Robust parallel lanczos methods for clustered eigenvalues

Abstract

In this paper two recently proposed single-vector Lanczos methods based on a simple restarting strategy are analysed and their suitability for the computation of closely clustered eigenvalues is evaluated. Both algorithms adopt an approach which yields a fixed fc-step restarting scheme in which one eigenpair at a time is computed using a deflation technique in which each Lanczos vector generated is orthogonal-ized against all previously converged eigenvectors. In the first algorithm each newly generated Lanczos vector is also orthogonalised with respect to all of its predecessors; in the second, a selective orthogonalisation strategy permits re-orthogonalization between the Lanczos vectors to be almost completely eliminated. 'Reverse communication' implementations of the algorithms on an MPP Connection Machine CM-200 with 8K processors are discussed. Advantages of the algorithms include the ease with which they cope with genuinely multiple eigenvalues, their guaranteed convergence and their fixed storage requirements.