Abstract
In this study, we propose a novel hybrid numerical optimization technique that combines iterative methods with a butterfly optimization scheme to solve nonlinear equations. The iterative methods, characterized by cubic convergence order, refine local solutions, while the butterfly optimization scheme enables global search. Our approach aims to improve efficiency and robustness by mitigating sensitivity to initial guesses. We conduct a local convergence analysis in Banach space and estimate convergence radii to guide the selection of initial values. The proposed technique is evaluated through engineering applications, demonstrating superior performance compared to classical methods and other optimization schemes such as particle swarm optimization, sperm swarm optimization, and ant line optimization.