Ball convergence for a fourth order iterative method

Abstract

Numerical iterative methods are shown to be convergent using hypotheses on higher order derivatives but these derivatives do not appear in the body structure of these methods. Therefore, the usage of them is limited although they may converge. In this article, we demonstrate the convergence but using hypotheses only on the function’s derivative of order one. In this way, we extend the usage of these methods. In addition, we present the computable radii of convergence of the considered scheme and error bounds in accordance with Lipschitz parameters. Moreover, we suggest one counter example where previous studies was not applicable but our results. Finally, we check our results on some other examples and also provide computable radii of convergence.