SOR-like iteration and FPI are consistent when they are equipped with certain optimal iterative parameters

TL;DR

This paper introduces new convergence analysis for SOR-like and fixed point iteration methods solving absolute value equations, deriving optimal parameters that unify both methods and improve convergence range.

Contribution

It provides the first analytical optimal parameters for both methods, showing their equivalence under these parameters, and offers new proofs for solution uniqueness conditions.

Findings

Derived analytical optimal iterative parameters for SOR-like and FPI methods.

Proved the equivalence of SOR-like iteration and FPI with optimal parameters.

Numerical results confirm the improved convergence and theoretical claims.

Abstract

Two common methods for solving absolute value equations (AVE) are SOR-like iteration method and fixed point iteration (FPI) method. In this paper, novel convergence analysis, which result wider convergence range, of the SOR-like iteration and the FPI are given. Based on the new analysis, a new optimal iterative parameter with a analytical form is obtained for the SOR-like iteration. In addition, an optimal iterative parameter with a analytical form is also obtained for FPI. Surprisingly, the SOR-like iteration and the FPI are the same whenever they are equipped with our optimal iterative parameters. As a by product, we give two new constructive proof for a well known sufficient condition such that AVE has a unique solution for any right hand side. Numerical results demonstrate our claims.