname">\PolSturmIsolateZeros{sturmname}</a> and +possibly refined afterwards.</p> +<p>The value is pre-formatted using <a class="reference internal" href="#poldectostring">\PolDecTostring</a>.</p> +</blockquote> +</div> +<div class="section" id="polsturmisolatedzeromultiplicity-sturmname-index"> +<span id="polsturmisolatedzeromultiplicity"></span><h3><a class="toc-backref" href="#id176"><span class="docutils literal"><span class="pre">\PolSturmIsolatedZeroMultiplicity{sturmname}{index}</span></span></a></h3> +<blockquote> +<p>Expands to the multiplicity of the unique root contained in the +<span class="docutils literal">index</span>-th interval.</p> +<div class="admonition attention"> +<p class="admonition-title">Attention!</p> +<p>A prior execution of <a class="reference internal" href="#id15">\PolSturmIsolateZeros*{sturmname}</a> is mandatory.</p> +</div> +<p>See <a class="reference internal" href="#the-degree-nine-polynomial-with-0-99-0-999-0-9999-as-triple-roots">The degree nine polynomial with 0.99, 0.999, 0.9999 as triple +roots</a> for an example of use.</p> +</blockquote> +</div> +<div class="section" id="polsturmnbofisolatedzeros-sturmname"> +<span id="polsturmnbofisolatedzeros"></span><h3><a class="toc-backref" href="#id177"><span class="docutils literal">\PolSturmNbOfIsolatedZeros{sturmname}</span></a></h3> +<blockquote> +<p>Expands to the number of real roots of the polynomial +<span class="docutils literal"><sturmname>_0</span>, i.e. the number of distinct real roots of the +polynomial originally used to create the Sturm chain via +<a class="reference internal" href="#poltosturm-polname-sturmname">\PolToSturm{polname}{sturmname}</a>.</p> +</blockquote> +<div class="admonition warning"> +<p class="admonition-title">Warning</p> +<p>The next few macros counting roots, with or without multiplicities, +less than or equal to some value, are under evaluation and may be +removed from the package if their utility is judged to be not high +enough. They can be re-coded at user level on the basis of the other +documented package macros anyway.</p> +</div> +</div> +<div class="section" id="polsturmnbofrootsof-sturmname-lessthanorequalto-value"> +<h3><a class="toc-backref" href="#id178"><span class="docutils literal"><span class="pre">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualTo{value}</span></span></a></h3> +<blockquote> +<p>Expands to the number of distinct roots (of the polynomial used to +create the Sturm chain) less than or equal to the <span class="docutils literal">value</span> (i.e. a +number of fraction recognizable by the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> macros).</p> +<div class="admonition attention"> +<p class="admonition-title">Attention!</p> +<p><a class="reference internal" href="#polsturmisolatezeros-sturmname">\PolSturmIsolateZeros{sturmname}</a> must have been executed +beforehand.</p> +<p>And the argument is a <span class="docutils literal">sturmname</span>, not a <span class="docutils literal">polname</span> (this is +why the macro contains Sturm in its name), simply to be reminded +of the above constraint.</p> +</div> +</blockquote> +</div> +<div class="section" id="polsturmnbofrootsof-sturmname-lessthanorequaltoexpr-expression"> +<h3><a class="toc-backref" href="#id179"><span class="docutils literal"><span class="pre">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualToExpr{expression}</span></span></a></h3> +<blockquote> +<p>Expands to the number of distinct roots (of the polynomial +used to create the Sturm chain) which are less than or equal to the +given <span class="docutils literal">expression</span>.</p> +<div class="admonition attention"> +<p class="admonition-title">Attention!</p> +<p><a class="reference internal" href="#polsturmisolatezeros-sturmname">\PolSturmIsolateZeros{sturmname}</a> must have been executed +beforehand.</p> +</div> +</blockquote> +</div> +<div class="section" id="polsturmnbwithmultofrootsof-sturmname-lessthanorequalto-value"> +<h3><a class="toc-backref" href="#id180"><span class="docutils literal"><span class="pre">\PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualTo{value}</span></span></a></h3> +<blockquote> +<p>Expands to the number counted with multiplicities of the roots (of +the polynomial used to create the Sturm chain) which are less than +or equal to the given <span class="docutils literal">value</span>.</p> +<div class="admonition attention"> +<p class="admonition-title">Attention!</p> +<p><a class="reference internal" href="#id15">\PolSturmIsolateZeros*{sturmname}</a> (or the double starred +variant) must have been executed beforehand.</p> +</div> +</blockquote> +</div> +<div class="section" id="polsturmnbwithmultofrootsof-sturmname-lessthanorequaltoexpr-expression"> +<h3><a class="toc-backref" href="#id181"><span class="docutils literal"><span class="pre">\PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualToExpr{expression}</span></span></a></h3> +<blockquote> +<p>Expands to the total number of roots (counted with multiplicities) +which are less than or equal to the given <span class="docutils literal">expression</span>.</p> +<div class="admonition attention"> +<p class="admonition-title">Attention!</p> +<p><a class="reference internal" href="#id15">\PolSturmIsolateZeros*{sturmname}</a> (or the double starred +variant) must have been executed beforehand.</p> +</div> +</blockquote> +</div> +<div class="section" id="polsturmnbofrationalroots-sturmname"> +<h3><a class="toc-backref" href="#id182"><span class="docutils literal">\PolSturmNbOfRationalRoots{sturmname}</span></a></h3> +<blockquote> +<p>Expands to the number of rational roots (without multiplicities).</p> +<div class="admonition attention"> +<p class="admonition-title">Attention!</p> +<p><a class="reference internal" href="#id17">\PolSturmIsolateZeros**{sturmname}</a> must have been executed +beforehand.</p> +</div> +</blockquote> +</div> +<div class="section" id="polsturmnbofrationalrootswithmultiplicities-sturmname"> +<h3><a class="toc-backref" href="#id183"><span class="docutils literal">\PolSturmNbOfRationalRootsWithMultiplicities{sturmname}</span></a></h3> +<blockquote> +<p>Expands to the number of rational roots (counted with multiplicities).