texlive[46318] Master/texmf-dist: polexpr (15jan18)
commits+karl at tug.org
commits+karl at tug.org
Mon Jan 15 23:20:09 CET 2018
Revision: 46318
http://tug.org/svn/texlive?view=revision&revision=46318
Author: karl
Date: 2018-01-15 23:20:09 +0100 (Mon, 15 Jan 2018)
Log Message:
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polexpr (15jan18)
Modified Paths:
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trunk/Master/texmf-dist/tex/latex/polexpr/polexpr.sty
Added Paths:
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trunk/Master/texmf-dist/doc/latex/polexpr/README.md
trunk/Master/texmf-dist/doc/latex/polexpr/polexpr.html
trunk/Master/texmf-dist/doc/latex/polexpr/polexpr.txt
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trunk/Master/texmf-dist/doc/latex/polexpr/README
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--*- fill-column: 72; mode: text; -*-
-
-Package polexpr
-===============
-
-License
--------
-
-Copyright (C) 2018 Jean-Francois Burnol
-
-See documentation of package 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-Francois Burnol.
-
-This Work consists of the package file polexpr.sty and this README.
-
-
-Abstract
---------
-
-The package provides "\poldef": a parser of polynomial expressions
-based upon the "\xintdeffunc" mechanism of package xintexpr.
-
-The syntax is
-
- \poldef <name>(x):=<expression in variable x>;
-
-where in place of "x" an arbitrary letter is authorized. The expression
-uses the operations of algebra (including composition of functions) with
-standard operators, fractional numbers (possibly in scientific notation)
-and previously defined polynomial functions or other constructs as
-recognized by the \xintexpr numerical parser.
-
-The so-defined name() \xintexpr-function is also known to the package
-via its polynomial coefficients, thus allowing dedicated macros to
-implement polynomial algorithmics.
-
-Examples
---------
-
-\poldef f(x):= 1-x+x^2;
-
-This defines polynomial "f". Polynomial names must start with a letter
-and may contain letters, digits, and underscores. The variable must be a
-single letter. The colon character is optional. The semi-colon at end of
-expression is mandatory.
-
-\PolDef{f}{1-x+x^2} does the same as \poldef f(x):= 1-x+x^2;
-To use another letter than x in the expression, one must pass it as
-an extra optional argument to \PolDef. Useful if the semi-colon has
-been assigned some non-standard catcode by some package.
-
-\PolLet{g}{f} saves a copy of "f" under name "g".
-
-\poldef f(z):= f(z)^2; redefines "f" in terms of itself.
-
-\poldef f(T):= f(f(T)); again redefines "f" in terms of its (new) self.
-
-\poldef k(z):= f(z)-g(g(z)^2)^2; should now define the zero
-polynomial... Let's check:
-\[ k(z) = \PolTypeset[z]{k} \]
-
-\PolDiff{f}{df_dx} sets "df_dx" to the derivative of "f".
-
-\PolDiff{df_dx}{f_xx} obtains second derivative
-
-\PolDiff[3]{f}{d3f_dx3} computes directly the third derivative
-
-$f(z) = \PolTypeset[z]{f} $\newline
-$f'(z) = \PolTypeset[z]{df_dx}$\newline
-$f''(z) = \PolTypeset[z]{f_xx}$\newline
-$f'''(z)= \PolTypeset[z]{d3f_dx3}$\par
-
-*Important*: the package does not currently know rational functions.
-and "/" in a parsed polynomial expression does the Euclidean quotient:
-
- (1-x^2)/(1-x) does give 1+x but (1/(1-x))*(1-x^2) evaluates to zero.
-
-*Attention*: "1/2 x" skips the space and is treated like "1/(2x)"
-because of the tacit multiplication rules of \xintexpr. But this means
-it gives zero! Thus one must use (1/2)x or 1/2*x or (1/2)*x for
-disambiguation.
-
-\poldef k(x):= (x-1)(x-2)(x-3)(x-4)/(x^2-5x+4);%
-
-\PolTypeset{k} gives the expected x^2-5x+6
-
-\poldef f1(x):= 25(x-1)(x^2-2)(x-3)(x-4)(x-5);%
-\poldef f2(x):= 37(x-1)(x^2-2)(x-6)(x-7)(x-8);%
-
-\PolGCD{f1}{f2}{k} sets "k" to the (unitary) GCD of "f1" and "f2".
-
-\PolToExpr{k} expandably gives 2-2*x^1-1*x^2+1*x^3 for console
-or file output (this is Maple-compatible input syntax).
-
-Non-expandable macros
----------------------
-
-\poldef name(letter):= polynomial expression using letter;
- This evaluates the polynomial expression and stores the
- coefficients in a private structure accessible later via other
- package macros, under the user-chosen "name". Of course
- previously defined polynomials are allowed in a new expression.
- Names must start with a letter and are constituted of letters,
- digits and underscore characters. See Examples above.
-
- As a side effect the function name() is recognized as a genuine
- \xintexpr...\relax function for (exact) numerical evaluation. It
- computes values not according to the original expression but via
- the Horner scheme corresponding to the polynomial coefficients.
-
- The original expression is lost after parsing, and in particular
- the package provides no way to typeset it. This has to be done
- manually, if needed.
-
-\PolDef{name}{P(x)}
- Does the same but the variable is assumed to be "x". To use another
- letter, pass it as first optional argument.
-
-\PolLet{g}{f}
- Makes a copy of already defined polynomial f to new one g.
- Same effect as \PolDef{g}{f(x)} but faster.
-
-\PolAssign{f}\toarray\Array
- Defines a one-argument expandable macro \Array{#1} which expands
- to the (raw) #1th polynomial coefficient.
-
- - Attention, coefficients here are indexed starting at 1.
-
- - With #1=-1, -2, ..., \Array{#1} returns leading coefficients.
-
- - With #1=0, returns the number of coefficients, i.e. 1+degree(f)
- for non-zero polynomials.
-
- - Out-of-range #1's return 0/1[0].
-
-\PolGet{f}\fromarray\Array
- Does the reverse operation to \PolAssign{f}\toarray\Array. No error
- checks on validity of coefficients as numbers. Each \Array{index}
- is expanded in an \edef before being assigned to a coefficient.
- Leading zero coefficients are removed from the polynomial.
-
- (contrived) Example: \xintAssignArray{1}{-2}{5}{-3}\to\foo
- \PolGet{f}\fromarray\foo
- This will define "f" as would have \poldef f(x):=1-2x+5x^2-3x^3;
- However the coefficients are still in their original form (i.e.
- they were not subjected to \xintRaw or similar xintfrac macro.)
-
-\PolFromCSV{f}{comma separated coefficients}
- Defines a polynomial directly from the comma separated list (or a
- macro expanding to such a list) of its coefficients, the constant
- term being the first item. No validity checks. Spaces from the list
- argument are trimmed. List items are expanded in an \edef, but
- currently they are left in their original form like e.g. 1.5e3
- which is not converted to 15/1[2] "raw" xintfrac format (this may
- change).
-
- Leading zero coefficients are removed:
- \PolFromCSV{J}{0, 0, 0, 0, 0, 0, 0, 0, 0, 0} defines the zero
- polynomial, which has only one (zero) coefficient.
-
- See also expandable macro \PolToCSV.
-
-\PolTypeset[x]{name}
- Typesets in descending powers in math mode using the specified
- variable (default x.) By default zero coefficients are skipped
- (issue \poltypesetalltrue to get all of them in output).
-
- Macros \PolTypesetCmd, \PolTypesetPlus, \PolTypesetMonomial
- can help configure the output. See the package code.
-
-\PolTypeset*[x]{name}
- Typesets in ascending powers.
-
-\PolDiff{f1}{f2}
- This sets f2 to the first derivative of f1. It is allowed to issue
- \PolDiff{f}{f}, effectively replacing f by f'.
-
- Coefficients of the result f2 are irreducible fractions
- (see `Technicalities`_ for the whole story.)
-
-\PolDiff[N]{f1}{f2}
- This sets f2 to the Nth derivative of f1. Identical arguments
- is allowed. With N=0, same effect as \PolLet{f2}{f1}.
- With negative N, switched to using \PolAntiDiff.
-
-\PolAntiDiff{f1}{f2}
- This sets f2 to the primitive of f1 vanishing at zero.
