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

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 $z\ge 0$ 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.