ABSTRACT
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 pseudo-optimal approximation. This includes the generalisation of previous results on
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.
Acknowledgements
The authors thank Prof. Serkan Gugercin for the fruitful discussion at the MODRED 2013 in Magdeburg, and the anonymous reviewers for their valuable remarks.
Disclosure statement
No potential conflict of interest was reported by the authors.
Notes
1. Available at http://portal.uni-freiburg.de/imteksimulation/downloads/benchmark.