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

Journal Article · 04 October 2024 · Numerical Algorithms

DOI: https://doi.org/10.1007/s11075-024-01937-2 · OSTI ID:2467396

Millwater, Harry  ^[1]; Balcer, Matthew Ryan  ^[2]; Millwater, Nathan  ^[3]

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1. Univ. of Texas at San Antonio, TX (United States)
2. Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
3. Southwest Research Institute, San Antonio, TX (United States)

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. Finally, source code examples using the Python, Julia, and Mathematica languages are provided as supplemental material.

• Research Organization: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
• Sponsoring Organization: USDOE National Nuclear Security Administration (NNSA)
• Grant/Contract Number: NA0004107; 89233218CNA000001
• OSTI ID: 2467396
• Report Number(s): LA-UR--24-20734
• Journal Information: Numerical Algorithms, Journal Name: Numerical Algorithms Journal Issue: 4 Vol. 99; ISSN 1017-1398
• Publisher: SpringerCopyright Statement
• Country of Publication: United States
• Language: English