Solving Large-Scale Discrete-Time Algebraic Riccati Equations by Doubling

We consider the solution of large-scale discrete-time algebraic Riccati equations with numerically low-ranked solutions. The structure-preserving doubling algorithm will be adapted, with the iterates for A not explicitly computed but in the recursive form Ak = A 2 k−1 − D (1) k S −1 k [D (2) k ] >, where D (1) k and D (2) k are low-ranked with S −1 k being small in dimension. With n being the dimension of the algebraic equations, the resulting algorithms are of an efficient O(n) complexity per iteration, without the need for any inner iterations, and essentially converge quadratically. Some numerical results will be presented. For instance in Section 4.2, Example 6, of dimension n = 79841 with 3.19 billion variables in the solution X, was solved using MATLAB on a MacBook Pro within 1,100 seconds to machine accuracy.