login  home  contents  what's new  discussion  bug reports help  links  subscribe  changes  refresh  edit

 Submitted by : (unknown) at: 2007-11-17T22:28:35-08:00 (12 years ago) Name : Axiom Version : default friCAS-20090114 Axiom-20050901 OpenAxiom-20091012 OpenAxiom-20110220 OpenAxiom-Release-141 Category : Axiom Aldor Interface Axiom Compiler Axiom Library Axiom Interpreter Axiom Documentation Axiom User Interface building Axiom from source lisp system MathAction Doyen CD Reduce Axiom on Windows Axiom on Linux Severity : critical serious normal minor wishlist Status : open closed rejected not reproducible fix proposed fixed somewhere duplicate need more info Optional subject :   Optional comment :

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


)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
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
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) -> %
++ a polynomial, x is a variable and n is a nonnegative integer.
monomial : (%,List VarSet,List NonNegativeInteger) -> %
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")
++ 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")
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
if R has PolynomialFactorizationExplicit then
PolynomialFactorizationExplicit
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
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
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)
 (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
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

 Subject:   Be Bold !! ( 15 subscribers )