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\documentclass{article}
\usepackage{open-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)
SUP ==> SparseUnivariatePolynomial
Exports ==> Join(ExpressionSpace, AlgebraicallyClosedField,
RetractableTo Z, RetractableTo Fraction Z,
LinearlyExplicitRingOver Z, RealConstant,
LinearlyExplicitRingOver Fraction Z,
CharacteristicZero,
ConvertibleTo Complex Float, DifferentialRing,
CoercibleFrom P) with
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
macro ALGOP == '%alg
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))$SparseUnivariatePolynomialFunctions2(P,%)
-- public
a,b:%
differentiate(x:%):% == 0
zero? a == zero? numer a
one? a == one? numer a and one? denom a
x:% / y:% == mainRatDenom(x /$Rep y)
x:% ** n:Integer ==
negative? n => 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,
CoercibleFrom P) with
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
zero? a == trueEqual(rep a, rep 0)
one? a == trueEqual(rep a, rep 1)
a=b == trueEqual(rep a - rep b,rep 0)
@
\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}
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