polexpr reference

Warning about unstability of the new syntax

Infix operators +, -, *, /, **, ^

As has been explained in the Syntax overview via examples section these infix operators have been made polynomial aware, not only in the \poldef context, but generally in any \xintexpr/\xinteval context, inclusive of \xintdeffunc.

Conversely functions declared via \xintdeffunc and making use of these operators will automatically be able to accept polynomials declared from \poldef as variables.

Usage of / for euclidean division of polynomials is deprecated. Only in case of a scalar denominator is it to be considered stable. Please use rather quo().

Experimental infix operators //, /:

Here is the tentative behaviour of A//B according to types:

• A non scalar and B non scalar: euclidean quotient,

• A scalar and B scalar: floored division,

• A scalar and B non scalar: produces zero,

• A non scalar and B scalar: coefficient per coefficient floored division.

This is an experimental overloading of the // and /: from \xintexpr.

The behaviour in the last case, but not only, is to be considerd unstable. The alternative would be for A//B with B scalar to act as quo(A,B). But, we have currently chosen to let //B for a scalar B act coefficient-wise on the numerator. Beware that it thus means it can be employed with the idea of doing euclidean division only by checking that B is non-scalar.

The /: operator provides the associated remainder so always A is reconstructed from (A//B)*B + A/:B.

If : is active character use /\string: (it is safer to use /\string : if it is not known if : has catcode other, letter, or is active, but note that /: is fine and needs no precaution if : has catcode letter, it is only an active : which is problematic, like for all other characters possibly used in an expression).

UNSTABLE

As explained above, there are (among other things) hesitations about behaviour with pol2 a scalar.

Comparison operators <, >, <=, >=, ==, !=

NOT YET IMPLEMENTED

As the internal representation by xintfrac and xintexpr of fractions does not currently require them to be in reduced terms, such operations would be a bit costly as they could not benefit from the \pdfstrcmp engine primitive. In fact xintexpr does not use it yet anywhere, even for normalized pure integers, although it could speed up signifcantly certain aspects of core arithmetic.

Equality of polynomials can currently be tested by computing the difference, which is a bit costly. And of course the deg() function allows comparing degrees. In this context note the following syntax:

(deg(Q)) ?? { zero } { non-zero scalar } { non-scalar }

for branching.

pol(<nutple expression>)

This converts a nutple [c0,c1,...,cN] into the polynomial variable having these coefficients. Attention that the square brackets are mandatory, except of course if the argument is actually an expression producing such a “nutple”.

Currently, this process will not normalize the coefficients (such as reducing to lowest terms), it only trims out the leading zero coefficients.

Inside \xintexpr, this is the only (allowed) way to create ex nihilo a polynomial variable; inside \poldef it is an alternative input syntax which is more efficient than the input c0 + c1 * x + c2 * x^2 + ....

lpol(<nutple expression>)

This converts a nutple [cN,...,c1,c0] into the polynomial variable having these coefficients, with leading coefficients coming first in the input. Attention that the square brackets are mandatory, except of course if the argument is actually an expression producing such a “nutple”.

Currently, this process will not normalize the coefficients (such as reducing to lowest terms), it only trims out the leading zero coefficients.

NAME UNSTABLE

It can be used in \poldef as an alternative input syntax, which is more efficient than using the algebraic notation with monomials.

(new with 0.8.1, an empty nutple will cause breakage)

\xinteval{<pol. expr.>}

This is documented here for lack of a better place: it evaluates the polynomial expression then outputs the “string” pol([c0, c1, ..., cN]) if the degree N is at least one (and the usual scalar output else).

The “pol” word uses letter catcodes, which is actually mandatory for this output to be usable as input, but it does not make sense to use this inside \poldef or \xintexpr at it means basically executing pol(coeffs(..expression..)) which is but a convoluted way to obtain the same result as (..expression..) (the parentheses delimiting the polynomial expression).

For example, \xinteval{(1+pol([0,1]))^10} expands (in two steps) to:

pol([1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1])

You do need loading polexpr for this, else of course pol([]) remains unknown to \xinteval{} as well as the polynomial algebra ! This example can also be done as \xinteval{subs((1+x)^10,x=pol([0,1]))}.

