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++ Author: Dave Barton, Barry Trager
++ Date Last Updated: 30 November 1994
++ Fix History: 30 Nov 94
++ Description:
++ This type is the basic representation of sparse recursive multivariate
++ polynomials. It is parameterized by the coefficient ring and the
++ variable set which may be infinite. The variable ordering is determined
++ by the variable set parameter. The coefficient ring may be non-commutative,
++ but the variables are assumed to commute.
SparseMultivariatePolynomial(R,VarSet) : SIG == CODE where
R : Ring
VarSet : OrderedSet
pgcd ==> PolynomialGcdPackage(IndexedExponents VarSet,VarSet,R,%)
SUP ==> SparseUnivariatePolynomial
SIG ==> PolynomialCategory(R,IndexedExponents(VarSet),VarSet)
CODE ==> add
--constants
--D := F(%) replaced by next line until compiler support completed
--representations
D := SparseUnivariatePolynomial(%)
VPoly:= Record(v:VarSet,ts:D)
Rep:= Union(R,VPoly)
--declarations
fn: R -> R
n: Integer
k: NonNegativeInteger
kp:PositiveInteger
k1:NonNegativeInteger
c: R
mvar: VarSet
val : R
var:VarSet
up: D
p,p1,p2,pval: %
Lval : List(R)
Lpval : List(%)
Lvar : List(VarSet)
--define
0 ==
0$R::%
1 ==
1$R::%
zero? p ==
p case R and zero?(p)$R
one? p ==
p case R and ((p) = 1)$R
-- a local function
red(p:%):% ==
p case R => 0
if ground?(reductum p.ts) then
leadingCoefficient(reductum p.ts) else [p.v,reductum p.ts]$VPoly
numberOfMonomials(p): NonNegativeInteger ==
p case R =>
zero?(p)$R => 0
1
+/[numberOfMonomials q for q in coefficients(p.ts)]
coerce(mvar):% ==
[mvar,monomial(1,1)$D]$VPoly
monomial? p ==
p case R => true
sup : D := p.ts
1 ^= numberOfMonomials(sup) => false
monomial? leadingCoefficient(sup)$D
-- local
moreThanOneVariable?: % -> Boolean
moreThanOneVariable? p ==
p case R => false
q:=p.ts
any?(x1+->not ground? x1 ,coefficients q) => true
false
-- if we already know we use this (slighlty) faster function
univariateKnown: % -> SparseUnivariatePolynomial R
univariateKnown p ==
p case R => (leadingCoefficient p) :: SparseUnivariatePolynomial(R)
monomial( leadingCoefficient p,degree p.ts)+ univariateKnown(red p)
univariate p ==
p case R =>(leadingCoefficient p) :: SparseUnivariatePolynomial(R)
moreThanOneVariable? p => error "not univariate"
monomial( leadingCoefficient p,degree p.ts)+ univariate(red p)
multivariate (u:SparseUnivariatePolynomial(R),var:VarSet) ==
ground? u => (leadingCoefficient u) ::%
[var,monomial(leadingCoefficient u,degree u)$D]$VPoly +
multivariate(reductum u,var)
univariate(p:%,mvar:VarSet):SparseUnivariatePolynomial(%) ==
p case R or mvar>p.v => monomial(p,0)$D
pt:=p.ts
mvar=p.v => pt
monomial(1,p.v,degree pt)*univariate(leadingCoefficient pt,mvar)+
univariate(red p,mvar)
-- a local functions, used in next definition
unlikeUnivReconstruct(u:SparseUnivariatePolynomial(%),mvar:VarSet):% ==
zero? (d:=degree u) => coefficient(u,0)
monomial(leadingCoefficient u,mvar,d)+
unlikeUnivReconstruct(reductum u,mvar)
multivariate(u:SparseUnivariatePolynomial(%),mvar:VarSet):% ==
ground? u => coefficient(u,0)
uu:=u
while not zero? uu repeat
cc:=leadingCoefficient uu
cc case R or mvar > cc.v => uu:=reductum uu
return unlikeUnivReconstruct(u,mvar)
[mvar,u]$VPoly
ground?(p:%):Boolean ==
p case R => true
false
monomial(p,mvar,k1) ==
zero? k1 or zero? p => p
p case R or mvar>p.v => [mvar,monomial(p,k1)$D]$VPoly
