Least squares based iterative algorithm for the coupled Sylvester matrix equations (Q1719221)

Summary: By analyzing the eigenvalues of the related matrices, the convergence analysis of the least squares based iteration is given for solving the coupled Sylvester equations \(\mathbf{A} \mathbf{X} + \mathbf{Y} \mathbf{B} = \mathbf{C}\) and \(\mathbf{D} \mathbf{X} + \mathbf{Y} \mathbf{E} = \mathbf{F}\) in this paper. The analysis shows that the optimal convergence factor of this iterative algorithm is 1. In addition, the proposed iterative algorithm can solve the generalized Sylvester equation \(\mathbf{A} \mathbf{X} \mathbf{B} + \mathbf{C} \mathbf{X} \mathbf{D} = \mathbf{F}\). The analysis demonstrates that if the matrix equation has a unique solution then the least squares based iterative solution converges to the exact solution for any initial values. A numerical example illustrates the effectiveness of the proposed algorithm.