We establish a set-oriented algorithm for the numerical approximation
of the rotation set of homeomorphisms of the two-torus homotopic to
the identity. A theoretical background is given by the concept of ε-rotation
sets. These are obtained by replacing orbits with ε-pseudo-orbits in the definition
of the Misiurewicz-Ziemian rotation set and are shown to converge to the
latter as ε decreases to zero. Based on this result, we prove the convergence
of the numerical approximations as precision and iteration time tend to infinity.
Further, we provide analytic error estimates for the algorithm under an
additional boundedness assumption, which is known to hold in many relevant
cases and in particular for non-empty interior rotation sets.