An efficient three-step method to solve system of nonlinear equations

In this paper, we suggest a sixth order convergence three-step method to solve system of nonlinear equations. Every iteration of the method requires two function evaluations, two first Fréchet derivative evaluations and two matrix inversions. Hence, the efficiency index is 61/(2n+6n2+43n3), which is better than that of other sixth order methods. The advantages of the method lie in the feature that this technique not only achieves an approximate solution with high accuracy, but also improves the calculation speed. Also, under several mild conditions the convergence analysis of the proposed method is provided. An efficient error estimation is presented for the approximate solution. Numerical examples are included to demonstrate the validity and applicability of the method and the comparisons are made with the existing results.