summaryrefslogtreecommitdiff
path: root/Master/texmf-dist/doc/latex/polexpr/polexpr.txt
blob: 2825128f0040882dac49e12d2950672f8c29604e (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
.. comment: -*- fill-column: 72; mode: rst; -*-

===============================
 Package polexpr documentation
===============================

0.3.1 (2018/01/18)
==================

.. contents::

First Examples
--------------

The syntax is::

    \poldef polname(x):= expression in variable x;

where in place of ``x`` an arbitrary *dummy variable* is authorized
(i.e. per default any of ``[a-z|A-Z]``; more letters can be declared
under Unicode engines.) One can also issue::

    \PolDef{polname}{expression in variable x}

which admits an optional first argument to modify the variable letter
from its default ``x``.

``\poldef f(x):= 1-x+x^2;``
    defines polynomial ``f``. Polynomial names must start with a
    letter and may contain letters, digits, 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``. Also usable without ``=``.

``\poldef f(z):= f(z)^2;``
    redefines ``f`` in terms of itself.

``\poldef f(T):= f(f(T));``
    again redefines ``f`` in terms of its (new) self.

``\poldef k(z):= f(z)-g(g(z)^2)^2;``
    should now define the zero polynomial... Let's check:
    ``\[ k(z) = \PolTypeset[z]{k} \]``

``\PolDiff{f}{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);

.. _warningtacit:

.. attention:: 

   ``1/2 x^2`` skips the space and is treated like ``1/(2*x^2)`` because
   of the tacit multiplication rules of \xintexpr. But this means it
   gives zero! Thus one must use ``(1/2)x^2`` or ``1/2*x^2`` or
   ``(1/2)*x^2`` for disambiguation: ``x - 1/2*x^2 + 1/3*x^3...``. It is
   even simpler to move the denominator to the right: ``x - x^2/2 +
   x^3/3 - ...``.

   It is worth noting that ``1/2(x-1)(x-2)`` suffers the same issue:
   xint_ tacit multiplication always "ties more", hence this gets
   interpreted as ``1/(2*(x-1)*(x-2))`` which gives zero by polynomial
   division. Thus, use one of ``(1/2)(x-1)(x-2)``, ``1/2*(x-1)(x-2)`` or
   ``(x-1)(x-2)/2``.

After::

  \poldef f_1(x):= 25(x-1)(x^2-2)(x-3)(x-4)(x-5);%
  \poldef f_2(x):= 37(x-1)(x^2-2)(x-6)(x-7)(x-8);%

the macro call ``\PolGCD{f_1}{f_2}{k}`` sets ``k`` to the (unitary) GCD of
``f_1`` and ``f_2`` (hence to the expansion of ``(x-1)(x^2-2)``.)

``\PolToExpr{k}``
    will (expandably) give in this case ``x^3-x^2-2*x+2``. This is
    useful for console or file output (the syntax is Maple- and
    PSTricks-compatible; the letter used in output can be
    (non-expandably) changed via a redefinition of `\\PolToExprVar`_.)

``\PolToExpr*{k}``
    gives ascending powers: ``2-2*x-x^2+x^3``.

Non-expandable macros
---------------------

.. _poldef;:

``\poldef polname(letter):= expression in letter;``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    This evaluates the *polynomial expression* and stores the coefficients
    in a private structure accessible later via other package macros,
    under the user-chosen ``polname``. Of course the *expression* can
    use other previously defined polynomials. Names must start with a
    letter and are constituted of letters, digits and underscore
    characters. The whole xintexpr_ syntax is authorized::

       \poldef sin(z) := add((-1)^i z^(2i+1)/(2i+1)!, i = 0..10);

    With fractional coefficients, beware the `tacit multiplication issue
    <warningtacit_>`_.

    As a side effect the function ``polname()`` is recognized as a
    genuine ``\xintexpr...\relax`` function for (exact) numerical
    evaluation (or within an ``\xintdefvar`` assignment.) It computes
    values not according to the original expression but via the Horner
    scheme corresponding to the polynomial coefficients.

