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\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra poly.spad}
\author{Dave Barton, James Davenport, Barry Trager, Patrizia Gianni, Marc Moreno Maza}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{domain FM FreeModule}
<<domain FM FreeModule>>=
)abbrev domain FM FreeModule
++ Author: Dave Barton, James Davenport, Barry Trager
++ Date Created:
++ Date Last Updated:
++ Basic Functions: BiModule(R,R)
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ A bi-module is a free module
++ over a ring with generators indexed by an ordered set.
++ Each element can be expressed as a finite linear combination of
++ generators. Only non-zero terms are stored.
FreeModule(R:Ring,S:OrderedType):
Join(BiModule(R,R),IndexedDirectProductCategory(R,S)) with
if R has CommutativeRing then Module(R)
== IndexedDirectProductAbelianGroup(R,S) add
--representations
Term:= Record(k:S,c:R)
Rep:= List Term
--declarations
x,y: %
r: R
n: Integer
f: R -> R
s: S
--define
if R has EntireRing then
r * x ==
zero? r => 0
one? r => x
--map(r*#1,x)
[[u.k,r*u.c] for u in x ]
else
r * x ==
zero? r => 0
one? r => x
--map(r*#1,x)
[[u.k,a] for u in x | (a:=r*u.c) ~= 0$R]
if R has EntireRing then
x * r ==
zero? r => 0
one? r => x
--map(r*#1,x)
[[u.k,u.c*r] for u in x ]
else
x * r ==
zero? r => 0
one? r => x
--map(r*#1,x)
[[u.k,a] for u in x | (a:=u.c*r) ~= 0$R]
if S has CoercibleTo OutputForm then
coerce(x) : OutputForm ==
null x => (0$R) :: OutputForm
le : List OutputForm := nil
for rec in reverse x repeat
rec.c = 1 => le := cons(rec.k :: OutputForm, le)
le := cons(rec.c :: OutputForm * rec.k :: OutputForm, le)
reduce("+",le)
@
\section{domain PR PolynomialRing}
<<domain PR PolynomialRing>>=
)abbrev domain PR PolynomialRing
++ Author: Dave Barton, James Davenport, Barry Trager
++ Date Created:
++ Date Last Updated: 14.08.2000. Improved exponentiation [MMM/TTT]
++ Basic Functions: Ring, degree, coefficient, monomial, reductum
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ This domain represents generalized polynomials with coefficients
++ (from a not necessarily commutative ring), and terms
++ indexed by their exponents (from an arbitrary ordered abelian monoid).
++ This type is used, for example,
++ by the \spadtype{DistributedMultivariatePolynomial} domain where
++ the exponent domain is a direct product of non negative integers.
PolynomialRing(R:Ring,E:OrderedAbelianMonoid): T == C
where
T == FiniteAbelianMonoidRing(R,E) with
--assertions
if R has IntegralDomain and E has CancellationAbelianMonoid then
fmecg: (%,E,R,%) -> %
++ fmecg(p1,e,r,p2) finds X : p1 - r * X**e * p2
if R has canonicalUnitNormal then canonicalUnitNormal
++ canonicalUnitNormal guarantees that the function
++ unitCanonical returns the same representative for all
++ associates of any particular element.
C == FreeModule(R,E) add
--representations
Term:= Record(k:E,c:R)
Rep:= List Term
--declarations
x,y,p,p1,p2: %
n: Integer
nn: NonNegativeInteger
np: PositiveInteger
e: E
r: R
--local operations
1 == [[0$E,1$R]]
characteristic == characteristic$R
numberOfMonomials x == (# x)$Rep
degree p == if null p then 0 else p.first.k
minimumDegree p == if null p then 0 else (last p).k
leadingCoefficient p == if null p then 0$R else p.first.c
leadingMonomial p == if null p then 0 else [p.first]
reductum p == if null p then p else p.rest
retractIfCan(p:%):Union(R,"failed") ==
null p => 0$R
not null p.rest => "failed"
zero?(p.first.k) => p.first.c
"failed"
coefficient(p,e) ==
for tm in p repeat
tm.k=e => return tm.c
tm.k < e => return 0$R
0$R
recip(p) ==
null p => "failed"
p.first.k > 0$E => "failed"
(u:=recip(p.first.c)) case "failed" => "failed"
(u::R)::%
coerce(r) == if zero? r then 0$% else [[0$E,r]]
coerce(n) == (n::R)::%
ground?(p): Boolean == empty? p or (empty? rest p and zero? degree p)
qsetrest!: (Rep, Rep) -> Rep
qsetrest!(l: Rep, e: Rep): Rep == RPLACD(l, e)$Lisp
entireRing? := R has EntireRing
--- term * polynomial
termTimes: (R, E, Term) -> Term
termTimes(r: R, e: E, tx:Term): Term == [e+tx.k, r*tx.c]
times(tco: R, tex: E, rx: %): % ==
if entireRing? then
map(termTimes(tco, tex, #1), rx::Rep)
else
[[tex + tx.k, r] for tx in rx::Rep | not zero? (r := tco * tx.c)]
-- local addm!
addm!: (Rep, R, E, Rep) -> Rep
-- p1 + coef*x^E * p2
-- `spare' (commented out) is for storage efficiency (not so good for
-- performance though.
