Solving System of Nonlinear Equations by using a New Three-Step Method

In this paper‎, ‎we suggest a fifth order convergence three-step method for solving system of nonlinear equations‎. ‎Each iteration of the method requires two function evaluations‎, ‎two first Fr\'{e}chet derivative evaluations and two matrix inversions‎. ‎Hence‎, ‎the efficiency index is ۵^{۱/({۲n+۴n^{۲}+\frac{۴}{۳}n^{۳}})}‎, ‎which is better than that of other three-step 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.

Keywords:

Nonlinear equations‎ , ‎Iterative method‎ , ‎Convergence order‎ , ‎Efficiency index