    Also, a function with the same name is created for use within
    ``\xintfloatexpr`` (or ``\xintdeffloatvar``.) This is indispensible
    for numerical algorithms as exact computations very quickly lead to
    very big fractions. Addition and multiplication steps of the Horner
    scheme will be executed as floating-point operations. The
    coefficients have already been rounded at time of definition,
    according to the then prevailing ``\xinttheDigits`` value.

    .. important::

       Package macros (such as derivatives or Euclidean division)
       operate with the "exact" polynomials; "floating point"
       polynomials are always obtained in a second step.

       To modifiy "in-place" the original coefficients of a polynomial
       and round them to float precision::

         \PolMapCoeffs{\xintFloat}{polname}
         % or \xintFloat[P] for precision P digits

       See `\\PolMapCoeffs{\\macro}{polname}`_.

    The original expression is lost after parsing, and in particular
    the package provides no way to typeset it. This has to be done
    manually, if needed.

.. _PolDef:

``\PolDef[letter]{polname}{expression in letter}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    Does the same in an undelimited macro format (thus avoiding
    potential problems with the catcode of the semi-colon in presence of
    some packages.) In absence of the ``[letter]`` optional argument,
    the variable is assumed to be ``x``.

``\PolLet{polname_2}={polname_1}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    Makes a copy of the already defined polynomial ``polname_1`` to a
    new one ``polname_2``. Same effect as
    ``\PolDef{polname_2}{polname_1(x)}`` but with less overhead. The
    ``=`` is optional.

``\PolAssign{polname}\toarray\macro``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    Defines a one-argument expandable macro ``\macro{#1}`` which expands
    to the (raw) #1th polynomial coefficient.

    - Attention, coefficients here are indexed starting at 1.

    - With #1=-1, -2, ..., ``\macro{#1}`` returns leading coefficients.

    - With #1=0, returns the number of coefficients, i.e. ``1 + deg f``
      for non-zero polynomials.

    - Out-of-range #1's return ``0/1[0]``.

    See also `\\PolNthCoeff{polname}{number}`_. The main difference is that
    with ``\PolAssign``, ``\macro`` is made a prefix to ``1 + deg f``
    already defined (hidden to user) macros holding individually the
    coefficients but `\\PolNthCoeff{polname}{number}`_ does each time the job
    to expandably recover the ``Nth`` coefficient, and due to
    expandability can not store it in a macro for future usage (of course,
    it can be an argument in an ``\edef``.) The other difference
    is the shift by one in indexing, mentioned above (negative
    indices act the same in both.)

``\PolGet{polname}\fromarray\macro``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    Does the converse operation to ``\PolAssign{polname}\toarray\macro``. No
    error checks on validity of coefficients as numbers. Each
    ``\macro{number}`` 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{polname}{<csv>}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    Defines a polynomial directly from the comma separated list of
    values (or a macro expanding to such a list) of its coefficients,
    the constant term being the first item. No validity checks. Spaces
    from the list argument are trimmed. List items are each 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{f}{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 <\\PolToCSV{polname}_>`_.

