D-RADI: A Low-rank ADI Algorithm for Solving Large-scale Discrete-time Algebraic Riccati Equations

Abstract

The low-rank alternating direction implicit (ADI) method is an efficient numerical technique for solving several types of large-scale matrix equations that admit low-rank solutions. The discrete-time algebraic Riccati equation (DARE) is an important matrix equation with applications in state estimation, controller design, and filter design. In the literature, the low-rank Cholesky factor ADI method for Stein equations has been used within Newton iterations to solve large-scale DAREs. However, no dedicated low-rank ADI solver is available for such DAREs. To address this gap, this paper presents a low-rank ADI solver for large-scale DAREs. We also propose an efficient approach to generate ADI shifts automatically, which makes the proposed solver fully autonomous for solving DAREs. The effectiveness of the proposed solver is compared with MATLAB's \texttt{idare} on a moderate-order problem. Efficiency and accuracy are further demonstrated on large-scale DAREs of order $10^6$. Numerical results confirm that the solver is efficient, accurate, and fully autonomous.