对称正定Toeplitz方程组的多级迭代求解

doi:10.12286/jssx.2012.4.397

计算数学 ›› 2012, Vol. 34 ›› Issue (4): 397-404.DOI: 10.12286/jssx.2012.4.397

对称正定Toeplitz方程组的多级迭代求解

刘仲云¹, 刘成志¹, 张育林²

1. 长沙理工大学数学与计算科学学院, 长沙 410004;
2. Minho大学数学系, Braga 4710-057, 葡萄牙

收稿日期:2012-02-01 出版日期:2012-11-15 发布日期:2012-11-12
基金资助:
湖南省教育厅重点资助项目(09A002 [2009]); 由 FEDER Funds through "Programa Operacional Factores de Competitividade COMPETE" and by Portuguese Funds through FCT within the Project PEst-C/MAT/UI0013/2011 and PTDC/MAT/112273/2009, Portugal资助.

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刘仲云, 刘成志, 张育林. 对称正定Toeplitz方程组的多级迭代求解[J]. 计算数学, 2012, 34(4): 397-404.

Liu Zhongyun, Liu Chengzhi, Zhang Yulin. THE MULTISTAGE METHODS FOR SOLVING SYMMETRIC POSITIVE DEFINITE TOEPLITZ SYSTEMS[J]. Mathematica Numerica Sinica, 2012, 34(4): 397-404.

THE MULTISTAGE METHODS FOR SOLVING SYMMETRIC POSITIVE DEFINITE TOEPLITZ SYSTEMS

Liu Zhongyun¹, Liu Chengzhi¹, Zhang Yulin²

1. School of Mathematics and Computing Science, Changsha University of Science and Technology, Changsha 410004, China;
2. Centro de Matem′atica, Universidade do Minho, Campus de Gualtar, 4710-057 Braga, Portugal

Received:2012-02-01 Online:2012-11-15 Published:2012-11-12

二级迭代法亦称内外迭代法. 多级迭代法由多个二级迭代嵌套而成.这些方法特别适合于并行计算,同时可以理解为古典迭代法的延伸或共轭梯度法的预处理子.本文讨论了对称正定Toeplitz线性方程组多级迭代法. 首先,基于Toeplitz矩阵的结构, 我们给出了多级块Jacobi分裂,然后证明了每一级分裂均为P-正则分裂, 并证明了当每一级内迭代次数均为偶数时,迭代法的收敛性. 最后通过数值实例验证了此方法的有效性.

关键词： Toeplitz矩阵, 多级迭代法, P-正则分裂, 收敛性

The two-stage iterative methods are also called inner/outer iterative methods. The multistage iteration is nested by several two-stage iterations. Those methods are especially suitable for parallel computation, and can be viewed as extensions of classical iterative methods or as preconditioners for conjugate gradient methods. In this paper, we consider the multistage iterative methods for solving symmetric positive definite Toeplitz systems. Based on the Toeplitz structure, we first construct a multistage block Jacobi splitting, then we prove that the corresponding splitting at each level is P-regular, and show that the resulting method is convergent when the number of iteration at each level is even. At the end, we give some numerical examples to illustrate the effectiveness of our methods.

Key words: Toeplitz matrices, multistage method, P-regular splitting, convergence

MR(2010)主题分类:

65F10
65F15

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摘要

对称正定Toeplitz方程组的多级迭代求解

THE MULTISTAGE METHODS FOR SOLVING SYMMETRIC POSITIVE DEFINITE TOEPLITZ SYSTEMS

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