How expensive are Jacobians, and are they worth it?
by
Andreas Griewank
Technical University Dresden

Jacobians, Hessians and other derivative matrices figure prominently in many if not most nonlinear calculations. Hence, one might reasonably ask how expensive they are to evaluate compared to the cost of evaluating the underlying gradients or more vector functions themselves. Upper bounds for this cost ratio are easily obtained, but they grow linearly with the number of independent or dependent variables. Morover, there are large families of problems, where a `suitable' representation of the Jacobians or Hessians can be obtained at the same cost as the function itself. In fact, if we define `suitability' sufficiently wide, this is true for all functions defined by a computer program for its evaluation. For some purposes, like the calculation of Newton-steps, explicitly evaluating all nonzero entries of Jacobians or Hessians may be a waste of computing resources, altogether.Thus the original question of how expensive it is to use exact derivatives in numerical methods becomes rather complicated. Specifically, one has to be very clear about the ultimate goal of the computation and the kind of representation of derivatives that are needed for that purpose. The talk will illustrate these claims by various examples and present some partial theoretical results.

Date received: March 19, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caen-50.