An extension of the Floater–Hormann family of barycentric rational interpolants

Klein, Georges

doi:10.1090/S0025-5718-2013-02688-9

An extension of the Floater–Hormann family of barycentric rational interpolants
HTML articles powered by AMS MathViewer

by Georges Klein PDF

Math. Comp. 82 (2013), 2273-2292 Request permission

Abstract:

The barycentric rational interpolants introduced by Floater and Hormann in $2007$ are “blends” of polynomial interpolants of fixed degree $d$. In some cases these rational functions achieve approximation of much higher quality than the classical polynomial interpolants, which, e.g., are ill-conditioned and lead to Runge’s phenomenon if the interpolation nodes are equispaced. For such nodes, however, the condition of Floater–Hormann interpolation deteriorates exponentially with increasing $d$. In this paper, an extension of the Floater–Hormann family with improved condition at equispaced nodes is presented and investigated. The efficiency of its applications such as the approximation of derivatives, integrals and antiderivatives of functions is compared to the corresponding results recently obtained with the original family of rational interpolants.

References

Bradley K. Alpert, High-order quadratures for integral operators with singular kernels, J. Comput. Appl. Math. 60 (1995), no. 3, 367–378. MR 1357317, DOI 10.1016/0377-0427(94)00040-8
R. Baltensperger and J.-P. Berrut, The errors in calculating the pseudospectral differentiation matrices for Čebyšev-Gauss-Lobatto points, Comput. Math. Appl. 37 (1999), no. 1, 41–48. MR 1664260, DOI 10.1016/S0898-1221(98)00240-5
Richard Baltensperger, Jean-Paul Berrut, and Benjamin Noël, Exponential convergence of a linear rational interpolant between transformed Chebyshev points, Math. Comp. 68 (1999), no. 227, 1109–1120. MR 1642809, DOI 10.1090/S0025-5718-99-01070-4
J.-P. Berrut, Rational functions for guaranteed and experimentally well-conditioned global interpolation, Comput. Math. Appl. 15 (1988), no. 1, 1–16. MR 937561, DOI 10.1016/0898-1221(88)90067-3
Jean-Paul Berrut and Richard Baltensperger, The linear rational pseudospectral method for boundary value problems, BIT 41 (2001), no. 5, suppl., 868–879. BIT 40th Anniversary Meeting. MR 2058847, DOI 10.1023/A:1021916623407
Jean-Paul Berrut, Michael S. Floater, and Georges Klein, Convergence rates of derivatives of a family of barycentric rational interpolants, Appl. Numer. Math. 61 (2011), no. 9, 989–1000. MR 2806141, DOI 10.1016/j.apnum.2011.05.001
J.-P. Berrut and H. D. Mittelmann, Lebesgue constant minimizing linear rational interpolation of continuous functions over the interval, Comput. Math. Appl. 33 (1997), no. 6, 77–86. MR 1449215, DOI 10.1016/S0898-1221(97)00034-5
Jean-Paul Berrut and Lloyd N. Trefethen, Barycentric Lagrange interpolation, SIAM Rev. 46 (2004), no. 3, 501–517. MR 2115059, DOI 10.1137/S0036144502417715
Len Bos, Stefano De Marchi, and Kai Hormann, On the Lebesgue constant of Berrut’s rational interpolant at equidistant nodes, J. Comput. Appl. Math. 236 (2011), no. 4, 504–510. MR 2843034, DOI 10.1016/j.cam.2011.04.004
Len Bos, Stefano De Marchi, Kai Hormann, and Georges Klein, On the Lebesgue constant of barycentric rational interpolation at equidistant nodes, Numer. Math. 121 (2012), no. 3, 461–471. MR 2929075, DOI 10.1007/s00211-011-0442-8
L. Brutman, On the Lebesgue function for polynomial interpolation, SIAM J. Numer. Anal. 15 (1978), no. 4, 694–704. MR 510554, DOI 10.1137/0715046
Michael S. Floater and Kai Hormann, Barycentric rational interpolation with no poles and high rates of approximation, Numer. Math. 107 (2007), no. 2, 315–331. MR 2328849, DOI 10.1007/s00211-007-0093-y
Bengt Fornberg, A practical guide to pseudospectral methods, Cambridge Monographs on Applied and Computational Mathematics, vol. 1, Cambridge University Press, Cambridge, 1996. MR 1386891, DOI 10.1017/CBO9780511626357
K. Hormann, G. Klein, S. De Marchi, Barycentric rational interpolation at quasi-equidistant nodes, Dolomites Res. Notes Approx. 5 (2012), 1–6.
Daan Huybrechs, On the Fourier extension of nonperiodic functions, SIAM J. Numer. Anal. 47 (2010), no. 6, 4326–4355. MR 2585189, DOI 10.1137/090752456
Georges Klein and Jean-Paul Berrut, Linear barycentric rational quadrature, BIT 52 (2012), no. 2, 407–424. MR 2931356, DOI 10.1007/s10543-011-0357-x
Georges Klein and Jean-Paul Berrut, Linear rational finite differences from derivatives of barycentric rational interpolants, SIAM J. Numer. Anal. 50 (2012), no. 2, 643–656. MR 2914279, DOI 10.1137/110827156
Rodrigo B. Platte, Lloyd N. Trefethen, and Arno B. J. Kuijlaars, Impossibility of fast stable approximation of analytic functions from equispaced samples, SIAM Rev. 53 (2011), no. 2, 308–318. MR 2806639, DOI 10.1137/090774707
M. J. D. Powell, Approximation theory and methods, Cambridge University Press, Cambridge-New York, 1981. MR 604014
Claus Schneider and Wilhelm Werner, Some new aspects of rational interpolation, Math. Comp. 47 (1986), no. 175, 285–299. MR 842136, DOI 10.1090/S0025-5718-1986-0842136-8
A. Schönhage, Fehlerfortpflanzung bei Interpolation, Numer. Math. 3 (1961), 62–71. MR 119389, DOI 10.1007/BF01386001
G. D. Smith, Numerical solution of partial differential equations, 3rd ed., Oxford Applied Mathematics and Computing Science Series, The Clarendon Press, Oxford University Press, New York, 1985. Finite difference methods. MR 827497
T. W. Tee, An Adaptive Rational Spectral Method for Differential Equations with Rapidly Varying Solutions, PhD thesis, University of Oxford, 2006.
T. W. Tee and Lloyd N. Trefethen, A rational spectral collocation method with adaptively transformed Chebyshev grid points, SIAM J. Sci. Comput. 28 (2006), no. 5, 1798–1811. MR 2272189, DOI 10.1137/050641296
L. N. Trefethen and others, Chebfun Version 4.0, The Chebfun Development Team, 2011, http://www.maths.ox.ac.uk/chebfun/.
L. N. Trefethen and J. A. C. Weideman, Two results on polynomial interpolation in equally spaced points, J. Approx. Theory 65 (1991), no. 3, 247–260. MR 1109406, DOI 10.1016/0021-9045(91)90090-W