Factorized solution of Lyapunov equations based on hierarchical matrix arithmetic
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.
scientific article; zbMATH DE number 5082635
scientific article