Automatic differentiation of numerical integration algorithms

Automatic differentiation (AD) is a technique for automatically augmenting computer programs with statements for the computation of derivatives. This article discusses the application of automatic differentiation to numerical integration algorithms for ordinary differential equations (ODEs), in part icular, the ramifications of the fact that AD is applied not only to the so lution of such an algorithm, but to the solution procedure itself. This sub tle issue can lead to surprising results when AD tools are applied to varia ble-stepsize, variable-order ODE integrators. The computation of the final 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.