Matrix Pencils: Proceedings of a Conference Held at Pite Havsbad, Sweden, March 22–24, 1982

299 Pages·1983·3.264 MB·English

by  James Demmel (auth.), Bo Kågström, Axel Ruhe (eds.)

Lecture Notes ni
Mathematics
Edited yb .A Dold dna .B nnamkcE
973
xirtaM Pencils
Proceedings of a Conference
Held ta Pite Havsbad, Sweden, March 22-24, 1982
Edited yb .B m6rtsg&K dna .A Ruhe
galreV-regnirpS
Berlin Heidelberg New kroY 1983
Editors
Bo K&gstr~m
Axel Ruhe
Institute of Information Processing
Department of Numerical Analysis, University of Ume&
901 87 Ume&, Sweden
AMS Subject Classifications 1980: 15A18, 15A42, 65-06, 65 F15, 65F 20,
65 F 25, 65 F 35, 65 L 02
ISBN 3-54041983-3 Springer-Verlag Berlin Heidelberg New York
ISBN 0-38?41983-3 Springer-Verlag New York Heidelberg Berlin
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Preface
A conference devoted to Matrix Pencils was held March 22-24, 1982 at
Hotel Larus, Pite Havsbad in Northern Sweden. It was organized jointly
by the Institute of Information Processing, Numerical Analysis depart-
ment at the University of Ume~, and the Swedish Institute of Applied
Mathematics (ITM).
The choice of topic reflects the research interest of the group in Ume~
since more than a decade, and the aim of the conference was to intro-
duce and discuss recent mathematical and numerical research in matrix
computations and ~ related areas, as well as the use of matrix computa-
tions in applied problems. More than fifty people from fourteen nations
participated at the conference; twenty-three talks were presented and
six short contributions. The present volume contains a selection of the
contributions to the conference, and here we give a short survey of the
contents.
The first group of papers deal with general A-IB pencils. Though the
mathematical theory for such pencils has been well understood for a
long time, we have only recently got any reliable numerical methods.
Pioneering work has been performed by Vera Nikolaevna Kublanovskaya
from Leningrad, and her talk at this conference gives an introduction
to that work. She presents the AB-algorithm and its modification as one
approach for handling spectral problems of linear pencils. Paul Van
Dooren treats essentially the same problem, with the emphasis on trans-
formations to block triangular form and finding reducing subspaces.
This new concept extends the noticn of deflating subspaces for regular
pencils, as introduced by G.W. Stewart, to the singular case. He also
points out the relevance of those concepts in the theory of dynamic
systems. Bo KAgstr~m extends his works on algorithms for the Jordan
Normal Form (JNF) of a matrix, to the pencil case, and shows how the
JNF software can be used for regular pencils. He also formulates a modi-
fication of the AB-algorithm for deflating zero and infinite eigenvalues
of a regular pencil, in terms of the Singular Value Decomposition. James
Demmel reports some results on the, condition of block diagonalization,
i.e. the part of an algorithm that preceeds the computation of reducing
subspaces. The bounds on the condition number of the transformation ma-
trix are expressed in terms of the projection matrices determined by the
partitioning.
VI
The general matrix pencil problem can be very hard to deal with numeri-
cally. One reason is that the JNF is not a continuous function of the
matrix elements. It is therefore necessary to see which numerical re-
sults that are relevant for the applied problem that gives rise to the
matrix pencil. One important class of such applied problems is systems
of ordinary differential equations F(t,y,y')=0. If these contain
algebraic equations, and ~F/~y' is singular, many difficulties arise.
C.W. (Bill) Gear and Linda Petzold deal with such differential alge-
braic equation (DAE) systems, and point out that they are considerably
different from ordinary differential systems, as we are used to know
them. Linear DAE systems with constant coefficients can be completely
characterized via the Kronecker canonical form of the matrix pencil
A-IB. However the linear nonconstant-coefficient case is not a simple
extension of the constant coefficient case when the nilpotency of
A-IB exceeds one.
In a special topic paper Kam-Moon Liu and Eduardo Ortiz are presenting
numerical results from the application of the Tau method to the solu-
tion of eigenvalue problems defined by ordinary differential equations.
