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--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
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--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.
)abbrev package EP EigenPackage
++ Author: P. Gianni
++ Date Created: summer 1986
++ Date Last Updated: October 1992
++ Basic Functions:
++ Related Constructors: NumericRealEigenPackage, NumericComplexEigenPackage,
++ RadicalEigenPackage
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ This is a package for the exact computation of eigenvalues and eigenvectors.
++ This package can be made to work for matrices with coefficients which are
++ rational functions over a ring where we can factor polynomials.
++ Rational eigenvalues are always explicitly computed while the
++ non-rational ones are expressed in terms of their minimal
++ polynomial.
-- Functions for the numeric computation of eigenvalues and eigenvectors
-- are in numeigen spad.
EigenPackage(R) : C == T
where
R : GcdDomain
P ==> Polynomial R
F ==> Fraction P
SE ==> Symbol()
SUP ==> SparseUnivariatePolynomial(P)
SUF ==> SparseUnivariatePolynomial(F)
M ==> Matrix(F)
NNI ==> NonNegativeInteger
ST ==> SuchThat(SE,P)
Eigenvalue ==> Union(F,ST)
EigenForm ==> Record(eigval:Eigenvalue,eigmult:NNI,eigvec : List M)
GenEigen ==> Record(eigval:Eigenvalue,geneigvec:List M)
C == with
characteristicPolynomial : (M,Symbol) -> P
++ characteristicPolynomial(m,var) returns the
++ characteristicPolynomial of the matrix m using
++ the symbol var as the main variable.
characteristicPolynomial : M -> P
++ characteristicPolynomial(m) returns the
++ characteristicPolynomial of the matrix m using
++ a new generated symbol symbol as the main variable.
eigenvalues : M -> List Eigenvalue
++ eigenvalues(m) returns the
++ eigenvalues of the matrix m which are expressible
++ as rational functions over the rational numbers.
eigenvector : (Eigenvalue,M) -> List M
++ eigenvector(eigval,m) returns the
++ eigenvectors belonging to the eigenvalue eigval
++ for the matrix m.
generalizedEigenvector : (Eigenvalue,M,NNI,NNI) -> List M
++ generalizedEigenvector(alpha,m,k,g)
++ returns the generalized eigenvectors
++ of the matrix relative to the eigenvalue alpha.
++ The integers k and g are respectively the algebraic and the
++ geometric multiplicity of tye eigenvalue alpha.
++ alpha can be either rational or not.
++ In the seconda case apha is the minimal polynomial of the
++ eigenvalue.
generalizedEigenvector : (EigenForm,M) -> List M
++ generalizedEigenvector(eigen,m)
++ returns the generalized eigenvectors
++ of the matrix relative to the eigenvalue eigen, as
++ returned by the function eigenvectors.
generalizedEigenvectors : M -> List GenEigen
++ generalizedEigenvectors(m)
++ returns the generalized eigenvectors
++ of the matrix m.
eigenvectors : M -> List(EigenForm)
++ eigenvectors(m) returns the eigenvalues and eigenvectors
++ for the matrix m.
++ The rational eigenvalues and the correspondent eigenvectors
++ are explicitely computed, while the non rational ones
++ are given via their minimal polynomial and the corresponding
++ eigenvectors are expressed in terms of a "generic" root of
++ such a polynomial.
T == add
PI ==> PositiveInteger
MF := GeneralizedMultivariateFactorize(SE,IndexedExponents SE,R,R,P)
UPCF2:= UnivariatePolynomialCategoryFunctions2(P,SUP,F,SUF)
---- Local Functions ----
tff : (SUF,SE) -> F
fft : (SUF,SE) -> F
charpol : (M,SE) -> F
intRatEig : (F,M,NNI) -> List M
intAlgEig : (ST,M,NNI) -> List M
genEigForm : (EigenForm,M) -> GenEigen
---- next functions needed for defining ModularField ----
reduction(u:SUF,p:SUF):SUF == u rem p
merge(p:SUF,q:SUF):Union(SUF,"failed") ==
p = q => p
p = 0 => q
q = 0 => p
"failed"
exactquo(u:SUF,v:SUF,p:SUF):Union(SUF,"failed") ==
val:=extendedEuclidean(v,p,u)
val case "failed" => "failed"
val.coef1
---- functions for conversions ----
fft(t:SUF,x:SE):F ==
n:=degree(t)
cf:=monomial(1,x,n)$P :: F
cf * leadingCoefficient t
tff(p:SUF,x:SE) : F ==
degree p=0 => leadingCoefficient p
r:F:=0$F
while not zero? p repeat
r:=r+fft(p,x)
p := reductum p
r
---- generalized eigenvectors associated to a given eigenvalue ---
genEigForm(eigen : EigenForm,A:M) : GenEigen ==
alpha:=eigen.eigval
