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The Loewner framework applied to Zolotarev sign and ratio problems

Athanasios C. Antoulas, Ion Victor Gosea, Charles Poussot-Vassal

TL;DR

This work investigates numerical rational approximation for Zolotarev sign ($Z4$) and ratio ($Z3$) problems using data-driven methods. It systematically compares the Loewner Framework (LF) with AAA and AAA-Lawson across diverse Zolotarev topologies, showing that LF is fast, non-iterative, and often yields highly accurate, symmetry-preserving approximants, sometimes surpassing near-optimal AAA-Lawson results at higher degrees. The study combines analytic benchmarks for the symmetric two-circle topology with extensive numerical experiments, demonstrating LF’s robustness, favorable runtime, and ability to recover the true pole/zero structure. A MATLAB package is provided to reproduce the results and facilitate application to similar data-driven rational-approximation tasks.

Abstract

In this work, we propose a numerical study concerning the approximation of functions associated with the 3rd and 4th Zolotarev problems. We compare various methods, in particular the Loewner framework, the standard AAA algorithm, and recently-proposed extensions of AAA (namely, the sign and Lawson variants). We show that the Loewner framework is fast and reliable, and provides approximants with a high level of accuracy. When the approximants are of a higher degree, Loewner approximants are often more accurate than near-optimal ones computed with AAA-Lawson. Last but not least, the Loewner framework is a direct method for which the running time is typically lower than that of the iterative AAA-Lawson variants. Moreover, for the latter, the running time increases substantially with the degree of the approximant, whereas for the Loewner method, it remains constant. These claims are supported by an extensive numerical treatment.

The Loewner framework applied to Zolotarev sign and ratio problems

TL;DR

This work investigates numerical rational approximation for Zolotarev sign () and ratio () problems using data-driven methods. It systematically compares the Loewner Framework (LF) with AAA and AAA-Lawson across diverse Zolotarev topologies, showing that LF is fast, non-iterative, and often yields highly accurate, symmetry-preserving approximants, sometimes surpassing near-optimal AAA-Lawson results at higher degrees. The study combines analytic benchmarks for the symmetric two-circle topology with extensive numerical experiments, demonstrating LF’s robustness, favorable runtime, and ability to recover the true pole/zero structure. A MATLAB package is provided to reproduce the results and facilitate application to similar data-driven rational-approximation tasks.

Abstract

In this work, we propose a numerical study concerning the approximation of functions associated with the 3rd and 4th Zolotarev problems. We compare various methods, in particular the Loewner framework, the standard AAA algorithm, and recently-proposed extensions of AAA (namely, the sign and Lawson variants). We show that the Loewner framework is fast and reliable, and provides approximants with a high level of accuracy. When the approximants are of a higher degree, Loewner approximants are often more accurate than near-optimal ones computed with AAA-Lawson. Last but not least, the Loewner framework is a direct method for which the running time is typically lower than that of the iterative AAA-Lawson variants. Moreover, for the latter, the running time increases substantially with the degree of the approximant, whereas for the Loewner method, it remains constant. These claims are supported by an extensive numerical treatment.

Paper Structure

This paper contains 27 sections, 3 theorems, 19 equations, 15 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The main approximation methods
  3. The Loewner framework
  4. The AAA algorithm
  5. Minimax algorithms and Lawson step extensions
  6. The Zolotarev problems
  7. Reminder and general procedure
  8. The case of two symmetric circles: configuration '1a' in trefethen2024computation
  9. Experimental setup and results
  10. Comments
  11. Preliminary conclusions
  12. More complete and accurate approximations
  13. Numerical experiments
  14. Matlab code package, functions and code sample
  15. Numerical test cases
  16. ...and 12 more sections

Key Result

Lemma 2.1

If $k=q$ the matrix pencil $({{{\mathbb L}_s}},\,{\mathbb L})$ is regular, then one can directly compute the interpolation function as below:

Figures (15)

  • Figure 1: Case '1a', $r=8$: zoom on zeros (left) and poles (right) of the approximants.
  • Figure 2: Case '1a', with $r=26$: different approximation methods: LF (top left), AAA (top right), AAA with damping (bottom left), and AAA-Lawson with damping and 200 iterations (bottom right). Each frame shows: the Zolotarev rational functions of degree $r$ for Z3 (${\mathbf h}_3$) and Z4 (${\mathbf h}_4$) defined by connected sets $E$ (on the left) and $F$ (on the right); the Z3 function approximate contours levels $\log_{10} |{\mathbf h}_3(z)|= 0,-1,-2,\cdots,-30$ between the two domains; black circles and dots respectively mark zeros and poles of the sign function ${\mathbf h}_4$; blue and red dots respectively denote the zeros and poles of ${\mathbf h}_3$. The minimum values $\sigma_r$ and computation time are given in the title, as well as the options of each AAA method, and give the number of singularities targeted (and obtained).
  • Figure 3: Case '1a': computation time and accuracy comparison for different approximation settings; we compare the standard LF with different AAA and AAA-Lawson settings.
  • Figure 4: Case '1b'. Top: same description as \ref{['fig:intro1']}. Bottom: same description as \ref{['fig:intro1_time']}.
  • Figure 5: Case '1c'. Top: same description as \ref{['fig:intro1']}. Bottom: same description as \ref{['fig:intro1_time']}.
  • ...and 10 more figures

Theorems & Definitions (4)

  • Lemma 2.1
  • Lemma 2.2
  • Remark 2.1
  • Theorem 3.1