Numerical improvements for solving Riccati equations

In this paper, we discuss some ideas for improving the efficiency and accuracy of numerical methods for solving algebraic Riccati equations (AREs) based on invariant or deflating subspace methods. The focus is on AREs for which symmetric solutions exist, and our methods apply to both standard linear-quadratic-Gaussian (or H₂) AREs and to so-called H_∞-type AREs arising from either continuous-time or discrete-time models. The first technique is a new symmetric representation of a symmetric Riccati solution computed from an orthonormal basis of a certain invariant or deflating subspace. The symmetric representation does not require sign definiteness of the Riccati solution. The second technique relates to improving algorithm efficiency. Using a pencil-based approach, the solution of a Riccati equation can always be reformulated so that the deflating subspace whose basis is being sought corresponds to eigenvalues outside the unit circle. Thus, the natural tendency of the QZ algorithm to deflate these eigenvalues last, and hence, to appear in the upper left blocks of the appropriate pencils, then reduces the amount of reordering that must be done to a Schur form