</p> +<div class="admonition attention"> +<p class="admonition-title">Attention!</p> +<p><a class="reference internal" href="#id17">\PolSturmIsolateZeros**{sturmname}</a> must have been executed +beforehand.</p> +</div> +</blockquote> +</div> +<div class="section" id="polsturmrationalroot-sturmname-k"> +<h3><a class="toc-backref" href="#id184"><span class="docutils literal"><span class="pre">\PolSturmRationalRoot{sturmname}{k}</span></span></a></h3> +<blockquote> +<p>Expands to the <span class="docutils literal">k</span>th rational root (they are ordered and indexed +starting at 1 for the most negative).</p> +<div class="admonition attention"> +<p class="admonition-title">Attention!</p> +<p><a class="reference internal" href="#id17">\PolSturmIsolateZeros**{sturmname}</a> must have been executed +beforehand.</p> +</div> +</blockquote> +</div> +<div class="section" id="polsturmrationalrootindex-sturmname-k"> +<h3><a class="toc-backref" href="#id185"><span class="docutils literal"><span class="pre">\PolSturmRationalRootIndex{sturmname}{k}</span></span></a></h3> +<blockquote> +<p>Expands to <span class="docutils literal">index</span> of the <span class="docutils literal">k</span>th rational root as part of the +ordered real roots (without multiplicities). I.e., above macro +<a class="reference internal" href="#polsturmrationalroot-sturmname-k">\PolSturmRationalRoot{sturmname}{k}</a> is equivalent to this +nested call:</p> +<pre class="literal-block">\PolSturmIsolatedZeroLeft{sturmname}{\PolSturmRationalRootIndex{sturmname}{k}}</pre> +<div class="admonition attention"> +<p class="admonition-title">Attention!</p> +<p><a class="reference internal" href="#id17">\PolSturmIsolateZeros**{sturmname}</a> must have been executed +beforehand.</p> +</div> +</blockquote> +</div> +<div class="section" id="polsturmrationalrootmultiplicity-sturmname-k"> +<h3><a class="toc-backref" href="#id186"><span class="docutils literal"><span class="pre">\PolSturmRationalRootMultiplicity{sturmname}{k}</span></span></a></h3> +<blockquote> +<p>Expands to the multiplicity of the <span class="docutils literal">k</span>th rational root.</p> +<div class="admonition attention"> +<p class="admonition-title">Attention!</p> +<p><a class="reference internal" href="#id17">\PolSturmIsolateZeros**{sturmname}</a> must have been executed +beforehand.</p> +</div> +</blockquote> +</div> +<div class="section" id="polintervalwidth-sturmname-index"> +<span id="polintervalwidth"></span><h3><a class="toc-backref" href="#id187"><span class="docutils literal"><span class="pre">\PolIntervalWidth{sturmname}{index}</span></span></a></h3> +<blockquote> +<p>The <span class="docutils literal">10^E</span> width of the current <span class="docutils literal">index</span>-th root localization +interval. Output is in <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> raw <span class="docutils literal">1/1[E]</span> format (if not zero).</p> +</blockquote> +</div> +</div> +<div class="section" id="expandable-macros-for-use-within-execution-of-polprintintervals"> +<h2><a class="toc-backref" href="#id188">Expandable macros for use within execution of <span class="docutils literal">\PolPrintIntervals</span></a></h2> +<p>These macros are for usage within custom user redefinitions of +<a class="reference internal" href="#polprintintervalsknownroot">\PolPrintIntervalsKnownRoot</a>, <a class="reference internal" href="#polprintintervalsunknownroot">\PolPrintIntervalsUnknownRoot</a>, or +in redefinitions of <a class="reference internal" href="#polprintintervalsprintexactzero">PolPrintIntervalsPrintExactZero</a> (used in the +default for the former) and of <a class="reference internal" href="#polprintintervalsprintleftendpoint">\PolPrintIntervalsPrintLeftEndPoint</a>, +<a class="reference internal" href="#polprintintervalsprintrightendpoint">\PolPrintIntervalsPrintRightEndPoint</a> (used in the default for the +latter).</p> +<div class="admonition attention"> +<p class="admonition-title">Attention!</p> +<p>Some macros formerly mentioned here got removed at 0.7: +<span class="docutils literal">\PolPrintIntervalsTheEndPoint</span>, +<span class="docutils literal"><span class="pre">\PolIfEndPointIsPositive{A}{B}</span></span>, +<span class="docutils literal"><span class="pre">\PolIfEndPointIsNegative{A}{B}</span></span>, +<span class="docutils literal"><span class="pre">\PolIfEndPointIsZero{A}{B}</span></span>.</p> +</div> +<div class="section" id="polprintintervalsthevar"> +<h3><a class="toc-backref" href="#id189"><span class="docutils literal">\PolPrintIntervalsTheVar</span></a></h3> +<blockquote> +<p>Expands to the name (default <span class="docutils literal">Z</span>) used for representing the roots, +which was passed as optional argument <span class="docutils literal">varname</span> to +<a class="reference internal" href="#polprintintervals-varname-sturmname">\PolPrintIntervals[varname]{sturmname}</a>.</p> +</blockquote> +</div> +<div class="section" id="polprintintervalstheindex"> +<h3><a class="toc-backref" href="#id190"><span class="docutils literal">\PolPrintIntervalsTheIndex</span></a></h3> +<blockquote> +<p>Expands to the index of the considered interval (indexing starting +at 1 for the leftmost interval).</p> +</blockquote> +</div> +<div class="section" id="polprintintervalsthesturmname"> +<h3><a class="toc-backref" href="#id191"><span class="docutils literal">\PolPrintIntervalsTheSturmName</span></a></h3> +<blockquote> +<p>Expands to the argument which was passed as <span class="docutils literal">sturmname</span> to +<a class="reference internal" href="#polprintintervals-varname-sturmname">\PolPrintIntervals[varname]{sturmname}</a>.</p> +</blockquote> +</div> +<div class="section" id="polprintintervalstheleftendpoint"> +<h3><a class="toc-backref" href="#id192"><span class="docutils literal">\PolPrintIntervalsTheLeftEndPoint</span></a></h3> +<blockquote> +<p>The left end point of the interval, as would be produced by +<a class="reference internal" href="#polsturmisolatedzeroleft">\PolSturmIsolatedZeroLeft</a> if it was +used with arguments the Sturm chain name and interval index returned +by <a class="reference internal" href="#polprintintervalsthesturmname">\PolPrintIntervalsTheSturmName</a> and +<a class="reference internal" href="#polprintintervalstheindex">\PolPrintIntervalsTheIndex</a>.