``\PolTypeset{polname}``
~~~~~~~~~~~~~~~~~~~~~~~~

    Typesets in descending powers in math mode. It uses letter ``x`` but
    this can be changed via an optional argument::

      \PolTypeset[z]{polname}

    By default zero coefficients are skipped (issue ``\poltypesetalltrue``
    to get all of them in output).

    These commands (whose meanings will be found in the package code)
    can be re-defined for customization. Their default definitions are
    expandable, but this is not a requirement.

``\PolTypesetCmd{raw_coeff}``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

    Basically will use ``\xintSignedFrac`` from xintfrac_, but checks if
    the coefficient is ``1`` or ``-1`` and then skips printing the
    ``1``, except for the constant term...

    One can do things such as for example: [#]_

    ::

      \renewcommand\PolTypesetCmd[1]{\num{\xintPFloat[5]{#1}}}
      \renewcommand\PolTypesetCmd[1]{\num{\xintRound{4}{#1}}}

    where e.g. we used the ``\num`` macro of ``siunitx`` as it
    understands floating point notation.

    .. [#] the difference in the syntaxes of ``\xintPFloat`` and
           ``\xintRound`` is explained from the fact that
           ``\xintPFloat`` by default uses the prevailing precision
           hence the extra argument like here ``5`` is an optional one.

``\PolTypesetCmdPrefix{raw_coeff}``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

    Expands to a ``+`` if the ``raw_coeff`` is zero or positive, and to
    nothing if ``raw_coeff`` is negative, as in latter case the
    ``\xintSignedFrac`` used by `\\PolTypesetCmd{raw_coeff}`_ will put
    the ``-`` sign in front of the fraction (if it is a fraction) and
    this will thus serve as separator in the typeset formula. Not used
    for the first term.

``\PolTypesetMonomialCmd``
^^^^^^^^^^^^^^^^^^^^^^^^^^

    This decides how a monomial (in variable ``\PolVar`` and with
    exponent ``\PolIndex``) is to be printed. The default does nothing
    for the constant term, ``\PolVar`` for the first degree and
    ``\PolVar^{\PolIndex}`` for higher degrees monomials. Beware that
    ``\PolIndex`` expands to digit tokens and needs termination in
    ``\ifnum`` tests.

``\PolTypeset*{polname}``
~~~~~~~~~~~~~~~~~~~~~~~~~

    Typesets in ascending powers. Use e.g. ``[h]`` optional argument
    (after the ``*``) to use letter ``h`` rather than ``x``.

``\PolDiff{polname_1}{polname_2}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    This sets ``polname_2`` to the first derivative of ``polname_1``. It
    is allowed to issue ``\PolDiff{f}{f}``, effectively replacing ``f``
    by ``f'``.

    Coefficients of the result ``polname_2`` are irreducible fractions
    (see `Technicalities`_ for the whole story.)

``\PolDiff[N]{polname_1}{polname_2}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    This sets ``polname_2`` to the ``N``-th derivative of ``polname_1``.
    Identical arguments is allowed. With ``N=0``, same effect as
    ``\PolLet{polname_2}={polname_1}``. With negative ``N``, switches to
    using ``\PolAntiDiff``.

``\PolAntiDiff{polname_1}{polname_2}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    This sets ``polname_2`` to the primitive of ``polname_1`` vanishing
    at zero.

    Coefficients of the result ``polname_2`` are irreducible fractions
    (see `Technicalities`_ for the whole story.)

``\PolAntiDiff[N]{polname_1}{polname_2}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    This sets ``polname_2`` to the result of ``N`` successive integrations on
    ``polname_1``. With negative ``N``, it switches to using ``\PolDiff``.

``\PolDivide{polname_1}{polname_2}{polname_Q}{polname_R}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    This sets ``polname_Q`` and ``polname_R`` to be the quotient and
    remainder in the Euclidean division of ``polname_1`` by
    ``polname_2``.

``\PolGCD{polname_1}{polname_2}{polname_GCD}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    This sets ``polname_GCD`` to be the G.C.D. It is a unitary
    polynomial except if both ``polname_1`` and ``polname_2`` vanish,
    then ``polname_GCD`` is the zero polynomial.

``\PolMapCoeffs{\macro}{polname}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    It modifies ('in-place': original coefficients get lost) each
    coefficient of the defined polynomial via the *expandable* macro
    ``\macro``. The degree is adjusted as necessary if some leading
    coefficients vanish after the operation. In replacement text of
    ``\macro``, ``\index`` expands to the coefficient index (which is
    defined to be zero for the constant term).

    Notice that ``\macro`` will have to handle inputs of the shape
    ``A/B[N]`` (xintfrac_ internal notation). This means that it probably
    will have to be expressed in terms of macros from xintfrac package.

    Example::

      \def\foo#1{\xintMul{#1}{\the\numexpr\index*\index\relax}}

    (or with ``\xintSqr{\index}``) to replace ``n``-th coefficient
    ``f_n`` by ``f_n*n^2``.

``\PolReduceCoeffs{polname}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    About the same as ``\PolMapCoeffs{\xintIrr}{polname}`` (but
    maintaining a ``[0]`` postfix for speedier xintfrac_ parsing when
    polynomial function is used for computations.) This is a
    one-argument macro, working 'in-place'.

