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Submitted by : (unknown) at: 2007-11-17T22:28:35-08:00 (9 years ago)
Name :
Axiom Version :
Category : Severity : Status :
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I must type the variable if I want the right result.

With theses types there is no problem :

axiom
coefficient (numer (12 * (sin x)^3 * z), (sin x)::Kernel Expression Integer, 3)

\label{eq1}{12}\  z(1)
Type: SparseMultivariatePolynomial(Integer,Kernel(Expression(Integer)))

But without type I get 0 :

axiom
coefficient (numer (12 * (sin x)^3 * z), sin z, 3)
There are 4 exposed and 0 unexposed library operations named coefficient having 3 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op coefficient to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation.
Cannot find a definition or applicable library operation named coefficient with argument type(s) SparseMultivariatePolynomial(Integer,Kernel(Expression(Integer))) Expression(Integer) PositiveInteger
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

A simpler example is:

axiom
coefficient (numer((sin z)^2), sin z::Kernel EXPR INT, 2)

\label{eq2}1(2)
Type: SparseMultivariatePolynomial(Integer,Kernel(Expression(Integer)))
axiom
coefficient (numer((sin z)^2), sin z, 2)
There are 4 exposed and 0 unexposed library operations named coefficient having 3 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op coefficient to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation.
Cannot find a definition or applicable library operation named coefficient with argument type(s) SparseMultivariatePolynomial(Integer,Kernel(Expression(Integer))) Expression(Integer) PositiveInteger
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

I modified POLYCAT as follows:

    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)

