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基于结构二次幂零矩阵的迭代法构造

发表时间:2017-10-13  浏览量:1632  下载量:245
全部作者: 张承平
作者单位: 海南热带海洋学院海洋信息工程学院
摘 要: 主要研究构造一类新的线性方程组的迭代方法,要求该方法具有较好的收敛速度和良好的并行性,适合并行计算。根据结构二次幂零矩阵的性质与特点,得出对角矩阵与结构二次幂零矩阵之和的逆矩阵计算公式。根据此公式,可将线性方程组的系数矩阵分裂为对角矩阵、结构二次幂零矩阵及剩余矩阵,因而得出新的迭代格式。通过三对角对称正定矩阵和五对角对称正定矩阵两个例子进行实验。实验结果表明,三对角对称正定矩阵的结构二次幂零矩阵迭代法比雅可比迭代法收敛速度快,迭代次数约为雅可比迭代法的一半,与高斯赛德尔迭代法次数相同;五对角对称正定矩阵的两种结构二次幂零矩阵迭代次数不相同,但比雅可比迭代法次数少。
关 键 词: 计算数学;并行迭代法;结构二次零幂矩阵;线性方程组
Title: Iterative construction based on structural 2-nilpotent matrix
Author: ZHANG Chengping
Organization: School of Marine Information Engineering, Hainan Tropical Ocean University
Abstract: In this paper, a new kind of iterative method for solving linear equations is studied. It is required that the method has good convergence speed and good parallelism property, which are suitable for parallel computing. According to the properties and characteristics of structural 2-nilpotent matrix, the formulas about the inverse matrix of the sum of diagonal matrix and structural 2-nilpotent matrix are gotten. The coefficient matrix of linear equations is split to the diagonal matrix, the structural 2-nilpotent matrix and the residual matrix. According to the formula, the iterative scheme matrix is gotten. By the experiment of two examples about the three-diagonally symmetric positive definite matrix and the five-diagonally symmetric positive definite matrix, the experimental results show that the structural 2-nilpotent matrix iterative method of the three-diagonally symmetric positive definite matrix converges faster than the Jacobian iterative method. The number of its iterations is about half of the Jacobi iterative method, and the same as to the Gauss-Seidel iterative method. Iterations numbers of the two kinds of structural 2-nilpotent matrix method of the five-diagonally symmetric positive definite matrix are not the same, but they are fewer than that of the Jacobian iterative method.
Key words: computational mathematics; parallel iterative method; structural 2-nilpotent matrix; linear equations
发表期数: 2017年10月第19期
引用格式: 张承平. 基于结构二次幂零矩阵的迭代法构造[J]. 中国科技论文在线精品论文,2017,10(19):2145-2150.
 
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