The convergence of Krylov subspace methods for large unsymmetric linear systems

The convergence problem of many Krylov subspace methods,e.g., FOM, GCR, GMRES and QMR, for solving large unsymmetric (non-Hermitian) linear systems is considered in a unified way when the coefficient matrixA is defective and its spectrum lies in the open right (left) half plane. Related theoretical error bounds are established and some intrinsic relationships between the convergence speed and the spectrum ofA are exposed. It is shown that these methods are likely to converge slowly once one of the three cases occurs:A is defective, the distribution of its spectrum is not favorable, or the Jordan basis ofA is ill conditioned. In the proof, some properties on the higher order derivatives of Chebyshev polynomials in an ellipse in the complex plane are derived, one of which corrects a result that has been used extensively in the literature. Supported by the China State Major Key Project for Basic Researches, the National Natural Science Foundation of China, the Doctoral Program of the Chinese National Educational Commission, the Foundation of Returned Scholars of China and Liaoning Province Natural Science Foundation.