Robust Preconditioning of Large, Sparse, Symmetric Eigenvalue Problems

Iterative methods for solving large, sparse, symmetric eigenvalue problems often encounter convergence diiculties because of ill-conditioning. The Generalized Davidson method is a well known technique which uses eigenvalue preconditioning to surmount these diiculties. Preconditioning the eigenvalue problem entails more subtleties than for linear systems. In addition, the use of an accurate conventional preconditioner (i.e., as used in linear systems) may cause deterioration of convergence or convergence to the wrong eigenvalue. The purpose of this paper is to assess the quality of eigenvalue preconditioning and to propose strategies to improve robustness. Numerical experiments for some ill-conditioned cases connrm the robustness of the approach.