path: root/Master/texmf-dist/doc/latex/polexpr/polexpr.txt

.. comment: -*- fill-column: 72; mode: rst; -*-

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

0.7.3 (2019/02/04)
==================

.. contents::

Basic syntax
------------

The syntax is::

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

where:

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

- ``polname`` consists of letters, digits, and the ``_`` and
  ``'`` characters. It must start with a letter.

.. attention::

   The ``'`` is authorized since ``0.5.1``. As a result some constructs
   recognized by the ``\xintexpr`` parser, such as ``var1 'and' var2``
   will get misinterpreted and cause errors. However these constructs
   are unlikely to be frequently needed in polynomial expressions, and
   the ``\xintexpr`` syntax offers alternatives, so it was deemed a
   small evil. Of course the ``\xintexpr`` parser is modified only
   temporarily during execution of ``\poldef``.

One can also issue::

    \PolDef{polname}{expression in variable x}

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

``\poldef f(x):= 1-x+x^2;``
    defines polynomial ``f``. Polynomial names must start with a
    letter and may contain letters, digits, underscores and the right
    tick character. The
    variable must be a single letter. The colon character is optional.
    The semi-colon at end of expression is mandatory.

``\PolDef{f}{1-x+x^2}``
    does the same as ``\poldef f(x):= 1-x+x^2;`` To use another letter
    than ``x`` in the expression, one must pass it as an extra optional
    argument to ``\PolDef``. Useful if the semi-colon has been assigned
    some non-standard catcode by some package.

``\PolLet{g}={f}``
    saves a copy of ``f`` under name ``g``. Also usable without ``=``.

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

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

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

``\PolDiff{f}{f'}``
    sets ``f'`` to the derivative of ``f``. The name doesn't have to be
    ``f'`` (in fact the ``'`` is licit only since ``0.5.1``).

.. important::

   This is not done automatically. If some new definition needs to use
   the derivative of some available polynomial, that derivative
   polynomial must have been defined via ``\PolDiff``: something like
   ``T'(x)^2`` will not work without a prior ``\PolDiff{T}{T'}``.

``\PolDiff{f'}{f''}``
    obtains second derivative.

``\PolDiff[3]{f}{f'''}``
    computes the third derivative.

::

  $f(z)   = \PolTypeset[z]{f}    $\newline
  $f'(z)  = \PolTypeset[z]{f'}   $\newline
  $f''(z) = \PolTypeset[z]{f''}  $\newline
  $f'''(z)= \PolTypeset[z]{f'''} $\par

.. important::

   The package does not currently know rational functions: ``/`` in
   a parsed polynomial expression does the Euclidean quotient::

     (1-x^2)/(1-x)

   does give ``1+x`` but ::

     (1/(1-x))*(1-x^2)

   evaluates to zero. This will work as expected::

     \poldef k(x):= (x-1)(x-2)(x-3)(x-4)/(x^2-5x+4);

.. _warningtacit:

.. attention::

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

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

After::

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

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

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

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

Examples of localization of roots
---------------------------------

- To make printed decimal numbers more enjoyable than via
  ``\xintSignedFrac``::

    \renewcommand\PolTypesetOne[1]{\PolDecToString{\xintREZ{#1}}}%

  ``\PolDecToString`` will use decimal notation to incorporate the power
  of ten part; and the ``\xintREZ`` will have the effect to suppress
  trailing zeros if present in raw numerator (if those digits end up
  after decimal mark.) Notice that the above are expandable macros and
  that one can also do::

    \renewcommand\PolToExprCmd[1]{\PolDecToString{\xintREZ{#1}}}%

  to modify output of `\\PolToExpr{polname}`_.

- For extra info in log file use ``\xintverbosetrue``.

- Only for some of these examples is the output included here.


A typical example
~~~~~~~~~~~~~~~~~

In this example the polynomial is square-free.

::

  \poldef f(x) := x^7 - x^6 - 2x + 1;

  \PolToSturm{f}{f}
  \PolSturmIsolateZeros{f}
  The \PolTypeset{f} polynomial has \PolSturmNbOfIsolatedZeros{f} distinct real
  roots which are located in the following intervals:
  \PolPrintIntervals{f}
  Here is the second root with ten more decimal digits:
  \PolRefineInterval[10]{f}{2}
  \[\PolSturmIsolatedZeroLeft{f}{2}<Z_2<\PolSturmIsolatedZeroRight{f}{2}\]
  And here is the first root with twenty digits after decimal mark:
  \PolEnsureIntervalLength{f}{1}{-20}
  \[\PolSturmIsolatedZeroLeft{f}{1}<Z_1<\PolSturmIsolatedZeroRight{f}{1}\]
  The first element of the Sturm chain has degree $\PolDegree{f_0}$. As
  this is the original degreee $\PolDegree{f}$ we know that $f$ is square free.
  Its derivative is up to a constant \PolTypeset{f_1} (in this example
  it is identical with it).
  \PolToSturm{f_1}{f_1}\PolSturmIsolateZeros{f_1}%
  The derivative has \PolSturmNbOfIsolatedZeros{f_1} distinct real
  roots:
  \PolPrintIntervals[W]{f_1}
  \PolEnsureIntervalLengths{f_1}{-10}%
  Here they are with ten digits after decimal mark:
  \PolPrintIntervals[W]{f_1}
  \PolDiff{f_1}{f''}
  \PolToSturm{f''}{f''}
  \PolSturmIsolateZeros{f''}
  The second derivative is \PolTypeset{f''}.
  It has \PolSturmNbOfIsolatedZeros{f''} distinct real
  roots:
  \PolPrintIntervals[X]{f''}
  Here is the positive one with 20 digits after decimal mark:
  \PolEnsureIntervalLength{f''}{2}{-20}%
  \[X_2 = \PolSturmIsolatedZeroLeft{f''}{2}\dots\]
  The more mathematically advanced among our dear readers will be able
  to give the exact value for $X_2$!

A degree four polynomial with nearby roots
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Notice that this example is a bit outdated as ``0.7`` release has
added ``\PolSturmIsolateZeros**{sturmname}`` which would find exactly
the roots. The steps here retain their interest when one is interested
in finding isolating intervals for example to prepare some demonstration
of dichotomy method.


::

  \PolDef{Q}{(x-1.050001)(x-1.105001)(x-1.110501)(x-1.111051)}
  \PolTypeset{Q}
  \PolToSturm{Q}{Q} % it is allowed to use same prefix for Sturm chain
  \PolSturmIsolateZeros{Q}
  \PolPrintIntervals{Q}
  % reports 1.0 < Z_1 < 1.1, 1.10 < Z_2 < 1.11, 1.110 < Z_3 < 1.111, and 1.111 < Z_4 < 1.112
  % but the above bounds do not allow minimizing separation between roots
  % so we refine:
  \PolRefineInterval*{Q}{1}
  \PolRefineInterval*{Q}{2}
  \PolRefineInterval*{Q}{3}
  \PolRefineInterval*{Q}{4}
  \PolPrintIntervals{Q}
  % reports 1.05 < Z_1 < 1.06, 1.105 < Z_2 < 1.106, 1.1105 < Z_3 < 1.1106,
  % and 1.11105 < Z_4 < 1.11106.
  \PolEnsureIntervalLengths{Q}{-6}
  \PolPrintIntervals{Q}
  % of course finds here all roots exactly


The degree nine polynomial with 0.99, 0.999, 0.9999 as triple roots
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

::

  % define a user command (xinttools is loaded automatically by polexpr)
  \newcommand\showmultiplicities[1]{% #1 = "sturmname"
  \xintFor* ##1 in {\xintSeq{1}{\PolSturmNbOfIsolatedZeros{#1}}}\do{%
      The multiplicity is \PolSturmIsolatedZeroMultiplicity{#1}{##1}
      \PolSturmIfZeroExactlyKnown{#1}{##1}%
      {at the root $x=\PolSturmIsolatedZeroLeft{#1}{##1}$}
      {for the root such that
      $\PolSturmIsolatedZeroLeft{#1}{##1}<x<\PolSturmIsolatedZeroRight{#1}{##1}$}
      \par
  }}%
  \PolDef{f}{(x-0.99)^3(x-0.999)^3(x-0.9999)^3}
  \renewcommand\PolTypesetOne[1]{\PolDecToString{\xintREZ{#1}}}
  \PolTypeset{f}\par
  \PolToSturm{f}{f}% it is allowed to use "polname" as "sturmname" too
  \PolSturmIsolateZerosAndGetMultiplicities{f}% use the "sturmname" here
  % or \PolSturmIsolateZeros*{f} which is exactly the same, but shorter..

  \showmultiplicities{f}

In this example, the output will look like this (but using math mode)::

  x^9 - 8.9667x^8 + 35.73400293x^7 - 83.070418400109x^6 + 124.143648875193123x^5
  - 123.683070924326075877x^4 + 82.149260397553075617891x^3 
  - 35.07602992699900159127007x^2 + 8.7364078733314648368671733x
  - 0.967100824643585986488103299

  The multiplicity is 3 at the root x = 0.99
  The multiplicity is 3 at the root x = 0.999
  The multiplicity is 3 at the root x = 0.9999

On first pass, these rational roots were found (due to their relative
magnitudes, using ``\PolSturmIsolateZeros**`` was not needed here). But
multiplicity computation works also with (decimal) roots not yet
identified or with non-decimal or irrational roots.

It is fun to modify only a tiny bit the polynomial and see if polexpr
survives::

  \PolDef{g}{f(x)+1e-27}
  \PolTypeset{g}\par
  \PolToSturm{g}{g}
  \PolSturmIsolateZeros*{g}

  \showmultiplicities{g}

This produces::

  x^9 - 8.9667x^8 + 35.73400293x^7 - 83.070418400109x^6 + 124.143648875193123x^5
  - 123.683070924326075877x^4 + 82.149260397553075617891x^3 
  - 35.07602992699900159127007x^2 + 8.7364078733314648368671733x
  - 0.967100824643585986488103298

  The multiplicity is 1 for the root such that 0.98 < x < 0.99
  The multiplicity is 1 for the root such that 0.9991 < x < 0.9992
  The multiplicity is 1 for the root such that 0.9997 < x < 0.9998

Which means that the multiplicity-3 roots each became a real and a pair of
complex ones. Let's see them better::

  \PolEnsureIntervalLengths{g}{-10}

  \showmultiplicities{g}

which produces::

  The multiplicity is 1 for the root such that 0.9899888032 < x < 0.9899888033
  The multiplicity is 1 for the root such that 0.9991447980 < x < 0.9991447981
  The multiplicity is 1 for the root such that 0.9997663986 < x < 0.9997663987

A degree five polynomial with three rational roots
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

::

  \poldef Q(x) :=  1581755751184441 x^5
                 -14907697165025339 x^4
                 +48415668972339336 x^3
                 -63952057791306264 x^2
                 +46833913221154895 x
                 -49044360626280925;

  \PolToSturm{Q}{Q}
  %\begin{flushleft}
    \renewcommand\PolTypesetCmdPrefix[1]{\allowbreak\xintiiifSgn{#1}{}{+}{+}}%
    $Q_0(x) = \PolTypeset{Q_0}$
  %\end{flushleft}
  \PolSturmIsolateZeros**{Q}
  \PolPrintIntervals{Q}

  $Q_{norr}(x) = \PolTypeset{Q_norr}$

Here, all real roots are rational::

  Z_1 = 833719/265381
  Z_2 = 165707065/52746197
  Z_3 = 355/113

  Q_norr(x) = x^2 + 1

And let's get their decimal expansion too::

  % print decimal expansion of the found roots
  \renewcommand\PolPrintIntervalsPrintExactZero
              {\xintTrunc{20}{\PolPrintIntervalsTheLeftEndPoint}\dots}
  \PolPrintIntervals{Q}

  Z_1 = 3.14159265358107777120...
  Z_2 = 3.14159265358979340254...
  Z_3 = 3.14159292035398230088...


A Mignotte type polynomial
~~~~~~~~~~~~~~~~~~~~~~~~~~

