# How AD Can Help Solve Differential-Algebraic Equations

## TL;DR

This paper demonstrates how algorithmic differentiation can efficiently solve differential-algebraic equations, improving index reduction and enabling direct solutions from Lagrangian formulations with practical examples.

## Contribution

It introduces the use of pure algorithmic differentiation for index reduction and direct solution of DAE systems, with theoretical and practical insights.

## Key findings

- Efficient index reduction via algorithmic differentiation.
- Successful application to complex mechanical systems.
- Demonstrated long-term integration of planetary models.

## Abstract

A characteristic feature of differential-algebraic equations is that one needs to find derivatives of some of their equations with respect to time, as part of so called index reduction or regularisation, to prepare them for numerical solution. This is often done with the help of a computer algebra system. We show in two significant cases that it can be done efficiently by pure algorithmic differentiation. The first is the Dummy Derivatives method, here we give a mainly theoretical description, with tutorial examples. The second is the solution of a mechanical system directly from its Lagrangian formulation. Here we outline the theory and show several non-trivial examples of using the "Lagrangian facility" of the Nedialkov-Pryce initial-value solver DAETS, namely: a spring-mass-multipendulum system, a prescribed-trajectory control problem, and long-time integration of a model of the outer planets of the solar system, taken from the DETEST testing package for ODE solvers.