Sharp error bounds of some Krylov subspace methods for non-Hermitian linear systems1

摘要

The convergence of the Krylov subspace methods, e.g., Full Orthogonal Method (FOM) and Generalized Minimal Residual Method (GMRES), etc., for solving large non-Hermitian linear systems is studied in a unified and detailed way when the coefficient matrix is defective; in particular, when its spectrum lies in the open right (left) half plane or is on the real axis. Related theoretical error bounds are established, which reveal some intrinsic relationships between the convergence properties and the eigen-characteristics of the coefficient matrix. These results not only generalize all the known ones for the diagonalizable matrices in the literature, but also sharp the corresponding estimates in Jia (Acta Mathematica Sinica (New Series) 14 (1998) 507–518).