p*[mvar,monomial(1,k1)$D]$VPoly
monomial(c:R,e:IndexedExponents(VarSet)):% ==
zero? e => (c::%)
monomial(1,leadingSupport e, leadingCoefficient e) *
monomial(c,reductum e)
coefficient(p:%, e:IndexedExponents(VarSet)) : R ==
zero? e =>
p case R => p::R
coefficient(coefficient(p.ts,0),e)
p case R => 0
ve := leadingSupport e
vp := p.v
ve < vp =>
coefficient(coefficient(p.ts,0),e)
ve > vp => 0
coefficient(coefficient(p.ts,leadingCoefficient e),reductum e)
coerce(n) ==
n::R::%
coerce(c) ==
c::%
characteristic ==
characteristic$R
recip(p) ==
p case R => (uu:=recip(p::R);uu case "failed" => "failed"; uu::%)
"failed"
- p ==
p case R => -$R p
[p.v, - p.ts]$VPoly
n * p ==
p case R => n * p::R
mvar:=p.v
up:=n*p.ts
if ground? up then leadingCoefficient(up) else [mvar,up]$VPoly
c * p ==
c = 1 => p
p case R => c * p::R
mvar:=p.v
up:=c*p.ts
if ground? up then leadingCoefficient(up) else [mvar,up]$VPoly
p1 + p2 ==
p1 case R and p2 case R => p1 +$R p2
p1 case R => [p2.v, p1::D + p2.ts]$VPoly
p2 case R => [p1.v, p1.ts + p2::D]$VPoly
p1.v = p2.v =>
mvar:=p1.v
up:=p1.ts+p2.ts
if ground? up then leadingCoefficient(up) else [mvar,up]$VPoly
p1.v < p2.v =>
[p2.v, p1::D + p2.ts]$VPoly
[p1.v, p1.ts + p2::D]$VPoly
p1 - p2 ==
p1 case R and p2 case R => p1 -$R p2
p1 case R => [p2.v, p1::D - p2.ts]$VPoly
p2 case R => [p1.v, p1.ts - p2::D]$VPoly
p1.v = p2.v =>
mvar:=p1.v
up:=p1.ts-p2.ts
if ground? up then leadingCoefficient(up) else [mvar,up]$VPoly
p1.v < p2.v =>
[p2.v, p1::D - p2.ts]$VPoly
[p1.v, p1.ts - p2::D]$VPoly
p1 = p2 ==
p1 case R =>
p2 case R => p1 =$R p2
false
p2 case R => false
p1.v = p2.v => p1.ts = p2.ts
false
p1 * p2 ==
p1 case R => p1::R * p2
p2 case R =>
mvar:=p1.v
up:=p1.ts*p2
if ground? up then leadingCoefficient(up) else [mvar,up]$VPoly
p1.v = p2.v =>
mvar:=p1.v
up:=p1.ts*p2.ts
if ground? up then leadingCoefficient(up) else [mvar,up]$VPoly
p1.v > p2.v =>
mvar:=p1.v
up:=p1.ts*p2
if ground? up then leadingCoefficient(up) else [mvar,up]$VPoly
--- p1.v < p2.v
mvar:=p2.v
up:=p1*p2.ts
if ground? up then leadingCoefficient(up) else [mvar,up]$VPoly
p ^ kp ==
p ** (kp pretend NonNegativeInteger)
p ** kp ==
p ** (kp pretend NonNegativeInteger )
p ^ k ==
p ** k
p ** k ==
p case R => p::R ** k
-- univariate special case
not moreThanOneVariable? p =>
multivariate( (univariateKnown p) ** k , p.v)
mvar:=p.v
up:=p.ts ** k
if ground? up then leadingCoefficient(up) else [mvar,up]$VPoly
if R has IntegralDomain then
UnitCorrAssoc ==> Record(unit:%,canonical:%,associate:%)
unitNormal(p) ==
u,c,a:R
p case R =>
(u,c,a):= unitNormal(p::R)$R
[u::%,c::%,a::%]$UnitCorrAssoc
(u,c,a):= unitNormal(leadingCoefficient(p))$R
[u::%,(a*p)::%,a::%]$UnitCorrAssoc
unitCanonical(p) ==
p case R => unitCanonical(p::R)$R
(u,c,a):= unitNormal(leadingCoefficient(p))$R
a*p
unit? p ==
p case R => unit?(p::R)$R
false
associates?(p1,p2) ==
p1 case R => p2 case R and associates?(p1,p2)$R
p2 case VPoly and p1.v = p2.v and associates?(p1.ts,p2.ts)
if R has approximate then
p1 exquo p2 ==
p1 case R and p2 case R =>
a:= (p1::R exquo p2::R)
if a case "failed" then "failed" else a::%
zero? p1 => p1
(p2 = 1) => p1
p1 case R or p2 case VPoly and p1.v < p2.v => "failed"
p2 case R or p1.v > p2.v =>
a:= (p1.ts exquo p2::D)
a case "failed" => "failed"
[p1.v,a]$VPoly::%
-- The next test is useful in the case that R has inexact
-- arithmetic (in particular when it is Interval(...)).
