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| author | dos-reis <gdr@axiomatics.org> | 2007-08-14 05:14:52 +0000 |
|---|---|---|
| committer | dos-reis <gdr@axiomatics.org> | 2007-08-14 05:14:52 +0000 |
| commit | ab8cc85adde879fb963c94d15675783f2cf4b183 (patch) | |
| tree | c202482327f474583b750b2c45dedfc4e4312b1d /src/algebra/fmod.spad.pamphlet | |
| download | open-axiom-ab8cc85adde879fb963c94d15675783f2cf4b183.tar.gz | |
Initial population.
Diffstat (limited to 'src/algebra/fmod.spad.pamphlet')
| -rw-r--r-- | src/algebra/fmod.spad.pamphlet | 143 |
1 files changed, 143 insertions, 0 deletions
diff --git a/src/algebra/fmod.spad.pamphlet b/src/algebra/fmod.spad.pamphlet new file mode 100644 index 00000000..57e6f206 --- /dev/null +++ b/src/algebra/fmod.spad.pamphlet @@ -0,0 +1,143 @@ +\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" + not (g = 1) => "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" + not (g = 1) => "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} |