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.
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).