On Iterative Solutions of Periodic Sylvester Matrix Equations

We propose a gradient based iterative algorithm with multiple iterative factors (MGI) to find the solutions of the Sylvester discrete-time periodic matrix equations AjXj + Xj+1Bj = Cj(j = 1, 2,⋯, T ). It is proved that the exact solution of the periodic matrix equations can be converged by the MGI method for any initial matrices. Then, we study the optimal convergence rate of gradient based iterative algorithm with single iterative factor (SGI). Nextly, we compare the convergence rate of the two algorithms, and find that MGI is faster than SGI when the appropriate convergence factors μj are selected. Finally, a numerical example is given to verify that MGI is superior to SGI in both speed and iterative steps.