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--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.
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-- - 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,
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)abbrev category OREPCAT UnivariateSkewPolynomialCategory
++ Author: Manuel Bronstein, Jean Della Dora, Stephen M. Watt
++ Date Created: 19 October 1993
++ Date Last Updated: 1 February 1994
++ Description:
++ This is the category of univariate skew polynomials over an Ore
++ coefficient ring.
++ The multiplication is given by \spad{x a = \sigma(a) x + \delta a}.
++ This category is an evolution of the types
++ MonogenicLinearOperator, OppositeMonogenicLinearOperator, and
++ NonCommutativeOperatorDivision
++ developped by Jean Della Dora and Stephen M. Watt.
UnivariateSkewPolynomialCategory(R:Ring):
Category == Join(Ring, BiModule(R, R), FullyRetractableTo R) with
degree: $ -> NonNegativeInteger
++ degree(l) is \spad{n} if
++ \spad{l = sum(monomial(a(i),i), i = 0..n)}.
minimumDegree: $ -> NonNegativeInteger
++ minimumDegree(l) is the smallest \spad{k} such that
++ \spad{a(k) ~= 0} if
++ \spad{l = sum(monomial(a(i),i), i = 0..n)}.
leadingCoefficient: $ -> R
++ leadingCoefficient(l) is \spad{a(n)} if
++ \spad{l = sum(monomial(a(i),i), i = 0..n)}.
reductum: $ -> $
++ reductum(l) is \spad{l - monomial(a(n),n)} if
++ \spad{l = sum(monomial(a(i),i), i = 0..n)}.
coefficient: ($, NonNegativeInteger) -> R
++ coefficient(l,k) is \spad{a(k)} if
++ \spad{l = sum(monomial(a(i),i), i = 0..n)}.
monomial: (R, NonNegativeInteger) -> $
++ monomial(c,k) produces c times the k-th power of
++ the generating operator, \spad{monomial(1,1)}.
coefficients: % -> List R
++ coefficients(l) returns the list of all the nonzero
++ coefficients of l.
apply: (%, R, R) -> R
++ apply(p, c, m) returns \spad{p(m)} where the action is
++ given by \spad{x m = c sigma(m) + delta(m)}.
if R has CommutativeRing then Algebra R
if R has IntegralDomain then
exquo: (%, R) -> Union(%, "failed")
++ exquo(l, a) returns the exact quotient of l by a,
++ returning \axiom{"failed"} if this is not possible.
monicLeftDivide: (%, %) -> Record(quotient: %, remainder: %)
++ monicLeftDivide(a,b) returns the pair \spad{[q,r]} such that
++ \spad{a = b*q + r} and the degree of \spad{r} is
++ less than the degree of \spad{b}.
++ \spad{b} must be monic.
++ This process is called ``left division''.
monicRightDivide: (%, %) -> Record(quotient: %, remainder: %)
++ monicRightDivide(a,b) returns the pair \spad{[q,r]} such that
++ \spad{a = q*b + r} and the degree of \spad{r} is
++ less than the degree of \spad{b}.
++ \spad{b} must be monic.
++ This process is called ``right division''.
if R has GcdDomain then
content: % -> R
++ content(l) returns the gcd of all the coefficients of l.
primitivePart: % -> %
++ primitivePart(l) returns l0 such that \spad{l = a * l0}
++ for some a in R, and \spad{content(l0) = 1}.
if R has Field then
leftDivide: (%, %) -> Record(quotient: %, remainder: %)
++ leftDivide(a,b) returns the pair \spad{[q,r]} such that
++ \spad{a = b*q + r} and the degree of \spad{r} is
++ less than the degree of \spad{b}.
++ This process is called ``left division''.
leftQuotient: (%, %) -> %
++ leftQuotient(a,b) computes the pair \spad{[q,r]} such that
++ \spad{a = b*q + r} and the degree of \spad{r} is
++ less than the degree of \spad{b}.
++ The value \spad{q} is returned.
leftRemainder: (%, %) -> %
++ leftRemainder(a,b) computes the pair \spad{[q,r]} such that
++ \spad{a = b*q + r} and the degree of \spad{r} is
++ less than the degree of \spad{b}.
