On the Homotopy Method for Perturbed Symmetric Generalized Eigenvalue Problems

Abstract

The generalized eigenvalue problem plays a significant role in many applications. Usually only a few smallest eigenpairs (i.e., eigenvalues and their corresponding eigenvectors) are desired. A frequently encountered problem is to solve a system slightly perturbed from the original system. If the perturbation is small, the new system can be solved by using Rayleigh quotient iteration (RQI); the initial Ritz vectors are provided by the eigenvectors from the original system. However, if the perturbation is relatively large, direct use of RQI will not be sufficient and, in many cases, will give inaccurate results, such as missing some of the eigenvalues. The homotopy method can be used to remedy this problem. In this paper, we first review the homotopy method and its theoretical background. The approach employed here is based on perturbation theory. We then discuss some algorithmic issues such as step size estimation and grouping clustered eigenvalues. Numerical examples are given to illustrate the potential applications of this method.