The Adi Iteration For Lyapunov Equations Implicitly Performs H-2 Pseudo-Optimal Model Order Reduction

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. We show that they are linked in the way that the ADI iteration can always be identified by a Petrov-Galerkin projection with rational block Krylov subspaces. Therefore, a unique Krylov-projected dynamical system can be associated with the ADI iteration, which is proven to be an H-2 pseudo-optimal approximation. This includes the generalisation of previous results on H-2 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.