Expandable macros
-----------------

All these macros expand completely in two steps except ``\PolToExpr``
and ``\PolToFloatExpr`` (and their auxiliaries) which need a
``\write``, ``\edef`` or a ``\csname...\endcsname`` context.

``\PolEval{polname}\At{numerical expression}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    It boils down to ``\xinttheexpr polname(numerical expression)\relax``.

    .. note::

       The ``0.2`` version stupidly tried to be clever and as a result
       of a misguided optimization choked if ``value`` was not a number
       but a numerical expression (a sum e.g.), but the more powerful
       behaviour has been reinstored at ``0.3`` release.

       The ``0.1`` and ``0.2`` version did a ``reduce`` which however is
       costly on big fractions and irrelevant if the output is served as
       argument of ``\xintRound`` or ``\xintFloat``. Thus ``reduce`` was
       removed, and former meaning is now available as
       `\\PolEvalReduced{polname}\\At{numerical expression}`_

``\PolEvalReduced{polname}\At{numerical expression}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    Boils down to ``\xinttheexpr reduce(polname(numerical expression))\relax``.

``\PolFloatEval{polname}\At{numerical expression}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    Boils down to ``\xintthefloatexpr polname(numerical expression)\relax``.

    This is done via a Horner Scheme (see `\\poldef <poldef;_>`_), with
    already rounded coefficients. [#]_ To use the *exact coefficients*
    (and *exact* additions and multiplications), just insert it in the
    float expression as in this example: [#]_

    ::

        \xintthefloatexpr 3.27*\xintexpr f(2.53)\relax^2\relax

    The ``f(2.53)`` is exactly computed then rounded at the time of
    getting raised to the power ``2``. Moving the ``^2`` inside, that
    operation would also be treated exactly.

    .. [#] Anyway each floating point operation starts by rounding its
           operands to the floating point precision.

    .. [#] The ``\xintexpr`` could be ``\xinttheexpr`` but that would be
           less efficient. Cf. xintexpr_ documentation about nested
           expressions.

``\PolNthCoeff{polname}{number}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    It expands to the raw ``N``-th coefficient (``0/1[0]`` if the index
    number is out of range). With ``N=-1``, ``-2``, ... expands to the
    leading coefficients.

``\PolDegree{polname}``
~~~~~~~~~~~~~~~~~~~~~~~

    It expands to the degree. This is ``-1`` if zero polynomial but this
    may change in future. Should it then expand to ``-\infty`` ?

``\PolToExpr{polname}``
~~~~~~~~~~~~~~~~~~~~~~~

    Expands [#]_ to ``coeff_N*x^N+...`` (descending powers.)

    .. [#] in a ``\write``, ``\edef``, or ``\csname...\endcsname``, but
           not under ``\romannumeral-`0``.

    By default zero coefficients are skipped (issue ``\poltoexpralltrue`` to
    get all of them in output).

    By default, no ``+`` sign before negative coefficients, for
    compliance with Maple input format (but see
    `\\PolToExprTermPrefix{raw_coeff}`_.) Also, like the default
    behaviour of `\\PolTypeset{polname}`_, does not print (for the non
    constant terms) coefficients equal to plus or minus one. The degree
    one monomial is output as ``x``, not ``x^1``. Complete customization is
    possible, see next macros.

    Of course ``\PolToExpr{f}`` can be inserted in a ``\poldef``, as the
    latter expands token by token, hence will force complete expansion
    of ``\PolToExpr{f}``, but a simple ``f(x)`` is more efficient for
    the identical result.

``\PolToExprOneTerm{raw_coeff}{number}``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

    This two argument expandable command takes care of the monomial and
    its coefficient. The default definition is done in order for
    coefficients of absolute value ``1`` not be printed explicitely
    (except of course for the constant term). Also by default, the
    monomial of degree one is ``x`` not ``x^1``, and ``x^0`` is skipped.

    For compatibility with Maple input requirements, by default a ``*``
    always precedes the ``x^number``, except if the coefficient is a one
    or a minus one. See `\\PolToExprTimes`_.

