Eighth-Order Iterative Algorithm for Multiple Roots of Unknown Multiplicity With Models in Civil, Control, and Biomedical Engineering

This study tackles the challenge of computing multiple roots of nonlinear equations with unknown multiplicity, a less explored area. Numerous iterative methods exist for known multiplicities, few optimal schemes address the unknown case. We present a novel eighth-order family derived from King’s method, using a univariate weight function and divided differences, requiring only one derivative and three function evaluations per iteration. Rigorous convergence analysis confirms optimal eighth-order convergence with an efficiency index of 1.6817. Simplified weight functions reduce complexity without sacrificing accuracy. Extensive numerical tests on real-world engineering problems, including structural beam deflection, control system root clustering, and biomechanical fracture prediction, validate the method’s superiority in absolute error, asymptotic error constant, and computational convergence order. Dynamical analysis using basins of attraction confirms robustness and stability, making the schemes suitable for diverse engineering applications and representing a significant contribution to the existing literature.