  In many applications, one is interested in classifying trajectories of
Hamiltonian systems as invariant tori, islands, or chaos. The convergence rate
of ergodic Birkhoff averages can be used to categorize these regions, but many
iterations of the return map are needed to implement this directly. Recently,
it has been shown that a weighted Birkhoff average can be used to accelerate
the convergence, resulting in a useful method for categorizing trajectories.
  In this paper, we show how a modified version the reduced rank extrapolation
method (named Birkhoff RRE) can also be used to find optimal weights for the
weighted average with a single linear least-squares solve.Using these, we
classify trajectories with fewer iterations of the map than the standard
weighted Birkhoff average. Furthermore, for the islands and invariant circles,
a subsequent eigenvalue problem gives the number of islands and the rotation
number. Using these numbers, we find Fourier parameterizations of invariant
circles and islands. We show examples of Birkhoff RRE on the standard map and
on magnetic field line dynamics.
