Contents

- Basic syntax
- Examples of localization of roots
- A typical example
- A degree four polynomial with nearby roots
- The degree nine polynomial with 0.99, 0.999, 0.9999 as triple roots
- A degree five polynomial with three rational roots
- A Mignotte type polynomial
- The Wilkinson polynomial
- The second Wilkinson polynomial
- The degree 41 polynomial with -2, -1.9, -1.8, ..., 0, 0.1, ..., 1.9, 2 as roots
- Roots of Chebyshev polynomials

- Non-expandable macros
`\poldef polname(letter):= expression in letter;``\PolDef[letter]{polname}{expression in letter}``\PolGenFloatVariant{polname}``\PolLet{polname_2}={polname_1}``\PolGlobalLet{polname_2}={polname_1}``\PolAssign{polname}\toarray\macro``\PolGet{polname}\fromarray\macro``\PolFromCSV{polname}{<csv>}``\PolTypeset{polname}``\PolTypeset*{polname}``\PolDiff{polname_1}{polname_2}``\PolDiff[N]{polname_1}{polname_2}``\PolAntiDiff{polname_1}{polname_2}``\PolAntiDiff[N]{polname_1}{polname_2}``\PolDivide{polname_1}{polname_2}{polname_Q}{polname_R}``\PolQuo{polname_1}{polname_2}{polname_Q}``\PolRem{polname_1}{polname_2}{polname_R}``\PolGCD{polname_1}{polname_2}{polname_GCD}``\PolToSturm{polname}{sturmname}``\PolToSturm*{polname}{sturmname}``\PolSetToSturmChainSignChangesAt{\macro}{sturmname}{fraction}``\PolSetToNbOfZerosWithin{\macro}{sturmname}{value_a}{value_b}``\PolSturmIsolateZeros{sturmname}``\PolSturmIsolateZeros*{sturmname}``\PolSturmIsolateZeros**{sturmname}``\PolSturmIsolateZerosAndGetMultiplicities{sturmname}``\PolSturmIsolateZerosGetMultiplicitiesAndRationalRoots{sturmname}``\PolSturmIsolateZerosAndFindRationalRoots{sturmname}``\PolRefineInterval*{sturmname}{index}``\PolRefineInterval[N]{sturmname}{index}``\PolEnsureIntervalLength{sturmname}{index}{E}``\PolEnsureIntervalLengths{sturmname}{E}``\PolPrintIntervals[varname]{sturmname}``\PolPrintIntervals*[varname]{sturmname}``\PolMapCoeffs{\macro}{polname}``\PolReduceCoeffs{polname}``\PolReduceCoeffs*{polname}``\PolMakeMonic{polname}``\PolMakePrimitive{polname}`

- Expandable macros
`\PolEval{polname}\AtExpr{numerical expression}``\PolEval{polname}\At{fraction}``\PolEvalReduced{polname}\AtExpr{numerical expression}``\PolEvalReduced{polname}\At{fraction}``\PolFloatEval{polname}\AtExpr{numerical expression}``\PolFloatEval{polname}\At{fraction}``\PolIfCoeffIsPlusOrMinusOne{A}{B}``\PolLeadingCoeff{polname}``\PolNthCoeff{polname}{number}``\PolDegree{polname}``\PolIContent{polname}``\PolToExpr{polname}``\PolToExpr*{polname}``\PolToFloatExpr{polname}``\PolToFloatExpr*{polname}``\PolToList{polname}``\PolToCSV{polname}``\PolSturmChainLength{sturmname}``\PolSturmIfZeroExactlyKnown{sturmname}{index}{A}{B}``\PolSturmIsolatedZeroLeft{sturmname}{index}``\PolSturmIsolatedZeroRight{sturmname}{index}``\PolSturmIsolatedZeroMultiplicity{sturmname}{index}``\PolSturmNbOfIsolatedZeros{sturmname}``\PolSturmNbOfRationalRoots{sturmname}``\PolSturmNbOfRationalRootsWithMultiplicities{sturmname}``\PolSturmRationalRoot{sturmname}{k}``\PolSturmRationalRootIndex{sturmname}{k}``\PolSturmRationalRootMultiplicity{sturmname}{k}``\PolIntervalWidth{sturmname}{index}`- Expandable macros for use within execution of
`\PolPrintIntervals` `\PolDecToString{decimal number}`

- Booleans (with default setting as indicated)
`\polexprsetup`- Technicalities
- CHANGE LOG
- Acknowledgments

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);

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`.

