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...

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...

Lanczos bidiagonalization is a competitive method for computing a partial singular value decomposition of a large sparse matrix, that is, when only a subset of the singular values and corresponding singular vectors are required. However, a straightforward implementation of the algorithm has the problem of loss of orthogonality between computed Lanczos vectors, and some reorthogonalization techn...

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...

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...

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 ...

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...

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...