Solving Large-scale Discrete-time Algebraic Riccati Equations by Doubling

This paper aims to the solution of large-scale discrete-time algebraic Riccati equation D(X) ≡ -X + A^T X(I + GX)^-1A + H = 0 with numerically low-ranked solutions. Based on the original structure-preserving doubling algorithm (SDA), we develop the large-scale doubling algorithm by applying Sherman-Morrison-Woodbury formula, using QR decomposition, SVD to compress and truncate the approximate solutions properly. The resulting large-scale doubling algorithm has an O(n) computational complexity and memory requirement per iteration and converges essentially quadratically. A detailed error analysis is given.