Rank deficiency gradient-based iterations for generalized coupled Sylvester matrix equations

In this paper, by constructing an objective function and using the gradient search, three gradient-based iterations are established for solving generalized coupled Sylvester matrix equations, when the related matrices are full-column rank, full-row rank or rank deficiency. It is proved that these three gradient-based iterative algorithms are convergent for any initial iterative values. By analyzing the spectral radius of the iterative matrices, we study the convergence properties and determine the optimal convergence factors of these iterations. We discuss the connection between the full-row rank iteration and the rank deficiency iteration. By using this connection, the computational efficiency increases greatly for a class of matrix equations. A numerical example is provided to illustrate the effectiveness of the proposed algorithms and testify the proposed conclusions in this paper.