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

Stefan Güttel, Shuai Shao

TL;DR

The authors develop a time-uniform, real-pole rational approximation framework for the matrix exponential acting on vectors over a positive time interval. They extend Andersson's classical real-pole results to a finite interval by concentrating poles at $-nq$ and derive an asymptotically optimal choice of $q$ that minimizes the time-uniform error, with practical guidance for selecting poles in practice. The work presents two numerical evaluation strategies—Rational Chebyshev interpolation and shift-and-invert Arnoldi—that leverage the shared-pole structure, and validates the approach through near-optimality experiments and a large-scale matrix-exponential application. Overall, the method enables efficient, scalable evaluation of $\exp(-tA)b$ across many time points by solving a small set of shifted linear systems and offers a pathway for integrating real-pole rational approximants into exponential integrators.

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.

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

TL;DR

The authors develop a time-uniform, real-pole rational approximation framework for the matrix exponential acting on vectors over a positive time interval. They extend Andersson's classical real-pole results to a finite interval by concentrating poles at and derive an asymptotically optimal choice of that minimizes the time-uniform error, with practical guidance for selecting poles in practice. The work presents two numerical evaluation strategies—Rational Chebyshev interpolation and shift-and-invert Arnoldi—that leverage the shared-pole structure, and validates the approach through near-optimality experiments and a large-scale matrix-exponential application. Overall, the method enables efficient, scalable evaluation of across many time points by solving a small set of shifted linear systems and offers a pathway for integrating real-pole rational approximants into exponential integrators.

Abstract

We propose an asympotically optimal choice of shared concentrated real poles of a family of rational approximants of time-dependent exponential functions for and in a positive time interval . 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 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.

Paper Structure

This paper contains 9 sections, 3 theorems, 38 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Optimal shared concentrated real poles
  3. Numerical evaluation of the approximants
  4. Rational Chebyshev interpolation
  5. Shift-and-invert Arnoldi method
  6. Numerical experiments
  7. Verifying near-optimality
  8. An application to matrix exponential function
  9. Conclusions

Key Result

theorem 2.1

For any $q > 0$, where and $\tilde{q}$ is the root in $\Im(z) \geq 0$ of the equation that has the smallest positive real part. The minimum of $\tilde{H}$ is $\sqrt{2}-1$, which is attained at $1/\sqrt{2}.$

Figures (4)

  • Figure 2.1: The plot of $\tilde{H}$ as a function of $q$.
  • Figure 4.1: Time-uniform error when using our choice of concentrated poles (marked by black circles) and numerically computed optimal choice of poles (marked by red circles), for different time ratios $\tau = t_{\max}/t_{\min}$. The degree is $n = 20$ in all cases.
  • Figure 4.2: Time-uniform error using our choice of poles and the numerically computed optimal poles for the ranges $T_0$, $T_1$, $T_2$, $T_3$, $T_4$, for degree $n$ from $1$ to $40$.
  • Figure 4.3: Plot of $\Vert \widehat{\bm{u}}(t) - \bm{u}(t)\Vert_2$ using rational Chebyshev interpolation (blue), shift-and-invert Arnoldi (red), and their approximate error bound (black), for the time points $T = \texttt{logspace(-3,0,41)}$ and degrees $n = 8, 14, 20, 26$.

Theorems & Definitions (6)

  • theorem 2.1
  • proof
  • theorem 2.2
  • proof
  • theorem 2.3
  • proof