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

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

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

0.5 (2018/04/08)
================

.. contents::

Basic 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``.

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

.. attention::

   At ``0.5``, `\\PolToSturm{P}{S} <PolToSturm_>`_  was modified to
   create a chain of primitive integer coefficients polynomials. This
   speeds up evaluations, hence localization of roots afterwards. Thus
   ``S_1`` will not necessarily be the derivative of ``S_0``, even if
   ``P`` is square-free.

- 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``.

- To make producing this documentation simpler, the results from execution
  of the code snippets are not included. Please try them out yourself...


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

In this example the polynomial is square-free; we can make sure of that by
comparing the degree of the first element of the Sturm chain with the
degree of the original polynomial. In such case the second element of
the Sturm chain is still the polynomial first derivative, because there
was no further reduction.

::

  \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 same as $\PolDegree{f}$ we know that the latter was square free.
  So the derivative is up to a constant \PolTypeset{f_1} (in fact here
  it is identical with it).
  \PolToSturm{f_1}{f_1}\PolSturmIsolateZeros{f_1}%
  It 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_xx}
  \PolToSturm{f_xx}{f_xx}
  \PolSturmIsolateZeros{f_xx}
  The second derivative is \PolTypeset{f_xx}.
  It has \PolSturmNbOfIsolatedZeros{f_xx} distinct real
  roots:
  \PolPrintIntervals[X]{f_xx}
  Here is the positive one with 20 digits after decimal mark:
  \PolEnsureIntervalLength{f_xx}{2}{-20}%
  \[X_2 = \PolSturmIsolatedZeroLeft{f_xx}{2}\dots\]
  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
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

::

  \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
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

::

  \PolDef{P}{(x-0.99)^3(x-0.999)^3(x-0.9999)^3}
  \PolTypeset{P}\par
  \PolToSturm{P}{P}%
  \PolLet{Psqfree}{P_0}\PolMakeMonic{Psqfree}\PolReduceCoeffs*{Psqfree}
  \par
  The monic square-free radical is \PolTypeset{Psqfree}.
  \PolSturmIsolateZeros{P}
  \par
  It has \PolSturmNbOfIsolatedZeros{P} real roots.
  \PolPrintIntervals{P}% all three roots found exactly

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 < Z_2 < 0.1, 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 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}
  \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)``, extremities assumed to not be roots.
   Thus here::

     \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.

   This will probably get replaced in future by the specification of
   a general interval.

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 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.

    .. 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 constant term being the first item. No validity checks. Spaces
    from the list argument are trimmed. List items are each expanded in
    an ``\edef`` and then put into normalized form via xintfrac_\ 's
    macro ``\xintRaw``.

    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}_>`_.

    .. 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, for example, ``polname`` being ``P`` and ``sturmname`` being
    ``S``, the macro starts by computing polynomials ``S_0 = P``, ``S_1
    = P'``, ..., with ``S_{n+1}`` the opposite of the remainder of
    euclidean division of ``S_{n-1}`` by ``S_{n}``.

    .. attention::

       Since ``0.5``, it further normalizes all these polynomials by making
       them primitive (see `\\PolMakePrimitive`_).

    The last non-zero remainder ``S_N`` is up to a factor the GCD of
    ``P`` and ``P'`` hence it is a constant (i.e. ``1`` or ``-1`` since
    ``0.5``) if and only if ``P`` is square-free.

    In case ``S_N`` is not a constant, the macro then goes on with
    dividing all ``S_k``'s with ``S_N`` (which then becomes ``1``).

    Thus ``S_0`` has exactly the same real and complex roots as
    polynomial ``polname``, but each root being now with multiplicity one.

.. _PolToSturm*:

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

    Does not divide the Sturm chain by its last element.

.. _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.

.. _PolSturmIsolateZeros:

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

    First, it evaluates using `Sturm theorem`_ the number of distinct
    real roots of ``sturmname_0``.

    .. important::

       The Sturm chain **must** be of the reduced type, i.e.
       as constructed via `\\PolToSturm{polname}{sturmname}`_.

    Then it locates, again using `Sturm theorem`_, as many disjoint
    intervals as there are roots. Some intervals reduce to singleton
    which are roots. Non-singleton intervals get refined to make sure
    none of their two limit points is a root: they contain each a single
    root, in their respective interiors.