::

  \PolDef{P}{x^10 - (10x-1)^2}%
  \PolTypeset{P}              % prints it in expanded form
  \PolToSturm{P}{P}           % we can use same prefix for Sturm chain
  \PolSturmIsolateZeros{P}    % finds 4 real roots
  This polynomial has \PolSturmNbOfIsolatedZeros{P} distinct real roots:
  \PolPrintIntervals{P}%
  % reports  -2 < Z_1 < -1, 0.09 < Z_2 < 0.10, 0.1 < Z_3 < 0.2, 1 < Z_4 < 2
  Let us refine the second and third intervals to separate the corresponding
  roots:
  \PolRefineInterval*{P}{2}% will refine to 0.0999990 < Z_2 < 0.0999991
  \PolRefineInterval*{P}{3}% will refine to 0.100001 < Z_3 < 0.100002
  \PolPrintIntervals{P}%
  Let us now get to know all roots with 10 digits after decimal mark:
  \PolEnsureIntervalLengths{P}{-10}%
  \PolPrintIntervals{P}% now all roots are known 10 decimal digits after mark
  Finally, we display 20 digits of the second root:
  \PolEnsureIntervalLength{P}{2}{-20}% makes Z_2 known with 20 digits after mark
  \[\PolSturmIsolatedZeroLeft{P}{2}<Z_2<\PolSturmIsolatedZeroRight{P}{2}\]

The last line produces::

