  Let $G(q)$ be a finite group of Lie type over a field with $q$ elements,
where $q$ is a prime power. The Green functions of $G(q)$, as defined by
Deligne and Lusztig, are known in \textit{almost} all cases by work of
Beynon--Spaltenstein, Lusztig und Shoji. Open cases exist for groups of
exceptional type ${^2\!E}_6$, $E_7$, $E_8$ in small characteristics. We propose
a general method for dealing with these cases, which procedes by a reduction to
the case where $q$ is a prime and then uses computer algebra techniques. In
this way, all open cases in type ${^2\!E}_6$, $E_7$ are solved, as well as at
least one particular open case in type $E_8$.
