Solving nonlinear systems based on improved bald eagle search algorithm

Abstract

Solving systems of nonlinear equations is an important problem in applied mathematics, engineering, and scientific computing. Traditional numerical methods often require derivative information and suitable initial approximations, which may reduce their effectiveness when dealing with complex nonlinear systems. This paper proposes an Improved Bald Eagle Search Algorithm (IBESA) for solving systems of nonlinear equations. The proposed method enhances the original Bald Eagle Search (BES) algorithm by integrating an archive strategy to preserve promising root approximations and a population diversification mechanism to prevent premature convergence. The nonlinear system is transformed into a global optimization problem by minimizing the Euclidean norm of the residual vector. The performance of IBESA was evaluated using six benchmark nonlinear systems representing polynomial, non-differentiable, trigonometric, and exponential models. Each problem was solved through 30 independent runs and compared with BES, Particle Swarm Optimization (PSO), Genetic Algorithm (GA), and Modified Firefly Algorithm (MODFA). Experimental results showed that IBESA successfully identified all known roots and achieved a 100% success rate across all test problems. The algorithm produced extremely small residual errors and demonstrated superior accuracy compared with BES, PSO, and GA, while also outperforming MODFA in terms of numerical precision. Statistical analyses further confirmed the robustness and effectiveness of the proposed approach. The results indicate that IBESA is a reliable and efficient derivative-free optimization method for solving nonlinear systems.