addm!(p1:Rep, coef:R, exp: E, p2:Rep): Rep ==
--local res, newend, last: Rep
res, newcell, endcell: Rep
spare: List Rep
res := empty()
endcell := empty()
--spare := empty()
while not empty? p1 and not empty? p2 repeat
tx := first p1
ty := first p2
exy := exp + ty.k
newcell := empty();
if tx.k = exy then
newcoef := tx.c + coef * ty.c
if not zero? newcoef then
tx.c := newcoef
newcell := p1
--else
-- spare := cons(p1, spare)
p1 := rest p1
p2 := rest p2
else if tx.k > exy then
newcell := p1
p1 := rest p1
else
newcoef := coef * ty.c
if not entireRing? and zero? newcoef then
newcell := empty()
--else if empty? spare then
-- ttt := [exy, newcoef]
-- newcell := cons(ttt, empty())
--else
-- newcell := first spare
-- spare := rest spare
-- ttt := first newcell
-- ttt.k := exy
-- ttt.c := newcoef
else
ttt := [exy, newcoef]
newcell := cons(ttt, empty())
p2 := rest p2
if not empty? newcell then
if empty? res then
res := newcell
endcell := res
else
qsetrest!(endcell, newcell)
endcell := newcell
if not empty? p1 then -- then end is const * p1
newcell := p1
else -- then end is (coef, exp) * p2
newcell := times(coef, exp, p2)
empty? res => newcell
qsetrest!(endcell, newcell)
res
pomopo! (p1, r, e, p2) == addm!(p1, r, e, p2)
p1 * p2 ==
xx := p1::Rep
empty? xx => p1
yy := p2::Rep
empty? yy => p2
zero? first(xx).k => first(xx).c * p2
zero? first(yy).k => p1 * first(yy).c
--if #xx > #yy then
-- (xx, yy) := (yy, xx)
-- (p1, p2) := (p2, p1)
xx := reverse xx
res : Rep := empty()
for tx in xx repeat res:=addm!(res,tx.c,tx.k,yy)
res
-- if R has EntireRing then
-- p1 * p2 ==
-- null p1 => 0
-- null p2 => 0
-- zero?(p1.first.k) => p1.first.c * p2
-- one? p2 => p1
-- +/[[[t1.k+t2.k,t1.c*t2.c]$Term for t2 in p2]
-- for t1 in reverse(p1)]
-- -- This 'reverse' is an efficiency improvement:
-- -- reduces both time and space [Abbott/Bradford/Davenport]
-- else
-- p1 * p2 ==
-- null p1 => 0
-- null p2 => 0
-- zero?(p1.first.k) => p1.first.c * p2
-- one? p2 => p1
-- +/[[[t1.k+t2.k,r]$Term for t2 in p2 | (r:=t1.c*t2.c) ~= 0]
-- for t1 in reverse(p1)]
-- -- This 'reverse' is an efficiency improvement:
-- -- reduces both time and space [Abbott/Bradford/Davenport]
if R has CommutativeRing then
p ** np == p ** (np pretend NonNegativeInteger)
p ** nn ==
null p => 0
zero? nn => 1
one? nn => p
empty? p.rest =>
zero?(cc:=p.first.c ** nn) => 0
[[nn * p.first.k, cc]]
binomThmExpt([p.first], p.rest, nn)
if R has Field then
unitNormal(p) ==
null p or (lcf:R:=p.first.c) = 1 => [1,p,1]
a := inv lcf
[lcf::%, [[p.first.k,1],:(a * p.rest)], a::%]
unitCanonical(p) ==
null p or (lcf:R:=p.first.c) = 1 => p
a := inv lcf
[[p.first.k,1],:(a * p.rest)]
else if R has IntegralDomain then
unitNormal(p) ==
null p or p.first.c = 1 => [1,p,1]
(u,cf,a):=unitNormal(p.first.c)
[u::%, [[p.first.k,cf],:(a * p.rest)], a::%]
unitCanonical(p) ==
null p or p.first.c = 1 => p
(u,cf,a):=unitNormal(p.first.c)
[[p.first.k,cf],:(a * p.rest)]
if R has IntegralDomain then
associates?(p1,p2) ==
null p1 => null p2
null p2 => false
p1.first.k = p2.first.k and
associates?(p1.first.c,p2.first.c) and
((p2.first.c exquo p1.first.c)::R * p1.rest = p2.rest)
p exquo r ==
[(if (a:= tm.c exquo r) case "failed"
then return "failed" else [tm.k,a])
for tm in p] :: Union(%,"failed")
if E has CancellationAbelianMonoid then
fmecg(p1:%,e:E,r:R,p2:%):% == -- p1 - r * X**e * p2
rout:%:= []
r:= - r
for tm in p2 repeat
e2:= e + tm.k
c2:= r * tm.c
c2 = 0 => "next term"
while not null p1 and p1.first.k > e2 repeat
(rout:=[p1.first,:rout]; p1:=p1.rest) --use PUSH and POP?
null p1 or p1.first.k < e2 => rout:=[[e2,c2],:rout]
if (u:=p1.first.c + c2) ~= 0 then rout:=[[e2, u],:rout]
p1:=p1.rest
NRECONC(rout,p1)$Lisp
if R has approximate then
p1 exquo p2 ==
null p2 => error "Division by 0"
p2 = 1 => p1
p1=p2 => 1
--(p1.lastElt.c exquo p2.lastElt.c) case "failed" => "failed"
rout:= []@List(Term)
while not null p1 repeat
(a:= p1.first.c exquo p2.first.c)
a case "failed" => return "failed"
ee:= subtractIfCan(p1.first.k, p2.first.k)
ee case "failed" => return "failed"
p1:= fmecg(p1.rest, ee, a, p2.rest)
rout:= [[ee,a], :rout]
null p1 => reverse(rout)::% -- nreverse?
"failed"
else -- R not approximate
p1 exquo p2 ==
null p2 => error "Division by 0"
p2 = 1 => p1
--(p1.lastElt.c exquo p2.lastElt.c) case "failed" => "failed"
rout:= []@List(Term)
while not null p1 repeat
(a:= p1.first.c exquo p2.first.c)
a case "failed" => return "failed"
ee:= subtractIfCan(p1.first.k, p2.first.k)
ee case "failed" => return "failed"
p1:= fmecg(p1.rest, ee, a, p2.rest)
rout:= [[ee,a], :rout]
null p1 => reverse(rout)::% -- nreverse?
