The authors’ thesis is that in order to obtain meaningful results when using numerical methods to solve various control problems, it is essential that the methods exploit and preserve any structure inherent in the problems. Their example of structure consists of the symmetry properties of Hamiltonian matrices that arise in linear-quadratic regulator (LQR) problems and H_∞ control problems.

Various stability results depend on all eigenvalues of a Hamiltonian matrix being purely imaginary. The authors assert that general methods (for example, methods not exploiting the Hamiltonian symmetry) make this determination impossible, due to round-off error. Building on some of their prior work, they describe an algorithm that, because it is structure-exploiting, perturbs imaginary results along the imaginary axis only, thereby giving correct stability results.

Furthermore, the described algorithm is expected to be 50 percent more efficient. The authors present some error and timing results for this algorithm and two comparison algorithms. One of the comparison algorithms incorrectly reported nonzero real parts for eigenvalues. The other, SQRED, did not have that failing, but showed larger relative error. On the other hand, SQRED was faster than the described algorithm. (The authors conclude, curiously, that “these new algorithms outperform standard approaches in every aspect.”)

There is a brief discussion of related techniques for computing invariant subspaces of Hamiltonian matrices. The authors allude to numerical results, but unfortunately do not include them, citing space limitations.

Readers interested in the theory supporting this work will have to refer to the bibliographic references, as little derivation is included.