Using $LDL^T$ factorizations in Newton's method for solving general large-scale algebraic Riccati equations
<|reference_start|>Using $LDL^T$ factorizations in Newton's method for solving general large-scale algebraic Riccati equations: 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.<|reference_end|>
@article{saak2024using,
    title={Using $LDL^{T}$ factorizations in Newton's method for solving general
  large-scale algebraic Riccati equations},
    author={Jens Saak, Steffen W. R. Werner},
    journal={Electron. Trans. Numer. Anal., 62:95-118, 2024},
    year={2024},
    doi={10.1553/etna_vol62s95},
    archivePrefix={arXiv},
    eprint={2402.06844},
    primaryClass={math.NA cs.NA math.OC}
}