$LDL^\top$ 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. This paper introduces a generalized RADI algorithm based on an factorization, which efficiently handles the general Riccati equations arising in important applications like state estimation and controller design. An efficient implementation is presented that avoids the Sherman-Morrison-Woodbury formula and instead uses a low-rank Cholesky factor ADI method as the base algorithm to compute low-rank factors of general-form Riccati equations. Sample MATLAB-based implementations of the proposed algorithm are also provided. An approach for automatically and efficiently generating ADI shifts is discussed. Numerical examples solving several Riccati equations of orders ranging from to accurately and efficiently are presented, demonstrating the effectiveness of the proposed algorithm.
Cite
@article{arxiv.2604.06556,
title = {$LDL^\top$ Factorization-based Generalized Low-rank ADI Algorithm for Solving Large-scale Algebraic Riccati Equations},
author = {Umair Zulfiqar},
journal= {arXiv preprint arXiv:2604.06556},
year = {2026}
}