I hesitated using as output the polynomial notation as produced by \PolToExpr{}, but finally opted for this.

evalp(<pol. expr.>, <pol. expr>)

Evaluates the first argument as a polynomial function of the second. Usually the second argument will be scalar, but this is not required:

\poldef K(x):= evalp(-3x^3-5x+1,-27x^4+5x-2);

If the first argument is an already declared polynomial P, use rather the functional form P() (which can accept a numerical as well as polynomial argument) as it is more efficient.

One can also use subs() syntax [1] (see xintexpr documentation):

\poldef K(x):= subs(-3y^3-5y+1, y = -27x^4+5x-2);

but the evalp() will use a Horner evaluation scheme which is usually more efficient.

name unstable

poleval? evalpol? peval? evalp? value? eval? evalat? eval1at2? evalat2nd?

Life is so complicated when one asks questions. Not everybody does, though, as is amply demonstrated these days.

syntax unstable

I am hesitating about permuting the order of the arguments.

deg(<pol. expr.>)

Computes the degree.

coeffs(<pol. expr.>)

Produces the nutple [c0,c1,...,cN] of coefficients. The highest degree coefficient is always non zero (except for the zero polynomial…).

name unstable

I am considering in particular using polcoeffs() to avoid having to overload coeffs() in future when matrix type will be added to xintexpr.

lcoeffs(<pol. expr.>)

Produces the nutple [cN,....,c1,c0] of coefficients, starting with the highest degree coefficient.

(new with 0.8.1)

coeff(<pol. expr.>, <num. expr.>)

As expected. Produces zero if the numerical index is negative or higher than the degree.

name, syntax and output unstable

I am hesitating with coeff(n,pol) syntax and also perhaps using polcoeff() in order to avoid having to overload coeff() when matrix type will be added to xintexpr.

The current behaviour is at odds with legacy \PolNthCoeff{<polname>}{<index>} regarding negative indices. Accessing leading or sub-leading coefficients can be done with other syntax, see lc(<pol. expr.>), and in some contexts it is useful to be able to rely on the fact that coefficients with negative indices do vanish, so I am for time being maintaining this.

lc(<pol. expr.>)

The leading coefficient. The same result can be obtained from coeffs(pol)[-1], which shows also how to generalize to access sub-leading coefficients. See the xintexpr documentation for Python-like indexing syntax.

monicpart(<pol. expr.>)

Divides by the leading coefficient, except that monicpart(0)==0.

unstable

Currently the coefficients are reduced to lowest terms (contrarily to legacy behaviour of \PolMakeMonic), and additionally the xintfrac \xintREZ macro is applied which extracts powers of ten from numerator or denominator and stores them internally separately. This is generally beneficial to efficiency of multiplication.

cont(<pol. expr.>)

The (fractional) greatest common divisor of the polynomial coefficients. It is always produced as an irreducible (non-negative) fraction. According to Gauss theorem the content of a product is the product of the contents.

name and syntax unstable

At 0.8 it was created as icontent() to match the legacy macro \PolIContent, whose name in 2018 was chosen in relation to Maple’s function icontent(), possibly because at that time I had not seen that Maple also had a content() function. Name changed at 0.8.1.

It will change syntax if in future multivariate polynomials are supported, and icontent() will then make a come-back.

primpart(<pol. expr.>)

The quotient (except for the zero polynomial) by cont(<pol. expr.>). This is thus a polynomial with integer coefficients having 1 as greatest common divisor. The sign of the leading coefficient is the same as in the original.

And primpart(0)==0.

The trailing zeros of the integer coefficients are extracted into a power of ten exponent part, in the internal representation.

quorem(<pol. expr.>, <pol. expr.>)

Produces a nutple [Q,R] with Q the euclidean quotient and R the remainder.

name unstable

poldiv()?

quo(<pol. expr.>, <pol. expr.>)

The euclidean quotient.

