# The stability of extended Floater–Hormann interpolants

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## Abstract

We present a new analysis of the stability of extended Floater–Hormann interpolants, in which both noisy data and rounding errors are considered. Contrary to what is claimed in the current literature, we show that the Lebesgue constant of these interpolants can grow exponentially with the parameters that define them, and we emphasize the importance of using the proper interpretation of the Lebesgue constant in order to estimate correctly the effects of noise and rounding errors. We also present a simple condition that implies the backward instability of the barycentric formula used to implement extended interpolants. Our experiments show that extended interpolants mentioned in the literature satisfy this condition and, therefore, the formula used to implement them is not backward stable. Finally, we explain that the extrapolation step is a significant source of numerical instability for extended interpolants based on extrapolation.

## Mathematics Subject Classification

65D05 65G30 41A20## Supplementary material

## References

- 1.Bochkanov, S.: AlgLib, an open source library for numerical computation. http://www.alglib.net
- 2.Berrut, J.-P., Klein, G.: Recent advances in linear barycentric rational interpolation. J. Comput. Appl. Math.
**259**(PART A), 95–107 (2014)MathSciNetCrossRefzbMATHGoogle Scholar - 3.Berrut, J.-P., Trefethen, L.N.: Barycentric lagrange interpolation. SIAM J. Numer. Anal.
**46**(3), 501–517 (2004)MathSciNetzbMATHGoogle Scholar - 4.Bos, L., De Marchi, S., Hormann, K., Klein, G.: On the Lebesgue constant of barycentric rational interpolation at equidistant nodes. Numer. Math.
**121**(3), 461–471 (2012)MathSciNetCrossRefzbMATHGoogle Scholar - 5.Brutman, L.: Lebesgue functions for polynomial interpolation—a survey. Ann. Numer. Math.
**4**, 111–127 (1997)MathSciNetzbMATHGoogle Scholar - 6.de Camargo, A.P.: On the numerical stability of Floater–Hormann’s rational interpolant. Numer. Algorithms
**72**(1), 131–152 (2016)MathSciNetCrossRefzbMATHGoogle Scholar - 7.Floater, M.S., Hormann, K.: Barycentric rational interpolation with no poles and high rates of approximation. Numer. Math.
**107**(2), 315–331 (2007)MathSciNetCrossRefzbMATHGoogle Scholar - 8.Fousse, L., Hanrot, G., Lefèvre, V., Pélissier, P., Zimmermann, P.: MPFR: a multiple-precision binary floating-point library with correct rounding. ACM Trans. Math. Softw.
**33**(2), 13 (2007)MathSciNetCrossRefGoogle Scholar - 9.Higham, N.J.: The numerical stability of barycentric Lagrange interpolation. IMA J. Numer. Anal.
**24**, 547–556 (2004)MathSciNetCrossRefzbMATHGoogle Scholar - 10.Higham, N.J.: Accuracy and Stability of Numerical Algorithms, 2nd edn. SIAM, Philadelphia (2002)CrossRefzbMATHGoogle Scholar
- 11.Klein, G.: An extension of the Floater–Hormann family of barycentric rational interpolants. Math. Compt.
**82**(284), 2273–2292 (2013)MathSciNetCrossRefzbMATHGoogle Scholar - 12.Klein, G., Berrut, J.-P.: Linear rational finite differences from derivatives of barycentric rational interpolants. SIAM J. Numer. Anal.
**50**(2), 643–656 (2012)MathSciNetCrossRefzbMATHGoogle Scholar - 13.Mascarenhas, W.F.: The stability of barycentric interpolation at the Chebyshev points of the second kind. Numer. Math. (2014). doi: 10.1007/s00211-014-0612-6
- 14.Mascarenhas, W.F., de Camargo, A.P.: On the backward stability of the second barycentric formula for interpolation. Dolomites Res. Notes Approx.
**7**, 1–12 (2014)CrossRefGoogle Scholar - 15.Mascarenhas, W.F., de Camargo, A.P.: The effects of rounding errors in the nodes on barycentric interpolation. (2013). arXiv:1309.7970v2 [math.NA] 4 Jul 2014, submitted to Numer. Math
- 16.Trefethen, L.N., Weideman, J.A.C.: Two results on polynomial interpolation in equally spaced points. J. Approx. Theory
**65**, 247–260 (1991)MathSciNetCrossRefzbMATHGoogle Scholar