General approach to constructing optimal multipoint families of iterative methods using Hermite's rational interpolation

We discuss accelerating convergence of multipoint iterative methods for solving scalar equations, using particular type rational interpolant. Both derivative-free and Newton-type methods are investigated simultaneously. As a conclusion a Theorem of König's type for multipoint iterations is stated. A new optimal multipoint family of methods based on rational interpolation is constructed. The iteration uses n function evaluations per cycle and O(j) operations in jth step of a single iteration to obtain 2n−1 order of convergence. Several equivalent forms of the obtained iterates and development techniques are presented.