arXiv subjects

K. B. Cantún-Avila

Publications and source records attributed to K. B. Cantún-Avila.

3 recordsLinked to original sources

arXiv2026

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

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.

math.NA↗

arXiv2026

Dual Numbers for Arbitrary Order Automatic Differentiation

Dual numbers are a well-established tool for computing derivatives and constitute the basis of forward-mode automatic differentiation. While the theoretical framework for computing derivatives of arbitrary order is well understood, practical and scalable implementations remain limited. Existing approaches based on nested dual numbers, such as those used in modern high-level languages, suffer from severe memory growth and poor scalability as the derivative order increases. In this work, we introduce DNAOAD, a Fortran-based automatic differentiation framework capable of computing derivatives of arbitrary order using dual numbers with a direct, non-nested representation. By avoiding recursive data structures, DNAOAD significantly reduces memory usage and enables the efficient computation of derivatives of very high order, overcoming key scalability limitations of existing methods and making it particularly well suited for high-performance scientific computing applications.

math.NA↗

arXiv2023

A dual number formulation to efficiently compute higher order directional derivatives

This contribution proposes a new formulation to efficiently compute directional derivatives of order one to fourth. The formulation is based on automatic differentiation implemented with dual numbers. Directional derivatives are particular cases of symmetric multilinear forms; therefore, using their symmetric properties and their coordinate representation, we implement functions to calculate mixed partial derivatives. Moreover, with directional derivatives, we deduce concise formulas for the velocity, acceleration, jerk, and jounce/snap vectors. The utility of our formulation is proved with three examples. The first example presents a comparison against the forward mode of finite differences to compute the fourth-order directional derivative of a scalar function. To this end, we have coded the finite differences method to calculate partial derivatives until the fourth order, to any order of approximation. The second example presents efficient computations of the velocity, acceleration, jerk, and jounce/snap. Finally, the third example is related to the computation of some partial derivatives. The implemented code of the proposed formulation and the finite differences method is proportioned as additional material to this article.

math.NA↗