The deprecated pol1/pol2 syntax computes the same polynomial.

rem(<pol. expr.>, <pol. expr.>)

The euclidean remainder. If pol2 is a (non-zero) scalar, this is zero.

There is no infix operator associated to this, for lack of evident notation. Please advise.

/: can be used if one is certain that pol2 is of degree at least one. But read the warning about it being unstable even in that case.

prem(<pol. expr. 1>, <pol. expr. 2>)

Produces a nutple [m, spR] where spR is the (special) pseudo Euclidean remainder. Its description is:

• the standard euclidean remainder R is spR/m

• m = b^f with b equal to the absolute value of the leading coefficient of pol2,

• f is the number of non-zero coefficients in the euclidean quotient, if deg(pol2)>0 (even if the remainder vanishes).

  If pol2 is a scalar however, the function outputs [1,0].

With these definitions one can show that if both pol1 and pol2 have integer coefficients, then this is also the case of spR, which makes its interest (and also m*Q has integer coefficients, with Q the euclidean quotient, if deg(pol2)>0). Also, prem() is computed faster than rem() for such integer coefficients polynomials.

Hint

If you want the euclidean quotient R evaluated via spR/m (which may be faster, even with non integer coefficients) use subs(last(x)/first(x),x=prem(P,Q)) syntax as it avoids computing prem(P,Q) twice. This does the trick both in \poldef or in \xintdefvar.

However, as is explained in the xintexpr documentation, using such syntax in an \xintdeffunc is (a.t.t.o.w) illusory, due to technicalities of how subs() gets converted into nested expandable macros. One needs an auxiliary function like this:

\xintdeffunc lastoverfirst(x):=last(x)/first(x);
\xintdeffunc myR(x)=lastoverfirst(prem(x));

Then, myR(pol1,pol2) will evaluate prem(pol1,pol2) only once and compute a polynomial identical to the euclidean remainder (internal representations of coefficients may differ).

In this case of integer coefficients polynomials, the polexpr internal representation of the integer coefficients in the pseudo remainder will be with unit denominators only if that was already the case for those of pol1 and pol2 (no automatic reduction to lowest terms is made prior or after computation).

Pay attention here that b is the absolute value of the leading coefficient of pol2. Thus the coefficients of the pseudo-remainder have the same signs as those of the standard remainder. This diverges from Maple’s function with the same name.

divmod(<pol. expr. 1>, <pol. expr. 2>)

Overloads the scalar divmod() and associates it with the experimental // and /: as extended to the polynomial type.

In particular when both pol1 and pol2 are scalars, this is the usual divmod() (as in Python) and for pol1 and pol2 non constant polynomials, this is the same as quorem().

Highly unstable overloading of \xinteval‘s divmod().

mod(<pol. expr. 1>, <pol. expr. 2>)

The R of the divmod() output. Same as R of quorem() when the second argument pol2 is of degree at least one.

Highly unstable overloading of \xinteval‘s mod().

polgcd(<pol. expr. 1>, <pol. expr. 2>, ...)

Evaluates to the greatest common polynomial divisor of all the polynomial inputs. The output is a primitive (in particular, with integer coefficients) polynomial. It is zero if and only if all inputs vanish.

Attention, there must be either at least two polynomial variables, or alternatively, only one argument which then must be a bracketed list or some expression or variable evaluating to such a “nutple” whose items are polynomials (see the documentation of the scalar gcd() in xintexpr).

The two variable case could (and was, during development) have been defined at user level like this:

\xintdeffunc polgcd_(P,Q):=
  (deg(Q))??{P}{1}{polgcd_(Q,primpart(last(prem(P,Q))))};
\xintdeffunc polgcd(P,Q):=polgcd_(primpart(P),primpart(Q));%

This is basically what is done internally for two polynomials, up to some internal optimizations.

UNSTABLE

I hesitate between returning a primitive or a monic polynomial. Maple returns a primitive polynomial if all inputs [2] have integer coefficients, else it returns a monic polynomial, but this is complicated technically for us to add such a check and would add serious overhead.

Internally, computations are done using primitive integer-coefficients polynomials (as can be seen in the function template above). So I decided finally to output a primitive polynomial, as one can always apply monicpart() to it.

