  We study the problem of estimating the maximal Lyapunov exponent of dominated
cocycles. In particular we are concerned with cocycles over Gibbs states on
shifts of finite type for which both the function defining the cocycle and the
potential defining the Gibbs state may depend on infinitely many coordinates
but are still very regular. We show that when the $n$th variation of both the
cocycle and the potential is $O(e^{-cn^{2}})$ for some $c>h_{\text{top}}$ then
using periodic points of period less then $n$ the Lyapunov exponent can be
approximated to an accuracy $O(n^{-kn})$ for some explicit $k>0$.
