New Highly Efficient Families of Higher-Order Methods for Simple Roots, Permittingf′(xn)=0

Construction of higher-order optimal and globally convergent methods for computing simple roots of nonlinear equations is an earliest and challenging problem in numerical analysis. Therefore, the aim of this paper is to present optimal and globally convergent families of King's method and Ostrowski's method having biquadratic and eight-order convergence, respectively, permittingf′(x)=0in the vicinity of the required root. Fourth-order King's family and Ostrowski's method can be seen as special cases of our proposed scheme. All the methods considered here are found to be more effective to the similar robust methods available in the literature. In their dynamical study, it has been observed that the proposed methods have equal or better stability and robustness as compared to the other methods.