The Golub–Kahan bidiagonalization algorithm has been widely used in solving leastsquares problems and in the computation of the SVD of rectangular matrices. Here we propose an algorithm based on the Golub–Kahan process for the solution of augmented systems that minimizes the norm of the error and, in particular, we propose a novel estimator of the error similar to the one proposed by Hestenes a...

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...

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 ...

We study an inexact inner–outer generalized Golub–Kahan algorithm for the solution of saddle-point problems with a two-times-two block structure. In each outer iteration, inner system has to be solved which in theory done exactly. Whenever is getting large, exact solver is, however, no longer efficient or even feasible and iterative methods must used. focus this article on numerical showing inf...

Abstract. Tikhonov regularization is one of the most popular approaches to solve discrete ill-posed problems with error-contaminated data. A regularization operator and a suitable value of a regularization parameter have to be chosen. This paper describes an iterative method, based on Golub-Kahan bidiagonalization, for solving large-scale Tikhonov minimization problems with a linear regularizat...

We describe a novel method for reducing a pair of large matrices {A,B} to a pair of small matrices {H,K}. The method is an extension of Golub–Kahan bidiagonalization to matrix pairs, and simplifies to the latter method when B is the identity matrix. Applications to Tikhonov regularization of large linear discrete ill-posed problems are described. In these problems the matrix A represents a disc...

We develop numerical algorithms for the efficient evaluation of quantities associated with generalized matrix functions [J. B. Hawkins and A. Ben-Israel, Linear and Multilinear Algebra, 1(2), 1973, pp. 163–171]. Our algorithms are based on Gaussian quadrature and Golub–Kahan bidiagonalization. Block variants are also investigated. Numerical experiments are performed to illustrate the effectiven...