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
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。
Abstract: 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.
文章引用：李嘉慧. 关于求解矩阵方程AXB = C的预处理Richardson迭代[J]. 应用数学进展, 2024, 13(7): 3130-3139. https://doi.org/10.12677/aam.2024.137298

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