Modified Jarratt Method without Memory with Twelfth-Order Convergence

Abstract

Reobtaining some old zero-finding iterative methods is not a rarity in numerical analysis. Routinely, most of the improvements of root solvers increase the order of convergence by adding a new evaluation of the function or its derivatives per iteration. In the present article, we give a simple way to develop the local order of convergence by using Jarratt method in the first step of a three-step cycle. The analysis of convergence illustrates that the proposed without memory method is a twelfth-order iterative scheme and its classical efficiency index is 1.644 which is bigger than that of Jarratt. Some numerical examples are provided to support and re-verify the novel method. Although the proposed technique is not optimal due to its 12th order convergence with five evaluations per full iteration, it consists of two evaluations of the first derivatives and three evaluations of the function; and more interestingly, there is no optimal method with 4 or 5 evaluations per iteration in literature up to now in which there is two derivatives evaluations per cycle. Moreover, the new method is very faster than the existing developments of the Jarratt method.