Reliable Computing 3: 259–268, 1997.
1997 Kluwer Academic Publishers. Printed in the Netherlands.
Fast Evaluation of Partial Derivatives and
INRIA—Sophia Antipolis, France; the current address is IIIA–CSIC, Campus UAB, 08193
Bellaterra, Spain, e-mail: firstname.lastname@example.org
(Received: 1 December 1996; accepted: 28 February 1997)
Abstract. For functions that share intermediate results, the computation of partial derivatives can be
modeled by node condensation on graphs. In this case, mixed evaluation strategies can outperform
either the backward or the forward mode of automatic differentiation. In this paper we present new
algorithms and heuristics to ﬁnd good evaluation strategies for partial derivatives. We show that these
techniques not only apply for interval derivatives but also for interval slopes.
The automatic computation of partial derivatives is usually done in a bottom-up
recursive fashion. This approach has a complexity of O(nF)wherenis the number
of function arguments, and F is the complexity of one function evaluation. The
computation of partial derivatives in the forward mode is not as efﬁcient as it could
be. It involves operations on vectors, many of which only need to be performed
once. This consideration led to the more efﬁcient top-down backward computation
of partial derivatives . The complexity of evaluating both the function value and
the partial derivatives reduces to CF , . A comparison between the different
approaches together with some practical applications of this technique can be found
The backward mode is usually more efﬁcient than the forward mode for the
evaluation of partial derivatives of a single function with respect to many input
variables. When manyfunctions share intermediate results but there is only one input
variable, the situation is reversed. In this case it is desirable to use the forward mode.
For multi-input, multi-output computational graphs, mixed evaluation strategies can
outperform either mode. In this paper (Section 3) we present new algorithms and
heuristics to ﬁnd good evaluation strategies for partial derivatives.
Most interval solution algorithms for nonlinear systems of equations are based on
interval derivatives. However, to ﬁnd global solutions it is preferable to use interval
slopes. Indeed, the width of interval slopes is approximately half of the width of
interval derivatives. This can have an important impact on the performance of the
algorithms far from the solution .