Suppose $\Gamma$ is a finitely generated subgroup of $SL(n,\mathbb{Z})$, given as a list of generators. We would like to (somewhat efficiently) try to compute the Zariski closure of $\Gamma$, which is a (real) algebraic group. The method should be computer assisted but rigorous.

In the cases we are considering, $\Gamma$ will usually be Zariski dense in $SL(n,\mathbb{R})$, so the algorithm we are looking for should be optimized for that case. Also we would like just to know the answer, so having the program run forever if $\Gamma$ is not Zariski dense in $SL(n,\mathbb{R})$ is fine for us: we will just analyze that example further.

We can probably come up with some ad-hoc method for doing this, but I was wondering if anyone on MO has some interesting ideas or references.