  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 {\epsilon}-rotation sets.
These are obtained by replacing orbits with {\epsilon}-pseudo-orbits in the
definition of the Misiurewicz-Ziemian rotation set and are shown to converge to
the latter as {\epsilon} 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.
