  The long-term dynamics of many dynamical systems evolve on an attracting,
invariant "slow manifold" that can be parameterized by a few observable
variables. Yet a simulation using the full model of the problem requires
initial values for all variables. Given a set of values for the observables
parameterizing the slow manifold, one needs a procedure for finding the
additional values such that the state is close to the slow manifold to some
desired accuracy. We consider problems whose solution has a singular
perturbation expansion, although we do not know what it is nor have any way to
compute it. We show in this paper that, under some conditions, computing the
values of the remaining variables so that their (m+1)-st time derivatives are
zero provides an estimate of the unknown variables that is an mth-order
approximation to a point on the slow manifold in sense to be defined. We then
show how this criterion can be applied approximately when the system is defined
by a legacy code rather than directly through closed form equations.
