$LDL^\top$ Factorization-based Generalized Low-rank ADI Algorithm for Solving Large-scale Algebraic Riccati Equations

Abstract: The low-rank alternating direction implicit (ADI) method is an efficient and effective solver for large-scale standard continuous-time algebraic Riccati equations that admit low-rank solutions. However, the existing low-rank ADI algorithm for Riccati equations (RADI) cannot be directly applied to general-form Riccati equations, such as those involving indefinite quadratic terms. This paper introduces a generalized RADI algorithm based on an $LDL^\top$ factorization, which efficiently handles the general Riccati equations arising in important applications like state estimation and controller design. An approach for automatically and efficiently generating ADI shifts is also discussed, along with a MATLAB implementation of the generalized RADI method. Numerical examples solving several Riccati equations of order $10^6$ accurately and efficiently are presented, demonstrating the effectiveness of the proposed algorithm.