Optimal eighth-order Steffensen-type iterative family for multiple roots with applications to nonlinear models

Several advanced iterative techniques for finding multiple roots of higher order, along with the evaluation of derivatives, have been extensively studied and documented in the literature. However, the development of higher order methods without derivatives remains a challenging task, resulting in a scarcity of such techniques in existing research. Motivated by this observation, we propose a novel eighth-order iteration function of the Traub-Steffensen type. The suggested family employs the first-order divided difference and weight functions of one and three variables, optimizing performance for multiple roots with known multiplicity. The iterative scheme requires four functional evaluations per iteration achieving optimal eighth-order convergence in the sense of the Kung-Traub conjecture with an efficiency index of 1.6818. A comprehensive convergence analysis is conducted to confirm the optimality of the proposed method. Extensive numerical testing demonstrates the stability of the theoretical predictions and the favorable convergence behavior of the new scheme. To validate its practical utility, we explore various real-world nonlinear problems involving multiple roots, such as modeling energy distribution in a blackbody radiation, root clustering, and other applications. These comparisons reveal the effectiveness of the proposed scheme relative to other eighth-order iterative methods in terms of computational order of convergence, residual error, and the difference between successive iterations. Furthermore, the stable convergence behavior of the proposed method analyzed through graphical analysis using polynomial and transcendental functions. Basins of attraction are plotted for the designed eighth-order algorithm and compared with similar methods in the field. These graphical representations highlight the superior convergence speed and overall performance of the proposed algorithm, demonstrating its robust competitiveness in solving nonlinear problems with multiple roots.