ON THE HOMOTOPY METHOD FOR PERTURBED SYMMETRICAL GENERALIZED EIGENVALUE PROBLEMS

The generalized eigenvalue problem plays a significant role in many ap plications. Usually only a few smallest eigenpairs (i.e., eigenvalues and their corresponding eigenvectors) are desired. A frequently encoun tered problem is to solve a system slightly perturbed from the origina l system. If the perturbation is small, the new system can be solved b y using Rayleigh quotient iteration (RQI); the initial Ritz vectors ar e provided by the eigenvectors from the original system. However, if t he perturbation is relatively large, direct use of RQI will not be suf ficient and, in many cases, will give inaccurate results, such as miss ing 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 per turbation theory. We then discuss some algorithmic issues such as step size estimation and grouping clustered eigenvalues. Numerical example s are given to illustrate the potential applications of this method.