Doubling Algorithms With Permuted Lagrangian Graph Bases
Please always quote using this URN:urn:nbn:de:0296-matheon-9067
- We derive a new representation of Lagrangian subspaces in the form % \[ {\mathrm Im}\Pi^T [I,X]^T, \] % where $\Pi$ is a symplectic matrix which is the product of a permutation matrix and a real orthogonal diagonal matrix, and $X$ satisfies % \[ \abs{X_{ij}} \leq \begin{cases}1 & \text{if $i=j$,}\\ \sqrt{2} & \text{if $i\neq j$.} \end{cases} \] % This representation allows to limit element growth in the context of doubling algorithms for the computation of Lagrangian subspaces and the solution of Riccati equations. It is shown that a simple doubling algorithm using this representation can reach full machine accuracy on a wide range of problems, obtaining invariant subspaces of the same quality as those computed by the state-of-the-art algorithms based on orthogonal transformations. The same idea carries over to representations of arbitrary subspaces and can be used for other types of structured pencils.
Author: | Volker Mehrmann, Federico Poloni |
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URN: | urn:nbn:de:0296-matheon-9067 |
Referee: | Fredi Tröltzsch |
Document Type: | Preprint, Research Center Matheon |
Language: | English |
Date of first Publication: | 2011/10/20 |
Release Date: | 2011/10/20 |
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Institute: | Technische Universität Berlin |
Project: | C Energy and Materials (Production) |
MSC-Classification: | 65-XX NUMERICAL ANALYSIS / 65Fxx Numerical linear algebra / 65F05 Direct methods for linear systems and matrix inversion |
65-XX NUMERICAL ANALYSIS / 65Fxx Numerical linear algebra | |
Preprint Number: | 821 |