Rational Interpolation Methods for Symmetric Sylvester Equations

Abstract

We discuss low-rank approximation methods for large-scale symmetric Sylvester equations. Follow- ing similar discussions for the Lyapunov case, we introduce an energy norm by the symmetric Sylvester operator. Given a rank nr, we derive necessary conditions for an approximation being optimal with respect to this norm. We show that the norm minimization problem is related to an objective function based on the H2-inner product for sym- metric state space systems. This objective function leads to first-order optimality conditions that are equivalent to the ones for the norm minimization problem. We further propose an iterative procedure and demonstrate its efficiency by means of some numerical examples.