Changes
http://wiki.axiom-developer.org/334MisrecognizedVariableInExpression/diff
--
??changed:
- -- coefficient(p,v,n) ==
- -- output(hconcat(["POLYCAT:", p::OutputForm, v::OutputForm,
n::OutputForm]))$OutputPackage
- -- output(hconcat(["POLYCAT:",
univariate(p,v)::OutputForm]))$OutputPackage
- -- coefficient(univariate(p,v),n)
-
-and obtained the following output::
-
- --(2) -> coefficient(numer(sin x)^2, (sin x)::Kernel EXPR INT, 2)
- -- 2
- -- POLYCAT:sin(x) sin(x)2
- -- 2
- -- POLYCAT:?
- --
- -- (2) 1
- -- Type: SparseMultivariatePolynomial(Integer,Kernel Expression
Integer)
- --(3) -> coefficient(numer(sin x)^2, (sin x), 2)
- -- 2
- -- POLYCAT:sin(x) sin(x)2
- -- 2
-[6 more lines...]
+ coefficient(p,v,n) ==
+ output(hconcat(["POLYCAT:", p::OutputForm, v::OutputForm,
n::OutputForm]))$OutputPackage
+ output(hconcat(["POLYCAT:",
univariate(p,v)::OutputForm]))$OutputPackage
+ coefficient(univariate(p,v),n)
+
+\begin{spad}
+)abbrev category POLYCAT PolynomialCategory
+++ Author:
+++ Date Created:
+++ Date Last Updated:
+++ Basic Functions: Ring, monomial, coefficient, differentiate, eval
+++ Related Constructors: Polynomial, DistributedMultivariatePolynomial
+++ Also See: UnivariatePolynomialCategory
+++ AMS Classifications:
+++ Keywords:
+++ References:
+++ Description:
+++ The category for general multi-variate polynomials over a ring
+++ R, in variables from VarSet, with exponents from the
+++ \spadtype{OrderedAbelianMonoidSup}.
+
+PolynomialCategory(R:Ring, E:OrderedAbelianMonoidSup, VarSet:OrderedSet):
+ Category ==
+ Join(PartialDifferentialRing VarSet, FiniteAbelianMonoidRing(R, E),
+ Evalable %, InnerEvalable(VarSet, R),
+ InnerEvalable(VarSet, %), RetractableTo VarSet,
+ FullyLinearlyExplicitRingOver R) with
+ -- operations
+ degree : (%,VarSet) -> NonNegativeInteger
+ ++ degree(p,v) gives the degree of polynomial p with respect to the
variable v.
+ degree : (%,List(VarSet)) -> List(NonNegativeInteger)
+ ++ degree(p,lv) gives the list of degrees of polynomial p
+ ++ with respect to each of the variables in the list lv.
+ coefficient: (%,VarSet,NonNegativeInteger) -> %
+ ++ coefficient(p,v,n) views the polynomial p as a univariate
+ ++ polynomial in v and returns the coefficient of the \spad{v**n} term.
+ coefficient: (%,List VarSet,List NonNegativeInteger) -> %
+ ++ coefficient(p, lv, ln) views the polynomial p as a polynomial
+ ++ in the variables of lv and returns the coefficient of the term
+ ++ \spad{lv**ln}, i.e. \spad{prod(lv_i ** ln_i)}.
+ monomials: % -> List %
+ ++ monomials(p) returns the list of non-zero monomials of polynomial p,
i.e.
+ ++ \spad{monomials(sum(a_(i) X^(i))) = [a_(1) X^(1),...,a_(n) X^(n)]}.
+ univariate : (%,VarSet) -> SparseUnivariatePolynomial(%)
+ ++ univariate(p,v) converts the multivariate polynomial p
+ ++ into a univariate polynomial in v, whose coefficients are still
+ ++ multivariate polynomials (in all the other variables).
+ univariate : % -> SparseUnivariatePolynomial(R)
+ ++ univariate(p) converts the multivariate polynomial p,
+ ++ which should actually involve only one variable,
+ ++ into a univariate polynomial
+ ++ in that variable, whose coefficients are in the ground ring.
