Explicitly restarted Lanczos algorithms in an MPP environment

Abstract

The Lanczos algorithm is one of the principal methods for the computation of a small part of the eigenspectrum of large, sparse, real symmetric matrices. A single-vector, explicitly restarted variant of the Lanczos method is proposed in this paper. The algorithm finds only one eigenpair at a time using a deflation technique in which the Lanczos factorization for the current eigenpair is generated in the null space of all previously computed eigenvectors. This approach yields a fixed k-step restarting scheme which permits short Lanczos factorizations and almost completely eliminates the reorthogonalization among the Lanczos vectors. The orthogonalization strategy developed falls naturally into the class of selective orthogonalization strategies as classified by Simon. `Reverse communication' software for the implementation of the proposed variant on a Connection Machine CM-200 with 8K processors and on a Cray T3D with 32 processors is discussed. Test results on the CM-200 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. The method has fixed, small storage requirements, copes easily with genuinely multiple eigenvalues and is guaranteed to converge to the desired eigenvalues.