</p> +</blockquote> +</div> +<div class="section" id="polprintintervalstherightendpoint"> +<h3><a class="toc-backref" href="#id193"><span class="docutils literal">\PolPrintIntervalsTheRightEndPoint</span></a></h3> +<blockquote> +<p>The right end point of the interval, as would be produced by +<a class="reference internal" href="#polsturmisolatedzeroright">\PolSturmIsolatedZeroRight</a> for +this Sturm chain name and index.</p> +</blockquote> +</div> +<div class="section" id="polprintintervalsthemultiplicity"> +<h3><a class="toc-backref" href="#id194"><span class="docutils literal">\PolPrintIntervalsTheMultiplicity</span></a></h3> +<blockquote> +<p>The multiplicity of the unique root within the interval of index +<a class="reference internal" href="#polprintintervalstheindex">\PolPrintIntervalsTheIndex</a>. Makes sense only if the starred (or +double-starred) variant of <a class="reference internal" href="#polsturmisolatezeros">\PolSturmIsolateZeros</a> was used earlier.</p> +</blockquote> +</div> +</div> +</div> +<div class="section" id="booleans-with-default-setting-as-indicated"> +<h1><a class="toc-backref" href="#id195">Booleans (with default setting as indicated)</a></h1> +<div class="section" id="xintverbosefalse"> +<h2><a class="toc-backref" href="#id196"><span class="docutils literal">\xintverbosefalse</span></a></h2> +<blockquote> +<p>This is actually an <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> configuration. Setting it to +<span class="docutils literal">true</span> triggers the writing of information to the log when new +polynomial or scalar variables are defined.</p> +<div class="admonition caution"> +<p class="admonition-title">Caution!</p> +<p>The macro and variable meanings as written to the log are to be +considered unstable and undocumented internal structures.</p> +</div> +</blockquote> +</div> +<div class="section" id="polnewpolverbosefalse"> +<h2><a class="toc-backref" href="#id197"><span class="docutils literal">\polnewpolverbosefalse</span></a></h2> +<blockquote> +<p>When <span class="docutils literal">\poldef</span> is used, both a variable and a function are +defined. The default <span class="docutils literal">\polnewpolverbosefalse</span> setting suppresses +the print-out to the log and terminal of the function macro meaning, +as it only duplicates the information contained in the variable +which is already printed out to the log and terminal.</p> +<p>However <a class="reference internal" href="#polgenfloatvariant-polname">\PolGenFloatVariant{polname}</a> does still print out the +information relative to the polynomial function it defines for use in +<span class="docutils literal">\xintfloateval{}</span> as there is no float polynomial variable, only the +function, and it is the only way to see its rounded coefficients +(<span class="docutils literal">\xintverbosefalse</span> suppresses also that info).</p> +<p>If set to <span class="docutils literal">true</span>, it overrides in both cases +<span class="docutils literal">\xintverbosefalse</span>. The setting only affects polynomial +declarations. Scalar variables such as those holding information on +roots obey only the <span class="docutils literal"><span class="pre">\xintverbose...</span></span> setting.</p> +<p>(new with <span class="docutils literal">0.8</span>)</p> +</blockquote> +</div> +<div class="section" id="poltypesetallfalse"> +<h2><a class="toc-backref" href="#id198"><span class="docutils literal">\poltypesetallfalse</span></a></h2> +<blockquote> +<p>If <span class="docutils literal">true</span>, <a class="reference internal" href="#poltypeset">\PolTypeset</a> will also typeset the vanishing +coefficients.</p> +</blockquote> +</div> +<div class="section" id="poltoexprallfalse"> +<h2><a class="toc-backref" href="#id199"><span class="docutils literal">\poltoexprallfalse</span></a></h2> +<blockquote> +<p>If <span class="docutils literal">true</span>, <a class="reference internal" href="#poltoexpr-pol-expr">\PolToExpr{<pol. expr.>}</a> and <a class="reference internal" href="#poltofloatexpr-pol-expr">\PolToFloatExpr{<pol. expr.>}</a> will +also include the vanishing coefficients in their outputs.</p> +</blockquote> +</div> +</div> +<div class="section" id="utilies"> +<h1><a class="toc-backref" href="#id200">Utilies</a></h1> +<div class="section" id="poldectostring-decimal-number"> +<span id="poldectostring"></span><h2><a class="toc-backref" href="#id201"><span class="docutils literal">\PolDecToString{decimal number}</span></a></h2> +<blockquote> +<p>This is a utility macro to print decimal numbers. It has been +backported to <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> (release <span class="docutils literal">1.3</span> of <span class="docutils literal">2018/03/01</span>) under +the name <span class="docutils literal">\xintDecToString</span>, and the <span class="docutils literal">polexpr</span> macro is simply +now an alias to it.</p> +<p>For example +<span class="docutils literal"><span class="pre">\PolDecToString{123.456e-8}</span></span> will expand to <span class="docutils literal">0.00000123456</span> +and <span class="docutils literal"><span class="pre">\PolDecToString{123.450e-8}</span></span> to <span class="docutils literal">0.00000123450</span> which +illustrates that trailing zeros are not trimmed. To trim trailing +zeroes, one can use <span class="docutils literal"><span class="pre">\PolDecToString{\xintREZ{#1}}</span></span>.</p> +<p>The precise behaviour of this macro may evolve in future releases of +<a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a>.