  0.09999900004999650028 < Z_2 < 0.09999900004999650029


The Wilkinson polynomial
~~~~~~~~~~~~~~~~~~~~~~~~

See `Wilkinson polynomial`_.

::

  \documentclass{article}
  \usepackage{polexpr}
  \begin{document}
  %\xintverbosetrue % for the curious...

  \poldef f(x) := mul((x - i), i = 1..20);

  \renewcommand\PolTypesetCmdPrefix[1]{\allowbreak\xintiiifSgn{#1}{}{+}{+}}%
  \renewcommand\PolTypesetOne[1]{\xintDecToString{#1}}%

  \noindent\PolTypeset{f}

  \PolToSturm{f}{f}
  \PolSturmIsolateZeros{f}
  \PolPrintIntervals{f}

  \clearpage

  \poldef g(x) := f(x) - 2**{-23} x**19;

  % be patient!
  \PolToSturm{g}{g}
  \noindent\PolTypeset{g_0}% integer coefficient primitive polynomial

  \PolSturmIsolateZeros{g}
  \PolEnsureIntervalLengths{g}{-10}

  \renewcommand\PolPrintIntervalsPrintMultiplicity{}
  \PolPrintIntervals*{g}

  \end{document}


The first polynomial::

  f(x) = x**20
  - 210 x**19
  + 20615 x**18
  - 1256850 x**17
  + 53327946 x**16
  - 1672280820 x**15
  + 40171771630 x**14
  - 756111184500 x**13
  + 11310276995381 x**12
  - 135585182899530 x**11
  + 1307535010540395 x**10
  - 10142299865511450 x**9
  + 63030812099294896 x**8
  - 311333643161390640 x**7
  + 1206647803780373360 x**6
  - 3599979517947607200 x**5
  + 8037811822645051776 x**4
  - 12870931245150988800 x**3
  + 13803759753640704000 x**2
  - 8752948036761600000 x
  + 2432902008176640000

is handled fast enough (a few seconds), but the modified one ``f(x) -
2**-23 x**19`` takes about 20x longer (the Sturm chain polynomials
have integer coefficients with up to 321 digits, whereas (surprisingly
perhaps) those of the Sturm chain polynomials derived from ``f`` never
have more than 21 digits ...).

Once the Sturm chain is computed and the zeros isolated, obtaining their
decimal digits is relatively faster. Here is for the ten real roots of
``f(x) - 2**-23 x**19`` as computed by the code above::

  Z_1 = 0.9999999999...
  Z_2 = 2.0000000000...
  Z_3 = 2.9999999999...
  Z_4 = 4.0000000002...
  Z_5 = 4.9999999275...
  Z_6 = 6.0000069439...
  Z_7 = 6.9996972339...
  Z_8 = 8.0072676034...
  Z_9 = 8.9172502485...
  Z_10 = 20.8469081014...

The second Wilkinson polynomial
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

::

  \documentclass{article}
  \usepackage{polexpr}
  \begin{document}
  \poldef f(x) := mul(x - 2^-i, i = 1..20);

  %\PolTypeset{f}

  \PolToSturm{f}{f}
  \PolSturmIsolateZeros**{f}
  \PolPrintIntervals{f}
  \end{document}

This takes more time than the polynomial with 1, 2, .., 20 as roots but
less than the latter modified by the ``2**-23`` change in one
coefficient.

Here is the output (with release 0.7.2)::

  Z_1  = 0.00000095367431640625
  Z_2  = 0.0000019073486328125
  Z_3  = 0.000003814697265625
  Z_4  = 0.00000762939453125
  Z_5  = 0.0000152587890625
  Z_6  = 0.000030517578125
  Z_7  = 0.00006103515625
  Z_8  = 0.0001220703125
  Z_9  = 1/4096
  Z_10 = 1/2048
  Z_11 = 1/1024
  Z_12 = 1/512
  Z_13 = 1/256
  Z_14 = 1/128
  Z_15 = 0.015625
  Z_16 = 0.03125
  Z_17 = 0.0625
  Z_18 = 0.125
  Z_19 = 0.25
  Z_20 = 0.5

There is some incoherence in output format which has its source in the
fact that some roots are found in branches which can only find decimal
roots, whereas some are found in branches which could find general
fractions and they use ``\xintIrr`` before storage of the found root.
This may evolve in future.


The degree 41 polynomial with -2, -1.9, -1.8, ..., 0, 0.1, ..., 1.9, 2 as roots
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

::

  \PolDef{P}{mul((x-i*1e-1), i=-20..20)}% i/10 is same but less efficient

In the defining expression we could have used ``i/10`` but this gives
less efficient internal form for the coefficients (the ``10``'s end up
in denominators). Using ``\PolToExpr{P}`` after having done

::

  \renewcommand\PolToExprCmd[1]{\PolDecToString{\xintREZ{#1}}}

we get this expanded form::

  x^41
  -28.7*x^39
  +375.7117*x^37
  -2975.11006*x^35
  +15935.28150578*x^33
  -61167.527674162*x^31
  +173944.259366417394*x^29
  -373686.963560544648*x^27
  +613012.0665016658846445*x^25
  -771182.31133138163125495*x^23
  +743263.86672885754888959569*x^21
  -545609.076599482896371978698*x^19
  +301748.325708943677229642930528*x^17
  -123655.8987669450434698869844544*x^15
  +36666.1782054884005855608205864192*x^13
  -7607.85821367459445649518380016128*x^11
  +1053.15135918687298508885950223794176*x^9
  -90.6380005918141132650786081964032*x^7
  +4.33701563847327366842552218288128*x^5
  -0.0944770968420804735498178265088*x^3
  +0.00059190121813899276854174416896*x

which shows coefficients with up to 36 significant digits...

Stress test: not a hard challenge to ``xint + polexpr``, but be a bit patient!

::

  \PolDef{P}{mul((x-i*1e-1), i=-20..20)}%
  \PolToSturm{P}{S}           % dutifully computes S_0, ..., S_{41}
  % the [1] optional argument limits the search to interval (-10,10)
  \PolSturmIsolateZeros[1]{S} % finds *exactly* (but a bit slowly) all 41 roots!
  \PolPrintIntervals{S}       % nice, isn't it?

.. note::

   Release ``0.5`` has *experimental* addition of optional argument
   ``E`` to ``\PolSturmIsolateZeros``. It instructs to search roots only
   in interval ``(-10^E, 10^E)``. Important: the extremities are
   *assumed to not be roots*. In this example, the ``[1]`` in
   ``\PolSturmIsolateZeros[1]{S}`` gives some speed gain; without it, it
   turns out in this case that ``polexpr`` would have started with
   ``(-10^6, 10^6)`` interval.

   Please note that this will probably get replaced in future by the
   specification of a general interval. Do not rely on meaning of this
   optional argument keeping the same.

Roots of Chebyshev polynomials
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

::

  \newcount\mycount
  \poldef T_0(x) := 1;
  \poldef T_1(x) := x;
  \mycount 2
  \xintloop
    \poldef T_\the\mycount(x) :=
            2x*T_\the\numexpr\mycount-1(x)
             - T_\the\numexpr\mycount-2(x);
  \ifnum\mycount<15
  \advance\mycount 1
  \repeat

  \[T_{15} = \PolTypeset[X]{T_15}\]
  \PolToSturm{T_15}{T_15}
  \PolSturmIsolateZeros{T_15}
  \PolEnsureIntervalLengths{T_15}{-10}
  \PolPrintIntervals{T_15}


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

.. _poldef;:

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

    This evaluates the *polynomial expression* and stores the coefficients
    in a private structure accessible later via other package macros,
    under the user-chosen ``polname``. Of course the *expression* can
    use other previously defined polynomials. Names must start with a
    letter and are constituted of letters, digits, underscores and
    (since ``0.5.1``) the right tick ``'``.
    The whole xintexpr_ syntax is authorized::

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

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

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

    .. attention::

       Release ``0.3`` also did the necessary set-up to let the
       polynomial be known to the ``\xintfloatexpr`` (or
       ``\xintdeffloatvar``) parser.

       Since ``0.4`` this isn't done automatically. Even more, a
       previously existing floating point variant of the same name will
       be let undefined again, to avoid hard to debug mismatches between
       exact and floating point polynomials. This also applies when the
       polynomial is produced not via ``\poldef`` or ``\PolDef`` but as
       a product of the other package macros.

       See `\\PolGenFloatVariant{polname}`_.

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

.. _PolDef:

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

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

.. _PolGenFloatVariant:

``\PolGenFloatVariant{polname}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    Makes the polynomial also usable in the ``\xintfloatexpr`` parser.
    It will therein evaluates via an Horner scheme with coefficients
    already pre-rounded to the float precision.

    See also `\\PolToFloatExpr{polname}`_.

    .. attention::

       Release ``0.3`` did this automatically on ``\PolDef`` and
       ``\poldef`` but this was removed at ``0.4`` for optimization.

       Any operation, for example generating the derivative polynomial,
       or dividing two polynomials or using the ``\PolLet``, **must** be
       followed by explicit usage of ``\PolGenFloatVariant{polname}`` if
       the new polynomial is to be used in ``\xintfloatexpr`` or alike
       context.

.. _PolLet:

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

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

.. _PolGlobalLet:

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

    Acts globally.

.. _PolAssign:

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

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

    - Attention, coefficients here are indexed starting at 1.

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

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

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

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

.. _PolGet:

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

    Does the converse operation to
    ``\PolAssign{polname}\toarray\macro``. Each individual
    ``\macro{number}`` gets expanded in an ``\edef`` and then normalized
    via xintfrac_\ 's macro ``\xintRaw``.

    The leading zeros are removed from the polynomial.

    (contrived) Example::

      \xintAssignArray{1}{-2}{5}{-3}\to\foo
      \PolGet{f}\fromarray\foo

    This will define ``f`` as would have ``\poldef f(x):=1-2x+5x^2-3x^3;``.

    .. note::

       Prior to ``0.5``, coefficients were not normalized via
       ``\xintRaw`` for internal storage.

.. _PolFromCSV:

``\PolFromCSV{polname}{<csv>}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    Defines a polynomial directly from the comma separated list of values
    (or a macro expanding to such a list) of its coefficients, the *first
    item* gives the constant term, the *last item* gives the leading
    coefficient, except if zero, then it is dropped (iteratively). List
    items are each expanded in an ``\edef`` and then put into normalized
    form via xintfrac_\ 's macro ``\xintRaw``.

    As leading zero coefficients are removed::

      \PolFromCSV{f}{0, 0, 0, 0, 0, 0, 0, 0, 0, 0}

    defines the zero polynomial, which holds only one coefficient.

    See also expandable macro `\\PolToCSV <\\PolToCSV{polname}_>`_.

    .. note::

       Prior to ``0.5``, coefficients were not normalized via
       ``\xintRaw`` for internal storage.

.. _PolTypeset:

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

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

      \PolTypeset[z]{polname}

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

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

.. _PolTypesetCmd:

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

    Checks if the coefficient is ``1`` or ``-1`` and then skips printing
    the ``1``, except for the constant term. Also it sets conditional
    `\\PolIfCoeffIsPlusOrMinusOne{A}{B}`_.

    The actual printing of the coefficients, when not equal to plus or
    minus one is handled by `\\PolTypesetOne{raw_coeff}`_.

.. _PolTypesetOne:

``\PolTypesetOne{raw_coeff}``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

    The default is ``\xintSignedFrac`` but this macro is annoying as it
    insists to use a power of ten, and not decimal notation.

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

    ::

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

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

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

    One can also give a try to using `\\PolDecToString{decimal number}`_
    which uses decimal notation (at least for the numerator part).

.. _PolTypesetMonomialCmd:

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

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

.. _PolTypesetCmdPrefix:

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

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

.. _PolTypeset*:

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

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

.. _PolDiff:

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

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

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

.. _PolDiff[N]:

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

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

.. _PolAntiDiff:

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

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

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

.. _PolAntiDiff[N]:

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

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

.. _PolDivide:

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

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

.. _PolQuo:

``\PolQuo{polname_1}{polname_2}{polname_Q}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

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

.. _PolRem:

``\PolRem{polname_1}{polname_2}{polname_R}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

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

.. _PolGCD:

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

    This sets ``polname_GCD`` to be the (monic) GCD of the two first
    polynomials. It is a unitary polynomial except if both ``polname_1``
    and ``polname_2`` vanish, then ``polname_GCD`` is the zero
    polynomial.

.. ``\PolIGCD{polname_1}{polname_2}{polname_iGCD}``
   ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    **NOT YET**

    This **assumes** that the two polynomials have integer coefficients.
    It then computes the greatest common divisor in the integer
    polynomial ring, normalized to have a positive leading coefficient
    (if the inputs are not both zero).

   ``\PolIContent{polname}``
   ~~~~~~~~~~~~~~~~~~~~~~~~~

    **NOT YET**

    This computes a positive rational number such that dividing the
    polynomial with it returns an integer coefficients polynomial with
    no common factor among the coefficients.

.. _PolToSturm:

``\PolToSturm{polname}{sturmname}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    With ``polname`` being for example  ``P``, the macro starts by
    computing polynomials ``P`` and ``P'``, then computes the (opposite
    of the) remainder in euclidean division, iteratively.

    The last non-zero remainder ``P_N_`` (where ``N`` is obtainable as
    `\\PolSturmChainLength{sturmname}`_) is up to a factor
    the GCD of ``P`` and ``P'`` hence it is a constant if and only if
    ``P`` is square-free.

    .. note::

       - Since ``0.5`` all these polynomials are divided by their rational
         content, so they have integer coefficients with no common factor,
         and the last one if a constant is either ``1`` or ``-1``.

       - After this normalization to primitive polynomials, they are
         stored internally as ``sturmname_k_``, ``k=0,1, ...``.

       - These polynomials are used internally only. To keep them as
         genuine declared polynomials also after the macro call, use the
         starred variant `PolToSturm*`_.

    .. note::

       It is perfectly allowed to use the polynomial name as Sturm chain name:
       ``\PolToSturm{f}(f}``.

    The macro then declares ``sturmname_0``, ``sturmname_1``, ..., which are
    the (non-declared) ``sturmname_k_`` divided by the last one. Division is
    not done if this last one is the constant ``1`` or ``-1``, i.e. if the
    original polynomial was square-free. These polynomials are primitive
    polynomials too, i.e. with integer coefficients having no common factor.

    Thus ``sturmname_0`` has exactly the same real and complex roots as
    polynomial ``polname``, but with each root now of multiplicity one:
    i.e. it is the "square-free part" of original polynomial ``polname``.

    Notice that ``sturmname_1`` isn't necessarily the derivative of
    ``sturmname_0`` due to the various normalizations.

    The polynomials ``sturmname_k`` main utility is for the execution of
    `\\PolSturmIsolateZeros{sturmname}`_. Be careful not to use these
    names ``sturmname_0``, ``sturmname_1``, etc... for defining other
    polynomials after having done ``\PolToSturm{polname}{sturmname}`` and
    before executing ``\PolSturmIsolateZeros{sturmname}`` else the
    latter will behave erroneously.

    `\\PolSturmChainLength{sturmname}`_ gives the index of the last
    element of the Sturm chain.

