THE EFFECTIVE AND MODIFIED JACOBI GRADIENT BASED ITERATIVE ALGORITHMS FOR THE DISCRETE-TIME PERIODIC SYLVESTER MATRIX EQUATIONS

In this paper, we discuss the new convergence properties of some gradient based iterative (GI) algorithms and propose two new GI-like algorithms for solving the discrete-time periodic Sylvester (DTPS) matrix equations and its generalized version, which often arise in the fields of physics, medicine and so forth. We first review the Jacobi GI (JGI) and accelerated JGI (AJGI) algorithms (Appl. Numer. Math., 168 (2021) 251-273) for the DTPS matrix equations, and establish the new and correct convergence conditions of these two algorithms. Then we apply a new update strategy to the JGI algorithm and develop the effective Jacobi gradient based iterative (EJGI) algorithm for solving the DTPS matrix equations, which is different from the AJGI one. Furthermore, based on the ideas of the JGI and the Gauss-Seidel (G-S) algorithms, we construct the modified Jacobi gradient based iterative (MJGI) algorithm for the generalized discrete-time periodic Sylvester (GDTPS) matrix equations. Compared with the JGI algorithm, the MJGI algorithm can make full use of the latest information to compute the next result and lead to a faster convergence rate. By utilizing the properties of the matrix norms, Kronecker product and techniques of inequalities, we prove that two proposed iterative algorithms are convergent under proper restrictions. Finally, some numerical examples are given to validate the efficiencies and advantages of the proposed EJGI and MJGI algorithms for DTPS and GDTPS matrix equations.