The Convergence of Krylov Subspace Methods for Large Unsymmetric Linear Systems

作     者：Jia Zhongxiao, Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, China 

作者机构：Department of Applied Mathematics Dalian University of Technology 116024 Dalian China 

出 版 物：《Acta Mathematica Sinica,English Series》 (数学学报（英文版）)

年 卷 期：1998年第14卷第4期

页      面：507-518页

核心收录：

学科分类：07[理学] 0701[理学-数学] 070101[理学-基础数学] 

主　　题：Unsymmetric linear systems Convergence Krylov subspace The Chebyshev polynomials Defective Derivatives 

摘      要：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 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 of its spectrum is not favorable, or the Jordan basis of A 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.