Attention that this is at odds with behaviour of the legacy \PolGCD (non expandable) macro.

resultant(<pol. expr. 1>, <pol. expr. 2>)

The resultant.

NOT YET IMPLEMENTED

disc(<pol. expr.>)

The discriminant.

NOT YET IMPLEMENTED

polpowmod(<pol. expr. 1>, <num. expr.>, <pol. expr. 2>)

Modular exponentiation: mod(pol1^N, pol2) in a more efficient manner than first computing pol1^N then reducing modulo pol2.

Attention that this is using the mod() operation, whose current experimental status is as follows:

• if deg(pol2)>0, the euclidean remainder operation,

• if pol2 is a scalar, coefficient-wise reduction modulo pol2.

UNSTABLE

This is currently implemented at high level via \xintdeffunc and recursive definitions, which were copied over from a scalar example in the xintexpr manual:

\xintdeffunc polpowmod_(P, m, Q) :=
       isone(m)?
           % m=1: return P modulo Q
           {   mod(P,Q)  }
           % m > 1: test if odd or even and do recursive call
           {   odd(m)? {  mod(P*sqr(polpowmod_(P, m//2, Q)), Q) }
                       {  mod(  sqr(polpowmod_(P, m//2, Q)), Q) }
            }
         ;%
\xintdeffunc polpowmod(P, m, Q) := (m)?{polpowmod_(P, m, Q)}{1};%

Negative exponents are not currently implemented.

For example:

\xinteval{subs(polpowmod(1+x,100,x^7),x=pol([0,1]))}
\xinteval{subs(polpowmod(1+x,20,10), x=pol([0,1]))}

produce respectively:

pol([1, 100, 4950, 161700, 3921225, 75287520, 1192052400])
pol([1, 0, 0, 0, 5, 4, 0, 0, 0, 0, 6, 0, 0, 0, 0, 4, 5, 0, 0, 0, 1])

rdcoeffs(<pol. expr.>)

This operates on the internal representation of the coefficients, reducing them to lowest terms.

name HIGHLY undecided

rdzcoeffs(<pol. expr.>)

This operates on the internal representation of the coefficients, reducing them to lowest terms then extracting from numerator or denominator the maximal power of ten to store as a decimal exponent.

This is sometimes favourable to more efficient polynomial algebra computations.

name HIGHLY undecided

diff1(<pol. expr.>)

The first derivative.

name UNSTABLE

This name may be used in future to be the partial derivative with respect to a first variable.

diff2(<pol. expr.>)

The second derivative.

name UNSTABLE

This name may be used in future to be the partial derivative with respect to a second variable.

diffn(<pol. expr. P>, <num. expr. n>)

The nth derivative of P. For n<0 computes iterated primitives vanishing at the origin.

The coefficients are not reduced to lowest terms.

name and syntax UNSTABLE

I am also considering reversing the order of the arguments.

antider(<pol. expr. P>)

The primitive of P with no constant term. Same as diffn(P,-1).

intfrom(<pol. expr. P>, <pol. expr. c>)

The primitive of P vanishing at c, i.e. \int_c^x P(t)dt.

Also c can be a polynomial… so if c is monomial x this will give zero!

UNSTABLE

Allowing general polynomial variable for c adds a bit of overhead to the case of a pure scalar. So I am hesitating maintaining this feature whose interest appears dubious.

Attention

As the two arguments are both allowed to be polynomials, if by inadvertance one exchanges the two, there is no error but the meaning of intfrom(c,P) is completely otherwise, as it produces c*(x - P) if c is a scalar:

>>> &pol
pol mode (i.e. function definitions use \poldef)
>>> P(x):=1+x^2;
P = x^2+1
    --> &GenFloat(P) lets P become usable as function in fp mode
    --> &ROOTS(P) (resp. &ROOTS(P,N)) finds all rational roots exactly and
        all irrational roots with at least 10 (resp. N) fractional digits
>>> intfrom(P,1);
@_1     pol([-4/3, 1, 0, 1/3])
>>> intfrom(1,P);
@_2     pol([-1, 1, -1])
>>> &bye

integral(<pol. expr. P>, [<pol. expr. a>, <pol. expr. b>])

\int_a^b P(t)dt.

