Finite diierence methods, like the mid-point rule, have been applied successfully to the numerical solution of ordinary and partial diierential equations. If such formulas are applied to observational data, in order to determine derivatives, the results can be disastrous. The reason for this is that measurement errors, and even rounding errors in computer approximations, are strongly ampliied i...

Automatic diierentiation (AD) tools can generate accurate and eecient derivative code for computer programs of arbitrary length. In some cases, however, the developer of the code to be diierentiated may be required to provide additional information to an AD tool to ensure the desired solution. We illustrate these issues with nondiierentiable language intrinsics such as max() in the context of c...

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 particular, the ramifications of the fact that AD is applied not only to the solution of such an algo...