.. _PolToSturm*:

``\PolToSturm*{polname}{sturmname}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    Does the same as `un-starred version <PolToSturm_>`_ and additionally it
    keeps for user usage the memory of the *un-normalized* Sturm chain
    polynomials ``sturmname_k_``, ``k=0,1, ..., N``, with
    ``N`` being `\\PolSturmChainLength{sturmname}`_.

    .. note::

       This behaviour was modified at  ``0.6``, anyhow the macro was
       broken at ``0.5``.

    .. hint::

       The square-free part of ``polname`` is ``sturmname_0``, and their
       quotient is the polynomial with name
       ``sturname_\PolSturmChainLength{sturmname}_``. It thus easy to
       set-up a loop iteratively computing the latter until the last one
       is a constant, thus obtaining the decomposition of an ``f`` as
       a product ``c f_1 f_2 f_3 ...`` of a constant and square-free (primitive)
       polynomials, where each ``f_i`` divides its predecessor.       

.. _PolSetToSturmChainSignChangesAt:

``\PolSetToSturmChainSignChangesAt{\macro}{sturmname}{fraction}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    Sets macro ``\macro`` to the number of sign changes in the Sturm
    chain with name prefix ``sturmname``, at location ``fraction``
    (which must be in format as acceptable by the xintfrac_ macros.)

    .. note::

       The author was lazy and did not provide rather an expandable
       variant, where one would do ``\edef\macro{\PolNbOf...}``.

       This will presumably get added in a future release.

       After some hesitation it was decided the macro would by default
       act globally. To make the scope of its macro definition local,
       use ``[\empty]`` as extra optional argument.

.. _PolSetToNbOfZerosWithin:

``\PolSetToNbOfZerosWithin{\macro}{sturmname}{value_a}{value_b}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    Applies the `Sturm Theorem`_ to set ``\macro`` to the exact number
    of **distinct** roots of ``sturmname_0`` in the interval ``(value_a,
    value_b]`` (the macro first re-orders the value for ``value_a <=
    value_b`` to hold).

    .. note::

       The author was lazy and did not provide rather an expandable
       variant, where one would do ``\edef\macro{\PolNbOf...}``.

       This will presumably get added in future.

       After some hesitation it was decided the macro would by default
       act globally. To make the scope of its macro definition local,
       use ``[\empty]`` as extra optional argument.

    See also the expandable
    `\\PolSturmNbOfRootsOf{sturmname}\\LessThanOrEqualTo{value}`_, from
    which it is immediate (with ``\numexpr``) to create an expandable
    variant of this macro. However the difference is that this macro
    requires only `\\PolToSturm <PolToSturm_>`_ to have been executed,
    whereas the expandable variant requires prior execution of
    `\\PolSturmIsolateZeros <PolSturmIsolateZeros_>`_.

    See also the expandable
    `\\PolSturmNbWithMultOfRootsOf{sturmname}\\LessThanOrEqualTo{value}`_
    which requires prior execution of
    `\\PolSturmIsolateZeros* <PolSturmIsolateZeros*_>`_.


.. _PolSturmIsolateZeros:

``\PolSturmIsolateZeros{sturmname}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    The macros locates, using `Sturm theorem`_, as many disjoint
    intervals as there are (real) roots.

    .. important::

       The Sturm chain must have been produced by an earlier
       `\\PolToSturm{polname}{sturmname}`_.

       Why does this macro ask for argument the name of Sturm chain,
       rather than the name of a polynomial? well this is mainly for
       legacy reason, and because it is accompanied by other macros for
       which it is simpler to assume the argument will be the name of an
       already computed Sturm chain.

       Notice that ``\PolToSturm{f}{f}`` is perfectly legal (the
       ``sturmname`` can be same as the ``polname``): it defines
       polynomials ``f_0``, ``f_1``, ... having ``f`` has name prefix.

       Such a prior call
       to ``\PolToSturm`` must have been made at any rate for
       ``\PolSturmIsolateZeros`` to be usable.
       
    After its execution they are two types of such intervals (stored in
    memory and accessible via macros or xintexpr_ variables, see below):

    - singleton ``{a}``: then ``a`` is a root, (necessarily a decimal
      number, but not all such decimal numbers are exactly identified yet).

    - open intervals ``(a,b)``: then there is exactly one root ``z``
      such that ``a < z < b``, and the end points are guaranteed to not
      be roots.

    The interval boundaries are decimal numbers, originating
    in iterated decimal subdivision from initial intervals
    ``(-10^E, 0)`` and ``(0, 10^E)`` with ``E`` chosen initially large
    enough so that all roots are enclosed; if zero is a root it is always
    identified as such. The non-singleton intervals are of the
    type ``(a/10^f, (a+1)/10^f)`` with ``a`` an integer, which is
    neither ``0`` nor ``-1``. Hence either ``a`` and ``a+1`` are both positive
    or they are both negative.

    One does not *a priori* know what will be the lengths of these
    intervals (except that they are always powers of ten), they
    vary depending on how many digits two successive roots have in
    common in their respective decimal expansions.

    .. important::

       If some two consecutive intervals share an end-point, no
       information is yet gained about the separation between the two
       roots which could at this stage be arbitrarily small.

       See `\\PolRefineInterval*{sturmname}{index}`_ which addresses
       this issue.

    .. This procedure is covariant
       with the independent variable ``x`` becoming ``-x``.
       Hmm, pas sûr et trop fatigué

    The interval boundaries (and exactly found roots) are made available
    for future computations in ``\xintexpr``-essions or polynomial
    definitions as variables ``<sturmname>L_1``,
    ``<sturmname>L_2``, etc..., for the left end-points and
    ``<sturmname>R_1``, ``<sturmname>R_2``, ..., for the right
    end-points.

    Thus for example, if ``sturmname`` is ``f``, one can use the
    xintexpr_ variables ``fL_1``, ``fL_2``, ... to refer in expressions
    to the left end-points (or to the exact root, if left and right end
    points coincide). Additionally, xintexpr_ variable ``fZ_1_isknown``
    will have value ``1`` if the root in the first interval is known,
    and ``0`` otherwise. And similarly for the other intervals.

    Also, macros `\\PolSturmIsolatedZeroLeft{sturmname}{index}`_ and
    `\\PolSturmIsolatedZeroRight{sturmname}{index}`_ are provided which
    expand to these same values, written in decimal notation (i.e.
    pre-processed by `\\PolDecToString <PolDecToString_>`_.) And there
    is also `\\PolSturmIfZeroExactlyKnown{sturmname}{index}{A}{B}`_.

    .. important::

       Trailing zeroes in the stored decimal numbers accessible via the
       macros are significant: they are also present in the decimal
       expansion of the exact root.

    These variables and macros are automatically updated when one next
    uses macros such as `\\PolRefineInterval*{sturmname}{index}`_.

    The start of decimal expansion of a positive ``k``-th root is given
    by `\\PolSturmIsolatedZeroLeft{sturmname}{k}
    <PolSturmIsolatedZeroLeft_>`_, and for a negative root it is given
    by `\PolSturmIsolatedZeroRight{sturmname}{k}
    <PolSturmIsolatedZeroRight_>`_. These two decimal
    numbers are either both zero or both of the same sign.

    The number of distinct roots is obtainable expandably as
    `\\PolSturmNbOfIsolatedZeros{sturmname}`_.

    Furthermore
    `\\PolSturmNbOfRootsOf{sturmname}\\LessThanOrEqualTo{value}`_ and
    `\\PolSturmNbOfRootsOf{sturmname}\\LessThanOrEqualToExpr{expression}`_.
    will expandably compute respectively the number of real roots at
    most equal to ``value`` or ``expression``, and the same but with
    multiplicities.

    .. note::

       In the current implementation the xintexpr_ variables
       and xinttools_ arrays are globally defined. On the
       other hand the Sturm sequence polynomials obey the current scope.

    .. note::

       As all computations are done *exactly* there can be no errors...
       apart those due to bad coding by author. The results are exact
       bounds for the mathematically exact real roots.

       Future releases will perhaps also provide macros based on Newton
       or Regula Falsi methods. Exact computations with such methods
       lead however quickly to very big fractions, and this forces usage
       of some rounding scheme for the abscissas if computation times
       are to remain reasonable. This raises issues of its own, which
       are studied in numerical mathematics.

.. _PolSturmIsolateZeros*:

``\PolSturmIsolateZeros*{sturmname}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    The macro does the same as `\\PolSturmIsolateZeros{sturmname}`_ and
    then in addition it does the extra work to determine all
    multiplicities (of the real roots):
    after executing this macro,
    `\\PolSturmIsolatedZeroMultiplicity{sturmname}{index}`_ will expand
    to the multiplicity of the root located in the ``index``\ -th
    interval (intervals are enumerated from left to right, with index
    starting at ``1``).

    Furthermore, if for example the ``sturmname`` is ``f``, xintexpr_
    variables ``fM_1``, ``fM_2``... hold the multiplicities thus
    computed.

    .. note::

       It is **not** necessary to have executed the `PolToSturm*`_ starred
       variant, as the non-starred variant keeps internally the memory of the
       original GCD (and even of the full non-normalized original Sturm
       chain), even though it does not make the declarations as *user-level*
       genuine polynomials.

    See `The degree nine polynomial with 0.99, 0.999, 0.9999 as triple
    roots`_ for an example.

.. _PolSturmIsolateZeros**:

``\PolSturmIsolateZeros**{sturmname}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    The macro does the same as `\\PolSturmIsolateZeros*{sturmname}`_ and
    in addition it does the extra work to determine all the *rational*
    roots.

    .. note::

       After execution of this macro, a root is "known" if and only if
       it is rational.