-
- Coefficients of the result f2 are irreducible fractions
- (see `Technicalities`_ for the whole story.)
-
-\PolAntiDiff[N]{f1}{f2}
- This sets f2 to the result of N successive integrations on f1.
- With negative N, it switches to using \PolDiff.
-
-\PolDivide{f1}{f2}{Q}{R}
- This sets Q and R to be the quotient and remainder in the Euclidean
- division of f1 by f2.
-
-\PolGCD{f}{g}{k}
- This sets k to be the G.C.D. It is a unitary polynomial except if
- both f and g vanish, then k is the zero polynomial.
-
-\PolMapCoeffs{\macro}{name}
- It modifies each coefficient of the defined polynomial via
- the *expandable* macro \macro. The degree is adjusted as necessary
- if some leading coefficients vanish after the operation.
- In replacement text of \macro, \index expands to the coefficient
- index (which is defined to be zero for the constant term).
-
- Notice that \macro will have to handle inputs of the shape A/B[N]
- (xintfrac internal notation). This means that it probably will
- have to be expressed in terms of macros from xintfrac package.
-
- Example: \def\foo#1{\xintMul{#1}{\the\numexpr\index^2\relax}}
- to replace nth coefficient f_n by f_n * n^2.
-
-\PolReduceCoeffs{name}
- About the same as \PolMapCoeffs{\xintIrr}{name} (but adds [0]
- postfix which speeds up xintfrac operations when evaluating.)
-
-Expandable macros
------------------
-
-All these macros expand completely in two steps except \PolToExpr
-which needs a \write, \edef or a \csname...\endcsname context.
-
-\PolEval{name}\At{value}
- It boils down to \xinttheexpr reduce(name(value))\relax.
-
-\PolNthCoeff{name}{N}
- It expands to the raw Nth coefficient (0/1[0] if index is out of
- range). With N=-1, -2, ... expands to the leading coefficients.
-
-\PolDegree{name}
- It expands to the degree. This is -1 if zero polynomial but this may
- change in future. Should it then expand to -\infty ?
-
-\PolToExpr{f}
- Expands to f_0 + f_1*x + f_2*x^2 + ... (ascending powers). [1, 2]
-
- [1] in a \write, \edef, or \csname...\endcsname, but not under
- \romannumeral-`0
-
- [2] the letter x is (in this release) not customizable.
-
- By default zero coefficients are skipped (issue \poltoexprtrue to
- get all of them in output).
-
- No + sign before negative coefficients, for compliance with Maple
- input format. This means though that parsing the result back via
- naive delimited macros is difficult, see \PolToList and \PolToCSV
- for more low-level formats making it easier to get expandably some
- output of one's choice, which may possibly be parsed later on by
- other macros of one's design, or from other packages.
-
- Of course "\PolToExpr{f}" can be inserted in a \poldef, as the
- latter expands token by token, hence will force complete expansion
- of \PolToExpr{f}, but simply "f(x)" will be more efficient for the
- identical result.
-
- \PolToExprCmd is the one-argument macro used by \PolToExpr for the
- coefficients, it defaults to \xintPRaw{\xintRawWithZeros{#1}}. One
- will have to redefine it to use \xintIrr{#1} in place of
- \xintRawWithZeros{#1} to get in output reduced coefficients.
-
-\PolToList{f}
- Expands to {f_0}{f_1}...{f_N} with N = degree of f (except zero
- polynomial which does give {0/1[0]} and not an empty output.)
-
-\PolToCSV{f}
- Expands to f_0, f_1, f_2, ....., f_N. Converse of \PolFromCSV.
-
-Technicalities
---------------
-
-- The catcode of the semi-colon is reset temporarily by \poldef macro in
- case some other package (for example the French babel module) may have
- made it active. This will fail though if the whole thing was already
- part of a macro argument, in such cases one can use \PolDef rather.
- The colon in := may be active with no consequences.
-
-- Beware the 1/2 x problem: as mentioned above, it will be give zero due
- to the tacit multiplication rules of \xintexpr and to the fact that
- the package will do the Euclidean division of 1 by polynomial 2x.
-
-- During execution of polynomial operations by \poldef (but not during
- the initial purely numerical parsing of the expression), the xintfrac
- macro \xintAdd is temporarily patched to always express a/b + c/d with
- L.C.M.(b,d) as denominator. Indeed the current (xint 1.2p) \xintAdd
- uses (ad+bc)/bd formula except if b divides d or d divides b, which
- quickly leads in real life to big denominators.
-
- It is probable that this convention will be backported as default
- behaviour of xintfrac's \xintAdd in a future xint release. When this
- change is merged, there will be an impact on coefficients computed by
- \poldef because the change will apply even to the pure numerical
- evaluations arising during the initial stage of the parsing. Of course
- the coefficients are still the same rational numbers, only
- representation as fractions may change.
-
-- As a consequence of previous rule, user-chosen common denominators
- survive addition and multiplications:
-
- \poldef P(x):= 1/2 + 2/2*x + 3/2*x^3 + 4/2*x^4;
- \poldef Q(x):= 1/3 + (2/3)x + (3/3)x^3 + (4/3)x^4;
- \poldef PQ(x):= P(x)*Q(x);
-
- gives the polynomial
-
- 1/6+4/6*x^1+4/6*x^2+6/6*x^3+20/6*x^4+16/6*x^5+9/6*x^6+24/6*x^7+16/6*x^8
-
- where all coefficients have the same denominator 6 (which in this
- example is the l.c.m of the denominators of the reduced coefficients.)
-
-- \PolDiff always applies \xintIrr to the resulting coefficients, except
- that the "decimal" part [N] (for example an input in scientific
- notation such as 1.23e5 gives 123/1[3] internally in xintfrac) is not
- taken into account in the reduction of the fraction. This is tentative
- and may change.
-
- Same remark for \PolAntiDiff.
-
-- If f was created from comma separated values by macro \PolFromCSV,
- then the exact same coefficients (except those zero coefficients
- beyond the leading monomial) will be in the output of \PolToList and
- \PolToCSV in their original input form: a 1.3e2 will again be a 1.3e2.
-
- In contrast when such coefficients are used in a \poldef (or \PolDef)
- expression, they get transformed during the parsing to the xintfrac
- "raw" format. This is an unavoidable consequence of usage by \poldef
- of \xintdeffunc which itself is based on \xintexpr. This "raw" format
- speeds up expansion of xintfrac macros for numerical evaluations.
-
-- Currently, the package does not as a result of \poldef add to the TeX
- memory an already pre-computed "array" structure for the polynomial
- coefficients, as would be constructed by \PolAssign{f}\toarray\Macro.
- Such structures are used, but for internal calculations in temporarily
- restricted scopes. Apart from the function f() known to the
- (numerical) \xintexpr parser (whose meaning can be found in the log
- file after \xintverbosetrue), the data is (currently) stored in a
- single other macro encapsulating the degree, and the coefficients as a
- list. This may evolve in future.
-
-- As is to be expected internal structures of the package are barely
- documented and unstable. Don't use them.
-
-
-CHANGE LOG
-----------
-
-- v0.1 (2018/01/11): initial release. Features:
-
- *. differentiation and anti-differentiation,
- *. Euclidean division and GCDs,
- *. various utilities such as \PolFromCSV, \PolToCSV, \PolToExpr.
-
- Only one-variable polynomials so far.
-
- Due to lack of available time I have not really yet set-up a
- sufficient enough test suite. Bug reports very welcome!
Added: trunk/Master/texmf-dist/doc/latex/polexpr/README.md
===================================================================
--- trunk/Master/texmf-dist/doc/latex/polexpr/README.md (rev 0)
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@@ -0,0 +1,67 @@
+Package polexpr README
+======================
+
+License
+-------
+
+Copyright (C) 2018 Jean-François Burnol
+
+See documentation of package [xint](http://www.ctan.org/pkg/xint) for
+contact information.
+
+This Work may be distributed and/or modified under the conditions of the
+LaTeX Project Public License version 1.3c. This version of this license
+is in
+
+> <http://www.latex-project.org/lppl/lppl-1-3c.txt>
+
+and version 1.3 or later is part of all distributions of LaTeX version
+2005/12/01 or later.
