*Article* **One-Point Optimal Family of Multiple Root Solvers of Second-Order**

#### **Deepak Kumar 1,\*, Janak Raj Sharma 1 and Clemente Cesarano 2,\***


Received: 6 June 2019; Accepted: 17 July 2019; Published: 21 July 2019

**Abstract:** This manuscript contains the development of a one-point family of iterative functions. The family has optimal convergence of a second-order according to the Kung-Traub conjecture. This family is used to approximate the multiple zeros of nonlinear equations, and is based on the procedure of weight functions. The convergence behavior is discussed by showing some essential conditions of the weight function. The well-known modified Newton method is a member of the proposed family for particular choices of the weight function. The dynamical nature of different members is presented by using a technique called the "basin of attraction". Several practical problems are given to compare different methods of the presented family.

**Keywords:** nonlinear equations; multiple roots; one-point methods; optimal convergence

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