++ The value \spad{r} is returned.
leftExactQuotient:(%, %) -> Union(%, "failed")
++ leftExactQuotient(a,b) computes the value \spad{q}, if it exists,
++ such that \spad{a = b*q}.
leftGcd: (%, %) -> %
++ leftGcd(a,b) computes the value \spad{g} of highest degree
++ such that
++ \spad{a = g*aa}
++ \spad{b = g*bb}
++ for some values \spad{aa} and \spad{bb}.
++ The value \spad{g} is computed using left-division.
leftExtendedGcd: (%, %) -> Record(coef1:%, coef2:%, generator:%)
++ leftExtendedGcd(a,b) returns \spad{[c,d]} such that
++ \spad{g = a * c + b * d = leftGcd(a, b)}.
rightLcm: (%, %) -> %
++ rightLcm(a,b) computes the value \spad{m} of lowest degree
++ such that \spad{m = a*aa = b*bb} for some values
++ \spad{aa} and \spad{bb}. The value \spad{m} is
++ computed using left-division.
rightDivide: (%, %) -> Record(quotient: %, remainder: %)
++ rightDivide(a,b) returns the pair \spad{[q,r]} such that
++ \spad{a = q*b + r} and the degree of \spad{r} is
++ less than the degree of \spad{b}.
++ This process is called ``right division''.
rightQuotient: (%, %) -> %
++ rightQuotient(a,b) computes the pair \spad{[q,r]} such that
++ \spad{a = q*b + r} and the degree of \spad{r} is
++ less than the degree of \spad{b}.
++ The value \spad{q} is returned.
rightRemainder: (%, %) -> %
++ rightRemainder(a,b) computes the pair \spad{[q,r]} such that
++ \spad{a = q*b + r} and the degree of \spad{r} is
++ less than the degree of \spad{b}.
++ The value \spad{r} is returned.
rightExactQuotient:(%, %) -> Union(%, "failed")
++ rightExactQuotient(a,b) computes the value \spad{q}, if it exists
++ such that \spad{a = q*b}.
rightGcd: (%, %) -> %
++ rightGcd(a,b) computes the value \spad{g} of highest degree
++ such that
++ \spad{a = aa*g}
++ \spad{b = bb*g}
++ for some values \spad{aa} and \spad{bb}.
++ The value \spad{g} is computed using right-division.
rightExtendedGcd: (%, %) -> Record(coef1:%, coef2:%, generator:%)
++ rightExtendedGcd(a,b) returns \spad{[c,d]} such that
++ \spad{g = c * a + d * b = rightGcd(a, b)}.
leftLcm: (%, %) -> %
++ leftLcm(a,b) computes the value \spad{m} of lowest degree
++ such that \spad{m = aa*a = bb*b} for some values
++ \spad{aa} and \spad{bb}. The value \spad{m} is
++ computed using right-division.
add
coerce(x:R):% == monomial(x, 0)
coefficients l ==
ans:List(R) := empty()
while l ~= 0 repeat
ans := concat(leadingCoefficient l, ans)
l := reductum l
ans
a:R * y:% ==
z:% := 0
while y ~= 0 repeat
z := z + monomial(a * leadingCoefficient y, degree y)
y := reductum y
z
retractIfCan(x:%):Union(R, "failed") ==
zero? x or zero? degree x => leadingCoefficient x
"failed"
if R has IntegralDomain then
l exquo a ==
ans:% := 0
while l ~= 0 repeat
(u := (leadingCoefficient(l) exquo a)) case "failed" =>
return "failed"
ans := ans + monomial(u::R, degree l)
l := reductum l
ans
if R has GcdDomain then
content l == gcd coefficients l
primitivePart l == (l exquo content l)::%
if R has Field then
leftEEA: (%, %) -> Record(gcd:%, coef1:%, coef2:%, lcm:%)
rightEEA: (%, %) -> Record(gcd:%, coef1:%, coef2:%, lcm:%)
ncgcd: (%, %, (%, %) -> %) -> %
nclcm: (%, %, (%, %) -> Record(gcd:%, coef1:%, coef2:%, lcm:%)) -> %
exactQuotient: Record(quotient:%, remainder:%) -> Union(%, "failed")
extended: (%, %, (%, %) -> Record(gcd:%, coef1:%, coef2:%, lcm:%)) ->
