A Four-Step High-Order Iterative Method for Nonlinear Equations with Scientific Applications

Abstract

In this paper, we propose a new four-step iterative method for solving nonlinear equations based on a predictor–corrector framework that combines Newton’s, Ostrowski’s, and Householder’s methods. To avoid explicit evaluation of higher derivatives, particularly the second derivative, polynomial interpolation is employed to approximate derivative information in the higher-order step, while retaining first-derivative evaluations where required. The resulting scheme attains an optimal convergence order of fourteen using six function evaluations per iteration. Numerical experiments on several benchmark functions and two classical application problems, namely the computation of libration points and a Fibonacci-type root-finding problem, demonstrate improved accuracy and robust convergence behavior. In the reported tests, the method achieves the expected computational order of convergence and typically converges within a small number of iterations. The convergence properties are further examined through residual errors, step differences, and the observed computational order of convergence.