"failed"
if R has Field then
x/r == inv(r)*x
@
\section{domain SUP SparseUnivariatePolynomial}
<<domain SUP SparseUnivariatePolynomial>>=
import NonNegativeInteger
import OutputForm
)abbrev domain SUP SparseUnivariatePolynomial
++ Author: Dave Barton, Barry Trager
++ Date Created:
++ Date Last Updated:
++ Basic Functions: Ring, monomial, coefficient, reductum, differentiate,
++ elt, map, resultant, discriminant
++ Related Constructors: UnivariatePolynomial, Polynomial
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ This domain represents univariate polynomials over arbitrary
++ (not necessarily commutative) coefficient rings. The variable is
++ unspecified so that the variable displays as \spad{?} on output.
++ If it is necessary to specify the variable name, use type \spadtype{UnivariatePolynomial}.
++ The representation is sparse
++ in the sense that only non-zero terms are represented.
++ Note: if the coefficient ring is a field, this domain forms a euclidean domain.
SparseUnivariatePolynomial(R:Ring): UnivariatePolynomialCategory(R) with
outputForm : (%,OutputForm) -> OutputForm
++ outputForm(p,var) converts the SparseUnivariatePolynomial p to
++ an output form (see \spadtype{OutputForm}) printed as a polynomial in the
++ output form variable.
fmecg: (%,NonNegativeInteger,R,%) -> %
++ fmecg(p1,e,r,p2) finds X : p1 - r * X**e * p2
== PolynomialRing(R,NonNegativeInteger)
add
--representations
Term := Record(k:NonNegativeInteger,c:R)
Rep := List Term
p:%
n:NonNegativeInteger
np: PositiveInteger
FP ==> SparseUnivariatePolynomial %
pp,qq: FP
lpp:List FP
-- for karatsuba
kBound: NonNegativeInteger := 63
upmp := UnivariatePolynomialMultiplicationPackage(R,%)
if R has FieldOfPrimeCharacteristic then
p ** np == p ** (np pretend NonNegativeInteger)
p ** n ==
null p => 0
zero? n => 1
one? n => p
empty? p.rest =>
zero?(cc:=p.first.c ** n) => 0
[[n * p.first.k, cc]]
-- not worth doing special trick if characteristic is too small
if characteristic$R < 3 then return expt(p,n pretend PositiveInteger)$RepeatedSquaring(%)
y:%:=1
-- break up exponent in qn * characteristic + rn
-- exponentiating by the characteristic is fast
rec := divide(n, characteristic$R)
qn:= rec.quotient
rn:= rec.remainder
repeat
if rn = 1 then y := y * p
if rn > 1 then y:= y * binomThmExpt([p.first], p.rest, rn)
zero? qn => return y
-- raise to the characteristic power
p:= [[t.k * characteristic$R , primeFrobenius(t.c)$R ]$Term for t in p]
rec := divide(qn, characteristic$R)
qn:= rec.quotient
rn:= rec.remainder
y
zero?(p): Boolean == empty?(p)
one?(p):Boolean == not empty? p and (empty? rest p and zero? first(p).k and one? first(p).c)
ground?(p): Boolean == empty? p or (empty? rest p and zero? first(p).k)
multiplyExponents(p,n) == [ [u.k*n,u.c] for u in p]
divideExponents(p,n) ==
null p => p
m:= (p.first.k :: Integer exquo n::Integer)
m case "failed" => "failed"
u:= divideExponents(p.rest,n)
u case "failed" => "failed"
[[m::Integer::NonNegativeInteger,p.first.c],:u]
karatsubaDivide(p, n) ==
zero? n => [p, 0]
lowp: Rep := p
highp: Rep := []
repeat
if empty? lowp then break
t := first lowp
if t.k < n then break
lowp := rest lowp
highp := cons([subtractIfCan(t.k,n)::NonNegativeInteger,t.c]$Term,highp)
[ reverse highp, lowp]
shiftRight(p, n) ==
[[subtractIfCan(t.k,n)::NonNegativeInteger,t.c]$Term for t in p]
shiftLeft(p, n) ==
[[t.k + n,t.c]$Term for t in p]
pomopo!(p1,r,e,p2) ==
rout:%:= []
for tm in p2 repeat
e2:= e + tm.k
c2:= r * tm.c
c2 = 0 => "next term"
while not null p1 and p1.first.k > e2 repeat
(rout:=[p1.first,:rout]; p1:=p1.rest) --use PUSH and POP?
