In this report we extend these results by describing algorithms which appear to be suitable for computer implementation.
Experimentation is needed to determine the range of input for which they are practical.
Our method is to first reduce each problem to the corresponding problem for triangularizable matrix groups. We then prove a structure theorem for triangularizable matrix groups that allows us to decide whether or not a matrix group is triangularizable and to reduce the problem of testing membership in a polycyclic, triangularizable matrix group to the corresponding problems for finitely generated abelian matrix groups and for finitely generated unitriangular matrix groups. In the case of an abelian matrix group we can find a presentation for the group, and our membership test can be made constructive.