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

TL;DR

This paper introduces an asymptotically optimal method for selecting concentrated real poles in rational approximations of the matrix exponential, improving uniform-in-time accuracy for various time ranges.

Contribution

It extends classical approximation results by proposing a new pole selection strategy that is near-optimal for time-dependent exponential functions.

Findings

Numerical experiments confirm near-optimal approximation accuracy.

The method performs well across different time intervals and approximation degrees.

Application to initial-value problems demonstrates practical effectiveness.

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.