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Automatic dierentiation (AD) is a technique for automatically augmenting computer programs with statements for the computation of deriva- tives. This article discusses the application of automatic dierentiation to nu- merical integration algorithms for ordinary dierential equations (ODEs), in particular, the ramications 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 investi- gate these issues using various integrators and suggest constructive approaches for obtaining the desired derivatives.

Automatic differentiation of numerical integration algorithms