Combining three themes---port-Hamiltonian energy-based modeling, circuit structural analysis, and structural analysis of differential-algebraic equations---we form a new model for electrical circui...

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 the so-called index reduction or regularization, 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-multi-pendulum 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.

The signature matrix structural analysis method developed by Pryce provides more structural information than the commonly used Pantelides method and applies to differential-algebraic equations (DAEs) of arbitrary order. It is useful to consider how existing methods using the Pantelides algorithm can benefit from such structural analysis. The dummy derivative method is a technique commonly used to solve DAEs that can benefit from such exploitation of underlying DAE structures and information found in the Signature Matrix method. This paper gives a technique to find structurally necessary dummy derivatives and how to use different block triangular forms effectively when performing the dummy derivative method and then provides a brief complexity analysis of the proposed approach. We finish by outlining an approach that can simplify the task of dummy pivoting.

This is a slightly expanded form of the author’s talk of the same title at SCAN 2014, Würzburg. Angled towards people who use interval numerical methods little or not at all, it briefly describes how interval arithmetic works, the mindset required to use it effectively, why an interval arithmetic standard was needed, the setting up of IEEE Working Group P1788 for the purpose, the structure of the standard it has produced, some difficulties we encountered, and the current state of the P1788 project. During production of these Proceedings the 1788 standard has been published, but the talk’s original title has been kept.

The Natural Coordinates (NCs) method for Lagrangian modelling and simulation of multi-body systems is valued for giving simple, sparse models. We describe our version of it (NPNCs) and compare with the classical ap- proach of Jalón and Bayo (JBNCs). NPNCs use the high-index differential- algebraic equation solver DAETS. Algorithmic differentiation, not symbolic algebra, forms the equations of motion from the Lagrangian. NPNCs give significantly smaller equation systems than JBNCs, at the cost of a non- constant mass matrix for fully 3D models—a minor downside in the DAETS context. A 2D and a 3D example are presented, with numerical results.