Estimating the largest singular values of large sparse matrices via modified moments (Q1186625)

The method in this paper determines a few of the largest singular values of large sparse matrices (numerical examples from latent semantic indexing up to size \(2\cdot 10^ 4\times 4\cdot 10^ 4\), densities 0.02\% to 2\%; from seismic tomography \(10^ 4\times 10^ 4\), density 1.3\%, approximately; done on a Cray Y-MP/4-64). A sequence of bidiagonal matrices whose singular values approximate those of the given matrix is obtained from a modification of a method by the second author and \textit{M. D. Kent} [Math. Comput. 53, No. 188, 619-626 (1989; Zbl 0675.65021)] using modified moments for estimating the eigenvalues of operators in the iterative solution of linear systems, and corresponding left and right singular vectors are obtained from a Lanczos recursion; here, attractive for parallel computers is the asynchronous computation of those bidiagonal matrices using modified moments with the iterations of an adapted Chebyshev semi-iteration. A comparison with a suitable Lanczos algorithm (with selective reorthogonalization) shows that the present method can be efficient for two of the largest singular values of the given matrix.