On obtaining convergence order of a fourth and sixth order method of Hueso et al. without using Taylor series expansion

Abstract

Hueso et al. (2015) studied the fourth and sixth order methods to approximate a solution of a nonlinear equation in Rn, where the convergence analysis needs the involved operator to be five times differentiable and seven times differentiable for fourth-order and sixth-order methods, respectively. Also, they found no error estimate for those methods, as the Taylor series expansion played a leading role in proving the convergence. In this paper, we extended the method in the Banach space settings and relaxed the higher order derivative of the involved operator so that the methods can be used in a bigger class of problems which were not covered by the analysis in Hueso et al. (2015). Also, we obtained an error estimate without Taylor series expansion. This error estimate helps to get the number of iterations to achieve a given accuracy. Moreover, new sixth-order method is introduced by small modification and numerical examples were discussed for all these methods to validate our theoretical results and to study the dynamics.