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\documentclass{article}
\usepackage{axiom}
\begin{document}
\title{\$SPAD/src/algebra integer.spad}
\author{James Davenport}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{package INTSLPE IntegerSolveLinearPolynomialEquation}
<<package INTSLPE IntegerSolveLinearPolynomialEquation>>=
)abbrev package INTSLPE IntegerSolveLinearPolynomialEquation
++ Author: Davenport
++ Date Created: 1991
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ This package provides the implementation for the
++ \spadfun{solveLinearPolynomialEquation}
++ operation over the integers. It uses a lifting technique
++ from the package GenExEuclid
IntegerSolveLinearPolynomialEquation(): C ==T
where
ZP ==> SparseUnivariatePolynomial Integer
C == with
solveLinearPolynomialEquation: (List ZP,ZP) -> Union(List ZP,"failed")
++ solveLinearPolynomialEquation([f1, ..., fn], g)
++ (where the fi are relatively prime to each other)
++ returns a list of ai such that
++ \spad{g/prod fi = sum ai/fi}
++ or returns "failed" if no such list of ai's exists.
T == add
oldlp:List ZP := []
slpePrime:Integer:=(2::Integer)
oldtable:Vector List ZP := empty()
solveLinearPolynomialEquation(lp,p) ==
if (oldlp ~= lp) then
-- we have to generate a new table
deg:= +/[degree u for u in lp]
ans:Union(Vector List ZP,"failed"):="failed"
slpePrime:=2147483647::Integer -- 2**31 -1 : a prime
-- a good test case for this package is
-- ([x**31-1,x-2],2)
while (ans case "failed") repeat
ans:=tablePow(deg,slpePrime,lp)$GenExEuclid(Integer,ZP)
if (ans case "failed") then
slpePrime:= prevPrime(slpePrime)$IntegerPrimesPackage(Integer)
oldtable:=(ans:: Vector List ZP)
answer:=solveid(p,slpePrime,oldtable)
answer
@
\section{domain INT Integer}
<<domain INT Integer>>=
)abbrev domain INT Integer
++ Author:
++ Date Created:
++ Change History:
++ Basic Operations:
++ Related Constructors:
++ Keywords: integer
++ Description: \spadtype{Integer} provides the domain of arbitrary precision
++ integers.
Integer: Join(IntegerNumberSystem, ConvertibleTo String, OpenMath) with
canonical
++ mathematical equality is data structure equality.
canonicalsClosed
++ two positives multiply to give positive.
noetherian
++ ascending chain condition on ideals.
infinite
++ nextItem never returns "failed".
== add
ZP ==> SparseUnivariatePolynomial %
ZZP ==> SparseUnivariatePolynomial Integer
x,y: %
n: NonNegativeInteger
writeOMInt(dev: OpenMathDevice, x: %): Void ==
if x < 0 then
OMputApp(dev)
OMputSymbol(dev, "arith1", "unary__minus")
OMputInteger(dev, (-x) pretend Integer)
OMputEndApp(dev)
else
OMputInteger(dev, x pretend Integer)
OMwrite(x: %): String ==
s: String := ""
sp := OM_-STRINGTOSTRINGPTR(s)$Lisp
dev: OpenMathDevice := OMopenString(sp pretend String, OMencodingXML)
OMputObject(dev)
writeOMInt(dev, x)
OMputEndObject(dev)
OMclose(dev)
s := OM_-STRINGPTRTOSTRING(sp)$Lisp pretend String
s
OMwrite(x: %, wholeObj: Boolean): String ==
s: String := ""
sp := OM_-STRINGTOSTRINGPTR(s)$Lisp
dev: OpenMathDevice := OMopenString(sp pretend String, OMencodingXML)
if wholeObj then
OMputObject(dev)
writeOMInt(dev, x)
if wholeObj then
OMputEndObject(dev)
OMclose(dev)
s := OM_-STRINGPTRTOSTRING(sp)$Lisp pretend String
s
OMwrite(dev: OpenMathDevice, x: %): Void ==
OMputObject(dev)
writeOMInt(dev, x)
OMputEndObject(dev)
OMwrite(dev: OpenMathDevice, x: %, wholeObj: Boolean): Void ==
if wholeObj then
OMputObject(dev)
writeOMInt(dev, x)
if wholeObj then
