Concentrated real-pole uniform-in-time approximation of the matrix exponential

Stefan Güttel; Shuai Shao

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arXiv · 2511.21858

Concentrated real-pole uniform-in-time approximation of the matrix exponential

Stefan Güttel·Shuai Shao

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Abstract

We propose an asympotically optimal choice of shared concentrated real poles of a family of rational approximants of time-dependent exponential functions $\exp(-tz)$ for $z \geq 0$ and $t$ in a positive time interval $T$. Our result extends a classical result by J.-E. Andersson [J. Approx. Theory, 32(2):85--95, 1981] on the asymptotic best rational approximation of $\exp(-z)$ with real poles. Numerical experiments demonstrate the near-optimality of our choice for various time ranges and for both small and large approximation degrees. An application of the uniform-in-time rational approximation using our proposed concentrated real poles to a linear constant-coefficient initial-value problem is also discussed.

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Stefan Güttel, Shuai Shao. 2025-11-26. Concentrated real-pole uniform-in-time approximation of the matrix exponential. https://arxiv.org/abs/2511.21858

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