null p1 or p1.first.k < e2 => rout:=[[e2,c2],:rout]
if (u:=p1.first.c + c2) ~= 0 then rout:=[[e2, u],:rout]
p1:=p1.rest
NRECONC(rout,p1)$Lisp
-- implementation using karatsuba algorithm conditionally
--
-- p1 * p2 ==
-- xx := p1::Rep
-- empty? xx => p1
-- yy := p2::Rep
-- empty? yy => p2
-- zero? first(xx).k => first(xx).c * p2
-- zero? first(yy).k => p1 * first(yy).c
-- (first(xx).k > kBound) and (first(yy).k > kBound) and (#xx > kBound) and (#yy > kBound) =>
-- karatsubaOnce(p1,p2)$upmp
-- xx := reverse xx
-- res : Rep := empty()
-- for tx in xx repeat res:= rep pomopo!( res,tx.c,tx.k,p2)
-- res
univariate(p:%) == p pretend SparseUnivariatePolynomial(R)
multivariate(sup:SparseUnivariatePolynomial(R),v:SingletonAsOrderedSet) ==
sup pretend %
univariate(p:%,v:SingletonAsOrderedSet) ==
zero? p => 0
monomial(leadingCoefficient(p)::%,degree p) +
univariate(reductum p,v)
multivariate(supp:SparseUnivariatePolynomial(%),v:SingletonAsOrderedSet) ==
zero? supp => 0
lc:=leadingCoefficient supp
degree lc > 0 => error "bad form polynomial"
monomial(leadingCoefficient lc,degree supp) +
multivariate(reductum supp,v)
if R has FiniteFieldCategory and R has PolynomialFactorizationExplicit then
RXY ==> SparseUnivariatePolynomial SparseUnivariatePolynomial R
squareFreePolynomial pp ==
squareFree(pp)$UnivariatePolynomialSquareFree(%,FP)
factorPolynomial pp ==
(generalTwoFactor(pp pretend RXY)$TwoFactorize(R))
pretend Factored SparseUnivariatePolynomial %
factorSquareFreePolynomial pp ==
(generalTwoFactor(pp pretend RXY)$TwoFactorize(R))
pretend Factored SparseUnivariatePolynomial %
gcdPolynomial(pp,qq) == gcd(pp,qq)$FP
factor p == factor(p)$DistinctDegreeFactorize(R,%)
solveLinearPolynomialEquation(lpp,pp) ==
solveLinearPolynomialEquation(lpp, pp)$FiniteFieldSolveLinearPolynomialEquation(R,%,FP)
else if R has PolynomialFactorizationExplicit then
import PolynomialFactorizationByRecursionUnivariate(R,%)
solveLinearPolynomialEquation(lpp,pp)==
solveLinearPolynomialEquationByRecursion(lpp,pp)
factorPolynomial(pp) ==
factorByRecursion(pp)
factorSquareFreePolynomial(pp) ==
factorSquareFreeByRecursion(pp)
if R has IntegralDomain then
if R has approximate then
p1:% exquo p2:% ==
null p2 => error "Division by 0"
p2 = 1 => p1
p1=p2 => 1
--(p1.lastElt.c exquo p2.lastElt.c) case "failed" => "failed"
rout:= []@List(Term)
while not null p1 repeat
(a:= p1.first.c exquo p2.first.c)
a case "failed" => return "failed"
ee:= subtractIfCan(p1.first.k, p2.first.k)
ee case "failed" => return "failed"
p1:= fmecg(p1.rest, ee, a, p2.rest)
rout:= [[ee,a], :rout]
null p1 => reverse(rout)::% -- nreverse?
"failed"
else -- R not approximate
p1:% exquo p2:% ==
null p2 => error "Division by 0"
p2 = 1 => p1
--(p1.lastElt.c exquo p2.lastElt.c) case "failed" => "failed"
rout:= []@List(Term)
while not null p1 repeat
(a:= p1.first.c exquo p2.first.c)
a case "failed" => return "failed"
ee:= subtractIfCan(p1.first.k, p2.first.k)
ee case "failed" => return "failed"
p1:= fmecg(p1.rest, ee, a, p2.rest)
rout:= [[ee,a], :rout]
null p1 => reverse(rout)::% -- nreverse?
"failed"
fmecg(p1,e,r,p2) == -- p1 - r * X**e * p2
rout:%:= []
r:= - r
for tm in p2 repeat
e2:= e + tm.k
c2:= r * tm.c
c2 = 0 => "next term"
while not null p1 and p1.first.k > e2 repeat
(rout:=[p1.first,:rout]; p1:=p1.rest) --use PUSH and POP?
null p1 or p1.first.k < e2 => rout:=[[e2,c2],:rout]
if (u:=p1.first.c + c2) ~= 0 then rout:=[[e2, u],:rout]
p1:=p1.rest
NRECONC(rout,p1)$Lisp
pseudoRemainder(p1,p2) ==
null p2 => error "PseudoDivision by Zero"
null p1 => 0
co:=p2.first.c;
e:=p2.first.k;
p2:=p2.rest;
e1:=max(p1.first.k:Integer-e+1,0):NonNegativeInteger
while not null p1 repeat
if (u:=subtractIfCan(p1.first.k,e)) case "failed" then leave
p1:=fmecg(co * p1.rest, u, p1.first.c, p2)
e1:= (e1 - 1):NonNegativeInteger
e1 = 0 => p1
co ** e1 * p1
toutput(t1:Term,v:OutputForm):OutputForm ==
t1.k = 0 => t1.c :: OutputForm
if t1.k = 1
then mon:= v
else mon := v ** t1.k::OutputForm
t1.c = 1 => mon
t1.c = -1 and
((t1.c :: OutputForm) = (-1$Integer)::OutputForm)@Boolean => - mon
t1.c::OutputForm * mon
outputForm(p:%,v:OutputForm) ==
l: List(OutputForm)
l:=[toutput(t,v) for t in p]
null l => (0$Integer)::OutputForm -- else FreeModule 0 problems
reduce("+",l)
coerce(p:%):OutputForm == outputForm(p, "?"::OutputForm)
elt(p:%,val:R) ==
null p => 0$R
co:=p.first.c
n:=p.first.k
for tm in p.rest repeat
co:= co * val ** (n - (n:=tm.k)):NonNegativeInteger + tm.c
n = 0 => co
co * val ** n
elt(p:%,val:%) ==
null p => 0$%
coef:% := p.first.c :: %
n:=p.first.k
for tm in p.rest repeat
coef:= coef * val ** (n-(n:=tm.k)):NonNegativeInteger+(tm.c::%)
n = 0 => coef
coef * val ** n
monicDivide(p1:%,p2:%) ==
