Optimal parameter of the SOR-like iteration method for solving absolute value equations

The SOR-like iteration method for solving the system of absolute value equations of finding a vector x such that Ax-|x|-b=0 with ν=‖A^-1‖₂<1 is investigated. The convergence conditions of the SOR-like iteration method proposed by Ke and Ma (Appl. Math. Comput., 311:195–202, 2017) are revisited and a new proof is given, which exhibits some insights in determining the convergent region and the optimal iteration parameter. Along this line, the optimal parameter which minimizes ‖T_ν(ω)‖₂ with (Formula presented.) and the approximate optimal parameter which minimizes an upper bound of ‖T_ν(ω)‖₂ are explored. The optimal and approximate optimal parameters are iteration-independent, and the bigger value of ν is, the smaller convergent region of the iteration parameter ω is. Numerical results are presented to demonstrate that the SOR-like iteration method with the optimal parameter is superior to that with the approximate optimal parameter proposed by Guo et al. (Appl. Math. Lett., 97:107–113, 2019).