Automatic Diierentiation of Numerical Integration Algorithms

Automatic diierentiation (AD) is a technique for automatically augmenting computer programs with statements for the computation of derivatives. This article discusses the application of automatic diierentiation to numerical integration algorithms for ordinary diierential equations (ODEs), in particular, the ramiications of the fact that AD is applied not only to the solution of such an algorithm, but to the solution procedure itself. This subtle issue can lead to surprising results when AD tools are applied to variable-stepsize, variable-order ODE integrators. The computation of the nal time step plays a special role in determining the computed derivatives. We investigate these issues using various integrators and suggest constructive approaches for obtaining the desired derivatives.