An efficient family of two-point sixth-order methods suitable for non-convergence cases

Construction of higher-order two-point and globally convergent methods for computing simple roots of nonlinear equations is one of the earliest and challenging problem of numerical analysis. The principle aim of this manuscript is to propose a new highly efficient two-point family of iterative methods having sixth-order convergence, permitting f'(x) = 0, in the vicinity of the required root. Each member of the proposed scheme is free from second-order derivative. A higher-order family of double-Newton methods with a bivariate weighting function proposed by Guem et. al (2015) is a special case of our proposed scheme. A variety of concrete numerical examples demonstrate that our proposed scheme perform better than existing sixth-order methods available in the literature. In their dynamical study, it has been observed that the proposed methods have better stability and robustness as compared to the other existing methods.