Reliability of Lanczos-type product methods from perturbation theory (Q5926517)

Large, sparse linear systems with a non-Hermitian matrix are solved by methods based on the Richardson iteration. The convergence of a vector sequence generated by the Richardson iteration can be accelerated using Padé methods. The relation between Lanczos polynomials and the denominators of certain vector Padé approximants is reviewed. Three product methods are presented. They solve a preconditioned large, sparse linear system and are based on a vector Padé approximation of a perturbation series. The methods are analogous to the method of \textit{H. A. Van der Vorst} [SIAM J. Sci. Stat. Comput. 13, No. 2, 631-644 (1992; Zbl 0761.65023)], \textit{M. H. Gutknecht} [SIAM J. Sci. Stat. Comput. 14, No. 5, 1020-1033 (1993; Zbl 0837.65031)], and \textit{S.-L. Zhang} [SIAM J. Sci. Stat. Comput. 18, No. 2, 537-551 (1997; Zbl 0872.65023)]. Two examples illustrate the convergence of Lanczos-type product methods.