A family of three-point methods of optimal order for solving nonlinear equations

A family of three-point iterative methods for solving nonlinear equations is constructed using a suitable parametric function and two arbitrary real parameters. It is proved that these methods have the convergence order eight requiring only four function evaluations per iteration. In this way it is demonstrated that the proposed class of methods supports the Kung–Traub hypothesis (1974) [3] on the upper bound 2 n of the order of multipoint methods based on n + 1 function evaluations. Consequently, this class of root solvers possesses very high computational efficiency. Numerical examples are included to demonstrate exceptional convergence speed with only few function evaluations.