b-d-Lawson: a method for the interpolation constrained rational minimax approximation

TL;DR

The paper addresses interpolation-constrained rational minimax approximation by introducing b-d-Lawson, a dual-based Lawson method that leverages a barycentric parameterization to naturally enforce interpolation conditions and improve numerical stability. It derives a max–min dual formulation and provides practical computation of the dual function via a small generalized eigenproblem, paving the way for an efficient iterative scheme. The authors establish weak and (under mild conditions) strong duality, discuss complementary slackness and extreme-point structure, and present a b-d-Lawson algorithm with adaptive support-point selection. Numerical experiments across discrete and interpolatory scenarios demonstrate competitive accuracy and clear dual convergence behavior, including challenging cases like the Zolotarev sign problem. They also acknowledge limitations and outline future work on convergence for general interpolation levels and constraint enhancements.

Abstract

In this paper, we propose a novel dual-based Lawson's method, termed b-d-Lawson, designed for addressing the rational minimax approximation under specific interpolation conditions. The b-d-Lawson approach incorporates two pivotal components that have been recently gained prominence in the realm of the rational approximations: the barycentric representation of the rational function and the dual framework for tackling minimax approximation challenges. The employment of barycentric formulae enables a streamlined parameterization of the rational function, ensuring natural satisfaction of interpolation conditions while mitigating numerical instability typically associated with Vandermonde basis matrices when monomial bases are utilized. This enhances both the accuracy and computational stability of the method. To address the bi-level min-max structure, the dual framework effectively transforms the challenge into a max-min dual problem, thereby facilitating the efficient application of Lawson's iteration. The integration of this dual perspective is crucial for optimizing the approximation process. We will discuss several applications of interpolation-constrained rational minimax approximation and illustrate numerical results to evaluate the performance of the b-d-Lawson method.

Figures (5)

• Figure 7.1: The top row: error curves from b-d-Lawson($40$), d-Lawson($40$) and AAA($40$) of the approximants of type $(6,6)$, respectively. Note that there are $11= 2n +2-\ell$ extreme points in the error curve from b-d-Lawson($40$), while $14= 2n +2$ extreme points for d-Lawson($40$) and AAA(40). The bottom row: (bottom-left) the function $f(x)$ in \ref{['eq:f2']}; (bottom-middle) the dual objective function values and maximum errors versus the iteration $k$ of b-d-Lawson; (bottom-right) the dual objective function values and maximum errors versus the iteration $k$ of d-Lawson.
• Figure 7.2: Error curves from b-d-Lawson($40$), d-Lawson($40$) and AAA($40$) of the approximants of type $(8,8)$ for Example \ref{['eg7:lne0']}. There are $18= 2n +2$ extreme points for d-Lawson($40$) and AAA(40), whereas $15= 2n +2-\ell$ extreme points for b-d-Lawson($40$).
• Figure 7.3: The right three subfigures are error curves from b-d-Lawson($40$), d-Lawson($40$) and AAA($40$) of $\xi\in \mathscr{R}_{(6)}$ in approximating the function \ref{['eq:f3']}, respectively.
• Figure 7.4: Approximation of the Riemann zeta function $\zeta(z)$: (left) the image of the computed approximant $\xi\in \mathscr{R}_{(40)}$ of the sampled nodes in $z = 0.5 + {\tt i} t, t\in [0, 50]$; (middle) errors associated with the sampled nodes in $z = 0.5 + {\tt i} t, t\in [-50, 50]$; (right) the phase portrait of $\xi(z)\approx \zeta(z)$ in a striped region.
• Figure 7.5: The (1,1)-subfigure: the sets $\mathbf{E}$ and $\mathbf{F}$; the (1,2)-subfigure: errors associated with the samples from b-d-Lawson($40$); the (1,3)-subfigure: errors associated with the samples from AAA without the option 'sign'; the (1,4)-subfigure: errors associated with the samples from AAA-Lawson(40) with the option 'sign'; the (2,2)-subfigure: the sets $\mathbf{E}$ and $\mathbf{F}$ with two interpolation nodes; the (2,3)-subfigure: errors associated with the samples from b-d-Lawson($40$) subject to two interpolation conditions. For each method, the circle (in black) with the radius $e(\xi)$ is also plotted for $\xi\in \mathscr{R}_{(15)}$.

Theorems & Definitions (21)

• Theorem 2.1
• Theorem 2.2
• proof
• Remark 2.1
• Lemma 4.1
• proof
• Proposition 4.1
• proof
• Remark 4.1
• Theorem 5.1