Diferenciaci\'on Autom\'atica Anidada. Un enfoque algebraico

TL;DR

This paper introduces a novel algebraic framework and a new structure called SuperAdouble for nested automatic differentiation, enabling accurate computation of derivatives within function evaluations using operator overloading.

Contribution

It presents an algebraic approach and a new data structure for nested automatic differentiation applicable with any operator-overloading AD library.

Findings

SuperAdouble guarantees correct nested derivative calculations.

Framework applicable to any AD library with operator overloading.

Provides an algebraic perspective on automatic differentiation.

Abstract

En este trabajo se presenta una propuesta para realizar Diferenciaci\'on Autom\'atica Anidada utilizando cualquier biblioteca de Diferenciaci\'on Autom\'atica que permita sobrecarga de operadores. Para calcular las derivadas anidadas en una misma evaluaci\'on de la funci\'on, la cual se asume que sea anal'itica, se trabaja con el modo forward utilizando una nueva estructura llamada SuperAdouble, que garantiza que se aplique correctamente la diferenciaci\'on autom\'atica y se calculen el valor y la derivada que se requiera. Tambi\'en se presenta un enfoque algebraico de la Diferenciaci\'on Autom\'atica y en particular del espacio de los SuperAdoubles. This paper proposes a framework to apply Nested Automatic Differentiation using any library of Automatic Differentiation which allows operator overloading. To compute nested derivatives of a function while it is being evaluated, which is assumed to be analytic, a new structure called SuperAdouble is used in the forward mode. This new class guarantees the correct application of Automatic Differentiation to calculate the value and derivative of a function where is required. Also, an Automatic Differentiation algebraic point of view is presented with particular emphasis in Nested Automatic Differentiation.