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

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.

This manuscript is the second half of the first version of arXiv 2507.08762. It includes in Section 5.5 a numerical example from the first version of arXiv 2507.08752