Robust preconditioning of large, sparse, symmetric eigenvalue problems

Iterative methods for solving large, sparse, symmetric eigenvalue problems often encounter convergence difficulties because of ill-conditioning. The generalized Davidson method is a well-known technique which uses eigenvalue preconditioning to surmount these difficulties. 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 confirm the robustness of the approach.