A Subspace Shift Technique for Nonsymmetric Algebraic Riccati Equations

TL;DR

This paper introduces a novel subspace shift technique to improve convergence in solving nonsymmetric algebraic Riccati equations, especially near critical cases where traditional methods struggle due to ill-conditioning.

Contribution

The paper proposes a new subspace shift method that accelerates convergence for near-critical nonsymmetric algebraic Riccati equations, extending techniques beyond the exactly critical case.

Findings

The new method effectively accelerates convergence in near-critical cases.

Numerical experiments confirm the efficiency of the proposed technique.

Theoretical analysis supports the method's robustness and applicability.

Abstract

The worst situation in computing the minimal nonnegative solution of a nonsymmetric algebraic Riccati equation associated with an M-matrix occurs when the corresponding linearizing matrix has two very small eigenvalues, one with positive and one with negative real part. When both these eigenvalues are exactly zero, the problem is called critical or null recurrent. While in this case the problem is ill-conditioned and the convergence of the algorithms based on matrix iterations is slow, there exist some techniques to remove the singularity and transform the problem to a well-behaved one. Ill-conditioning and slow convergence appear also in close-to-critical problems, but when none of the eigenvalues is exactly zero the techniques used for the critical case cannot be applied. In this paper, we introduce a new method to accelerate the convergence properties of the iterations also in close-to-critical cases, by working on the invariant subspace associated with the problematic eigenvalues as a whole. We present a theoretical analysis and several numerical experiments which confirm the efficiency of the new method.