Analysis of an Iteration Method for the Algebraic Riccati Equation

We consider a recently published method for solving algebraic Riccati equations. We present a new perspective on this method in terms of the underlying linear-quadratic optimal control problem: we prove that the matrix obtained by this method expresses the optimal cost for a projected optimal control problem. The projection is determined by the so-called shift parameters of the method. Our representation in terms of the optimal control problem gives rise to a simple and very general convergence analysis.