On the modified Hermitian and skew-Hermitian splitting iteration methods for a class of weakly absolute value equations

Cui-Xia Li

doi:10.1186/s13660-016-1202-1

Abstract

In this paper, based on the modified Hermitian and skew-Hermitian splitting (MHSS) iteration method, the nonlinear MHSS-like iteration method is presented to solve a class of the weakly absolute value equations (AVE). By using a smoothing approximate function, the convergence properties of the nonlinear MHSS-like iteration method are presented. Numerical experiments are reported to illustrate the feasibility, robustness, and effectiveness of the proposed method.

Highlights

Consider the following weakly absolute value equations (AVE): Ax – |x| = b, ( . )where b ∈ Rn, | · | denotes the componentwise absolute value, A = W + iT where W ∈ Rn×n is a symmetric p√ositive definite matrix and T ∈ Rn×n is a symmetric positive semidefinite matrix, and i = – denotes the imaginary unit
The AVE ( . ) is a class of the important nonlinear linear systems, and it often comes from the fact that linear programs, quadratic programs, and bimatrix games can all be reduced to a linear complementarity problem (LCP) [ – ]
Compared with the nonlinear HSS-like iteration method, the potential advantage of the nonlinear modified Hermitian and skew-Hermitian splitting (MHSS)-like iteration method is that only two linear subsystems with coefficient matrices αI + W and αI + T, both being real and symmetric positive definite, need to be solved at each step. This shows that the nonlinear MHSS-like iteration method can avoid a shifted skew-Hermitian linear subsystem with coefficient matrix αI + iT

Summary

Introduction

This shows that the nonlinear MHSS-like iteration method can avoid a shifted skew-Hermitian linear subsystem with coefficient matrix αI + iT. The convergent conditions of the nonlinear MHSS-like iteration method are obtained by using a smoothing approximate function. We establish the following MHSS-like iteration method for solving the AVE

Results

Conclusion