OMputEndObject(dev)
zero? x == ZEROP(x)$Lisp
one? x == x = 1
0 == 0$Lisp
1 == 1$Lisp
base() == 2$Lisp
copy x == x
inc x == x + 1
dec x == x - 1
hash x == SXHASH(x)$Lisp
negative? x == MINUSP(x)$Lisp
coerce(x):OutputForm == outputForm(x pretend Integer)
coerce(m:Integer):% == m pretend %
convert(x:%):Integer == x pretend Integer
length a == INTEGER_-LENGTH(a)$Lisp
addmod(a, b, p) ==
(c:=a + b) >= p => c - p
c
submod(a, b, p) ==
(c:=a - b) < 0 => c + p
c
mulmod(a, b, p) == (a * b) rem p
convert(x:%):Float == coerce(x pretend Integer)$Float
convert(x:%):DoubleFloat == coerce(x pretend Integer)$DoubleFloat
convert(x:%):InputForm == convert(x pretend Integer)$InputForm
convert(x:%):String == string(x pretend Integer)$String
latex(x:%):String ==
s : String := string(x pretend Integer)$String
(-1 < (x pretend Integer)) and ((x pretend Integer) < 10) => s
concat("{", concat(s, "}")$String)$String
positiveRemainder(a, b) ==
negative?(r := a rem b) =>
negative? b => r - b
r + b
r
reducedSystem(m:Matrix %):Matrix(Integer) ==
m pretend Matrix(Integer)
reducedSystem(m:Matrix %, v:Vector %):
Record(mat:Matrix(Integer), vec:Vector(Integer)) ==
[m pretend Matrix(Integer), vec pretend Vector(Integer)]
abs(x) == ABS(x)$Lisp
random() == random()$Lisp
random(x) == RANDOM(x)$Lisp
x = y == EQL(x,y)$Lisp
x < y == (x<y)$Lisp
x > y == (x > y)$Lisp -- Don't rely on default; help the inliner
x <= y == (x <= y)$Lisp -- Ditto
x >= y == (x >= y)$Lisp -- Ditto
- x == (-x)$Lisp
x + y == (x+y)$Lisp
x - y == (x-y)$Lisp
x * y == (x*y)$Lisp
(m:Integer) * (y:%) == (m*y)$Lisp -- for subsumption problem
x ** n == EXPT(x,n)$Lisp
odd? x == ODDP(x)$Lisp
max(x,y) == MAX(x,y)$Lisp
min(x,y) == MIN(x,y)$Lisp
divide(x,y) == DIVIDE2(x,y)$Lisp
x quo y == QUOTIENT2(x,y)$Lisp
x rem y == REMAINDER2(x,y)$Lisp
shift(x, y) == ASH(x,y)$Lisp
recip(x) == if one? x or x=-1 then x else "failed"
gcd(x,y) == GCD(x,y)$Lisp
UCA ==> Record(unit:%,canonical:%,associate:%)
unitNormal x ==
x < 0 => [-1,-x,-1]$UCA
[1,x,1]$UCA
unitCanonical x == abs x
solveLinearPolynomialEquation(lp:List ZP,p:ZP):Union(List ZP,"failed") ==
solveLinearPolynomialEquation(lp pretend List ZZP,
p pretend ZZP)$IntegerSolveLinearPolynomialEquation pretend
Union(List ZP,"failed")
squareFreePolynomial(p:ZP):Factored ZP ==
squareFree(p)$UnivariatePolynomialSquareFree(%,ZP)
factorPolynomial(p:ZP):Factored ZP ==
-- GaloisGroupFactorizer doesn't factor the content
-- so we have to do this by hand
pp:=primitivePart p
leadingCoefficient pp = leadingCoefficient p =>
factor(p)$GaloisGroupFactorizer(ZP)
mergeFactors(factor(pp)$GaloisGroupFactorizer(ZP),
map(#1::ZP,
factor((leadingCoefficient p exquo
leadingCoefficient pp)
::%))$FactoredFunctions2(%,ZP)
)$FactoredFunctionUtilities(ZP)
factorSquareFreePolynomial(p:ZP):Factored ZP ==
factorSquareFree(p)$GaloisGroupFactorizer(ZP)
gcdPolynomial(p:ZP, q:ZP):ZP ==
zero? p => unitCanonical q
zero? q => unitCanonical p
gcd([p,q])$HeuGcd(ZP)
-- myNextPrime: (%,NonNegativeInteger) -> %
-- myNextPrime(x,n) ==
-- nextPrime(x)$IntegerPrimesPackage(%)
-- TT:=InnerModularGcd(%,ZP,67108859 pretend %,myNextPrime)
-- gcdPolynomial(p,q) == modularGcd(p,q)$TT
@
\section{domain NNI NonNegativeInteger}
<<domain NNI NonNegativeInteger>>=
)abbrev domain NNI NonNegativeInteger
++ Author:
++ Date Created:
++ Change History:
++ Basic Operations:
++ Related Constructors:
++ Keywords: integer
++ Description: \spadtype{NonNegativeInteger} provides functions for non
++ negative integers.