Warning

The brackets here are not denoting an optional argument but a mandatory nutple argument [a, b] with two items. No real recoverable-from error check is done on the input syntax. The input can be an xintexpr variable which happens to be a nutple with two items, or any expression which evaluates to such a nutple.

a and b are not restricted to be scalars, they are allowed to be themselves polynomial variables or even polynomial expressions.

To compute \int_{x-1}^x P(t)dt it is more efficient to use intfrom(x-1).

Similary to compute \int_x^{x+1} P(t)dt, use -intfrom(x+1).

UNSTABLE

Am I right to allow general polynomials a and b hence add overhead to the pure scalar case ?

\poldef polname(letter):= expression using the letter as indeterminate;

This evaluates the polynomial expression and stores the coefficients in a private structure accessible later via other package macros, used with argument polname. Of course the expression can make use of previously defined polynomials.

Polynomial names must start with a letter and are constituted of letters, digits, underscores, the @ (see Technicalities) and the right tick '.

The whole xintexpr syntax is authorized, as long as the final result is of polynomial type:

\poldef polname(z) := add((-1)^i z^(2i+1)/(2i+1)!, i = 0..10);

With fractional coefficients, beware the tacit multiplication issue.

Furthermore:

• a variable polname is defined which can be used in \poldef as well as in \xinteval for algebraic computations or as argument to polynomial aware functions,

• a function polname() is defined which can be used in \poldef as well as in \xinteval. It accepts there as argument scalars and also other polynomials (via their names, thanks to previous item).

Any function defined via \xintdeffunc and only algebraic operations, as well as ople indexing or slicing operations, should work fine in \xintexpr/\xinteval with such polynomial names as argument.

In the case of a constant polynomial, the xintexpr variable (not the internal data structure on which the package macros operate) associated to it is indistinguishable from a scalar, it is actually a scalar and has lost all traces from its origins as a polynomial (so for example can be used as argument to the cos() function). The function on the other hand remains a one-argument function, which simply has a constant value.

Attention

The function polname() is defined only for \xintexpr/\xinteval context. It will be unknown to \xintfloateval.

Worse, a previously existing floating point function of the same name will be made 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 result of usage of the other package macros.

See \PolGenFloatVariant{<polname>} to generate a function usable in \xintfloateval.

Attention

Using the variable mypol inside \xintfloateval will generate low-level errors because the infix operators there are not polynomial-aware, and the polynomial specific functions such as deg() are only defined for usage inside \xintexpr.

In short, currently polynomials defined via polexpr can be used in floating point context only for numerical evaluations, via functions obtained from \PolGenFloatVariant{<polname>} usage.

Changes to the original polynomial via package macros are not automatically mapped to the numerical floating point evaluator which must be manually updated as necessary when the original rational coefficient polynomial is modified.

The original expression is lost after parsing, and in particular the package provides no way to typeset it (of course the package provides macros to typeset the computed polynomial). Typesetting the original expression has to be done manually, if needed.

\PolDef[<letter>]{<polname>}{<expr. using the letter as indeterminate>}

Does the same as \poldef in an undelimited macro format, the main interest is to avoid potential problems with the catcode of the semi-colon in presence of some packages. In absence of a [<letter>] optional argument, the variable is assumed to be x.

\PolGenFloatVariant{}

Syntax: \PolGenFloatVariant{<polname>}

Makes the polynomial also usable in the \xintfloatexpr/\xintfloateval parser. It will therein evaluates via an Horner scheme using polynomial coefficients already pre-rounded to the float precision.

See also \PolToFloatExpr{<pol. expr.>}.

Attention

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 \xintfloateval.

\PolTypeset[]{}

Syntax: \PolTypeset[<letter>]{<pol. expr.>}

Typesets in descending powers, switching to math mode if in text mode, after evaluating the polynomial expression:

\PolTypeset{mul(x-i,i=1..5)}% possible since polexpr 0.8

The letter used in the input is by default assumed to be x, but can be modified by a redefinition of \PolToExprInVar.

