Fourth-Order Derivative-Free Optimal Families of King's and Ostrowski's Methods

In this paper, we present several new fourth-order optimal schemes that do not require any derivative evaluation for solving nonlinear equations, numerically. The presented approach of deriving these families is based on approximating derivatives by finite difference and weight function approach. The fourth-order derivative-free optimal families of King's and Ostrowski's methods are the main findings of the present work. Further, we have also shown that the families of fourth-order methods proposed by Petkovic et al., Appl Math Comput 217: 1887-1895 (2010) [12] and Kung-Traub, J ACM 21: 643-651 (1974) [8] are special cases of our proposed schemes. The proposed methods are compared with their closest competitors in a series of numerical experiments. All the methods considered here are found to be more effective to similar robust methods available in the literature.