arXiv subjects

Kelong Zhao

Publications and source records attributed to Kelong Zhao.

2 recordsLinked to original sources

arXiv2024

Barycentric Interpolation Based on Equilibrium Potential

We present a novel barycentric interpolation algorithm designed for analytic functions $f\in\mathcal{A}(E)$ defined on the complex plane. The algorithm, which encompasses both polynomial and rational interpolation, is tailored to handle singularities near $E$. Our method is applicable to regions $E$ bounded by piecewise smooth Jordan curves, and it imposes no connectivity restrictions on the region. The key feature of our approach lies in efficiently computing discrete points via the numerical solution of Symm's integral equation, enabling the construction of polynomial or rational barycentric interpolants. Furthermore, our method provides relevant parameters for the equilibrium potential, such as Robin's constant, which can be used to estimate convergence rates. Numerical experiments demonstrate the convergence rate achieved by our method in comparison to the theoretical convergence rate.

math.NA↗

arXiv2024

Polynomial and rational interpolation: potential, barycentric weights, and Lebesgue constants

In this paper, we focus on barycentric weights and Lebesgue constants for Lagrange interpolation of arbitrary node distributions on $[-1,1]$. The following three main works are included: estimates of upper and lower bounds on the barycentric weights are given in terms of the logarithmic potential function; for interpolation of non-equilibrium potentials, lower bounds with exponentially growing parts of Lebesgue constants are given; and for interpolation consistent with equilibrium potentials, non-exponentially growing upper bounds on their Lebesgue constants are given. Based on the work in this paper, we can discuss the behavior of the Lebesgue constant and the existence of exponential convergence in a unified manner in the framework of potential theory.

math.NA↗