+ ++ Error: if polynomial is genuinely multivariate
+ mainVariable : % -> Union(VarSet,"failed")
+ ++ mainVariable(p) returns the biggest variable which actually
+ ++ occurs in the polynomial p, or "failed" if no variables are
+ ++ present.
+ ++ fails precisely if polynomial satisfies ground?
+ minimumDegree : (%,VarSet) -> NonNegativeInteger
+ ++ minimumDegree(p,v) gives the minimum degree of polynomial p
+ ++ with respect to v, i.e. viewed a univariate polynomial in v
+ minimumDegree : (%,List(VarSet)) -> List(NonNegativeInteger)
+ ++ minimumDegree(p, lv) gives the list of minimum degrees of the
+ ++ polynomial p with respect to each of the variables in the list lv
+ monicDivide : (%,%,VarSet) -> Record(quotient:%,remainder:%)
+ ++ monicDivide(a,b,v) divides the polynomial a by the polynomial b,
+ ++ with each viewed as a univariate polynomial in v returning
+ ++ both the quotient and remainder.
+ ++ Error: if b is not monic with respect to v.
+ monomial : (%,VarSet,NonNegativeInteger) -> %
+ ++ monomial(a,x,n) creates the monomial \spad{a*x**n} where \spad{a} is
+ ++ a polynomial, x is a variable and n is a nonnegative integer.
+ monomial : (%,List VarSet,List NonNegativeInteger) -> %
+ ++ monomial(a,[v1..vn],[e1..en]) returns \spad{a*prod(vi**ei)}.
+ multivariate : (SparseUnivariatePolynomial(R),VarSet) -> %
+ ++ multivariate(sup,v) converts an anonymous univariable
+ ++ polynomial sup to a polynomial in the variable v.
+ multivariate : (SparseUnivariatePolynomial(%),VarSet) -> %
+ ++ multivariate(sup,v) converts an anonymous univariable
+ ++ polynomial sup to a polynomial in the variable v.
+ isPlus: % -> Union(List %, "failed")
+ ++ isPlus(p) returns \spad{[m1,...,mn]} if polynomial \spad{p = m1 +
... + mn} and
+ ++ \spad{n >= 2} and each mi is a nonzero monomial.
+ isTimes: % -> Union(List %, "failed")
+ ++ isTimes(p) returns \spad{[a1,...,an]} if polynomial \spad{p = a1
... an}
+ ++ and \spad{n >= 2}, and, for each i, ai is either a nontrivial
constant in R or else of the
+ ++ form \spad{x**e}, where \spad{e > 0} is an integer and x in a
member of VarSet.
+ isExpt: % -> Union(Record(var:VarSet, exponent:NonNegativeInteger),_
+ "failed")
+ ++ isExpt(p) returns \spad{[x, n]} if polynomial p has the form
\spad{x**n} and \spad{n > 0}.
+ totalDegree : % -> NonNegativeInteger
+ ++ totalDegree(p) returns the largest sum over all monomials
+ ++ of all exponents of a monomial.
+ totalDegree : (%,List VarSet) -> NonNegativeInteger
+ ++ totalDegree(p, lv) returns the maximum sum (over all monomials of
polynomial p)
+ ++ of the variables in the list lv.
+ variables : % -> List(VarSet)
+ ++ variables(p) returns the list of those variables actually
+ ++ appearing in the polynomial p.
+ primitiveMonomials: % -> List %
+ ++ primitiveMonomials(p) gives the list of monomials of the
+ ++ polynomial p with their coefficients removed.
+ ++ Note: \spad{primitiveMonomials(sum(a_(i) X^(i))) =
[X^(1),...,X^(n)]}.
