A uniform error bound for the overrelaxation methods

作     者：Li, XZ 

作者机构：Department of Mathematics Computer Science Georgia Southern University Statesboro Georgia 30460 USA 

出 版 物：《LINEAR ALGEBRA AND ITS APPLICATIONS》 (线性代数及其应用)

年 卷 期：1997年第254卷第0期

页      面：315-333页

核心收录：

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

摘      要：Let Ax = b be a system of linear equations where A is symmetric and positive definite. Suppose that the associated block Jacobi matrix B is consistently ordered, weekly cyclic of index 2, and convergent [i.e., mu(1) := rho(B) 1]. Consider using the overrelaxation methods (SOR, AOR, MSOR, SSOR, or USSOR), x(n+1) = T(omega)x(n) + c(omega) for n greater than or equal to 0, to solve the system. We derive a uniform error bound for the overrelaxation methods, \\x - x(n)\\(2) less than or equal to 1/[1 + s(mu(1)(2)) + t(mu(1)(2))](2) x [(t(0) + \t(1)\mu(1)(2))(2)\\delta(n)\\(2) - 2t(0) [delta(n), delta(n+1)] + \t(1)\mu(1)(2)\\delta(n)\\ \\delta(n + 1)\\ + \\delta(n + 1)\\(2)], where \\.\\ = \\.\\(2), delta(n) = x(n) - x(n - 1), and s(mu(2)) and t(mu(2)) := t(0) + t(1) mu(2) are two coefficients of the corresponding functional equation connecting the eigenvalues lambda of T-omega to the eigenvalues mu of B, As special cases of the uniform error bound, we will give two error bounds for the SSOR and USSOR methods. (C) Elsevier Science Inc., 1997.