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\documentclass[a4paper,svgnames,dvipsnames,dvipdfmx]{article}
\usepackage{geometry}
\usepackage{shortvrb}
\usepackage{xcolor}
\usepackage{graphicx}
\usepackage{polexpr}
\usepackage{hyperref}
\usepackage{bookmark}
% \usepackage{amsmath}
\usepackage{framed}
\usepackage{newtxtext,newtxmath}
\title{\pkg{polexpr} root localization examples}
\author{Jean-FranĂ§ois Burnol}
\date{To access the reference documentation:\\
% faut-il vraiment le ./ ?
      \texttt{texdoc} \href{run:./polexpr.html}{polexpr.html}}
%\usepackage{parskip}
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\begin{document}
\maketitle


The package provides a parser |\poldef| of algebraic polynomial
expressions.

Once defined, a polynomial is usable by its name either as a numerical
function in |\xintexpr/\xinteval|, or for additional polynomial
definitions, or as argument to the package macros.

%   The localization of
% real roots to arbitrary precision as well as the determination of all
% rational roots is implemented via such macros.

% Since release |0.8|, polexpr extends the \pkg{xintexpr}
% syntax to recognize
% polynomials as a new variable type (and not only as functions).
% Functionality which previously was implemented via macros such as the
% computation of a greatest common divisor is now available directly in
% |\xintexpr|, |\xinteval| or |\poldef| via infix or functional
% syntax.

This document illustrates root localization via usage of macros such as
|\PolToSturm| and |\PolSturmIsolateZeros| which implement the
\href{https://en.wikipedia.org/wiki/Sturm%27s_theorem}{Sturm theorem}:
\begin{itemize}
\item Root localization based on
  \href{https://en.wikipedia.org/wiki/Sturm%27s_theorem}{Sturm theorem} was
  added at release |0.4| (2018/02/16).
\item Ability to find all rational roots was added at release |0.7.2|
  (2018/12/09).
\end{itemize}
As of |0.8| (2021/03/29), \pkg{polexpr} is usable with Plain \TeX\ and not
only with \LaTeX. The examples here use most of the time a syntax which works
with both.

Copying-pasting from |pdf| the example source may lose formatting.
Formerly, they were included verbatim in the |html| documentation.  Here
they are both rendered verbatim and got executed during the \LaTeX\ run
which created this |pdf| file, with the output shown after the source code.

% Perhaps future releases will implement other approaches, which are known
% to be generically computationally more efficient, at least in high
% degrees, than the \href{https://en.wikipedia.org/wiki/Sturm%27s_theorem}{Sturm theorem} based approach.  This is not
% immediate priority though (perhaps support of multivariate polynomials
% would be more important feature; or localization of complex roots).

Regarding how polynomial coefficients are printed on the typeset page by
|\PolTypeset|:
\begin{itemize}\def\everbatimtop {\MacroFont}% sans le \small
\item The default for |\PolTypesetOne| is to use |\xintTeXsignedFrac| with
  \LaTeX, |\xintTeXsignedOver| with Plain.  See the \pkg{xintexpr}
  documentation for a description of what these macros do.  A sensible
  definition is: 
\begin{everbatim}
  \def\PolTypesetOne#1{\PolDecToString{\xintREZ{#1}}}%
\end{everbatim}
% le \smallskip est ennuyeux ici
  It means to use decimal notation, with perhaps a trailing denominator if the
  argument is a fraction, and will suppress trailing zeros after the decimal
  mark.

\item
  As these are expandable macros, they are usable to redefine
  |\PolToExprCmd| as well:
\begin{everbatim}
  \def\PolToExprCmd#1{\PolDecToString{\xintREZ{#1}}}%
\end{everbatim}
% le \smallskip est ennuyeux ici
  This will customize the output of |\PolToExpr| (which a priori is
  destined for writes to external files but may also be used on the typeset
  page).
\end{itemize}

With |\xintverbosetrue| in the \TeX\ source extra information relative
to the internal data manipulated by the macros will be written to the |.log|
file.

