Asymptotic condition numbers for linear ordinary differential equations: the generic real case
Provides a theoretical characterization of sensitivity in real linear ODEs, which is incremental for researchers in numerical analysis and dynamical systems.
This paper extends the analysis of asymptotic condition numbers for linear ODEs from the complex case to the generic real case, determining the long-time behavior of directional and pointwise condition numbers under relative error perturbations.
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.