  We study how the Newton-GMRES iteration can enable dynamic simulators
(time-steppers) to perform fixed-point and path-following computations.For a
class of dissipative problems, whose dynamics are characterized by a slow
manifold, the Jacobian matrices in such computations are compact perturbations
of the identity. We examine the number of GMRES iterations required for each
nonlinear iteration as a function of the dimension of the slow subspace and the
time-stepper reporting horizon. In a path-following computation, only a small
number (one or two) of additional GMRES iterations is required.
