阶矩阵及其在传统预处理方法中的应用

本文应用矩阵元素阶和阶矩阵概念,讨论了ICCG和MICCG这两种传统的预处理方法在实用中的一些问题。为什么ICCG(s,t)在s+t固定时取(s,t)=(1,1),(1,2),(1,3),(2,4),(3,5),…有较高的收敛速度?为什么MICCG(m)当m>3时迭代次数不变?ICCG和MICCG的填入方式如何系统化?MICCG是否总比ICCG收敛速度高?本文拟作一个初步的讨论。通过LU分解的阶矩阵,本文给出了按阶递增的填入原则,将ICCG和MICCG系统化为P阶ICCG和P阶MICCG,并讨论了MICCG原有填入方式存在的问题。应用误差阵的阶矩阵,本文讨沦了MICCG迭代参数选取中存在的问题,给出了合理的参数选取方法。通过不同算例,本文还比较了ICCG和MICCG的计算效率。

ORDER MATRIX AND ITS APPLICATION TO SEVERAL TRADITIONAL PRECONDITIONING METHODS

Using the concepts of element order and order matrix, some practical problems are discussed in which the traditional preconditioning methods ICCG and MICCG are adopted. If the fill-in number is fixed, why the method of ICCG(s,t) becomes the most efficient when (s,t) is successively (1,1), (1,2), (1,3), (2, 4), (3, 5),..? Why the number of iterations didn't decrease when m is larger than 3 for MICCG(m)? Is it possible to improve the fill-in method of MICCG? Is it always true that MICCG is better than ICCG? It tries to give a preliminary discussion on these problems in here. From the way of high order approximate LU decomposition, a method is introduced which improves and systematizes the ICCG and MICCG. An estimation of the condition number of ICCG is given based on the discussion of the order matrix for the error matrix. It is also pointed out that there was a trouble in selecting the parameter for MICCG. A reasonable way to select the parameter is given. Thus the number of iterations of MICCG decreases when the order of MICCG increases.