Lars Grasedyck:

Existence of a low rank or H-matrix approximant to the solution of a Sylvester equation

We consider the Sylvester equation

where the complex valued n x m matrix C is of low rank and the spectra of the complex valued n x n matrix A and m x m matrix B are separated by a line. We prove that the singular values of the solution X decay exponentially, that means for any epsilon in (0,1) there exists a matrix X' of rank k = O(log(1/epsilon)) such that ¦¦ X - X' ¦¦₂ is less or equal to epsilon ¦¦ X ¦¦₂. As a generalisation we prove that if A, B, C are hierarchical matrices then the solution X can be approximated by the hierarchical matrix format. The blockwise rank of the approximation is again proportional to log(1/epsilon).

Mathematics Subject Classification (1991): 15A24, 49N10, 65F50

Keywords: Data-sparse approximation, Sylvester equation, low rank approximation, singular value bounds, hierarchical matrices

Mail an Jens Burmeister

[Thu Feb 19 18:56:36 2009]

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