Sylvester Equations and the Factorization of the Error System in Krylov Subspace Methods

This paper presents a factorization of the error system that arises in model reduction of linear time invariant systems by Krylov subspace methods. The factorization is introduced for reduced models that match moments and/or Markov parameters of the original system with multiple inputs and outputs. Furthermore, dual results are given for the reduction with input and output Krylov subspaces. To this end, this work constitutes a generalization of [Wolf et al. (2011)] where the factorization was first presented for a special case. The results emerge from an investigation of the Sylvester equations that arise in the context of Krylov subspaces. To that effect, previous results on Sylvester equations are revised and extended in this paper. The theoretic results on Krylov subspaces that are presented here can be useful in error analysis and in the selection of expansion points in Krylov-based model order reduction.