A Low-rank ADI Algorithm for Solving Large-scale Non-symmetric Algebraic Riccati Equations
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911622800343040 |
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| author | Zulfiqar, Umair |
| author_facet | Zulfiqar, Umair |
| contents | This paper considers large-scale nonsymmetric continuous-time algebraic Riccati equations (NAREs) that admit low-rank solutions. Low-rank alternating direction implicit (ADI) methods have proven to be an efficient approach for solving several matrix equations, including Lyapunov equations, Sylvester equations, and symmetric Riccati equations. Although a low-rank algorithm for the Sylvester equation has been used as an inner loop in computing low-rank solutions of NAREs, no low-rank ADI algorithm currently exists for NAREs themselves. This paper fills this gap by developing a low-rank ADI algorithm for large-scale NAREs that admit a low-rank solution. Since Lyapunov equations, Sylvester equations, and symmetric Riccati equations are special cases of the NARE, the existing low-rank ADI methods in the literature are special cases of the more general low-rank ADI method proposed here. An automatic and computationally efficient method for shift generation is also discussed, and a subspace-accelerated projection approach is presented to generate shifts for subsequent iterations without user intervention. Once initialized with arbitrary shifts, the proposed algorithm solves large-scale NAREs autonomously, generating its own shifts. Numerical results are presented using benchmark example of order $10^6$, demonstrating the computational efficiency and accuracy of the proposed algorithm. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_23208 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Low-rank ADI Algorithm for Solving Large-scale Non-symmetric Algebraic Riccati Equations Zulfiqar, Umair Numerical Analysis Systems and Control This paper considers large-scale nonsymmetric continuous-time algebraic Riccati equations (NAREs) that admit low-rank solutions. Low-rank alternating direction implicit (ADI) methods have proven to be an efficient approach for solving several matrix equations, including Lyapunov equations, Sylvester equations, and symmetric Riccati equations. Although a low-rank algorithm for the Sylvester equation has been used as an inner loop in computing low-rank solutions of NAREs, no low-rank ADI algorithm currently exists for NAREs themselves. This paper fills this gap by developing a low-rank ADI algorithm for large-scale NAREs that admit a low-rank solution. Since Lyapunov equations, Sylvester equations, and symmetric Riccati equations are special cases of the NARE, the existing low-rank ADI methods in the literature are special cases of the more general low-rank ADI method proposed here. An automatic and computationally efficient method for shift generation is also discussed, and a subspace-accelerated projection approach is presented to generate shifts for subsequent iterations without user intervention. Once initialized with arbitrary shifts, the proposed algorithm solves large-scale NAREs autonomously, generating its own shifts. Numerical results are presented using benchmark example of order $10^6$, demonstrating the computational efficiency and accuracy of the proposed algorithm. |
| title | A Low-rank ADI Algorithm for Solving Large-scale Non-symmetric Algebraic Riccati Equations |
| topic | Numerical Analysis Systems and Control |
| url | https://arxiv.org/abs/2604.23208 |