Improved conjugate gradient method for nonlinear system of equations

In this paper, we propose a hybrid conjugate gradient (CG) method based on the approach of convex combination of Fletcher–Reeves (FR) and Polak–Ribiere–Polyak (PRP) parameters, and Quasi-Newton’s update. This is made possible by using self-scaling memory-less Broyden’s update together with a hybrid direction consisting of two CG parameters. However, an important property of the new algorithm is that, it generates a descent search direction via non-monotone type line search. The global convergence of the algorithm is established under appropriate conditions. Finally, numerical experiments on some benchmark test problems, demonstrate the effectiveness of the proposed algorithm over some existing alternatives.