  A new version of the alternating directions implicit (ADI) iteration for the
solution of large-scale Lyapunov equations is introduced. It generalizes the
hitherto existing iteration, by incorporating tangential directions in the way
they are already available for rational Krylov subspaces. Additionally, first
strategies to adaptively select shifts and tangential directions in each
iteration are presented. Numerical examples emphasize the potential of the new
results.
