A novel Approach for Real-World Problems Based on the Hermite Interpolation Technique and Analysis Using Basins of Attraction

Abstract

This paper presents a novel ninth-order iterative scheme based on the Hermite interpolation technique for solving nonlinear equations arises in Real-World models of the form . In contrast to traditional methods, this approach does not use second derivatives, instead relying on three function evaluations and two first derivative evaluations per iteration. Existing iterative methods frequently suffer from slow convergence and the requirement for higher-order derivatives, which can be computationally costly. The proposed method addresses these limitations by providing a faster convergence without the computational burden of second derivatives. The Taylor series expansion is used to conduct a detailed convergence analysis of the proposed method. The method's effectiveness and stability are further validated by comparisons with existing approaches in the literature.