Payame Noor University (PNU)Control and Optimization in Applied Mathematics2383-31301220161201Solving System of Nonlinear Equations by using a New Three-Step Methodحل دستگاه معادلات غیرخطی با استفاده از یک روش سه گامی جدید53623398ENMehdiAhmadiDepartment of Mathematics, Malayer University, Malayer, Iran.HamidEsmaeiliDepartment of Mathematics, Bu-Ali Sina University, Hamedan, Iran.RErfanifarDepartment of Mathematics, Bu-Ali Sina University, Hamedan, Iran.Journal Article20151113In 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 $5^{1/({2n+4n^{2}+\frac{4}{3}n^{3}})}$, 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.https://mathco.journals.pnu.ac.ir/article_3398_cee896e67e39e179b12bcb4a52ca7cc3.pdf