The symmetric case (A symmetric and B positive definite) is well under-
stood, and here the emphasis is on development of efficient algorithms
for large sparse problems, and their use in applications. Finite element
computations is big business, and the papers by Liv Aasland and Petter
Bj~rstad, Eric Carnoy and M. Geradin, and Thomas Ericsson deal mainly
with usage of the Lanczos algorithm in such computations. The Lanczos
algorithm is now on the verge of replacing simultaneous iteration in
commercial finite element packages. The symmetric (A-IB)-problem with
B positive definite is also discussed in the papers by D.J. Evans and
Alan Jennings. Their topics are preconditioned iterative methods for
the determination of extreme eigenpairs and bounds for the eigenvalues
when a Cholesky factorization of B is available, respectively.
Generalizations of the Lanczos algorithm for nonsymmetric problems are
discussed by Saad and Ruhe. Youcef Saad gives a unified view, in terms
of projections of the most commonly used algorithms for large sparse
problems, including the nonsymmetric Lanczos algorithm, the Arnoldi
method and the subspace iteration. He also presents some ~ priori error
bounds for these methods, in terms of the distance from the exact eigen-
vector to the approximating subspace. In the paper by Axel Ruhe the
two-sided Arnoldi method is presented. The algorithm computes both
left and right eigenvector approximations, by applying the Arnoldi
method both to the matrix and its transpose. For nonsymmetric problems
much work remains to be done, and it is not yet clear which of the
generalizations of Lanczos that is the winner.
The other major group of papers deal with singular value problems, and
applications to statistical data analysis. Gene Golub gave a review of
statistical computations and generalized eigenvalue problems. He dis-
cussed estimates from linear mode3Ls and compared the variance of a
weighted least squares estimate, with that of a minimum variance esti-
mate. This is important when using weighting to obtain better behaved
estimates, and gives guidelines on how to choose weights and compute
estimates.
Charles van Loan was the first to introduce the generalized singular
value decomposition (GSVD). Here he uses the GSVD to analyse some
weighting methods for least squares problems with equality constraints.
So far it has not existed any stable algorithm to actually compute the
GSVD. G.W. (Pete) Stewart describes how the problem of computing the
GSVD can be reduced to that of computing a CS-decomposition (Cosine-
Sinus) of a matrix with orthonormal columns, and suggests an algorithm,
that is made stable by a reorthogonalization technique.
Two papers deal with theoretical aspects. Jiguang Sun from Beijing
gives perturbation results both for the generalized eigenvalue and the
generalized singular value problems. As an example he generalizes some
classical perturbation theorems like the Bauer-Fike, Hoffman-Wielandt,
Weyl-Lidskii and the Davis-Kahan theorems. Per Ake Wedin studies angles
between subspaces. By utilizing the singular value decomposition and
orthogonal projections, he shows how angles between two subspaces of
C n are related to the principal angles between certain invariant two
dimensional subspaces. Further these concepts are used to point out
that whenever relevant perturbation identities are known, it is easy to
use an angle function to get perturbation bounds. Svante Wold, in the
last local contribution, deals with applications to chemical data ana-
lysis.
Finally we would like to take this opportunity of thanking all speakers,
chairmen and participants for their contributions. In particular we are
IV
indebted to Christina Holmstr6m and Inga-Lena Olsson for their care-
ful typing of various documents associated with the conference, and
some of the typing in this volume.
The financial support from ITM is gratefully acknowledged.
