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., v. 24) and has practical consequences for computation with rational functions.

The stability of barycentric interpolation at the Chebyshev points of the second kind

“…When f k is not small, if the rounding errors are very small and if t is very close to x k then α n,k (x), δ n,k (t) and δ n,k (x k ) are very small and Lemma 2, through equations (21) and (22), allows us to replace β n,k,i (t,x) by 1 and κ n,k,i (t,x) by γ i f i /γ k f k in (20). We can then neglect the second-order terms (t − x k ) (x i − x i ), and write…”

“…The literature does not pay due attention to the case of simplified weights in (4) and t ∈ [−1, 1] that we consider. For instance, [21] is concerned with extrapolation and not interpolation whereas [11] considers generic nodes. Unfortunately, as we show in section 3 below, the situation is less favourable for the first barycentric formula.…”

Fractional pseudospectral integration matrices for solving fractional differential, integral, and integro-differential equations

“…(22) are scale-invariant, and thus, avoid any problems of underflow and overflow [51]. Furthermore, they are numerically stable for evaluating the polynomial interpolants at points τ ∈ [−1, +1] through any set of interpolating points with a small Lebesgue constant [52,53]. However, direct calculation of the barycentric weights using Eq.…”

Computing Complex Singularities of Differential Equations with Chebfun

“…We can see in Figure 4.1 for example, that a polynomial can be extrapolated throughout the Bernstein ellipse, but outside of that it is useless as an approximation of the underlying function, so if we cannot stably evaluate our rational approximant outside of the Bernstein ellipse we are doing no better than we can with a polynomial! More detail can be found in a short article written with Trefethen and Gonnet [22]. ratinterp has since been corrected to evaluate the rational function as the quotient of two polynomials, each evaluated by a numerically stable version of polynomial barycentric interpolation formula.…”