Summary: We propose a simple modification over Chun’s method for constructing iterative methods with at least cubic convergence [Ch. Chun
, Appl. Math. Comput. 189, No. 2, 1384–1392 (2007; Zbl 1122.65326
)]. Using iteration formulas of order two, we now obtain several new interesting families of cubically or quartically convergent iterative methods. The fourth-order family of Ostrowski’s method is the main finding of the present work. Per iteration, this family of Ostrowski’s method requires two evaluations of the function and one evaluation of its first-order derivative. Therefore, the efficiency index of this Ostrowski’s family is , which is better than those of most third-order iterative methods and Newton’s method. The performance of Ostrowski’s family is compared with its closest competitors, namely Ostrowski’s method, Jarratt’s method and King’s family in a series of numerical experiments.