THE CONVERGENCE OF KRYLOV SUBSPACE METHODS FOR LARGE UNSYMMETRIC LINEAR-SYSTEMS

The convergence problem of many Krylov subspace methods, e.g., FOM, GC R, GMRES and QMR, for solving large unsymmetric (non-Hermitian) linear systems is considered in a unified way when the coefficient matrix A 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 of A 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 o f its spectrum is not favorable, or the Jordan basis of A is ill condi tioned. In the proof, some properties on the higher order derivatives of Chebyshev polynomials in an ellipse in the complex plane are derive d, one of which corrects a result that has been used extensively in th e literature.