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

TL;DR

This paper introduces a novel approach using LDL^T factorizations within Newton's method to efficiently solve large-scale sparse algebraic Riccati equations, extending applicability beyond small dense cases.

Contribution

It reformulates the Newton-Kleinman iteration with LDL^T factorizations for large sparse matrices, enabling effective solutions where traditional methods struggle.

Findings

Convergence results for various Riccati formulations.

Numerical evidence of the method's effectiveness.

Applicability to large-scale sparse problems.

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.