预条件下Gauss-Seidel迭代法的收敛性
Convergence of Preconditioned Gauss-Seidel Iterative Method
DOI: 10.12677/PM.2020.1011119, PDF, HTML, 下载: 584  浏览: 1,003 
作者: 黄江玲：陕西师范大学数学与信息科学学院，陕西 西安
关键词: 非奇异M矩阵；Gauss-Seidel迭代法；谱半径；预条件；收敛性；Non-Singular M Matrix； Gauss-Seidel Iterative Method； Spectral Radius； Preconditioned； Convergence
摘要: 假设在线性方程组Ax=b的系数矩阵A是非奇异M-阵，通过比较预条件Gauss-Seidel迭代法的迭代矩阵的谱半径与经典Gauss-Seidel迭代法的迭代矩阵谱半径大小，得到了预条件下Gauss-Seidel迭代法的收敛速度要快于经典的Gauss-Seidel迭代法的收敛速度，从而得出该预条件处理的有效性，最后用一个数值算例验证了该结论。
Abstract: Assuming under the condition that the coefficient matrix 𝐴 in linear equations Ax=b is an nonsingular 𝑀-matrix, by comparing the spectral radius of the iterative matrix of the Gauss-Seidel iterative method under preconditions with the spectral radius of the iterative matrix of the classical Gauss-Seidel iterative method, it is obtained that the convergence speed of the Gauss-Seidel iterative method under preconditions is faster than that of the classical Gauss-Seidel iterative method, and the effectiveness of the preconditioning is obtained. Finally, a numerical example is used to verify the conclusion.
文章引用：黄江玲. 预条件下Gauss-Seidel迭代法的收敛性[J]. 理论数学, 2020, 10(11): 1007-1014. https://doi.org/10.12677/PM.2020.1011119

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