Long-time relative error analysis for linear ordinary differential equations with perturbed initial value
Abstract
We investigate the propagation of initial value perturbations along the solution of a linear ordinary differential equation . This propagation is analyzed using the relative error rather than the absolute error. Our focus is on the long-term behavior of this relative error, which differs significantly from that of the absolute error. The present paper is a practical sequel to the theoretical papers \cite{M1,M2} on the long-time behavior of the relative error: it includes applicative examples and important issues not addressed in \cite{M1,M2}. In addition, the present paper shows that understanding the long-term behavior provides insights into the growth of the relative error over all times, not just at large times. Therefore, it represents a crucial and fundamental aspect of the conditioning of linear ordinary differential equations, with applications in, for example, non-normal dynamics.
Cite
@article{arxiv.2507.08752,
title = {Long-time relative error analysis for linear ordinary differential equations with perturbed initial value},
author = {Stefano Maset},
journal= {arXiv preprint arXiv:2507.08752},
year = {2026}
}
Comments
This new version of the manuscript is reduced in length, with changes in Section 2 and 3. Moreover the example in 7.6 is moved to the arXiv manuscript "Asymptotic condition numbers for linear ordinary differential equations: the generic real case" and the example in 7.7 to the new version of arXive 2507.08762