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块三对角矩阵局部块分解及其在预条件中的应用
引用本文:吴建平,李晓梅.块三对角矩阵局部块分解及其在预条件中的应用[J].计算机学报,2002,25(8):823-829.
作者姓名:吴建平  李晓梅
作者单位:1. 国防科学技术大学并行与分布处理重点实验室,长沙,410073
2. 总装备部指挥技术学院,北京,101416
摘    要:该文利用块三对阵角阵分解因子的估值分析了其局部依赖性,并用其构了一类不完全分解型预条件子,给出了五点差分矩阵预条件后的条件数估计,并比较了条件数估计值与实际值,表明了估计值的准确性与预备件的有效性,在具体实现时,考虑了预条件的6个串行实现方案并提出了一个有效的并行化方法,该并行算法具有通信量少的特点,最后在由4中微机通过高速以太网连成的机群系统上作了大量数值实验,并将其与其它较效的预条件方法进行了。结果表明该预条件方法效果较好,尤其适用于并行计算。

关 键 词:块三对角矩阵  局部块分解  预条件  并行算法  数值解  微机
修稿时间:2001年2月15日

Local Factorization of Block Tridiagonal Matrices and Its Application to Preconditioners
WU Jian-Ping,LI Xiao-Mei.Local Factorization of Block Tridiagonal Matrices and Its Application to Preconditioners[J].Chinese Journal of Computers,2002,25(8):823-829.
Authors:WU Jian-Ping  LI Xiao-Mei
Affiliation:WU Jian-Ping 1) LI Xiao-Mei 2) 1)
Abstract:The core of many physical applications is how to solve systems of sparse linear equations efficiently. In general direct methods lead to large requirements both in storage and computation. Moreover, when the condition number of the coefficient matrix is large, direct methods are poor in stability. Based on these considerations, more and more researchers focused on iterative methods in recent years. But iterative methods may converge very slowly, or even diverge. One solution to this problem is to construct preconditioners. Preconditioners based on incomplete decomposition of the coefficient matrix have been proved to be efficient. But this kind of preconditioner is hard to be parallelized. In this paper, we consider block tridiagonal matrices. For this kind of matrices, their factorizations have local dependence in some sense, which is analyzed with the help of the evaluation of the actual factors. Then this kind of localization is exploited to construct a type of preconditioners. To analyze this kind of preconditioner, we first give a theorem about the condition number of a preconditioned symmetric M matrix, and then the condition numbers of the preconditioned model matrix are evaluated. With the comparison of these evaluations to the actual ones that are computed with the help of MATLAB, we can conclude that the evaluation is very accurate. Further the condition numbers are small, which means that the constructed preconditioners are effective. To study the efficiency of the preconditioner further, six implementations of the preconditioner are given in this paper. At the same time, there also considers an efficient parallelization, which has the advantage of low communication requirements. Then lots of experiments are done on a cluster of 4 PCs connected with Fast Ethernet to test the provided algorithms. The results show that the preconditioners constructed in this paper are comparable to the known effective ones in serial implementation, but they are more appropriate in parallel computing.
Keywords:M matrix  block LU decomposition  incomplete factorization  preconditioner  parallel algorithm
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