Application of artificial bee colony and other swarm intelligence algorithms for solving nonlinear equations

Abstract

Solving nonlinear equations, both in scalar and multivariate forms, is a critical computational challenge encountered across various scientific and engineering domains. Classical numerical techniques such as Newton-Raphson and secant methods often suffer from convergence issues, dependence on initial guesses, and difficulty handling complex landscapes. To address these limitations, this study explores the application of Swarm Intelligence (SI) algorithms—namely Artificial Bee Colony (ABC), Particle Swarm Optimization (PSO), Firefly Algorithm (FA), Grey Wolf Optimizer (GWO), and Ant Colony Optimization (ACO)—for solving nonlinear equations by transforming them into global optimization problems. A comprehensive experimental framework was employed, evaluating the performance of these algorithms on five nonlinear benchmark equations and two nonlinear systems involving two and three variables. Metrics such as Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), CPU time, average iterations to convergence, and success rate were analyzed over 50 independent trials. GWO and FA consistently achieved superior accuracy, faster convergence, and higher success rates. PSO and ABC showed moderate performance but exhibited sensitivity to parameter settings and problem topology. ACO demonstrated relatively lower efficiency and scalability. The study further includes graphical comparisons, residual trends, and a performance suitability matrix to guide algorithm selection. The findings reinforce the effectiveness of bio-inspired solvers in nonlinear root-finding tasks and highlight emerging trends like hybridization, adaptive control, and metaheuristic ensembles. This work provides a practical reference for researchers and practitioners aiming to implement robust, derivative-free methods for solving nonlinear equations in real-world scenarios.