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 introduce a randomized algorithm for overdetermined linear least-squares regression. Given an arbitrary full-rank m x n matrix A with m >/= n, any m x 1 vector b, and any positive real number epsilon, the procedure computes an n x 1 vector x such that x minimizes the Euclidean norm ||Ax - b || to relative precision epsilon. The algorithm typically requires ((log(n)+log(1/epsilon))mn+n(3)) fl...

We introduce a randomized algorithm for computing the minimal-norm solution to an underdetermined system of linear equations. Given an arbitrary full-rank matrix Am×n with m < n, any vector bm×1, and any positive real number ε less than 1, the procedure computes a vector xn×1 approximating to relative precision ε or better the vector pn×1 of minimal Euclidean norm satisfying Am×n pn×1 = bm×1. T...

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. ...