Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-22T07:39:15.830Z Has data issue: false hasContentIssue false
  • English
  • Français

An application of the homotopy method to the generalised symmetric eigenvalue problem

Published online by Cambridge University Press:  17 February 2009

Wen-Wei Lin  and
Gerhard Lutzer
Wen-Wei Lin
Affiliation:
Institute of Applied Mathematics, National Tsing Hua University, Hsinchu, Taiwan 30043, R.O.C.
Gerhard Lutzer
Affiliation:
Institute of Applied Mathematics, National Tsing Hua University, Hsinchu, Taiwan 30043, R.O.C.
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.

The homotopy method is used to find all eigenpairs of a generalised symmetric eigenvalue problem Ax = λBx with positive definite B. The determination of n eigenpairs (x, λ) is reduced to curve tracing of n disjoint smooth curves in Rn × R × [0, 1]. The method might be attractive if A and B are sparse symmetric. In this paper it is shown that the method will work for almost all symmetric tridiagonal matrices A and B.

Type
Research Article
Information
The ANZIAM Journal , Volume 30 , Issue 2 , October 1988 , pp. 230 - 249
Copyright
Copyright © Australian Mathematical Society 1988

References

[1]Chan, T. F., “Deflated decomposition of solutions of nearly singular systems”, SIAM J. Numer. Anal. 21 (1984) 738–754.CrossRefGoogle Scholar
[2]Chan, T. F., “On the existence and computation of LU-factorization with small pivots”, Math. Comp. 42 (1984) 535–547.Google Scholar
[3]Chu, M. T., “A simple application of the homotopy method to symmetric eigenvalue problem”, Linear Algebra Appl. 59 (1984) 85–90.CrossRefGoogle Scholar
[4]Li, T. Y. and Rhee, N., “Homotopy algorithm for symmetric eigenvalue problems”, (to appear).Google Scholar
[5]Li, T. Y. and Sauer, T., “Homotopy method for generalized eigenvalue problems”, Linear Algebra Appl. 91 (1987) 65–74.CrossRefGoogle Scholar
[6]Osborne, M. R., “Numerical methods for hydrodynamic stability problems”, SIAM J. Appl. Math. 15 (1967) 539–557.CrossRefGoogle Scholar
[7]Shampine, L. F. and Gordon, M. K., Computer solution of ordinary differential equations (W. H. Freeman and Company, San Francisco 1975).Google Scholar
[8]Wilkinson, J. H. and Reinch, C., Linear Algebra (Springer-Verlag, New York, Heidelberg, Berlin 1971).CrossRefGoogle Scholar