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.

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 $X_2$!

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

% 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

\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...

\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

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...

\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.

\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.

\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}

This evaluates the

polynomial expressionand stores the coefficients in a private structure accessible later via other package macros, under the user-chosenpolname. Of course theexpressioncan use other previously defined polynomials. Names must start with a letter and are constituted of letters, digits, underscores and (since0.5.1) 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.

As a side effect the function

polname()is recognized as a genuine\xintexpr...\relaxfunction for (exact) numerical evaluation (or within an\xintdefvarassignment.) It computes values not according to the original expression but via the Horner scheme corresponding to the polynomial coefficients.Attention!

Release

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

0.4this 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\poldefor\PolDefbut as a product of the other package macros.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.

Does the same as \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 bex.

Makes the polynomial also usable in the

\xintfloatexprparser. It will therein evaluates via an Horner scheme with coefficients already pre-rounded to the float precision.See also \PolToFloatExpr{polname}.

Attention!

Release

0.3did this automatically on\PolDefand\poldefbut this was removed at0.4for optimization.Any operation, for example generating the derivative polynomial, or dividing two polynomials or using the

\PolLet,mustbe followed by explicit usage of\PolGenFloatVariant{polname}if the new polynomial is to be used in\xintfloatexpror alike context.

Makes a copy of the already defined polynomialpolname_1to a new onepolname_2. Same effect as\PolDef{polname_2}{polname_1(x)}but with less overhead. The=is optional.

Acts globally.

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 ffor non-zero polynomials.- Out-of-range #1's return
0/1[0].See also \PolNthCoeff{polname}{number}. The main difference is that with

\PolAssign,\macrois made a prefix to1 + deg falready defined (hidden to user) macros holding individually the coefficients but \PolNthCoeff{polname}{number} does each time the job to expandably recover theNthcoefficient, 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.)

Does the converse operation to

\PolAssign{polname}\toarray\macro. Each individual\macro{number}gets expanded in an\edefand 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\fooThis will define

fas would have\poldef f(x):=1-2x+5x^2-3x^3;.Note

Prior to

0.5, coefficients were not normalized via\xintRawfor internal storage.

Defines a polynomial directly from the comma separated list of values (or a macro expanding to such a list) of its coefficients, the

first itemgives the constant term, thelast itemgives the leading coefficient, except if zero, then it is dropped (iteratively). List items are each expanded in an\edefand 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.

Note

Prior to

0.5, coefficients were not normalized via\xintRawfor internal storage.

Typesets in descending powers in math mode. It uses letter

xbut this can be changed via an optional argument:\PolTypeset[z]{polname}By default zero coefficients are skipped (issue

\poltypesetalltrueto 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.

Checks if the coefficient is

1or-1and then skips printing the1, 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}.

The default is

\xintSignedFracbut 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: [1]

\renewcommand\PolTypesetOne[1]{\num{\xintPFloat[5]{#1}}} \renewcommand\PolTypesetOne[1]{\num{\xintRound{4}{#1}}}where e.g. we used the

\nummacro ofsiunitxas it understands floating point notation.

[1] the difference in the syntaxes of \xintPFloatand\xintRoundis explained from the fact that\xintPFloatby default uses the prevailing precision hence the extra argument like here5is an optional one.One can also give a try to using \PolDecToString{decimal number} which uses decimal notation (at least for the numerator part).

