AI 中文总结
该研究提出基于广义对偶数的框架，可高效计算多方向任意阶方向导数，支持混合导数与运动学量计算，相关代码为兼容Fortran包管理器的开源实现。
AI 中文摘要
针对标量值与向量值函数，本文通过广义对偶数公式实现沿多个（可能互不相同）方向的任意阶方向导数计算。所提框架将广义对偶评估与对称多线性形式的容斥重构相结合，可通过重复方向评估重构通用多方向导数，无需显式构建高阶导数张量；混合方向导数与混合偏导数作为该公式的特殊情况自然产生。该方法还可系统计算任意阶运动学量，并通过自动生成的时间导数为常微分方程组构建泰勒级数方法。数值算例包括高维函数的高阶方向导数、混合偏导数、任意阶运动学量及泰勒级数积分方法的计算，实现代码采用现代Fortran开发，属于兼容Fortran包管理器生态的开源框架。
英文摘要
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.
