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Testing a change from:

      differentiate(p: %,v: OV) ==
            multivariate(differentiate(univariate(p,v)),v)

to:

      differentiate(p: %,v: OV) ==
            multivariate(differentiate(univariate(p,v),v),v)

i.e. differentiate should be extended from the underlying Ring.

spad
)abbrev domain GDMP GeneralDistributedMultivariatePolynomial
++ Author: Barry Trager
++ Date Created:
++ Date Last Updated:
++ Basic Functions: Ring, degree, eval, coefficient, monomial, differentiate,
++ resultant, gcd, leadingCoefficient
++ Related Constructors: DistributedMultivariatePolynomial,
++ HomogeneousDistributedMultivariatePolynomial
++ Also See: Polynomial
++ AMS Classifications:
++ Keywords: polynomial, multivariate, distributed
++ References:
++ Description:
++   This type supports distributed multivariate polynomials
++ whose variables are from a user specified list of symbols.
++ The coefficient ring may be non commutative,
++ but the variables are assumed to commute.
++ The term ordering is specified by its third parameter.
++ Suggested types which define term orderings include: \spadtype{DirectProduct},
++ \spadtype{HomogeneousDirectProduct}, \spadtype{SplitHomogeneousDirectProduct}
++ and finally \spadtype{OrderedDirectProduct} which accepts an arbitrary user
++ function to define a term ordering.
GeneralDistributedMultivariatePolynomial(vl,R,E): public == private where vl: List Symbol R: Ring E: DirectProductCategory(#vl,NonNegativeInteger) OV ==> OrderedVariableList(vl) SUP ==> SparseUnivariatePolynomial NNI ==> NonNegativeInteger
public == PolynomialCategory(R,E,OV) with reorder: (%,List Integer) -> % ++ reorder(p, perm) applies the permutation perm to the variables ++ in a polynomial and returns the new correctly ordered polynomial univariate:(%,OV) -> SUP(R)
private == PolynomialRing(R,E) add --representations Term := Record(k:E,c:R) Rep := List Term n := #vl Vec ==> Vector(NonNegativeInteger) zero?(p : %): Boolean == null(p::Rep)
totalDegree p == zero? p => 0 "max"/[reduce("+",(t.k)::(Vector NNI), 0) for t in p]
monomial(p:%, v: OV,e: NonNegativeInteger):% == locv := lookup v p*monomial(1, directProduct [if z=locv then e else 0 for z in 1..n]$Vec)
coerce(v: OV):% == monomial(1,v,1)
listCoef(p : %): List R == rec : Term [rec.c for rec in (p::Rep)]
mainVariable(p: %) == zero?(p) => "failed" for v in vl repeat vv := variable(v)::OV if degree(p,vv)>0 then return vv "failed"
ground?(p) == mainVariable(p) case "failed"
retract(p : %): R == not ground? p => error "not a constant" leadingCoefficient p
retractIfCan(p : %): Union(R,"failed") == ground?(p) => leadingCoefficient p "failed"
degree(p: %,v: OV) == degree(univariate(p,v)) minimumDegree(p: %,v: OV) == minimumDegree(univariate(p,v)) if R has PartialDifferentialRing(Symbol) then
differentiate(p : %, v : OV) == pu := univariate(p) dpu := map((x : R) : R +-> differentiate(x, convert(v)@Symbol), pu) multivariate(differentiate(pu) + dpu, v) else
differentiate(p : %, v : OV) == multivariate(differentiate(univariate(p,v)),v)
degree(p: %,lv: List OV) == [degree(p,v) for v in lv] minimumDegree(p: %,lv: List OV) == [minimumDegree(p,v) for v in lv]
numberOfMonomials(p:%) == l : Rep := p::Rep null(l) => 1 #l
monomial?(p : %): Boolean == l : Rep := p::Rep null(l) or null rest(l)
if R has OrderedRing then maxNorm(p : %): R == l : List R := nil r,m : R m := 0 for r in listCoef(p) repeat if r > m then m := r else if (-r) > m then m := -r m
--trailingCoef(p : %) == -- l : Rep := p : Rep -- null l => 0
-- r : Term := last l -- r.c
--leadingPrimitiveMonomial(p : %) == -- ground?(p) => 1$% -- r : Term := first(p:Rep) -- r := [r.k,1$R]$Term -- new cell
-- list(r)$Rep :: %
-- The following 2 defs are inherited from PolynomialRing
--leadingMonomial(p : %) == -- ground?(p) => p -- r : Term := first(p:Rep) -- r := [r.k,r.c]$Term -- new cell -- list(r)$Rep :: %
--reductum(p : %): % == -- ground? p => 0$% -- (rest(p:Rep)):%
if R has Field then (p : %) / (r : R) == inv(r) * p
variables(p: %) == maxdeg:Vector(NonNegativeInteger) := new(n,0) while not zero?(p) repeat tdeg := degree p p := reductum p for i in 1..n repeat maxdeg.i := max(maxdeg.i, tdeg.i) [index(i::PositiveInteger) for i in 1..n | maxdeg.i ~= 0]
reorder(p: %,perm: List Integer):% == #perm ~= n => error "must be a complete permutation of all vars" q := [[directProduct [term.k.j for j in perm]$Vec,term.c]$Term for term in p] sort((x, y) +-> x.k > x.k, q)
--coerce(dp:DistributedMultivariatePolynomial(vl,R)):% == -- q:=dp:List(Term) -- sort(#1.k > #2.k,q):%
univariate(p: %,v: OV):SUP(%) == zero?(p) => 0 exp := degree p locv := lookup v deg:NonNegativeInteger := 0 nexp := directProduct [if i=locv then (deg :=exp.i;0) else exp.i for i in 1..n]$Vec monomial(monomial(leadingCoefficient p,nexp),deg)+ univariate(reductum p,v)
eval(p: %,v: OV,val:%):% == univariate(p,v)(val)
eval(p: %,v: OV,val:R):% == eval(p,v,val::%)$%
eval(p: %,lv: List OV,lval: List R):% == lv = [] => p eval(eval(p,first lv,(first lval)::%)$%, rest lv, rest lval)$%
-- assume Lvar are sorted correctly evalSortedVarlist(p: %,Lvar: List OV,Lpval: List %):% == v := mainVariable p v case "failed" => p pv := v:: OV Lvar=[] or Lpval=[] => p mvar := Lvar.first mvar > pv => evalSortedVarlist(p,Lvar.rest,Lpval.rest) pval := Lpval.first pts:SUP(%):= map(evalSortedVarlist(#1,Lvar,Lpval),univariate(p,pv)) mvar=pv => pts(pval) multivariate(pts,pv)
eval(p:%,Lvar:List OV,Lpval:List %) == nlvar:List OV := sort((x : OV, y : OV) : Boolean +-> x > y, Lvar) nlpval := Lvar = nlvar => Lpval nlpval := [Lpval.position(mvar,Lvar) for mvar in nlvar] evalSortedVarlist(p,nlvar,nlpval)
multivariate(p1:SUP(%),v: OV):% == 0=p1 => 0 degree p1 = 0 => leadingCoefficient p1 leadingCoefficient(p1)*(v::%)^degree(p1) + multivariate(reductum p1,v)
univariate(p: %):SUP(R) == (v := mainVariable p) case "failed" => monomial(leadingCoefficient p,0) q := univariate(p,v:: OV) ans:SUP(R) := 0 while q ~= 0 repeat ans := ans + monomial(ground leadingCoefficient q,degree q) q := reductum q ans
multivariate(p:SUP(R),v: OV):% == 0=p => 0 (leadingCoefficient p)*monomial(1,v,degree p) + multivariate(reductum p,v)
if R has GcdDomain then content(p: %):R == zero?(p) => 0 "gcd"/[t.c for t in p]
if R has EuclideanDomain and not(R has FloatingPointSystem) then gcd(p: %,q:%):% == gcd(p,q)$PolynomialGcdPackage(E,OV,R,%)
else gcd(p: %,q:%):% == r : R (pv := mainVariable(p)) case "failed" => (r := leadingCoefficient p) = 0$R => q gcd(r,content q)::% (qv := mainVariable(q)) case "failed" => (r := leadingCoefficient q) = 0$R => p gcd(r,content p)::% pv<qv => gcd(p,content univariate(q,qv)) qv<pv => gcd(q,content univariate(p,pv)) multivariate(gcd(univariate(p,pv),univariate(q,qv)),pv)
coerce(p: %) : OutputForm == zero?(p) => (0$R) :: OutputForm l,lt : List OutputForm lt := nil vl1 := [v::OutputForm for v in vl] for t in reverse p repeat l := nil for i in 1..#vl1 repeat t.k.i = 0 => l t.k.i = 1 => l := cons(vl1.i,l) l := cons(vl1.i ^ t.k.i ::OutputForm,l) l := reverse l if (t.c ~= 1) or (null l) then l := cons(t.c :: OutputForm,l) 1 = #l => lt := cons(first l,lt) lt := cons(reduce("*",l),lt) 1 = #lt => first lt reduce("+",lt)
spad
   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/6560064616104104256-25px001.spad
      using old system compiler.
   GDMP abbreviates domain GeneralDistributedMultivariatePolynomial 
------------------------------------------------------------------------
   initializing NRLIB GDMP for GeneralDistributedMultivariatePolynomial 
   compiling into NRLIB GDMP 
   processing macro definition Vec ==> Vector NonNegativeInteger 
   compiling exported zero? : $ -> Boolean
****** comp fails at level 1 with expression: ******
error in function zero? 
((|null| (|::| |p| |Rep|))) ****** level 1 ****** $x:= (null (:: p Rep)) $m:= (Boolean) $f:= ((((|p| # #) (|Vec| #) (|n| #) (* #) ...)))
>> Apparent user error: cannot compile (null (:: p Rep))

