Asymptotic condition numbers for linear ordinary differential equations: the generic real case

TL;DR

This paper investigates the long-term behavior of condition numbers for real linear ODEs, extending previous complex case studies to the generic real case with detailed asymptotic analysis.

Contribution

It provides a comprehensive analysis of the asymptotic behavior of condition numbers for real linear ODEs, building on prior complex case results with new depth.

Findings

Asymptotic behaviors of condition numbers are characterized in the real case.

The analysis reveals differences between real and complex cases in long-time behavior.

Provides insights into stability and sensitivity of solutions to real linear ODEs.

Abstract

The paper \cite{M0} studied, for a \emph{complex} linear ordinary differential equation $y^\prime(t)=Ay(t)$, the long-time propagation to the solution $y(t)$ of a perturbation of the initial value. By measuring the perturbations with relative errors, this paper introduced a directional pointwise condition number, defined for a specific initial value and for a specific direction of perturbation of this initial value, and a pointwise condition number, defined for a specific initial value and the worst-case scenario for the direction of perturbation. The asymptotic (long-time) behaviors of these two condition numbers were determined. The present paper analyzes such asymptotic behaviors in depth, for a \emph{real} linear ordinary differential equation in a generic case.