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) == output(hconcat(["POLYCAT:", p::OutputForm, v::OutputForm, n::OutputForm]))$OutputPackage output(hconcat(["POLYCAT:", univariate(p,v)::OutputForm]))$OutputPackage coefficient(univariate(p,v),n) coefficient(p,lv,ln) == empty? lv => empty? ln => p error "mismatched lists in coefficient" empty? ln => error "mismatched lists in coefficient" coefficient(coefficient(univariate(p,first lv),first ln), rest lv,rest ln) monomial(p,lv,ln) == empty? lv => empty? ln => p error "mismatched lists in monomial" empty? ln => error "mismatched lists in monomial" monomial(monomial(p,first lv, first ln),rest lv, rest ln) retract(p:%):VarSet == q := mainVariable(p)::VarSet q::% = p => q error "Polynomial is not a single variable" retractIfCan(p:%):Union(VarSet, "failed") == ((q := mainVariable p) case VarSet) and (q::VarSet::% = p) => q "failed" mkPrim(p:%):% == monomial(1,degree p) primitiveMonomials p == [mkPrim q for q in monomials p] totalDegree p == ground? p => 0 u := univariate(p, mainVariable(p)::VarSet) d: NonNegativeInteger := 0 while u ^= 0 repeat d := max(d, degree u + totalDegree leadingCoefficient u) u := reductum u d totalDegree(p,lv) == ground? p => 0 u := univariate(p, v:=(mainVariable(p)::VarSet)) d: NonNegativeInteger := 0 w: NonNegativeInteger := 0 if member?(v, lv) then w:=1 while u ^= 0 repeat d := max(d, w*(degree u) + totalDegree(leadingCoefficient u,lv)) u := reductum u d
if R has CommutativeRing then resultant(p1,p2,mvar) == resultant(univariate(p1,mvar),univariate(p2,mvar)) discriminant(p,var) == discriminant(univariate(p,var))
if R has IntegralDomain then allMonoms(l:List %):List(%) == removeDuplicates_! concat [primitiveMonomials p for p in l] P2R(p:%, b:List E, n:NonNegativeInteger):Vector(R) == w := new(n, 0)$Vector(R) for i in minIndex w .. maxIndex w for bj in b repeat qsetelt_!(w, i, coefficient(p, bj)) w eq2R(l:List %, b:List E):Matrix(R) == matrix [[coefficient(p, bj) for p in l] for bj in b] reducedSystem(m:Matrix %):Matrix(R) == l := listOfLists m b := removeDuplicates_! concat [allMonoms r for r in l]$List(List(%)) d := [degree bj for bj in b] mm := eq2R(first l, d) l := rest l while not empty? l repeat mm := vertConcat(mm, eq2R(first l, d)) l := rest l mm reducedSystem(m:Matrix %, v:Vector %): Record(mat:Matrix R, vec:Vector R) == l := listOfLists m r := entries v b : List % := removeDuplicates_! concat(allMonoms r, concat [allMonoms s for s in l]$List(List(%))) d := [degree bj for bj in b] n := #d mm := eq2R(first l, d) w := P2R(first r, d, n) l := rest l r := rest r while not empty? l repeat mm := vertConcat(mm, eq2R(first l, d)) w := concat(w, P2R(first r, d, n)) l := rest l r := rest r [mm, w]
if R has PolynomialFactorizationExplicit then -- we might be in trouble if its actually only -- a univariate polynomial category - have to remember to -- over-ride these in UnivariatePolynomialCategory PFBR ==>PolynomialFactorizationByRecursion(R,E,VarSet,%) gcdPolynomial(pp,qq) == gcdPolynomial(pp,qq)$GeneralPolynomialGcdPackage(E,VarSet,R,%) solveLinearPolynomialEquation(lpp,pp) == solveLinearPolynomialEquationByRecursion(lpp,pp)$PFBR factorPolynomial(pp) == factorByRecursion(pp)$PFBR factorSquareFreePolynomial(pp) == factorSquareFreeByRecursion(pp)$PFBR factor p == v:Union(VarSet,"failed"):=mainVariable p v case "failed" => ansR:=factor leadingCoefficient p makeFR(unit(ansR)::%, [[w.flg,w.fctr::%,w.xpnt] for w in factorList ansR]) up:SparseUnivariatePolynomial %:=univariate(p,v) ansSUP:=factorByRecursion(up)$PFBR makeFR(multivariate(unit(ansSUP),v), [[ww.flg,multivariate(ww.fctr,v),ww.xpnt] for ww in factorList ansSUP]) if R has CharacteristicNonZero then mat: Matrix % conditionP mat == ll:=listOfLists transpose mat -- hence each list corresponds to a -- column, i.e. to one variable llR:List List R := [ empty() for z in first ll] monslist:List List % := empty() ch:=characteristic()$% for l in ll repeat mons:= "setUnion"/[primitiveMonomials u for u in l] redmons:List % :=[] for m in mons repeat vars:=variables m degs:=degree(m,vars) deg1:List NonNegativeInteger deg1:=[ ((nd:=d:Integer exquo ch:Integer) case "failed" => return "failed" ; nd::Integer::NonNegativeInteger) for d in degs ] redmons:=[monomial(1,vars,deg1),:redmons] llR:=[[ground coefficient(u,vars,degs),:v] for u in l for v in llR] monslist:=[redmons,:monslist] ans:=conditionP transpose matrix llR ans case "failed" => "failed" i:NonNegativeInteger:=0 [ +/[m*(ans.(i:=i+1))::% for m in mons ] for mons in monslist]