The letter used in the output is also by default x. This one can be changed on-the-fly via the optional <letter>:

\PolTypeset[z]{polname or polynomial expression}

By default zero coefficients are skipped (use \poltypesetalltrue to get all of them in output).

The following macros (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*[]{}

Syntax: \PolTypeset*[<letter>]{<pol. expr.>}

Typesets in ascending powers. The <letter> optional argument (after the *) declares the letter to use in the output. As for \PolTypeset, it defaults to x.

To modify the expected x in the input, see \PolToExprInVar.

Extended at 0.8 to accept general expressions and not only polynomial names.

\PolLet{}={}

Syntax: \PolLet{<polname_2>}={<polname_1>}

Makes a copy of the already defined polynomial polname_1 to a new one polname_2. This has the same effect as \PolDef{<polname_2>}{<polname_1>(x)} or (better) \PolDef{<polname_2>}{<polname_1>} but with less overhead. The = is optional.

\PolGlobalLet{}={}

Syntax: \PolGlobalLet{<polname_2>}={<polname_1>}

Acts globally.

\PolAssign{}\toarray{}

Syntax: \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. This is an unfortunate legacy situation related to the original indexing convention in xinttools arrays.

• 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>}{<index>}.

\PolGet{}\fromarray{}

Syntax: \PolGet{<polname>}\fromarray{<\macro>}

Does the converse operation to \PolAssign{<polname>}\toarray\macro. Each individual \macro{<value>} 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;.

\PolFromCSV{}{}

Syntax: \PolFromCSV{<polname>}{<csv>}

Defines a polynomial directly from the comma separated list of values (or a macro expanding to such a list) of its coefficients, the first item gives the constant term, the last item gives the leading coefficient, except if zero, then it is dropped (iteratively). List items are each expanded in an \edef and then put into normalized form via xintfrac‘s macro \xintRaw.

As leading zero coefficients are removed:

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

defines the zero polynomial, which holds only one coefficient.

See also expandable macro \PolToCSV{<polname>}.

\PolMapCoeffs{}{}

Syntax: \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 the replacement text of \macro, \index expands to the coefficient index (starting at zero for the constant term).

Notice that \macro will have to handle inputs in the xintfrac internal format. This means that it probably will have to be expressed in terms of macros from the 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{}

Syntax: \PolReduceCoeffs{<polname>}

Reduces the internal representations of the coefficients to their lowest terms.

\PolReduceCoeffs*{}

Syntax: \PolReduceCoeffs*{<polname>}

Reduces the internal representations of the coefficients to their lowest terms, but ignoring a possible separated “power of ten part”.

For example, xintfrac stores an 30e2/50 input as 30/50 with a separate 10^2 part. This will thus get replaced by 3e^2/5 (or rather whatever xintfrac uses for internal representation), and not by 60 as would result from complete reduction.

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 generally smaller.

\PolMakeMonic{}

Syntax: \PolMakeMonic{<polname>}

Divides by the leading coefficient. It is recommended to execute \PolReduceCoeffs*{<polname>} immediately afterwards. This is not done automatically, in case the original polynomial had integer coefficients and the user wants to keep the leading one as common denominator for typesetting purposes.

\PolMakePrimitive{}

Syntax: \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.

\PolDiff{}{}

Syntax: \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[]{}{}

Syntax: \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{}{}

Syntax: \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[]{}{}

Syntax: \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{}{}{}{}

Syntax: \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{}{}{}

Syntax: \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{}{}{}

Syntax: \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{}{}{}

Syntax: \PolGCD{<polname_1>}{<polname_2>}{<polname_GCD>}

This sets polname_GCD to be the (monic) GCD of polname_1 and polname_2. It is a unitary polynomial except if both polname_1 and polname_2 vanish, then polname_GCD is the zero polynomial.