``\PolToExprOneTermStyleB{raw_coeff}{number}``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

    For output in this style::

      2*x^11/3+3*x^8/7-x^5−x^4/4−x^3−x^2/2−2*x+1

    issue ``\let\PolToExprOneTerm\PolToExprOneTermStyleB`` before using
    ``\PolToExpr``. Note that then ``\PolToExprCmd`` isn't used at all.

    To suppress the ``*``'s, cf. `\\PolToExprTimes`_.

``\PolToExprCmd{raw_coeff}``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^

    It is the one-argument macro used by the package definition of
    ``\PolToExprOneTerm`` for the coefficients themselves (when not
    equal to plus or minus one), and it defaults to
    ``\xintPRaw{\xintRawWithZeros{#1}}``. One will have to redefine it
    to ``\xintIrr{#1}`` or to ``\xintPRaw{\xintIrr{#1}}`` to obtain in the
    output forcefully reduced coefficients.

``\PolToExprTermPrefix{raw_coeff}``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

    Defined identically as `\\PolTypesetCmdPrefix{raw_coeff}`_. It
    prefixes with a plus sign for non-negative coefficients, because
    they don't carry one by themselves.

``\PolToExprVar``
^^^^^^^^^^^^^^^^^

    This expands to the variable to use in output (it does not have to
    be a single letter, may be an expandable macro.) Initial definition
    is ``x``.

``\PolToExprTimes``
^^^^^^^^^^^^^^^^^^^

    This expands to the symbol used for multiplication of an
    ``x^{number}`` by the corresponding coefficient. The default is
    ``*``. Redefine the macro to expand to nothing to get rid of it (but
    this will give output incompatible with some professional computer
    algebra software).

``\PolToExpr*{polname}``
~~~~~~~~~~~~~~~~~~~~~~~~

    Expands to ``coeff_0+coeff_1*x+coeff_2*x^2+...`` (ascending powers).
    Customizable like `\\PolToExpr{polname}`_ via the same macros.

``\PolToFloatExpr{polname}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    Similar to `\\PolToExpr{polname}`_ but uses `\\PolToFloatExprCmd
    <\\PolToFloatExprCmd{raw_coeff}>`_
    which by default rounds and converts the coefficients to floating
    point format.

    .. note::

       The polynomial function for usage in ``\xintfloatexpr`` is
       already prepared with the rounded coefficients, but the latter
       are not easily recoverable (and especially not expandably) from
       this. Thus ``\PolToFloatExprCmd`` operates from the *exact*
       coefficients anew. This means though that if the prevailing float
       precision was changed with ``\xintDigits:=P;`` syntax, the output
       will obey this precision ``P``, but the polynomial function was
       defined earlier and operates on floating point numbers with
       coefficients which were rounded at time of definition.

       This may change in future, if the pre-rounded coefficients are
       stored in a more easily accessible data structure.

``\PolToFloatExprOneTerm{raw_coeff}{number}``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

    Similar to `\\PolToExprOneTerm
    <\\PolToExprOneTerm{raw_coeff}{number}>`_. But does not treat
    especially coefficients equal to plus or minus one.

``\PolToFloatExprCmd{raw_coeff}``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

    It is the one-argument macro used by ``\PolToFloatExprOneTerm``.
    Its package definition is ``\xintFloat{#1}``.

    .. caution::

       Currently (xint_ ``1.2p``) ``\xintFloat{0}`` outputs ``0.e0``
       which is perfectly acceptable input for Python, but not for
       Maple. Thus, one should better leave the `\\poltoexprallfalse`_
       toggle to its default ``\iffalse`` state, if one intends to use
       the output in a Maple worksheet.

       But even then the zero polynomial will cause a problem. Workaround::

         \renewcommand\PolToFloatExprCmd[1]{\xintiiifZero{#1}{0.0}{\xintFloat{#1}}}

       Usage of ``\xintiiifZero`` and not ``\xintifZero`` is only for
       optimization (I can't help it) because ``#1`` is known to be
       in ``xintfrac`` raw format.

``\PolToFloatExpr*{polname}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    Typesets in ascending powers.

``\PolToList{polname}``
~~~~~~~~~~~~~~~~~~~~~~~

    Expands to ``{coeff_0}{coeff_1}...{coeff_N}`` with ``N`` = degree
    (except zero polynomial which does give ``{0/1[0]}`` and not an
    empty output.)

