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

✍ Scribed by L. Grasedyck

Publisher
John Wiley and Sons
Year
2004
Tongue
English
Weight
190 KB
Volume
11
Category
Article
ISSN
1070-5325

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✦ Synopsis

Abstract

We consider the Sylvester equation AX−XB+C=0 where the matrix C∈ℂ^n×m^ is of low rank and the spectra of A∈ℂ^n×n^ and B∈ℂ^m×m^ are separated by a line. We prove that the singular values of the solution X decay exponentially, that means for any ε∈(0,1) there exists a matrix X̃ of rank k=O(log(1/ε)) such that ∥X−X̃∥~2~⩽ε∥X∥~2~. As a generalization we prove that if A,B,C are hierarchical matrices then the solution X can be approximated by the hierarchical matrix format described in Hackbusch (Computing 2000; 62: 89–108). The blockwise rank of the approximation is again proportional to log(1/ε). Copyright © 2004 John Wiley & Sons, Ltd.

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