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

Abstract

We present a new analysis of the stability of the first and second barycentric formulae for interpolation at the Chebyshev points of the second kind. Our theory shows that the second formula is more stable than previously thought and our experiments confirm its stability in practice. We also extend our current understanding regarding the accuracy problems of the first barycentric formula.

The effects of rounding errors in the nodes on barycentric interpolation

“…Finally, we note that [12] also presents bounds for Step III applicable in Salzer's case. On the one hand, some bounds in that article involve the Lipschitz constant, on the other hand, they do not have O(log n) factors.…”

“…and picked 2000 equally spaced points in each of these intervals. The first formula was evaluated at these points as in subsection 3.1 of [12].…”

Moore: Interval Arithmetic in C++20

“…among all t ∈ [−1, 1], for a given vector x of nodes, and I use interval arithmetic to find such minimizers and validate them. The first experiment timed the evaluation of the Lebesgue function for 257 Chebyshev nodes of the second kind [12], with interval weights, at a million points t. I obtained the normalized times in Table 1 (the time for the Moore library was taken as the unit.) This table indicates that for the arithmetic operations involved in the evaluation of the Lebesgue function (1) the Moore library is more efficient that the boost, Filib and libieeep1788 libraries.…”

A Robust and Scalable Implementation of the Parks-McClellan Algorithm for Designing FIR Filters

“…This second reference shows how the modified Lagrange formula (5) is backward stable in general, whereas (6) is shown to be forward stable for vectors of points x ∈ R n+2 having a small Lebesgue constant 3 . Despite a less favorable numerical behavior of (6) in general, for sets of points having a small Lebesgue constant, like Chebyshev nodes, using the second barycentric formula is preferable [8,33]. More recently, [34] argues that (6) is backward stable if the relevant Lebesgue constant associated with the interpolation vector x is small.…”