The relaxed gradient-based iterative algorithms for a class of generalized coupled Sylvester-conjugate matrix equations

In this paper, two relaxed gradient-based iterative algorithms for solving a class of generalized coupled Sylvester-conjugate matrix equations are proposed. The proposed algorithm is different from the gradient-based iterative algorithm and the modified gradient-based iterative algorithm that are recently available in the literature. With the real representation of a complex matrix as a tool, the sufficient and necessary condition for the convergence factor is determined to guarantee that the iterative solution given by the proposed algorithms converge to the exact solution for any initial matrices. Moreover, some sufficient convergence conditions for the suggested algorithms are presented. Finally, numerical example is provided to illustrate the effectiveness of the proposed algorithms and testify the conclusions suggested in this paper.