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

In this paperý, ýwe suggest a fifth order convergence three-step method for solving system of nonlinear equationsý. ýEach iteration of the method requires two function evaluationsý, ýtwo first Fr'{e}chet derivative evaluations and two matrix inversionsý. ýHenceý, ýthe efficiency index is $5^{1/({2n^{2}\frac{4}{3}n^{3}})}$ý, ýwhich is better than that of other three-step methodsý. ýThe advantages of the method lie in the feature that this technique not only achieves an approximate solution with high accuracyý, ýbut also improves the calculation speedý. ýAlsoý, ýunder several mild conditions the convergence analysis of the proposed method is providedý. ýAn efficient error estimation is presented for the approximate solutioný. ýNumerical examples are included to demonstrate the validity and applicability of the method and the comparisons are made with the existing results.