NonNegativeInteger: Join(OrderedAbelianMonoidSup,Monoid) with
quo : (%,%) -> %
++ a quo b returns the quotient of \spad{a} and b, forgetting
++ the remainder.
rem : (%,%) -> %
++ a rem b returns the remainder of \spad{a} and b.
gcd : (%,%) -> %
++ gcd(a,b) computes the greatest common divisor of two
++ non negative integers \spad{a} and b.
divide: (%,%) -> Record(quotient:%,remainder:%)
++ divide(a,b) returns a record containing both
++ remainder and quotient.
exquo: (%,%) -> Union(%,"failed")
++ exquo(a,b) returns the quotient of \spad{a} and b, or "failed"
++ if b is zero or \spad{a} rem b is zero.
shift: (%, Integer) -> %
++ shift(a,i) shift \spad{a} by i bits.
commutative("*")
++ commutative("*") means multiplication is commutative : \spad{x*y = y*x}.
== SubDomain(Integer,#1 >= 0) add
x,y:%
sup(x,y) == MAX(x,y)$Lisp
shift(x:%, n:Integer):% == ASH(x,n)$Lisp
subtractIfCan(x, y) ==
c:Integer := rep x - rep y
c < 0 => "failed"
per c
@
\section{domain PI PositiveInteger}
<<domain PI PositiveInteger>>=
)abbrev domain PI PositiveInteger
++ Author:
++ Date Created:
++ Change History:
++ Basic Operations:
++ Related Constructors:
++ Keywords: positive integer
++ Description: \spadtype{PositiveInteger} provides functions for
++ positive integers.
PositiveInteger: Join(OrderedAbelianSemiGroup,Monoid) with
gcd: (%,%) -> %
++ gcd(a,b) computes the greatest common divisor of two
++ positive integers \spad{a} and b.
commutative("*")
++ commutative("*") means multiplication is commutative : x*y = y*x
== SubDomain(NonNegativeInteger,#1 > 0)
@
\section{domain ROMAN RomanNumeral}
<<domain ROMAN RomanNumeral>>=
)abbrev domain ROMAN RomanNumeral
++ Author:
++ Date Created:
++ Change History:
++ Basic Operations:
++ convert, roman
++ Related Constructors:
++ Keywords: roman numerals
++ Description: \spadtype{RomanNumeral} provides functions for converting
++ integers to roman numerals.
RomanNumeral(): Join(IntegerNumberSystem,ConvertibleFrom Symbol) with
canonical
++ mathematical equality is data structure equality.
canonicalsClosed
++ two positives multiply to give positive.
noetherian
++ ascending chain condition on ideals.
roman : Symbol -> %
++ roman(n) creates a roman numeral for symbol n.
roman : Integer -> %
++ roman(n) creates a roman numeral for n.
== Integer add
import NumberFormats()
roman(n:Integer) == n::%
roman(sy:Symbol) == convert sy
convert(sy:Symbol):% == ScanRoman(string sy)::%
coerce(r:%):OutputForm ==
n := convert(r)@Integer
-- okay, we stretch it
zero? n => n::OutputForm
negative? n => - ((-r)::OutputForm)
FormatRoman(n::PositiveInteger)::Symbol::OutputForm
@
\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>>
<<package INTSLPE IntegerSolveLinearPolynomialEquation>>
<<domain INT Integer>>
<<domain NNI NonNegativeInteger>>
<<domain PI PositiveInteger>>
<<domain ROMAN RomanNumeral>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
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