The use of homotopy analysis method for solving generalized Sylvester matrix equation with applications

Abstract

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.