%0 Journal Article
%T 关于求解矩阵方程<i>AXB </i>= <i>C</i>的预处理Richardson迭代<br>On the Preprocessed Richardson Iteration for Solving Matrix Equation <i>AXB</i> = <i>C</i>
%A 李嘉慧
%J Advances in Applied Mathematics
%P 3130-3139
%@ 2324-8009
%D 2024
%I Hans Publishing
%R 10.12677/aam.2024.137298
%X 本文类比于求解线性方程<i>Ax</i> = <i>b</i>的Richardson迭代，通过引入可调参数<i>ω</i>，提出了求解矩阵方程<i>AXB</i> = <i>C</i>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<i>ω</i>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。<br />
In this paper, analogous to Richardson iteration for solving linear equation <i>Ax </i>= <i>b</i>, Richardson iteration and preprocessed Jacobi and Gauss-Seidel iteration are proposed for solving the matrix equation <i>AXB</i> =<i> C</i> by introducing a tunable parameter <i>ω</i>, and their convergence properties are analyzed in detail. Moreover, the optimal choices of the parameter <i>ω</i> 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.
%K Richardson迭代，Jacobi预处理子，Gauss-Seidel预处理子，收敛性，最优参数<br>Richardson Iteration
%K Jacobi Preprocessor
%K Gauss-Seidel Preprocessor
%K Convergence
%K Optimal Parameter
%U http://www.hanspub.org/journal/PaperInformation.aspx?PaperID=91176