A new class of methods with higher order of convergence for solving systems of nonlinear equations

By studying the commonness of some fifth order methods modified from third order ones for solving systems of nonlinear equations, we propose a new class of three-step methods of convergence order five by modifying a class of two-step methods with cubic convergence. Next, for a given method of order p ≥ 2 which uses the extended Newton iteration yk = xk − aF′(xk)−1F(xk) as a predictor, a new method of order p + 2 is proposed. For example, we construct a class of m + 2-step methods of convergence order 2m + 3 by introducing only one evaluation of the function to each of the last m steps for any positive integer m. In this paper, we mainly focus on the class of fifth order methods when m = 1. Computational efficiency in the general form is considered. Several examples for numerical tests are given to show the asymptotic behavior and the computational efficiency of these higher order methods.