-- In the case where the test succeeds, empirical evidence
-- suggests that it can speed up the computation several times,
-- but in other cases where there are a lot of variables
-- p1 and p2 differ only in the low order terms
-- (for example, p1=p2+1)
-- it slows exquo down by about 15-20%.
p1 = p2 => 1
a:= p1.ts exquo p2.ts
a case "failed" => "failed"
mvar:=p1.v
up:SUP %:=a
if ground? (up) then
leadingCoefficient(up) else [mvar,up]$VPoly::%
else
p1 exquo p2 ==
p1 case R and p2 case R =>
a:= (p1::R exquo p2::R)
if a case "failed" then "failed" else a::%
zero? p1 => p1
(p2 = 1) => p1
p1 case R or p2 case VPoly and p1.v < p2.v => "failed"
p2 case R or p1.v > p2.v =>
a:= (p1.ts exquo p2::D)
a case "failed" => "failed"
[p1.v,a]$VPoly::%
a:= p1.ts exquo p2.ts
a case "failed" => "failed"
mvar:=p1.v
up:SUP %:=a
if ground? up then leadingCoefficient(up) else [mvar,up]$VPoly::%
map(fn,p) ==
p case R => fn(p)
mvar:=p.v
up:=map(x1+->map(fn,x1),p.ts)
if ground? up then leadingCoefficient(up) else [mvar,up]$VPoly
if R has Field then
(p : %) / (r : R) ==
inv(r) * p
if R has GcdDomain then
content(p) ==
p case R => p
c :R :=0
up:=p.ts
while not(zero? up) and not(c = 1) repeat
c:=gcd(c,content leadingCoefficient(up))
up := reductum up
c
if R has EuclideanDomain and
R has CharacteristicZero and
not(R has FloatingPointSystem) then
content(p,mvar) ==
p case R => p
gcd(coefficients univariate(p,mvar))$pgcd
gcd(p1,p2) ==
gcd(p1,p2)$pgcd
gcd(lp:List %) ==
gcd(lp)$pgcd
gcdPolynomial(a:SUP $,b:SUP $):SUP $ ==
gcd(a,b)$pgcd
else if R has GcdDomain then
content(p,mvar) ==
p case R => p
content univariate(p,mvar)
gcd(p1,p2) ==
p1 case R =>
p2 case R => gcd(p1,p2)$R::%
zero? p1 => p2
gcd(p1, content(p2.ts))
p2 case R =>
zero? p2 => p1
gcd(p2, content(p1.ts))
p1.v < p2.v => gcd(p1, content(p2.ts))
p1.v > p2.v => gcd(content(p1.ts), p2)
mvar:=p1.v
up:=gcd(p1.ts, p2.ts)
if ground? up then leadingCoefficient(up) else [mvar,up]$VPoly
if R has FloatingPointSystem then
-- eventually need a better notion of gcd's over floats
-- this essentially computes the gcds of the monomial contents
gcdPolynomial(a:SUP $,b:SUP $):SUP $ ==
ground? (a) =>
zero? a => b
gcd(leadingCoefficient a, content b)::SUP $
ground?(b) =>
zero? b => b
gcd(leadingCoefficient b, content a)::SUP $
conta := content a
mona:SUP $ := monomial(conta, minimumDegree a)
if mona ^= 1 then
a := (a exquo mona)::SUP $
contb := content b
monb:SUP $ := monomial(contb, minimumDegree b)
if monb ^= 1 then
b := (b exquo monb)::SUP $
mong:SUP $ := monomial(gcd(conta, contb),
min(degree mona, degree monb))
degree(a) >= degree b =>
not((a exquo b) case "failed") =>
mong * b
mong
not((b exquo a) case "failed") => mong * a
mong
coerce(p):OutputForm ==
p case R => (p::R)::OutputForm
outputForm(p.ts,p.v::OutputForm)
coefficients p ==
p case R => list(p :: R)$List(R)
"append"/[coefficients(p1)$% for p1 in coefficients(p.ts)]
retract(p:%):R ==
p case R => p :: R
error "cannot retract nonconstant polynomial"
retractIfCan(p:%):Union(R, "failed") ==
p case R => p::R
