Dual Numbers for Arbitrary Order Automatic Differentiation

Abstract: Dual numbers are a well-known tool for computing derivatives of functions. While the theoretical framework for calculating derivatives of arbitrary order is well established, practical implementations remain less developed. One notable implementation is available in the Julia programming language where dual numbers are designed to be nested, enabling the computation of derivatives to arbitrary order. However, this approach has a significant drawback as it struggles with scalability for high-order derivatives. The nested structure quickly consumes memory, making it challenging to compute derivatives of higher orders. In this study, we introduce DNAOAD, a Fortran-based implementation of automatic differentiation capable of handling derivatives of arbitrary order using dual numbers. This implementation employs a direct approach to represent dual numbers without relying on recursive or nested structures. As a result, DNAOAD facilitates the efficient computation of derivatives of very high orders while addressing the memory limitations of existing methods.