Factorized solution of Lyapunov equations based on hierarchical matrix arithmetic
type: Item
instance of: scholarly article
title: Factorized solution of Lyapunov equations based on hierarchical matrix arithmetic
zbMATH Open document ID: 1111.65039
author: Ulrike Baur
author: P. Benner
published in: Computing
publication date: 2007-01-08T00:00:00Z
review text: To obtain the symmetric positive definite solution \(X\) of a Lyapunov equation \(AX+XA^T+BB^T=0\), it is best to compute its factor \(Y\) such that \(X=YY^T\). An approximation to \(Y\) can be obtained using the sign method [\textit{J. D. Roberts}, Int. J. Control 32, 677--687 (1980, Zbl 0463.93050)]. In some applications (finite element method; boundary element method) the matrices have a hierarchical structure of blocks of low rank (so called \(\mathcal H\) matrices). In this paper a method is proposed that combines the \(\mathcal H\) matrix structure with the sign method to compute a \(Y\)-factor of the solution of the Lyapunov equation. An extension to more general Lyapunov equations (such as those appearing in descriptor systems) is also included.
Mathematics Subject Classification ID: 65F30
Mathematics Subject Classification ID: 65F10
Mathematics Subject Classification ID: 15A24
Mathematics Subject Classification ID: 93B40
zbMATH DE Number: 5082635
zbMATH Keywords: H-matrix
zbMATH Keywords: hierarchical matrix
zbMATH Keywords: rank structured matrix
zbMATH Keywords: sign function
zbMATH Keywords: model reduction
zbMATH Keywords: Lyapunov equation
zbMATH Keywords: descriptor system
zbMATH Keywords: sign method
reviewed by: Adhemar Bultheel
describes a project that uses: mctoolbox
describes a project that uses: toms/782
describes a project that uses: SLICOT
describes a project that uses: hlib
describes a project that uses: RRQR
MaRDI profile type: Publication
full work available at URL: s00607-006-0178-y
OpenAlex ID: W2091078728
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DOI: 10.1007/S00607-006-0178-Y
description: scientific article; zbMATH DE number 5082635