Generating function method for constructing new iterations

In this paper we propose a generating function method for constructing new two- and three-point iterations with p (3 ≤ p ≤ 8) order of convergence. This approach allows us to derive a new family of the optimal order iterative methods that include well known methods as special cases. The necessary and sufficient conditions for pth order convergence of the proposed iterations are given in terms of parameters τn and αn. We also propose some generating functions for τn and αn. We give the extension of a class of optimal fourth-order Jarratt's type iterations with a≠23. We develop a unified representation of all optimal eighth-order methods. Several numerical results are given to demonstrate the efficiency and the performance of the presented methods and compare them with some other existing methods.