Volume 62, pp. 95-118, 2024.

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

Jens Saak and Steffen W. R. Werner

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.

Full Text (PDF) [449 KB], BibTeX

Key words

Riccati equation, Newton's method, large-scale sparse matrices, low-rank factorization, indefinite terms

AMS subject classifications

15A24, 49M15, 65F45, 65H10, 93A15

Links to the cited ETNA articles

[1]	Vol. 1 (1993), pp. 33-48 Gregory Ammar, Peter Benner, and Volker Mehrmann: A multishift algorithm for the numerical solution of algebraic Riccati equations
[20]	Vol. 43 (2014-2015), pp. 142-162 Peter Benner, Patrick Kürschner, and Jens Saak: Self-generating and efficient shift parameters in ADI methods for large Lyapunov and Sylvester equations
[37]	Vol. 33 (2008-2009), pp. 53-62 M. Heyouni and K. Jbilou: An extended block Arnoldi algorithm for large-scale solutions of the continuous-time algebraic Riccati equation
[66]	Vol. 30 (2008), pp. 187-202 Tatjana Stykel: Low-rank iterative methods for projected generalized Lyapunov equations

< Back