  Two approaches for approximating the solution of large-scale Lyapunov
equations are considered: the alternating direction implicit (ADI) iteration
and projective methods by Krylov subspaces. A link between them is presented by
showing that the ADI iteration can always be identified by a Petrov-Galerkin
projection with rational block Krylov subspaces. Then a unique Krylov-projected
dynamical system can be associated with the ADI iteration, which is proven to
be an H2 pseudo-optimal approximation. This includes the generalization of
previous results on H2 pseudo-optimality to the multivariable case.
Additionally, a low-rank formulation of the residual in the Lyapunov equation
is presented, which is well-suited for implementation, and which yields a
measure of the "obliqueness" that the ADI iteration is associated with.
