Functional Diierentiation of Computer Programs

We present two purely functional implementations of the computational diierentiation tools { the well known numeric (not symbolic!) techniques which permit to compute point-wise derivatives of functions deened by computer programs economically and exactly. We show how the co-recursive (lazy) algorithm formulation permits to construct in a transparent and elegant manner the entire innnite tower of derivatives of higher order for any expressions present in the program , and we present a purely functional variant of the reverse (or adjoint) mode of computational diierentiation, using a chain of delayed evaluations represented by closures. Some concrete applications are also discussed.