+
+This Work has the LPPL maintenance status author-maintained.
+
+The Author of this Work is Jean-François Burnol.
+
+This Work consists of the package file polexpr.sty, this README.md and the
+documentation file polexpr.txt.
+
+Abstract
+--------
+
+The package provides `\poldef`. This a parser of polynomial expressions
+based upon the `\xintdeffunc` mechanism of xintexpr.
+
+The parsed expressions use the operations of algebra (inclusive of
+composition of functions) with standard operators, fractional numbers
+(possibly in scientific notation) and previously defined polynomial
+functions or other constructs as recognized by the `\xintexpr` numerical
+parser.
+
+The polynomials are then not only genuine `\xintexpr` numerical
+functions but additionally are also known to the package via their
+coefficients. This allows dedicated macros to implement polynomial
+algorithmics.
+
+CHANGE LOG
+----------
+
+- v0.1 (2018/01/11): initial release. Features:
+ - The `\poldef` parser itself,
+ - Differentiation and anti-differentiation,
+ - Euclidean division and GCDs,
+ - Various utilities such as `\PolFromCSV`, `\PolMapCoeffs`,
+ `\PolToCSV`, `\PolToExpr`, ...
+ Only one-variable polynomials so far.
+
+- v0.2 (2018/01/14)
+ - Fix: `"README thinks \numexpr recognizes ^ operator"`.
+ - Convert README to reStructuredText markup.
+ - Move main documentation from README to separate `polexpr.txt`
+ file.
+ - Provide `polexpr.html` as obtained via
+ [DocUtils](http://docutils.sourceforge.net/docs/index.html)
+ `rst2html.py`.
+ - Convert README to (CTAN compatible) Markdown markup.
+ Due to lack of available time the test suite might not be extensive
+ enough. Bug reports are very welcome!
+
Property changes on: trunk/Master/texmf-dist/doc/latex/polexpr/README.md
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+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
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+<style type="text/css">
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+ margin-left: 1px }
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+
+h1 tt.docutils, h2 tt.docutils, h3 tt.docutils,
+h4 tt.docutils, h5 tt.docutils, h6 tt.docutils {
+ font-size: 100% }
+
+ul.auto-toc {
+ list-style-type: none }
+
+</style>
+</head>
+<body>
+<div class="document" id="package-polexpr-documentation">
+<h1 class="title">Package polexpr documentation</h1>
+
+<!-- comment: -*- fill-column: 72; mode: rst; -*- -->
+<div class="section" id="first-examples">
+<h1>First Examples</h1>
+<p>The syntax is:</p>
+<pre class="literal-block">
+\poldef <name>(x):=<expression in variable x>;
+</pre>
+<p>where in place of <tt class="docutils literal">x</tt> an arbitrary <em>dummy variable</em> is authorized
+(i.e. per default any of <tt class="docutils literal"><span class="pre">[a..z|A..Z]</span></tt>; more letters can be declared
+under Unicode engines.) One can also issue:</p>
+<pre class="literal-block">
+\PolDef{name}{expression in variable x}
+</pre>
+<p>which admits an optional first argument to modify the variable letter
+from its default <tt class="docutils literal">x</tt>.</p>
+<dl class="docutils">
+<dt><tt class="docutils literal">\poldef <span class="pre">f(x):=</span> <span class="pre">1-x+x^2;</span></tt></dt>
+<dd>defines polynomial <tt class="docutils literal">f</tt>. Polynomial names must start with a letter
+and may contain letters, digits, and underscores. The variable must
+be a single letter. The colon character is optional. The semi-colon
+at end of expression is mandatory.</dd>
+<dt><tt class="docutils literal"><span class="pre">\PolDef{f}{1-x+x^2}</span></tt></dt>
+<dd>does the same as <tt class="docutils literal">\poldef <span class="pre">f(x):=</span> <span class="pre">1-x+x^2;</span></tt> To use another letter
+than <tt class="docutils literal">x</tt> in the expression, one must pass it as an extra optional
+argument to <tt class="docutils literal">\PolDef</tt>. Useful if the semi-colon has been assigned
+some non-standard catcode by some package.</dd>
+<dt><tt class="docutils literal"><span class="pre">\PolLet{g}{f}</span></tt></dt>
+<dd>saves a copy of <tt class="docutils literal">f</tt> under name <tt class="docutils literal">g</tt>.</dd>
+<dt><tt class="docutils literal">\poldef <span class="pre">f(z):=</span> <span class="pre">f(z)^2;</span></tt></dt>
+<dd>redefines <tt class="docutils literal">f</tt> in terms of itself.</dd>
+<dt><tt class="docutils literal">\poldef <span class="pre">f(T):=</span> <span class="pre">f(f(T));</span></tt></dt>
+<dd>again redefines <tt class="docutils literal">f</tt> in terms of its (new) self.</dd>
+<dt><tt class="docutils literal">\poldef <span class="pre">k(z):=</span> <span class="pre">f(z)-g(g(z)^2)^2;</span></tt></dt>
+<dd>should now define the zero polynomial... Let's check:
+<tt class="docutils literal">\[ k(z) = <span class="pre">\PolTypeset[z]{k}</span> \]</tt></dd>
+<dt><tt class="docutils literal"><span class="pre">\PolDiff{f}{df_dx}</span></tt></dt>
+<dd>sets <tt class="docutils literal">df_dx</tt> to the derivative of <tt class="docutils literal">f</tt>.</dd>
+<dt><tt class="docutils literal"><span class="pre">\PolDiff{df_dx}{f_xx}</span></tt></dt>
+<dd>obtains second derivative.</dd>
+<dt><tt class="docutils literal"><span class="pre">\PolDiff[3]{f}{d3f_dx3}</span></tt></dt>
+<dd>computes directly the third derivative. Its name does not have to be
+chosen so complicated <tt class="docutils literal">:)</tt>, but the right quote <tt class="docutils literal">'</tt> is not
+allowed in polynomial names (currently).</dd>
+</dl>
+<pre class="literal-block">
+$f(z) = \PolTypeset[z]{f} $\newline
+$f'(z) = \PolTypeset[z]{df_dx}$\newline
+$f''(z) = \PolTypeset[z]{f_xx}$\newline
+$f'''(z)= \PolTypeset[z]{d3f_dx3}$\par
+</pre>
+<div class="admonition important">
+<p class="first admonition-title">Important</p>
+<p>The package does not currently know rational functions: <tt class="docutils literal">/</tt> in
+a parsed polynomial expression does the Euclidean quotient:</p>
+<pre class="literal-block">
+(1-x^2)/(1-x)
+</pre>
+<p>does give <tt class="docutils literal">1+x</tt> but</p>
+<pre class="literal-block">
+(1/(1-x))*(1-x^2)
+</pre>
+<p>evaluates to zero. This will work as expected:</p>
+<pre class="last literal-block">
+\poldef k(x):= (x-1)(x-2)(x-3)(x-4)/(x^2-5x+4);
+</pre>
+</div>
+<div class="admonition attention">
+<p class="first admonition-title">Attention!</p>
+<p class="last"><tt class="docutils literal">1/2 x</tt> skips the space and is treated like <tt class="docutils literal"><span class="pre">1/(2x)</span></tt> because of
+the tacit multiplication rules of xintexpr. But this means it gives
+zero! Thus one must use <tt class="docutils literal">(1/2)x</tt> or <tt class="docutils literal">1/2*x</tt> or <tt class="docutils literal"><span class="pre">(1/2)*x</span></tt> for
+disambiguation: <tt class="docutils literal"><span class="pre">x-1/2*x^2+1/3*x^3...</span></tt></p>
+</div>
+<p>After:</p>
+<pre class="literal-block">
+\poldef f1(x):= 25(x-1)(x^2-2)(x-3)(x-4)(x-5);%
+\poldef f2(x):= 37(x-1)(x^2-2)(x-6)(x-7)(x-8);%
+</pre>
+<p>the macro call <tt class="docutils literal"><span class="pre">\PolGCD{f1}{f2}{k}</span></tt> sets <tt class="docutils literal">k</tt> to the (unitary) GCD of
+<tt class="docutils literal">f1</tt> and <tt class="docutils literal">f2</tt>.</p>
+<dl class="docutils">
+<dt><tt class="docutils literal">\PolToExpr{k}</tt></dt>
+<dd>will thus (expandably) give in this case <tt class="docutils literal"><span class="pre">2-2*x^1-1*x^2+1*x^3</span></tt>.
