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#### Keywords

- matrix pencil (2)
- optimal control (2)
- BVD pencil (1)
- Cayley transformation (1)
- Hamiltonian matrix (1)
- Lagrangian subspace (1)
- Lure equation (1)
- VARMA(1,1) model (1)
- deflating subspace (1)
- disk function method (1)

We propose a generalization of the Structured Doubling Algorithm (SDA) to compute invariant subspaces
of structured matrix pencils
that arise in the context of solving linear quadratic optimal control problems.
The new algorithm is
designed to attain better accuracy when the classical Riccati equation approach for the solution of the optimal control problem is not well suited because
the stable and unstable invariant subspaces are not well separated (due to eigenvalues near or on the imaginary
axis) or in the case when the Riccati solution does not exist at all. We analyze the convergence
of the method and compare the new method with the classical SDA algorithm as well as some recent structured QR-methods.

We introduce a~numerical method for the numerical solution of the Lur'e matrix equations that arise, for instance, in linear-quadratic infinite time horizon optimal control. The method is based on the characterization of the solutions in terms of deflating subspaces of a suitable even matrix pencil. Via a Cayley transformation, the problem is transformed to the discrete-time case. This leaves us with a symplectic problem with several Jordan blocks of eigenvalue 1 and even size, which arise from the remaining eigenvalues at infinity of the original problem. For the solution of this modified problem, we use the {\em structure-preserving doubling algorithm} (SDA), an iterative scheme for the solution of dense continuous- and discrete-time algebraic Riccati equations. Unlike other iterative schemes, this algorithm converges also when the pencil has eigenvalues on the unit circle, as is the case in our problem. Implementation issues such as the choice of the parameter $\gamma$ in the Cayley transform are discussed. The numerical examples presented confirm the effectiveness of this method.

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.

Enforcing solvability of a nonlinear matrix equation and estimation of multivariate ARMA time series
(2013)

The matrix equation $X+AX^{-1}A^T=B$, arising in parameter estimation of certain time series models,
is solvable only for certain values of the matrices $A,B$.
We present a numerical method to modify $A,B$ in order to make the matrix equation solvable.
Since solvability depends on the location of the eigenvalues of the palindromic matrix polynomial $\lambda^2 A+\lambda B+A^T$,
our method works by moving those eigenvalues to specified locations using first order spectral perturbation theory.
The method is heuristic but works in practice, as is supported by several compelling numerical examples.
These examples arise from parameter estimation of a common time series model, the multivariate ARMA(1,1).