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关于求解矩阵方程AXB = C的预处理Richardson迭代
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Abstract:
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.
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