Doubling Algorithms with Permuted Lagrangian Graph Bases

Abstract

We derive a new representation of Lagrangian subspaces in the form ${\rm Im} \Pi^T\big[\begin{smallmatrix}I \\ X\end{smallmatrix}\big]$, where $\Pi$ is a symplectic matrix which is the product of a permutation matrix and a real orthogonal diagonal matrix, and $X$ satisfies $\left\vert X_{ij}\right\vert \leq \begin{cases}1 & \text{if $i=j$,}\\ \sqrt{2} & \text{if $i\neq j$.} \end{cases}$ 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.