+This is useful for console or file output (the syntax is Maple- and
+PSTricks-compatible; currently the letter <tt class="docutils literal">x</tt> in output is not
+customizable, but this can easily be added if requested from author.)</dd>
+</dl>
+</div>
+<div class="section" id="non-expandable-macros">
+<h1>Non-expandable macros</h1>
+<dl class="docutils">
+<dt><tt class="docutils literal">\poldef <span class="pre">name(letter):=</span> polynomial expression using letter;</tt></dt>
+<dd><p class="first">This evaluates the polynomial expression and stores the coefficients
+in a private structure accessible later via other package macros,
+under the user-chosen <tt class="docutils literal">name</tt>. Of course previously defined
+polynomials are allowed in a new expression. Names must start with a
+letter and are constituted of letters, digits and underscore
+characters. See Examples above.</p>
+<p>As a side effect the function <tt class="docutils literal">name()</tt> is recognized as a genuine
+<tt class="docutils literal"><span class="pre">\xintexpr...\relax</span></tt> function for (exact) numerical evaluation. It
+computes values not according to the original expression but via
+the Horner scheme corresponding to the polynomial coefficients.</p>
+<p class="last">The original expression is lost after parsing, and in particular
+the package provides no way to typeset it. This has to be done
+manually, if needed.</p>
+</dd>
+<dt><tt class="docutils literal"><span class="pre">\PolDef{name}{P(x)}</span></tt></dt>
+<dd>Does the same but the variable is assumed to be <tt class="docutils literal">x</tt>. To use
+another letter, pass it as first optional argument:
+<tt class="docutils literal"><span class="pre">\PolDef[X]{name}{P(X)}</span></tt>.</dd>
+<dt><tt class="docutils literal"><span class="pre">\PolLet{g}{f}</span></tt></dt>
+<dd>Makes a copy of already defined polynomial f to new one g. Same
+effect as <tt class="docutils literal"><span class="pre">\PolDef{g}{f(x)}</span></tt> but faster.</dd>
+<dt><tt class="docutils literal"><span class="pre">\PolAssign{f}\toarray\Array</span></tt></dt>
+<dd><p class="first">Defines a one-argument expandable macro <tt class="docutils literal"><span class="pre">\Array{#1}</span></tt> which expands
+to the (raw) #1th polynomial coefficient.</p>
+<ul class="last simple">
+<li>Attention, coefficients here are indexed starting at 1.</li>
+<li>With #1=-1, -2, ..., <tt class="docutils literal"><span class="pre">\Array{#1}</span></tt> returns leading coefficients.</li>
+<li>With #1=0, returns the number of coefficients, i.e. <tt class="docutils literal">1 + deg f</tt>
+for non-zero polynomials.</li>
+<li>Out-of-range #1's return <tt class="docutils literal">0/1[0]</tt>.</li>
+</ul>
+</dd>
+<dt><tt class="docutils literal"><span class="pre">\PolGet{f}\fromarray\Array</span></tt></dt>
+<dd><p class="first">Does the converse operation to <tt class="docutils literal"><span class="pre">\PolAssign{f}\toarray\Array</span></tt>. No
+error checks on validity of coefficients as numbers. Each
+<tt class="docutils literal">\Array{index}</tt> is expanded in an <tt class="docutils literal">\edef</tt> before being assigned
+to a coefficient. Leading zero coefficients 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 class="last">This will define <tt class="docutils literal">f</tt> as would have <tt class="docutils literal">\poldef <span class="pre">f(x):=1-2x+5x^2-3x^3;</span></tt>.
+However the coefficients are still in their original form (i.e.
+they were not subjected to <tt class="docutils literal">\xintRaw</tt> or similar xintfrac macro.)</p>
+</dd>
+<dt><tt class="docutils literal"><span class="pre">\PolFromCSV{f}{comma</span> separated coefficients}</tt></dt>
+<dd><p class="first">Defines a polynomial directly from the comma separated list (or a
+macro expanding to such a list) of its coefficients, the constant
+term being the first item. No validity checks. Spaces from the list
+argument are trimmed. List items are expanded in an <tt class="docutils literal">\edef</tt>, but
+currently left in their original form like e.g. <tt class="docutils literal">1.5e3</tt> which is
+not converted to <tt class="docutils literal">15/1[2]</tt> <em>raw</em> xintfrac format (this may
+change).</p>
+<p>Leading zero coefficients are removed:</p>
+<pre class="literal-block">
+\PolFromCSV{J}{0, 0, 0, 0, 0, 0, 0, 0, 0, 0}
+</pre>
+<p>defines the zero polynomial, which has only one (zero) coefficient.</p>
+<p class="last">See also expandable macro <tt class="docutils literal">\PolToCSV</tt>.</p>
+</dd>
+<dt><tt class="docutils literal">\PolTypeset{name}</tt></dt>
+<dd><p class="first">Typesets in descending powers in math mode. It uses letter <tt class="docutils literal">x</tt> but
+this can be changed via an optional argument:</p>
+<pre class="literal-block">
+\PolTypeset[z]{name}
+</pre>
+<p>By default zero coefficients are skipped (issue <tt class="docutils literal">\poltypesetalltrue</tt>
+to get all of them in output).</p>
+<p class="last">Macros <tt class="docutils literal">\PolTypesetCmd</tt>, <tt class="docutils literal">\PolTypesetPlus</tt>, <tt class="docutils literal">\PolTypesetMonomial</tt>
+can help configure the output. See the package code.</p>
+</dd>
+<dt><tt class="docutils literal"><span class="pre">\PolTypeset*{name}</span></tt></dt>
+<dd>Typesets in ascending powers. Change the letter from its default
+<tt class="docutils literal">x</tt> by optional argument.</dd>
+<dt><tt class="docutils literal"><span class="pre">\PolDiff{f1}{f2}</span></tt></dt>
+<dd><p class="first">This sets f2 to the first derivative of <tt class="docutils literal">f1</tt>. It is allowed to
+issue <tt class="docutils literal"><span class="pre">\PolDiff{f}{f}</span></tt>, effectively replacing <tt class="docutils literal">f</tt> by <tt class="docutils literal">f'</tt>.</p>
+<p class="last">Coefficients of the result <tt class="docutils literal">f2</tt> are irreducible fractions
+(see <a class="reference internal" href="#technicalities">Technicalities</a> for the whole story.)</p>
+</dd>
+<dt><tt class="docutils literal"><span class="pre">\PolDiff[N]{f1}{f2}</span></tt></dt>
+<dd>This sets <tt class="docutils literal">f2</tt> to the <tt class="docutils literal">N</tt>-th derivative of <tt class="docutils literal">f1</tt>. Identical
+arguments is allowed. With <tt class="docutils literal">N=0</tt>, same effect as
+<tt class="docutils literal"><span class="pre">\PolLet{f2}{f1}</span></tt>. With negative <tt class="docutils literal">N</tt>, switches to using
+<tt class="docutils literal">\PolAntiDiff</tt>.</dd>
+<dt><tt class="docutils literal"><span class="pre">\PolAntiDiff{f1}{f2}</span></tt></dt>
+<dd><p class="first">This sets <tt class="docutils literal">f2</tt> to the primitive of <tt class="docutils literal">f1</tt> vanishing at zero.</p>
+<p class="last">Coefficients of the result <tt class="docutils literal">f2</tt> are irreducible fractions
+(see <a class="reference internal" href="#technicalities">Technicalities</a> for the whole story.)</p>
+</dd>
+<dt><tt class="docutils literal"><span class="pre">\PolAntiDiff[N]{f1}{f2}</span></tt></dt>
