Volume 42, pp. 147-164, 2014.

Rational interpolation methods for symmetric Sylvester equations

Peter Benner and Tobias Breiten


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.

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Key words

Sylvester equations, rational interpolation, energy norm

AMS subject classifications

15A24, 37M99

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