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

Zhong Zhou
Vol. 6, No. 10, pp. 26-37, 2026 DOI: 10.54691/be6ztx03 October 21, 2026
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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.
Keywords. Halley’s Method; Systems of Nonlinear Equations; High-Precision.
APA Citation Zhong Zhou (2026). High-Precision Solution of Nonlinear Equations by Halley’s Method. Frontiers in Science and Engineering, 6(10), 26-37. https://doi.org/10.54691/be6ztx03

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