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modring.spad
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modring.spad
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)abbrev domain MODRING ModularRing
++ Author: P.Gianni, B.Trager
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ These domains are used for the factorization and gcds
++ of univariate polynomials over the integers in order to work modulo
++ different primes.
++ See \spadtype{EuclideanModularRing} , \spadtype{ModularField}
ModularRing(R, Mod, reduction : (R, Mod) -> R,
merge:(Mod,Mod) -> Union(Mod,"failed"),
exactQuo : (R,R,Mod) -> Union(R,"failed")) : C == T
where
R : CommutativeRing
Mod : AbelianMonoid
C == Ring with
modulus : % -> Mod
++ modulus(x) \undocumented
coerce : % -> R
++ coerce(x) \undocumented
reduce : (R, Mod) -> %
++ reduce(r, m) \undocumented
exQuo : (%,%) -> Union(%,"failed")
++ exQuo(x, y) \undocumented
recip : % -> Union(%,"failed")
++ recip(x) \undocumented
inv : % -> %
++ inv(x) \undocumented
T == add
--representation
Rep := Record(val : R, modulo : Mod)
--declarations
x, y : %
--define
modulus(x) == x.modulo
coerce(x : %) : R == x.val
coerce(i : Integer) : % == [i::R, 0]$Rep
i : Integer * x : % == (i::%)*x
coerce(x) : OutputForm == (x.val)::OutputForm
reduce (a : R, m : Mod) == [reduction(a, m), m]$Rep
characteristic() : NonNegativeInteger == characteristic()$R
0 == [0$R, 0$Mod]$Rep
1 == [1$R, 0$Mod]$Rep
zero? x == zero? x.val
one? x == (x.val = 1)
newmodulo(m1 : Mod, m2 : Mod) : Mod ==
r := merge(m1, m2)
r case "failed" => error "incompatible moduli"
r
x = y ==
x.val = y.val => true
x.modulo = y.modulo => false
(x-y).val = 0
x+y == reduce((x.val +$R y.val), newmodulo(x.modulo, y.modulo))
x-y == reduce((x.val -$R y.val), newmodulo(x.modulo, y.modulo))
-x == reduce ((-$R x.val), x.modulo)
x*y == reduce((x.val *$R y.val), newmodulo(x.modulo, y.modulo))
exQuo(x, y) ==
xm := x.modulo
if xm ~=$Mod y.modulo then xm := newmodulo(xm, y.modulo)
r := exactQuo(x.val, y.val, xm)
r case "failed"=> "failed"
[r@R, xm]$Rep
--if R has EuclideanDomain then
recip x ==
r := exactQuo(1$R, x.val, x.modulo)
r case "failed" => "failed"
[r, x.modulo]$Rep
inv x ==
if (u := recip x) case "failed" then error("not invertible")
else u
)abbrev domain EMR EuclideanModularRing
++ Description:
++ These domains are used for the factorization and gcds
++ of univariate polynomials over the integers in order to work modulo
++ different primes.
++ See \spadtype{ModularRing}, \spadtype{ModularField}
EuclideanModularRing(S, R, Mod, reduction : (R, Mod) -> R,
merge:(Mod,Mod) -> Union(Mod,"failed"),
exactQuo : (R,R,Mod) -> Union(R,"failed")) : C == T
where
S : CommutativeRing
R : UnivariatePolynomialCategory S
Mod : AbelianMonoid
C == EuclideanDomain with
modulus : % -> Mod
++ modulus(x) \undocumented
coerce : % -> R
++ coerce(x) \undocumented
reduce : (R, Mod) -> %
++ reduce(r, m) \undocumented
exQuo : (%,%) -> Union(%,"failed")
++ exQuo(x, y) \undocumented
recip : % -> Union(%,"failed")
++ recip(x) \undocumented
inv : % -> %
++ inv(x) \undocumented
elt : (%, R) -> R
++ elt(x, r) or x.r \undocumented
T == ModularRing(R, Mod, reduction, merge, exactQuo) add
--representation
Rep := Record(val : R, modulo : Mod)
--declarations
x, y, z : %
divide(x, y) ==
t := merge(x.modulo, y.modulo)
t case "failed" => error "incompatible moduli"
xm := t@Mod
yv := y.val
invlcy : R
if (leadingCoefficient yv = 1) then invlcy := 1
else
invlcy := (inv reduce((leadingCoefficient yv)::R, xm)).val
yv := reduction(invlcy*yv, xm)
r := monicDivide(x.val, yv)
[reduce(invlcy*r.quotient, xm), reduce(r.remainder, xm)]
if R has fmecg : (R, NonNegativeInteger, S, R) -> R
then x rem y ==
t := merge(x.modulo, y.modulo)
t case "failed" => error "incompatible moduli"
xm := t@Mod
yv := y.val
invlcy : R
if not (leadingCoefficient yv = 1) then
invlcy := (inv reduce((leadingCoefficient yv)::R, xm)).val
yv := reduction(invlcy*yv, xm)
dy := degree yv
xv := x.val
while (d := degree xv - dy) >= 0 repeat
xvl := reduction(leadingMonomial(xv), xm)
xvl = 0 => xv := reductum(xv)
xv := xvl + reductum(xv)
xv := fmecg(xv, d::NonNegativeInteger,
leadingCoefficient xv, yv)
xv = 0 => return [xv, xm]$Rep
[reduction(xv, xm), xm]$Rep
else x rem y ==
t := merge(x.modulo, y.modulo)
t case "failed" => error "incompatible moduli"
xm := t@Mod
yv := y.val
invlcy : R
if not (leadingCoefficient yv = 1) then
invlcy := (inv reduce((leadingCoefficient yv)::R, xm)).val
yv := reduction(invlcy*yv, xm)
r := monicDivide(x.val, yv)
reduce(r.remainder, xm)
euclideanSize x == degree x.val
unitCanonical x ==
zero? x => x
degree(x.val) = 0 => 1
(leadingCoefficient(x.val) = 1) => x
invlcx : % := inv reduce((leadingCoefficient(x.val))::R, x.modulo)
invlcx * x
unitNormal x ==
zero?(x) or ((leadingCoefficient(x.val)) = 1) => [1, x, 1]
lcx := reduce((leadingCoefficient(x.val))::R, x.modulo)
invlcx := inv lcx
degree(x.val) = 0 => [lcx, 1, invlcx]
[lcx, invlcx * x, invlcx]
elt(x : %, s : R) : R == reduction(elt(x.val, s), x.modulo)
)abbrev domain MODFIELD ModularField
++ These domains are used for the factorization and gcds
++ of univariate polynomials over the integers in order to work modulo
++ different primes.
++ See \spadtype{ModularRing}, \spadtype{EuclideanModularRing}
ModularField(R, Mod, reduction : (R, Mod) -> R,
merge:(Mod,Mod) -> Union(Mod,"failed"),
exactQuo : (R,R,Mod) -> Union(R,"failed")) : C == T
where
R : CommutativeRing
Mod : AbelianMonoid
C == Field with
modulus : % -> Mod
++ modulus(x) \undocumented
coerce : % -> R
++ coerce(x) \undocumented
reduce : (R, Mod) -> %
++ reduce(r, m) \undocumented
exQuo : (%,%) -> Union(%,"failed")
++ exQuo(x, y) \undocumented
T == ModularRing(R, Mod, reduction, merge, exactQuo)
--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
--All rights reserved.
--
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--
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--
-- - Redistributions in binary form must reproduce the above copyright
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-- 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,
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