Chebyshev acceleration for SOR-like method

There are several proposals for the generalization of Young's successive over-relaxation (SOR) method to solve the saddle-point problem or augmented system. The most practical version is the SOR-like method (G.H. Golub et al., BIT, 41, 71-85, 2001), which was further studied by Li et al. (Int. J. Comput. Math., 81, 749-765, 2004) who found that the iteration matrix of the SOR-like method has no complex eigenvalues only under certain conditions. Motivated by the results of Li and co-authors, we consider the Chebyshev acceleration of the SOR-like method (GSOR-SI). First, the convergence of the GSOR-SI method is given. Secondly, it is shown that the asymptotic rate of the convergence of the GSOR-SI method is much larger than that of the SOR-like method, which indicates that the GSOR-SI method has a faster rate of convergence than the SOR-like method. Finally, numerical comparisons are given which show the GSOR-SI method is indeed faster than the SOR-like method.