Higher-order efficient class of Chebyshev–Halley type methods

Construction of two-point sixth-order methods for simple root is an ambitious and challenging task in numerical analysis. Therefore, the main aim of this paper is to introduce a new highly efficient two-point sixth-order class of Chebyshev–Halley type methods free from second-order derivative for the first time. Each member of the proposed class requires only four functional evaluations (viz. two evaluations of function f and two of first-order derivative f ′) per full iteration. A variety of concrete numerical examples illustrate that our proposed methods are more efficient and perform better than existing two-point/three-point sixth-order methods available in the literature. From their dynamical study, it has been observed that our proposed methods have better stability and robustness as compared to the other existing methods.