Algorithm 755: ADOL-C: a package for the automatic differentiation of algorithms written in C/C++
ACM Transactions on Mathematical Software (TOMS)
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This paper deals with the calculation of partial derivatives (w.r.t.the independent variables, x) of a vec of dependentvariables y which satisfy a system of nonlinear equationsg(u(x), y) = 0. A number ofauthors have suggested that the forward accumulation method of automaticdifferentiation can be applied to a suitable iterative scheme for solvingthe nonlinear system with a view to giving simultaneous convergence both tothe correct value y and also to its Jacobian matrixy_x. It is known, however, that convergence of thederivatives may not occur at the same rate as the convergence of they values. In this paper we avoid both the difficulty andthe potential cost of iterating the gradient part of the calculation tosufficient accuracy. We do this by observing that forward accumulation needonly be applied to the functions g after thedependent variables, y, have been computed in standardreal arithmetic usin g any appropriate method. This so-called Post-Differentiation (PD) technique is shown, on a number ofexamples, to have an advantage in terms of both accuracy and speed overapproaches where forward accumulation is applied over the entire iterativeprocess. Moreover, the PD technique can be implemented in such a way as toprovide a friendly interface for non-specialist users.