**7. Conclusions**

In the foregoing study, we have developed a fifth order iterative method for approximating solution of systems of nonlinear equations. The methodology is based on third order Traub-Steffensen method and further developed by using derivative free modification of classical Chebyshev's method. The iterative scheme is totally derivative-free and so particularly suitable to those problems where derivatives are lengthy to compute. To prove the local fifth order of convergence for the new method, a development of first-order divided difference operator and direct computation by Taylor's expansion are used.

We have examined the computational efficiency of the new method. A comparison of efficiencies with that of the existing most efficient methods is also performed. It is proved that, in general, the new algorithm is more efficient. Numerical experiments are performed and the performance is compared with existing derivative-free methods. From numerical results it has been observed that the proposed method has equal or better convergence compared to existing methods. Theoretical results related to convergence order and computational efficiency have also been verified in the considered numerical problems. Similar numerical tests, performed for a variety of other different problems, have confirmed the above drawn conclusions to a good extent.

**Author Contributions:** Methodology, J.R.S.; writing, review and editing, J.R.S.; investigation, D.K.; data curation, D.K.; conceptualization, L.J; formal analysis, L.J.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.