Ume~, September 1982
Bo K~gstr~m Axel Ruhe
Contents
A. GENERAL (A-IB)-PENCILS
A.I CANONICAL REDUCTIONS - THEORY AND ALGORITHMS . . . . . . . . I
J. DEMMEL: The condition number of equivalence trans-
formations that block diagonalize matrix
pencils . . . . . . . . . . . . . . . . . . . . . . 2
~' V.N. KUBLANOVSKAYA: An approach to solving the spectral
problem of A-IB . . . . . . . . . . . . . . . . . 17
\ B. KAGSTROM: On computing the Kronecker canonical form
of regular (A-lB)-pencils . . . . . . . . . . . . 30
P. VAN DOOREN: Reducing subspaces - definitions, proper-
ties and algorithms . . . . . . . . . . . . . . . 58
A.2 ASPECTS FROM DIFFERENTIAL EQUATIONS . . . . . . . . . . . . 74
C.W. GEAR*, L.R. PETZOLD*: Differential/algebraic sys-
tems and matrix pencils . . . . . . . . . . . . . 75
K-M. LIU, E.L. ORTIZ*: Approximation of eigenvalues de-
fined by ordinary differential equations
with the Tau method . . . . . . . . . . . . . . . 90
A.3 ALGORITHMS FOR LARGE SPARSE UNSYMMETRIC (A-II)-PROBLEMS . 103
,'J A. RUHE: The two-sided Arnoldi algorithm for nonsymmetric
eigenvalue problems . . . . . . . . . . . . . . . 104
Y. SAAD: Projection methods for solving large sparse
eigenvalue problems . . . . . . . . . . . . . . . 121
B. SYMMETRIC (A-IB)-PENCILS AND APPLICATIONS . . . . . . . . . 145
L. AASLAND, P. BJ#RSTAD: The generalized eigenvalue
problem in shipdesign and offshore industry
- a comparison of traditional methods with
the Lanczos process . . . . . . . . . . . . . . . 146
E.G. CARNOY, M. GERADIN: On the practical use of the
Lanczos algorithm in finite element applica-
tions to vibration and bifurication problems. 156
llIV
~' T. ERICSSON: Implementation and applications of
the spectral transformation Lanczos
algorithm . . . . . . . . . . . . . . . . . . . . 177
D.J. EVANS: Preconditioned iterative methods for the
generalized eigenvalue problem . . . . . . . . . . 189
A. JENNINGS: On bounds for symmetric eigenvalue
problems . . . . . . . . . . . . . . . . . . . . . 195
C. GENERALIZED SINGULAR VALUES AND DATA ANALYSIS . . . . . . . 206
I~ G.W. STEWART: A method for computing the generalized
singular value decomposition. . . . . . . . . . 207
~ J-G. SUN: Perturbation analysis for the generalized
eigenvalue and the generalized singular
value problem . . . . . . . . . . . . . . . . . . 221
\i C. VAN LOAN: A generalized SVD analysis of some
weighting methods for equality constrained
least squares . . . . . . . . . . . . . . . . . . 245
' P-A. WEDIN: On angles between subspaces . . . . . . . . . . . 263
S. WOLD, H. MARTENS, H. WOLD: The multivariate cali-
bration problem in chemistry solved by the
PLS method . . . . . . . . . . . . . . . . . . . . 286
In addition the following talks were presented at the
conference:
Germund Dahlquist, Royal Institute of Technology, Stockholm,
Sweden.
Some reflections on transformation of time
dependent matrices to block diagonal form.
Emeric Deutsch, Polytechnic Institute of New York, New York,
U.S.A.
Lower bounds for the Perron root of a non-
negative matrix.
Gene Golub, Stanford University, Stanford, California, U.S.A.
The generalized eigenvalue problem and statis-
tical computations.
Kresimir Veselic, Fernuniversit~t Hagen, Hagen, Germany
A Jacobi-like algorithm for the problem
Ax=lBx with symmetric indefinite A and B.
Olof Widlund, Courant Institute, New York, U.S.A.
Iterative solution of large generalized
eigenvalue problems.
ADDRESSES OF THE AUTHORS
Liv Aasland, Petter Bj~rstad: Det Norske Veritas
FDIV 40
Postboks 300
N-1322 H@VIK, NORGE
Eric Carnoy, M. Geradin: L.T.A.S.
Dynamique des constructions m~caniques
Universit~ de Liege
Rue Ernest Solvay, 12
B-4000 Liege
BELGIQUE
James Demmel: Computer Science Division/EECS Dept
University of California
Berkely, CA 94720
U.S.A.
Thomas Ericsson: Institute of Information Processing
Department of Numerical Analysis
University of Ume~
S-901 87 UMEA, SWEDEN
David Evans: Department of Computer Studies
University of Technology
Loughborough, Leicestershire LE 11 3 TU
UNITED KINGDOM
C.W. Gear: Department'of Computer Science
University of Illinois
1304 W Springfield
URBANA IL 61801
U.S.A.
Alan Jennings: Civil Engineering Department
Queens University
Belfast, BTG 5PG
UNITED KINGDOM
Vera Kublanovskaya: USSR Leningrad D-II
Fontanka, 25
Inst of Mathematics Acad. of Sciences
Bo K~gstr6m: Institute of Information Processing
Department of Numerical Analysis
University of Ume~
S-901 87 UMEA, SWEDEN
Eduardo Ortiz, K-M. Liu: Imperial College of Science and Tech-
nology
Department of Mathematics
Huxley Building
Queen's Gate, London SW7 2BZ
UNITED KINGDOM