+<dd>This sets <tt class="docutils literal">f2</tt> to the result of <tt class="docutils literal">N</tt> successive integrations on
+<tt class="docutils literal">f1</tt>. With negative <tt class="docutils literal">N</tt>, it switches to using <tt class="docutils literal">\PolDiff</tt>.</dd>
+<dt><tt class="docutils literal"><span class="pre">\PolDivide{f1}{f2}{Q}{R}</span></tt></dt>
+<dd>This sets <tt class="docutils literal">Q</tt> and <tt class="docutils literal">R</tt> to be the quotient and remainder in the
+Euclidean division of <tt class="docutils literal">f1</tt> by <tt class="docutils literal">f2</tt>.</dd>
+<dt><tt class="docutils literal"><span class="pre">\PolGCD{f}{g}{k}</span></tt></dt>
+<dd>This sets <tt class="docutils literal">k</tt> to be the G.C.D. It is a unitary polynomial except
+if both <tt class="docutils literal">f</tt> and <tt class="docutils literal">g</tt> vanish, then <tt class="docutils literal">k</tt> is the zero polynomial.</dd>
+<dt><tt class="docutils literal"><span class="pre">\PolMapCoeffs{\macro}{name}</span></tt></dt>
+<dd><p class="first">It modifies each coefficient of the defined polynomial via the
+<em>expandable</em> macro <tt class="docutils literal">\macro</tt>. The degree is adjusted as necessary
+if some leading coefficients vanish after the operation. In
+replacement text of <tt class="docutils literal">\macro</tt>, <tt class="docutils literal">\index</tt> expands to the
+coefficient index (which is defined to be zero for the constant
+term).</p>
+<p>Notice that <tt class="docutils literal">\macro</tt> will have to handle inputs of the shape
+<tt class="docutils literal">A/B[N]</tt> (xintfrac internal notation). This means that it probably
+will have to be expressed in terms of macros from xintfrac package.</p>
+<p>Example:</p>
+<pre class="literal-block">
+\def\foo#1{\xintMul{#1}{\the\numexpr\index*\index\relax}}
+</pre>
+<p class="last">(or with <tt class="docutils literal"><span class="pre">\xintSqr{\xindex}</span></tt>) to replace <tt class="docutils literal">n</tt>-th coefficient
+<tt class="docutils literal">f_n</tt> by <tt class="docutils literal">f_n*n^2</tt>.</p>
+</dd>
+<dt><tt class="docutils literal">\PolReduceCoeffs{name}</tt></dt>
+<dd>About the same as <tt class="docutils literal"><span class="pre">\PolMapCoeffs{\xintIrr}{name}</span></tt> (but adds
+<tt class="docutils literal">[0]</tt> postfix which speeds up xintfrac operations when
+evaluating.)</dd>
+</dl>
+</div>
+<div class="section" id="expandable-macros">
+<h1>Expandable macros</h1>
+<p>All these macros expand completely in two steps except <tt class="docutils literal">\PolToExpr</tt>
+which needs a <tt class="docutils literal">\write</tt>, <tt class="docutils literal">\edef</tt> or a <tt class="docutils literal"><span class="pre">\csname...\endcsname</span></tt> context.</p>
+<dl class="docutils">
+<dt><tt class="docutils literal"><span class="pre">\PolEval{name}\At{value}</span></tt></dt>
+<dd>It boils down to <tt class="docutils literal">\xinttheexpr <span class="pre">reduce(name(value))\relax</span></tt>.</dd>
+<dt><tt class="docutils literal"><span class="pre">\PolNthCoeff{name}{N}</span></tt></dt>
+<dd>It expands to the raw <tt class="docutils literal">N</tt>-th coefficient (<tt class="docutils literal">0/1[0]</tt> if index is
+out of range). With <tt class="docutils literal"><span class="pre">N=-1</span></tt>, <tt class="docutils literal"><span class="pre">-2</span></tt>, ... expands to the leading
+coefficients.</dd>
+<dt><tt class="docutils literal">\PolDegree{name}</tt></dt>
+<dd>It expands to the degree. This is <tt class="docutils literal"><span class="pre">-1</span></tt> if zero polynomial but this
+may change in future. Should it then expand to <tt class="docutils literal"><span class="pre">-\infty</span></tt> ?</dd>
+<dt><tt class="docutils literal">\PolToExpr{f}</tt></dt>
+<dd><p class="first">Expands to <tt class="docutils literal">f_0 + f_1*x + f_2*x^2 + ...</tt> (ascending powers). <a class="footnote-reference" href="#id3" id="id1">[1]</a>,
+<a class="footnote-reference" href="#id4" id="id2">[2]</a></p>
+<table class="docutils footnote" frame="void" id="id3" rules="none">
+<colgroup><col class="label" /><col /></colgroup>
+<tbody valign="top">
+<tr><td class="label"><a class="fn-backref" href="#id1">[1]</a></td><td>in a <tt class="docutils literal">\write</tt>, <tt class="docutils literal">\edef</tt>, or <tt class="docutils literal"><span class="pre">\csname...\endcsname</span></tt>, but
+not under <tt class="docutils literal"><span class="pre">\romannumeral-`0</span></tt>.</td></tr>
+</tbody>
+</table>
+<table class="docutils footnote" frame="void" id="id4" rules="none">
+<colgroup><col class="label" /><col /></colgroup>
+<tbody valign="top">
+<tr><td class="label"><a class="fn-backref" href="#id2">[2]</a></td><td>the letter <tt class="docutils literal">x</tt> is (in this release) not customizable.</td></tr>
+</tbody>
+</table>
+<p>By default zero coefficients are skipped (issue <tt class="docutils literal">\poltoexprtrue</tt> to
+get all of them in output).</p>
+<p>No <tt class="docutils literal">+</tt> sign before negative coefficients, for compliance with Maple
+input format. This means though that parsing the result back via
+naive delimited macros is difficult, see <tt class="docutils literal">\PolToList</tt> and <tt class="docutils literal">\PolToCSV</tt>
+for more low-level formats making it easier to get expandably some
+output of one's choice, which may possibly be parsed later on by
+other macros of one's design, or from other packages.</p>
+<p>Of course <tt class="docutils literal">\PolToExpr{f}</tt> can be inserted in a poldef, as the
+latter expands token by token, hence will force complete expansion
+of <tt class="docutils literal">\PolToExpr{f}</tt>, but simply <tt class="docutils literal">f(x)</tt> will be more efficient for the
+identical result.</p>
+<p class="last"><tt class="docutils literal">\PolToExprCmd</tt> is the one-argument macro used by <tt class="docutils literal">\PolToExpr</tt> for the
+coefficients, it defaults to <tt class="docutils literal"><span class="pre">\xintPRaw{\xintRawWithZeros{#1}}</span></tt>. One
+will have to redefine it to use <tt class="docutils literal"><span class="pre">\xintIrr{#1}</span></tt> in place of
+<tt class="docutils literal"><span class="pre">\xintRawWithZeros{#1}</span></tt> to get in output reduced coefficients.</p>
+</dd>
+<dt><tt class="docutils literal">\PolToList{f}</tt></dt>
+<dd>Expands to <tt class="docutils literal"><span class="pre">{f_0}{f_1}...{f_N}</span></tt> with <tt class="docutils literal">N</tt> = degree of f (except
+zero polynomial which does give <tt class="docutils literal">{0/1[0]}</tt> and not an empty
+output.)</dd>
+<dt><tt class="docutils literal">\PolToCSV{f}</tt></dt>
+<dd>Expands to <tt class="docutils literal">f_0, f_1, f_2, <span class="pre">.....,</span> f_N</tt>. Converse to
+<tt class="docutils literal">\PolFromCSV</tt>.</dd>
+</dl>
+</div>
+<div class="section" id="technicalities">
+<h1>Technicalities</h1>
+<ul>
+<li><p class="first">The catcode of the semi-colon is reset temporarily by <tt class="docutils literal">\poldef</tt> 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 <tt class="docutils literal">\PolDef</tt> rather.
