The Study Of Numerical Methods For Absolute Value Equation

Absolute value equations often appear in the scientific computing and engineering application,for example,optimization problems,such as solving the linear programming,convex quadratic programming,double matrix countermeasure and the linear complementary problems.Usually,these problems can be transformed into an absolute value equation.Therefore,the study of effective numerical methods for absolute value equation has important theoretical and practical application value.In this paper,we mainly consider the numerical methods for solving the absolute value equation:Ax+B|x+=b,A?B?Rn×n,b?Rn.We proposed two iterative algorithms,namely,modulus-based matrix splitting algorithms and SOR-like iteration method,and the convergence analysis has been established.Finally,some numerical experiments have been presented to illustrate the feasibility and effectiveness of the new proposed methods.This paper is divided into four chapters and organized as follows:In the first chapter,we mainly introduce the research background,the current research status and the existing numerical methods for the absolute value equations.Then some preliminary theoretical knowledge has been presented.In the second chapter,based on the equivalence between the absolute value equations and the linear complementary problems,a modulus-based matrix splitting iteration method for solving the absolute value equation has been proposed,and the convergence of the algorithm has been proved.Finally,several numerical examples have been given to show the validity of the new method.In the third chapter,we propose a SOR-like iteration method for solving the absolute value equations.By rewritten the absolute value equations into a nonlinear equation,we proposed an iterative method.In addition,we proved that the new algorithm will converge when the appropriate parameters are chosen.Finally,some numerical experiments have been presented to illustrate the numerical performance.In the last section,the main research work of this dissertation is summarized,and the further research directions about the absolute value equations have been proposed.