\begin{framed}
  Package macros related to root localization create (user-level) new
  polynomials, or numeric variables, via a naming scheme using the given
  |&lt;sturmname&gt;| as prefix. It is thus advisable to keep this
  |&lt;sturmname&gt;| name-space separate from the one used to name polynomial or
  scalar variables.
\end{framed}

\begin{framed}
  Regrettably all examples here use the condemnable |\PolToSturm{f}{f}|
  practice which means that internally defined polynomials will use as prefix
  the original polynomial name.  This merge of namespaces may cause
  overwriting previously defined data and may lead to hard-to-debug problems.
\end{framed}


\section{A first example}

In this example the polynomial is square-free.
\begin{everbatim*}
  \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}
\end{everbatim*}

\begin{everbatim*}
  Here is the second root with ten more decimal digits:
  \PolRefineInterval[10]{f}{2}
  $$\PolSturmIsolatedZeroLeft{f}{2}&lt;Z_2&lt;\PolSturmIsolatedZeroRight{f}{2}$$
\end{everbatim*}

\begin{everbatim*}
  And here is the first root with twenty digits after decimal mark:
  \PolEnsureIntervalLength{f}{1}{-20}
  $$\PolSturmIsolatedZeroLeft{f}{1}&lt;Z_1&lt;\PolSturmIsolatedZeroRight{f}{1}$$
\end{everbatim*}

\begin{everbatim*}
  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}
\end{everbatim*}

\begin{everbatim*}
  \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}%
  $$X_2 = \PolSturmIsolatedZeroLeft{f''}{2}\dots$$
\end{everbatim*}
%The more mathematically advanced among our dear readers will be able
%  to give the exact value for $X_2$!

\section{A degree four polynomial with nearby roots}


Notice that this example is a bit outdated as |0.7| release has
added |\PolSturmIsolateZeros**{&lt;sturmname&gt;}| 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.


\begin{everbatim*}
  \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}
\end{everbatim*}

\begin{everbatim*}
  \PolRefineInterval*{Q}{1}
  \PolRefineInterval*{Q}{2}
  \PolRefineInterval*{Q}{3}
  \PolRefineInterval*{Q}{4}
  \PolPrintIntervals{Q}
\end{everbatim*}

\begin{everbatim*}
  \PolEnsureIntervalLengths{Q}{-6}
  \PolPrintIntervals{Q}
  % finds here all roots exactly
\end{everbatim*}

\section{The degree nine polynomial with 0.99, 0.999, 0.9999 as triple roots}

Define a user command (\pkg[xint]{xinttools} is loaded automatically by
\pkg{polexpr}):

\begin{everbatim*}
  \def\showmultiplicities#1{% #1 = "sturmname"
  \xintFor* ##1 in {\xintSeq{1}{\PolSturmNbOfIsolatedZeros{#1}}}\do{%
      The multiplicity is \PolSturmIsolatedZeroMultiplicity{#1}{##1}
      \PolSturmIfZeroExactlyKnown{#1}{##1}%
      {at the root $x=\PolSturmIsolatedZeroLeft{#1}{##1}$}
      {for the root such that
      $\PolSturmIsolatedZeroLeft{#1}{##1}&lt;x&lt;\PolSturmIsolatedZeroRight{#1}{##1}$}
      \par
  }}%
\end{everbatim*}

\begin{everbatim*}
  \PolDef{f}{(x-0.99)^3(x-0.999)^3(x-0.9999)^3}
  \def\PolTypesetOne#1{\PolDecToString{\xintREZ{#1}}}
  \PolTypeset{f}\par
\end{everbatim*}

\begin{everbatim*}
  \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}
\end{everbatim*}
% 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:

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

  \showmultiplicities{g}
\end{everbatim*}
% 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 &lt; x &lt; 0.99
%   The multiplicity is 1 for the root such that 0.9991 &lt; x &lt; 0.9992
%   The multiplicity is 1 for the root such that 0.9997 &lt; x &lt; 0.9998

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

\begin{everbatim*}
  \PolEnsureIntervalLengths{g}{-10}

  \showmultiplicities{g}
\end{everbatim*}
% which produces::

%   The multiplicity is 1 for the root such that 0.9899888032 &lt; x &lt; 0.9899888033
%   The multiplicity is 1 for the root such that 0.9991447980 &lt; x &lt; 0.9991447981
%   The multiplicity is 1 for the root such that 0.9997663986 &lt; x &lt; 0.9997663987

