  We study the iteration of block spin transformations in the O(3) symmetric
non-linear sigma-model on a two-dimensional square lattice with help of the
Monte Carlo method. In contrast to the classical Monte Carlo Renormalization
Group approach, we do attempt to explicitly compute the block spin effective
actions. Using two different methods for the determination of effective
couplings, we study the renormalization group flow for various parametrization
and truncation schemes. The largest ansatz for the effective action contains
thirteen coupling constants.
  Actions on the renormalized trajectory should describe theories with no
lattice artefacts, even at small correlation length. However, tests with the
step scaling function of Luescher et al. reveal that our truncated effective
actions show sizable scaling violations indicating that the ansaetze are still
too small.
