Several numerical methods for the solution of large linear ill-posed problems combine Tikhonov regularization with an iterative method based on partial Lanczos bidiagonalization of the operator. This paper discusses the determination of the regularization parameter and the dimension of the Krylov subspace for this kind of methods. A method that requires a Krylov subspace of minimal dimension is...

In this paper we propose a fast structure-preserving algorithm for computing the singular value decomposition of quaternion matrices. The algorithm is based on the structurepreserving bidiagonalization of the real counterpart for quaternion matrices by applying orthogonal JRS-symplectic matrices. The algorithm is efficient and numerically stable. 2014 Elsevier Inc. All rights reserved.

in this work‎, ‎an iterative method based on a matrix form of lsqr algorithm is constructed for solving the linear operator equation $mathcal{a}(x)=b$‎ ‎and the minimum frobenius norm residual problem $||mathcal{a}(x)-b||_f$‎ ‎where $xin mathcal{s}:={xin textsf{r}^{ntimes n}~|~x=mathcal{g}(x)}$‎, ‎$mathcal{f}$ is the linear operator from $textsf{r}^{ntimes n}$ onto $textsf{r}^{rtimes s}$‎, ‎$ma...

We develop probabilistic upper bounds for the matrix two-norm, the largest singular value. These bounds, which are true upper bounds with a userchosen high probability, are derived with a number of different polynomials that implicitly arise in the Lanczos bidiagonalization process. Since these polynomials are adaptively generated, the bounds typically give very good results. They can be comput...

This paper discusses the implementation and evaluation of the reduction of a dense matrix to bidiagonal form on the Trident processor. The standard Golub and Kahan Householder bidiagonalization algorithm, which is rich in matrix-vector operations, and the LAPACK subroutine _GEBRD, which is rich in a mixture of vector, matrix-vector, and matrix operations, are simulated on the Trident processor....

The matrix-form LSQR method is presented in this paper for solving the least squares problem of the matrix equation AXB = C with tridiagonal matrix constraint. Based on a matrix-form bidiagonalization procedure, the least squares problem associated with the tridiagonal constrained matrix equation AXB = C reduces to a unconstrained least squares problem of linear system, which can be solved by u...

Many numerical methods for the solution of linear ill-posed problems apply Tikhonov regularization. This paper presents a modification of a numerical method proposed by Golub and von Matt for quadratically constrained least-squares problems and applies it to Tikhonov regularization of large-scale linear discrete ill-posed problems. The method is based on partial Lanczos bidiagonalization and Ga...

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...