\section{A degree five polynomial with three rational roots}

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

  \PolToSturm{Q}{Q}
    \def\PolTypesetCmdPrefix#1{\allowbreak\xintiiifSgn{#1}{}{+}{+}}%
    $Q_0(x) = \PolTypeset{Q_0}$
  \PolSturmIsolateZeros**{Q}
  \PolPrintIntervals{Q}

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

Here, all real roots are rational. Let's get their decimal expansion too:
\begin{everbatim*}
\begingroup
  % print decimal expansion of the found roots
  \def\PolPrintIntervalsPrintExactZero
              {\xintTrunc{20}{\PolPrintIntervalsTheLeftEndPoint}\dots}
  \PolPrintIntervals{Q}
\endgroup % we localized the modified \PolPrintIntervalsPrintExactZero
\end{everbatim*}


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

\section{A Mignotte type polynomial}


\begin{everbatim*}
  \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}%
\end{everbatim*}
Let us refine the second and third intervals to separate the corresponding
roots:
\begin{everbatim*}
  \PolRefineInterval*{P}{2}% will refine to 0.0999990 &lt; Z_2 &lt; 0.0999991
  \PolRefineInterval*{P}{3}% will refine to 0.100001 &lt; Z_3 &lt; 0.100002
  \PolPrintIntervals{P}%
\end{everbatim*}
Let us now get to know all roots with 10 digits after decimal mark:
\begin{everbatim*}
  \PolEnsureIntervalLengths{P}{-10}%
  \PolPrintIntervals{P}% now all roots are known 10 decimal digits after mark
\end{everbatim*}
Finally, we display 20 digits of the second root:
\begin{everbatim*}
  \PolEnsureIntervalLength{P}{2}{-20}% makes Z_2 known with 20 digits after mark
  $$\PolSturmIsolatedZeroLeft{P}{2}&lt;Z_2&lt;\PolSturmIsolatedZeroRight{P}{2}$$
\end{everbatim*}
% The last line produces::

%   0.09999900004999650028 &lt; Z_2 &lt; 0.09999900004999650029

\section{The Wilkinson polynomial}


See \url{https://en.wikipedia.org/wiki/Wilkinson%27s_polynomial}.


\begin{everbatim*}
  %\xintverbosetrue % for the curious...

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

  \def\PolTypesetCmdPrefix#1{\allowbreak\xintiiifSgn{#1}{}{+}{+}}%
  \def\PolTypesetOne#1{\xintDecToString{#1}}%

  \noindent\PolTypeset{f}

  \PolToSturm{f}{f}
  \PolSturmIsolateZeros{f}
  \PolPrintIntervals{f}
  % \vfill\eject

  % This page is commented out because it takes about 30s on a 2GHz CPU
  % \poldef g(x) := f(x) - 2**{-23} x**19;

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

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

  % \PolPrintIntervals*{g}
\end{everbatim*}

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, but the modified one |f(x) -
2**-23 x**19| takes about 20x longer.

Its 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 are the ten real roots of
|f(x) - 2**-23 x**19| which would be computed by the commented-out code above:
\begin{everbatim}
  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...
\end{everbatim}

\section{The second Wilkinson polynomial}


\begin{everbatim*}
  \poldef f(x) := mul(x - 2^-i, i = 1..20);

  %\PolTypeset{f}

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

This takes more time than the polynomial with 1, 2, .., 20 as roots but
less than the latter modified by the |2**-23| tiny change to one of its
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.


\section{The degree 41 polynomial with -2, -1.9, -1.8, ..., 0, 0.1, ..., 1.9, 2 as roots}



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

\begin{everbatim*}
\begingroup
  \def\PolToExprCmd#1{\PolDecToString{\xintREZ{#1}}}
  \def\PolToExprTermPrefix#1{\newline\xintiiifSgn{#1}{}{+}{+}}
  \def\PolToExprTimes{${}\cdot{}$}
\ttfamily
\PolToExpr{P}
\endgroup
\end{everbatim*}
  % 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 \pkg{xint} + \pkg{polexpr}, but be a bit
patient!

\begin{everbatim*}
  \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?
% Unfortunately \PolPrintIntervals uses a non-breakable array environment
% But see next section on how to customize \PolPrintIntervals and let it
% allow pagebreaks
\end{everbatim*}

\begin{quote}
  Release |0.5| has \emph{experimental} addition of optional argument |E| to
  |\PolSturmIsolateZeros|. It instructs to search roots only in interval
  |(-10^E, 10^E)|. Important: the extremities are \emph{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 \pkg{polexpr} would have
  started with |(-10^6, 10^6)| interval.

