Doubling Algorithms with Permuted Lagrangian Graph Bases

Abstract

We derive a new representation of Lagrangian subspaces in the form ImΠT [ I X ] , where Π is a symplectic matrix which is the product of a permutation matrix and a real orthogonal diagonal matrix, and X satisfies |Xij | ≤ { 1 if i = j, √ 2 if i = j. This representation allows us 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.

DOI: 10.1137/110850773

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@article{Mehrmann2012DoublingAW, title={Doubling Algorithms with Permuted Lagrangian Graph Bases}, author={Volker Mehrmann and Federico Poloni}, journal={SIAM J. Matrix Analysis Applications}, year={2012}, volume={33}, pages={780-805} }