Numerical Stability of Barycentric Lagrange Interpolation

Abstract

This paper presents a refined study on the numerical stability of barycentric Lagrange interpolation constructed using Legendre Gauss Lobatto (LGL) nodes, which are the zeros of $(1-x^2) P^{\prime}_{n-1}(x)$, where $P_{n-1}(x)$ is the Legendre polynomial of degree $(n-1)$. Unlike prior work focused on Chebyshev nodes, our formulation provides a detailed error analysis tailored to this specific node set. We derive the barycentric weights from the structure of the underlying node generating polynomial and establish new interpolation error bounds supported by asymptotic estimates of barycentric weights. Extensive numerical comparisons between classical Lagrange and barycentric interpolation across various challenging test functions demonstrate that the barycentric formulation maintains machine level accuracy and robust stability even at high degrees (up to $n = 550$), while classical Lagrange interpolation suffers from catastrophic numerical error. These results underline the practical superiority of barycentric interpolation at LGL nodes and provide a rigorous theoretical foundation for its use in high-accuracy applications.