Higher Order Automatic Differentiation with Dual Numbers

In engineering applications, we often need the derivatives of functions defined by a program. The approach chosen for derivative computation must be algebraic to allow computer implementation. A particular solution to obtain first derivatives is the application of dual numbers. This paper proposes simple and compact generalizations of this idea to obtain derivatives of arbitrary order for single or multi-variate functions and the automatic handling of 0/0 ambiguities in the calculations. We also provide the C++ code that takes advantage of operator overloading and recursion. The method is demonstrated by path animation, Gaussian curvature computation, and curve fairing.