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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)). [ABSTRACT FROM AUTHOR]

A class of preconditioned iterative methods, i.e., preconditioned generalized accelerated overrelaxation (GAOR) methods, is proposed to solve linear systems based on a class of weighted linear least squares problems. The convergence and comparison results are obtained. The comparison results show that the convergence rate of the preconditioned iterative methods is better than that of the original methods. Furthermore, the effectiveness of the proposed methods is shown in the numerical experiment. [ABSTRACT FROM AUTHOR]