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\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra lindep.spad}
\author{Manuel Bronstein}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{package LINDEP LinearDependence}
<<package LINDEP LinearDependence>>=
)abbrev package LINDEP LinearDependence
++ Test for linear dependence
++ Author: Manuel Bronstein
++ Date Created: ???
++ Date Last Updated: 14 May 1991
++ Description: Test for linear dependence.
LinearDependence(S, R): Exports == Implementation where
S: IntegralDomain
R: Join(Ring,LinearlyExplicitRingOver S)
Q ==> Fraction S
Exports ==> with
linearlyDependent?: Vector R -> Boolean
++ \spad{linearlyDependent?([v1,...,vn])} returns true if
++ the vi's are linearly dependent over S, false otherwise.
linearDependence : Vector R -> Union(Vector S, "failed")
++ \spad{linearDependence([v1,...,vn])} returns \spad{[c1,...,cn]} if
++ \spad{c1*v1 + ... + cn*vn = 0} and not all the ci's are 0,
++ "failed" if the vi's are linearly independent over S.
if S has Field then
solveLinear: (Vector R, R) -> Union(Vector S, "failed")
++ \spad{solveLinear([v1,...,vn], u)} returns \spad{[c1,...,cn]}
++ such that \spad{c1*v1 + ... + cn*vn = u},
++ "failed" if no such ci's exist in S.
else
solveLinear: (Vector R, R) -> Union(Vector Q, "failed")
++ \spad{solveLinear([v1,...,vn], u)} returns \spad{[c1,...,cn]}
++ such that \spad{c1*v1 + ... + cn*vn = u},
++ "failed" if no such ci's exist in the quotient field of S.
Implementation ==> add
aNonZeroSolution: Matrix S -> Union(Vector S, "failed")
aNonZeroSolution m ==
every?(zero?, v := first nullSpace m) => "failed"
v
linearlyDependent? v ==
zero?(n := #v) => true
one? n => zero?(v(minIndex v))
positive? nullity reducedSystem transpose v
linearDependence v ==
zero?(n := #v) => empty()
one? n =>
zero?(v(minIndex v)) => new(1, 1)
"failed"
aNonZeroSolution reducedSystem transpose v
if S has Field then
solveLinear(v:Vector R, c:R):Union(Vector S, "failed") ==
zero? c => new(#v, 0)
empty? v => "failed"
sys := reducedSystem(transpose v, new(1, c))
particularSolution(sys.mat, sys.vec)$LinearSystemMatrixPackage(S,
Vector S, Vector S, Matrix S)
else
solveLinear(v:Vector R, c:R):Union(Vector Q, "failed") ==
zero? c => new(#v, 0)
empty? v => "failed"
sys := reducedSystem(transpose v, new(1, c))
particularSolution(map(#1::Q, sys.mat)$MatrixCategoryFunctions2(S,
Vector S,Vector S,Matrix S,Q,Vector Q,Vector Q,Matrix Q),
map(#1::Q, sys.vec)$VectorFunctions2(S, Q)
)$LinearSystemMatrixPackage(Q,
Vector Q, Vector Q, Matrix Q)
@
\section{package ZLINDEP IntegerLinearDependence}
<<package ZLINDEP IntegerLinearDependence>>=
)abbrev package ZLINDEP IntegerLinearDependence
++ Test for linear dependence over the integers
++ Author: Manuel Bronstein
++ Date Created: ???
++ Date Last Updated: 14 May 1991
++ Description: Test for linear dependence over the integers.
IntegerLinearDependence(R): Exports == Implementation where
R: Join(Ring,LinearlyExplicitRingOver Integer)
Z ==> Integer
Exports ==> with
linearlyDependentOverZ?: Vector R -> Boolean
++ \spad{linearlyDependentOverZ?([v1,...,vn])} returns true if the
++ vi's are linearly dependent over the integers, false otherwise.
linearDependenceOverZ : Vector R -> Union(Vector Z, "failed")
++ \spad{linearlyDependenceOverZ([v1,...,vn])} returns
++ \spad{[c1,...,cn]} if
++ \spad{c1*v1 + ... + cn*vn = 0} and not all the ci's are 0, "failed"
++ if the vi's are linearly independent over the integers.
solveLinearlyOverQ : (Vector R, R) ->
Union(Vector Fraction Z, "failed")
++ \spad{solveLinearlyOverQ([v1,...,vn], u)} returns \spad{[c1,...,cn]}
++ such that \spad{c1*v1 + ... + cn*vn = u},
++ "failed" if no such rational numbers ci's exist.
Implementation ==> add
import LinearDependence(Z, R)
linearlyDependentOverZ? v == linearlyDependent? v
linearDependenceOverZ v == linearDependence v
solveLinearlyOverQ(v, c) == solveLinear(v, c)
@
\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 LINDEP LinearDependence>>
<<package ZLINDEP IntegerLinearDependence>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
|
