Efficient Computation of Arbitrary-Order Directional Derivatives in Multiple Directions via Generalized Dual Numbers

Abstract

Arbitrary-order directional derivatives along multiple (possibly distinct) directions are computed through a generalized dual-number formulation for both scalar- and vector-valued functions. The proposed framework combines generalized dual evaluations with an inclusion--exclusion reconstruction of symmetric multilinear forms, allowing general multidirectional derivatives to be reconstructed from repeated-direction evaluations without explicitly constructing higher-order derivative tensors. Mixed directional and mixed partial derivatives arise naturally as particular cases of the formulation. The methodology further enables the systematic computation of arbitrary-order kinematic quantities and the construction of Taylor-series methods for systems of ordinary differential equations through automatically generated time derivatives. Numerical examples include the computation of high-order directional derivatives in high-dimensional functions, mixed partial derivatives, arbitrary-order kinematic quantities, and Taylor-series integration methods. The implementation is developed in modern Fortran within an open-source framework compatible with the Fortran Package Manager ecosystem.