  We consider the dissipative spin-orbit problem in Celestial Mechanics, which
describes the rotational motion of a triaxial satellite moving on a Keplerian
orbit subject to tidal forcing and "drift".
  Our goal is to construct quasi-periodic solutions with fixed frequency,
satisfying appropriate conditions.
  With the goal of applying rigorous KAM theory, we compute such quasi-periodic
solution with very high precision. To this end, we have developed a very
efficient algorithm. The first step is to compute very accurately the return
map to a surface of section (using a high order Taylor's method with extended
precision). Then, we find an invariant curve for the return map using recent
algorithms that take advantage of the geometric features of the problem. This
method is based on a rapidly convergent Newton's method which is guaranteed to
converge if the initial error is small enough. So, it is very suitable for a
continuation algorithm.
  The resulting algorithm is quite efficient. We only need to deal with a one
dimensional function. If this function is discretized in $N$ points, the
algorithm requires $O(N \log N) $ operations and $O(N) $ storage. The most
costly step (the numerical integration of the equation along a turn) is trivial
to parallelize.
  The main goal of the paper is to present the algorithms, implementation
details and several sample results of runs.
  We also present both a rigorous and a numerical comparison of the results of
averaged and not averaged models.
