An Empirical Analysis of Convergence Rates in...

An Empirical Analysis of Convergence Rates in Newton's Method for Solving Nonlinear Equations: A Case Study in Numerical Analysis

Author Name : M. A. Pathan

ABSTRACT This research investigates the convergence rates of Newton's Method when applied to various nonlinear equations, ranging from simple polynomials to more complex transcendental and piecewise functions. The study systematically explores the impact of initial guesses, function complexity, and derivative behavior on the method's performance. Newton's Method, known for its quadratic convergence under ideal conditions, demonstrated rapid convergence for well-behaved functions like quadratic and cubic equations. However, the method's sensitivity to initial guesses and function characteristics was evident in cases involving transcendental functions, such as sine and exponential functions, where slower convergence or non-convergence was observed. The piecewise function analysis further highlighted the method's limitations in handling discontinuities, with varying convergence behavior across different segments of the function. These findings underscore the importance of selecting appropriate initial conditions and understanding the function's nature when applying Newton's Method in practice. The research also suggests potential improvements through alternative iterative methods and strategies for optimizing initial guesses. The study's implications extend to practitioners in numerical analysis, where Newton's Method remains a powerful tool, albeit with context-dependent effectiveness. Future research could focus on enhancing the method's computational efficiency and exploring its application to real-world problems, particularly those involving complex or empirical data