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authordos-reis <gdr@axiomatics.org>2007-08-14 05:14:52 +0000
committerdos-reis <gdr@axiomatics.org>2007-08-14 05:14:52 +0000
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downloadopen-axiom-ab8cc85adde879fb963c94d15675783f2cf4b183.tar.gz
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+\documentclass{article}
+\usepackage{axiom}
+\begin{document}
+\title{\$SPAD/src/algebra constant.spad}
+\author{Manuel Bronstein, James Davenport}
+\maketitle
+\begin{abstract}
+\end{abstract}
+\eject
+\tableofcontents
+\eject
+\section{domain IAN InnerAlgebraicNumber}
+<<domain IAN InnerAlgebraicNumber>>=
+)abbrev domain IAN InnerAlgebraicNumber
+++ Algebraic closure of the rational numbers
+++ Author: Manuel Bronstein
+++ Date Created: 22 March 1988
+++ Date Last Updated: 4 October 1995 (JHD)
+++ Description: Algebraic closure of the rational numbers.
+++ Keywords: algebraic, number.
+InnerAlgebraicNumber(): Exports == Implementation where
+ Z ==> Integer
+ FE ==> Expression Z
+ K ==> Kernel %
+ P ==> SparseMultivariatePolynomial(Z, K)
+ ALGOP ==> "%alg"
+ SUP ==> SparseUnivariatePolynomial
+
+ Exports ==> Join(ExpressionSpace, AlgebraicallyClosedField,
+ RetractableTo Z, RetractableTo Fraction Z,
+ LinearlyExplicitRingOver Z, RealConstant,
+ LinearlyExplicitRingOver Fraction Z,
+ CharacteristicZero,
+ ConvertibleTo Complex Float, DifferentialRing) with
+ coerce : P -> %
+ ++ coerce(p) returns p viewed as an algebraic number.
+ numer : % -> P
+ ++ numer(f) returns the numerator of f viewed as a
+ ++ polynomial in the kernels over Z.
+ denom : % -> P
+ ++ denom(f) returns the denominator of f viewed as a
+ ++ polynomial in the kernels over Z.
+ reduce : % -> %
+ ++ reduce(f) simplifies all the unreduced algebraic numbers
+ ++ present in f by applying their defining relations.
+ trueEqual : (%,%) -> Boolean
+ ++ trueEqual(x,y) tries to determine if the two numbers are equal
+ norm : (SUP(%),Kernel %) -> SUP(%)
+ ++ norm(p,k) computes the norm of the polynomial p
+ ++ with respect to the extension generated by kernel k
+ norm : (SUP(%),List Kernel %) -> SUP(%)
+ ++ norm(p,l) computes the norm of the polynomial p
+ ++ with respect to the extension generated by kernels l
+ norm : (%,Kernel %) -> %
+ ++ norm(f,k) computes the norm of the algebraic number f
+ ++ with respect to the extension generated by kernel k
+ norm : (%,List Kernel %) -> %
+ ++ norm(f,l) computes the norm of the algebraic number f
+ ++ with respect to the extension generated by kernels l
+ Implementation ==> FE add
+
+ Rep := FE
+
+ -- private
+ mainRatDenom(f:%):% ==
+ ratDenom(f::Rep::FE)$AlgebraicManipulations(Integer, FE)::Rep::%
+-- mv:= mainVariable denom f
+-- mv case "failed" => f
+-- algv:=mv::K
+-- q:=univariate(f, algv, minPoly(algv))$PolynomialCategoryQuotientFunctions(IndexedExponents K,K,Integer,P,%)
+-- q(algv::%)
+
+ findDenominator(z:SUP %):Record(num:SUP %,den:%) ==
+ zz:=z
+ while not(zz=0) repeat
+ dd:=(denom leadingCoefficient zz)::%
+ not(dd=1) =>
+ rec:=findDenominator(dd*z)
+ return [rec.num,rec.den*dd]
+ zz:=reductum zz
+ [z,1]
+ makeUnivariate(p:P,k:Kernel %):SUP % ==
+ map(#1::%,univariate(p,k))$SparseUnivariatePolynomialFunctions2(P,%)
+ -- public
+ a,b:%
+ differentiate(x:%):% == 0
+ zero? a == zero? numer a
+-- one? a == one? numer a and one? denom a
+ one? a == (numer a = 1) and (denom a = 1)
+ x:% / y:% == mainRatDenom(x /$Rep y)
+ x:% ** n:Integer ==
+ n < 0 => mainRatDenom (x **$Rep n)
+ x **$Rep n
+ trueEqual(a,b) ==
+ -- if two algebraic numbers have the same norm (after deleting repeated
+ -- roots, then they are certainly conjugates. Note that we start with a
+ -- monic polynomial, so don't have to check for constant factors.
+ -- this will be fooled by sqrt(2) and -sqrt(2), but the = in
+ -- AlgebraicNumber knows what to do about this.
