A typical example

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

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

\PolToSturm{f}{f}
\PolSturmIsolateZeros{f}
The \PolTypeset{f} polynomial has \PolSturmNbOfIsolatedZeros{f} distinct real
roots which are located in the following intervals:
\PolPrintIntervals{f}
Here is the second root with ten more decimal digits:
\PolRefineInterval[10]{f}{2}
\[\PolSturmIsolatedZeroLeft{f}{2}<Z_2<\PolSturmIsolatedZeroRight{f}{2}\]
And here is the first root with twenty digits after decimal mark:
\PolEnsureIntervalLength{f}{1}{-20}
\[\PolSturmIsolatedZeroLeft{f}{1}<Z_1<\PolSturmIsolatedZeroRight{f}{1}\]
The first element of the Sturm chain has degree $\PolDegree{f_0}$. As
this same as $\PolDegree{f}$ we know that the latter was square free.
So the derivative is up to a constant \PolTypeset{f_1} (in fact here
it is identical with it).
\PolToSturm{f_1}{f_1}\PolSturmIsolateZeros{f_1}%
It has \PolSturmNbOfIsolatedZeros{f_1} distinct real
roots:
\PolPrintIntervals[W]{f_1}
\PolEnsureIntervalLengths{f_1}{-10}%
Here they are with ten digits after decimal mark:
\PolPrintIntervals[W]{f_1}
\PolDiff{f_1}{f''}
\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

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

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

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

A Mignotte type polynomial

\PolDef{P}{x^10 - (10x-1)^2}%
\PolTypeset{P}              % prints it in expanded form
\PolToSturm{P}{P}           % we can use same prefix for Sturm chain
\PolSturmIsolateZeros{P}    % finds 4 real roots
This polynomial has \PolSturmNbOfIsolatedZeros{P} distinct real roots:
\PolPrintIntervals{P}%
% reports  -2 < Z_1 < -1, 0 < Z_2 < 0.1, 0.1 < Z_3 < 0.2, 1 < Z_4 < 2
Let us refine the second and third intervals to separate the corresponding
roots:
\PolRefineInterval*{P}{2}% will refine to 0.0999990 < Z_2 < 0.0999991
\PolRefineInterval*{P}{3}% will refine to 0.100001 < Z_3 < 0.100002
\PolPrintIntervals{P}%
Let us now get to know all roots with 10 digits after decimal mark:
\PolEnsureIntervalLengths{P}{-10}%
\PolPrintIntervals{P}% now all roots are known 10 decimal digits after mark
Finally, we display 20 digits of the second root:
\PolEnsureIntervalLength{P}{2}{-20}% makes Z_2 known with 20 digits after mark
\[\PolSturmIsolatedZeroLeft{P}{2}<Z_2<\PolSturmIsolatedZeroRight{P}{2}\]

The last line produces:

0.09999900004999650028 < Z_2 < 0.09999900004999650029

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

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

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

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

we get this expanded form:

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

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

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

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

\PolSturmIsolateZeros[1]{S}

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}

\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.

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[letter]{polname}{expression in letter}

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 be x.

\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{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{polname_2}={polname_1}

Acts globally.

\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{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{polname}{<csv>}

Defines a polynomial directly from the comma separated list of values (or a macro expanding to such a list) of its coefficients, the constant term being the first item. No validity checks. Spaces from the list argument are trimmed. List items are each expanded in an \edef and then put into normalized form via xintfrac's macro \xintRaw.

Leading zero coefficients are removed:

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

defines the zero polynomial, which has only one (zero) coefficient.

See also expandable macro \PolToCSV.

Note

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

\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.

\PolTypeset*{polname}

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

\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]{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{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]{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{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{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{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{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.

\PolToSturm{polname}{sturmname}

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

Attention!

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

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

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

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

\PolToSturm*{polname}{sturmname}

Does not divide the Sturm chain by its last element.

\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{\macro}{sturmname}{value_a}{value_b}

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

Note

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

This will presumably get added in future.

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

\PolSturmIsolateZeros{sturmname}

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

Important

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

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

The interval boundaries are decimal numbers, originating in iterated decimal subdivision from initial intervals (-10^E, 0) and (0, 10^E); if zero is a root it is always identified individually. The non-singleton intervals are of the type (a/10^f, (a+1)/10^f) with a an integer, which is neither 0 nor -1. Hence a and a+1 are both positive or both negative.

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

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

Important

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

Note

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

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

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

Note

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

Note

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

Note

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

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

\PolRefineInterval*{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 kth root is then strictly separated from the other roots.

\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{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 kth root will then be known exactly.

\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[varname]{sturmname}

This is a convenience macro which prints the bounds for the roots Z_1, Z_2, ... (the optional argument varname allows to specify a replacement for the default Z). This will be done in a math mode array, one interval per row, and pattern rcccl, where the second and fourth column hold the < sign, except when the interval reduces to a singleton, which means the root is known exactly. The user is invited to renewcommand the macro if some other type of tabular environment for example is wanted.

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

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

\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{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*{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 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.

\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{polname}

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.

\PolEval{polname}\AtExpr{numerical expression}

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

\PolEval{polname}\At{fraction}

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

Attention!

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

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

\PolEvalReduced{polname}\AtExpr{numerical expression}

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

\PolEvalReduced{polname}\At{fraction}

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

Attention!

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

\PolFloatEval{polname}\AtExpr{numerical expression}

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 coefficients with exactly executed additions and multiplications, just insert it in the float expression as in this example: [3]

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

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

Attention!

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

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

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

\PolFloatEval{polname}\At{fraction}

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

Attention!

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

\PolIfCoeffIsPlusOrMinusOne{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{polname}

Expands to the leading coefficient.

\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{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{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.

\PolToExpr{polname}

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 \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.

\PolToExpr*{polname}

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

\PolToFloatExpr{polname}

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 \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}.

\PolToFloatExpr*{polname}

Typesets in ascending powers.

\PolToList{polname}

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

\PolToCSV{polname}

Expands to coeff_0, coeff_1, coeff_2, ....., coeff_N. Converse to \PolFromCSV.

\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{sturmname}{index}{A}{B}

Executes A if the indexth interval reduces to a singleton, i.e. the root is known exactly, else B.

Note

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

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

\PolSturmIsolatedZeroLeft{sturmname}{index}

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

\PolSturmIsolatedZeroRight{sturmname}{index}

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

\PolSturmNbOfIsolatedZeros{sturmname}

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

\PolIntervalWidth{sturmname}{index}

The 10^E width of the current indexth root localization interval. Output is in xintfrac raw 1/1[E] format (if not zero).

Macros for use within execution of \PolPrintIntervals

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

\PolDecToString{decimal number}

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

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

The exact behaviour of this macro may evolve in future releases of xint.

\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.