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On the restriction to unitarity for rational approximations to the exponential function

Tobias Jawecki

TL;DR

The paper analyzes rational approximations to $e^{i\omega x}$ on $x\in[-1,1]$ with degree $\le n$, comparing unconstrained Chebyshev approximants in $\mathcal{R}_n$ to unitary-constrained best approximants in $\mathcal{U}_n$ (where $|r(i x)|=1$). It proves that the unitary best approximant is not a Chebyshev approximant for $\omega\in(0,(n+1)\pi)$, and establishes the sharp inequality $E_{n,\omega}^u/2 \le E_{n,\omega}^c < E_{n,\omega}^u$ for $\omega>0$, with distinct behavior in the large-frequency regime $\omega\ge (n+1)\pi$ where $E_{n,\omega}^c=1$ and $E_{n,\omega}^u=2$. Both errors share the same leading small-$\omega$ asymptotics $E_{n,\omega}^u=E_{n,\omega}^c=\frac{2^{-2n}(n!)^2}{(2n)!(2n+1)!}\,\omega^{2n+1}+O(\omega^{2n+2})$, indicating the unitarity restriction is not severely detrimental near $\omega=0$. The results have practical implications for numerical methods related to the matrix exponential and unitary-preserving time integration, showing that enforcing unitarity incurs at most a factor of two penalty in worst-case error and becomes asymptotically negligible as the domain shrinks to the origin.

Abstract

In the present work we consider rational best approximations to the exponential function that minimize a uniform error on a subset of the imaginary axis. Namely, Chebyshev approximation and unitary best approximation where the latter is subject to further restriction to unitarity, i.e., requiring that the imaginary axis is mapped to the unit circle. We show that Chebyshev approximants are not unitary, and consequently, distinct to unitary best approximants. However, unitary best approximation attains at most twice the error of Chebyshev approximation, and thus, the restriction to unitarity is not a severe restriction in a practical setting. Moreover, Chebyshev approximation and unitary best approximation attain the same asymptotic error as the underlying domain of approximation shrinks to the origin.

On the restriction to unitarity for rational approximations to the exponential function

TL;DR

The paper analyzes rational approximations to on with degree , comparing unconstrained Chebyshev approximants in to unitary-constrained best approximants in (where ). It proves that the unitary best approximant is not a Chebyshev approximant for , and establishes the sharp inequality for , with distinct behavior in the large-frequency regime where and . Both errors share the same leading small- asymptotics , indicating the unitarity restriction is not severely detrimental near . The results have practical implications for numerical methods related to the matrix exponential and unitary-preserving time integration, showing that enforcing unitarity incurs at most a factor of two penalty in worst-case error and becomes asymptotically negligible as the domain shrinks to the origin.

Abstract

In the present work we consider rational best approximations to the exponential function that minimize a uniform error on a subset of the imaginary axis. Namely, Chebyshev approximation and unitary best approximation where the latter is subject to further restriction to unitarity, i.e., requiring that the imaginary axis is mapped to the unit circle. We show that Chebyshev approximants are not unitary, and consequently, distinct to unitary best approximants. However, unitary best approximation attains at most twice the error of Chebyshev approximation, and thus, the restriction to unitarity is not a severe restriction in a practical setting. Moreover, Chebyshev approximation and unitary best approximation attain the same asymptotic error as the underlying domain of approximation shrinks to the origin.

Paper Structure

This paper contains 3 sections, 4 theorems, 44 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Proof of Theorem \ref{['thm:errinequals']}

Key Result

Theorem 1

Provided $\omega >0$, the uniform errors $E_{n,\omega}^u$ and $E_{n,\omega}^c$ of the unitary best approximation and the Chebyshev approximation, respectively, satisfy This includes the case $\omega\geq (n+1)\pi$ for which the Chebyshev approximant corresponds to $r^c \equiv 0$ with $E_{n,\omega}^c=1$, and all $r^u\in\mathcal{U}_n$ are unitary best approximants with $E_{n,\omega}^u=2$. Moreover,

Figures (1)

  • Figure 1: The errors of the unitary best approximation $E_{n,\omega}^u$ (solid line) and the Chebyshev approximation $E_{n,\omega}^c$ (dashed line) for degree $n=0$ plotted over $\omega$.

Theorems & Definitions (11)

  • Theorem 1
  • proof
  • Proposition 2
  • Remark 3
  • proof : Proof of Proposition \ref{['prop:asymerr']}
  • Example 4
  • Proposition 5
  • proof : Proof of Proposition \ref{['prop:ruisnotrc']}
  • Proposition 6
  • proof
  • ...and 1 more