Optimal parameters of GSOR-like methods for solving the augmented linear systems

For the augmented system of linear equations, Golub et al. [G.H. Golub, X. Wu, J.-Y. Yuan, SOR-like methods for augmented systems, BIT 41 (2001) 71–85] studied an SOR-like method, by further accelerating it with another parameter, Bai et al. [Z.-Z. Bai, B.N. Parlett, Z.-Q. Wang, On generalized successive overrelaxation methods for augmented linear systems Numer. Math. 102 (2005) 1–38] gave out a generalized SOR method. By considering a new splitting of the coefficient matrix, this paper presents another generalization of the SOR-like method (GSOR-like) which is different from the method in the Bai et al.’s paper (2005), and mainly discusses the selection of the optimal parameters. Theoretical analyses show that the convergence region for the relaxation parameter ωω in our method properly contains that of the Golub et al.’s paper (2001) and our method has the same optimal asymptotic convergence rate with the method in the Bai et al.’s paper (2005). Further, the numerical example given shows the superiority of the GSOR-like method.