Record(coef1:%, coef2:%, generator:%)
leftQuotient(a, b) == leftDivide(a,b).quotient
leftRemainder(a, b) == leftDivide(a,b).remainder
leftExtendedGcd(a, b) == extended(a, b, leftEEA)
rightLcm(a, b) == nclcm(a, b, leftEEA)
rightQuotient(a, b) == rightDivide(a,b).quotient
rightRemainder(a, b) == rightDivide(a,b).remainder
rightExtendedGcd(a, b) == extended(a, b, rightEEA)
leftLcm(a, b) == nclcm(a, b, rightEEA)
leftExactQuotient(a, b) == exactQuotient leftDivide(a, b)
rightExactQuotient(a, b) == exactQuotient rightDivide(a, b)
rightGcd(a, b) == ncgcd(a, b, rightRemainder)
leftGcd(a, b) == ncgcd(a, b, leftRemainder)
exactQuotient qr == (zero?(qr.remainder) => qr.quotient; "failed")
-- returns [g = leftGcd(a, b), c, d, l = rightLcm(a, b)]
-- such that g := a c + b d
leftEEA(a, b) ==
a0 := a
u0:% := v:% := 1
v0:% := u:% := 0
while b ~= 0 repeat
qr := leftDivide(a, b)
(a, b) := (b, qr.remainder)
(u0, u):= (u, u0 - u * qr.quotient)
(v0, v):= (v, v0 - v * qr.quotient)
[a, u0, v0, a0 * u]
ncgcd(a, b, ncrem) ==
zero? a => b
zero? b => a
degree a < degree b => ncgcd(b, a, ncrem)
while b ~= 0 repeat (a, b) := (b, ncrem(a, b))
a
extended(a, b, eea) ==
zero? a => [0, 1, b]
zero? b => [1, 0, a]
degree a < degree b =>
rec := eea(b, a)
[rec.coef2, rec.coef1, rec.gcd]
rec := eea(a, b)
[rec.coef1, rec.coef2, rec.gcd]
nclcm(a, b, eea) ==
zero? a or zero? b => 0
degree a < degree b => nclcm(b, a, eea)
rec := eea(a, b)
rec.lcm
-- returns [g = rightGcd(a, b), c, d, l = leftLcm(a, b)]
-- such that g := a c + b d
rightEEA(a, b) ==
a0 := a
u0:% := v:% := 1
v0:% := u:% := 0
while b ~= 0 repeat
qr := rightDivide(a, b)
(a, b) := (b, qr.remainder)
(u0, u):= (u, u0 - qr.quotient * u)
(v0, v):= (v, v0 - qr.quotient * v)
[a, u0, v0, u * a0]
)abbrev package APPLYORE ApplyUnivariateSkewPolynomial
++ Author: Manuel Bronstein
++ Date Created: 7 December 1993
++ Date Last Updated: 1 February 1994
++ Description:
++ \spad{ApplyUnivariateSkewPolynomial} (internal) allows univariate
++ skew polynomials to be applied to appropriate modules.
ApplyUnivariateSkewPolynomial(R:Ring, M: LeftModule R,
P: UnivariateSkewPolynomialCategory R): with
apply: (P, M -> M, M) -> M
++ apply(p, f, m) returns \spad{p(m)} where the action is given
++ by \spad{x m = f(m)}.
++ \spad{f} must be an R-pseudo linear map on M.
== add
apply(p, f, m) ==
w:M := 0
mn:M := m
for i in 0..degree p repeat
w := w + coefficient(p, i) * mn
mn := f mn
w
import Integer
import NonNegativeInteger
)abbrev domain AUTOMOR Automorphism
++ Author: Manuel Bronstein
++ Date Created: 31 January 1994
++ Date Last Updated: 31 January 1994
++ References:
++ Description:
++ Automorphism R is the multiplicative group of automorphisms of R.
-- In fact, non-invertible endomorphism are allowed as partial functions.
-- This domain is noncanonical in that f*f^{-1} will be the identity
-- function but won't be equal to 1.
Automorphism(R:Ring): Join(Group, Eltable(R, R)) with
morphism: (R -> R) -> %
++ morphism(f) returns the non-invertible morphism given by f.
morphism: (R -> R, R -> R) -> %
++ morphism(f, g) returns the invertible morphism given by f, where
++ g is the inverse of f..
morphism: ((R, Integer) -> R) -> %
++ morphism(f) returns the morphism given by \spad{f^n(x) = f(x,n)}.
