Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-25T10:07:40.706Z Has data issue: false hasContentIssue false
  • English
  • Français

Homotopy continuation method for the numerical solutions of generalised symmetric eigenvalue problems

Published online by Cambridge University Press:  17 February 2009

D. C. Dzeng  and
W. W. Lin
W. W. Lin
Affiliation:
Institute of Applied Maths., Tsing Hua University, Hsinchu, Taiwan, Republic of China.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider a generalised symmetric eigenvalue problem Ax = λMx, where A and M are real n by n symmetric matrices such that M is positive semidefinite. The purpose of this paper is to develop an algorithm based on the homotopy methods in [9, 11] to compute all eigenpairs, or a specified number of eigenvalues, in any part of the spectrum of the eigenvalue problem Ax = λMx. We obtain a special Kronecker structure of the pencil A − λM, and give an algorithm to compute the number of eigenvalues in a prescribed interval. With this information, we can locate the lost eigenpair by using the homotopy algorithm when multiple arrivals occur. The homotopy maintains the structures of the matrices A and M (if any), and the homotopy curves are n disjoint smooth curves. This method can be used to find all/some isolated eigenpairs for large sparse A and M on SIMD machines.

Type
Research Article
Information
The ANZIAM Journal , Volume 32 , Issue 4 , April 1991 , pp. 437 - 456
Copyright
Copyright © Australian Mathematical Society 1991

References

[1] Bunch, J. R. and Kaufman, L., “Some stable methods for calculating inertia and solving symmetric linear systems”, Math. Comp. J. 31 (1977) 163–179.CrossRefGoogle Scholar
[2] Bunse-Gerstner, A., “An algorithm for the symmetric generalized eigenvalue problem”, Lin. Alg. Appl. 58 (1984) 43–68.CrossRefGoogle Scholar
[3] Chu, M. T., “A simple application of the homotopy method to symmetric eigenvalue problems”, Lin. Alg. Appl. 59 (1984) 85–90.CrossRefGoogle Scholar
[4] Chu, M. T., Li, T. Y. and Sauer, T., “Homotopy method for general λ-matrix problem”, SIAM J. Matrix Anal. Appl. 9 (1988) 528–536.CrossRefGoogle Scholar
[5] Fix, G. and Heiberger, R., “An algorithm for the ill-conditioned generalized eigenvalue problem”, SIAM J. Numer. Anal. 9 (1972) 78–88.CrossRefGoogle Scholar
[6] Geltner, P. B., “General Rayleigh quotient iteration”, SIAM J. Numer. Anal. 18 (1981) 839–843.CrossRefGoogle Scholar
[7] Kalaba, R., Spingarn, K., and Tesfatsion, L., “Variational equations for the eigenvalues and eigenvectors of non-symmetric matrices”, J. Optimiz. Theory and Appl. 33 (1981) 1–8.CrossRefGoogle Scholar
[8] Kalaba, R., Spingarn, K., and Tesfatsion, L., “Individual tracking of an eigenvalue and eigenvector of a parameterize matrix”, Nonlinear Analysis, Theory, Methods and Applications 5 (1981) 337–340.Google Scholar
[9] Li, T. Y. and Rhee, N., “Homotopy algorithm for symmetric eigenvalue problems”, Numer. Math. 55 (1989) 265–280.CrossRefGoogle Scholar
[10] Li, T. Y. and Sauer, T., “Homotopy method for generalized eigenvalue problems”, Lin. Alg. Appl. 91 (1987) 65–74.CrossRefGoogle Scholar
[11] Lin, W. W. and Lutzer, G., “An application of the homotopy method to the generalized symmetric eigenvalue problem”, J. Austral. Math. Soc. Ser. B, 30 (1987) 232–249.Google Scholar
[12] Milnor, J., Topology from the differentiate viewpoint, (Univ. of Virginia Press, 1965).Google Scholar
[13] Moler, C. B. and Stewart, G. W., “An algorithm for generalised matrix eigenvalue problems”, SIAM J. Numer. Anal. 10 (1973) 241–256.CrossRefGoogle Scholar
[14] Parlett, B. N., The Symmetric Eigenvalue Problem, (Prentice-Hall, Inc., Englewood Cliffs, 1980).Google Scholar
[15] Peters, G. and Wilkinson, J. H., “Inverse iteration, ill-conditioned equations and Newton's method”, SIAM Review, 21 (1979) 339–360.CrossRefGoogle Scholar
[16] Stoer, J. and Bulirsch, R., “Introduction to numerical analysis”, Translated by Bartels, R., Gautschi, W. and Witzgall, C. (1980).CrossRefGoogle Scholar