A Hybrid of the Newton-GMRES and Electromagnetic Meta-Heuristic Methods for Solving Systems of Nonlinear Equations

Solving systems of nonlinear equations is perhaps one of the most difficult problems in all numerical computation. Although numerous methods have been developed to attack this class of numerical problems, one of the simplest and oldest methods, Newton’s method is arguably the most commonly used. As is well known, the convergence and performance characteristics of Newton’s method can be highly sensitive to the initial guess of the solution supplied to the method. In this paper a hybrid scheme is proposed, in which the Electromagnetic Meta-Heuristic method (EM) is used to supply a good initial guess of the solution to the finite difference version of the Newton-GMRES method (NG) for solving a system of nonlinear equations. Numerical examples are given in order to compare the performance of the hybrid of the EM and NG methods. Empirical results show that the proposed method is an efficient approach for solving systems of nonlinear equations.