    Furthermore, primitive polynomial ``sturmname_sqf_norr`` is created
    to match the (square-free) ``sturmname_0`` from which all rational
    roots have been removed (see `\\polexprsetup`_ for customizing this
    name). The number of distinct rational roots is thus the difference
    between the degrees of these two polynomials (see also
    `\\PolSturmNbOfRationalRoots{sturmname}`_).

    And ``sturmname_norr`` is ``sturmname_0_`` from which all rational
    roots have been removed (see `\\polexprsetup`_), i.e. it contains
    the irrational roots of the original polynomial, with the same
    multiplicities.

    See `A degree five polynomial with three rational
    roots`_ for an example.

.. _PolSturmIsolateZerosAndGetMultiplicities:

``\PolSturmIsolateZerosAndGetMultiplicities{sturmname}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    This is another name for `\\PolSturmIsolateZeros*{sturmname}`_.

.. _PolSturmIsolateZerosGetMultiplicitiesAndRationalRoots:

``\PolSturmIsolateZerosGetMultiplicitiesAndRationalRoots{sturmname}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    This is another name for `\\PolSturmIsolateZeros**{sturmname}`_.


``\PolSturmIsolateZerosAndFindRationalRoots{sturmname}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    This works exactly like `\\PolSturmIsolateZeros**{sturmname}`_
    (inclusive of declaring the polynomials ``sturmname_sqf_norr`` and
    ``sturmname_norr`` with no rational roots) except that it does *not*
    compute the multiplicities of the *non-rational* roots.

    .. note::

       There is no macro to find the rational roots but not compute
       their multiplicities at the same time.

    .. attention::

       This macro does *not* define xintexpr_ variables
       ``sturmnameM_1``, ``sturmnameM_2``, ... holding the
       multiplicities and it leaves the multiplicity array (whose accessor
       is `\\PolSturmIsolatedZeroMultiplicity{sturmname}{index}`_) into
       a broken state, as all non-rational roots will supposedly have
       multiplicity one. This means that the output of
       `\\PolPrintIntervals* <PolPrintIntervals*_>`_ for example will be
       erroneous for the intervals with irrational roots.

       I decided to document it because finding multiplicities of the
       non rational roots is somewhat costly, and one may be interested
       only into finding the rational roots (of course random
       polynomials with integer coefficients will not have *any*
       rational root anyhow).


.. _PolRefineInterval*:

``\PolRefineInterval*{sturmname}{index}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    The ``index``\ -th interval (starting indexing at one) is further
    subdivided as many times as is necessary in order for the newer
    interval to have both its end-points distinct from the end-points of
    the original interval. This means that the ``k``\ th root is then
    strictly separated from the other roots.

.. _PolRefineInterval[N]:

``\PolRefineInterval[N]{sturmname}{index}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    The ``index``\ -th interval (starting count at one) is further
    subdivided once, reducing its length by a factor of 10. This is done
    ``N`` times if the optional argument ``[N]`` is present.

.. _PolEnsureIntervalLength:

``\PolEnsureIntervalLength{sturmname}{index}{E}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    The ``index``\ -th interval is subdivided until its length becomes at
    most ``10^E``. This means (for ``E<0``) that the first ``-E`` digits
    after decimal mark of the ``k``\ th root will then be known exactly.

.. _PolEnsureIntervalLengths:

``\PolEnsureIntervalLengths{sturmname}{E}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    The intervals as obtained from ``\PolSturmIsolateZeros`` are (if
    necessary) subdivided further by (base 10) dichotomy in order for
    each of them to have length at most ``10^E`` (length will be shorter
    than ``10^E`` in output only if it did not change or became zero.)

    This means that decimal expansions of all roots will be known with
    ``-E`` digits (for ``E<0``) after decimal mark.

.. _PolPrintIntervals:

``\PolPrintIntervals[varname]{sturmname}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    This is a convenience macro which prints the bounds for the roots
    ``Z_1``, ``Z_2``, ... (the optional argument ``varname`` allows to
    specify a replacement for the default ``Z``). This will be done (by
    default) in a
    math mode ``array``, one interval per row, and pattern ``rcccl``,
    where the second and fourth column hold the ``<`` sign, except when
    the interval reduces to a singleton, which means the root is known
    exactly.

    .. attention::

       This macro was refactored at 0.7, its default output remained
       identical but the ways to customize it got completely
       modified.

    See next macros which govern its output.

``\PolPrintIntervalsNoRealRoots``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

    Executed in place of an ``array`` environment, when there are no
    real roots. Default definition::

      \newcommand\PolPrintIntervalsNoRealRoots{}

``\PolPrintIntervalsBeginEnv``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

    Default definition::

      \newcommand\PolPrintIntervalsBeginEnv{\[\begin{array}{rcccl}}

``\PolPrintIntervalsEndEnv``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^

    Default definition::

      \newcommand\PolPrintIntervalsEndEnv{\end{array}\]}

``\PolPrintIntervalsKnownRoot``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

    Default definition::

      \newcommand\PolPrintIntervalsKnownRoot{%
        &&\PolPrintIntervalsTheVar_{\PolPrintIntervalsTheIndex}%
        &=&\PolPrintIntervalsPrintExactZero
      }

``\PolPrintIntervalsUnknownRoot``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

    Default definition::

      \newcommand\PolPrintIntervalsUnknownRoot{%
        \PolPrintIntervalsPrintLeftEndPoint&<&%
        \PolPrintIntervalsTheVar_{\PolPrintIntervalsTheIndex}&<&%
        \PolPrintIntervalsPrintRightEndPoint
      }


.. _PolPrintIntervalsPrintExactZero:

``\PolPrintIntervalsPrintExactZero``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

    Default definition::

      \newcommand\PolPrintIntervalsPrintExactZero{\PolPrintIntervalsTheLeftEndPoint}


.. _PolPrintIntervalsPrintLeftEndPoint:

``\PolPrintIntervalsPrintLeftEndPoint``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

    Default definition::

      \newcommand\PolPrintIntervalsPrintLeftEndPoint{\PolPrintIntervalsTheLeftEndPoint}

.. _PolPrintIntervalsPrintRightEndPoint:

``\PolPrintIntervalsPrintRightEndPoint``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

    Default definition is::

      \newcommand\PolPrintIntervalsPrintRightEndPoint{\PolPrintIntervalsTheRightEndPoint}

.. _PolPrintIntervals*:

``\PolPrintIntervals*[varname]{sturmname}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    This starred variant produces an alternative output (which
    displays the root multiplicity), and is provided as an
    example of customization.

    As replacement for `\\PolPrintIntervalsKnownRoot`_,
    `\\PolPrintIntervalsPrintExactZero`_,
    `\\PolPrintIntervalsUnknownRoot`_ it uses its own
    ``\POL@@PrintIntervals...`` macros. We only reproduce here one
    definition::

      \newcommand\POL@@PrintIntervalsPrintExactZero{%
         \displaystyle
         \xintSignedFrac{\PolPrintIntervalsTheLeftEndPoint}%
      }%
    
    Multiplicities are printed using this auxiliary macro:

``\PolPrintIntervalsPrintMultiplicity``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

    whose default definition is::

      \newcommand\PolPrintIntervalsPrintMultiplicity{(\mbox{mult. }\PolPrintIntervalsTheMultiplicity)}


.. _PolMapCoeffs:

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

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

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

    Example::

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

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

.. _PolReduceCoeffs:

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

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

.. _PolReduceCoeffs*:

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

    This starred variant leaves un-touched the decimal exponent in the
    internal representation of the fractional coefficients, i.e. if a
    coefficient is internally ``A/B[N]``, then ``A/B`` is reduced to
    smallest terms, but the ``10^N`` part is kept as is. Note: if the
    polynomial is freshly defined directly via `\\PolFromCSV
    <PolFromCSV_>`_ its coefficients might still be internally in some
    format like ``1.5e7``; the macro will anyhow always first do the
    needed conversion to strict format ``A/B[N]``.

    Evaluations with polynomials treated by this can be much faster than
    with those handled by the non-starred variant
    `\\PolReduceCoeffs{polname}`_: as the numerators and denominators
    remain smaller, this proves very beneficial in favorable cases
    (especially when the coefficients are decimal numbers) to the
    expansion speed of the xintfrac_ macros used internally by
    `\\PolEval <PolEvalAt_>`_.