</p> +</blockquote> +</div> +<div class="section" id="polexprsetup"> +<h2><a class="toc-backref" href="#id202"><span class="docutils literal">\polexprsetup</span></a></h2> +<blockquote> +<p>Serves to customize the package. Currently only two keys are +recognized:</p> +<ul class="simple"> +<li><p><span class="docutils literal">norr</span>: the postfix that <a class="reference internal" href="#id17">\PolSturmIsolateZeros**{sturmname}</a> +should append to <span class="docutils literal">sturmname</span> to declare the primitive polynomial +obtained from original one after removal of all rational roots. +The default value is <span class="docutils literal">_norr</span> (standing for “no rational roots”).</p></li> +<li><p><span class="docutils literal">sqfnorr</span>: the postfix that <a class="reference internal" href="#id17">\PolSturmIsolateZeros**{sturmname}</a> +should append to <span class="docutils literal">sturmname</span> to declare the primitive polynomial +obtained from original one after removal of all rational roots and +suppression of all multiplicities. +The default value is <span class="docutils literal">_sqf_norr</span> (standing for “square-free with +no rational roots”).</p></li> +</ul> +<p>The package executes <span class="docutils literal">\polexprsetup{norr=_norr, sqfnorr=_sqf_norr}</span> as default.</p> +</blockquote> +</div> +</div> +<div class="section" id="technicalities"> +<h1><a class="toc-backref" href="#id203">Technicalities</a></h1> +<ul> +<li><p>The catcode of the semi-colon is reset temporarily by <a class="reference internal" href="#poldef">\poldef</a> macro in case some other package (for example the French +babel module) may have made it active. This will fail though if the +whole thing was already part of a macro argument, in such cases one +can use <a class="reference internal" href="#id8">\PolDef{f}{P(x)}</a> +rather. The colon in <span class="docutils literal">:=</span> may be active with no consequences.</p></li> +<li><p>As a consequence of <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> addition and subtraction always using +least common multiples for the denominators <a class="footnote-reference brackets" href="#id40" id="id39">7</a>, user-chosen common +denominators survive additions and multiplications. For example, this:</p> +<pre class="literal-block">\poldef P(x):= 1/2 + 2/2*x + 3/2*x^3 + 4/2*x^4; +\poldef Q(x):= 1/3 + (2/3)x + (3/3)x^3 + (4/3)x^4; +\poldef PQ(x):= P*Q;</pre> +<p>gives internally the polynomial:</p> +<pre class="literal-block">1/6+4/6*x^1+4/6*x^2+6/6*x^3+20/6*x^4+16/6*x^5+9/6*x^6+24/6*x^7+16/6*x^8</pre> +<p>where all coefficients have the same denominator 6. Notice though that +<span class="docutils literal">\PolToExpr{PQ}</span> outputs the <span class="docutils literal">6/6*x^3</span> as <span class="docutils literal">x^3</span> because (by +default) it recognizes and filters out coefficients equal to one or +minus one (since release <span class="docutils literal">0.3</span>). One can use for example +<span class="docutils literal">\PolToCSV{PQ}</span> to see the internally stored coefficients.</p> +<dl class="footnote brackets"> +<dt class="label" id="id40"><span class="brackets"><a class="fn-backref" href="#id39">7</a></span></dt> +<dd><p>prior to <span class="docutils literal">0.4.1</span>, <span class="docutils literal">polexpr</span> used to temporarily patch +during the parsing of polynomials the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> macros. This +patch was backported to <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> at release <span class="docutils literal">1.3</span>.</p> +</dd> +</dl> +</li> +<li><p><a class="reference internal" href="#poldiff-polname-1-polname-2">\PolDiff{polname_1}{polname_2}</a> always applies <span class="docutils literal">\xintIrr</span> to the +resulting coefficients, except that the <em>power of ten</em> part <span class="docutils literal">[N]</span> +(for example an input in scientific notation such as <span class="docutils literal">1.23e5</span> gives +<span class="docutils literal">123/1[3]</span> internally in xintfrac) is not taken into account in the +reduction of the fraction. This is tentative and may change.</p> +<p>Same remark for <a class="reference internal" href="#polantidiff-polname-1-polname-2">\PolAntiDiff{polname_1}{polname_2}</a>.</p> +</li> +<li><p>Currently, the package stores all coefficients from index <span class="docutils literal">0</span> to +index equal to the polynomial degree inside a single macro, as a list. +This data structure is obviously very inefficient for polynomials of +high degree and few coefficients (as an example with <span class="docutils literal">\poldef <span class="pre">f(x):=x^1000</span> + x^500;</span> the subsequent definition <span class="docutils literal">\poldef <span class="pre">g(x):=</span> <span class="pre">f(x)^2;</span></span> will do of the order of 1,000,000 multiplications and +additions involvings only zeroes... which does take time). This +may change in the future.</p></li> +<li><p>As is to be expected internal structures of the package are barely +documented and unstable. Don't use them.</p></li> +</ul> +</div> +<div class="section" id="change-log"> +<h1><a class="toc-backref" href="#id204">CHANGE LOG</a></h1> +<ul> +<li><p>v0.1 (2018/01/11): initial release. Features:</p> +<ul class="simple"> +<li><p>The <a class="reference internal" href="#poldef">\poldef</a> parser itself,</p></li> +<li><p>Differentiation and anti-differentiation,</p></li> +<li><p>Euclidean division and GCDs,</p></li> +<li><p>Various utilities such as <a class="reference internal" href="#polfromcsv">\PolFromCSV</a>, +<a class="reference internal" href="#polmapcoeffs">\PolMapCoeffs</a>, +<a class="reference internal" href="#poltocsv">\PolToCSV</a>, <a class="reference internal" href="#poltoexpr">\PolToExpr</a>, ...</p></li> +</ul> +<p>Only one-variable polynomials so far.</p> +</li> +<li><p>v0.2 (2018/01/14)</p> +<ul class="simple"> +<li><p>Fix: <span class="docutils literal">"README thinks \numexpr recognizes ^ operator"</span>.</p></li> +<li><p>Convert README to reStructuredText markup.</p></li> +<li><p>Move main documentation from README to separate <span class="docutils literal">polexpr.txt</span> file.</p></li> +<li><p>Provide <span class="docutils literal">polexpr.html</span> as obtained via <a class="reference external" href="http://docutils.sourceforge.net/docs/index.html">DocUtils</a> <span class="docutils literal">rst2html.py</span>.