null p2 => error "monicDivide: division by 0"
leadingCoefficient p2 ~= 1 => error "Divisor Not Monic"
p2 = 1 => [p1,0]
null p1 => [0,0]
degree p1 < (n:=degree p2) => [0,p1]
rout:Rep := []
p2 := p2.rest
while not null p1 repeat
(u:=subtractIfCan(p1.first.k, n)) case "failed" => leave
rout:=[[u, p1.first.c], :rout]
p1:=fmecg(p1.rest, rout.first.k, rout.first.c, p2)
[reverse!(rout),p1]
if R has IntegralDomain then
discriminant(p) == discriminant(p)$PseudoRemainderSequence(R,%)
-- discriminant(p) ==
-- null p or zero?(p.first.k) => error "cannot take discriminant of constants"
-- dp:=differentiate p
-- corr:= p.first.c ** ((degree p - 1 - degree dp)::NonNegativeInteger)
-- (-1)**((p.first.k*(p.first.k-1)) quo 2):NonNegativeInteger
-- * (corr * resultant(p,dp) exquo p.first.c)::R
subResultantGcd(p1,p2) == subResultantGcd(p1,p2)$PseudoRemainderSequence(R,%)
-- subResultantGcd(p1,p2) == --args # 0, non-coef, prim, ans not prim
-- --see algorithm 1 (p. 4) of Brown's latest (unpublished) paper
-- if p1.first.k < p2.first.k then (p1,p2):=(p2,p1)
-- p:=pseudoRemainder(p1,p2)
-- co:=1$R;
-- e:= (p1.first.k - p2.first.k):NonNegativeInteger
-- while not null p and p.first.k ~= 0 repeat
-- p1:=p2; p2:=p; p:=pseudoRemainder(p1,p2)
-- null p or p.first.k = 0 => "enuf"
-- co:=(p1.first.c ** e exquo co ** max(0, (e-1))::NonNegativeInteger)::R
-- e:= (p1.first.k - p2.first.k):NonNegativeInteger; c1:=co**e
-- p:=[[tm.k,((tm.c exquo p1.first.c)::R exquo c1)::R] for tm in p]
-- if null p then p2 else 1$%
resultant(p1,p2) == resultant(p1,p2)$PseudoRemainderSequence(R,%)
-- resultant(p1,p2) == --SubResultant PRS Algorithm
-- null p1 or null p2 => 0$R
-- 0 = degree(p1) => ((first p1).c)**degree(p2)
-- 0 = degree(p2) => ((first p2).c)**degree(p1)
-- if p1.first.k < p2.first.k then
-- (if odd?(p1.first.k) then p1:=-p1; (p1,p2):=(p2,p1))
-- p:=pseudoRemainder(p1,p2)
-- co:=1$R; e:=(p1.first.k-p2.first.k):NonNegativeInteger
-- while not null p repeat
-- if not odd?(e) then p:=-p
-- p1:=p2; p2:=p; p:=pseudoRemainder(p1,p2)
-- co:=(p1.first.c ** e exquo co ** max(e:Integer-1,0):NonNegativeInteger)::R
-- e:= (p1.first.k - p2.first.k):NonNegativeInteger; c1:=co**e
-- p:=(p exquo ((leadingCoefficient p1) * c1))::%
-- degree p2 > 0 => 0$R
-- (p2.first.c**e exquo co**((e-1)::NonNegativeInteger))::R
if R has GcdDomain then
content(p) == if null p then 0$R else "gcd"/[tm.c for tm in p]
--make CONTENT more efficient?
primitivePart(p) ==
null p => p
ct :=content(p)
unitCanonical((p exquo ct)::%)
-- exquo present since % is now an IntegralDomain
gcd(p1,p2) ==
gcdPolynomial(p1 pretend SparseUnivariatePolynomial R,
p2 pretend SparseUnivariatePolynomial R) pretend %
if R has Field then
divide( p1, p2) ==
zero? p2 => error "Division by 0"
one? p2 => [p1,0]
ct:=inv(p2.first.c)
n:=p2.first.k
p2:=p2.rest
rout:=empty()$List(Term)
while p1 ~= 0 repeat
(u:=subtractIfCan(p1.first.k, n)) case "failed" => leave
rout:=[[u, ct * p1.first.c], :rout]
p1:=fmecg(p1.rest, rout.first.k, rout.first.c, p2)
[reverse!(rout),p1]
p / co == inv(co) * p
@
\section{package SUP2 SparseUnivariatePolynomialFunctions2}
<<package SUP2 SparseUnivariatePolynomialFunctions2>>=
)abbrev package SUP2 SparseUnivariatePolynomialFunctions2
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ This package lifts a mapping from coefficient rings R to S to
++ a mapping from sparse univariate polynomial over R to
++ a sparse univariate polynomial over S.
++ Note that the mapping is assumed
++ to send zero to zero, since it will only be applied to the non-zero
++ coefficients of the polynomial.
SparseUnivariatePolynomialFunctions2(R:Ring, S:Ring): with
map:(R->S,SparseUnivariatePolynomial R) -> SparseUnivariatePolynomial S
++ map(func, poly) creates a new polynomial by applying func to
++ every non-zero coefficient of the polynomial poly.
== add
map(f, p) == map(f, p)$UnivariatePolynomialCategoryFunctions2(R,
SparseUnivariatePolynomial R, S, SparseUnivariatePolynomial S)
@
\section{domain UP UnivariatePolynomial}
<<domain UP UnivariatePolynomial>>=
)abbrev domain UP UnivariatePolynomial
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions: Ring, monomial, coefficient, reductum, differentiate,
++ elt, map, resultant, discriminant
++ Related Constructors: SparseUnivariatePolynomial, MultivariatePolynomial
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ This domain represents univariate polynomials in some symbol
++ over arbitrary (not necessarily commutative) coefficient rings.
++ The representation is sparse
++ in the sense that only non-zero terms are represented.
++ Note: if the coefficient ring is a field, then this domain forms a euclidean domain.