k:=eigen.eigmult
g:=#(eigen.eigvec)
k = g => [alpha,eigen.eigvec]
[alpha,generalizedEigenvector(alpha,A,k,g)]
---- characteristic polynomial ----
charpol(A:M,x:SE) : F ==
dimA :PI := (nrows A):PI
dimA ~= ncols A => error " The matrix is not square"
B:M:=zero(dimA,dimA)
for i in 1..dimA repeat
for j in 1..dimA repeat B(i,j):=A(i,j)
B(i,i) := B(i,i) - monomial(1$P,x,1)::F
determinant B
-------- EXPORTED FUNCTIONS --------
---- characteristic polynomial of a matrix A ----
characteristicPolynomial(A:M):P ==
x:SE:=new()$SE
numer charpol(A,x)
---- characteristic polynomial of a matrix A ----
characteristicPolynomial(A:M,x:SE) : P == numer charpol(A,x)
---- Eigenvalues of the matrix A ----
eigenvalues(A:M): List Eigenvalue ==
x:=new()$SE
pol:= charpol(A,x)
lrat:List F :=empty()
lsym:List ST :=empty()
for eq in solve(pol,x)$SystemSolvePackage(R) repeat
alg:=numer lhs eq
degree(alg, x)=1 => lrat:=cons(rhs eq,lrat)
lsym:=cons([x,alg],lsym)
append([lr::Eigenvalue for lr in lrat],
[ls::Eigenvalue for ls in lsym])
---- Eigenvectors belonging to a given eigenvalue ----
---- the eigenvalue must be exact ----
eigenvector(alpha:Eigenvalue,A:M) : List M ==
alpha case F => intRatEig(alpha::F,A,1$NNI)
intAlgEig(alpha::ST,A,1$NNI)
---- Eigenvectors belonging to a given rational eigenvalue ----
---- Internal function -----
intRatEig(alpha:F,A:M,m:NNI) : List M ==
n:=nrows A
B:M := zero(n,n)$M
for i in 1..n repeat
for j in 1..n repeat B(i,j):=A(i,j)
B(i,i):= B(i,i) - alpha
[v::M for v in nullSpace(B**m)]
---- Eigenvectors belonging to a given algebraic eigenvalue ----
------ Internal Function -----
intAlgEig(alpha:ST,A:M,m:NNI) : List M ==
n:=nrows A
MM := ModularField(SUF,SUF,reduction,merge,exactquo)
AM:=Matrix MM
x:SE:=lhs alpha
pol:SUF:=unitCanonical map(coerce,univariate(rhs alpha,x))$UPCF2
alg:MM:=reduce(monomial(1,1),pol)
B:AM := zero(n,n)
for i in 1..n repeat
for j in 1..n repeat B(i,j):=reduce(A(i,j)::SUF,pol)
B(i,i):= B(i,i) - alg
sol: List M :=empty()
for vec in nullSpace(B**m) repeat
w:M:=zero(n,1)
for i in 1..n repeat w(i,1):=tff((vec.i)::SUF,x)
sol:=cons(w,sol)
sol
---- Generalized Eigenvectors belonging to a given eigenvalue ----
generalizedEigenvector(alpha:Eigenvalue,A:M,k:NNI,g:NNI) : List M ==
alpha case F => intRatEig(alpha::F,A,(1+k-g)::NNI)
intAlgEig(alpha::ST,A,(1+k-g)::NNI)
---- Generalized Eigenvectors belonging to a given eigenvalue ----
generalizedEigenvector(eigen :EigenForm,A:M) : List M ==
generalizedEigenvector(eigen.eigval,A,eigen.eigmult,# eigen.eigvec)
---- Generalized Eigenvectors -----
generalizedEigenvectors(A:M) : List GenEigen ==
n:= nrows A
leig:=eigenvectors A
[genEigForm(leg,A) for leg in leig]
---- eigenvectors and eigenvalues ----
eigenvectors(A:M):List(EigenForm) ==
n:=nrows A
x:=new()$SE
p:=numer charpol(A,x)
MM := ModularField(SUF,SUF,reduction,merge,exactquo)
AM:=Matrix(MM)
ratSol : List EigenForm := empty()
algSol : List EigenForm := empty()
lff:=factors factor p
for fact in lff repeat
pol:=fact.factor
degree(pol,x)=1 =>
vec:F :=-coefficient(pol,x,0)/coefficient(pol,x,degree(pol,x))
ratSol:=cons([vec,fact.exponent :: NNI,
intRatEig(vec,A,1$NNI)]$EigenForm,ratSol)
alpha:ST:=[x,pol]
algSol:=cons([alpha,fact.exponent :: NNI,
intAlgEig(alpha,A,1$NNI)]$EigenForm,algSol)
append(ratSol,algSol)
)abbrev package CHARPOL CharacteristicPolynomialPackage
++ Author: Barry Trager
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ This package provides a characteristicPolynomial function
++ for any matrix over a commutative ring.
CharacteristicPolynomialPackage(R:CommutativeRing):C == T where
PI ==> PositiveInteger
M ==> Matrix R
C == with
characteristicPolynomial: (M, R) -> R
++ characteristicPolynomial(m,r) computes the characteristic
++ polynomial of the matrix m evaluated at the point r.
++ In particular, if r is the polynomial 'x, then it returns
++ the characteristic polynomial expressed as a polynomial in 'x.
T == add
---- characteristic polynomial ----
characteristicPolynomial(A:M,v:R) : R ==
dimA :PI := (nrows A):PI
dimA ~= ncols A => error " The matrix is not square"
B:M:=zero(dimA,dimA)
for i in 1..dimA repeat
for j in 1..dimA repeat B(i,j):=A(i,j)
B(i,i) := B(i,i) - v
determinant B
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