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\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra radeigen.spad}
\author{Patrizia Gianni}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{package REP RadicalEigenPackage}
<<package REP RadicalEigenPackage>>=
)abbrev package REP RadicalEigenPackage
++ Author: P.Gianni
++ Date Created: Summer 1987
++ Date Last Updated: October 1992
++ Basic Functions:
++ Related Constructors: EigenPackage, RadicalSolve
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ Package for the computation of eigenvalues and eigenvectors.
++ This package works for matrices with coefficients which are
++ rational functions over the integers.
++ (see \spadtype{Fraction Polynomial Integer}).
++ The eigenvalues and eigenvectors are expressed in terms of radicals.
RadicalEigenPackage() : C == T
where
R ==> Integer
P ==> Polynomial R
F ==> Fraction P
RE ==> Expression R
SE ==> Symbol()
M ==> Matrix(F)
MRE ==> Matrix(RE)
ST ==> SuchThat(SE,P)
NNI ==> NonNegativeInteger
EigenForm ==> Record(eigval:Union(F,ST),eigmult:NNI,eigvec:List(M))
RadicalForm ==> Record(radval:RE,radmult:Integer,radvect:List(MRE))
C == with
radicalEigenvectors : M -> List(RadicalForm)
++ radicalEigenvectors(m) computes
++ the eigenvalues and the corresponding eigenvectors of the
++ matrix m;
++ when possible, values are expressed in terms of radicals.
radicalEigenvector : (RE,M) -> List(MRE)
++ radicalEigenvector(c,m) computes the eigenvector(s) of the
++ matrix m corresponding to the eigenvalue c;
++ when possible, values are
++ expressed in terms of radicals.
radicalEigenvalues : M -> List RE
++ radicalEigenvalues(m) computes the eigenvalues of the matrix m;
++ when possible, the eigenvalues are expressed in terms of radicals.
eigenMatrix : M -> Union(MRE,"failed")
++ eigenMatrix(m) returns the matrix b
++ such that \spad{b*m*(inverse b)} is diagonal,
++ or "failed" if no such b exists.
normalise : MRE -> MRE
++ normalise(v) returns the column
++ vector v
++ divided by its euclidean norm;
++ when possible, the vector v is expressed in terms of radicals.
gramschmidt : List(MRE) -> List(MRE)
++ gramschmidt(lv) converts the list of column vectors lv into
++ a set of orthogonal column vectors
++ of euclidean length 1 using the Gram-Schmidt algorithm.
orthonormalBasis : M -> List(MRE)
++ orthonormalBasis(m) returns the orthogonal matrix b such that
++ \spad{b*m*(inverse b)} is diagonal.
++ Error: if m is not a symmetric matrix.
T == add
PI ==> PositiveInteger
RSP := RadicalSolvePackage R
import EigenPackage R
---- Local Functions ----
evalvect : (M,RE,SE) -> MRE
innerprod : (MRE,MRE) -> RE
---- eval a vector of F in a radical expression ----
evalvect(vect:M,alg:RE,x:SE) : MRE ==
n:=nrows vect
xx:=kernel(x)$Kernel(RE)
w:MRE:=zero(n,1)$MRE
for i in 1..n repeat
v:=eval(vect(i,1) :: RE,xx,alg)
setelt(w,i,1,v)
w
---- inner product ----
innerprod(v1:MRE,v2:MRE): RE == (((transpose v1)* v2)::MRE)(1,1)
---- normalization of a vector ----
normalise(v:MRE) : MRE ==
normv:RE := sqrt(innerprod(v,v))
normv = 0$RE => v
(1/normv)*v
---- Eigenvalues of the matrix A ----
radicalEigenvalues(A:M): List(RE) ==
x:SE :=new()$SE
pol:= characteristicPolynomial(A,x) :: F
radicalRoots(pol,x)$RSP
---- Eigenvectors belonging to a given eigenvalue ----
---- expressed in terms of radicals ----
radicalEigenvector(alpha:RE,A:M) : List(MRE) ==
n:=nrows A
B:MRE := zero(n,n)$MRE
for i in 1..n repeat
for j in 1..n repeat B(i,j):=(A(i,j))::RE
B(i,i):= B(i,i) - alpha
[v::MRE for v in nullSpace B]
---- eigenvectors and eigenvalues ----
radicalEigenvectors(A:M) : List(RadicalForm) ==
leig:List EigenForm := eigenvectors A
n:=nrows A
sln:List RadicalForm := empty()
veclist: List MRE
for eig in leig repeat
eig.eigval case F =>
veclist := empty()
for ll in eig.eigvec repeat
m:MRE:=zero(n,1)
for i in 1..n repeat m(i,1):=(ll(i,1))::RE
veclist:=cons(m,veclist)
sln:=cons([(eig.eigval)::F::RE,eig.eigmult,veclist]$RadicalForm,sln)
sym := eig.eigval :: ST
xx:= lhs sym
lval : List RE := radicalRoots((rhs sym) :: F ,xx)$RSP
for alg in lval repeat
nsl:=[alg,eig.eigmult,
[evalvect(ep,alg,xx) for ep in eig.eigvec]]$RadicalForm
sln:=cons(nsl,sln)
sln
---- orthonormalization of a list of vectors ----
---- Grahm - Schmidt process ----
gramschmidt(lvect:List(MRE)) : List(MRE) ==
lvect=[] => []
v:=lvect.first
n := nrows v
RMR:=RectangularMatrix(n:PI,1,RE)
orth:List(MRE):=[(normalise v)]
for v: local in lvect.rest repeat
pol:=((v:RMR)-(+/[(innerprod(w,v)*w):RMR for w in orth])):MRE
orth:=cons(normalise pol,orth)
orth
---- The matrix of eigenvectors ----
eigenMatrix(A:M) : Union(MRE,"failed") ==
lef:List(MRE):=[:eiv.radvect for eiv in radicalEigenvectors(A)]
n:=nrows A
#lef <n => "failed"
d:MRE:=copy(lef.first)
for v in lef.rest repeat d:=(horizConcat(d,v))::MRE
d
---- orthogonal basis for a symmetric matrix ----
orthonormalBasis(A:M):List(MRE) ==
not symmetric?(A) => error "the matrix is not symmetric"
basis:List(MRE):=[]
lvec:List(MRE) := []
alglist:List(RadicalForm):=radicalEigenvectors(A)
n:=nrows A
for alterm in alglist repeat
if (lvec:=alterm.radvect)=[] then error "sorry "
if #(lvec)>1 then
lvec:= gramschmidt(lvec)
basis:=[:lvec,:basis]
else basis:=[normalise(lvec.first),:basis]
basis
@
\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 REP RadicalEigenPackage>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
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