New Developments and Extensions of Newton-Type Methods

Two- and three-point iteration methods for solving nonlinear equations are presented. They contain iteration parameters, and the suitable choice of these parameters allows increasing the convergence order. Necessary and sufficient conditions for the convergence of Newton-type iterations are suggested. The convergence order of any existing methods can be established using the proposed sufficient condition. Moreover, these conditions allow constructing new iterations. In particular, we propose a generating function method for constructing new two- and three-point iterations. Based on this approach, we design a new family of optimal two-point iterations that includes well-known methods as special cases. New developments and extensions of some optimal three-point iterations are also driven.