The Convergence of Krylov Subspace Methods for Large Unsymmetric Linear Systems

The convergence problem of Krylov subspace methods, e.g. FOM, GMRES, GCR and many others, for solving large unsymmetric linear systems has been intensively investigated. There are many results in the literature, mainly for the case where the coefficient matrix A is diagonalizable and its spectrum lies in the open right (left) half plane. In this paper, we focus on a convergence analysis of those Krylov subspace methods which give rise to a similar minimax problem in the case where the coefficient matrix A is defective. When the spectrum of A either lies in the open right (left) half plane or is on the real axis, we establish the related theoretical error bounds and reveal some intrinsic relationships between the convergence speed and the spectrum of A. The results show that these methods are likely to converge slowly whenever one of three cases occurs: A is defective, the distribution of its spectrum is not favorable, or the Jordan basis of A is ill-conditioned. Keywords. unsymmetric..