Asymptotic condition numbers for linear ordinary differential equations
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913131262902272 |
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| author | Maset, Stefano |
| author_facet | Maset, Stefano |
| contents | We are interested in the relative
conditioning of the problem $y_0\mapsto \mathrm{e}^{tA}y_0$, i.e., the
relative conditioning of the action of the matrix exponential $\mathrm{e}%
^{tA}$ on a vector with respect to perturbations of this vector. The present paper is a qualitative study of the long-time behavior of this
conditioning. In other words, we are interested in studying the propagation to the solution $y(t)$ of perturbations of the initial value for a linear ordinary differential equation $y^\prime(t)=Ay(t)$, by measuring these perturbations with relative errors. We introduce three condition numbers: the first considers a specific initial value and a specific direction of perturbation; the second considers a specific initial value and the worst case by varying the direction of perturbation; and the third considers the worst case by varying both the initial value
and the direction of perturbation. The long-time behaviors of
these three condition numbers are studied. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_08762 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotic condition numbers for linear ordinary differential equations Maset, Stefano Numerical Analysis We are interested in the relative conditioning of the problem $y_0\mapsto \mathrm{e}^{tA}y_0$, i.e., the relative conditioning of the action of the matrix exponential $\mathrm{e}% ^{tA}$ on a vector with respect to perturbations of this vector. The present paper is a qualitative study of the long-time behavior of this conditioning. In other words, we are interested in studying the propagation to the solution $y(t)$ of perturbations of the initial value for a linear ordinary differential equation $y^\prime(t)=Ay(t)$, by measuring these perturbations with relative errors. We introduce three condition numbers: the first considers a specific initial value and a specific direction of perturbation; the second considers a specific initial value and the worst case by varying the direction of perturbation; and the third considers the worst case by varying both the initial value and the direction of perturbation. The long-time behaviors of these three condition numbers are studied. |
| title | Asymptotic condition numbers for linear ordinary differential equations |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2507.08762 |