UnivariatePolynomial(x:Symbol, R:Ring):
Join(UnivariatePolynomialCategory(R),CoercibleFrom Variable x) with
fmecg: (%,NonNegativeInteger,R,%) -> %
++ fmecg(p1,e,r,p2) finds X : p1 - r * X**e * p2
== SparseUnivariatePolynomial(R) add
Rep:=SparseUnivariatePolynomial(R)
coerce(p:%):OutputForm == outputForm(p, outputForm x)
coerce(v:Variable(x)):% == monomial(1, 1)
@
\section{package UP2 UnivariatePolynomialFunctions2}
<<package UP2 UnivariatePolynomialFunctions2>>=
)abbrev package UP2 UnivariatePolynomialFunctions2
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ This package lifts a mapping from coefficient rings R to S to
++ a mapping from \spadtype{UnivariatePolynomial}(x,R) to
++ \spadtype{UnivariatePolynomial}(y,S). Note that the mapping is assumed
++ to send zero to zero, since it will only be applied to the non-zero
++ coefficients of the polynomial.
UnivariatePolynomialFunctions2(x:Symbol, R:Ring, y:Symbol, S:Ring): with
map: (R -> S, UnivariatePolynomial(x,R)) -> UnivariatePolynomial(y,S)
++ map(func, poly) creates a new polynomial by applying func to
++ every non-zero coefficient of the polynomial poly.
== add
map(f, p) == map(f, p)$UnivariatePolynomialCategoryFunctions2(R,
UnivariatePolynomial(x, R), S, UnivariatePolynomial(y, S))
@
\section{package POLY2UP PolynomialToUnivariatePolynomial}
<<package POLY2UP PolynomialToUnivariatePolynomial>>=
)abbrev package POLY2UP PolynomialToUnivariatePolynomial
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ This package is primarily to help the interpreter do coercions.
++ It allows you to view a polynomial as a
++ univariate polynomial in one of its variables with
++ coefficients which are again a polynomial in all the
++ other variables.
PolynomialToUnivariatePolynomial(x:Symbol, R:Ring): with
univariate: (Polynomial R, Variable x) ->
UnivariatePolynomial(x, Polynomial R)
++ univariate(p, x) converts the polynomial p to a one of type
++ \spad{UnivariatePolynomial(x,Polynomial(R))}, ie. as a member of \spad{R[...][x]}.
== add
univariate(p, y) ==
q:SparseUnivariatePolynomial(Polynomial R) := univariate(p, x)
map(#1, q)$UnivariatePolynomialCategoryFunctions2(Polynomial R,
SparseUnivariatePolynomial Polynomial R, Polynomial R,
UnivariatePolynomial(x, Polynomial R))
@
\section{package UPSQFREE UnivariatePolynomialSquareFree}
<<package UPSQFREE UnivariatePolynomialSquareFree>>=
)abbrev package UPSQFREE UnivariatePolynomialSquareFree
++ Author: Dave Barton, Barry Trager
++ Date Created:
++ Date Last Updated:
++ Basic Functions: squareFree, squareFreePart
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ This package provides for square-free decomposition of
++ univariate polynomials over arbitrary rings, i.e.
++ a partial factorization such that each factor is a product
++ of irreducibles with multiplicity one and the factors are
++ pairwise relatively prime. If the ring
++ has characteristic zero, the result is guaranteed to satisfy
++ this condition. If the ring is an infinite ring of
++ finite characteristic, then it may not be possible to decide when
++ polynomials contain factors which are pth powers. In this
++ case, the flag associated with that polynomial is set to "nil"
++ (meaning that that polynomials are not guaranteed to be square-free).
UnivariatePolynomialSquareFree(RC:IntegralDomain,P):C == T
where
fUnion ==> Union("nil", "sqfr", "irred", "prime")
FF ==> Record(flg:fUnion, fctr:P, xpnt:Integer)
P:Join(UnivariatePolynomialCategory(RC),IntegralDomain) with
gcd: (%,%) -> %
++ gcd(p,q) computes the greatest-common-divisor of p and q.
C == with
squareFree: P -> Factored(P)
++ squareFree(p) computes the square-free factorization of the
++ univariate polynomial p. Each factor has no repeated roots, and the
++ factors are pairwise relatively prime.
squareFreePart: P -> P
++ squareFreePart(p) returns a polynomial which has the same
++ irreducible factors as the univariate polynomial p, but each
++ factor has multiplicity one.
BumInSepFFE: FF -> FF
++ BumInSepFFE(f) is a local function, exported only because
++ it has multiple conditional definitions.
T == add
if RC has CharacteristicZero then
squareFreePart(p:P) == (p exquo gcd(p, differentiate p))::P
else
squareFreePart(p:P) ==
unit(s := squareFree(p)$%) * */[f.factor for f in factors s]
if RC has FiniteFieldCategory then
BumInSepFFE(ffe:FF) ==
["sqfr", map(charthRoot,ffe.fctr), characteristic$P*ffe.xpnt]
else if RC has CharacteristicNonZero then
BumInSepFFE(ffe:FF) ==
np := multiplyExponents(ffe.fctr,characteristic$P:NonNegativeInteger)
(nthrp := charthRoot(np)) case nothing =>
["nil", np, ffe.xpnt]
["sqfr", nthrp, characteristic$P*ffe.xpnt]
else
BumInSepFFE(ffe:FF) ==
["nil",
multiplyExponents(ffe.fctr,characteristic$P:NonNegativeInteger),
ffe.xpnt]
if RC has CharacteristicZero then
squareFree(p:P) == --Yun's algorithm - see SYMSAC '76, p.27
--Note ci primitive is, so GCD's don't need to %do contents.
--Change gcd to return cofctrs also?
ci:=p; di:=differentiate(p); pi:=gcd(ci,di)
degree(pi)=0 =>
(u,c,a):=unitNormal(p)
makeFR(u,[["sqfr",c,1]])
i:NonNegativeInteger:=0; lffe:List FF:=[]
lcp := leadingCoefficient p
while degree(ci)~=0 repeat
ci:=(ci exquo pi)::P
di:=(di exquo pi)::P - differentiate(ci)
pi:=gcd(ci,di)
i:=i+1
degree(pi) > 0 =>
lcp:=(lcp exquo (leadingCoefficient(pi)**i))::RC
lffe:=[["sqfr",pi,i],:lffe]
makeFR(lcp::P,lffe)
else
squareFree(p:P) == --Musser's algorithm - see SYMSAC '76, p.27
--p MUST BE PRIMITIVE, Any characteristic.
