Tests for Monte Carlo renormalization studies on Ising models

In this expanded version of an earlier letter, we consider many computational details that were omitted for want of space. For d = 2 Ising spins with up to 13 different short-range interactions, we construct the critical surface in the vicinity of (Onsager's) nearest-neighbor (nn) critical point by using the body of the available information on the solvable nn case. We then see if the Monte Carlo renormalization group flows generated from this point do indeed lie on this surface and quantify the errors if they do not.