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.