Geodesic
Rational interpolation methods for symmetric Sylvester equations
Electronic transactions on numerical analysis, Tome 42 (2014), pp. 147-164
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.
Classification : 15A24, 37M99
Keywords: Sylvester equations, rational interpolation, energy norm 
@article{ETNA_2014__42__a2,
     author = {Benner,  Peter and Breiten,  Tobias},
     title = {Rational interpolation methods for symmetric {Sylvester} equations},
     journal = {Electronic transactions on numerical analysis},
     pages = {147--164},
     year = {2014},
     volume = {42},
     zbl = {1312.65067},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ETNA_2014__42__a2/}
}
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AU  - Breiten,  Tobias
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VL  - 42
UR  - http://geodesic.mathdoc.fr/item/ETNA_2014__42__a2/
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%A Breiten,  Tobias
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%J Electronic transactions on numerical analysis
%D 2014
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%U http://geodesic.mathdoc.fr/item/ETNA_2014__42__a2/
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%F ETNA_2014__42__a2
Benner,  Peter; Breiten,  Tobias. Rational interpolation methods for symmetric Sylvester equations. Electronic transactions on numerical analysis, Tome 42 (2014), pp. 147-164. http://geodesic.mathdoc.fr/item/ETNA_2014__42__a2/