Stability of Barycentric Interpolation Formulas for Extrapolation

Abstract

The barycentric interpolation formula defines a stable algorithm for evaluation at points in $[-1,1]$ of polynomial interpolants through data on Chebyshev grids. Here it is shown that for evaluation at points in the complex plane outside $[-1,1]$, the algorithm becomes unstable and should be replaced by the alternative modified Lagrange or “first barycentric” formula dating to Jacobi in 1825. This difference in stability confirms the theory published by N. J. Higham in 2004 [IMA J. Numer. Anal., 24 (2004), pp. 547--556] and has practical consequences for computation with rational functions.