  Given two elements of a vector space acted on by a reductive group, we ask
whether they lie in the same orbit, and if not, whether one lies in the orbit
closure of the other. We develop techniques to optimize the orbit and orbit
closure algorithms and apply these to give a partial classification of orbit
closure containments in the case of cubic surfaces with infinitely many
singular points, which are known to fall into 13 normal forms. We also discuss
the computational obstructions to completing this classification, and discuss
tools for future work in this direction.
