Delayed over-relaxation in iterative schemes to solve rank deficient linear system of (matrix) equations

Recently in [Journal of Computational Physics, 321 (2016), 829-907], an approach has been developed for solving linear system of equations with nonsingular coefficient matrix. The method is derived by using a delayed over-relaxation step (DORS) in a generic (convergent) basic stationary iterative method. In this paper, we first prove semi-convergence of iterative methods with DORS to solve singular linear system of equations. In particular, we propose applying the DORS in the Modified HSS (MHSS) to solve singular complex symmetric systems and in the Richardson method to solve normal equations. Moreover, based on the obtained results, an algorithm is developed for solving coupled matrix equations. It is seen that the proposed method outperforms the relaxed gradient-based (RGB) method [Comput. Math. Appl. 74 (2017), no. 3, 597-604] for solving coupled matrix equations. Numerical results are examined to illustrate the validity of the established results and applicability of the presented algorithms.