Matrix Pencils: Proceedings of a Conference Held at Pite Havsbad, Sweden, March 22–24, 1982 PDF
Preview Matrix Pencils: Proceedings of a Conference Held at Pite Havsbad, Sweden, March 22–24, 1982
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 This work si tcejbus to .thgirypoc All rights era ,devreser whether the whole or part of the lairetam si ,denrecnoc specifically those of ,noitalsnart reprinting, esu-er of ,snoitartsulli ,gnitsacdaorb noitcudorper yb photocopying enihcam or similar ,snaem dna storage ni data .sknab rednU § 54 of the namreG Copyright waL where copies era made for other than private ,esu a fee si elbayap to tfahcsllesegsgnutrewreV" ,"troW .hcinuM © yb galreV-regnirpS Berlin Heidelberg 3891 detnirP ni ynamreG Printing dna binding: Beltz ,kcurdtesffO .rtsgreB/hcabsmeH 012345-0413/6412 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
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