Optimal Iterative Algorithm for Solving Nonlinear Equations with Applications

The rising demand for accurate and efficient solutions to complex nonlinear models, driven by developments in various research and engineering fields, emphasizes the importance of overcoming this challenge. This study presents and examines a novel three-step optimal eighth-order iterative method for root-finding by merging two different existing techniques. Based on the computational cost, the proposed method acquires optimal eight-order convergence with four functional evaluations (three evaluations for the function and one computation of first derivative). Furthermore, the proposed scheme supports the Kung–Traub conjecture with efficiency index of \(8^\frac{1}{4}=1.682\) . The utilization of finite difference is applied for the estimation of derivatives, and, simultaneously, the incorporation of a weight function on the opposite side is employed to enhance both convergence and efficiency. We also established the convergence criteria developed for the root-finding technique and demonstrate the fact that the proposed approach is eighth-order convergent. In order to demonstrate the efficacy as well as application of the constructed root-finding technique, we addressed a few practical engineering as well as biological- based application problems. In contrast to several existing approaches, this particular method converges more quickly.