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An inverse-free ADI algorithm for computing Lagrangian invariant subspaces

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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