## Rational interpolation methods for symmetric Sylvester equations

Peter Benner and Tobias Breiten

### Abstract

We discuss low-rank approximation methods for large-scale symmetric Sylvester equations. Following similar discussions for the Lyapunov case, we introduce an energy norm by the symmetric Sylvester operator. Given a rank $n_r,$ 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 $\mathcal{H}_2$-inner product for symmetric 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.

Full Text (PDF) [680 KB]

### Key words

Sylvester equations, rational interpolation, energy norm

15A24, 37M99

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