Root localization routines via the Sturm Theorem

As \PolToSturm{<polname>}{<sturmname>} and \PolSturmIsolateZeros{<sturmname>} and variants declare additional polynomial or scalar variables with names based on <sturmname> as prefix, it is advisable to keep the <sturmname> namespace separate from the one applying to \xintexpr variables generally, or to polynomials.

Displaying the found roots: \PolPrintIntervals[<varname>]{}

Syntax: \PolPrintIntervals[<varname>]{<sturmname>}

This is a convenience macro which prints the bounds for the roots Z_1, Z_2, … (the optional argument varname allows to specify a replacement for the default Z). This will be done (by default) in a math mode array, one interval per row, and pattern rcccl, where the second and fourth column hold the < sign, except when the interval reduces to a singleton, which means the root is known exactly.

Note

The explanations here and in this section are for LaTeX. With other TeX macro formats, the LaTeX syntax such as for example \begin{array}{rcccl} which appears in the documentation here is actually replaced with quasi-equivalent direct use of TeX primitives.

The next macros which govern its output.

\PolToExpr{}

Syntax: \PolToExpr{<pol. expr.>}

Produces expandably [3] the string coeff_N*x^N+..., i.e. the polynomial is using descending powers.

Since 0.8 the input is not restricted to be a polynomial name but is allowed to be an arbitrary expression. Then x is expected as indeterminate but this can be customized via \PolToExprInVar.

The output uses the letter x by default, this is customizable via \PolToExprVar. The default output is compatible both with

• the Maple’s input format,

• and the PSTricks \psplot[algebraic] input format.

Attention that it is not compatible with Python, see further \PolToExprCaret in this context.

The following applies:

• vanishing coefficients are skipped (issue \poltoexpralltrue to override this and produce output such as x^3+0*x^2+0*x^1+0),

• negative coefficients are not prefixed by a + sign (else, Maple would not be happy),

• coefficients numerically equal to 1 (or -1) are present only via their sign,

• the letter x is used and the degree one monomial is output as x, not as x^1.

• (0.8) the caret ^ is of catcode 12. This means that one can for convenience typeset in regular text mode, for example using \texttt (in LaTeX). But TeX will not know how to break the expression across end-of-lines anyhow. Formerly ^ was suitable for math mode but as the exponent is not braced this worked only for polynomials of degrees at most 9. Anyhow this is not supposed to be a typesetting macro.

Complete customization is possible, see the next macros. Any user redefinition must maintain the expandability property.

\PolToFloatExpr{}

Syntax: \PolToFloatExpr{<pol. expr.>}

Similar to \PolToExpr{<pol. expr.>} but using \PolToFloatExprCmd{<raw_coeff>} which by default rounds and converts the coefficients to floating point format.

Note

This is unrelated to \PolGenFloatVariant{<polname>}: \PolToFloatExprCmd{<raw_coeff>} operates on the exact coefficients anew (and may thus produce something else than the coefficients of the polynomial function acting in \xintfloateval if the floating point precision was changed in between).

Extended at 0.8 to accept general expressions as input.

\PolToExpr*{}

Syntax: \PolToExpr*{<pol. expr.>}

Ascending powers: coeff_0+coeff_1*x+coeff_2*x^2+....

Extended at 0.8 to accept general expressions as input.

Customizable with the same macros as for \PolToExpr{<pol. expr.>}.

\PolToFloatExpr*{}

Syntax: \PolToFloatExpr*{<pol. expr.>}

Ascending powers.

Extended at 0.8 to accept general expressions as input.

\PolNthCoeff{}{}

Syntax: \PolNthCoeff{<polname>}{<index>}

It expands to the raw N-th coefficient (N=0 corresponds to the constant coefficient). If N is out of range, zero (in its default xintfrac format 0/1[0]) is returned.

Negative indices N=-1, -2, … return the leading coefficient, sub-leading coefficient, …, and finally 0/1[0] for N<-1-degree.

\PolLeadingCoeff{}

Syntax: \PolLeadingCoeff{<polname>}

Expands to the leading coefficient.

\PolDegree{}

Syntax: \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{}

Syntax: \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.

