On the efficient computation of high-order derivatives for implicitly defined functions

TL;DR

This paper introduces an algorithmic differentiation method for accurately computing high-order derivatives of implicitly defined functions, especially useful when analytical solutions are infeasible, demonstrated on a quantum field theory model.

Contribution

The paper presents a novel algorithmic differentiation technique that efficiently computes high-order derivatives of implicit functions, overcoming limitations of numerical approximation methods.

Findings

Method achieves higher accuracy than standard numerical derivatives.

Applicable to complex models with implicit dependencies.

Demonstrated effectiveness on a quantum field theoretical model.

Abstract

Scientific studies often require the precise calculation of derivatives. In many cases an analytical calculation is not feasible and one resorts to evaluating derivatives numerically. These are error-prone, especially for higher-order derivatives. A technique based on algorithmic differentiation is presented which allows for a precise calculation of higher-order derivatives. The method can be widely applied even for the case of only numerically solvable, implicit dependencies which totally hamper a semi-analytical calculation of the derivatives. As a demonstration the method is applied to a quantum field theoretical physical model. The results are compared with standard numerical derivative methods.