We consider algorithms for going from a “full” matrix to a condensed “band bidiagonal” form using orthogonal transformations. We use the framework of “algorithms by tiles”. Within this framework, we study: (i) the tiled bidiagonalization algorithm BiDiag, which is a tiled version of the standard scalar bidiagonalization algorithm; and (ii) the R-bidiagonalization algorithm R-BiDiag, which is a ...

The harmonic Lanczos bidiagonalization method can be used to compute the smallest singular triplets of a large matrix A. We prove that for good enough projection subspaces harmonic Ritz values converge if the columns of A are strongly linearly independent. On the other hand, harmonic Ritz values may miss some desired singular values when the columns of A are almost linearly dependent. Furthermo...

LSQR uses the Golub-Kahan bidiagonalization process to solve sparse least-squares problems with and without regularization. In some cases, projections of the right-hand side vector are required, rather than the least-squares solution itself. We show that projections may be obtained from the bidiagonalization as linear combinations of (theoretically) orthogonal vectors. Even the least-squares so...

New restarted Lanczos bidiagonalization methods for the computation of a few of the largest or smallest singular values of a large matrix are presented. Restarting is carried out by augmentation of Krylov subspaces that arise naturally in the standard Lanczos bidiagonalization method. The augmenting vectors are associated with certain Ritz or harmonic Ritz vectors. Computed examples show the ne...

Many applications require the solution of large nonsymmetric linear systems with multiple right-hand sides. Instead of applying an iterative method to each of these systems individually, it is often more eecient to use a block version of the method that generates iterates for all the systems simultaneously. In this paper, we propose block versions of Galerkin/minimal residual pair of bidiagonal...

We present a class of algorithms based on rational Krylov methods to compute the action generalized matrix function vector. These incorporate existing Golub-Kahan bidiagonalization as special case. By exploiting quasiseparable structure projected matrices, we show that basis vectors can be updated using short recurrence, which seen generalization case bidiagonalization. also prove error bounds ...

This paper proposes a harmonic Lanczos bidiagonalization method for computing some interior singular triplets of large matrices. It is shown that the approximate singular triplets are convergent if a certain Rayleigh quotient matrix is uniformly bounded and the approximate singular values are well separated. Combining with the implicit restarting technique, we develop an implicitly restarted ha...

We describe an extended bidiagonalization scheme designed to compute low-rank approximations of very large data matrices. Its goal is identical to that of the truncated singular value decomposition, but it is significantly cheaper. It consists in an extension of the standard Lanczos bidiagonalization that improves its approximation capabilities, while keeping the computational cost reasonable. ...