The PRP Type Methods For Solving Symmetric Nonlinear Equations

The linear conjugate gradient method is a very efficient algorithm for solving symmetric positive linear equations. It possesses the remarkable quadratic termination property, that is, it converges to the solution of the equations within finite iterates. This method has been extended to solving general nonlinear optimization, which are called the nonlinear conjugate gradient methods. These methods are very popular for solving large-scale unconstrained optimization problems due to their simplicity, low memory storage and fast convergence.The classical PRP method has been regarded as one of the most efficient nonlinear CG methods in numerical perspective. However, its convergence properties are not so satisfactory. The main difficulty lies in that the search direction generated by the PRP method may be not a descent method even for strongly convex functions with the strong Wolfe line search. Thus, to guarantee its global convergence, some modifications to it are often required. Recently, Zhou wei jun showed that the original PRP method converges globally for nonconvex optimization by utilizing a nonmonotone line search.The purpose of this paper is to extend the original PRP method to solving relative-ly large-scale symmetric nonlinear equations without computing Jacobian and exact gradient of the metric function. The paper is organized as follows.In Chapter 1, we simply introduce the background of the problem, which we will consider, and the related preliminary knowledge.In Chapter 2, by utilizing the symmetric structure of the problem sufficiently, based on some kind of approximate gradient, we propose two PRP type methods for solving symmetric nonlinear equations. One is based on the alternative direc-tion method which we call inexact PRP method 1; the other is an approximate norm descent method which we call inexact PRP method 2. We show that the two methods are well-defined and the residual sequences generated by both algorithms converge.In Chapter 3, we prove that the proposed methods have global convergence prop-erty for solving symmetric nonlinear equations under suitable conditions. Moreover, we establish the R-linear convergence of the inexact PRP method 1.In Chapter 4, we do some numerical experiments and report some numerical re-sults, which indicate that the proposed methods are efficient.