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arXiv · 2402.06844

Using $LDL^{T}$ factorizations in Newton's method for solving general large-scale algebraic Riccati equations

Jens Saak·Steffen W. R. Werner
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Abstract

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.

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Jens Saak, Steffen W. R. Werner. 2024-02-10. Using $LDL^{T}$ factorizations in Newton's method for solving general large-scale algebraic Riccati equations. https://doi.org/10.1553/etna_vol62s95

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