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-.. comment: -*- fill-column: 72; mode: rst; -*-
-
-===============================
- Package polexpr documentation
-===============================
-
-0.7.5 (2020/01/31)
-==================
-
-.. 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)
-
- * bugfix:
-
- - 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).
-
-- v0.7.4 (2019/02/12)
-
- * bugfix:
-
- - 20000000000 is too big for ``\numexpr``, shouldn't I know that?
- Thanks to Jürgen Gilg for report.
-
-- v0.7.5 (2020/01/31)
-
- Synced with xint 1.4. Requires it.
-
-
-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