Calculation of machine precision second order derivatives using dual-complex numbers

• Published in: Numerical algorithms Vol. 99; no. 4; pp. 1925 - 1941
• Main Authors: Millwater, H., Balcer, M., Millwater, N.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.08.2025; Springer Nature B.V; Springer
• Subjects: Algebra; Algorithms; automatic differentiation; Complex numbers; Complex Taylor Series Expansion; Complexity; Computer Science; Derivatives; dual numbers; Libraries; Mechanics; Methods; Numeric Computing; Numerical Analysis; Numerical differentiation; numerical methods; Python; sensitivity methods; Source code; Taylor series; Theory of Computation; Complex Taylor Series Expansion; Sensitivity methods; Dual numbers; Numerical methods; Automatic differentiation
• ISSN: 1017-1398; 1572-9265; 1572-9265
• DOI: 10.1007/s11075-024-01937-2

It is well known that both complex and dual numbers can be employed to obtain machine precision first-order derivatives; however, neither, on their own, can compute machine precision 2nd order derivatives. To address this limitation, it is demonstrated in this paper that combined dual-complex numbers can be used to compute machine precision 1st and 2nd order derivatives. The dual-complex approach is simpler than utilizing multicomplex or hyper-dual numbers as existing dual libraries can be used as is or easily augmented to accept complex numbers, and the complexity of developing, integrating, and deploying multicomplex or hyper-dual libraries is avoided. The efficacy of this approach is demonstrated for both univariate and multivariate functions. Source code examples using the Python, Julia, and Mathematica languages are provided as supplemental material. Article Highlights A new step-size-independent method using dual-complex numbers for numerical differentiation is presented that provides machine precision results for first and second order derivatives. The method combines the first order Complex Taylor Series and dual number methods. Existing dual libraries can be used as is or easily extended.