$LDL^\top$ Factorization-based Generalized Low-rank ADI Algorithm for Solving Large-scale Algebraic Riccati Equations
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915937620328448 |
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| author | Zulfiqar, Umair |
| author_facet | Zulfiqar, Umair |
| contents | 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 $LDL^\top$ 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 $10^6$ to $10^7$ accurately and efficiently are presented, demonstrating the effectiveness of the proposed algorithm. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_06556 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | $LDL^\top$ Factorization-based Generalized Low-rank ADI Algorithm for Solving Large-scale Algebraic Riccati Equations Zulfiqar, Umair Numerical Analysis Systems and Control 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 $LDL^\top$ 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 $10^6$ to $10^7$ accurately and efficiently are presented, demonstrating the effectiveness of the proposed algorithm. |
| title | $LDL^\top$ Factorization-based Generalized Low-rank ADI Algorithm for Solving Large-scale Algebraic Riccati Equations |
| topic | Numerical Analysis Systems and Control |
| url | https://arxiv.org/abs/2604.06556 |