Long-time relative error analysis for linear ordinary differential equations with perturbed initial value

TL;DR

This paper analyzes how initial perturbations propagate in linear ODEs over long times using relative error, providing practical insights into the conditioning and growth of errors in such systems.

Contribution

It extends theoretical understanding of long-term relative error behavior in linear ODEs with practical examples and addresses issues not covered in previous work.

Findings

Long-term relative error behaves differently from absolute error.

Understanding relative error growth aids in analyzing non-normal dynamics.

Provides practical examples illustrating theoretical concepts.

Abstract

We investigate the propagation of initial value perturbations along the solution of a linear ordinary differential equation \( y'(t) = Ay(t) \). 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.