(untitled) (European Journal of Pure and Applied Mathematics, 2025)
The increasing demand for accurate and effective approaches to complicated nonlinear models, driven by progress in diverse research and engineering fields, underscores the importance of tackling this challenge. This investigation introduces and analyses an innovative sixteenth-order iterative approach. This method originates from a weighted family akin to the Steffensen-Ostrowski type and is tailored for addressing nonlinear equations. Through the computational prowess of Maple 16, we assess the effectiveness of the suggested approach, in accordance with the Kung-Traub conjecture. This scheme’s formulation encompasses a bivariate weight function utilized in the third step, succeeded by Lagrange interpolation in the fourth step. The theoretical predictions have been verified through numerical computations, showing encouraging convergence characteristics of the novel approach. To demonstrate its efficacy, we apply it to practical scenarios, including Kepler’s celestial motion, an ideally mixed reactor, and models of predator-prey dynamics, neural activation, and periodic ecosystem growth. The numerical results reveal a notable edge of the method over current optimal four-point iterative approaches, as evidenced by its computational order of convergence and the variations observed between successive iterations. Additionally, we employ graphical analysis to explore complex polynomials and illustrate the basins of attraction for the suggested sixteenth-order algorithm, contrasting its efficacy with alternative methodologies in the domain. The visual representations affirm the method’s convergence rate and overall effectiveness.