``\PolToCSV{polname}``
~~~~~~~~~~~~~~~~~~~~~~

    Expands to ``coeff_0, coeff_1, coeff_2, ....., coeff_N``. Converse
    to `\\PolFromCSV <\\PolFromCSV{polname}{\<csv\>}_>`_.

Booleans (with default setting as indicated)
--------------------------------------------

``\xintverbosefalse``
~~~~~~~~~~~~~~~~~~~~~

    This is actually an xintexpr_ configuration. Setting it to
    ``true`` triggers the writing of information to the log when new
    polynomials are defined.

    .. caution::

       The macro meanings as written to the log are to be considered
       unstable and undocumented internal structures.

``\poltypesetallfalse``
~~~~~~~~~~~~~~~~~~~~~~~

    If ``true``, `\\PolTypeset{polname}`_ will also typeset the vanishing
    coefficients.


``\poltoexprallfalse``
~~~~~~~~~~~~~~~~~~~~~~

    If ``true``, `\\PolToExpr{polname}`_ and `\\PolToFloatExpr{polname}`_ will
    also include the vanishing coefficients in their outputs.


Technicalities
--------------

- The catcode of the semi-colon is reset temporarily by `\\poldef
  <poldef;_>`_ macro in case some other package (for example the French
  babel module) may have made it active. This will fail though if the
  whole thing was already part of a macro argument, in such cases one
  can use `\\PolDef{f}{P(x)} <PolDef_>`_
  rather. The colon in ``:=`` may be active with no consequences.

- 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{polname_1}{polname_2}`_ always applies ``\xintIrr`` to the
  resulting coefficients, except that the *power of ten* part ``[N]``
  (for example an input in scientific notation such as ``1.23e5`` gives
  ``123/1[3]`` internally in xintfrac) is not taken into account in the
  reduction of the fraction. This is tentative and may change.

  Same remark for `\\PolAntiDiff{polname_1}{polname_2}`_.

- If ``f`` was created from comma separated values by macro
  `\\PolFromCSV{polname}{\<csv\>}`_, then the coefficients will be in
  the output of `\\PolToList{polname}`_ and `\\PolToCSV{polname}`_ 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 *raw* format speeds up expansion of xintfrac macros for numerical
  evaluations.

- Currently, the package stores all coefficients from index ``0`` to
  index equal to the polynomial degree inside a single macro, as a list.
  This data structure is obviously very inefficient for polynomials of
  high degree and few coefficients (as an example with ``\poldef
  f(x):=x^1000 + x^500;`` the subsequent definition ``\poldef g(x):=
  f(x)^2;`` will do of the order of 1,000,000 multiplications and
  additions involvings only zeroes... which does take time). This
  may change in the future.

- Tests have been made with Newton's iteration (for which computing
  exactly the derivative is precisely what this package is made for) or
  Regula Falsi method for locating roots: using exact computations leads
  quickly to gigantic fractions (but dichotomy method much less so). It
  is thus recommended to use ``\xintdeffloatvar`` or
  ``\xintthefloatexpr`` contexts for any kind of numerical mathematics.
  Of course, exact computations are invaluable for number theory or
  combinatorics...

- 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}.

- 0.3 (2018/01/17)
  
  Make polynomials known to ``\xintfloatexpr`` and improve
  documentation.

- 0.3.1 (2018/01/18)

  Fix two typos in documentation.

Files of 0.3.1 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 (there were
some breaking changes from 0.2 to 0.3).

Acknowledgments
---------------

Thanks to Jürgen Gilg whose question about xint_ usage for
differentiating polynomials was the initial trigger leading to this
package, and to Jürgen Gilg and Thomas Söll for testing it on some
concrete problems.

.. _xintfrac:
.. _xintexpr:
.. _xint: http://www.ctan.org/pkg/xint