An Inexact PRP Conjugate Gradient Method for Symmetric Nonlinear Equations

In this article, without computing exact gradient and Jacobian, we proposed a derivative-free Polak-Ribière-Polyak (PRP) method for solving nonlinear equations whose Jacobian is symmetric. This method is a generalization of the classical PRP method for unconstrained optimization problems. By utilizing the symmetric structure of the system sufficiently, we prove global convergence of the proposed method with some backtracking type line search under suitable assumptions. Moreover, we extend the proposed method to nonsmooth equations by adopting the smoothing technique. We also report some numerical results to show its efficiency.