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

Expands to a+if theraw_coeffis zero or positive, and to nothing ifraw_coeffis negative, as in latter case the\xintSignedFracused 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.

Typesets in ascending powers. Use e.g.[h]optional argument (after the*) to use letterhrather thanx.

This sets

polname_2to the first derivative ofpolname_1. It is allowed to issue\PolDiff{f}{f}, effectively replacingfbyf'.Coefficients of the result

polname_2are irreducible fractions (see Technicalities for the whole story.)

This setspolname_2to theN-th derivative ofpolname_1. Identical arguments is allowed. WithN=0, same effect as\PolLet{polname_2}={polname_1}. With negativeN, switches to using\PolAntiDiff.

This sets

polname_2to the primitive ofpolname_1vanishing at zero.Coefficients of the result

polname_2are irreducible fractions (see Technicalities for the whole story.)

This setspolname_2to the result ofNsuccessive integrations onpolname_1. With negativeN, it switches to using\PolDiff.

This setspolname_Qandpolname_Rto be the quotient and remainder in the Euclidean division ofpolname_1bypolname_2.

This setspolname_Qto be the quotient in the Euclidean division ofpolname_1bypolname_2.

This setspolname_Rto be the remainder in the Euclidean division ofpolname_1bypolname_2.

This setspolname_GCDto be the (monic) GCD of the two first polynomials. It is a unitary polynomial except if bothpolname_1andpolname_2vanish, thenpolname_GCDis the zero polynomial.

With

polnamebeing for exampleP, the macro starts by computing polynomialsPandP', then computes the (opposite of the) remainder in euclidean division, iteratively.The last non-zero remainder

P_N_(whereNis obtainable as \PolSturmChainLength{sturmname}) is up to a factor the GCD ofPandP'hence it is a constant if and only ifPis square-free.Note

- Since
0.5all 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 either1or-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 constant1or-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_0has exactly the same real and complex roots as polynomialpolname, but with each root now of multiplicity one: i.e. it is the "square-free part" of original polynomialpolname.Notice that

sturmname_1isn't necessarily the derivative ofsturmname_0due to the various normalizations.The polynomials

sturmname_kmain utility is for the execution of \PolSturmIsolateZeros{sturmname}. Be careful not to use these namessturmname_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.

Does the same as un-starred version and additionally it keeps for user usage the memory of the

un-normalizedSturm chain polynomialssturmname_k_,k=0,1, ..., N, withNbeing \PolSturmChainLength{sturmname}.Note

This behaviour was modified at

0.6, anyhow the macro was broken at0.5.Hint

The square-free part of

polnameissturmname_0, and their quotient is the polynomial with namesturname_\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 anfas a productc f_1 f_2 f_3 ...of a constant and square-free (primitive) polynomials, where eachf_idivides its predecessor.

Sets macro

\macroto the number of sign changes in the Sturm chain with name prefixsturmname, at locationfraction(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.

Applies the Sturm Theorem to set

\macroto the exact number ofdistinctroots ofsturmname_0in the interval(value_a, value_b](the macro first re-orders the value forvalue_a <= value_bto 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 to have been executed, whereas the expandable variant requires prior execution of \PolSturmIsolateZeros.See also the expandable \PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualTo{value} which requires prior execution of \PolSturmIsolateZeros*.

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 (thesturmnamecan be same as thepolname): it defines polynomialsf_0,f_1, ... havingfhas name prefix.Such a prior call to

\PolToSturmmust have been made at any rate for\PolSturmIsolateZerosto 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}: thenais 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 rootzsuch thata < 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)withEchosen 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)withaan integer, which is neither0nor-1. Hence eitheraanda+1are both positive or they are both negative.One does not

a prioriknow 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.

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

sturmnameisf, one can use the xintexpr variablesfL_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 variablefZ_1_isknownwill have value1if the root in the first interval is known, and0otherwise. 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.) 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}, and for a negative root it is given by PolSturmIsolatedZeroRight{sturmname}{k}. These two decimal numbers are either both zero or both of the same sign.The number of distinct roots is obtainable 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

valueorexpression, 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

exactlythere 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.

