Two New Derivative Free Methods of Seventh and Eighth Order with Convergence Analysis

Abstract

Nonlinear equations represent many difficulties in the realms of science and engineering, but efficiently solving them is not an easy feat. In this study, two new derivative free iterative schemes, namely the scheme of seventh order and the scheme of eighth order, are proposed, which can be used to find solutions of nonlinear equations. These new schemes take advantage of finite difference approximations and the divided difference technique, eliminating the need for derivative computation and yet providing very fast convergence towards the solution. To test our new schemes, we consider several nonlinear equations and compare their performance with established high order methods. We observe that our schemes outperform other methods in terms of accuracy and efficiency, with efficiency index values of 1.79 and 1.86, respectively. Our proposed schemes emerge as highly efficient and effective.