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

Jens Saak, Steffen W. R. Werner

TL;DR

This work advances the numerical solution of general continuous-time algebraic Riccati equations for large-scale sparse systems by reformulating the Newton-Kleinman iteration with indefinite symmetric $LDL^{\mathsf{T}}$ low-rank factorizations of the stabilizing solution. It establishes convergence results for prominent CARE realizations with positive or negative definite quadratic terms, introduces an exact line-search and an inexact Newton variant, and extends the method to non-invertible $E$ via projected Riccati equations. Numerical experiments across dense and large-scale sparse problems demonstrate high accuracy and robustness of the proposed approach, though performance can be problem-dependent relative to RADI-based methods. Overall, the paper provides a principled, scalable framework for solving general CAREs that broadens applicability to practical large-scale control and model-reduction tasks.

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.

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

TL;DR

This work advances the numerical solution of general continuous-time algebraic Riccati equations for large-scale sparse systems by reformulating the Newton-Kleinman iteration with indefinite symmetric low-rank factorizations of the stabilizing solution. It establishes convergence results for prominent CARE realizations with positive or negative definite quadratic terms, introduces an exact line-search and an inexact Newton variant, and extends the method to non-invertible via projected Riccati equations. Numerical experiments across dense and large-scale sparse problems demonstrate high accuracy and robustness of the proposed approach, though performance can be problem-dependent relative to RADI-based methods. Overall, the paper provides a principled, scalable framework for solving general CAREs that broadens applicability to practical large-scale control and model-reduction tasks.

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.
Paper Structure (19 sections, 1 theorem, 55 equations, 1 figure, 10 tables, 1 algorithm)

This paper contains 19 sections, 1 theorem, 55 equations, 1 figure, 10 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Example equations from the literature
  3. Linear-quadratic regulator problems
  4. Linear-quadratic-Gaussian control and unstable model order reduction
  5. H-infinity control and robust model order reduction
  6. Passivity, contractivity and spectral factorizations
  7. Low-rank inexact Newton-Kleinman iteration with line search
  8. Derivation of the low-rank Newton-Kleinman scheme
  9. An equivalent reformulation via low-rank updates
  10. Convergence results
  11. Inexact Newton with exact line search
  12. Non-invertible E matrices and projected Riccati equations
  13. Numerical experiments
  14. Experimental setup
  15. Convergence behavior for indefinite terms
  16. ...and 4 more sections

Key Result

Theorem 1

Assume eqn:riccati has a unique stabilizing solution $X_{\ast}$, let $K_{0}$ be a feedback matrix such that the eigenvalues of $\lambda E - (A - B K_{0})$ lie in the open left complex half-plane and let either $R > 0$ or $R < 0$ be true. Then, for the exact solutions $X_{k} = L_{k} D_{k} L_{k}^{\mke

Figures (1)

  • Figure 1: Convergence of NEWTON and RI for all example equations with the $\texttt{rail}_{\texttt{(6)}}$ data set: We can see that in all examples where it was applicable RI obtains its final approximation significantly faster than NEWTON. This comes from the use of RADI as underlying solver. However, the shown implicit residual computed by RI is not accurate as shown in \ref{['tab:sparse_res']}.

Theorems & Definitions (4)

  • Remark 1
  • Remark 2
  • Theorem 1
  • proof