Comparative Analysis of Halley and Hybrid Methods for Numerically Solving the Roots of Non-Linear Equations

Abstract

Abstract: Finding the roots of nonlinear equations is a fundamental problem in numerical analysis with wide applications in engineering, science, and applied mathematics. The purpose of this study is to conduct a comparative analysis between Halley's method and a selected hybrid method for numerically solving the roots of nonlinear equations. Halley's method is a third-order iterative technique known for its fast convergence when provided with a good initial guess. On the other hand, hybrid methods are designed to combine the strengths of multiple numerical algorithms to enhance accuracy, stability, and robustness against different function characteristics. This study employs four test functions—polynomial, trigonometric, exponential, and logarithmic—to evaluate the performance of both methods in terms of convergence speed, computational efficiency, and sensitivity to initial guesses. The results indicate that Halley's method performs better in terms of speed under ideal conditions, while the hybrid method is more reliable in handling diverse nonlinear behaviors. Therefore, the appropriate method selection should consider both the nature of the function and the need for speed or stability in the computation.