The Picard–HSS iteration method for absolute value equations

Abstract

Recently Bai and Yang in (Appl Numer Math 59:2923–2936, 2009) proposed the Picard–Hermitian and skew-Hermitian splitting (HSS) iteration method to solve the system of nonlinear equations $Ax=\varphi (x)$, where $A\in \mathbb {C}^{n \times n}$ is a non-Hermitian positive definite matrix and $\varphi :\mathbb {D}\subset \mathbb {C}^n \rightarrow \mathbb {C}^n$ is continuously differentiable function defined on the open complex domain $\mathbb {D}$ in the $n$-dimensional complex linear space $\mathbb {C}^n$. In this paper, we focus our attention to the absolute value equation (AVE) $Ax=\varphi (x)$ where $\varphi (x)=|x|+b$, where $b\in \mathbb {C}^n$. Since the function $\varphi$ in AVE is not continuously differentiable function the convergence analysis of the Picard–HSS iteration method for this problem needs to be investigated. We give sufficient conditions for the convergence of the Picard–HSS iteration method for AVE. Some numerical experiments are given to show the effectiveness of the method and to compare with two available methods.