\PolToList{}

Syntax: \PolToList{<polname>}

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

\PolToCSV{}

Syntax: \PolToCSV{<polname>}

Expands to coeff_0, coeff_1, coeff_2, ....., coeff_N, starting with constant term and ending with leading coefficient. Converse to \PolFromCSV{<polname>}{<csv>}.

\PolEval{}\AtExpr{}

Syntax: \PolEval{<polname>}\AtExpr{<num. expr.>}

Same output as \xinteval{polname(numerical expression)}.

\PolEval{}\At{}

Syntax: \PolEval{<polname>}\At{<value>}

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

\PolEvalReduced{}\AtExpr{}

Syntax: \PolEvalReduced{<polname>}\AtExpr{<num. expr.>}

Same output as \xinteval{reduce(polname(numerical expression))}.

\PolEvalReduced{}\At{}

Syntax: \PolEvalReduced{<polname>}\At{<value>}

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

\PolFloatEval{}\AtExpr{}

Syntax: \PolFloatEval{<polname>}\AtExpr{<num. expr.>}

Same output as \xintfloateval{polname(numerical expression)}.

Attention

\PolGenFloatVariant must have been issued before.

To use the exact coefficients with exactly executed additions and multiplications and do the rounding only as the final last step, the following syntax can be used: [4]

\xintfloateval{3.27*\xintexpr f(2.53)\relax^2}

\PolFloatEval{}\At{}

Syntax: \PolFloatEval{<polname>}\At{<value>}

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

Expandable macros in relation to root localization via Sturm Theorem

Expandable macros for use within execution of \PolPrintIntervals

These macros are for usage within custom user redefinitions of \PolPrintIntervalsKnownRoot, \PolPrintIntervalsUnknownRoot, or in redefinitions of PolPrintIntervalsPrintExactZero (used in the default for the former) and of \PolPrintIntervalsPrintLeftEndPoint, \PolPrintIntervalsPrintRightEndPoint (used in the default for the latter).

\xintverbosefalse

This is actually an xintexpr configuration. Setting it to true triggers the writing of information to the log when new polynomial or scalar variables are defined.

Caution

The macro and variable meanings as written to the log are to be considered unstable and undocumented internal structures.

\polnewpolverbosefalse

When \poldef is used, both a variable and a function are defined. The default \polnewpolverbosefalse setting suppresses the print-out to the log and terminal of the function macro meaning, as it only duplicates the information contained in the variable which is already printed out to the log and terminal.

However \PolGenFloatVariant{<polname>} does still print out the information relative to the polynomial function it defines for use in \xintfloateval{} as there is no float polynomial variable, only the

function, and it is the only way to see its rounded coefficients (\xintverbosefalse suppresses also that info).

If set to true, it overrides in both cases \xintverbosefalse. The setting only affects polynomial declarations. Scalar variables such as those holding information on roots obey only the \xintverbose... setting.

(new with 0.8)

\poltypesetallfalse

If true, \PolTypeset will also typeset the vanishing coefficients.

\poltoexprallfalse

If true, \PolToExpr{<pol. expr.>} and \PolToFloatExpr{<pol. expr.>} will also include the vanishing coefficients in their outputs.

\PolDecToString{}

Syntax: \PolDecToString{decimal number}

This is a utility macro to print decimal numbers. It is an alias for \xintDecToString.

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}}.

Attention that a.t.t.o.w. if the argument is for example 1/5, the macro does not identify that this is in fact a number with a finite decimal expansion and it outputs 1/5. See current xintfrac documentation.

\polexprsetup{key=val,...}

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

• norr: the postfix that \PolSturmIsolateZeros**{<sturmname>} should append to <sturmname> to declare the primitive polynomial obtained from original one after removal of all rational roots. The default value is _norr (standing for “no rational roots”).

• sqfnorr: the postfix that \PolSturmIsolateZeros**{<sturmname>} should append to <sturmname> to declare the primitive polynomial obtained from original one after removal of all rational roots and suppression of all multiplicities. The default value is _sqf_norr (standing for “square-free with no rational roots”).

The package executes \polexprsetup{norr=_norr, sqfnorr=_sqf_norr} as default.