Computing the smallest singular triplets of a large matrix

Abstract

In this paper we present a new type of restarted Krylov methods for calculating the smallest singular triplets of a large sparse matrix, A. The new framework avoids the Lanczos bidiagonalization process and the use of polynomial filtering. This simplifies the restarting mechanism and allows the introduction of several modifications. Convergence is assured by a monotonicity property that pushes the computed Ritz values toward their limits.Other innovations regard the construction of improved Krylov subspaces, which are generated by (ATA)−1, the inverse of the cross-product matrix, or by approximation of this matrix. The approximate inverse is computed by applying an iterative method to solve the related linear system. Numerical experiments illustrate the usefulness of the proposed approach. Keywords: Computing the smallest singular values, Large sparse matrices, Restarted Krylov methods, Inexact inversions, Inner-outer methods, JEL classification: 65F15, 65F25, 65F30, 65F50