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 at1).Furthermore, if for example the

sturmnameisf, xintexpr variablesfM_1,fM_2... hold the multiplicities thus computed.Note

It is

notnecessary 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 asuser-levelgenuine polynomials.See The degree nine polynomial with 0.99, 0.999, 0.9999 as triple roots for an example.

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

rationalroots.Note

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

Furthermore, primitive polynomial

sturmname_sqf_norris created to match the (square-free)sturmname_0from 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_norrissturmname_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.

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

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

This works exactly like \PolSturmIsolateZeros**{sturmname} (inclusive of declaring the polynomials

sturmname_sqf_norrandsturmname_norrwith no rational roots) except that it doesnotcompute the multiplicities of thenon-rationalroots.Note

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

Attention!

This macro does

notdefine xintexpr variablessturmnameM_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* 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

anyrational root anyhow).

Theindex-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 thekth root is then strictly separated from the other roots.

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

Theindex-th interval is subdivided until its length becomes at most10^E. This means (forE<0) that the first-Edigits after decimal mark of thekth root will then be known exactly.

The intervals as obtained from

\PolSturmIsolateZerosare (if necessary) subdivided further by (base 10) dichotomy in order for each of them to have length at most10^E(length will be shorter than10^Ein output only if it did not change or became zero.)This means that decimal expansions of all roots will be known with

-Edigits (forE<0) after decimal mark.

This is a convenience macro which prints the bounds for the roots

Z_1,Z_2, ... (the optional argumentvarnameallows to specify a replacement for the defaultZ). This will be done (by default) in a math modearray, one interval per row, and patternrcccl, 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.

Executed in place of an

arrayenvironment, when there are no real roots. Default definition:\newcommand\PolPrintIntervalsNoRealRoots{}

Default definition:

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

Default definition:

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

Default definition:

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

Default definition:

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

Default definition:

\newcommand\PolPrintIntervalsPrintExactZero{\PolPrintIntervalsTheLeftEndPoint}

Default definition:

\newcommand\PolPrintIntervalsPrintLeftEndPoint{\PolPrintIntervalsTheLeftEndPoint}

Default definition is:

\newcommand\PolPrintIntervalsPrintRightEndPoint{\PolPrintIntervalsTheRightEndPoint}

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:

whose default definition is:

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

It modifies ('in-place': original coefficients get lost) each coefficient of the defined polynomial via the

expandablemacro\macro. The degree is adjusted as necessary if some leading coefficients vanish after the operation. In replacement text of\macro,\indexexpands to the coefficient index (which is defined to be zero for the constant term).Notice that

\macrowill have to handle inputs of the shapeA/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 replacen-th coefficientf_nbyf_n*n^2.

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'.

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], thenA/Bis reduced to smallest terms, but the10^Npart is kept as is. Note: if the polynomial is freshly defined directly via \PolFromCSV its coefficients might still be internally in some format like1.5e7; the macro will anyhow always first do the needed conversion to strict formatA/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.

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.

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

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.

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

Evaluates the polynomial at valuefractionwhich must be in (or expand to) a format acceptable to the xintfrac macros.

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

Evaluates the polynomial at valuefractionwhich must be in (or expand to) a format acceptable to the xintfrac macros, and produce an irreducible fraction.

Boils down to

\xintthefloatexpr polname(numerical expression)\relax.This is done via a Horner Scheme (see \poldef and \PolGenFloatVariant{polname}), with already rounded coefficients. [2] To use the

exact coefficientswithexactly executedadditions and multiplications, just insert it in the float expression as in this example: [3]\xintthefloatexpr 3.27*\xintexpr f(2.53)\relax^2\relaxThe

f(2.53)is exactly computed then rounded at the time of getting raised to the power2. Moving the^2inside, that operation would also be treated exactly.