.. _PolMakeMonic:

``\PolMakeMonic{polname}``
~~~~~~~~~~~~~~~~~~~~~~~~~~

    Divides by the leading coefficient. It is recommended to execute
    `\\PolReduceCoeffs*{polname}`_ immediately afterwards. This is not
    done automatically, due to the case the original polynomial had integer
    coefficients and we want to keep the leading one as common
    denominator.

.. _PolMakePrimitive:

``\PolMakePrimitive{polname}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    Divides by the integer content see (`\\PolIContent
    <PolIContent_>`_). This thus produces a polynomial with integer
    coefficients having no common factor. The sign of the leading
    coefficient is not modified.

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

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

.. _PolEvalAtExpr:

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

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

.. _PolEvalAt:

``\PolEval{polname}\At{fraction}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    Evaluates the polynomial at value ``fraction`` which must be in (or
    expand to) a format acceptable to the xintfrac_ macros.

.. _PolEvalReducedAtExpr:

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

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

.. _PolEvalReducedAt:

``\PolEvalReduced{polname}\At{fraction}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    Evaluates the polynomial at value ``fraction`` which must be in (or
    expand to) a format acceptable to the xintfrac_ macros, and produce
    an irreducible fraction.

.. _PolFloatEvalAtExpr:

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

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

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

    ::

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

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


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

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

.. _PolFloatEvalAt:

``\PolFloatEval{polname}\At{fraction}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    Evaluates the polynomial at value ``fraction`` which must be in (or
    expand to) a format acceptable to the xintfrac_ macros, and produces
    a floating point number.

.. _PolIfCoeffIsPlusOrMinusOne:

``\PolIfCoeffIsPlusOrMinusOne{A}{B}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    This macro is a priori undefined.

    It is defined via the default `\\PolTypesetCmd{raw_coeff}`_ to be
    used if needed in the execution of `\\PolTypesetMonomialCmd`_,
    e.g. to insert a ``\cdot`` in front of ``\PolVar^{\PolIndex}`` if
    the coefficient is not plus or minus one.

    The macro will execute ``A`` if the coefficient has been found to be
    plus or minus one, and ``B`` if not.

.. _PolLeadingCoeff:

``\PolLeadingCoeff{polname}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    Expands to the leading coefficient.

.. _PolNthCoeff:

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

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

.. _PolDegree:

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

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

.. _PolIContent:

``\PolIContent{polname}``
~~~~~~~~~~~~~~~~~~~~~~~~~

    It expands to the contents of the polynomial, i.e. to the positive
    fraction such that dividing by this fraction produces a polynomial
    with integer coefficients having no common prime divisor.

    See `\\PolMakePrimitive <PolMakePrimitive_>`_.

.. _PolToExpr:

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

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

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

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

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

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

.. _PolToExprOneTerm:

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

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

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

.. _PolToExprOneTermStyleA:

``\PolToExprOneTermStyleA{raw_coeff}{number}``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

    Holds the default package meaning of
    `\\PolToExprOneTerm{raw_coeff}{number}`_.

.. _PolToExprOneTermStyleB:

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

    For output in this style::

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

    issue ``\let\PolToExprOneTerm\PolToExprOneTermStyleB`` before usage of
    ``\PolToExpr``. Note that then ``\PolToExprCmd`` isn't used at all.
    To revert to package default, issue
    ``\let\PolToExprOneTerm\PolToExprOneTermStyleA``.

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

.. _PolToExprCmd:

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

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

.. _PolToExprTermPrefix:

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

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

.. _PolToExprVar:

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

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

.. _PolToExprTimes:

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

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

.. _PolToExpr*:

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

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

.. _PolToFloatExpr:

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

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

    .. note::

       It is not necessary to have issued
       `\\PolGenFloatVariant{polname}`_. The rounded coefficients are
       not easily recoverable from the ``\xintfloatexpr`` polynomial
       function hence ``\PolToFloatExprCmd`` operates from the *exact*
       coefficients anew.

       Attention that both macros obey the prevailing float precision.
       If it is changed between those macro calls, then a mismatch
       exists between the coefficients as used in ``\xintfloatexpr`` and
       those output by ``\PolToFloatExpr{polname}``.

.. _PolToFloatExprOneTerm:

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

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

.. _PolToFloatExprCmd:

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

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

    .. caution::

       Currently (xint_ ``1.3c``) ``\xintFloat{0}`` outputs ``0.e0``
       which is perfectly acceptable input for Python, but not for
       Maple. Thus, one should better leave the `\\poltoexprallfalse`_
       toggle to its default ``\iffalse`` state, if one intends to use
       the output in a Maple worksheet.

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

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

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

.. _PolToFloatExpr*:

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

    Typesets in ascending powers.

.. _PolToList:

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

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

.. _PolToCSV:

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

    Expands to ``coeff_0, coeff_1, coeff_2, ....., coeff_N``, starting
    with constant term and ending with leading coefficient. Converse
    to `\\PolFromCSV <\\PolFromCSV{polname}{\<csv\>}_>`_.

.. _PolSturmChainLength:

``\PolSturmChainLength{sturmname}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    Returns the integer ``N`` such that ``sturmname_N`` is the last one
    in the Sturm chain ``sturmname_0``, ``sturmname_1``, ...

    See `\\PolToSturm{polname}{sturmname}`_.

.. _PolSturmIfZeroExactlyKnown:

``\PolSturmIfZeroExactlyKnown{sturmname}{index}{A}{B}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    Executes ``A`` if the ``index``\ -th interval reduces to a singleton,
    i.e. the root is known exactly, else ``B``.

    .. note::

       ``index`` is allowed to be something like ``1+2*3`` as it is fed
       to ``\the\numexpr...\relax``.

.. _PolSturmIsolatedZeroLeft:

``\PolSturmIsolatedZeroLeft{sturmname}{index}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    Expands to the left end-point for the ``index``\ -th interval, as
    computed by some earlier `\\PolSturmIsolateZeros{sturmname}`_.

    .. note::

       Of course, this is kept updated by macros such as
       `\\PolRefineInterval{sturmname}{index} <PolRefineInterval[N]_>`_.

    The value is pre-formatted using `\\PolDecTostring
    <PolDecToString_>`_.

.. _PolSturmIsolatedZeroRight:

``\PolSturmIsolatedZeroRight{sturmname}{index}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    Expands to the right end-point for the ``index``\ -th interval as
    computed by some earlier `\\PolSturmIsolateZeros{sturmname}`_ and
    possibly refined afterwards.

    The value is pre-formatted using `\\PolDecTostring
    <PolDecToString_>`_.

.. _PolSturmIsolatedZeroMultiplicity:

``\PolSturmIsolatedZeroMultiplicity{sturmname}{index}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    Expands to the multiplicity of the unique root contained in the
    ``index``\ -th interval.

    .. attention::

       A prior execution of `\\PolSturmIsolateZeros*{sturmname}`_ is mandatory.

    See `The degree nine polynomial with 0.99, 0.999, 0.9999 as triple
    roots`_ for an example of use.

.. _PolSturmNbOfIsolatedZeros:

``\PolSturmNbOfIsolatedZeros{sturmname}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    Expands to the number of real roots of the polynomial
    ``<sturmname>_0``, i.e. the number of distinct real roots of the
    polynomial originally used to create the Sturm chain via
    `\\PolToSturm{polname}{sturmname}`_.

.. warning::

   The next few macros counting roots, with or without multiplicities,
   less than or equal to some value, are under evaluation and may be
   removed from the package if their utility is judged to be not high
   enough. They can be re-coded at user level on the basis of the other
   documented package macros anyway.

``\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualTo{value}``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

    Expands to the number of distinct roots (of the polynomial used to
    create the Sturm chain) less than or equal to the ``value`` (i.e. a
    number of fraction recognizable by the xintfrac_ macros).

    .. attention::

       `\\PolSturmIsolateZeros{sturmname}`_ must have been executed
       beforehand.

       And the argument is a ``sturmname``, not a ``polname`` (this is
       why the macro contains Sturm in its name), simply to be reminded
       of the above constraint.

``\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualToExpr{expression}``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

    Expands to the number of distinct roots (of the polynomial 
    used to create the Sturm chain) which are less than or equal to the
    given ``expression``.

    .. attention::

       `\\PolSturmIsolateZeros{sturmname}`_ must have been executed
       beforehand.

``\PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualTo{value}``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

    Expands to the number counted with multiplicities of the roots (of
    the polynomial used to create the Sturm chain) which are less than
    or equal to the given ``value``.

    .. attention::

       `\\PolSturmIsolateZeros*{sturmname}`_ (or the double starred
       variant) must have been executed beforehand.

``\PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualToExpr{expression}``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

    Expands to the total number of roots (counted with multiplicities)
    which are less than or equal to the given ``expression``.

    .. attention::

       `\\PolSturmIsolateZeros*{sturmname}`_ (or the double starred
       variant) must have been executed beforehand.

``\PolSturmNbOfRationalRoots{sturmname}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    Expands to the number of rational roots (without multiplicities).

    .. attention::

       `\\PolSturmIsolateZeros**{sturmname}`_ must have been executed
       beforehand.

``\PolSturmNbOfRationalRootsWithMultiplicities{sturmname}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    Expands to the number of rational roots (counted with multiplicities).

    .. attention::

       `\\PolSturmIsolateZeros**{sturmname}`_ must have been executed
       beforehand.

``\PolSturmRationalRoot{sturmname}{k}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    Expands to the ``k``\ th rational root (they are ordered and indexed
    starting at 1 for the most negative).

    .. attention::

       `\\PolSturmIsolateZeros**{sturmname}`_ must have been executed
       beforehand.

``\PolSturmRationalRootIndex{sturmname}{k}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    Expands to ``index`` of the ``k``\ th rational root as part of the
    ordered real roots (without multiplicities). I.e., above macro
    `\\PolSturmRationalRoot{sturmname}{k}`_ is equivalent to this
    nested call::

      \PolSturmIsolatedZeroLeft{sturmname}{\PolSturmRationalRootIndex{sturmname}{k}}

    .. attention::

       `\\PolSturmIsolateZeros**{sturmname}`_ must have been executed
       beforehand.

``\PolSturmRationalRootMultiplicity{sturmname}{k}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    Expands to the multiplicity of the ``k``\ th rational root.

    .. attention::

       `\\PolSturmIsolateZeros**{sturmname}`_ must have been executed
       beforehand.

.. _PolIntervalWidth:

``\PolIntervalWidth{sturmname}{index}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    The ``10^E`` width of the current ``index``\ -th root localization
    interval. Output is in xintfrac_ raw ``1/1[E]`` format (if not zero).