  Please note that this feature may be removed or modified.
\end{quote}

\section{Roots of a Chebyshev polynomial}


\begin{everbatim*}
  \poldef T_0(x) := 1;
  \poldef T_1(x) := x;
  \catcode`@ 11
  \count@ 2
  \xintloop
    \poldef T_\the\count@(x) :=
            2x*T_\the\numexpr\count@-1\relax
             - T_\the\numexpr\count@-2\relax;
  \ifnum\count@&lt;15
  \advance\count@ 1
  \repeat
  \catcode`@ 12

  $$T_{15} = \PolTypeset[X]{T_15}$$
  \PolToSturm{T_15}{T_15}
  \PolSturmIsolateZeros*{T_15}% "*" as we will want to confirm multiplicity one
  % takes time (each next decimal digit is obtained by dichotomy)
  \PolEnsureIntervalLengths{T_15}{-20}% ensure 20 decimal digits for each root
\end{everbatim*}

Here is now an example of customization.  Indeed
|\PolPrintIntervals| default uses |array| and thus does not allow page breaks.
And it uses |Left &lt; Z &lt; Right| as presentation of roots and we would like here
rather |Z = decimal expansion...|.
\begin{everbatim*}
% 0.8.6 adds an internal patch which would allow usage of amsmath environments
% like this:
% \def\PolPrintIntervalsBeginEnv{\begin{align*}}
% \def\PolPrintIntervalsEndEnv{\end{align*}}
% (the problem was that align evaluates twice its contents so global variables
%  need a reset at the end of first pass, which is what 0.8.6 took care of)
%
% Let's simply do this:
\def\PolPrintIntervalsBeginEnv{\begingroup\leftskip3cm\relax}
\def\PolPrintIntervalsEndEnv{\par\endgroup}
%
% The rows are separated by \PolPrintIntervalsRowSeparator which defaults to \\
% with LaTeX and \cr with Plain. (prior to 0.8.6 it was hardcoded)
\def\PolPrintIntervalsRowSeparator{\\[\jot]}
% And we enter math mode manually at each row, copying pasting from package
% defaults with some added mathon/mathoff:
\def\PolPrintIntervalsKnownRoot{%
   $\PolPrintIntervalsPrintMultiplicity\quad
   \PolPrintIntervalsTheVar_{\PolPrintIntervalsTheIndex}=%
   \PolPrintIntervalsPrintExactZero
   $
}%
\def\PolPrintIntervalsUnknownRoot{%
   $\PolPrintIntervalsPrintMultiplicity\quad
   \xintifSgn{\PolPrintIntervalsTheLeftEndPoint}%
     {\xintifSgn{\PolPrintIntervalsTheRightEndPoint}
       {\PolPrintIntervalsTheVar_{\PolPrintIntervalsTheIndex}=%
        \PolPrintIntervalsPrintRightEndPoint\dots}%
      {0&gt;\PolPrintIntervalsTheVar_{\PolPrintIntervalsTheIndex}&gt;%
        \PolPrintIntervalsPrintLeftEndPoint}%
      {\impossibleA}}%
    {\xintifSgn{\PolPrintIntervalsTheRightEndPoint}
      {\impossibleB}%
      {\impossibleC}%
      {0&lt;\PolPrintIntervalsTheVar_{\PolPrintIntervalsTheIndex}&lt;%
        \PolPrintIntervalsPrintRightEndPoint}}%
    {\xintifSgn{\PolPrintIntervalsTheRightEndPoint}
      {\impossibleD}%
      {\impossibleE}%
      {\PolPrintIntervalsTheVar_{\PolPrintIntervalsTheIndex}=%
        \PolPrintIntervalsPrintLeftEndPoint\dots}}%
   $
}%
\PolPrintIntervals{T_15}
\end{everbatim*}
\end{document}

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