</p></li> +<li><p>Convert README to (CTAN compatible) Markdown markup.</p></li> +</ul> +<p>Due to lack of available time the test suite might not be extensive +enough. Bug reports are very welcome!</p> +</li> +<li><p>v0.3 (2018/01/17)</p> +<ul> +<li><p>bug fixes:</p> +<ul> +<li><p>the <span class="docutils literal">0.1</span> <a class="reference internal" href="#polevalat">\PolEval</a> accepted expressions for its second +argument, but this was removed by mistake at <span class="docutils literal">0.2</span>. Restored.</p> +<p><strong>Attention</strong>: at <span class="docutils literal">0.4</span> this has been reverted again, and +<a class="reference internal" href="#polevalatexpr">\PolEval{P}\AtExpr{foo}</a> syntax is needed for +using expressions in the second argument.</p> +</li> +</ul> +</li> +<li><p>incompatible or breaking changes:</p> +<ul class="simple"> +<li><p><a class="reference internal" href="#poltoexpr">\PolToExpr</a> now by default uses <em>descending</em> +powers (it also treats differently coefficients equal to 1 or -1.) +Use <a class="reference internal" href="#id33">\PolToExpr*</a> for <em>ascending</em> powers.</p></li> +<li><p><a class="reference internal" href="#polevalat">\PolEval</a> reduced the output to smallest terms, +but as this is costly with big fractions and not needed if e.g. +wrapped in an <span class="docutils literal">\xintRound</span> or <span class="docutils literal">\xintFloat</span>, this step has been +removed; the former meaning is available as <a class="reference internal" href="#polevalreducedat">\PolEvalReduced</a>.</p></li> +</ul> +</li> +<li><p>new (or newly documented) macros:</p> +<ul class="simple"> +<li><p><a class="reference internal" href="#poltypesetcmd">\PolTypesetCmd</a></p></li> +<li><p><a class="reference internal" href="#poltypesetcmdprefix">\PolTypesetCmdPrefix</a></p></li> +<li><p><a class="reference internal" href="#poltypesetmonomialcmd">\PolTypesetMonomialCmd</a></p></li> +<li><p><a class="reference internal" href="#polevalreducedat">\PolEvalReducedAt</a></p></li> +<li><p><a class="reference internal" href="#poltofloatexpr">\PolToFloatExpr</a></p></li> +<li><p><a class="reference internal" href="#poltoexproneterm">\PolToExprOneTerm</a></p></li> +<li><p><a class="reference internal" href="#poltofloatexproneterm">\PolToFloatExprOneTerm</a></p></li> +<li><p><a class="reference internal" href="#poltoexprcmd">\PolToExprCmd</a></p></li> +<li><p><a class="reference internal" href="#id36">\PolToFloatExprCmd</a></p></li> +<li><p><a class="reference internal" href="#poltoexprtermprefix">\PolToExprTermPrefix</a></p></li> +<li><p><a class="reference internal" href="#poltoexprvar">\PolToExprVar</a></p></li> +<li><p><a class="reference internal" href="#poltoexprtimes">\PolToExprTimes</a></p></li> +</ul> +</li> +<li><p>improvements:</p> +<ul> +<li><p>documentation has a table of contents, internal hyperlinks, +standardized signature notations and added explanations.</p></li> +<li><p>one can do <span class="docutils literal"><span class="pre">\PolLet{g}={f}</span></span> or <span class="docutils literal"><span class="pre">\PolLet{g}{f}</span></span>.</p></li> +<li><p><span class="docutils literal">\PolToExpr{f}</span> is highly customizable.</p></li> +<li><p><a class="reference internal" href="#poldef">\poldef</a> and other defining macros prepare the polynomial +functions for usage within <span class="docutils literal">\xintthefloatexpr</span> (or +<span class="docutils literal">\xintdeffloatvar</span>). Coefficients are pre-rounded to the +floating point precision. Indispensible for numerical algorithms, +as exact fractions, even reduced, quickly become very big. See the +documentation about how to use the exact polynomials also in +floating point context.</p> +<p><strong>Attention</strong>: this has been reverted at <span class="docutils literal">0.4</span>. The macro +<a class="reference internal" href="#polgenfloatvariant">\PolGenFloatVariant</a> must be used for +generation floating point polynomial functions.</p> +</li> +</ul> +</li> +</ul> +</li> +<li><p>v0.3.1 (2018/01/18)</p> +<p>Fixes two typos in example code included in the documentation.</p> +</li> +<li><p>v0.4 (2018/02/16)</p> +<ul> +<li><p>bug fixes:</p> +<ul class="simple"> +<li><p>when Euclidean division gave a zero remainder, the internal +representation of this zero polynomial could be faulty; this +could cause mysterious bugs in conjunction with other package +macros such as <a class="reference internal" href="#polmapcoeffs">\PolMapCoeffs</a>.</p></li> +<li><p><a class="reference internal" href="#polgcd">\PolGCD</a> was buggy in case of first polynomial being +of lesser degree than the second one.</p></li> +</ul> +</li> +<li><p>breaking changes:</p> +<ul> +<li><p>formerly <a class="reference internal" href="#polevalat">\PolEval{P}\At{foo}</a> allowed <span class="docutils literal">foo</span> to +be an expression, which was transparently handled via +<span class="docutils literal">\xinttheexpr</span>. Now, <span class="docutils literal">foo</span> must be a fraction (or a macro +expanding to such) in the format acceptable by <span class="docutils literal">xintfrac.sty</span> +macros. Use <a class="reference internal" href="#polevalatexpr">\PolEval{P}\AtExpr{foo}</a> for more +general arguments using expression syntax. E.g., if <span class="docutils literal">foo</span> is the +name of a variable known to <span class="docutils literal">\xintexpr</span>.</p> +<p>The same holds for <a class="reference internal" href="#polevalreducedat">\PolEvalReduced</a> +and <a class="reference internal" href="#polfloatevalat">\PolFloatEval</a>.</p> +</li> +<li><p>the <span class="docutils literal">3.0</span> automatic generation of floating point variants has +been reverted. Not only do <em>not</em> the package macros automatically +generate floating point variants of newly created polynomials, +they actually make pre-existing such variant undefined.</p> +<p>See <a class="reference internal" href="#polgenfloatvariant">\PolGenFloatVariant</a>.