Computation of Zolotarev rational functions

TL;DR

The paper addresses the problem of computing Zolotarev rational functions $r_n^*$ of degree $n$ that minimize the ratio of their size on two disjoint sets in the complex plane. The approach reduces Z3 to Z4 via a mathematical equivalence and solves Z4 by AAA-Lawson approximations to a sign function on the two sets, followed by a straightforward transformation to obtain $r_n^*$. The authors also introduce two practical algorithmic enhancements to AAA for sign problems and to Lawson iterations, and demonstrate robust, fast computation across diverse geometries with clear connections to potential theory. They discuss substantial applications to ADI-type matrix iterations and spectral slicing, illustrating the practical impact of numerically accessible Zolotarev functions and the potential for new quadrature rules.

Abstract

An algorithm is presented to compute Zolotarev rational functions, that is, rational functions $r_n^*$ of a given degree that are as small as possible on one set $E\subseteq\complex\cup\{\infty\}$ relative to their size on another set $F\subseteq\complex\cup\{\infty\}$ (the third Zolotarev problem). Along the way we also approximate the sign function relative to $E$ and $F$ (the fourth Zolotarev problem).

Figures (7)

• Figure 1: \newlabelfig10 Six examples of degree $n=12$ Zolotarev rational functions $r_n^*$ defined by connected sets $E$ (on the left in each image) and $F$ (on the right). The solutions plotted are near-optimal but not exactly so. The Zolotarev function $r_n^*$ satisfies $|r_n^*(z)| \approx \min_{t\in F} |r_n^*(t)| = 1$ for $z$ on the boundary of $F$, and contours show levels $\log_{10} |r_n^*(z)| = -1, -2, \dots$ between the two domains. Blue dots mark the zeros of $r_n^*$ and red dots mark the poles. Black circles and dots mark zeros and poles of the sign function $\hat{r}_n^{}$ to be introduced in section $2$. The minimum values $\sigma_n = \|r_n^*\|_E^{}$ are listed in the titles. Details of the geometries are given in the appendix.
• Figure 1: \newlabelfig40 Zeros ( black circles) and poles ( black dots) of the functions $\hat{r}_n^{}$ of Problem Z4 for the examples of Figures \ref{['fig1']}c and \ref{['fig2']}c, together with contour lines showing distances to $+1$ (red) and $-1$ (green). Specifically, on the left, the red contours show $\log_{10} |r_n^{}(z)-1| = -1, -2, \dots, -6$ and the green contours show $\log_{10} |r_n^{}(z)+1| = -1, -2, \dots, -6$; the pattern on the right is the same except with levels $-1/2, -1, -3/2,\dots, -3$. Images like this show that although Problem Z4 is posed just on the domains $E$ and $F$, $\hat{r}_n^{}(z)$ defines an approximation to a sign function throughout the complex plane, with its poles and zeros delineating an approximate branch cut. Computation of $\hat{r}_n^{}$ to solve Problem Z4 is the first step in our solution of Problem Z3.
• Figure 1: \newlabelfig50 AAA convergence for degree $n=13$ approximation $\hat{r}_n^{} \approx \hbox{\rm sign}_{E/F}$ with $E = [-1.5,-0.5]$ and $F = [ 0.5,1.5]$, with each interval approximated by $200$ Chebyshev points. On the left, the original AAA iteration reveals difficulties typical of many $\hbox{\rm sign}_{E/F}$ examples. The " blending of singular values" adjustment described in the text produces the better results on the right. (This would then further improve to an equioscillatory error curve with the AAA-Lawson iteration with damping, as illustrated for a more difficult problem in Figure $6$.) The sawtoothed convergence curve reflects poles falling between sample points at every other iteration, a common phenomenon in approximation of even or odd functions on real domains.
• Figure 2: \newlabelfig20 The same as in Figure \ref{['fig1']}, now for four problems where $E$ and/or $F$ are disconnected.
• Figure 2: \newlabelfig60 First row: AAA convergence for degree $15$ approximation of $\hbox{\rm sign}_{E/F}$ with $E = [-1.8,-0.2]$ and $F = [ 0.5,1.5]$, with each interval approximated by $100$ Chebyshev points. On the left, the result of AAA followed by $150$ steps of AAA-Lawson iteration. On the right, the same but with AAA-Lawson applied with damping factor $\delta = 0.8$. Second row: a similar comparison for degree $n=4$ rational approximation of $\hbox{\rm ReLU}(x) = \max(x,0)$, sampled in $200$ Chebyshev points, with damping factor $\delta = 0.5$.

Theorems & Definitions (1)

• Theorem 1