+ if R has OrderedSet then OrderedSet
+ -- OrderedRing view removed to allow EXPR to define abs
+ --if R has OrderedRing then OrderedRing
+ if (R has ConvertibleTo InputForm) and
+ (VarSet has ConvertibleTo InputForm) then
+ ConvertibleTo InputForm
+ if (R has ConvertibleTo Pattern Integer) and
+ (VarSet has ConvertibleTo Pattern Integer) then
+ ConvertibleTo Pattern Integer
+ if (R has ConvertibleTo Pattern Float) and
+ (VarSet has ConvertibleTo Pattern Float) then
+ ConvertibleTo Pattern Float
+ if (R has PatternMatchable Integer) and
+ (VarSet has PatternMatchable Integer) then
+ PatternMatchable Integer
+ if (R has PatternMatchable Float) and
+ (VarSet has PatternMatchable Float) then
+ PatternMatchable Float
+ if R has CommutativeRing then
+ resultant : (%,%,VarSet) -> %
+ ++ resultant(p,q,v) returns the resultant of the polynomials
+ ++ p and q with respect to the variable v.
+ discriminant : (%,VarSet) -> %
+ ++ discriminant(p,v) returns the disriminant of the polynomial p
+ ++ with respect to the variable v.
+ if R has GcdDomain then
+ GcdDomain
+ content: (%,VarSet) -> %
+ ++ content(p,v) is the gcd of the coefficients of the polynomial p
+ ++ when p is viewed as a univariate polynomial with respect to the
+ ++ variable v.
+ ++ Thus, for polynomial 7*x**2*y + 14*x*y**2, the gcd of the
+ ++ coefficients with respect to x is 7*y.
+ primitivePart: % -> %
+ ++ primitivePart(p) returns the unitCanonical associate of the
+ ++ polynomial p with its content divided out.
+ primitivePart: (%,VarSet) -> %
+ ++ primitivePart(p,v) returns the unitCanonical associate of the
+ ++ polynomial p with its content with respect to the variable v
+ ++ divided out.
+ squareFree: % -> Factored %
+ ++ squareFree(p) returns the square free factorization of the
+ ++ polynomial p.
+ squareFreePart: % -> %
+ ++ squareFreePart(p) returns product of all the irreducible factors
+ ++ of polynomial p each taken with multiplicity one.
+
+ -- assertions
+ if R has canonicalUnitNormal then canonicalUnitNormal
+ ++ we can choose a unique representative for each
+ ++ associate class.
+ ++ This normalization is chosen to be normalization of
+ ++ leading coefficient (by default).
+ if R has PolynomialFactorizationExplicit then
+ PolynomialFactorizationExplicit
+ add
+ p:%
+ v:VarSet
+ ln:List NonNegativeInteger
+ lv:List VarSet
+ n:NonNegativeInteger
+ pp,qq:SparseUnivariatePolynomial %
+ eval(p:%, l:List Equation %) ==
+ empty? l => p
+ for e in l repeat
+ retractIfCan(lhs e)@Union(VarSet,"failed") case "failed" =>
+ error "cannot find a variable to evaluate"
+ lvar:=[retract(lhs e)@VarSet for e in l]
+ eval(p, lvar,[rhs e for e in l]$List(%))
+ monomials p ==
+-- zero? p => empty()
+-- concat(leadingMonomial p, monomials reductum p)
+-- replaced by sequential version for efficiency, by WMSIT, 7/30/90
+ ml:= empty$List(%)
+ while p ^= 0 repeat
+ ml:=concat(leadingMonomial p, ml)
+ p:= reductum p
+ reverse ml
+ isPlus p ==
+ empty? rest(l := monomials p) => "failed"
+ l
+ isTimes p ==
+ empty?(lv := variables p) or not monomial? p => "failed"
+ l := [monomial(1, v, degree(p, v)) for v in lv]
+-- one?(r := leadingCoefficient p) =>
+ ((r := leadingCoefficient p) = 1) =>
+ empty? rest lv => "failed"
+ l
+ concat(r::%, l)
+ isExpt p ==
+ (u := mainVariable p) case "failed" => "failed"
+ p = monomial(1, u::VarSet, d := degree(p, u::VarSet)) =>
+ [u::VarSet, d]
+ "failed"
+ -- coefficient(p,v,n) == coefficient(univariate(p,v),n)
+ coefficient(p,v,n) ==
+[231 more lines...]
--
forwarded from http://wiki.axiom-developer.org/[EMAIL PROTECTED]
