The Picard-HSS iteration method for absolute value equations

Recently Bai and Yang in [On HSS-based iteration methods for weakly nonlinear systems, Appl. Numer. Math. 59 (2009) 2923–2936.] proposed the Picard-HSS iteration method to solve the system of nonlinear equations Ax = φ(x), where A ∈ Cn×n is a nonHermitian positive definite matrix and φ : D ⊂ C → C is continuously differentiable function defined on the open complex domain D in the n-dimensional complex linear space C. In this paper, we focus our attention to the absolute value equation (AVE) Ax = φ(x) where φ(x) = |x|+ b, where b ∈ C. Since the function φ 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.