On the relaxed gradient-based iterative methods for the generalized coupled Sylvester-transpose matrix equations

In this paper, we propose the full-rank and reduced-rank relaxed gradient-based iterative algorithms for solving the generalized coupled Sylvester-transpose matrix equations. We provide analytically the necessary and sufficient condition for the convergence of the proposed iterative algorithm and give explicitly the optimal step size such that the convergence rate of the algorithm is maximized. Some numerical examples are examined to confirm the feasibility and efficiency of the proposed algorithms.