Gradient Algorithmic Diierentiation in Maple

Many scientiic applications require computation of the derivatives of a function f : IR n ! IR m as well as the function values of f itself. All computer algebra systems can diierentiate functions represented by formulae. But not all functions can be described by formulae. And formulae are not always the most eeective means for representing functions and derivatives. In this paper we describe the algorithms used by the Maple 2] routine GRADIENT that accepts as input a Maple procedure for the computation of f and outputs a new Maple procedure that computes the gradient of f. The design of the GRADIENT routine is such that it is also trivial to generate Maple procedures for the computation Jacobians and Hessians.