== add
Rep == (R, Integer) -> R
err: R -> R
ident: (R, Integer) -> R
iter: (R -> R, NonNegativeInteger, R) -> R
iterat: (R -> R, R -> R, Integer, R) -> R
apply: (Rep, R, Integer) -> R
1 == per ident
err r == error "Morphism is not invertible"
ident(r, n) == r
f = g == %peq(f,g)$Foreign(Builtin)
elt(f, r) == apply(rep f, r, 1)
inv f == per apply(rep f, #1, - #2)
(f: %) ** (n: Integer) == per apply(rep f, #1, n * #2)
coerce(f:%):OutputForm == message("R -> R")
morphism(f:(R, Integer) -> R):% == per f
morphism(f:R -> R):% == morphism(f, err)
morphism(f, g) == per iterat(f, g, #2, #1)
apply(f, r, n) == f(r, n)
iterat(f, g, n, r) ==
negative? n => iter(g, (-n)::NonNegativeInteger, r)
iter(f, n::NonNegativeInteger, r)
iter(f, n, r) ==
for i in 1..n repeat r := f r
r
f * g ==
f = g => f**2
per iterat(f g #1, (inv g)(inv f) #1, #2, #1)
)abbrev package OREPCTO UnivariateSkewPolynomialCategoryOps
++ Author: Manuel Bronstein
++ Date Created: 1 February 1994
++ Date Last Updated: 1 February 1994
++ Description:
++ \spad{UnivariateSkewPolynomialCategoryOps} provides products and
++ divisions of univariate skew polynomials.
-- Putting those operations here rather than defaults in OREPCAT allows
-- OREPCAT to be defined independently of sigma and delta.
-- MB 2/94
UnivariateSkewPolynomialCategoryOps(R, C): Exports == Implementation where
R: Ring
C: UnivariateSkewPolynomialCategory R
N ==> NonNegativeInteger
MOR ==> Automorphism R
QUOREM ==> Record(quotient: C, remainder: C)
Exports ==> with
times: (C, C, MOR, R -> R) -> C
++ times(p, q, sigma, delta) returns \spad{p * q}.
++ \spad{\sigma} and \spad{\delta} are the maps to use.
apply: (C, R, R, MOR, R -> R) -> R
++ apply(p, c, m, sigma, delta) returns \spad{p(m)} where the action
++ is given by \spad{x m = c sigma(m) + delta(m)}.
if R has IntegralDomain then
monicLeftDivide: (C, C, MOR) -> QUOREM
++ monicLeftDivide(a, b, sigma) returns the pair \spad{[q,r]}
++ such that \spad{a = b*q + r} and the degree of \spad{r} is
++ less than the degree of \spad{b}.
++ \spad{b} must be monic.
++ This process is called ``left division''.
++ \spad{\sigma} is the morphism to use.
monicRightDivide: (C, C, MOR) -> QUOREM
++ monicRightDivide(a, b, sigma) returns the pair \spad{[q,r]}
++ such that \spad{a = q*b + r} and the degree of \spad{r} is
++ less than the degree of \spad{b}.
++ \spad{b} must be monic.
++ This process is called ``right division''.
++ \spad{\sigma} is the morphism to use.
if R has Field then
leftDivide: (C, C, MOR) -> QUOREM
++ leftDivide(a, b, sigma) returns the pair \spad{[q,r]} such
++ that \spad{a = b*q + r} and the degree of \spad{r} is
++ less than the degree of \spad{b}.
++ This process is called ``left division''.
++ \spad{\sigma} is the morphism to use.
rightDivide: (C, C, MOR) -> QUOREM
++ rightDivide(a, b, sigma) returns the pair \spad{[q,r]} such
++ that \spad{a = q*b + r} and the degree of \spad{r} is
++ less than the degree of \spad{b}.
++ This process is called ``right division''.
++ \spad{\sigma} is the morphism to use.
