Dual Numbers and Automatic Differentiation to Efficiently Compute Velocities and Accelerations

• Published in: Acta applicandae mathematicae Vol. 170; no. 1; pp. 649 - 659
• Main Authors: Peñuñuri, F., Peón, R., González–Sánchez, D., Escalante Soberanis, M. A.
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.12.2020; Springer Nature B.V
• Subjects: Applications of Mathematics; Approximation; Calculus of Variations and Optimal Control; Optimization; Computational Mathematics and Numerical Analysis; Derivatives; Differentiation; Hessian matrices; Mathematical analysis; Mathematics; Mathematics and Statistics; Partial Differential Equations; Probability Theory and Stochastic Processes; Programming languages; Velocity; 65Z05; 65D25; Velocities and accelerations; Dual numbers; Automatic differentiation
• ISSN: 0167-8019; 1572-9036
• DOI: 10.1007/s10440-020-00351-9

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.