Analysis of the rational Krylov subspace projection method for large-scale algebraic Riccati equations

In the numerical solution of the algebraic Riccati equation $A^* X + X A - X BB^* X + C^* C =0$, where $A$ is large, sparse, and stable, and $B$, $C$ have low rank, projection methods have recently emerged as a possible alternative to the more established Newton--Kleinman iteration. In spite of convincing numerical experiments, a systematic matrix analysis of this class of methods is still lacking. We derive new relations for the approximate solution, the residual, and the error matrices, giving new insights into the role of the matrix $A-BB^*X$ and of its approximations in the numerical procedure. In the context of linear-quadratic regulator problems, we show that the Riccati approximate solution is related to the optimal value of the reduced cost functional, thus completely justifying the projection method from a model order reduction point of view. Finally, the new results provide theoretical ground for recently proposed modifications of projection methods onto rational Krylov subspaces.