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

    The interval boundaries are decimal numbers, originating
    in iterated decimal subdivision from initial intervals
    ``(-10^E, 0)`` and ``(0, 10^E)``; if zero is a root it is always
    identified individually. 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 ``a`` and ``a+1`` are both positive
    or both negative.

    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.

    Also two macro arrays (in the sense of
    xinttools_'s ``\xintAssignArray``) are created for holding the
    interval end-points written out in standard decimal notation
    (see `\\PolDecToString{decimal number}`_).
    To access these values, macros
    `\\PolSturmIsolatedZeroLeft{sturmname}{index}`_ and
    `\\PolSturmIsolatedZeroRight{sturmname}{index}`_ are provided.

    .. important::

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

    .. note::

       The actual array macros are ``\POL_ZeroInt<sturmname>L`` and
       ``\POL_ZeroInt<sturmname>R`` but as these names use the
       non-letter character ``_`` and possibly also digits from
       ``sturmname``, the accessor macros above have been made part of
       the package.

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

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

    .. note::

       In the current implementation the ``<sturmname>...`` variables
       and the ``\POL_ZeroInt...`` arrays are globally defined. On the
       other hand the Sturm sequence polynomials obey the current scope.

    .. note::

       When two successive roots are located in adjacent intervals, the
       separation between them is not lower bounded. See
       `\\PolRefineInterval*{sturmname}{index}`_.

    .. 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.

.. _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 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. The user is invited to renewcommand the macro if some other
    type of tabular environment for example is wanted.

    In each array cell the corresponding interval end-point (which may
    be an exactly known root) is available as macro
    `\\PolPrintIntervalsTheEndPoint`_ (in decimal notation). And the
    corresponding interval index is available as
    `\\PolPrintIntervalsTheIndex`_.

    These values may be tested to decide some on-the-fly customization
    (color for example), via the following auxiliaries which can be
    modified by user. Furthermore these auxiliaries can also use the
    following conditionals: `\\PolIfEndPointIsPositive{A}{B}`_,
    `\\PolIfEndPointIsNegative{A}{B}`_, `\\PolIfEndPointIsZero{A}{B}`_.

.. _PolPrintIntervalsPrintExactZero:

``\PolPrintIntervalsPrintExactZero``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
    This is provided to help customize how an exactly known root is
    printed in the right most column of the array. The package
    definition is::

      \newcommand\PolPrintIntervalsPrintExactZero{\PolPrintIntervalsTheEndPoint}%

    Recall that this is expanded in an array cell.

    If for example you want to print in red the third root, known
    exactly, the macro could make a test for the value of
    `\\PolPrintIntervalsTheIndex`_  and act accordingly.


.. _PolPrintIntervalsPrintLeftEndPoint:

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

    Package definition is::

      \newcommand\PolPrintIntervalsPrintLeftEndPoint{\PolPrintIntervalsTheEndPoint}%

.. _PolPrintIntervalsPrintRightEndPoint:

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

    Package definition is::

      \newcommand\PolPrintIntervalsPrintRightEndPoint{\PolPrintIntervalsTheEndPoint}%

.. _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.

    .. attention::

       Meaning was changed at ``0.4``. Formerly ``\PolEval{P}\At{foo}``
       accepted for ``foo`` an expression which was handled by
       ``\xintexpr``. See `\\PolEval{polname}\\AtExpr{numerical
       expression}`_.

       In particular, to use an ``\xintexpr`` user-declared variable (or
       e.g. the variables as defined by `\\PolSturmIsolateZeros
       <PolSturmIsolateZeros>`_) one **must** use the ``\AtExpr`` syntax.

.. _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.

    .. attention::

       Meaning was changed at ``0.4``. Formerly the evaluation point
       could be given as an expression.

.. _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.

    .. attention::

       At ``polexpr 0.3``, polynoms were automatically also prepared for
       use in floating point contexts. This got dropped at ``0.4`` for
       optimization purposes. See `\\PolGenFloatVariant{polname}`_.

    .. [#] 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.

    .. attention::

       Meaning was changed at ``0.4``. Formerly the evaluation point
       could be given as an expression.

.. _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_ ``1.2p``) ``\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*:

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

    Typesets in ascending powers.