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

[3] The \xintexprhere could be\xinttheexprbut that would be less efficient. Cf. xintexpr documentation about nested expressions.

Evaluates the polynomial at valuefractionwhich must be in (or expand to) a format acceptable to the xintfrac macros, and produces a floating point number.

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

\cdotin front of\PolVar^{\PolIndex}if the coefficient is not plus or minus one.The macro will execute

Aif the coefficient has been found to be plus or minus one, andBif not.

Expands to the leading coefficient.

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

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

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.

Expands [4] to

coeff_N*x^N+...(descending powers.)

[4] in a \write,\edef, or\csname...\endcsname, but not under\romannumeral-`0.By default zero coefficients are skipped (issue

\poltoexpralltrueto 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 asx, notx^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 simplef(x)is more efficient for the identical result.

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

1not be printed explicitely (except of course for the constant term). Also by default, the monomial of degree one isxnotx^1, andx^0is skipped.For compatibility with Maple input requirements, by default a

*always precedes thex^number, except if the coefficient is a one or a minus one. See \PolToExprTimes.

Holds the default package meaning of \PolToExprOneTerm{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+1issue

\let\PolToExprOneTerm\PolToExprOneTermStyleBbefore usage of\PolToExpr. Note that then\PolToExprCmdisn't used at all. To revert to package default, issue\let\PolToExprOneTerm\PolToExprOneTermStyleA.To suppress the

*'s, cf. \PolToExprTimes.

It is the one-argument macro used by the package definition of\PolToExprOneTermfor 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.

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

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 isx.

This expands to the symbol used for multiplication of anx^{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).

Expands tocoeff_0+coeff_1*x+coeff_2*x^2+...(ascending powers). Customizable like \PolToExpr{polname} via the same macros.

Similar to \PolToExpr{polname} but uses \PolToFloatExprCmd 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

\xintfloatexprpolynomial function hence\PolToFloatExprCmdoperates from theexactcoefficients 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

\xintfloatexprand those output by\PolToFloatExpr{polname}.

Similar to \PolToExprOneTerm. But does not treat especially coefficients equal to plus or minus one.

It is the one-argument macro used by

\PolToFloatExprOneTerm. Its package definition is\xintFloat{#1}.Caution!

Currently (xint

1.3c)\xintFloat{0}outputs0.e0which is perfectly acceptable input for Python, but not for Maple. Thus, one should better leave the \poltoexprallfalse toggle to its default\iffalsestate, 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

\xintiiifZeroand not\xintifZerois only for optimization (I can't help it) because#1is known to be inxintfracraw format.

Typesets in ascending powers.

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

Expands tocoeff_0, coeff_1, coeff_2, ....., coeff_N, starting with constant term and ending with leading coefficient. Converse to \PolFromCSV.

Returns the integer

Nsuch thatsturmname_Nis the last one in the Sturm chainsturmname_0,sturmname_1, ...

Executes

Aif theindex-th interval reduces to a singleton, i.e. the root is known exactly, elseB.Note

indexis allowed to be something like1+2*3as it is fed to\the\numexpr...\relax.

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}.

The value is pre-formatted using \PolDecTostring.

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.

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.

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.

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 apolname(this is why the macro contains Sturm in its name), simply to be reminded of the above constraint.

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.

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.

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.

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

Attention!

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

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

Attention!

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

Expands to the

kth rational root (they are ordered and indexed starting at 1 for the most negative).Attention!

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

Expands to

indexof thekth 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.

Expands to the multiplicity of the

kth rational root.Attention!

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

The10^Ewidth of the currentindex-th root localization interval. Output is in xintfrac raw1/1[E]format (if not zero).

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}`.

Expands to the name (defaultZ) used for representing the roots, which was passed as optional argumentvarnameto \PolPrintIntervals[varname]{sturmname}.

