Reaserch On The Iterative Algorithm Of The Nonlinear Absolute Value Equations

Absolute value equations?AVE?(has the following formAx+B|x|-b,A,B?C^n×nx,b?Cⁿ) is a special nonlinear problem and a NP-hard problem.The problem arises widely in computational mathematics,operation research,economics and engineering,mainly stem from linear complementarity problem,interval linear equations,quadratic programming problem and bimatrix game problem.Therefore,the numerical solution of nonlinear absolute value equations has many important applications and scientific value.In this paper,we mainly study several kinds of iterative methods for the numerical solution of absolute value equations,and establish their convergence theory.The numerical experiments show that these methods are very effective.The paper is organized as follows:In Chapter 1,we introduced the research background of the nonlinear absolute value equations and the source of the problem,and analyzed the existing mature methods to solve this problem.Some preliminary notations and results are given in Chapter 2,including symbols,definitions and lemmas which are used in this paper.In Chapter 3,we present two sufficient conditions for the existence and uniqueness of the solution of the AVE problem,an iterative method for the problem is constructed and its convergence theorem is given.In Chapter 4,some iterative methods are proposed including Jacobi-like,Gauss-Seidel-like,successive overrelaxation-like?SOR-like?,accelerated Overrelaxation-like?AOR-like?,symmetric successive overrelaxation-like?SOOR-like?and symmetric accelerated Overrelaxation-like?SAOR-like?iterative methods for nonlinear AVE,and their convergence theorems are established.The numerical experiments show that these methods are feasible.In Chapter 5,the factors which affect the computation time and the number of iterations are analyzed,and the relaxation factor is adjusted to improve the algorithm.Numerical experiments show that our methods are more effective than the existing iterative method.In the end,conclusions and further work in the future are given in Chapter 6.