Order of Convergence and Dynamics of Newton–Gauss-Type Methods

Abstract

On the basis of the new iterative technique designed by Zhongli Liu in 2016 with convergence orders of three and five, an extension to order six can be found in this paper. The study of high-convergence-order iterative methods under weak conditions is of extreme importance, because higher order means that fewer iterations are carried out to achieve a predetermined error tolerance. In order to enhance the practicality of these methods by Zhongli Liu, the convergence analysis is carried out without the application of Taylor expansion and requires the operator to be only two times differentiable, unlike the earlier studies. A semilocal convergence analysis is provided. Furthermore, numerical experiments verifying the convergence criteria, comparative studies and the dynamics are discussed for better interpretation. © 2023 by the authors.