Concentrated real-pole uniform-in-time approximation of the matrix exponential
This provides a practical, near-optimal rational approximation method for matrix exponential in time-dependent problems, though it is an incremental extension of existing theory.
The paper proposes an asymptotically optimal choice of concentrated real poles for rational approximants of exp(-tz) that achieve uniform-in-time approximation, extending Andersson's classical result. Numerical experiments confirm near-optimality across various time ranges and approximation degrees.
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.