Computing smallest singular triplets with implicitly restarted Lanczos bidiagonalization
This paper brings together Lanczos bidiagonalization, implicit restarts, deflation, and harmonic Ritz values for computing a few of the smallest singular values and vectors of a large sparse, possibly rectangular matrix. Although all these concepts are well known on their own, this approach is among the first to apply them together in the context of singular value problems. Several numerical experiments demonstrate that harmonic Ritz values and deflations are particularly important if the smallest singular values tend to be clustered. Moreover, it is shown that the proposed approach is, for all matrices under consideration, more efficient than some other methods. Meanwhile, however, the hunt on singular values has continued, and new, promising methods have been developed, for example by \textit{M. E. Hochstenbach} [Harmonic and refined extraction methods for the singular value problem, with applications in least squares problems, Preprint 1263, Dept. of Math., Utrecht University (2004)] as well as by \textit{J. Baglama} and \textit{L. Reichel} [Augmented implicitly restarted Lanczos bidiagonalization methods, Preprint (2004)].
scientific article; zbMATH DE number 2081655
scientific article