Automatic Differentiation and Implicit Differential Equations ∗

Many physical processes are most naturally and easily modeled as mixed systems of differential and algebraic equations (DAEs). There has been an increased interest in several areas in exploiting the advantages of working directly with these implicit models. Differentiation plays an important role in both the analysis and numerical solution of DAEs. Automatic differentiation can have a significant impact on what is considered a practical approach and what types of problems can be solved. However, working with DAEs places special demands on automatic differentiation codes. More is required than just computing a gradient quickly. This paper will begin with a brief introduction to DAEs and how differentiation is important when working with DAEs. Then the requirements in terms of both information and performance that DAEs make of automatic differentiation software will be presented. Some of our own experience in using automatic differentiation software will be mentioned. It will be seen that automatic differentiation software has a significant role to play in the future for DAEs but that not all of the demands that the numerical solution of DAEs places on automatic differentiation software are currently being met.