A Grey Wolf–Newton-Raphson Hybrid Algorithm for Solving Systems of Nonlinear Equations

Abstract

Background: Solving systems of nonlinear equations (SNLEs) remains a fundamental challenge in computational mathematics, with applications spanning structural mechanics, financial modeling, and engineering design. While the Newton-Raphson method offers rapid local convergence, it is highly sensitive to initial guesses and often fails in high-dimensional or ill-conditioned systems. Metaheuristic algorithms, such as the Grey Wolf Optimizer (GWO), excel in global exploration but lack the precision and speed required for high-accuracy solutions. This study proposes a novel hybrid algorithm that synergistically integrates GWO with a curvature-aware, damped Newton-Raphson method to overcome the limitations of standalone approaches.

Methods: We developed a two-phase hybrid framework: (1) GWO performs global exploration to identify promising solution basins, and (2) a damped Newton-Raphson method, guided by Jacobian conditioning and adaptive damping, refines the solution locally. The transition between phases is criterion-based, triggered only when proximity to a root (∥F(x)∥ < 10⁻⁴) and numerical stability (Jacobian condition number κ < 10⁶) are ensured. The algorithm was rigorously evaluated on benchmark functions (CEC2020, Rosenbrock, Rastrigin) across dimensions ranging from n=2 to n=1000. Performance was statistically validated against classical solvers (fsolve) and pure metaheuristics (GWO, PSO) using Wilcoxon signed-rank tests and ANOVA (p < 0.05).

Results: The hybrid GWO-Newton algorithm achieved a 98% success rate across all benchmarks, significantly outperforming fsolve (72%), standalone GWO (65%), and PSO (70%). It reduced the average number of iterations by 43% compared to fsolve and delivered solutions with a mean residual error of 8.3×10⁻⁹ — three orders of magnitude more precise than metaheuristic alternatives. The adaptive damping and population resizing mechanisms proved critical for maintaining stability in ill-conditioned and high-dimensional problems.

Conclusions: The proposed hybrid framework successfully bridges the gap between global exploration and local precision, offering a robust, scalable, and highly accurate solver for complex nonlinear systems. The integration of adaptive, curvature-informed controls ensures reliable convergence where traditional methods fail. This approach redefines numerical root-finding by harmonizing stochastic and deterministic principles, making it a powerful tool for real-world scientific and engineering applications.