On obtaining order of convergence of Jarratt-like method without using Taylor series expansion

Abstract

In 2014, Sharma and Arora introduced two efficient Jarratt-like methods for solving systems of non-linear equations which are of convergence order four and six. To prove the respective convergence order, they used Taylor expansion which demands existence of derivative of the function up to order seven. In this paper, we obtain the respective convergence order for these methods using assumptions only on first three derivatives of the function. Other problems with this approach are: the lack of computable a priori estimates on the error distances involved as well as isolation of the solution results. These concerns constitute our motivation for this article. One extension of the fourth order method is presented which is of convergence order eight and the same is proved without any extra assumptions on the higher order derivatives. All the results are proved in a general Banach space setting. Numerical examples and dynamics of the methods are studied to analyse the performance of the method. © The Author(s) under exclusive licence to Sociedade Brasileira de Matemática Aplicada e Computacional 2024.