</p> +</li> +</ul> +</li> +<li><p>new non-expandable macros:</p> +<ul class="simple"> +<li><p><a class="reference internal" href="#polgenfloatvariant">\PolGenFloatVariant</a></p></li> +<li><p><a class="reference internal" href="#polgloballet">\PolGlobalLet</a></p></li> +<li><p><a class="reference internal" href="#poltypesetone">\PolTypesetOne</a></p></li> +<li><p><a class="reference internal" href="#polquo">\PolQuo</a></p></li> +<li><p><a class="reference internal" href="#polrem">\PolRem</a></p></li> +<li><p><a class="reference internal" href="#poltosturm">\PolToSturm</a></p></li> +<li><p><a class="reference internal" href="#id12">\PolToSturm*</a></p></li> +<li><p><a class="reference internal" href="#polsettosturmchainsignchangesat">\PolSetToSturmChainSignChangesAt</a></p></li> +<li><p><a class="reference internal" href="#polsettonbofzeroswithin">\PolSetToNbOfZerosWithin</a></p></li> +<li><p><a class="reference internal" href="#polsturmisolatezeros">\PolSturmIsolateZeros</a></p></li> +<li><p><a class="reference internal" href="#polrefineinterval">\PolRefineInterval*</a></p></li> +<li><p><a class="reference internal" href="#polrefineinterval-n">\PolRefineInterval[N]</a></p></li> +<li><p><a class="reference internal" href="#polensureintervallength">\PolEnsureIntervalLength</a></p></li> +<li><p><a class="reference internal" href="#polensureintervallengths">\PolEnsureIntervalLengths</a></p></li> +<li><p><a class="reference internal" href="#polprintintervals">\PolPrintIntervals</a></p></li> +<li><p><a class="reference internal" href="#polprintintervalsprintexactzero">\PolPrintIntervalsPrintExactZero</a></p></li> +<li><p><a class="reference internal" href="#polprintintervalsprintleftendpoint">\PolPrintIntervalsPrintLeftEndPoint</a></p></li> +<li><p><a class="reference internal" href="#polprintintervalsprintrightendpoint">\PolPrintIntervalsPrintRightEndPoint</a></p></li> +<li><p><a class="reference internal" href="#id23">\PolReduceCoeffs*</a></p></li> +<li><p><a class="reference internal" href="#polmakemonic">\PolMakeMonic</a></p></li> +</ul> +</li> +<li><p>new expandable macros:</p> +<ul class="simple"> +<li><p><a class="reference internal" href="#poltoexpronetermstylea">\PolToExprOneTermStyleA</a></p></li> +<li><p><a class="reference internal" href="#polifcoeffisplusorminusone">\PolIfCoeffIsPlusOrMinusOne</a></p></li> +<li><p><a class="reference internal" href="#polleadingcoeff">\PolLeadingCoeff</a></p></li> +<li><p><a class="reference internal" href="#polsturmchainlength">\PolSturmChainLength</a></p></li> +<li><p><a class="reference internal" href="#polsturmnbofisolatedzeros">\PolSturmNbOfIsolatedZeros</a></p></li> +<li><p><a class="reference internal" href="#polsturmifzeroexactlyknown">\PolSturmIfZeroExactlyKnown</a></p></li> +<li><p><a class="reference internal" href="#polsturmisolatedzeroleft">\PolSturmIsolatedZeroLeft</a></p></li> +<li><p><a class="reference internal" href="#polsturmisolatedzeroright">\PolSturmIsolatedZeroRight</a></p></li> +<li><p><span class="docutils literal">\PolPrintIntervalsTheEndPoint</span> (removed at 0.7)</p></li> +<li><p><a class="reference internal" href="#polprintintervalstheindex">\PolPrintIntervalsTheIndex</a></p></li> +<li><p><span class="docutils literal">\PolIfEndPointIsPositive</span> (removed at 0.7)</p></li> +<li><p><span class="docutils literal">\PolIfEndPointIsNegative</span> (removed at 0.7)</p></li> +<li><p><span class="docutils literal">\PolIfEndPointIsZero</span> (removed at 0.7)</p></li> +<li><p><a class="reference internal" href="#polintervalwidth">\PolIntervalWidth</a></p></li> +<li><p><a class="reference internal" href="#poldectostring">\PolDecToString</a></p></li> +</ul> +</li> +<li><p>improvements:</p> +<p>The main new feature is implementation of the <a class="reference external" href="https://en.wikipedia.org/wiki/Sturm%27s_theorem">Sturm algorithm</a> +for localization of the real roots of polynomials.</p> +</li> +</ul> +</li> +<li><p>v0.4.1 (2018/03/01)</p> +<p>Synced with xint 1.3.</p> +</li> +<li><p>v0.4.2 (2018/03/03)</p> +<p>Documentation fix.</p> +</li> +<li><p>v0.5 (2018/04/08)</p> +<ul class="simple"> +<li><p>bug fixes:</p> +<ul> +<li><p><a class="reference internal" href="#polget-polname-fromarray-macro">\PolGet{polname}\fromarray\macro</a> crashed when <span class="docutils literal">\macro</span> was +an <a class="reference external" href="http://www.ctan.org/pkg/xint">xinttools</a> array macro with no items. It now produces the zero +polynomial.</p></li> +</ul> +</li> +<li><p>breaking changes:</p> +<ul> +<li><p><a class="reference internal" href="#poltosturm">\PolToSturm</a> creates primitive integer coefficients polynomials. +This speeds up localization of roots via +<a class="reference internal" href="#polsturmisolatezeros">\PolSturmIsolateZeros</a>. In case of user protests the author +will make available again the code producing the bona fide Sturm +polynomials as used formerly.</p></li> +<li><p>polynomials created from <a class="reference internal" href="#polfromcsv">\PolFromCSV</a> or <a class="reference internal" href="#polget">\PolGet</a> +get their coefficients normalized via <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a>'s <span class="docutils literal">\xintRaw</span>.</p></li> +</ul> +</li> +<li><p>experimental change:</p> +<ul> +<li><p>optional argument to <a class="reference internal" href="#polsturmisolatezeros">\PolSturmIsolateZeros</a> (see <a class="reference internal" href="#the-degree-41-polynomial-with-2-1-9-1-8-0-0-1-1-9-2-as-roots">The +degree 41 polynomial with -2, -1.9, -1.8, ..., 0, 0.1, ..., 1.9, 2 +as roots</a> for usage). It will presumably be replaced in future by +an interval specification.</p></li> +</ul> +</li> +<li><p>new non-expandable macro:</p> +<ul> +<li><p><a class="reference internal" href="#polmakeprimitive">\PolMakePrimitive</a></p></li> +</ul> +</li> +<li><p>new expandable macro:</p> +<ul> +<li><p><a class="reference internal" href="#policontent">\PolIContent</a></p></li> +</ul> +</li> +</ul> +</li> +<li><p>v0.5.1 (2018/04/22)</p> +<ul class="simple"> +<li><p>new feature:</p> +<ul> +<li><p>the character <span class="docutils literal">'</span> can be used in polynomial names.