A class of three-point root-solvers of optimal order of convergence

Abstract

The construction of a class of three-point methods for solving nonlinear equations of the eighth order is presented. These methods are developed by combining fourth order methods from the class of optimal two-point methods and a modified Newton’s method in the third step, obtained by a suitable approximation of the first derivative based on interpolation by a nonlinear fraction. It is proved that the new three-step methods reach the eighth order of convergence using only four function evaluations, which supports the Kung–Traub conjecture on the optimal order of convergence. Numerical examples for the selected special cases of two-step methods are given to demonstrate very fast convergence and a high computational efficiency of the proposed multipoint methods. Some computational aspects and the comparison with existing methods are also included.