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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.
A new implicitly-restarted Krylov subspace method
for real symmetric/skew-symmetric generalized eigenvalue problems
is presented. The new method improves and generalizes the SHIRA method
to the case where the skew symmetric matrix is singular.
It computes a few eigenvalues and eigenvectors of the matrix pencil
close to a given target point. Several applications from control theory are
presented and the properties of the new method are illustrated by benchmark
examples.
A generalization of the method of Chu, Liu and Mehrmann
for the computation of the Hamiltonian real Schur form is presented.
The new method avoids some of the difficulties that may arise when
a Hamiltonian matrix has tightly clustered groups of eigenvalues.
A detailed analysis of the method is presented and several numerical examples demonstrate the superior behavior of the method.
Canonical forms for matrix triples $(A,G,\hat G)$, where
$A$ is arbitrary rectangular and $G$, $\hat G$ are either real symmetric
or skew symmetric, or complex Hermitian or skew Hermitian, are derived.
These forms generalize classical product Schur forms as well as
singular value decompositions.
An new proof for the complex case is given, where there is no need to
distinguish whether $G$ and $\hat G$ are Hermitian or skew Hermitian.
This proof is independent from the results in Bolschakov/Reichstein 1995, where
a similar canonical form has been obtained for the complex case,
and it allows generalization to the real case. Here,
the three cases, i.e., that
$G$ and $\hat G$ are both symmetric, both skew symmetric or one each,
are treated separately.
Lagrangian invariant subspaces for symplectic matrices play an important role in the numerical solution of discrete time, robust and optimal control problems. The sensitivity (perturbation) analysis of these subspaces, however, is a difficult problem, in particular, when the eigenvalues are on or close to some critical regions in the complex plane, such as the unit circle.
We present a detailed perturbation analysis for several different cases of real and complex symplectic matrices. We analyze stability and conditional stability
as well as the index of stability for these subspaces.