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MATHEMATICAL ANALYSIS AND NUMERICAL ANALYSIS OF SEVERAL KINDS OF DIFFERENTIAL EQUATIONS.

几种微分方程的数学分析和数值分析。

基本信息

批准号：
06640335
负责人：
MUROYA Yoshiaki
金额：
$ 1.41万
依托单位：
WASEDA UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1994
资助国家：
日本
起止时间：
1994 至 1996
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-06640335/
关键词：
IMPROVED SOR METHOD MULTIPLE RELAXATION PARAMETERS ADAPTIVE IMPROVED SOR METHOD IMPROVED SSOR METHOD 改良SOR法反復解法 SOR法 -

项目摘要

To solve non-symmetric linear systems derived from the discretization of singular pertur-bation problems, we propose a generalized SOR method with multiple relaxation parameters, that is the improved SOR method with orderings and study its theory and practical use.In the case of tridiagonal matrices, optimal choices of the parameters are examined : It is shown that the spectral radius of the iterative matrix is reduced to zero for a pair of parameter values which are computed from the pivots of the Gaussian elimination applied to the system. A proper choice of orderings and starting vectors for the iteration is also proposed.We apply the above method to two-dimensional cases, and propose the "adaptive improved block SOR method with orderings" for block tridiafonal matrices. The point of this method is to change the multiple relaxation parameters not only for each block but also for each iteration. If special multiple relaxation parameters are selected and used with this method for an n * n block tridiagonal matrix whose block matrices are all n * n matrices, then this iterative method converges at most n^2 iterations.We also proposed the improved SSOR method with orderings, which converges at most only one iteration for a tridiagonal system, and n iterations for a block tridiagonal system.The generalized convergence theorems to the improved SOR method with orderings are also considered, and we study necessary and sufficient conditions for a matrix to be a generalized diagonally dominant.Using the notation 'basic LUL factorization' of matrices, we give some techniques to obtain special multiple relaxation parameters such that the spectral radius of the iterative matrix is zero for the Hessenberg matrices and a class of matrices.

为了求解由奇异摄动问题离散化得到的非对称线性方程组，本文提出了一种带多松弛参数的广义SOR方法，即带序的改进SOR方法，并研究了它的理论和实际应用.在三对角矩阵的情况下，讨论了参数的最优选择问题.它示出的迭代矩阵的谱半径减少到零的一对参数值，这是从应用到系统的高斯消去的枢轴计算。本文将上述方法应用于二维情形，提出了块三对角矩阵的“带排序的自适应改进块SOR方法”。该方法的要点是不仅改变每个块的多重松弛参数，而且改变每次迭代的多重松弛参数。对于n × n块三对角矩阵，若选取特殊的多重松弛参数，则此迭代方法至多收敛n^2次，并提出了改进的带序SSOR方法，该方法对三对角系统至多收敛一次，和n次迭代，并讨论了带序的改进SOR方法的广义收敛性定理，研究了矩阵是广义对角占优矩阵的充要条件，利用矩阵的基本LUL分解，本文给出了一些技巧，使Hessenberg矩阵和一类矩阵的迭代矩阵的谱半径为零。