Round-Off and Truncation Error Trade-Off in Finite-Difference Numerical Differentiation: Optimal Step-Size Determination

Authors

  • Ping-Yun Tung Department of Bio-industrial Mechatronics Engineering, National Chung-Hsing University, Taichung 402202, Taiwan, Taiwan, Province of China
  • Anggara Trisna Nugraha Department of Marine Electrical Engineering, Politeknik Perkapalan Negeri Surabaya, Surabaya 60111, Indonesia, Indonesia
  • Yu-Min Lee Department of Bio-industrial Mechatronics Engineering, National Chung-Hsing University, Taichung 402202, Taiwan, Taiwan, Province of China

DOI:

https://doi.org/10.31004/jestm.v6i3.449

Keywords:

numerical differentiation, finite difference, round-off error, truncation error, optimal step size

Abstract

Numerical differentiation by finite differences is an inherently ill-conditioned operation in which two antagonistic error sources, truncation and round-off, govern the attainable accuracy. The objective of this study is to quantify this trade-off and to determine the step size that minimises the total error of the centred finite-difference operator. Using the smooth test function f(x) = -0.1x4 - 0.15x3 - 0.5x2 - 0.25x + 1.2 evaluated at x0 = 0.5, the first derivative was approximated for step sizes from 100 to 10-10 in IEEE-754 double precision, and the true, truncation, round-off, and total errors were measured and compared with a Taylor-series error model. The results reproduce the characteristic U-shaped error curve: truncation dominates for large h with second-order slope, round-off dominates for small h with slope minus one, and the minimum true error of 5.42x10-12 occurs at h = 10-6, in close agreement with the analytical optimum hopt = (3ε|f|/M3)1/3. The centred scheme is shown to be markedly more accurate than the forward scheme. The study concludes that an a-priori optimal step size can be reliably predicted, providing practical guidance for derivative-based engineering computation.

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Published

2026-10-26

How to Cite

Tung, P.-Y., Nugraha, A. T., & Lee, Y.-M. (2026). Round-Off and Truncation Error Trade-Off in Finite-Difference Numerical Differentiation: Optimal Step-Size Determination. Journal of Engineering Science and Technology Management (JES-TM), 6(3), 707–713. https://doi.org/10.31004/jestm.v6i3.449

Issue

Vol. 6 No. 3 (2026): September-November 2026

Section

Articles

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Copyright (c) 2026 Anggara Trisna Nugraha, Yu-Min Lee, Ping-Yun Tung

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