求解大规模非Hermite线性方程组的Krylov子空间型方法的收敛性分析

本文用统一的方式研究了当系数矩阵Ａ亏损且其谱位于右（左）半开平面时很多求解大规模非Ｈｅｒｍｉｔｅ线性方程组的Ｋｒｙｌｏｖ子空间型方法的收敛性，建立了有关的理论收敛界，揭示了收敛速度和Ａ的谱之间的内在联系．结果证明，当如下三种情形之一出现时，这些方法的收敛速度将会减慢：Ａ亏损，其谱的分布不理想，或Ａ的Ｊｏｒｄａｎ基病态．在证明中，我们给出了Ｃｈｅｂｙｓｈｅｖ多项式的高阶导数在复平面中某椭圆域上的若干新性质，其中之一修正了文献中广泛使用的一个结果．

A Convergence Analysis of Some Krylov Subspace Methods for Large Non Hermitian Linear Systems

The convergence problem of some Krylov subspace methods, e.g. FOM, GMRES, GCR and many others, for solving large non Hermitian linear systems is considered 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 revealed. 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. In the proof, some important results and properties on the Chebyshev polynomials and their higher derivatives in an ellipse in the complex plane are derived, one of which corrects awrong result given by Manteuffel T. in 1975 that has been used extensively in the literature.