High-Precision Solution of Nonlinear Equations by Halley’s Method

Abstract

Although Newton’s method is widely used for solving nonlinear equations and systems of nonlinear equations due to its quadratic convergence, it may become inefficient in settings where a single iteration is computationally expensive or where solutions must be computed at precisions far beyond standard 64-bit double precision. In such cases, Halley’s method, which achieves cubic convergence, can offer better overall efficiency. This work investigates Halley-type methods for both scalar nonlinear equations and multivariate nonlinear systems, establishes several lemmas and corollaries, and provides high-precision implementations based on the Python Mpmath library. Numerical experiments show that, despite the higher per-step cost, Halley’s method can require fewer iterations and less total computation time than Newton’s method when high-precision solutions are required.