/
radeigen.spad
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/
radeigen.spad
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)abbrev package REP RadicalEigenPackage
++ Author: P.Gianni
++ Date Created: Summer 1987
++ 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
SY ==> Symbol()
M ==> Matrix(F)
MRE ==> Matrix(RE)
ST ==> SuchThat(SY, 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 from EigenPackage R
---- Local Functions ----
evalvect : (M, RE, SY) -> MRE
innerprod : (MRE, MRE) -> RE
---- eval a vector of F in a radical expression ----
evalvect(vect : M, alg : RE, x : SY) : 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$RE => v
(1/normv)*v
---- Eigenvalues of the matrix A ----
radicalEigenvalues(A : M) : List(RE) ==
x : SY := new()$SY
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 in lvect.rest repeat
pol := ((v pretend RMR)-(+/[(innerprod(w, v)*w) pretend 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)
for alterm in alglist repeat
if (lvec := alterm.radvect)=[] then error "sorry "
if #(lvec)>1 then
lvec := gramschmidt(lvec)
basis := [:lvec, :basis]
else basis := cons(normalise(lvec.first), basis)
basis
--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
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--
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--
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-- 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
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