+The colon in <tt class="docutils literal">:=</tt> may be active with no consequences.</p>
+</li>
+<li><p class="first">Beware the <tt class="docutils literal">1/2 x</tt> problem: as mentioned above, it will be give zero due
+to the tacit multiplication rules of <tt class="docutils literal">\xintexpr</tt> and to the fact that
+the package will do the Euclidean division of <tt class="docutils literal">1</tt> by polynomial <tt class="docutils literal">2x</tt>.</p>
+</li>
+<li><p class="first">During execution of polynomial operations by <tt class="docutils literal">\poldef</tt> (but not
+during the initial purely numerical parsing of the expression), the
+xintfrac macro <tt class="docutils literal">\xintAdd</tt> is temporarily patched to always express
+<tt class="docutils literal">a/b + c/d</tt> with <tt class="docutils literal"><span class="pre">l.c.m.(b,d)</span></tt> as denominator. Indeed the current
+(xint 1.2p) <tt class="docutils literal">\xintAdd</tt> uses <tt class="docutils literal"><span class="pre">(ad+bc)/bd</span></tt> formula except if <tt class="docutils literal">b</tt>
+divides <tt class="docutils literal">d</tt> or <tt class="docutils literal">d</tt> divides <tt class="docutils literal">b</tt>, which quickly leads in real life
+to big denominators.</p>
+<p>It is probable that this convention will be backported as default
+behaviour of xintfrac's <tt class="docutils literal">\xintAdd</tt> in a future xint release. When this
+change is merged, there will be an impact on coefficients computed by
+<tt class="docutils literal">\poldef</tt> because the change will apply even to the pure numerical
+evaluations arising during the initial stage of the parsing. Of course
+the coefficients are still the same rational numbers, only
+representation as fractions may change.</p>
+</li>
+<li><p class="first">As a consequence of previous rule, user-chosen common denominators
+survive addition and multiplications:</p>
+<pre class="literal-block">
+\poldef P(x):= 1/2 + 2/2*x + 3/2*x^3 + 4/2*x^4;
+\poldef Q(x):= 1/3 + (2/3)x + (3/3)x^3 + (4/3)x^4;
+\poldef PQ(x):= P(x)Q(x);
+</pre>
+<p>gives 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 (which in this
+example is the <tt class="docutils literal">l.c.m</tt> of the denominators of the reduced coefficients.)</p>
+</li>
+<li><p class="first"><tt class="docutils literal">\PolDiff</tt> always applies <tt class="docutils literal">\xintIrr</tt> to the resulting coefficients, except
+that the <em>power of ten</em> part <tt class="docutils literal">[N]</tt> (for example an input in scientific
+notation such as <tt class="docutils literal">1.23e5</tt> gives <tt class="docutils literal">123/1[3]</tt> internally in xintfrac) is not
+taken into account in the reduction of the fraction. This is tentative
+and may change.</p>
+<p>Same remark for <tt class="docutils literal">\PolAntiDiff</tt>.</p>
+</li>
+<li><p class="first">If <tt class="docutils literal">f</tt> was created from comma separated values by macro PolFromCSV,
+then the coefficients will be in the output of <tt class="docutils literal">\PolToList</tt> and
+<tt class="docutils literal">\PolToCSV</tt> in the same format as originally in input: a <tt class="docutils literal">1.3e2</tt>
+will again be a <tt class="docutils literal">1.3e2</tt>.</p>
+<p>In contrast when such coefficients are used in a <tt class="docutils literal">\poldef</tt> (or
+<tt class="docutils literal">\PolDef</tt>) expression, they get transformed during the parsing to
+the xintfrac <em>raw</em> format. This is an unavoidable consequence of usage
+by poldef of <tt class="docutils literal">\xintdeffunc</tt> which itself is based on <tt class="docutils literal">\xintexpr.</tt>
+This <em>raw</em> format speeds up expansion of xintfrac macros for numerical
+evaluations.</p>
+</li>
+<li><p class="first">Currently, the package does not as a result of <tt class="docutils literal">\poldef</tt> add to the TeX
+memory an already pre-computed <em>array</em> structure for the polynomial
+coefficients, as would be constructed by <tt class="docutils literal"><span class="pre">\PolAssign{f}\toarray\macro</span></tt>.
+Such structures are used, but for internal calculations in temporarily
+restricted scopes. Apart from the function <tt class="docutils literal">f()</tt> known to the
+(numerical) <tt class="docutils literal">\xintexpr</tt> parser (whose meaning can be found in the log
+file after xintverbosetrue), the data is (currently) stored in a
+single other macro encapsulating the degree, and the coefficients as a
+list. This may evolve in future.</p>
+</li>
+<li><p class="first">As is to be expected internal structures of the package are barely
+documented and unstable. Don't use them.</p>
+</li>
+</ul>
+</div>
+<div class="section" id="releases">
+<h1>RELEASES</h1>
+<ul class="simple">
+<li>0.1 (2018/01/11): initial release (files README, polexpr.sty).</li>
+<li>0.2 (2018/01/14): documentation moved to polexpr.{txt,html}.</li>
+</ul>
+<p>Files of 0.2 release:</p>
+<ul class="simple">
+<li>README.md,</li>
+<li>polexpr.sty (package file),</li>
+<li>polexpr.txt (documentation),</li>
+<li>polexpr.html (conversion via <a class="reference external" href="http://docutils.sourceforge.net/docs/index.html">DocUtils</a> rst2html.py)</li>
+</ul>
+<p>See README.md for the License and the change log.</p>
+</div>
+</div>
+</body>
+</html>
Property changes on: trunk/Master/texmf-dist/doc/latex/polexpr/polexpr.html
___________________________________________________________________
Added: svn:eol-style
## -0,0 +1 ##
+native
\ No newline at end of property
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===================================================================
--- trunk/Master/texmf-dist/doc/latex/polexpr/polexpr.txt (rev 0)
+++ trunk/Master/texmf-dist/doc/latex/polexpr/polexpr.txt 2018-01-15 22:20:09 UTC (rev 46318)
@@ -0,0 +1,394 @@
+.. comment: -*- fill-column: 72; mode: rst; -*-
+
+Package polexpr documentation
+=============================
+
+First Examples
+--------------
+
+The syntax is::
+
+ \poldef <name>(x):=<expression in variable x>;
+
+where in place of ``x`` an arbitrary *dummy variable* is authorized
+(i.e. per default any of ``[a..z|A..Z]``; more letters can be declared
+under Unicode engines.) One can also issue::
+
+ \PolDef{name}{expression in variable x}
+
+which admits an optional first argument to modify the variable letter
+from its default ``x``.
+
+``\poldef f(x):= 1-x+x^2;``
+ defines polynomial ``f``. Polynomial names must start with a letter
+ and may contain letters, digits, and underscores. The variable must
+ be a single letter. The colon character is optional. The semi-colon
+ at end of expression is mandatory.
+
+``\PolDef{f}{1-x+x^2}``
+ does the same as ``\poldef f(x):= 1-x+x^2;`` To use another letter
+ than ``x`` in the expression, one must pass it as an extra optional
+ argument to ``\PolDef``. Useful if the semi-colon has been assigned
+ some non-standard catcode by some package.
+
+``\PolLet{g}{f}``
+ saves a copy of ``f`` under name ``g``.
+
+``\poldef f(z):= f(z)^2;``
+ redefines ``f`` in terms of itself.
+
+``\poldef f(T):= f(f(T));``
+ again redefines ``f`` in terms of its (new) self.
+
+``\poldef k(z):= f(z)-g(g(z)^2)^2;``
+ should now define the zero polynomial... Let's check:
+ ``\[ k(z) = \PolTypeset[z]{k} \]``
+
+``\PolDiff{f}{df_dx}``
+ sets ``df_dx`` to the derivative of ``f``.
+
+``\PolDiff{df_dx}{f_xx}``
+ obtains second derivative.
+
+``\PolDiff[3]{f}{d3f_dx3}``
+ computes directly the third derivative. Its name does not have to be
+ chosen so complicated ``:)``, but the right quote ``'`` is not
+ allowed in polynomial names (currently).
+
+::
+
+ $f(z) = \PolTypeset[z]{f} $\newline
+ $f'(z) = \PolTypeset[z]{df_dx}$\newline
+ $f''(z) = \PolTypeset[z]{f_xx}$\newline
+ $f'''(z)= \PolTypeset[z]{d3f_dx3}$\par
+
+.. important::
+
+ The package does not currently know rational functions: ``/`` in
+ a parsed polynomial expression does the Euclidean quotient::
+
+ (1-x^2)/(1-x)
+
+ does give ``1+x`` but ::
+
+ (1/(1-x))*(1-x^2)
+
+ evaluates to zero. This will work as expected::
+
+ \poldef k(x):= (x-1)(x-2)(x-3)(x-4)/(x^2-5x+4);
+
+.. attention::
+
+ ``1/2 x`` skips the space and is treated like ``1/(2x)`` because of
+ the tacit multiplication rules of \xintexpr. But this means it gives
+ zero! Thus one must use ``(1/2)x`` or ``1/2*x`` or ``(1/2)*x`` for
+ disambiguation: ``x-1/2*x^2+1/3*x^3...``
+
+After::
+
+ \poldef f1(x):= 25(x-1)(x^2-2)(x-3)(x-4)(x-5);%
+ \poldef f2(x):= 37(x-1)(x^2-2)(x-6)(x-7)(x-8);%
+
+the macro call ``\PolGCD{f1}{f2}{k}`` sets ``k`` to the (unitary) GCD of
+``f1`` and ``f2``.
