\documentclass{article} \usepackage{open-axiom} \begin{document} \title{\$SPAD/src/algebra integer.spad} \author{James Davenport} \maketitle \begin{abstract} \end{abstract} \eject \tableofcontents \eject \section{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} <>= )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 \spad{nothing}. == add ZP ==> SparseUnivariatePolynomial % ZZP ==> SparseUnivariatePolynomial Integer import %icst0: % from Foreign Builtin import %icst1: % from Foreign Builtin import %ineg: % -> % from Foreign Builtin import %iabs: % -> % from Foreign Builtin import %irandom: % -> % from Foreign Builtin import %iodd?: % -> Boolean from Foreign Builtin import %hash: % -> SingleInteger from Foreign Builtin import %iadd: (%,%) -> % from Foreign Builtin import %isub: (%,%) -> % from Foreign Builtin import %imul: (%,%) -> % from Foreign Builtin import %irem: (%,%) -> % from Foreign Builtin import %iquo: (%,%) -> % from Foreign Builtin import %imax: (%,%) -> % from Foreign Builtin import %imin: (%,%) -> % from Foreign Builtin import %igcd: (%,%) -> % from Foreign Builtin import %ieq: (%,%) -> Boolean from Foreign Builtin import %ilt: (%,%) -> Boolean from Foreign Builtin import %ile: (%,%) -> Boolean from Foreign Builtin import %igt: (%,%) -> Boolean from Foreign Builtin import %ige: (%,%) -> Boolean from Foreign Builtin import %ilength: % -> % from Foreign Builtin import %i2s: % -> String from Foreign Builtin import %strconc: (String,String) -> String from Foreign Builtin x,y: % n: NonNegativeInteger writeOMInt(dev: OpenMathDevice, x: %): Void == if negative? x 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: String := OM_-STRINGTOSTRINGPTR(s)$Lisp dev: OpenMathDevice := OMopenString(sp, OMencodingXML()) OMputObject(dev) writeOMInt(dev, x) OMputEndObject(dev) OMclose(dev) OM_-STRINGPTRTOSTRING(sp)$Lisp OMwrite(x: %, wholeObj: Boolean): String == s: String := "" sp: String := OM_-STRINGTOSTRINGPTR(s)$Lisp dev: OpenMathDevice := OMopenString(sp, OMencodingXML()) if wholeObj then OMputObject(dev) writeOMInt(dev, x) if wholeObj then OMputEndObject(dev) OMclose(dev) OM_-STRINGPTRTOSTRING(sp)$Lisp 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 == x = 0@% one? x == x = 1@% 0 == %icst0 1 == %icst1 base() == 2 pretend % copy x == x inc x == x + 1@% dec x == x - 1@% hash x == %hash x negative? x == x < 0@% positive? x == 0 < x coerce(x):OutputForm == outputForm(x pretend Integer) coerce(m:Integer):% == m pretend % convert(x:%):Integer == x pretend Integer length a == %ilength a addmod(a, b, p) == c := %iadd(a,b) c >= p => c - p c submod(a, b, p) == c := %isub(a,b) negative? c => c + p c mulmod(a, b, p) == %imul(a,b) rem p convert(x:%):Float == coerce(x)$Float convert(x:%):DoubleFloat == coerce(x)$DoubleFloat convert(x:%):InputForm == convert(x)$InputForm convert(x:%):String == %i2s x latex(x:%):String == s : String := convert x -1 < x and x < 10 => s %strconc("{", %strconc(s, "}")) positiveRemainder(a, b) == negative?(r := a rem b) => negative? b => r - b r + b r reducedSystem(m:Matrix %):Matrix(Integer) == m pretend Matrix(Integer) reducedSystem(m:Matrix %, v:Vector %): Record(mat:Matrix(Integer), vec:Vector(Integer)) == [m pretend Matrix(Integer), vec pretend Vector(Integer)] abs(x) == %iabs x random() == random()$Lisp random(x) == %irandom x x = y == %ieq(x,y) x < y == %ilt(x,y) x > y == %igt(x,y) x <= y == %ile(x,y) x >= y == %ige(x,y) - x == %ineg x x + y == %iadd(x,y) x - y == %isub(x,y) x * y == %imul(x,y) (m:Integer) * (y:%) == %imul(m,y) -- for subsumption problem x ** n == %ipow(x,n)$Foreign(Builtin) odd? x == %iodd? x max(x,y) == %imax(x,y) min(x,y) == %imin(x,y) divide(x,y) == %idivide(x,y)$Foreign(Builtin) x quo y == %iquo(x,y) x rem y == %irem(x,y) shift(x, y) == ASH(x,y)$Lisp recip(x) == if one? x or x=-1 then x else "failed" gcd(x,y) == %igcd(x,y) UCA ==> Record(unit:%,canonical:%,associate:%) unitNormal x == negative? x => [-1@%,-x,-1@%]$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)$TT @ \section{domain NNI NonNegativeInteger} <>= )abbrev domain NNI NonNegativeInteger ++ Author: ++ Date Created: ++ Change History: ++ Basic Operations: ++ Related Constructors: ++ Keywords: integer ++ Description: \spadtype{NonNegativeInteger} provides functions for non ++ negative integers. NonNegativeInteger: Join(OrderedAbelianMonoidSup,Monoid) with quo : (%,%) -> % ++ a quo b returns the quotient of \spad{a} and b, forgetting ++ the remainder. rem : (%,%) -> % ++ a rem b returns the remainder of \spad{a} and b. gcd : (%,%) -> % ++ gcd(a,b) computes the greatest common divisor of two ++ non negative integers \spad{a} and b. divide: (%,%) -> Record(quotient:%,remainder:%) ++ divide(a,b) returns a record containing both ++ remainder and quotient. exquo: (%,%) -> Union(%,"failed") ++ exquo(a,b) returns the quotient of \spad{a} and b, or "failed" ++ if b is zero or \spad{a} rem b is zero. shift: (%, Integer) -> % ++ shift(a,i) shift \spad{a} by i bits. random : % -> % ++ random(n) returns a random integer from 0 to \spad{n-1}. commutative("*") ++ commutative("*") means multiplication is commutative : \spad{x*y = y*x}. == SubDomain(Integer,#1 >= 0) add x,y:% sup(x,y) == %imax(x,y)$Foreign(Builtin) 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} <>= )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} <>= )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} <>= --Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd. --All rights reserved. --Copyright (C) 2007-2010, Gabriel Dos Reis. --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. @ <<*>>= <> <> <> <> <> <> @ \eject \begin{thebibliography}{99} \bibitem{1} nothing \end{thebibliography} \end{document}