Expandable macros for use within execution of ``\PolPrintIntervals``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

These macros are for usage within custom user redefinitions of
`\\PolPrintIntervalsKnownRoot`_, `\\PolPrintIntervalsUnknownRoot`_, or
in redefinitions of `\PolPrintIntervalsPrintExactZero`_ (used in the
default for the former) and of `\\PolPrintIntervalsPrintLeftEndPoint`_,
`\\PolPrintIntervalsPrintRightEndPoint`_ (used in the default for the
latter).

.. attention::

   Some macros formerly mentioned here got removed at 0.7:
   ``\PolPrintIntervalsTheEndPoint``,
   ``\PolIfEndPointIsPositive{A}{B}``,
   ``\PolIfEndPointIsNegative{A}{B}``,
   ``\PolIfEndPointIsZero{A}{B}``.

``\PolPrintIntervalsTheVar``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^

    Expands to the name (default ``Z``) used for representing the roots,
    which was passed as optional argument ``varname`` to
    `\\PolPrintIntervals[varname]{sturmname}`_.

``\PolPrintIntervalsTheIndex``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

    Expands to the index of the considered interval (indexing starting
    at 1 for the leftmost interval).

``\PolPrintIntervalsTheSturmName``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

    Expands to the argument which was passed as ``sturmname`` to
    `\\PolPrintIntervals[varname]{sturmname}`_.

``\PolPrintIntervalsTheLeftEndPoint``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

    The left end point of the interval, as would be produced by
    `\\PolSturmIsolatedZeroLeft <PolSturmIsolatedZeroLeft_>`_ if it was
    used with arguments the Sturm chain name and interval index returned
    by `\\PolPrintIntervalsTheSturmName`_ and
    `\\PolPrintIntervalsTheIndex`_.

``\PolPrintIntervalsTheRightEndPoint``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

    The right end point of the interval, as would be produced by
    `\\\PolSturmIsolatedZeroRight <PolSturmIsolatedZeroRight_>`_ for
    this Sturm chain name and index.

``\PolPrintIntervalsTheMultiplicity``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

    The multiplicity of the unique root within the interval of index
    `\\PolPrintIntervalsTheIndex`_. Makes sense only if the starred (or
    double-starred) variant of `\\PolSturmIsolateZeros
    <PolSturmIsolateZeros_>`_ was used earlier.

.. _PolDecToString:

``\PolDecToString{decimal number}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    This is a utility macro to print decimal numbers. It has been
    backported to xintfrac_ (release ``1.3`` of ``2018/03/01``) under
    the name ``\xintDecToString``, and the ``polexpr`` macro is simply
    now an alias to it.

    For example
    ``\PolDecToString{123.456e-8}`` will expand to ``0.00000123456``
    and ``\PolDecToString{123.450e-8}`` to ``0.00000123450`` which
    illustrates that trailing zeros are not trimmed. To trim trailing
    zeroes, one can use ``\PolDecToString{\xintREZ{#1}}``.

    The precise behaviour of this macro may evolve in future releases of
    xint_.

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

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

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

    .. caution::

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

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

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


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

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

``\polexprsetup``
-----------------

    Serves to customize the package. Currently only two keys are
    recognized:

    - ``norr``: the postfix that `\\PolSturmIsolateZeros**{sturmname}`_
      should append to ``sturmname`` to declare the primitive polynomial
      obtained from original one after removal of all rational roots.
      The default value is ``_norr`` (standing for “no rational roots”).

    - ``sqfnorr``: the postfix that `\\PolSturmIsolateZeros**{sturmname}`_
      should append to ``sturmname`` to declare the primitive polynomial
      obtained from original one after removal of all rational roots and
      suppression of all multiplicities.
      The default value is ``_sqf_norr`` (standing for “square-free with
      no rational roots”).

    The package executes ``\polexprsetup{norr=_norr,
    sqfnorr=_sqf_norr}`` as default.

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

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

- As a consequence of xintfrac_ addition and subtraction always using
  least common multiples for the denominators [#]_, user-chosen common
  denominators survive additions and multiplications. For example, this::

    \poldef P(x):= 1/2 + 2/2*x + 3/2*x^3 + 4/2*x^4;
    \poldef Q(x):= 1/3 + (2/3)x + (3/3)x^3 + (4/3)x^4;
    \poldef PQ(x):= P(x)Q(x);

  gives internally the polynomial::

    1/6+4/6*x^1+4/6*x^2+6/6*x^3+20/6*x^4+16/6*x^5+9/6*x^6+24/6*x^7+16/6*x^8

  where all coefficients have the same denominator 6. Notice though that
  ``\PolToExpr{PQ}`` outputs the ``6/6*x^3`` as ``x^3`` because (by
  default) it recognizes and filters out coefficients equal to one or
  minus one (since release ``0.3``). One can use for example
  ``\PolToCSV{PQ}`` to see the internally stored coefficients.

  .. [#] prior to ``0.4.1``, ``polexpr`` used to temporarily patch
         during the parsing of polynomials the xintfrac_ macros. This
         patch was backported to xint_ at release ``1.3``.

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

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

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

- As is to be expected internal structures of the package are barely
  documented and unstable. Don't use them.


CHANGE LOG
----------

- v0.1 (2018/01/11): initial release. Features:

  * The `\\poldef <poldef;_>`_ parser itself,
  * Differentiation and anti-differentiation,
  * Euclidean division and GCDs,
  * Various utilities such as `\\PolFromCSV <PolFromCSV_>`_,
    `\\PolMapCoeffs <PolMapCoeffs_>`_,
    `\\PolToCSV <PolToCSV_>`_, `\\PolToExpr <PolToExpr_>`_, ...

  Only one-variable polynomials so far.

- v0.2 (2018/01/14)

  * Fix: ``"README thinks \numexpr recognizes ^ operator"``.
  * Convert README to reStructuredText markup.
  * Move main documentation from README to separate ``polexpr.txt`` file.
  * Provide ``polexpr.html`` as obtained via DocUtils_ ``rst2html.py``.
  * Convert README to (CTAN compatible) Markdown markup.

  Due to lack of available time the test suite might not be extensive
  enough. Bug reports are very welcome!

- v0.3 (2018/01/17)

  * bug fixes:

    - the ``0.1`` `\\PolEval <PolEvalAt_>`_ accepted expressions for its second
      argument, but this was removed by mistake at ``0.2``. Restored.

      **Attention**: at ``0.4`` this has been reverted again, and
      `\\PolEval{P}\\AtExpr{foo} <PolEvalAtExpr_>`_ syntax is needed for
      using expressions in the second argument.
  * incompatible or breaking changes:

    - `\\PolToExpr <PolToExpr_>`_ now by default uses *descending*
      powers (it also treats differently coefficients equal to 1 or -1.)
      Use `\\PolToExpr* <PolToExpr*_>`_ for *ascending* powers.
    - `\\PolEval <PolEvalAt_>`_ reduced the output to smallest terms,
      but as this is costly with big fractions and not needed if e.g.
      wrapped in an ``\xintRound`` or ``\xintFloat``, this step has been
      removed; the former meaning is available as `\\PolEvalReduced
      <PolEvalReducedAt_>`_.
  * new (or newly documented) macros:

    - `\\PolTypesetCmd <PolTypesetCmd_>`_
    - `\\PolTypesetCmdPrefix <PolTypesetCmdPrefix_>`_
    - `\\PolTypesetMonomialCmd <PolTypesetMonomialCmd_>`_
    - `\\PolEvalReducedAt <PolEvalReducedAt_>`_
    - `\\PolToFloatExpr <PolToFloatExpr_>`_
    - `\\PolToExprOneTerm <PolToExprOneTerm_>`_
    - `\\PolToFloatExprOneTerm <PolToFloatExprOneTerm_>`_
    - `\\PolToExprCmd <PolToExprCmd_>`_
    - `\\PolToFloatExprCmd <PolToFloatExprCmd_>`_
    - `\\PolToExprTermPrefix <PolToExprTermPrefix_>`_
    - `\\PolToExprVar <PolToExprVar_>`_
    - `\\PolToExprTimes <PolToExprTimes_>`_
  * improvements:

    - documentation has a table of contents, internal hyperlinks,
      standardized signature notations and added explanations.
    - one can do ``\PolLet{g}={f}`` or ``\PolLet{g}{f}``.
    - ``\PolToExpr{f}`` is highly customizable.
    - `\\poldef <poldef;_>`_ and other defining macros prepare the polynomial
      functions for usage within ``\xintthefloatexpr`` (or
      ``\xintdeffloatvar``). Coefficients are pre-rounded to the
      floating point precision. Indispensible for numerical algorithms,
      as exact fractions, even reduced, quickly become very big. See the
      documentation about how to use the exact polynomials also in
      floating point context.

      **Attention**: this has been reverted at ``0.4``. The macro
      `\\PolGenFloatVariant <PolGenFloatVariant_>`_ must be used for
      generation floating point polynomial functions.

- v0.3.1 (2018/01/18)

  Fixes two typos in example code included in the documentation.

- v0.4 (2018/02/16)

  * bug fixes:

    - when Euclidean division gave a zero remainder, the internal
      representation of this zero polynomial could be faulty; this
      could cause mysterious bugs in conjunction with other package
      macros such as `\\PolMapCoeffs <PolMapCoeffs_>`_.
    - `\\PolGCD <PolGCD_>`_ was buggy in case of first polynomial being
      of lesser degree than the second one.
  * breaking changes:

    - formerly `\\PolEval{P}\\At{foo} <PolEvalAt_>`_ allowed ``foo`` to
      be an expression, which was transparently handled via
      ``\xinttheexpr``. Now, ``foo`` must be a fraction (or a macro
      expanding to such) in the format acceptable by ``xintfrac.sty``
      macros. Use `\\PolEval{P}\\AtExpr{foo} <PolEvalAtExpr_>`_ for more
      general arguments using expression syntax. E.g., if ``foo`` is the
      name of a variable known to ``\xintexpr``.

