Author(s) :
Volker Mehrmann
,
Federico Poloni
Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 14-2014
MSC 2000
- 65F15 Eigenvalues, eigenvectors
-
65F50 Sparse matrices
Abstract :
The numerical computation of Lagrangian invariant subspaces of large scale Hamiltonian matrices is discussed in the context of the solution of Lyapunov and Riccati equations. A new version of the low-rank alternating direction implicit method is introduced, which in order to avoid numerical difficulties with solutions that are of very large norm, uses an inverse-free representation of the subspace and avoids inverses of ill-conditioned matrices. It is shown that this prevents large growth of the elements of the solution which may destroy a low-rank approximation of the solution. A partial error analysis is presented and the behavior of the method is demonstrated via several numerical examples.
Keywords :
Lagrangian subspace, permuted Lagrangian subspace, Lyapunov equation, Riccati equation, low-rank ADI method, inverse-free arithmetic, permuted graph basis