Implementation ==> add
termPoly: (R, N, C, MOR, R -> R) -> C
localLeftDivide : (C, C, MOR, R) -> QUOREM
localRightDivide: (C, C, MOR, R) -> QUOREM
times(x, y, sigma, delta) ==
zero? y => 0
z:C := 0
while x ~= 0 repeat
z := z + termPoly(leadingCoefficient x, degree x, y, sigma, delta)
x := reductum x
z
termPoly(a, n, y, sigma, delta) ==
zero? y => 0
(u := subtractIfCan(n, 1)) case "failed" => a * y
n1 := u::N
z:C := 0
while y ~= 0 repeat
m := degree y
b := leadingCoefficient y
z := z + termPoly(a, n1, monomial(sigma b, m + 1), sigma, delta)
+ termPoly(a, n1, monomial(delta b, m), sigma, delta)
y := reductum y
z
apply(p, c, x, sigma, delta) ==
w:R := 0
xn:R := x
for i in 0..degree p repeat
w := w + coefficient(p, i) * xn
xn := c * sigma xn + delta xn
w
-- localLeftDivide(a, b) returns [q, r] such that a = q b + r
-- b1 is the inverse of the leadingCoefficient of b
localLeftDivide(a, b, sigma, b1) ==
zero? b => error "leftDivide: division by 0"
zero? a or
(n := subtractIfCan(degree(a),(m := degree b))) case "failed" =>
[0,a]
q := monomial((sigma**(-m))(b1 * leadingCoefficient a), n::N)
qr := localLeftDivide(a - b * q, b, sigma, b1)
[q + qr.quotient, qr.remainder]
-- localRightDivide(a, b) returns [q, r] such that a = q b + r
-- b1 is the inverse of the leadingCoefficient of b
localRightDivide(a, b, sigma, b1) ==
zero? b => error "rightDivide: division by 0"
zero? a or
(n := subtractIfCan(degree(a),(m := degree b))) case "failed" =>
[0,a]
q := monomial(leadingCoefficient(a) * (sigma**n) b1, n::N)
qr := localRightDivide(a - q * b, b, sigma, b1)
[q + qr.quotient, qr.remainder]
if R has IntegralDomain then
monicLeftDivide(a, b, sigma) ==
unit?(u := leadingCoefficient b) =>
localLeftDivide(a, b, sigma, recip(u)::R)
error "monicLeftDivide: divisor is not monic"
monicRightDivide(a, b, sigma) ==
unit?(u := leadingCoefficient b) =>
localRightDivide(a, b, sigma, recip(u)::R)
error "monicRightDivide: divisor is not monic"
if R has Field then
leftDivide(a, b, sigma) ==
localLeftDivide(a, b, sigma, inv leadingCoefficient b)
rightDivide(a, b, sigma) ==
localRightDivide(a, b, sigma, inv leadingCoefficient b)
)abbrev domain ORESUP SparseUnivariateSkewPolynomial
++ Author: Manuel Bronstein
++ Date Created: 19 October 1993
++ Date Last Updated: September, 2008
++ Description:
++ This is the domain of sparse univariate skew polynomials over an Ore
++ coefficient field.
++ The multiplication is given by \spad{x a = \sigma(a) x + \delta a}.
SparseUnivariateSkewPolynomial(R:Ring, sigma:Automorphism R, delta: R -> R):
UnivariateSkewPolynomialCategory R with
outputForm: (%, OutputForm) -> OutputForm
++ outputForm(p, x) returns the output form of p using x for the
++ otherwise anonymous variable.
== SparseUnivariatePolynomial R add
import UnivariateSkewPolynomialCategoryOps(R, %)
x:% * y:% == times(x, y, sigma, delta)
apply(p, c, r) == apply(p, c, r, sigma, delta)
x:% ** n:PositiveInteger == expt(x,n)$RepeatedSquaring(%)
x:% ** n:NonNegativeInteger ==
zero? n => 1
expt(x,n::PositiveInteger)$RepeatedSquaring(%)
if R has IntegralDomain then
monicLeftDivide(a, b) == monicLeftDivide(a, b, sigma)
monicRightDivide(a, b) == monicRightDivide(a, b, sigma)
if R has Field then
leftDivide(a, b) == leftDivide(a, b, sigma)
rightDivide(a, b) == rightDivide(a, b, sigma)
)abbrev domain OREUP UnivariateSkewPolynomial
++ Author: Manuel Bronstein
++ Date Created: 19 October 1993
++ Date Last Updated: 1 February 1994
++ Description:
++ This is the domain of univariate skew polynomials over an Ore
++ coefficient field in a named variable.
++ The multiplication is given by \spad{x a = \sigma(a) x + \delta a}.
UnivariateSkewPolynomial(x:Symbol, R:Ring, sigma:Automorphism R, delta: R -> R):
Join(UnivariateSkewPolynomialCategory R,CoercibleFrom Variable x)
== SparseUnivariateSkewPolynomial(R, sigma, delta) add
Rep := SparseUnivariateSkewPolynomial(R, sigma, delta)
coerce(v:Variable(x)):% == monomial(1, 1)
coerce(p:%):OutputForm == outputForm(p, outputForm x)$Rep
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