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\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra cden.spad}
\author{Manuel Bronstein}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{package ICDEN InnerCommonDenominator}
<<package ICDEN InnerCommonDenominator>>=
)abbrev package ICDEN InnerCommonDenominator
--% InnerCommonDenominator
++ Author: Manuel Bronstein
++ Date Created: 2 May 1988
++ Date Last Updated: 22 Nov 1989
++ Description: InnerCommonDenominator provides functions to compute
++ the common denominator of a finite linear aggregate of elements
++ of the quotient field of an integral domain.
++ Keywords: gcd, quotient, common, denominator.
InnerCommonDenominator(R, Q, A, B): Exports == Implementation where
R: IntegralDomain
Q: QuotientFieldCategory R
A: FiniteLinearAggregate R
B: FiniteLinearAggregate Q
Exports ==> with
commonDenominator: B -> R
++ commonDenominator([q1,...,qn]) returns a common denominator
++ d for q1,...,qn.
clearDenominator : B -> A
++ clearDenominator([q1,...,qn]) returns \spad{[p1,...,pn]} such that
++ \spad{qi = pi/d} where d is a common denominator for the qi's.
splitDenominator : B -> Record(num: A, den: R)
++ splitDenominator([q1,...,qn]) returns
++ \spad{[[p1,...,pn], d]} such that
++ \spad{qi = pi/d} and d is a common denominator for the qi's.
Implementation ==> add
import FiniteLinearAggregateFunctions2(Q, B, R, A)
clearDenominator l ==
d := commonDenominator l
map(numer(d * #1), l)
splitDenominator l ==
d := commonDenominator l
[map(numer(d * #1), l), d]
if R has GcdDomain then
commonDenominator l == reduce(lcm, map(denom, l),1)
else
commonDenominator l == reduce("*", map(denom, l), 1)
@
\section{package CDEN CommonDenominator}
<<package CDEN CommonDenominator>>=
)abbrev package CDEN CommonDenominator
--% CommonDenominator
++ Author: Manuel Bronstein
++ Date Created: 2 May 1988
++ Date Last Updated: 22 Nov 1989
++ Description: CommonDenominator provides functions to compute the
++ common denominator of a finite linear aggregate of elements of
++ the quotient field of an integral domain.
++ Keywords: gcd, quotient, common, denominator.
CommonDenominator(R, Q, A): Exports == Implementation where
R: IntegralDomain
Q: QuotientFieldCategory R
A: FiniteLinearAggregate Q
Exports ==> with
commonDenominator: A -> R
++ commonDenominator([q1,...,qn]) returns a common denominator
++ d for q1,...,qn.
clearDenominator : A -> A
++ clearDenominator([q1,...,qn]) returns \spad{[p1,...,pn]} such that
++ \spad{qi = pi/d} where d is a common denominator for the qi's.
splitDenominator : A -> Record(num: A, den: R)
++ splitDenominator([q1,...,qn]) returns
++ \spad{[[p1,...,pn], d]} such that
++ \spad{qi = pi/d} and d is a common denominator for the qi's.
Implementation ==> add
clearDenominator l ==
d := commonDenominator l
map(numer(d * #1)::Q, l)
splitDenominator l ==
d := commonDenominator l
[map(numer(d * #1)::Q, l), d]
if R has GcdDomain then
qlcm: (Q, Q) -> Q
qlcm(a, b) == lcm(numer a, numer b)::Q
commonDenominator l == numer reduce(qlcm, map(denom(#1)::Q, l), 1)
else
commonDenominator l == numer reduce("*", map(denom(#1)::Q, l), 1)
@
\section{package UPCDEN UnivariatePolynomialCommonDenominator}
<<package UPCDEN UnivariatePolynomialCommonDenominator>>=
)abbrev package UPCDEN UnivariatePolynomialCommonDenominator
--% UnivariatePolynomialCommonDenominator
++ Author: Manuel Bronstein
++ Date Created: 2 May 1988
++ Date Last Updated: 22 Feb 1990
++ Description: UnivariatePolynomialCommonDenominator provides
++ functions to compute the common denominator of the coefficients of
++ univariate polynomials over the quotient field of a gcd domain.
++ Keywords: gcd, quotient, common, denominator, polynomial.
UnivariatePolynomialCommonDenominator(R, Q, UP): Exports == Impl where
R : IntegralDomain
Q : QuotientFieldCategory R
UP: UnivariatePolynomialCategory Q
Exports ==> with
commonDenominator: UP -> R
++ commonDenominator(q) returns a common denominator d for
++ the coefficients of q.
clearDenominator : UP -> UP
++ clearDenominator(q) returns p such that \spad{q = p/d} where d is
++ a common denominator for the coefficients of q.
splitDenominator : UP -> Record(num: UP, den: R)
++ splitDenominator(q) returns \spad{[p, d]} such that \spad{q = p/d} and d
++ is a common denominator for the coefficients of q.
Impl ==> add
import CommonDenominator(R, Q, List Q)
commonDenominator p == commonDenominator coefficients p
clearDenominator p ==
d := commonDenominator p
map(numer(d * #1)::Q, p)
splitDenominator p ==
d := commonDenominator p
[map(numer(d * #1)::Q, p), d]
@
\section{package MCDEN MatrixCommonDenominator}
<<package MCDEN MatrixCommonDenominator>>=
)abbrev package MCDEN MatrixCommonDenominator
--% MatrixCommonDenominator
++ Author: Manuel Bronstein
++ Date Created: 2 May 1988
++ Date Last Updated: 20 Jul 1990
++ Description: MatrixCommonDenominator provides functions to
++ compute the common denominator of a matrix of elements of the
++ quotient field of an integral domain.
++ Keywords: gcd, quotient, matrix, common, denominator.
MatrixCommonDenominator(R, Q): Exports == Implementation where
R: IntegralDomain
Q: QuotientFieldCategory R
VR ==> Vector R
VQ ==> Vector Q
Exports ==> with
commonDenominator: Matrix Q -> R
++ commonDenominator(q) returns a common denominator d for
++ the elements of q.
clearDenominator : Matrix Q -> Matrix R
++ clearDenominator(q) returns p such that \spad{q = p/d} where d is
++ a common denominator for the elements of q.
splitDenominator : Matrix Q -> Record(num: Matrix R, den: R)
++ splitDenominator(q) returns \spad{[p, d]} such that \spad{q = p/d} and d
++ is a common denominator for the elements of q.
Implementation ==> add
import ListFunctions2(Q, R)
import MatrixCategoryFunctions2(Q,VQ,VQ,Matrix Q,R,VR,VR,Matrix R)
clearDenominator m ==
d := commonDenominator m
map(numer(d * #1), m)
splitDenominator m ==
d := commonDenominator m
[map(numer(d * #1), m), d]
if R has GcdDomain then
commonDenominator m == lcm map(denom, parts m)
else
commonDenominator m == reduce("*",map(denom, parts m),1)$List(R)
@
\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 ICDEN InnerCommonDenominator>>
<<package CDEN CommonDenominator>>
<<package UPCDEN UnivariatePolynomialCommonDenominator>>
<<package MCDEN MatrixCommonDenominator>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
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