An efficient derivative free family of fourth order methods for solving systems of nonlinear equations

We present a derivative free two-step family of fourth order methods for solving systems of nonlinear equations using the well-known Traub–Steffensen method in the first step. In order to determine the local convergence order, we apply the first-order divided difference operator for functions of several variables and direct computation by Taylor’s expansion. Computational efficiencies of the methods of new family are considered and compared with existing methods of similar structure. It is showed that the new family is especially efficient in solving large systems. Four numerical examples are given to compare the proposed methods with existing methods and to confirm the theoretical results.