--Note ci primitive, so GCD's don't need to %do contents.
--Change gcd to return cofctrs also?
ci := gcd(p,differentiate(p))
degree(ci)=0 =>
(u,c,a):=unitNormal(p)
makeFR(u,[["sqfr",c,1]])
di := (p exquo ci)::P
i:NonNegativeInteger:=0; lffe:List FF:=[]
dunit : P := 1
while degree(di)~=0 repeat
diprev := di
di := gcd(ci,di)
ci:=(ci exquo di)::P
i:=i+1
degree(diprev) = degree(di) =>
lc := (leadingCoefficient(diprev) exquo leadingCoefficient(di))::RC
dunit := lc**i * dunit
pi:=(diprev exquo di)::P
lffe:=[["sqfr",pi,i],:lffe]
dunit := dunit * di ** (i+1)
degree(ci)=0 => makeFR(dunit*ci,lffe)
redSqfr:=squareFree(divideExponents(ci,characteristic$P)::P)
lsnil:= [BumInSepFFE(ffe) for ffe in factorList redSqfr]
lffe:=append(lsnil,lffe)
makeFR(dunit*(unit redSqfr),lffe)
@
\section{package PSQFR PolynomialSquareFree}
<<package PSQFR PolynomialSquareFree>>=
)abbrev package PSQFR PolynomialSquareFree
++ Author:
++ Date Created:
++ Date Last Updated: November 1993, (P.Gianni)
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ This package computes square-free decomposition of multivariate
++ polynomials over a coefficient ring which is an arbitrary gcd domain.
++ The requirement on the coefficient domain guarantees that the \spadfun{content} can be
++ removed so that factors will be primitive as well as square-free.
++ Over an infinite ring of finite characteristic,it may not be possible to
++ guarantee that the factors are square-free.
PolynomialSquareFree(VarSet:OrderedSet,E,RC:GcdDomain,P):C == T where
E:OrderedAbelianMonoidSup
P:PolynomialCategory(RC,E,VarSet)
C == with
squareFree : P -> Factored P
++ squareFree(p) returns the square-free factorization of the
++ polynomial p. Each factor has no repeated roots, and the
++ factors are pairwise relatively prime.
T == add
SUP ==> SparseUnivariatePolynomial(P)
NNI ==> NonNegativeInteger
fUnion ==> Union("nil", "sqfr", "irred", "prime")
FF ==> Record(flg:fUnion, fctr:P, xpnt:Integer)
finSqFr : (P,List VarSet) -> Factored P
pthPower : P -> Factored P
pPolRoot : P -> P
putPth : P -> P
chrc:=characteristic$RC
if RC has CharacteristicNonZero then
-- find the p-th root of a polynomial
pPolRoot(f:P) : P ==
lvar:=variables f
empty? lvar => f
mv:=first lvar
uf:=univariate(f,mv)
uf:=divideExponents(uf,chrc)::SUP
uf:=map(pPolRoot,uf)
multivariate(uf,mv)
-- substitute variables with their p-th power
putPth(f:P) : P ==
lvar:=variables f
empty? lvar => f
mv:=first lvar
uf:=univariate(f,mv)
uf:=multiplyExponents(uf,chrc)::SUP
uf:=map(putPth,uf)
multivariate(uf,mv)
-- the polynomial is a perfect power
pthPower(f:P) : Factored P ==
proot : P := 0
isSq : Boolean := false
if (g:=charthRoot f) case nothing then proot:=pPolRoot(f)
else
proot := g :: P
isSq := true
psqfr:=finSqFr(proot,variables f)
isSq =>
makeFR((unit psqfr)**chrc,[[u.flg,u.fctr,
(u.xpnt)*chrc] for u in factorList psqfr])
makeFR((unit psqfr),[["nil",putPth u.fctr,u.xpnt]
for u in factorList psqfr])
-- compute the square free decomposition, finite characteristic case
finSqFr(f:P,lvar:List VarSet) : Factored P ==
empty? lvar => pthPower(f)
mv:=first lvar
lvar:=lvar.rest
differentiate(f,mv)=0 => finSqFr(f,lvar)
uf:=univariate(f,mv)
cont := content uf
cont1:P:=1
uf := (uf exquo cont)::SUP
squf := squareFree(uf)$UnivariatePolynomialSquareFree(P,SUP)
pfaclist:List FF :=[]
for u in factorList squf repeat
uexp:NNI:=(u.xpnt):NNI
u.flg = "sqfr" => -- the square free factor is OK
pfaclist:= cons([u.flg,multivariate(u.fctr,mv),uexp],
pfaclist)
--listfin1:= finSqFr(multivariate(u.fctr,mv),lvar)
listfin1:= squareFree multivariate(u.fctr,mv)
flistfin1:=[[uu.flg,uu.fctr,uu.xpnt*uexp]
for uu in factorList listfin1]
cont1:=cont1*((unit listfin1)**uexp)
pfaclist:=append(flistfin1,pfaclist)
cont:=cont*cont1
cont ~= 1 =>
sqp := squareFree cont
pfaclist:= append (factorList sqp,pfaclist)
makeFR(unit(sqp)*coefficient(unit squf,0),pfaclist)
makeFR(coefficient(unit squf,0),pfaclist)
squareFree(p:P) ==
mv:=mainVariable p
mv case "failed" => makeFR(p,[])$Factored(P)
characteristic$RC ~=0 => finSqFr(p,variables p)
up:=univariate(p,mv)
cont := content up
up := (up exquo cont)::SUP
squp := squareFree(up)$UnivariatePolynomialSquareFree(P,SUP)
pfaclist:List FF :=
[[u.flg,multivariate(u.fctr,mv),u.xpnt]
for u in factorList squp]
cont ~= 1 =>
sqp := squareFree cont
makeFR(unit(sqp)*coefficient(unit squp,0),
append(factorList sqp, pfaclist))
makeFR(coefficient(unit squp,0),pfaclist)
@
\section{package UPMP UnivariatePolynomialMultiplicationPackage}
<<package UPMP UnivariatePolynomialMultiplicationPackage>>=
)abbrev package UPMP UnivariatePolynomialMultiplicationPackage
++ Author: Marc Moreno Maza
++ Date Created: 14.08.2000
++ Description:
++ This package implements Karatsuba's trick for multiplying
++ (large) univariate polynomials. It could be improved with
++ a version doing the work on place and also with a special
++ case for squares. We've done this in Basicmath, but we
++ believe that this out of the scope of AXIOM.
