The Effect of Numerical Differentiation Precision on Newton’s Method: When Can Finite Difference Derivatives Outperform Exact Derivatives?

Newton’s method is traditionally regarded as most effective when exact derivative information is available, yielding quadratic convergence near a solution. In practice, however, derivatives are frequently approximated numerically due to model complexity, noise, or computational constraints. This paper presents a comprehensive numerical and analytical investigation of how numerical differentiation precision influences the convergence and stability of Newton’s method. We demonstrate that, for ill-conditioned or noise-sensitive problems, finite difference approximations can outperform exact derivatives by inducing an implicit regularization effect. Theoretical error expansions, algorithmic formulations, and extensive numerical experiments are provided. The results challenge the prevailing assumption that exact derivatives are always preferable and offer practical guidance for selecting finite difference step sizes in Newton-type methods. Additionally, we explore extensions to multidimensional systems, discuss adaptive step size strategies, and provide theoretical convergence guarantees under derivative approximation errors.