+
+``\PolToExpr{k}``
+ will thus (expandably) give in this case ``2-2*x^1-1*x^2+1*x^3``.
+ This is useful for console or file output (the syntax is Maple- and
+ PSTricks-compatible; currently the letter ``x`` in output is not
+ customizable, but this can easily be added if requested from author.)
+
+Non-expandable macros
+---------------------
+
+``\poldef name(letter):= polynomial expression using letter;``
+ This evaluates the polynomial expression and stores the coefficients
+ in a private structure accessible later via other package macros,
+ under the user-chosen ``name``. Of course previously defined
+ polynomials are allowed in a new expression. Names must start with a
+ letter and are constituted of letters, digits and underscore
+ characters. See Examples above.
+
+ As a side effect the function ``name()`` is recognized as a genuine
+ ``\xintexpr...\relax`` function for (exact) numerical evaluation. It
+ computes values not according to the original expression but via
+ the Horner scheme corresponding to the polynomial coefficients.
+
+ The original expression is lost after parsing, and in particular
+ the package provides no way to typeset it. This has to be done
+ manually, if needed.
+
+``\PolDef{name}{P(x)}``
+ Does the same but the variable is assumed to be ``x``. To use
+ another letter, pass it as first optional argument:
+ ``\PolDef[X]{name}{P(X)}``.
+
+``\PolLet{g}{f}``
+ Makes a copy of already defined polynomial f to new one g. Same
+ effect as ``\PolDef{g}{f(x)}`` but faster.
+
+``\PolAssign{f}\toarray\Array``
+ Defines a one-argument expandable macro ``\Array{#1}`` which expands
+ to the (raw) #1th polynomial coefficient.
+
+ - Attention, coefficients here are indexed starting at 1.
+
+ - With #1=-1, -2, ..., ``\Array{#1}`` returns leading coefficients.
+
+ - With #1=0, returns the number of coefficients, i.e. ``1 + deg f``
+ for non-zero polynomials.
+
+ - Out-of-range #1's return ``0/1[0]``.
+
+``\PolGet{f}\fromarray\Array``
+ Does the converse operation to ``\PolAssign{f}\toarray\Array``. No
+ error checks on validity of coefficients as numbers. Each
+ ``\Array{index}`` is expanded in an ``\edef`` before being assigned
+ to a coefficient. Leading zero coefficients are removed from the
+ polynomial.
+
+ (contrived) Example::
+
+ \xintAssignArray{1}{-2}{5}{-3}\to\foo
+ \PolGet{f}\fromarray\foo
+
+ This will define ``f`` as would have ``\poldef f(x):=1-2x+5x^2-3x^3;``.
+ However the coefficients are still in their original form (i.e.
+ they were not subjected to ``\xintRaw`` or similar xintfrac macro.)
+
+``\PolFromCSV{f}{comma separated coefficients}``
+ Defines a polynomial directly from the comma separated list (or a
+ macro expanding to such a list) of its coefficients, the constant
+ term being the first item. No validity checks. Spaces from the list
+ argument are trimmed. List items are expanded in an ``\edef``, but
+ currently left in their original form like e.g. ``1.5e3`` which is
+ not converted to ``15/1[2]`` *raw* xintfrac format (this may
+ change).
+
+ Leading zero coefficients are removed::
+
+ \PolFromCSV{J}{0, 0, 0, 0, 0, 0, 0, 0, 0, 0}
+
+ defines the zero polynomial, which has only one (zero) coefficient.
+
+ See also expandable macro ``\PolToCSV``.
+
+``\PolTypeset{name}``
+ Typesets in descending powers in math mode. It uses letter ``x`` but
+ this can be changed via an optional argument::
+
+ \PolTypeset[z]{name}
+
+ By default zero coefficients are skipped (issue ``\poltypesetalltrue``
+ to get all of them in output).
+
+ Macros ``\PolTypesetCmd``, ``\PolTypesetPlus``, ``\PolTypesetMonomial``
+ can help configure the output. See the package code.
+
+``\PolTypeset*{name}``
+ Typesets in ascending powers. Change the letter from its default
+ ``x`` by optional argument.
+
+``\PolDiff{f1}{f2}``
+ This sets f2 to the first derivative of ``f1``. It is allowed to
+ issue ``\PolDiff{f}{f}``, effectively replacing ``f`` by ``f'``.
+
+ Coefficients of the result ``f2`` are irreducible fractions
+ (see `Technicalities`_ for the whole story.)
+
+``\PolDiff[N]{f1}{f2}``
+ This sets ``f2`` to the ``N``-th derivative of ``f1``. Identical
+ arguments is allowed. With ``N=0``, same effect as
+ ``\PolLet{f2}{f1}``. With negative ``N``, switches to using
+ ``\PolAntiDiff``.
+
+``\PolAntiDiff{f1}{f2}``
+ This sets ``f2`` to the primitive of ``f1`` vanishing at zero.
+
+ Coefficients of the result ``f2`` are irreducible fractions
+ (see `Technicalities`_ for the whole story.)
+
+``\PolAntiDiff[N]{f1}{f2}``
+ This sets ``f2`` to the result of ``N`` successive integrations on
+ ``f1``. With negative ``N``, it switches to using ``\PolDiff``.
+
+``\PolDivide{f1}{f2}{Q}{R}``
+ This sets ``Q`` and ``R`` to be the quotient and remainder in the
+ Euclidean division of ``f1`` by ``f2``.
+
+``\PolGCD{f}{g}{k}``
+ This sets ``k`` to be the G.C.D. It is a unitary polynomial except
+ if both ``f`` and ``g`` vanish, then ``k`` is the zero polynomial.
+
+``\PolMapCoeffs{\macro}{name}``
+ It modifies each coefficient of the defined polynomial via the
+ *expandable* macro ``\macro``. The degree is adjusted as necessary
+ if some leading coefficients vanish after the operation. In
+ replacement text of ``\macro``, ``\index`` expands to the
+ coefficient index (which is defined to be zero for the constant
+ term).
+
+ Notice that ``\macro`` will have to handle inputs of the shape
+ ``A/B[N]`` (xintfrac internal notation). This means that it probably
+ will have to be expressed in terms of macros from xintfrac package.
+
+ Example::
+
+ \def\foo#1{\xintMul{#1}{\the\numexpr\index*\index\relax}}
+
+ (or with ``\xintSqr{\xindex}``) to replace ``n``-th coefficient
+ ``f_n`` by ``f_n*n^2``.
+
+``\PolReduceCoeffs{name}``
+ About the same as ``\PolMapCoeffs{\xintIrr}{name}`` (but adds
+ ``[0]`` postfix which speeds up xintfrac operations when
+ evaluating.)
+
+Expandable macros
+-----------------
+
+All these macros expand completely in two steps except ``\PolToExpr``
+which needs a ``\write``, ``\edef`` or a ``\csname...\endcsname`` context.
+
+``\PolEval{name}\At{value}``
+ It boils down to ``\xinttheexpr reduce(name(value))\relax``.
+
+``\PolNthCoeff{name}{N}``
+ It expands to the raw ``N``-th coefficient (``0/1[0]`` if index is
+ out of range). With ``N=-1``, ``-2``, ... expands to the leading
+ coefficients.
+
+``\PolDegree{name}``
+ It expands to the degree. This is ``-1`` if zero polynomial but this
+ may change in future. Should it then expand to ``-\infty`` ?
+
+``\PolToExpr{f}``
+ Expands to ``f_0 + f_1*x + f_2*x^2 + ...`` (ascending powers). [1]_,
+ [2]_
+
+ .. [1] in a ``\write``, ``\edef``, or ``\csname...\endcsname``, but
+ not under ``\romannumeral-`0``.
