A modified fixed point iteration method for solving the system of absolute value equations

Abstract

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<∥A−1∥<22, excluding the possible case that 22≤∥A−1∥<1. To complete the gap, we develop a modified FPI (MFPI) method for solving the AVE with 0<∥A−1∥<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<∥A−1∥<1, demonstrating its convergence; and compare it with FPI when 0<∥A−1∥<22, illustrating its superiority.