New modified optimal families of King's and Traub–Ostrowski's method

Based on quadratically convergent Schröder's method, we derive many new interesting families of fourth-order multipoint iterative methods without memory for obtaining simple roots of nonlinear equations by using the weight function approach. The classical King's family of fourth-order methods and Traub–Ostrowski's method are obtained as special cases. According to the Kung–Traub conjecture, these methods have the maximal efficiency index because only three functional values are needed per step. Therefore, the fourth-order family of King's method and Traub–Ostrowski's method are the main findings of the present work. The performance of proposed multipoint methods is compared with their closest competitors, namely, King's family, Traub–Ostrowski's method, and Jarratt's method in a series of numerical experiments. All the methods considered here are found to be effective and comparable to the similar robust methods available in the literature.