The Lanczos algorithm (LA) is a useful iterative method for the reduction of a large matrix to tridiagonal form. It is a storage efficient procedure requiring only the preceding two Lanczos vectors to compute the next. The quasi-minimal residual (QMR) method is a powerful method for the solution of linear equation systems, Ax = b. In this report we provide another application of the QMR method:...

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 total least squares (TLS) techniques, also called orthogonal regression and errors-in-variables modeling, see [15, 16], have been developed independently in several areas. For a given linear (orthogonally invariant) approximation problem AX ≈ B, where A ∈ Rm×n, B ∈ Rm×d, X ∈ Rn×d, the TLS formulation aims at a solution of a modified problem (A + E)X = B + G such that min ‖[G,E]‖F . The alge...

A convergence analysis for the nonsymmetric Lanczos algorithm is presented. By using a tridiagonal structure of the algorithm, some identities concerning Ritz values and Ritz vectors are established and used to derive approximation bounds. In particular, the analysis implies the classical results for the symmetric Lanczos algorithm.

The Biorthogonal Lanczos and the Biconjugate Gradients methods have been proposed as iterative methods to approximate the solution of nonsymmetric and indefinite linear systems. Sonneveld [19] obtained the Conjugate Gradient Squared by squaring the matrix polynomials of the Biconjugate Gra­ dients method. Here we square the Biorthogonal Lanczos, the Biconjugate Residual and the Biconjugate Orth...

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 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. In this paper, we investigate a method for restarting the Lanczos method for approximating the matrix-vector product f(A)b, where A ∈ Rn×n is a symmetric matrix. For analytic f we derive a novel restart function that identifies the error in the Lanczos approximation. The restart procedure is then generated by a restart formula using a sequence of these restart functions. We present an...