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

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

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

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

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

This is a utility macro to print decimal numbers. It has been backported to xintfrac (release

1.3of2018/03/01) under the name\xintDecToString, and thepolexprmacro is simply now an alias to it.For example

\PolDecToString{123.456e-8}will expand to0.00000123456and\PolDecToString{123.450e-8}to0.00000123450which 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.

This is actually an xintexpr configuration. Setting it to

truetriggers 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.

Iftrue, \PolTypeset{polname} will also typeset the vanishing coefficients.

Iftrue, \PolToExpr{polname} and \PolToFloatExpr{polname} will also include the vanishing coefficients in their outputs.

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

norr: the postfix that \PolSturmIsolateZeros**{sturmname} should append tosturmnameto 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 tosturmnameto 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.

The catcode of the semi-colon is reset temporarily by \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)} 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 [5], 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.[5] 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.

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

- The \poldef parser itself,
- Differentiation and anti-differentiation,
- Euclidean division and GCDs,
- Various utilities such as \PolFromCSV, \PolMapCoeffs, \PolToCSV, \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!

- Fix:
v0.3 (2018/01/17)

bug fixes:

the

`0.1`\PolEval 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} syntax is needed for using expressions in the second argument.

incompatible or breaking changes:

- \PolToExpr now by default uses
*descending*powers (it also treats differently coefficients equal to 1 or -1.) Use \PolToExpr* for*ascending*powers. - \PolEval 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.

- \PolToExpr now by default uses
new (or newly documented) macros:

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 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 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.
- \PolGCD was buggy in case of first polynomial being of lesser degree than the second one.

breaking changes:

formerly \PolEval{P}\At{foo} 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} 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 and \PolFloatEval.

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.

new non-expandable macros:

- \PolGenFloatVariant
- \PolGlobalLet
- \PolTypesetOne
- \PolQuo
- \PolRem
- \PolToSturm
- \PolToSturm*
- \PolSetToSturmChainSignChangesAt
- \PolSetToNbOfZerosWithin
- \PolSturmIsolateZeros
- \PolRefineInterval*
- \PolRefineInterval[N]
- \PolEnsureIntervalLength
- \PolEnsureIntervalLengths
- \PolPrintIntervals
- \PolPrintIntervalsPrintExactZero
- \PolPrintIntervalsPrintLeftEndPoint
- \PolPrintIntervalsPrintRightEndPoint
- \PolReduceCoeffs*
- \PolMakeMonic

new expandable macros:

- \PolToExprOneTermStyleA
- \PolIfCoeffIsPlusOrMinusOne
- \PolLeadingCoeff
- \PolSturmChainLength
- \PolSturmNbOfIsolatedZeros
- \PolSturmIfZeroExactlyKnown
- \PolSturmIsolatedZeroLeft
- \PolSturmIsolatedZeroRight
`\PolPrintIntervalsTheEndPoint`(removed at 0.7)- \PolPrintIntervalsTheIndex
`\PolIfEndPointIsPositive`(removed at 0.7)`\PolIfEndPointIsNegative`(removed at 0.7)`\PolIfEndPointIsZero`(removed at 0.7)- \PolIntervalWidth
- \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.

- \PolGet{polname}\fromarray\macro crashed when
- 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
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:
- new expandable macro:

- bug fixes:
v0.5.1 (2018/04/22)

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

- the character

- new feature:
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 with a constant polynomial caused a division by zero error.

- new macro:
- \PolSturmIsolateZeros* acts like the non-starred variant 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}

- bugfix:
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}`.

- although \PolPrintIntervals[varname]{sturmname} default output
remains the same, some auxiliary macros for user-customization
have been removed:
- 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*
- \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

- breaking changes:
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).

- Debugging information not destined to user showed in log if root
finding was done under

- bugfix:
v0.7.4 (2019/02/12)

- bugfix:
- 20000000000 is too big for
`\numexpr`, shouldn't I know that? Thanks to Jürgen Gilg for report.

- 20000000000 is too big for

- bugfix:
v0.7.5 (2020/01/31)

Synced with xint 1.4. Requires it.

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.