摘要
其中,G是一个依赖于A的N阶矩阵,h是一个依赖于A和b的N阶向量。 方法(2)收敛的充要条件是迭代矩阵G的谱半径小于1。这个结论适合于任一线性定常迭代方法。但对非定常迭代方法,收敛性问题比较复杂,一般很难运用谱半径进行收敛性分析,Young给出的一个例子(见[1pp.298])便说明了这一点。 然而,对一种特殊的非定常迭代方法——契比雪夫半迭代法(下文简称CSI方法),却可以提供基于谱半径的收敛性条件。这正是本文的核心内容。
In this paper, we proved a sufficient condition for convergence of Cheby shev semi-iterative (CSI) methods applied to the numerical solution of algebraic linear systems, which depends on the bounds on the eigenvalues of a particular matrix and which is given by N.R.Santos and O.L.Linhares in 1986 with a wrong proof. In addition, we discuss the case where the eigenvalues of iterative matrices are complex and establish some sufficient and necessary conditions for convergence of Chebyshev CSI methods.