      The same holds for `\\PolEvalReduced <PolEvalReducedAt_>`_
      and `\\PolFloatEval <PolFloatEvalAt_>`_.
    - the ``3.0`` automatic generation of floating point variants has
      been reverted. Not only do *not* the package macros automatically
      generate floating point variants of newly created polynomials,
      they actually make pre-existing such variant undefined.

      See `\\PolGenFloatVariant <PolGenFloatVariant_>`_.
  * new non-expandable macros:

    - `\\PolGenFloatVariant <PolGenFloatVariant_>`_
    - `\\PolGlobalLet <PolGlobalLet_>`_
    - `\\PolTypesetOne <PolTypesetOne_>`_
    - `\\PolQuo <PolQuo_>`_
    - `\\PolRem <PolRem_>`_
    - `\\PolToSturm <PolToSturm_>`_
    - `\\PolToSturm\* <PolToSturm*_>`_
    - `\\PolSetToSturmChainSignChangesAt <PolSetToSturmChainSignChangesAt_>`_
    - `\\PolSetToNbOfZerosWithin <PolSetToNbOfZerosWithin_>`_
    - `\\PolSturmIsolateZeros <PolSturmIsolateZeros_>`_
    - `\\PolRefineInterval* <PolRefineInterval*_>`_
    - `\\PolRefineInterval[N] <PolRefineInterval[N]_>`_
    - `\\PolEnsureIntervalLength <PolEnsureIntervalLength_>`_
    - `\\PolEnsureIntervalLengths <PolEnsureIntervalLengths_>`_
    - `\\PolPrintIntervals <PolPrintIntervals_>`_
    - `\\PolPrintIntervalsPrintExactZero <PolPrintIntervalsPrintExactZero_>`_
    - `\\PolPrintIntervalsPrintLeftEndPoint <PolPrintIntervalsPrintLeftEndPoint_>`_
    - `\\PolPrintIntervalsPrintRightEndPoint <PolPrintIntervalsPrintRightEndPoint_>`_
    - `\\PolReduceCoeffs* <PolReduceCoeffs*_>`_
    - `\\PolMakeMonic <PolMakeMonic_>`_
  * new expandable macros:

    - `\\PolToExprOneTermStyleA <PolToExprOneTermStyleA_>`_
    - `\\PolIfCoeffIsPlusOrMinusOne <PolIfCoeffIsPlusOrMinusOne_>`_
    - `\\PolLeadingCoeff <PolLeadingCoeff_>`_
    - `\\PolSturmChainLength <PolSturmChainLength_>`_
    - `\\PolSturmNbOfIsolatedZeros <PolSturmNbOfIsolatedZeros_>`_
    - `\\PolSturmIfZeroExactlyKnown <PolSturmIfZeroExactlyKnown_>`_
    - `\\PolSturmIsolatedZeroLeft <PolSturmIsolatedZeroLeft_>`_
    - `\\PolSturmIsolatedZeroRight <PolSturmIsolatedZeroRight_>`_
    - ``\PolPrintIntervalsTheEndPoint`` (removed at 0.7)
    - `\\PolPrintIntervalsTheIndex`_
    - ``\PolIfEndPointIsPositive`` (removed at 0.7)
    - ``\PolIfEndPointIsNegative`` (removed at 0.7)
    - ``\PolIfEndPointIsZero`` (removed at 0.7)
    - `\\PolIntervalWidth <PolIntervalWidth_>`_
    - `\\PolDecToString <PolDecToString_>`_
  * improvements:

    The main new feature is implementation of the `Sturm algorithm`_
    for localization of the real roots of polynomials.

- v0.4.1 (2018/03/01)

  Synced with xint 1.3.

- v0.4.2 (2018/03/03)

  Documentation fix.

- v0.5 (2018/04/08)

  * bug fixes:

    - `\\PolGet{polname}\\fromarray\\macro`_ crashed when ``\macro`` was
      an xinttools_ array macro with no items. It now produces the zero
      polynomial.
  * breaking changes:

    - `\\PolToSturm`_ creates primitive integer coefficients polynomials.
      This speeds up localization of roots via
      `\\PolSturmIsolateZeros`_. In case of user protests the author
      will make available again the code producing the bona fide Sturm
      polynomials as used formerly.
    - polynomials created from `\\PolFromCSV`_ or `\\PolGet <PolGet_>`_
      get their coefficients normalized via xintfrac_\ 's ``\xintRaw``.
  * experimental change:

    - optional argument to `\\PolSturmIsolateZeros`_ (see `The
      degree 41 polynomial with -2, -1.9, -1.8, ..., 0, 0.1, ..., 1.9, 2
      as roots`_ for usage). It will presumably be replaced in future by
      an interval specification.
  * new non-expandable macro:

    - `\\PolMakePrimitive`_
  * new expandable macro:

    - `\\PolIContent`_
      
- v0.5.1 (2018/04/22)

  * new feature:

    - the character ``'`` can be used in polynomial names.

- v0.6 (2018/11/20)

  * bugfix:

    - the starred variant `\\PolToSturm*{polname}{sturmname}`_ was
      broken. On the occasion of the fix, its meaning has been modified,
      see its documentation.

    - using `\\PolToSturm <PolToSturm_>`_ with a constant polynomial
      caused a division by zero error.

  * new macro:

    - `\\PolSturmIsolateZeros* <PolSturmIsolateZeros*_>`_
      acts like the `non-starred variant
      <PolSturmIsolateZeros_>`_ then computes all the multiplicities.

  * new expandable macros:

    - `\\PolSturmIsolatedZeroMultiplicity{sturmname}{index}`_
    - `\\PolSturmNbOfRootsOf{sturmname}\\LessThanOrEqualTo{value}`_
    - `\\PolSturmNbOfRootsOf{sturmname}\\LessThanOrEqualToExpr{expression}`_
    - `\\PolSturmNbWithMultOfRootsOf{sturmname}\\LessThanOrEqualTo{value}`_
    - `\\PolSturmNbWithMultOfRootsOf{sturmname}\\LessThanOrEqualToExpr{expression}`_

- v0.7 (2018/12/08), v0.7.1 (bugfix), v0.7.2 (2nd bugfix) (2018/12/09)

  * breaking changes:

    - although `\\PolPrintIntervals[varname]{sturmname}`_ default output
      remains the same, some auxiliary macros for user-customization
      have been removed: ``\PolPrintIntervalsTheEndPoint``,
      ``\PolIfEndPointIsPositive{A}{B}``,
      ``\PolIfEndPointIsNegative{A}{B}``, and
      ``\PolIfEndPointIsZero{A}{B}``.

  * bugfix:

    - it could happen that, contrarily to documentation, an interval
      computed by `\\PolSturmIsolateZeros{sturmname}`_ had zero as an
      endpoint,
    - `\\PolEnsureIntervalLength{sturmname}{index}{E}`_ could under
      certain circumstances erroneously replace a non-zero root by
      zero,
    - `\\PolEnsureIntervalLengths{sturmname}{E}`_ crashed when used with
      a polynomial with no real roots, hence for which no isolation intervals
      existed (thanks to Thomas Söll for report).

  * new macros:

    - `\\PolSturmIsolateZeros**{sturmname}`_
    - `\\PolSturmIsolateZerosGetMultiplicitiesAndRationalRoots{sturmname}`_
    - `\\PolSturmIsolateZerosAndFindRationalRoots{sturmname}`_
    - `\\polexprsetup`_
    - `\\PolPrintIntervals* <PolPrintIntervals*_>`_
    - `\\PolPrintIntervalsNoRealRoots`_
    - `\\PolPrintIntervalsBeginEnv`_
    - `\\PolPrintIntervalsEndEnv`_
    - `\\PolPrintIntervalsKnownRoot`_
    - `\\PolPrintIntervalsUnknownRoot`_
    - `\\PolPrintIntervalsPrintMultiplicity`_

  * new expandable macros:

    - `\\PolSturmNbOfRationalRoots{sturmname}`_
    - `\\PolSturmNbOfRationalRootsWithMultiplicities{sturmname}`_
    - `\\PolSturmRationalRoot{sturmname}{k}`_
    - `\\PolSturmRationalRootIndex{sturmname}{k}`_
    - `\\PolSturmRationalRootMultiplicity{sturmname}{k}`_
    - `\\PolPrintIntervalsTheVar`_
    - `\\PolPrintIntervalsTheSturmName`_
    - `\\PolPrintIntervalsTheMultiplicity`_

- v0.7.3 (2019/02/04)

  * bug fixes:

    - Debugging information not destined to user showed in log if root
      finding was done under ``\xintverbosetrue`` regime.
    - `\\PolPrintIntervalsTheVar`_ remained defined after
      `\\PolPrintIntervals`_ but was left undefined after
      `\\PolPrintIntervals*`_ (reported by Jürgen Gilg). Now remains
      defined in both cases, and `\\PolPrintIntervalsTheSturmName`_
      also.
    - Polynomial names ending in digits caused errors (reported by Thomas
      Söll).


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

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

Renewed thanks to them on occasion of the ``0.6`` and ``0.7`` releases for their
continued interest.

See README.md for the License.

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

.. _Wilkinson polynomial: https://en.wikipedia.org/wiki/Wilkinson%27s_polynomial

.. _Sturm algorithm:
.. _Sturm Theorem: https://en.wikipedia.org/wiki/Sturm%27s_theorem

.. _DocUtils: http://docutils.sourceforge.net/docs/index.html