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\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra modmon.spad}
\author{The Axiom Team}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{domain MODMON ModMonic}
<<domain MODMON ModMonic>>=
)abbrev domain MODMON ModMonic
++ Description:
++ This package \undocumented
ModMonic(R,P): C == T
where
R: Ring
P: UnivariatePolynomialCategory(R)
C == Join(UnivariatePolynomialCategory(R),CoercibleFrom P) with
--operations
setPoly : P -> P
++ setPoly(x) \undocumented
modulus : -> P
++ modulus() \undocumented
reduce: P -> %
++ reduce(x) \undocumented
lift: % -> P --reduce lift = identity
++ lift(x) \undocumented
Vectorise: % -> Vector(R)
++ Vectorise(x) \undocumented
UnVectorise: Vector(R) -> %
++ UnVectorise(v) \undocumented
An: % -> Vector(R)
++ An(x) \undocumented
pow : -> PrimitiveArray(%)
++ pow() \undocumented
computePowers : -> PrimitiveArray(%)
++ computePowers() \undocumented
if R has FiniteFieldCategory then
frobenius: % -> %
++ frobenius(x) \undocumented
--LinearTransf: (%,Vector(R)) -> SquareMatrix<deg> R
--assertions
if R has Finite then Finite
T == add
Rep := P
--constants
m:Rep := monomial(1,1)$Rep --| degree(m) > 0 and LeadingCoef(m) = R$1
d := degree(m)$Rep
d1 := (d-1):NonNegativeInteger
twod := 2*d1+1
frobenius?:Boolean := R has FiniteFieldCategory
--VectorRep:= DirectProduct(d:NonNegativeInteger,R)
--declarations
x,y: %
d,n: Integer
e,k1,k2: NonNegativeInteger
c: R
--vect: Vector(R)
power: PrimitiveArray(%) := new(0,0)
frobeniusPower: PrimitiveArray(%) := new(0,0)
computeFrobeniusPowers : () -> PrimitiveArray(%)
--representations
--mutable m --take this out??
--define
setPoly (mon : P) ==
mon =$Rep m => mon
oldm := m
not one? leadingCoefficient mon => error "polynomial must be monic"
-- following copy code needed since FFPOLY can modify mon
copymon:Rep:= 0
while not zero? mon repeat
copymon := monomial(leadingCoefficient mon, degree mon)$Rep + copymon
mon := reductum mon
m := copymon
d := degree(m)$Rep
d1 := (d-1)::NonNegativeInteger
twod := 2*d1+1
power := computePowers()
if frobenius? then
degree(oldm)>1 and not((oldm exquo$Rep m) case "failed") =>
for i in 1..d1 repeat
frobeniusPower(i) := reduce lift frobeniusPower(i)
frobeniusPower := computeFrobeniusPowers()
m
modulus == m
if R has Finite then
size == d * size()$R
random == UnVectorise([random()$R for i in 0..d1])
0 == 0$Rep
1 == 1$Rep
c * x == c *$Rep x
n * x == (n::R) *$Rep x
coerce(c:R):% == monomial(c,0)$Rep
coerce(x:%):OutputForm == coerce(x)$Rep
coefficient(x,e):R == coefficient(x,e)$Rep
reductum(x) == reductum(x)$Rep
leadingCoefficient x == (leadingCoefficient x)$Rep
degree x == (degree x)$Rep
lift(x) == x pretend Rep
reduce(p) == (monicDivide(p,m)$Rep).remainder
coerce(p: P): % == reduce(p)
x = y == x =$Rep y
x + y == x +$Rep y
- x == -$Rep x
x * y ==
p := x *$Rep y
ans:=0$Rep
while (n:=degree p)>d1 repeat
ans:=ans + leadingCoefficient(p)*power.(n-d)
p := reductum p
ans+p
Vectorise(x) == [coefficient(lift(x),i) for i in 0..d1]
UnVectorise(vect) ==
reduce(+/[monomial(vect.(i+1),i) for i in 0..d1]@%::Rep)
computePowers ==
mat : PrimitiveArray(%):= new(d,0)
mat.0:= reductum(-m)$Rep
w: % := monomial$Rep (1,1)
for i in 1..d1 repeat
mat.i := w *$Rep mat.(i-1)
if degree mat.i=d then
mat.i:= reductum mat.i + leadingCoefficient mat.i * mat.0
mat
if frobenius? then
computeFrobeniusPowers() ==
mat : PrimitiveArray(%):= new(d,1)
mat.1:= mult := monomial(1, size()$R)$%
for i in 2..d1 repeat
mat.i := mult * mat.(i-1)
mat
frobenius(a:%):% ==
aq:% := 0
while not zero? a repeat
aq:= aq + leadingCoefficient(a)*frobeniusPower(degree a)
a := reductum a
aq
pow == power
monomial(c,e)==
if e<d then monomial(c,e)$Rep
else
if e<=twod then
c * power.(e-d)
else
k1:=e quo twod
k2 := (e-k1*twod)::NonNegativeInteger
reduce(((power.d1 **k1)*monomial(c,k2))::Rep)
if R has Field then
(x:% exquo y:%):Union(%, "failed") ==
uv := extendedEuclidean(y, modulus(), x)$Rep
uv case "failed" => "failed"
return reduce(uv.coef1::Rep)
recip(y:%):Union(%, "failed") == 1 exquo y
divide(x:%, y:%) ==
(q := (x exquo y)) case "failed" => error "not divisible"
[q, 0]
-- An(MM) == Vectorise(-(reduce(reductum(m))::MM))
-- LinearTransf(vect,MM) ==
-- ans:= 0::SquareMatrix<d>(R)
-- for i in 1..d do setelt(ans,i,1,vect.i)
-- for j in 2..d do
-- setelt(ans,1,j, elt(ans,d,j-1) * An(MM).1)
-- for i in 2..d do
-- setelt(ans,i,j, elt(ans,i-1,j-1) + elt(ans,d,j-1) * An(MM).i)
-- ans
@
\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 MODMON ModMonic>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
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