A family of optimal three-point methods for solving nonlinear equations using two parametric functions

Using an interactive approach which combines symbolic computation and Taylor's series, a wide family of three-point iterative methods for solving nonlinear equations is constructed. These methods use two suitable parametric functions at the second and third step and reach the eighth order of convergence consuming only four function evaluations per iteration. This means that the proposed family supports the Kung-Traub hypothesis (1974) on the upper bound 2m of the order of multipoint methods based on m + 1 function evaluations, providing very high computational efficiency. Different methods are obtained by taking specific parametric functions. The presented numerical examples demonstrate exceptional convergence speed with only few function evaluations.