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

Jamilu Sabi’U,Kazeem Olalekan Aremu,Abdullah Shah,Kanikar Muangchoo,Auwal Bala Abubakar

doi:10.3390/sym13101829

Abstract

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–Ribiére–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.

Highlights

Thereafter, some efficient conjugate gradient (CG) methods for unconstrained optimization problems were incorporated with the approximate gradient relation to solve large scale symmetric nonlinear equations
This paper presented an inexact optimal hybrid CG algorithm for solving a system of symmetric nonlinear equations
Some mild assumptions are used to prove the global convergence of the method

Summary

Introduction

F ( x ), means that the Jacobian of F ( x ) is symmetric Such a class of problems could be defined from the gradient mapping of an unconstrained optimization problem, the Karush–. The following descent modification of the PRP parameter has been proposed by Babaie-Kafaki and Ghanbari [10] based on the Dai–Liao approach [13] as β DPRP k Fk−1 k. The reported numerical experiments illustrated that their algorithm is promising in handling large-scale symmetric nonlinear equations This result in further studies on conjugate gradient methods for solving symmetric nonlinear equations were inspired. Thereafter, some efficient CG methods for unconstrained optimization problems were incorporated with the approximate gradient relation to solve large scale symmetric nonlinear equations. The proposed method is derivative-free and matrix-free, and could sufficiently handle large-scale symmetric nonlinear systems efficiently. The section is the derivation and details of the proposed method, followed by a convergence analysis, numerical experiment, and conclusion

A Class of Optimal Hybrid CG Method

The First Choice

The Second Choice

Global Convergence

Numerical Experiment

Conclusions