"failed"
mymerge:(List VarSet,List VarSet) ->List VarSet
mymerge(l:List VarSet,m:List VarSet):List VarSet ==
empty? l => m
empty? m => l
first l = first m =>
empty? rest l =>
setrest!(l,rest m)
l
empty? rest m => l
setrest!(l, mymerge(rest l, rest m))
l
first l > first m =>
empty? rest l =>
setrest!(l,m)
l
setrest!(l, mymerge(rest l, m))
l
empty? rest m =>
setrest!(m,l)
m
setrest!(m,mymerge(l,rest m))
m
variables p ==
p case R => empty()
lv:List VarSet:=empty()
q := p.ts
while not zero? q repeat
lv:=mymerge(lv,variables leadingCoefficient q)
q := reductum q
cons(p.v,lv)
mainVariable p ==
p case R => "failed"
p.v
eval(p,mvar,pval) ==
univariate(p,mvar)(pval)
eval(p,mvar,val) ==
univariate(p,mvar)(val)
evalSortedVarlist(p,Lvar,Lpval):% ==
p case R => p
empty? Lvar or empty? Lpval => p
mvar := Lvar.first
mvar > p.v => evalSortedVarlist(p,Lvar.rest,Lpval.rest)
pval := Lpval.first
pts := map(x1+->evalSortedVarlist(x1,Lvar,Lpval),p.ts)
mvar=p.v =>
pval case R => pts (pval::R)
pts pval
multivariate(pts,p.v)
eval(p,Lvar,Lpval) ==
empty? rest Lvar => evalSortedVarlist(p,Lvar,Lpval)
sorted?((x1,x2) +-> x1 > x2, Lvar) => evalSortedVarlist(p,Lvar,Lpval)
nlvar := sort((x1,x2) +-> x1 > x2,Lvar)
nlpval :=
Lvar = nlvar => Lpval
nlpval := [Lpval.position(mvar,Lvar) for mvar in nlvar]
evalSortedVarlist(p,nlvar,nlpval)
eval(p,Lvar,Lval) ==
eval(p,Lvar,[val::% for val in Lval]$(List %)) -- kill?
degree(p,mvar) ==
p case R => 0
mvar= p.v => degree p.ts
mvar > p.v => 0 -- might as well take advantage of the order
max(degree(leadingCoefficient p.ts,mvar),degree(red p,mvar))
degree(p,Lvar) ==
[degree(p,mvar) for mvar in Lvar]
degree p ==
p case R => 0
degree(leadingCoefficient(p.ts)) + monomial(degree(p.ts), p.v)
minimumDegree p ==
p case R => 0
md := minimumDegree p.ts
minimumDegree(coefficient(p.ts,md)) + monomial(md, p.v)
minimumDegree(p,mvar) ==
p case R => 0
mvar = p.v => minimumDegree p.ts
md:=minimumDegree(leadingCoefficient p.ts,mvar)
zero? (p1:=red p) => md
min(md,minimumDegree(p1,mvar))
minimumDegree(p,Lvar) ==
[minimumDegree(p,mvar) for mvar in Lvar]
totalDegree(p, Lvar) ==
ground? p => 0
null setIntersection(Lvar, variables p) => 0
u := univariate(p, mv := mainVariable(p)::VarSet)
weight:NonNegativeInteger := (member?(mv,Lvar) => 1; 0)
tdeg:NonNegativeInteger := 0
while u ^= 0 repeat
termdeg:NonNegativeInteger := weight*degree u +
totalDegree(leadingCoefficient u, Lvar)
tdeg := max(tdeg, termdeg)
u := reductum u
tdeg
if R has CommutativeRing then
differentiate(p,mvar) ==
p case R => 0
mvar=p.v =>
up:=differentiate p.ts
if ground? up then leadingCoefficient(up) else [mvar,up]$VPoly
up:=map(x1 +-> differentiate(x1,mvar),p.ts)
if ground? up then leadingCoefficient(up) else [p.v,up]$VPoly
leadingCoefficient(p) ==
p case R => p
leadingCoefficient(leadingCoefficient(p.ts))
leadingMonomial p ==
p case R => p
monomial(leadingMonomial leadingCoefficient(p.ts),
p.v, degree(p.ts))
reductum(p) ==
p case R => 0
p - leadingMonomial p
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