+
+ .. [2] the letter ``x`` is (in this release) not customizable.
+
+ By default zero coefficients are skipped (issue ``\poltoexprtrue`` to
+ get all of them in output).
+
+ No ``+`` sign before negative coefficients, for compliance with Maple
+ input format. This means though that parsing the result back via
+ naive delimited macros is difficult, see ``\PolToList`` and ``\PolToCSV``
+ for more low-level formats making it easier to get expandably some
+ output of one's choice, which may possibly be parsed later on by
+ other macros of one's design, or from other packages.
+
+ Of course ``\PolToExpr{f}`` can be inserted in a \poldef, as the
+ latter expands token by token, hence will force complete expansion
+ of ``\PolToExpr{f}``, but simply ``f(x)`` will be more efficient for the
+ identical result.
+
+ ``\PolToExprCmd`` is the one-argument macro used by ``\PolToExpr`` for the
+ coefficients, it defaults to ``\xintPRaw{\xintRawWithZeros{#1}}``. One
+ will have to redefine it to use ``\xintIrr{#1}`` in place of
+ ``\xintRawWithZeros{#1}`` to get in output reduced coefficients.
+
+``\PolToList{f}``
+ Expands to ``{f_0}{f_1}...{f_N}`` with ``N`` = degree of f (except
+ zero polynomial which does give ``{0/1[0]}`` and not an empty
+ output.)
+
+``\PolToCSV{f}``
+ Expands to ``f_0, f_1, f_2, ....., f_N``. Converse to
+ ``\PolFromCSV``.
+
+Technicalities
+--------------
+
+- The catcode of the semi-colon is reset temporarily by ``\poldef`` macro in
+ case some other package (for example the French babel module) may have
+ made it active. This will fail though if the whole thing was already
+ part of a macro argument, in such cases one can use ``\PolDef`` rather.
+ The colon in ``:=`` may be active with no consequences.
+
+- Beware the ``1/2 x`` problem: as mentioned above, it will be give zero due
+ to the tacit multiplication rules of ``\xintexpr`` and to the fact that
+ the package will do the Euclidean division of ``1`` by polynomial ``2x``.
+
+- During execution of polynomial operations by ``\poldef`` (but not
+ during the initial purely numerical parsing of the expression), the
+ xintfrac macro ``\xintAdd`` is temporarily patched to always express
+ ``a/b + c/d`` with ``l.c.m.(b,d)`` as denominator. Indeed the current
+ (xint 1.2p) ``\xintAdd`` uses ``(ad+bc)/bd`` formula except if ``b``
+ divides ``d`` or ``d`` divides ``b``, which quickly leads in real life
+ to big denominators.
+
+ It is probable that this convention will be backported as default
+ behaviour of xintfrac's ``\xintAdd`` in a future xint release. When this
+ change is merged, there will be an impact on coefficients computed by
+ ``\poldef`` because the change will apply even to the pure numerical
+ evaluations arising during the initial stage of the parsing. Of course
+ the coefficients are still the same rational numbers, only
+ representation as fractions may change.
+
+- As a consequence of previous rule, user-chosen common denominators
+ survive addition and multiplications::
+
+ \poldef P(x):= 1/2 + 2/2*x + 3/2*x^3 + 4/2*x^4;
+ \poldef Q(x):= 1/3 + (2/3)x + (3/3)x^3 + (4/3)x^4;
+ \poldef PQ(x):= P(x)Q(x);
+
+ gives the polynomial::
+
+ 1/6+4/6*x^1+4/6*x^2+6/6*x^3+20/6*x^4+16/6*x^5+9/6*x^6+24/6*x^7+16/6*x^8
+
+ where all coefficients have the same denominator 6 (which in this
+ example is the ``l.c.m`` of the denominators of the reduced coefficients.)
+
+- ``\PolDiff`` always applies ``\xintIrr`` to the resulting coefficients, except
+ that the *power of ten* part ``[N]`` (for example an input in scientific
+ notation such as ``1.23e5`` gives ``123/1[3]`` internally in xintfrac) is not
+ taken into account in the reduction of the fraction. This is tentative
+ and may change.
+
+ Same remark for ``\PolAntiDiff``.
+
+- If ``f`` was created from comma separated values by macro \PolFromCSV,
+ then the coefficients will be in the output of ``\PolToList`` and
+ ``\PolToCSV`` in the same format as originally in input: a ``1.3e2``
+ will again be a ``1.3e2``.
+
+ In contrast when such coefficients are used in a ``\poldef`` (or
+ ``\PolDef``) expression, they get transformed during the parsing to
+ the xintfrac *raw* format. This is an unavoidable consequence of usage
+ by \poldef of ``\xintdeffunc`` which itself is based on ``\xintexpr.``
+ This *raw* format speeds up expansion of xintfrac macros for numerical
+ evaluations.
+
+- Currently, the package does not as a result of ``\poldef`` add to the TeX
+ memory an already pre-computed *array* structure for the polynomial
+ coefficients, as would be constructed by ``\PolAssign{f}\toarray\macro``.
+ Such structures are used, but for internal calculations in temporarily
+ restricted scopes. Apart from the function ``f()`` known to the
+ (numerical) ``\xintexpr`` parser (whose meaning can be found in the log
+ file after \xintverbosetrue), the data is (currently) stored in a
+ single other macro encapsulating the degree, and the coefficients as a
+ list. This may evolve in future.
+
+- As is to be expected internal structures of the package are barely
+ documented and unstable. Don't use them.
+
+RELEASES
+--------
+
+- 0.1 (2018/01/11): initial release (files README, polexpr.sty).
+- 0.2 (2018/01/14): documentation moved to polexpr.{txt,html}.
+
+Files of 0.2 release:
+
+- README.md,
+- polexpr.sty (package file),
+- polexpr.txt (documentation),
+- polexpr.html (conversion via `DocUtils`__ rst2html.py)
+
+ __ http://docutils.sourceforge.net/docs/index.html
+
+See README.md for the License and the change log.
Property changes on: trunk/Master/texmf-dist/doc/latex/polexpr/polexpr.txt
___________________________________________________________________
Added: svn:eol-style
## -0,0 +1 ##
+native
\ No newline at end of property
Modified: trunk/Master/texmf-dist/tex/latex/polexpr/polexpr.sty
===================================================================
--- trunk/Master/texmf-dist/tex/latex/polexpr/polexpr.sty 2018-01-15 22:19:53 UTC (rev 46317)
+++ trunk/Master/texmf-dist/tex/latex/polexpr/polexpr.sty 2018-01-15 22:20:09 UTC (rev 46318)
@@ -1,7 +1,7 @@
% author: Jean-François Burnol
% License: LPPL 1.3c (author-maintained)
\ProvidesPackage{polexpr}%
- [2018/01/11 v0.1 Polynomial expressions with rational coefficients (JFB)]%
+ [2018/01/14 v0.2 Polynomial expressions with rational coefficients (JFB)]%
\RequirePackage{xintexpr}[2016/03/19]% xint 1.2g (or 1.2c 2015/11/16 at least)
\edef\POL at restorecatcodes
{\catcode`\noexpand\_ \the\catcode`\_ \catcode0 \the\catcode0\relax}%
@@ -572,7 +572,7 @@
%% CORE ALGEBRA MACROS
%% We do this non-expandably, but in a nestable way... this is the whole
-%% point because \xintdeffunc as used by \PolNewPol creates a big nested macro.
+%% point because \xintdeffunc as used by \poldef creates a big nested macro.
%% The idea is to execute it with another meaning given to \xintAdd etc..,
%% so that it operates on "polynomials". This is a mixture of expandable
%% and non-expandable techniques.
@@ -920,7 +920,10 @@
%% EXPANDABLE MACROS
\newcommand\PolEval{}%
-\def\PolEval#1#2\At#3{\romannumeral`^^@\xinttheexpr reduce(#1(#3))\relax}%
+\def\PolEval#1#2\At#3{%
+ \romannumeral0\xintpraw % only serves to not print denominator if = 1
+ {\xintIrr{\csname XINT_expr_userfunc_#1\endcsname {#3},}[0]}%
+}%
%
\newcommand\PolNthCoeff[2]{\romannumeral`^^@%
\expandafter\POL at nthcoeff
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