COMPUTING EXTREMAL SINGULAR TRIPLETS OF SPARSE MATRICES ON A SHARED-MEMORY MULTIPROCESSOR

Abstract

In this paper, we develop four numerical methods for computing the singular value decomposition (SVD) of large sparse matrices on a shared-memory multiprocessor architecture. We particularly consider the SVD of unstructured sparse matrices in which the number of rows may be substantially larger or smaller than the number of columns. We emphasize Lanczos, block-Lanczos, subspace iteration and the trace minimization methods for determining a select number of smallest singular triplets (singular values and corresponding left- and right-singular vectors) for sparse matrices. The target architectures for implementations of such methods include the Alliant FX/80 and the Cray-2S/4–128. This algorithmic research is particularly motivated by recent information-retrieval techniques in which approximations to large sparse term-document matrices are needed, and by nonlinear inverse problems arising from seismic reflection tomography applications.