.. _PolToList:

``\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:

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

    Expands to ``coeff_0, coeff_1, coeff_2, ....., coeff_N``. 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`` may be a TeX count, or a ``\value{latexcounter}``, or a
       numerical expression as parsable by ``\numexpr``: it does not
       have to be given via explicit digits.

       This remark applies also to the other package macros with
       ``index`` being the name of the argument in this documentation.
       There is also an out-of-range check done for some reasonable
       error message (right before everything goes haywire).

.. _PolSturmIsolatedZeroLeft:

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

    Expands to the left end-point for the ``index``\ th interval
    obtained via `\\PolSturmIsolateZeros{sturmname}`_ and possibly
    refined afterwards.

.. _PolSturmIsolatedZeroRight:

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

    Expands to the right end-point for the ``index``\ th interval
    obtained via `\\PolSturmIsolateZeros{sturmname}`_ and possibly
    refined afterwards.

.. _PolSturmNbOfIsolatedZeros:

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

    Expands to the number of real roots of the polynomial
    ``<sturmname>_0`` (which is the number of distinct real roots of the
    polynomial used to create the Sturm chain via
    `\\PolToSturm{polname}{sturmname}`_.

.. _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).

Macros for use within execution of ``\PolPrintIntervals``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

More precisely, they can be used within the replacement texts of the
`\\PolPrintIntervalsPrintLeftEndPoint`_, etc, macros.


.. _PolPrintIntervalsTheEndPoint:

``\PolPrintIntervalsTheEndPoint``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

    Within a custom `\\PolPrintIntervalsPrintLeftEndPoint`_, custom
    `\\PolPrintIntervalsPrintRightEndPoint`_, or custom
    `\\PolPrintIntervalsPrintExactZero`_ this macro expands to the left
    or right end point of the considered interval. Serves as default
    replacement for `\\PolPrintIntervalsPrintLeftEndPoint`_ , etc...

.. _PolPrintIntervalsTheIndex:

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

    Within a custom `\\PolPrintIntervalsPrintLeftEndPoint`_, custom
    `\\PolPrintIntervalsPrintRightEndPoint`_, or custom
    `\\PolPrintIntervalsPrintExactZero`_ this macro expands to the index
    of the considered interval. For example if user wants to print the
    corresponding end points in red, the index value can thus be tested
    in the replacement text of `\\PolPrintIntervalsPrintLeftEndPoint`_ and
    the other two similar macros.

.. _PolIfEndPointIsPositive:

``\PolIfEndPointIsPositive{A}{B}``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

    Within a custom `\\PolPrintIntervalsPrintLeftEndPoint`_, custom
    `\\PolPrintIntervalsPrintRightEndPoint`_, or custom
    `\\PolPrintIntervalsPrintExactZero`_ this macro executes ``A`` if
    the considered interval end-point is positive, else ``B``.

.. _PolIfEndPointIsNegative:

``\PolIfEndPointIsNegative{A}{B}``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

    Within a custom `\\PolPrintIntervalsPrintLeftEndPoint`_, custom
    `\\PolPrintIntervalsPrintRightEndPoint`_, or custom
    `\\PolPrintIntervalsPrintExactZero`_ this macro executes ``A`` if
    the considered interval end-point is negative, else ``B``.

.. _PolIfEndPointIsZero:

``\PolIfEndPointIsZero{A}{B}``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

    Within a custom `\\PolPrintIntervalsPrintLeftEndPoint`_, custom
    `\\PolPrintIntervalsPrintRightEndPoint`_, or custom
    `\\PolPrintIntervalsPrintExactZero`_ this macro executes ``A`` if
    the considered interval end-point is zero, else ``B``.

.. _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 exact 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.


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 <PolPrintIntervalsTheEndPoint_>`_
    - `\\PolPrintIntervalsTheIndex <PolPrintIntervalsTheIndex_>`_
    - `\\PolIfEndPointIsPositive <PolIfEndPointIsPositive_>`_
    - `\\PolIfEndPointIsNegative <PolIfEndPointIsNegative_>`_
    - `\\PolIfEndPointIsZero <PolIfEndPointIsZero_>`_
    - `\\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 macros:

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

    - `\\PolIContent`_
      


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 on occasion of ``0.4`` release!

See README.md for the License.

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

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

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