Modified iterative method for augmented system

The successive overrelaxation-like (SOR-like) method with the real parameters ω is considered for solving the augmented system. The new method is called the modified SOR-like (MSOR-like) method. The functional equation between the parameters and the eigenvalues of the iteration matrix of the MSOR-like method is given. Therefore, the necessary and sufficient condition for the convergence of the MSOR-like method is derived. The optimal iteration parameter ω of the MSOR-like method is derived. Finally, the proof of theorem and numerical computation based on a particular linear system are given, which clearly show that the MSOR-like method outperforms the SOR-like (Li, C. J., Li, B. J., and Evans, D. J. Optimum accelerated parameter for the GSOR method. Neural, Parallel & Scientific Computations, 7(4), 453–462 (1999)) and the modified symmetric SOR-like (MSSOR-like) methods (Wu, S. L., Huang, T. Z., and Zhao, X. L. A modified SSOR iterative method for augmented systems. Journal of Computational and Applied Mathematics, 228(4), 424–433 (2009)).