Structured condition numbers and small sample condition estimation of symmetric algebraic Riccati equations

TL;DR

This paper analyzes the sensitivity of symmetric algebraic Riccati equations to structured perturbations, deriving improved condition number bounds and proposing a statistical estimation method based on small sample techniques.

Contribution

It introduces new upper bounds for structured condition numbers by leveraging symmetry, enhancing previous perturbation analysis of symmetric algebraic Riccati equations.

Findings

Condition numbers accurately estimate solution changes due to data perturbations.

Proposed statistical algorithm effectively estimates condition numbers in practice.

Numerical experiments confirm the improved accuracy of the new bounds.

Abstract

This paper is devoted to a structured perturbation analysis of the symmetric algebraic Riccati equations by exploiting the symmetry structure. Based on the analysis, the upper bounds for the structured normwise, mixed and componentwise condition numbers are derived. Due to the exploitation of the symmetry structure, our results are improvements of the previous work on the perturbation analysis and condition numbers of the symmetric algebraic Riccati equations. Our preliminary numerical experiments demonstrate that our condition numbers provide accurate estimates for the change in the solution caused by the perturbations on the data. Moreover, by applying the small sample condition estimation method, we propose a statistical algorithm for practically estimating the condition numbers of the symmetric algebraic Riccati equations.