UnivariatePolynomialMultiplicationPackage(R: Ring, U: UnivariatePolynomialCategory(R)): C == T
where
HL ==> Record(quotient:U,remainder:U)
C == with
noKaratsuba: (U, U) -> U
++ \spad{noKaratsuba(a,b)} returns \spad{a*b} without
++ using Karatsuba's trick at all.
karatsubaOnce: (U, U) -> U
++ \spad{karatsuba(a,b)} returns \spad{a*b} by applying
++ Karatsuba's trick once. The other multiplications
++ are performed by calling \spad{*} from \spad{U}.
karatsuba: (U, U, NonNegativeInteger, NonNegativeInteger) -> U;
++ \spad{karatsuba(a,b,l,k)} returns \spad{a*b} by applying
++ Karatsuba's trick provided that both \spad{a} and \spad{b}
++ have at least \spad{l} terms and \spad{k > 0} holds
++ and by calling \spad{noKaratsuba} otherwise. The other
++ multiplications are performed by recursive calls with
++ the same third argument and \spad{k-1} as fourth argument.
T == add
noKaratsuba(a,b) ==
zero? a => a
zero? b => b
zero?(degree(a)) => leadingCoefficient(a) * b
zero?(degree(b)) => a * leadingCoefficient(b)
lu: List(U) := reverse monomials(a)
res: U := 0;
for u in lu repeat
res := pomopo!(res, leadingCoefficient(u), degree(u), b)
res
karatsubaOnce(a:U,b:U): U ==
da := minimumDegree(a)
db := minimumDegree(b)
if not zero? da then a := shiftRight(a,da)
if not zero? db then b := shiftRight(b,db)
d := da + db
n: NonNegativeInteger := min(degree(a),degree(b)) quo 2
rec: HL := karatsubaDivide(a, n)
ha := rec.quotient
la := rec.remainder
rec := karatsubaDivide(b, n)
hb := rec.quotient
lb := rec.remainder
w: U := (ha - la) * (lb - hb)
u: U := la * lb
v: U := ha * hb
w := w + (u + v)
w := shiftLeft(w,n) + u
zero? d => shiftLeft(v,2*n) + w
shiftLeft(v,2*n + d) + shiftLeft(w,d)
karatsuba(a:U,b:U,l:NonNegativeInteger,k:NonNegativeInteger): U ==
zero? k => noKaratsuba(a,b)
degree(a) < l => noKaratsuba(a,b)
degree(b) < l => noKaratsuba(a,b)
numberOfMonomials(a) < l => noKaratsuba(a,b)
numberOfMonomials(b) < l => noKaratsuba(a,b)
da := minimumDegree(a)
db := minimumDegree(b)
if not zero? da then a := shiftRight(a,da)
if not zero? db then b := shiftRight(b,db)
d := da + db
n: NonNegativeInteger := min(degree(a),degree(b)) quo 2
k := subtractIfCan(k,1)::NonNegativeInteger
rec: HL := karatsubaDivide(a, n)
ha := rec.quotient
la := rec.remainder
rec := karatsubaDivide(b, n)
hb := rec.quotient
lb := rec.remainder
w: U := karatsuba(ha - la, lb - hb, l, k)
u: U := karatsuba(la, lb, l, k)
v: U := karatsuba(ha, hb, l, k)
w := w + (u + v)
w := shiftLeft(w,n) + u
zero? d => shiftLeft(v,2*n) + w
shiftLeft(v,2*n + d) + shiftLeft(w,d)
@
\section{License}
<<license>>=
--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
--All rights reserved.
--
--Redistribution and use in source and binary forms, with or without
--modification, are permitted provided that the following conditions are
--met:
--
-- - Redistributions of source code must retain the above copyright
-- notice, this list of conditions and the following disclaimer.
--
-- - Redistributions in binary form must reproduce the above copyright
-- notice, this list of conditions and the following disclaimer in
-- the documentation and/or other materials provided with the
-- distribution.
--
-- - Neither the name of The Numerical ALgorithms Group Ltd. nor the
-- names of its contributors may be used to endorse or promote products
-- derived from this software without specific prior written permission.
--
--THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS
--IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
--TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
--PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
--OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
--EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
--PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
--PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
--LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
--NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
@
<<*>>=
<<license>>
<<domain FM FreeModule>>
<<domain PR PolynomialRing>>
<<package UPSQFREE UnivariatePolynomialSquareFree>>
<<package PSQFR PolynomialSquareFree>>
<<package UPMP UnivariatePolynomialMultiplicationPackage>>
<<domain SUP SparseUnivariatePolynomial>>
<<package SUP2 SparseUnivariatePolynomialFunctions2>>
<<domain UP UnivariatePolynomial>>
<<package UP2 UnivariatePolynomialFunctions2>>
<<package POLY2UP PolynomialToUnivariatePolynomial>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
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