</p></li> +</ul> +</li> +</ul> +</li> +<li><p>v0.6 (2018/11/20)</p> +<ul class="simple"> +<li><p>bugfix:</p> +<ul> +<li><p>the starred variant <a class="reference internal" href="#id13">\PolToSturm*{polname}{sturmname}</a> was +broken. On the occasion of the fix, its meaning has been modified, +see its documentation.</p></li> +<li><p>using <a class="reference internal" href="#poltosturm">\PolToSturm</a> with a constant polynomial +caused a division by zero error.</p></li> +</ul> +</li> +<li><p>new macro:</p> +<ul> +<li><p><a class="reference internal" href="#id14">\PolSturmIsolateZeros*</a> +acts like the <a class="reference internal" href="#polsturmisolatezeros">non-starred variant</a> then computes all the multiplicities.</p></li> +</ul> +</li> +<li><p>new expandable macros:</p> +<ul> +<li><p><a class="reference internal" href="#polsturmisolatedzeromultiplicity-sturmname-index">\PolSturmIsolatedZeroMultiplicity{sturmname}{index}</a></p></li> +<li><p><a class="reference internal" href="#polsturmnbofrootsof-sturmname-lessthanorequalto-value">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualTo{value}</a></p></li> +<li><p><a class="reference internal" href="#polsturmnbofrootsof-sturmname-lessthanorequaltoexpr-expression">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualToExpr{expression}</a></p></li> +<li><p><a class="reference internal" href="#polsturmnbwithmultofrootsof-sturmname-lessthanorequalto-value">\PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualTo{value}</a></p></li> +<li><p><a class="reference internal" href="#polsturmnbwithmultofrootsof-sturmname-lessthanorequaltoexpr-expression">\PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualToExpr{expression}</a></p></li> +</ul> +</li> +</ul> +</li> +<li><p>v0.7 (2018/12/08), v0.7.1 (bugfix), v0.7.2 (2nd bugfix) (2018/12/09)</p> +<ul class="simple"> +<li><p>breaking changes:</p> +<ul> +<li><p>although <a class="reference internal" href="#polprintintervals-varname-sturmname">\PolPrintIntervals[varname]{sturmname}</a> default output +remains the same, some auxiliary macros for user-customization +have been removed: <span class="docutils literal">\PolPrintIntervalsTheEndPoint</span>, +<span class="docutils literal"><span class="pre">\PolIfEndPointIsPositive{A}{B}</span></span>, +<span class="docutils literal"><span class="pre">\PolIfEndPointIsNegative{A}{B}</span></span>, and +<span class="docutils literal"><span class="pre">\PolIfEndPointIsZero{A}{B}</span></span>.</p></li> +</ul> +</li> +<li><p>bugfix:</p> +<ul> +<li><p>it could happen that, contrarily to documentation, an interval +computed by <a class="reference internal" href="#polsturmisolatezeros-sturmname">\PolSturmIsolateZeros{sturmname}</a> had zero as an +endpoint,</p></li> +<li><p><a class="reference internal" href="#polensureintervallength-sturmname-index-e">\PolEnsureIntervalLength{sturmname}{index}{E}</a> could under +certain circumstances erroneously replace a non-zero root by +zero,</p></li> +<li><p><a class="reference internal" href="#polensureintervallengths-sturmname-e">\PolEnsureIntervalLengths{sturmname}{E}</a> crashed when used with +a polynomial with no real roots, hence for which no isolation intervals +existed (thanks to Thomas Söll for report).</p></li> +</ul> +</li> +<li><p>new macros:</p> +<ul> +<li><p><a class="reference internal" href="#id17">\PolSturmIsolateZeros**{sturmname}</a></p></li> +<li><p><a class="reference internal" href="#polsturmisolatezerosgetmultiplicitiesandrationalroots-sturmname">\PolSturmIsolateZerosGetMultiplicitiesAndRationalRoots{sturmname}</a></p></li> +<li><p><a class="reference internal" href="#polsturmisolatezerosandfindrationalroots-sturmname">\PolSturmIsolateZerosAndFindRationalRoots{sturmname}</a></p></li> +<li><p><a class="reference internal" href="#polexprsetup">\polexprsetup</a></p></li> +<li><p><a class="reference internal" href="#id21">\PolPrintIntervals*</a></p></li> +<li><p><a class="reference internal" href="#polprintintervalsnorealroots">\PolPrintIntervalsNoRealRoots</a></p></li> +<li><p><a class="reference internal" href="#polprintintervalsbeginenv">\PolPrintIntervalsBeginEnv</a></p></li> +<li><p><a class="reference internal" href="#polprintintervalsendenv">\PolPrintIntervalsEndEnv</a></p></li> +<li><p><a class="reference internal" href="#polprintintervalsknownroot">\PolPrintIntervalsKnownRoot</a></p></li> +<li><p><a class="reference internal" href="#polprintintervalsunknownroot">\PolPrintIntervalsUnknownRoot</a></p></li> +<li><p><a class="reference internal" href="#polprintintervalsprintmultiplicity">\PolPrintIntervalsPrintMultiplicity</a></p></li> +</ul> +</li> +<li><p>new expandable macros:</p> +<ul> +<li><p><a class="reference internal" href="#polsturmnbofrationalroots-sturmname">\PolSturmNbOfRationalRoots{sturmname}</a></p></li> +<li><p><a class="reference internal" href="#polsturmnbofrationalrootswithmultiplicities-sturmname">\PolSturmNbOfRationalRootsWithMultiplicities{sturmname}</a></p></li> +<li><p><a class="reference internal" href="#polsturmrationalroot-sturmname-k">\PolSturmRationalRoot{sturmname}{k}</a></p></li> +<li><p><a class="reference internal" href="#polsturmrationalrootindex-sturmname-k">\PolSturmRationalRootIndex{sturmname}{k}</a></p></li> +<li><p><a class="reference internal" href="#polsturmrationalrootmultiplicity-sturmname-k">\PolSturmRationalRootMultiplicity{sturmname}{k}</a></p></li> +<li><p><a class="reference internal" href="#polprintintervalsthevar">\PolPrintIntervalsTheVar</a></p></li> +<li><p><a class="reference internal" href="#polprintintervalsthesturmname">\PolPrintIntervalsTheSturmName</a></p></li> +<li><p><a class="reference internal" href="#polprintintervalsthemultiplicity">\PolPrintIntervalsTheMultiplicity</a></p></li> +</ul> +</li> +</ul> +</li> +<li><p>v0.7.3 (2019/02/04)</p> +<ul class="simple"> +<li><p>bugfix:</p> +<ul> +<li><p>Debugging information not destined to user showed in log if root +finding was done under <span class="docutils literal">\xintverbosetrue</span> regime.