if R has CharacteristicNonZero then charthRootlv: (%,List VarSet,NonNegativeInteger) -> Union(%,"failed") charthRoot p == vars:= variables p empty? vars => ans := charthRoot ground p ans case "failed" => "failed" ans::R::% ch:=characteristic()$% charthRootlv(p,vars,ch) charthRootlv(p,vars,ch) == empty? vars => ans := charthRoot ground p ans case "failed" => "failed" ans::R::% v:=first vars vars:=rest vars d:=degree(p,v) ans:% := 0 while (d>0) repeat (dd:=(d::Integer exquo ch::Integer)) case "failed" => return "failed" cp:=coefficient(p,v,d) p:=p-monomial(cp,v,d) ansx:=charthRootlv(cp,vars,ch) ansx case "failed" => return "failed" d:=degree(p,v) ans:=ans+monomial(ansx,v,dd::Integer::NonNegativeInteger) ansx:=charthRootlv(p,vars,ch) ansx case "failed" => return "failed" return ans+ansx
monicDivide(p1,p2,mvar) == result:=monicDivide(univariate(p1,mvar),univariate(p2,mvar)) [multivariate(result.quotient,mvar), multivariate(result.remainder,mvar)]
if R has GcdDomain then if R has EuclideanDomain and R has CharacteristicZero then squareFree p == squareFree(p)$MultivariateSquareFree(E,VarSet,R,%) else squareFree p == squareFree(p)$PolynomialSquareFree(VarSet,E,R,%) squareFreePart p == unit(s := squareFree p) * */[f.factor for f in factors s] content(p,v) == content univariate(p,v) primitivePart p == unitNormal((p exquo content p) ::%).canonical primitivePart(p,v) == unitNormal((p exquo content(p,v)) ::%).canonical if R has OrderedSet then p:% < q:% == (dp:= degree p) < (dq := degree q) => (leadingCoefficient q) > 0 dq < dp => (leadingCoefficient p) < 0 leadingCoefficient(p - q) < 0 if (R has PatternMatchable Integer) and (VarSet has PatternMatchable Integer) then patternMatch(p:%, pat:Pattern Integer, l:PatternMatchResult(Integer, %)) == patternMatch(p, pat, l)$PatternMatchPolynomialCategory(Integer,E,VarSet,R,%) if (R has PatternMatchable Float) and (VarSet has PatternMatchable Float) then patternMatch(p:%, pat:Pattern Float, l:PatternMatchResult(Float, %)) == patternMatch(p, pat, l)$PatternMatchPolynomialCategory(Float,E,VarSet,R,%)
if (R has ConvertibleTo Pattern Integer) and (VarSet has ConvertibleTo Pattern Integer) then convert(x:%):Pattern(Integer) == map(convert, convert, x)$PolynomialCategoryLifting(E,VarSet,R,%,Pattern Integer) if (R has ConvertibleTo Pattern Float) and (VarSet has ConvertibleTo Pattern Float) then convert(x:%):Pattern(Float) == map(convert, convert, x)$PolynomialCategoryLifting(E, VarSet, R, %, Pattern Float) if (R has ConvertibleTo InputForm) and (VarSet has ConvertibleTo InputForm) then convert(p:%):InputForm == map(convert, convert, p)$PolynomialCategoryLifting(E,VarSet,R,%,InputForm)
spad
   Compiling FriCAS source code from file 
      /var/zope2/var/LatexWiki/5427536097468235958-25px004.spad using 
      old system compiler.
   POLYCAT abbreviates category PolynomialCategory 
------------------------------------------------------------------------
   initializing NRLIB POLYCAT for PolynomialCategory 
   compiling into NRLIB POLYCAT 
;;; *** |PolynomialCategory| REDEFINED Time: 0.13 SEC.
POLYCAT- abbreviates domain PolynomialCategory& ------------------------------------------------------------------------ initializing NRLIB POLYCAT- for PolynomialCategory& compiling into NRLIB POLYCAT- compiling exported eval : (S,List Equation S) -> S
;;; *** |POLYCAT-;eval;SLS;1| REDEFINED Time: 0.52 SEC.
compiling exported monomials : S -> List S ****** comp fails at level 3 with expression: ****** error in function monomials
(SEQ (LET |ml| (|elt| (|List| S) |empty|)) (REPEAT (WHILE | << | (^= |p| 0) | >> |) (SEQ (LET |ml| (|concat| (|leadingMonomial| |p|) |ml|)) (|exit| 1 (LET |p| (|reductum| |p|))))) (|exit| 1 (|reverse| |ml|))) ****** level 3 ****** $x:= (^= p (Zero)) $m:= (Boolean) $f:= ((((|ml| #) (|p| # #) (* #) (+ #) ...)))
>> Apparent user error: WHILE operand: (^= p (Zero)) is not Boolean valued

and obtained the following output:

axiom
coefficient(numer(sin x)^2, (sin x)::Kernel EXPR INT, 2)

\label{eq3}1(3)
Type: SparseMultivariatePolynomial(Integer,Kernel(Expression(Integer)))
axiom
coefficient(numer(sin x)^2, (sin x), 2)
There are 4 exposed and 0 unexposed library operations named coefficient having 3 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op coefficient to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation.
Cannot find a definition or applicable library operation named coefficient with argument type(s) SparseMultivariatePolynomial(Integer,Kernel(Expression(Integer))) Expression(Integer) PositiveInteger
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

Note also, that the result types are different.

Status: open => not reproducible




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