The Broyden's Family Method For Solving Symmetric Nonlinear Equations

Quasi-Newton methods are a class of very efficient algorithms for solving uncons-trained optimization problems and nonlinear equations.Among all quasi-Newton methods,the Broyden's family method are the most representative algorithms,which contains the famous BFGS method.There are many important results on global and local convergence of quasi-Newton methods for unconstrained optimization.Although Great progress has been made in local convergence of quasi-Newton methods for nonlinear equations,study on their global convergence is very rare,the main reason is that the quasi-Newton direction may be not a descent direction of some metric functions.Griewank(1986)and Li and Fukushima(2000)respectively proved global convergence of the Broyden Rank 1 method for solving nonlinear equations under different conditions.Based on the Gauss-Newton method,Li and Fukushima(1999)introduced a BFGS type method for solving symmetric nonlinear equations and established its global convergence.This paper discusses quasi-Newton methods for solving symmetric nonlinear equations,in fact,we extend the BFGS method proposed by Li and Fukushima(1999)to the Broyden's family method.The paper is organized as follows:In Chapter 1,we simply introduce the background and related preparatory knowle-dge.In Chapter 2,we first give the motivation of the method and specific algorithm.Based on the Gauss-Newton method,we present the Broyden's family method method by the use of the symmetric structure of the underlying problem,which generalizes the results of those obtained by Li and Fukushima(1999).In Chapter 3,we investigate the convergence properties of the proposed method.Under suitable conditions,we show that the method possesses global convergence and locally superlinear convergence rate.In Chapter 4,we do some numerical experiments.Numerical results show that,by choosing suitable parameters,the proposed method performs better than the BFGS method proposed by Li and Fukushima(1999).