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.

微分是数值计算最常见的主题之一。梯度和粗麻布用于物理和工程科学的许多问题。当导数计算的准确性很重要时，通常使用自动微分（AD）。实施AD后，不会出现截断或取消错误。因此，以可用的机器精度计算导数。在这项研究中，通过使用双数实​​现AD的正向模式，以开发有效的方法来计算速度和加速度。众所周知，AD的反向模式比AD的正向模式更有效地计算梯度和Hessian。但是，在计算速度和加速度时并不需要直接使用渐变和Hessian。然而，需要方向导数和Hessian运算符对特定矢量的作用。可以通过使用双数来有效地计算这两个运算。