A modified fixed point iteration method for solving the system of absolute value equations. (4th March 2022) - Yu, Dongmei
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 A x − | x | = b is interesting for its simplicity and efficiency. However, its convergence is only guaranteed for the case that 0 < ∥ A − 1 ∥ < 2 2, excluding the possible case that 2 2 ≤ ∥ 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 ∥ < 2 2, illustrating its superiority.