On the numerical stability of the second barycentric formula for trigonometric interpolation in shifted equispaced points

We consider the numerical stability of the second barycentric formula for evaluation at points in [0, 2 pi] of trigonometric interpolants in an odd number of equispaced points in that interval. We show that, contrary to the prevailing view, which claims that this formula is always stable, it actually possesses a subtle instability that seems not to have been noticed before. This instability can be corrected by modifying the formula. We establish the forward stability of the resulting algorithm by using techniques that mimic those employed previously by Higham (2004, The numerical stability of barycentric Lagrange interpolation. IMA J. Numer. Anal., 24, 547-556) to analyse the second barycentric formula for polynomial interpolation. We show how these results can be extended to interpolation on other intervals of length-2p in many cases. Finally, we investigate the formula for an even number of points and show that, in addition to the instability that affects the odd-length formula, it possesses another instability that is more difficult to correct.