/usr/share/axiom-20170501/src/algebra/PR.spad is in axiom-source 20170501-3.
This file is owned by root:root, with mode 0o644.
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++ Author: Dave Barton, James Davenport, Barry Trager
++ Date Created:
++ Date Last Updated: 14.08.2000. Improved exponentiation [MMM/TTT]
++ 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,E) : SIG == CODE where
R : Ring
E : OrderedAbelianMonoid
SIG ==> 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.
CODE ==> 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
times!: (R, %) -> %
times: (R, E, %) -> %
entireRing? := R has EntireRing
times!(r: R, x: %): % ==
res, endcell, newend, xx: Rep
if entireRing? then
for tx in x repeat tx.c := r*tx.c
else
xx := x
res := empty()
while not empty? xx repeat
tx := first xx
tx.c := r * tx.c
if zero? tx.c then
xx := rest xx
else
newend := xx
xx := rest xx
if empty? res then
res := newend
endcell := res
else
qsetrest!(endcell, newend)
endcell := newend
res;
--- 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(x1+->termTimes(tco, tex, x1), 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()
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
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
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 CommutativeRing then
p ** np == p ** (np pretend NonNegativeInteger)
p ^ np == p ** (np pretend NonNegativeInteger)
p ^ nn == p ** nn
p ** nn ==
null p => 0
zero? nn => 1
(nn = 1) => 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
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
|