Cairong Chen

The SOR-like iteration method for solving the 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 $\eta_{\nu}(\omega) =\max\{|1-\omega|,\nu\omega^2\}$ 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]). In some situation, the SOR-like itration method with the optimal parameter performs better, in terms of CPU time, than the generalized Newton method (Mangasarian, [{\em Optim. Lett.}, 3:101--108, 2009]) for solving the AVE.

The fixed point iteration (FPI) method proposed by Ke [Appl Math Lett. 2020;99:105990] for solving the absolute value equations (AVE) with the form $Ax-|x|=b$ is interesting for its simplicity and efficiency. However, its convergence is only guaranteed for the case that $0<\parallel A^{-1}\parallel <\frac{\sqrt{2}}{2}$, excluding the possible case that $\frac{\sqrt{2}}{2}\le \parallel A^{-1}\parallel <1$. To complete the gap, we develop a modified FPI (MFPI) method for solving the AVE with $0<\parallel A^{-1}\parallel <1$, which, besides keeping the simplicity of FPI, improves its efficiency by judiciously choosing the involving parameter. Under mild conditions, we prove its linear convergence. We present some preliminary numerical results for $0<\parallel A^{-1}\parallel <1$, demonstrating its convergence; and compare it with FPI when $0<\parallel A^{-1}\parallel <\frac{\sqrt{2}}{2}$, illustrating its superiority.