A highly parallel explicitly restarted Lanczos algorithm

Abstract

The Lanczos algorithm is one of the principal methods for the computation of a few of the extreme eigenvalues and their corresponding eigenvectors of large, usually sparse, real symmetric matrices. In this paper a single-vector Lanczos method based on a simple restarting strategy is proposed. The algorithm finds one eigenpair at a time using a deflation technique in which each Lanczos vector generated is orthogonalized against all previously converged eigenvectors. The approach taken yields a fixed k-step restarting scheme which permits the reorthogonalization between the Lanczos vectors to be almost completely eliminated. The orthogonalization strategy developed falls naturally into the class of selective orthogonalization strategies as described by Simon.A ’reverse communication’ implementation of the algorithm on an MPP Connection Machine CM-200 with 8K processors is discussed. Test results using examples from the Harwell-Boeing collection of sparse matrices show the method to be very effective when compared with Sorensen's state of the art routine taken from the ARPACK library. Advantages of the algorithm include its guaranteed convergence, the ease with which it copes with genuinely multiple eigenvalues and fixed storage requirements.