*Article* **On a Class of Optimal Fourth Order Multiple Root Solvers without Using Derivatives**

**Janak Raj Sharma 1,\*, Sunil Kumar 1 and Lorentz Jäntschi 2,3,\***


Received: 23 October 2019; Accepted: 20 November 2019; Published: 26 November 2019

**Abstract:** Many optimal order multiple root techniques involving derivatives have been proposed in literature. On the contrary, optimal order multiple root techniques without derivatives are almost nonexistent. With this as a motivational factor, here we develop a family of optimal fourth-order derivative-free iterative schemes for computing multiple roots. The procedure is based on two steps of which the first is Traub–Steffensen iteration and second is Traub–Steffensen-like iteration. Theoretical results proved for particular cases of the family are symmetric to each other. This feature leads us to prove the general result that shows the fourth-order convergence. Efficacy is demonstrated on different test problems that verifies the efficient convergen<sup>t</sup> nature of the new methods. Moreover, the comparison of performance has proven the presented derivative-free techniques as good competitors to the existing optimal fourth-order methods that use derivatives.

**Keywords:** iterative function; multiple root; composite method; derivative-free method; optimal convergence

**MSC:** 65H05; 41A25; 49M15
