**5. Conclusions**

In this paper, we propose a family of fourth-order derivative-free numerical methods for obtaining multiple roots of nonlinear equations. Analysis of the convergence was carried out, which proved the order four under standard assumptions of the function whose zeros we are looking for. In addition, our designed scheme also satisfies the Kung–Traub hypothesis of optimal order of convergence. Some special cases are established. These are employed to solve nonlinear equations including those arising in practical problems. The new methods are compared with existing techniques of same order. Testing of the numerical results shows that the presented derivative-free methods are good competitors to the existing optimal fourth-order techniques that require derivative evaluations in the algorithm. We conclude the work with a remark that derivative-free methods are good alternatives to Newton-type schemes in the cases when derivatives are expensive to compute or difficult to obtain.

**Author Contributions:** Methodology, J.R.S.; writing, review and editing, J.R.S.; investigation, S.K.; data curation, S.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.
