External selective orthogonalization for the Lanczos algorithm in distributed memory environments

The k-step explicit restart Lanczos algorithm, LExpRes, for the computation of a few of the extreme eigenpairs of large, usually sparse, symmetric matrices, computes one eigenpair at a time using a deflation technique in which each Lanczos vector generated is orthogonalized against all previously converged eigenvectors. The computation of the inner products associated with this external orthogonalization often creates a bottleneck in parallel distributed memory environments. In this paper methods are proposed which significantly reduce this computational overhead in LExpRes, thereby effectively improving its efficiency. The performances of these methods on the Cray-T3D and the Cray-T3E are assessed and critically compared with that of the original algorithm.