+ ka:=reverse tower a
+ kb:=reverse tower b
+ empty? ka and empty? kb => retract(a)@Fraction Z = retract(b)@Fraction Z
+ pa,pb:SparseUnivariatePolynomial %
+ pa:=monomial(1,1)-monomial(a,0)
+ pb:=monomial(1,1)-monomial(b,0)
+ na:=map(retract,norm(pa,ka))$SparseUnivariatePolynomialFunctions2(%,Fraction Z)
+ nb:=map(retract,norm(pb,kb))$SparseUnivariatePolynomialFunctions2(%,Fraction Z)
+ (sa:=squareFreePart(na)) = (sb:=squareFreePart(nb)) => true
+ g:=gcd(sa,sb)
+ (dg:=degree g) = 0 => false
+ -- of course, if these have a factor in common, then the
+ -- answer is really ambiguous, so we ought to be using Duval-type
+ -- technology
+ dg = degree sa or dg = degree sb => true
+ false
+ norm(z:%,k:Kernel %): % ==
+ p:=minPoly k
+ n:=makeUnivariate(numer z,k)
+ d:=makeUnivariate(denom z,k)
+ resultant(n,p)/resultant(d,p)
+ norm(z:%,l:List Kernel %): % ==
+ for k in l repeat
+ z:=norm(z,k)
+ z
+ norm(z:SUP %,k:Kernel %):SUP % ==
+ p:=map(#1::SUP %,minPoly k)$SparseUnivariatePolynomialFunctions2(%,SUP %)
+ f:=findDenominator z
+ zz:=map(makeUnivariate(numer #1,k),f.num)$SparseUnivariatePolynomialFunctions2( %,SUP %)
+ zz:=swap(zz)$CommuteUnivariatePolynomialCategory(%,SUP %,SUP SUP %)
+ resultant(p,zz)/norm(f.den,k)
+ norm(z:SUP %,l:List Kernel %): SUP % ==
+ for k in l repeat
+ z:=norm(z,k)
+ z
+ belong? op == belong?(op)$ExpressionSpace_&(%) or has?(op, ALGOP)
+
+ convert(x:%):Float ==
+ retract map(#1::Float, x pretend FE)$ExpressionFunctions2(Z,Float)
+
+ convert(x:%):DoubleFloat ==
+ retract map(#1::DoubleFloat,
+ x pretend FE)$ExpressionFunctions2(Z, DoubleFloat)
+
+ convert(x:%):Complex(Float) ==
+ retract map(#1::Complex(Float),
+ x pretend FE)$ExpressionFunctions2(Z, Complex Float)
+
+@
+\section{domain AN AlgebraicNumber}
+<<domain AN AlgebraicNumber>>=
+)abbrev domain AN AlgebraicNumber
+++ Algebraic closure of the rational numbers
+++ Author: James Davenport
+++ Date Created: 9 October 1995
+++ Date Last Updated: 10 October 1995 (JHD)
+++ Description: Algebraic closure of the rational numbers, with mathematical =
+++ Keywords: algebraic, number.
+AlgebraicNumber(): Exports == Implementation where
+ Z ==> Integer
+ P ==> SparseMultivariatePolynomial(Z, Kernel %)
+ SUP ==> SparseUnivariatePolynomial
+
+ Exports ==> Join(ExpressionSpace, AlgebraicallyClosedField,
+ RetractableTo Z, RetractableTo Fraction Z,
+ LinearlyExplicitRingOver Z, RealConstant,
+ LinearlyExplicitRingOver Fraction Z,
+ CharacteristicZero,
+ ConvertibleTo Complex Float, DifferentialRing) with
+ coerce : P -> %
+ ++ coerce(p) returns p viewed as an algebraic number.
+ numer : % -> P
+ ++ numer(f) returns the numerator of f viewed as a
+ ++ polynomial in the kernels over Z.
+ denom : % -> P
+ ++ denom(f) returns the denominator of f viewed as a
+ ++ polynomial in the kernels over Z.
+ reduce : % -> %
+ ++ reduce(f) simplifies all the unreduced algebraic numbers
+ ++ present in f by applying their defining relations.
+ norm : (SUP(%),Kernel %) -> SUP(%)
+ ++ norm(p,k) computes the norm of the polynomial p
+ ++ with respect to the extension generated by kernel k
+ norm : (SUP(%),List Kernel %) -> SUP(%)
+ ++ norm(p,l) computes the norm of the polynomial p
+ ++ with respect to the extension generated by kernels l
+ norm : (%,Kernel %) -> %
+ ++ norm(f,k) computes the norm of the algebraic number f
+ ++ with respect to the extension generated by kernel k
+ norm : (%,List Kernel %) -> %
+ ++ norm(f,l) computes the norm of the algebraic number f
+ ++ with respect to the extension generated by kernels l
+ Implementation ==> InnerAlgebraicNumber add
+ Rep:=InnerAlgebraicNumber
+ a,b:%
+ zero? a == trueEqual(a::Rep,0::Rep)
+ one? a == trueEqual(a::Rep,1::Rep)
+ a=b == trueEqual((a-b)::Rep,0::Rep)
+
+@
+\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 IAN InnerAlgebraicNumber>>
+<<domain AN AlgebraicNumber>>
+@
+\eject
+\begin{thebibliography}{99}
+\bibitem{1} nothing
+\end{thebibliography}
+\end{document}