Local convergence and a chemical application of derivative free root finding methods with one parameter based on interpolation

Abstract

We present a local convergence analysis of a derivative free fourth order method with one parameter based on rational interpolation in order to approximate a locally unique root of a function. The method is optimal in the sense of Traub. In earlier studies such as Steffensen (Scand Actuar J 16(1):64–72, 1933) and Zafer et al. (Sci World J, 2015. doi:10.1155/2015/934260) the convergence was based on hypotheses on the third derivative or even higher. We extend the applicability of theses methods using only the first derivative. Moreover, we provide computable radii and error bounds based on Lipschitz constants. Furthermore, the dynamics of this method are studied in order to find the best choice of the parameter in terms of convergence. An application is also presented in this study.