A Recursive Approach to Multivariate Automatic Di erentiation

In one approach to automatic di erentiation, the range of a function is generalized from a single real value to an aggregate representing the values of the function and one or more derivatives. The operations and functions of elementary analysis are extended to these aggregates so as to preserve the validity of the derivatives. In this paper we develop a recursive approach to de ning the necessary operations in the context of functions of several variables. Formally, the de nitions are essentially the same as those needed in the single variable case. The resulting system provides automatic propagation of values of all partial derivatives up to a prespeci ed order for a function of several variables.