A Derivative Free Fourth-Order Optimal Scheme for Applied Science Problems

Abstract

We suggest a new and cost-effective iterative scheme for nonlinear equations. The main features of the presented scheme are that it does not involve any derivative in the structure, achieves an optimal convergence of fourth-order factors, has more flexibility for obtaining new members, and is two-point, cost-effective, more stable and yields better numerical results. The derivation of our scheme is based on the weight function technique. The convergence order is studied in three main theorems. We have demonstrated the applicability of our methods on four numerical problems. Out of them, two are real-life cases, while the third one is a root clustering problem and the fourth one is an academic problem. The obtained numerical results illustrate preferable outcomes as compared to the existing ones in terms of absolute residual errors, CPU timing, approximated zeros and absolute error difference between two consecutive iterations.