Using $LDL^{T}$ factorizations in Newton's method for solving general large-scale algebraic Riccati equations
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916397448167424 |
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| author | Saak, Jens Werner, Steffen W. R. |
| author_facet | Saak, Jens Werner, Steffen W. R. |
| contents | Continuous-time algebraic Riccati equations can be found in many disciplines in different forms. In the case of small-scale dense coefficient matrices, stabilizing solutions can be computed to all possible formulations of the Riccati equation. This is not the case when it comes to large-scale sparse coefficient matrices. In this paper, we provide a reformulation of the Newton-Kleinman iteration scheme for continuous-time algebraic Riccati equations using indefinite symmetric low-rank factorizations. This allows the application of the method to the case of general large-scale sparse coefficient matrices. We provide convergence results for several prominent realizations of the equation and show in numerical examples the effectiveness of the approach. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_06844 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Using $LDL^{T}$ factorizations in Newton's method for solving general large-scale algebraic Riccati equations Saak, Jens Werner, Steffen W. R. Numerical Analysis Optimization and Control 15A24, 49M15, 65F45, 65H10, 93A15 Continuous-time algebraic Riccati equations can be found in many disciplines in different forms. In the case of small-scale dense coefficient matrices, stabilizing solutions can be computed to all possible formulations of the Riccati equation. This is not the case when it comes to large-scale sparse coefficient matrices. In this paper, we provide a reformulation of the Newton-Kleinman iteration scheme for continuous-time algebraic Riccati equations using indefinite symmetric low-rank factorizations. This allows the application of the method to the case of general large-scale sparse coefficient matrices. We provide convergence results for several prominent realizations of the equation and show in numerical examples the effectiveness of the approach. |
| title | Using $LDL^{T}$ factorizations in Newton's method for solving general large-scale algebraic Riccati equations |
| topic | Numerical Analysis Optimization and Control 15A24, 49M15, 65F45, 65H10, 93A15 |
| url | https://arxiv.org/abs/2402.06844 |