Total (Quasi) Minimal Backward Perturbation Methods For Linear Systems

This paper mainly studies the backward perturbation methods in solving the linear systems, and total minimal backward perturbation (TMINBACK) method for symmetric linear systems is presented. This new method is composed of Lanczos method and the technique which minimized the normwise of the total backward perturbation. In order to reduce the computations and the memories, restarted version is used in it. By using the norm of the total backward perturbation as the termination of the algorithm, the new method makes up the shortcomings of using residual norm, and then the restarted total minimal backward perturbation (RTMINBACK) method is proposed. This paper emphasizes that the new method uses the augmented Krylov subspace technique. The deflated restarted total minimization backward perturbation method (DRTMINBACK) is presented.When solving the nonsymmetric linear systems, the paper makes the total backward perturbation analysis to QMR method, and uses the total quasi-minimal backward perturbation (TQMBACK) technique. So the restarted total quasi-minimal backward perturbation (RTQMBACK) method is given.The numerical experiments show the new methods are greatly improved not only in the convergence but the computations.