An Inexact Optimal Hybrid Conjugate Gradient Method for Solving Symmetric Nonlinear Equations

This article presents an inexact optimal hybrid conjugate gradient (CG) method for solving symmetric nonlinear systems. The method is a convex combination of the optimal Dai–Liao (DL) and the extended three-term Polak–Ribiere–Polyak (PRP) CG methods. However, two different formulas for selecting the convex parameter are derived by using the conjugacy condition and also by combining the proposed direction with the default Newton direction. The proposed method is again derivative-free, therefore the Jacobian information is not required throughout the iteration process. Furthermore, the global convergence of the proposed method is shown using some appropriate assumptions. Finally, the numerical performance of the method is demonstrated by solving some examples of symmetric nonlinear problems and comparing them with some existing symmetric nonlinear equations CG solvers.