Test:

fricas
P := DMP([x,y], EXPR INT)

\label{eq1}\hbox{\axiomType{DistributedMultivariatePolynomial}\ } ([ x , y ] , \hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }))(1)
Type: Type
fricas
a :P := x

\label{eq2}x(2)
Type: DistributedMultivariatePolynomial?([x,y],Expression(Integer))
fricas
b := a/x

\label{eq3}{1 \over x}\  x(3)
Type: DistributedMultivariatePolynomial?([x,y],Expression(Integer))
fricas
univariate(b)

\label{eq4}{1 \over x}\  ?(4)
Type: SparseUnivariatePolynomial?(Expression(Integer))
fricas
differentiate(b,x)

\label{eq5}1 \over x(5)
Type: DistributedMultivariatePolynomial?([x,y],Expression(Integer))

Instructions: What is the output for each of the following? Do the problem only one at a time. You should write down your answer, its type, or possible errors, and with explanations. Then check it (using )set mess bot off), revise your explanation if necessary, or explain Axiom's output. Finally, use )set mess bot on to see exactly what the Interpreter did. Then and only then, advance to the next problem.
  variables (2*x+1/x)$DMP([x], EXPR INT)
  variables (2*y+1/y)$DMP([y], INT)
  a:=(2*x+1/x)$DMP([x], EXPR INT); variables a
  b:=(2*y+1/y)$DMP([y], INT); variables b
  x:DMP([x], EXPR INT):=x; variables (2*x+1/x)
  y:DMP([y], INT):=y; variables (2*y+1/y)
Answer: see SandBoxPolynomialQuiz




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