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

The SOR-like iteration method for solvingthe system of absolute value equations~(AVE) of finding a vector $x$ such that $Ax - |x| - b = 0$ with $\nu = \|A^{-1}\|_2 < 1$ is investigated. The convergence conditions of the SOR-like iteration method proposed by Ke and Ma ([{\em 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 $\|T_\nu(\omega)\|_2$ with$$T_\nu(\omega) = \left(\begin{array}{cc} |1-\omega| & \omega^2\nu \\ |1-\omega| & |1-\omega| +\omega^2\nu \end{array}\right)$$and the approximate optimal parameter which minimizes an upper bound of $\|T_\nu(\omega)\|_2$are explored. The optimal and approximate optimal parameters are iteration-independent and the bigger value of $\nu$ is, the smaller convergent region of the iteration parameter $\omega$ 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, Wu and Li ([{\em Appl. Math. Lett.}, 97:107--113, 2019]).