</p></li> +<li><p><a class="reference internal" href="#polprintintervalsthevar">\PolPrintIntervalsTheVar</a> remained defined after +<a class="reference internal" href="#polprintintervals">\PolPrintIntervals</a> but was left undefined after +<a class="reference internal" href="#id21">\PolPrintIntervals*</a> (reported by Jürgen Gilg). Now remains +defined in both cases, and <a class="reference internal" href="#polprintintervalsthesturmname">\PolPrintIntervalsTheSturmName</a> +also.</p></li> +<li><p>Polynomial names ending in digits caused errors (reported by Thomas +Söll).</p></li> +</ul> +</li> +</ul> +</li> +<li><p>v0.7.4 (2019/02/12)</p> +<ul class="simple"> +<li><p>bugfix:</p> +<ul> +<li><p>20000000000 is too big for <span class="docutils literal">\numexpr</span>, shouldn't I know that? +Thanks to Jürgen Gilg for report.</p></li> +</ul> +</li> +</ul> +</li> +<li><p>v0.7.5 (2020/01/31)</p> +<p>Synced with xintexpr 1.4. Requires it.</p> +</li> +<li><p>v0.8 (2021/03/29)</p> +<p>Synced with xintexpr 1.4d. Requires it.</p> +<ul class="simple"> +<li><p>breaking changes:</p> +<ul> +<li><p>As the usability of character <span class="docutils literal">'</span> in names has been extended +from <span class="docutils literal">\poldef</span> to also generally <span class="docutils literal">\xintexpr</span>, <span class="docutils literal">\xintdefvar</span>, +and <span class="docutils literal">\xintdeffunc</span>, it breaks there the infix operators +<span class="docutils literal">'and'</span>, <span class="docutils literal">'or'</span>, <span class="docutils literal">'xor'</span> and <span class="docutils literal">'mod'</span>. See the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> +documentation for the <span class="docutils literal">&&</span>, <span class="docutils literal">||</span>, <span class="docutils literal">xor()</span> and <span class="docutils literal">/:</span> +alternatives.</p></li> +<li><p><a class="reference internal" href="#poltoexpr">\PolToExpr</a> by default uses a catcode 12 +<span class="docutils literal">^</span>. See its documentation and the new configuration +<a class="reference internal" href="#poltoexprcaret">\PolToExprCaret</a>.</p></li> +</ul> +</li> +<li><p>deprecated:</p> +<ul> +<li><p>Usage of <span class="docutils literal">P/Q</span> for the euclidean quotient of two polynomials is +deprecated. Start using <span class="docutils literal">quo(P,Q)</span> in its place.</p></li> +</ul> +</li> +<li><p>bugfix:</p> +<ul> +<li><p>The <span class="docutils literal">\xintglobaldefstrue</span> setting was obeyed only partially +by the polexpr macros defining polynomials.</p></li> +<li><p>The <span class="docutils literal">\xintexpr</span> variables storing the values of the extremities +of the intervals as found by <a class="reference internal" href="#polsturmisolatezeros">\PolSturmIsolateZeros</a> were not updated at 0.7.5 to the +xintexpr 1.4 format and thus caused low-level TeX errors if used.</p></li> +<li><p>Attempting to use in <span class="docutils literal">\poldef</span> a function previously declared +via <span class="docutils literal">\xintdeffunc</span> which made usage of the indexing or slicing +"ople" syntax typically caused <span class="docutils literal">TeX capacity exceeded</span> error. +Indeed 0.7.5 only partially made polexpr able to cope with the +extended possibilities for xintexpr 1.4 user-declared functions. +Hopefully <span class="docutils literal">0.8</span> achieves full functionality in this context.</p></li> +</ul> +</li> +<li><p>new macros:</p> +<ul> +<li><p><a class="reference internal" href="#polnewpolverbosefalse">\polnewpolverbosefalse</a></p></li> +<li><p><a class="reference internal" href="#poltoexprcaret">\PolToExprCaret</a></p></li> +<li><p><a class="reference internal" href="#poltoexprinvar">\PolToExprInVar</a></p></li> +<li><p>alongside the major new functionalities described in the next item +<a class="reference internal" href="#poltypeset">\PolTypeset</a> and <a class="reference internal" href="#poltoexpr">\PolToExpr</a> have +been enhanced to accept as argument a general expression and not +only a pre-declared polynomial name.</p></li> +</ul> +</li> +<li><p>new features:</p> +<ul> +<li><p>The package is usable under Plain and probably most any TeX format, +and not only under LaTeX.</p></li> +<li><p>The core of the package has been rewritten entirely in order to +start letting <span class="docutils literal">\xintexpr</span> recognize a polynomial type as a genuine +variable. This has allowed:</p> +<ul> +<li><p>to solve the reduced inter-operability problems between polexpr +and <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> which arose as consequences to the deep <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> <span class="docutils literal">1.4</span> +evolution,</p></li> +<li><p>to make available most of the functionality associated to +expandable macros directly in the <span class="docutils literal">\xinteval</span> syntax as +operators or functions,</p></li> +<li><p>to provide (expandable) functional interface in <span class="docutils literal">\xinteval</span> to +features previously available only via (for some, non-expandable) +macro interface such as gcd computations.</p></li> +</ul> +</li> +</ul> +</li> +</ul> +<p>See the updated <a class="reference internal" href="#quick-syntax-overview">Quick syntax overview</a> and then <a class="reference internal" href="#polexpr08">the extended syntax +description</a>.</p> +</li> +</ul> +</div> +<div class="section" id="acknowledgments"> +<h1><a class="toc-backref" href="#id205">Acknowledgments</a></h1> +<p>Thanks to Jürgen Gilg whose question about <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> usage for +differentiating polynomials was the initial trigger leading to this +package, and to Jürgen Gilg and Thomas Söll for testing it on some +concrete problems.</p> +<p>Renewed thanks to them on occasion of the <span class="docutils literal">0.6</span>, <span class="docutils literal">0.7</span>, and <span class="docutils literal">0.8</span> +releases for their continued interest.</p> +<p>See README.md for the License.</p> +</div> +</div> +</body> +</html> |