Precondition Krylov Subspace Algorithms for COmputing Eigenvalues of Large Matrices

This project will develop preconditioned Krylov subspace methods with analysis for computing a few eigenvalues of a large generalized eigenvalue problem Ax = lambda Bx, and study their robust implementations with a long term goal to develop a specialized package for public distributions. The resulting algorithms should inherit desirable characteristics of the existing Krylov subspace methods, but extend their capability for efficient preconditioning. The feasibility of this objective has been demonstrated by a preliminary study that led to an algorithm of this type for computing the smallest eigenvalue of a symmetric definite problem. In this project, this preliminary work will be strengthened and its idea further developed and generalized to produce algorithms that are capable of delivering extreme as well as interior eigenvalues for symmetric as well as nonsymmetric problems alike. This project builds upon the PI's research expertise and contributions over the past decade, to significantly advance the state-of-the-art of numerical methods for large matrix eigenvalue problems. Unifying several existing ideas and concepts and bringing new approaches, the resulting methods would be an ideal topics for classroom learning and thesis research. Moreover, the preliminary study indicates that they would be well suited for black-box implementations and thus have the potential to reach a broader application community.