On Schröder’s families of root-finding methods

Schröder’s methods of the first and second kind for solving a nonlinear equation f ( x ) = 0 , originally derived in 1870, are of great importance in the theory and practice of iteration processes. They were rediscovered several times and expressed in different forms during the last 130 years. It was proved in the paper of Petković and Herceg (1999) [7] that even seven families of iteration methods for solving nonlinear equations are mutually equivalent. In this paper we show that these families are also equivalent to another four families of iteration methods and find that all of them have the origin in Schröder’s generalized method (of the second kind) presented in 1870. In the continuation we consider Smale’s open problem from 1994 about possible link between Schröder’s methods of the first and second kind and state the link in a simple way.