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\documentclass{article}
\usepackage{axiom}
\begin{document}
\title{\$SPAD/src/algebra fmod.spad}
\author{The Axiom Team}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{domain ZMOD IntegerMod}
<<domain ZMOD IntegerMod>>=
)abbrev domain ZMOD IntegerMod
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ IntegerMod(n) creates the ring of integers reduced modulo the integer
++ n.
IntegerMod(p:PositiveInteger):
Join(CommutativeRing, Finite, ConvertibleTo Integer, StepThrough) == add
size() == p
characteristic() == p
lookup x == (zero? x => p; (convert(x)@Integer) :: PositiveInteger)
-- Code is duplicated for the optimizer to kick in.
if p <= convert(max()$SingleInteger)@Integer then
Rep:= SingleInteger
q := p::SingleInteger
bloodyCompiler: Integer -> %
bloodyCompiler n == positiveRemainder(n, p)$Integer :: Rep
convert(x:%):Integer == convert(x)$Rep
coerce(x):OutputForm == coerce(x)$Rep
coerce(n:Integer):% == bloodyCompiler n
0 == 0$Rep
1 == 1$Rep
init == 0$Rep
nextItem(n) ==
m:=n+1
m=0 => "failed"
m
x = y == x =$Rep y
x:% * y:% == mulmod(x, y, q)
n:Integer * x:% == mulmod(bloodyCompiler n, x, q)
x + y == addmod(x, y, q)
x - y == submod(x, y, q)
random() == random(q)$Rep
index a == positiveRemainder(a::%, q)
- x == (zero? x => 0; q -$Rep x)
x:% ** n:NonNegativeInteger ==
n < p => powmod(x, n::Rep, q)
powmod(convert(x)@Integer, n, p)$Integer :: Rep
recip x ==
(c1, c2, g) := extendedEuclidean(x, q)$Rep
not one? g => "failed"
positiveRemainder(c1, q)
else
Rep:= Integer
convert(x:%):Integer == convert(x)$Rep
coerce(n:Integer):% == positiveRemainder(n::Rep, p)
coerce(x):OutputForm == coerce(x)$Rep
0 == 0$Rep
1 == 1$Rep
init == 0$Rep
nextItem(n) ==
m:=n+1
m=0 => "failed"
m
x = y == x =$Rep y
x:% * y:% == mulmod(x, y, p)
n:Integer * x:% == mulmod(positiveRemainder(n::Rep, p), x, p)
x + y == addmod(x, y, p)
x - y == submod(x, y, p)
random() == random(p)$Rep
index a == positiveRemainder(a::Rep, p)
- x == (zero? x => 0; p -$Rep x)
x:% ** n:NonNegativeInteger == powmod(x, n::Rep, p)
recip x ==
(c1, c2, g) := extendedEuclidean(x, p)$Rep
not one? g => "failed"
positiveRemainder(c1, p)
@
\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 ZMOD IntegerMod>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
|
