In this paper, we introduce a new iterative algorithm for solving a generalized Sylvester matrix equation of the form \(\sum_{t=1}^{p}A_{t}XB_{t}=C\) which includes a class of linear matrix equations. The objective of the algorithm is to minimize an error at each iteration by the idea of gradient-descent. We show that the proposed algorithm is widely applied to any problems with any initial matrices as long as such problem has a unique solution. The convergence rate and error estimates are given in terms of the condition number of the associated iteration matrix. Furthermore, we apply the proposed algorithm to sparse systems arising from discretizations of the one-dimensional heat equation and the two-dimensional Poisson’s equation. Numerical simulations illustrate the capability and effectiveness of the proposed algorithm comparing to well-known methods and recent methods.

在本文中，我们介绍了一种新的迭代算法，用于求解形式为\（\ sum_ {t = 1} ^ {p} A_ {t} XB_ {t} = C \）的广义Sylvester矩阵方程其中包括一类线性矩阵方程。该算法的目的是通过梯度下降的思想使每次迭代的误差最小。我们表明，所提出的算法可广泛应用于任何初始矩阵的任何问题，只要此类问题具有唯一的解决方案即可。收敛速度和误差估计是根据相关迭代矩阵的条件数给出的。此外，我们将提出的算法应用于一维热方程和二维泊松方程离散化产生的稀疏系统。数值仿真表明，与已知方法和最新方法相比，该算法的能力和有效性。