In this research, we introduce and analyze homotopy analysis method (HAM) for solving approximately linear matrix equation \( \sum \limits \limits _{i=1}^{s}A_iXB_i+C=\mathbf{0} \), where \( A_i,\;B_i \;(i=1,\ldots ,s), \; C \in \mathbb {C}^{n \times n} \) and \(X\in \mathbb {C}^{n \times n} \) must be determined. In this method we consider a convergence control parameter \( \delta \), and then we determine the optimum value of \( \delta \) for obtaining fast convergence method. Moreover, we obtain the corresponding spectral radius of convergence factor of HAM method. Finally, we will apply this method to solve some test problems to support the theoretical results.

在这项研究中，我们介绍并分析了用于解决近似线性矩阵方程\（\ sum \ limits \ limits _ {i = 1} ^ {s} A_iXB_i + C = \ mathbf {0} \）的同伦分析方法（HAM ），其中\（A_i，\; B_i \;（i = 1，\ ldots，s），\; C \ in \ mathbb {C} ^ {n \ times n} \）和\（X \ in \ mathbb {C } ^ {n \ times n} \）必须确定。在这种方法中，我们考虑收敛控制参数\（\ delta \），然后确定\（\ delta \）的最佳值以获得快速收敛方法。此外，我们获得了相应的HAM方法的收敛因子谱半径。最后，我们将应用此方法解决一些测试问题以支持理论结果。