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Dual Numbers and Automatic Differentiation to Efficiently Compute Velocities and Accelerations

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Abstract

Differentiation is one of the most common subjects of numerical calculations. Gradients and Hessians are used in many problems of the physical and engineering sciences. Automatic differentiation (AD) is usually employed when the accuracy in derivatives calculations is important. When AD is implemented, there are no truncation or cancellation errors. Therefore, the derivatives are calculated with the available machine precision. In this study, the forward mode of AD by using dual numbers is implemented to develop efficient methods for computing velocities and accelerations. It is known that the reverse mode of AD is more efficient than the forward mode of AD to compute gradients and Hessians. Nonetheless, gradients and Hessians are not directly required for the calculation of velocities and accelerations. However, directional derivatives and the action of the Hessian operator on specific vectors are required. Both operations can be efficiently computed through the use of dual numbers.

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Authors and Affiliations

  1. Departamento de Ingeniería Física, Universidad Autónoma de Yucatán, A.P. 150, Cordemex, Mérida, Yucatán, Mexico

    F. Peñuñuri

  2. Departamento de Ingeniería Mecatrónica, Universidad Autónoma de Yucatán, A.P. 150, Cordemex, Mérida, Yucatán, Mexico

    R. Peón

  3. CONACYT–Departamento de Matemáticas, Universidad de Sonora, C.P 83000, Hermosillo, Sonora, Mexico

    D. González–Sánchez

  4. Departamento de Ingeniería en Energías Renovables, Universidad Autónoma de Yucatán, A.P. 150, Cordemex, Mérida, Yucatán, Mexico

    M. A. Escalante Soberanis

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  1. F. Peñuñuri
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  2. R. Peón
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  3. D. González–Sánchez
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  4. M. A. Escalante Soberanis
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Correspondence to F. Peñuñuri.

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Peñuñuri, F., Peón, R., González–Sánchez, D. et al. Dual Numbers and Automatic Differentiation to Efficiently Compute Velocities and Accelerations. Acta Appl Math 170, 649–659 (2020). https://doi.org/10.1007/s10440-020-00351-9

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  • DOI: https://doi.org/10.1007/s10440-020-00351-9

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