J Austral Math Soc Ser B 32 pp437--456, 1991.

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

D. C. Dzeng and W. W. Lin

Abstract

We consider a generalised symmetric eigenvalue problem Ax = lMx, 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 = lMx. We obtain a special Kronecker structure of the pencil A - lM, 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.

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Authors

D. C. Dzeng

W. W. Lin

Institute of Applied Maths., Tsing Hua University, Hsinchu, Taiwan, Republic of China.