关于求解矩阵方程AXB = C的预处理Richardson迭代  

On the Preprocessed Richardson Iteration for Solving Matrix Equation AXB = C

作　　者：李嘉慧 

机构地区：[1]长沙理工大学数学与统计学院,湖南 长沙

出　　处：《应用数学进展》2024年第7期3130-3139,共10页Advances in Applied Mathematics

摘　　要：本文类比于求解线性方程Ax = b的Richardson迭代,通过引入可调参数ω,提出了求解矩阵方程AXB = C的Richardson迭代及其Jacobi和Gauss-Seidel预处理迭代,并详细分析了它们的收敛性。此外,对于一些特殊情况,可以得到参数ω的最优选择,使得迭代矩阵的谱半径达到最小。最后,通过数值实验,我们验证了所提算法的有效性。In this paper, analogous to Richardson iteration for solving linear equation Ax = b, Richardson iteration and preprocessed Jacobi and Gauss-Seidel iteration are proposed for solving the matrix equation AXB = C by introducing a tunable parameter ω, and their convergence properties are analyzed in detail. Moreover, the optimal choices of the parameter ω to minimize the spectral radius of the iteration matrix are also obtained for some special cases. Finally, numerical experiments are carried out to illustrate the effectiveness of the proposed algorithms.

关 键 词：Richardson迭代 Jacobi预处理子 Gauss-Seidel预处理子 收敛性 最优参数 

分 类 号：O15[理学—数学]