BIT Numerical Mathematics

, Volume 52, Issue 2, pp 407–424 | Cite as

Linear barycentric rational quadrature



Linear interpolation schemes very naturally lead to quadrature rules. Introduced in the eighties, linear barycentric rational interpolation has recently experienced a boost with the presentation of new weights by Floater and Hormann. The corresponding interpolants converge in principle with arbitrary high order of precision. In the present paper we employ them to construct two linear rational quadrature rules. The weights of the first are obtained through the direct numerical integration of the Lagrange fundamental rational functions; the other rule, based on the solution of a simple boundary value problem, yields an approximation of an antiderivative of the integrand. The convergence order in the first case is shown to be one unit larger than that of the interpolation, under some restrictions. We demonstrate the efficiency of both approaches with numerical tests.


Rational interpolation Barycentric form Quadrature 

Mathematics Subject Classification (2000)

65D05 65D32 41A05 41A20 41A25 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Baltensperger, R., Berrut, J.P.: The errors in calculating the pseudospectral differentiation matrices for Čebyšev–Gauss–Lobatto points. Comput. Math. Appl. 37, 41–48 (1999). Corrigenda in Comput. Math. Appl. 38, 119 (1999) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Baltensperger, R., Berrut, J.P., Noël, B.: Exponential convergence of a linear rational interpolant between transformed Chebyshev points. Math. Comput. 68, 1109–1120 (1999) MATHCrossRefGoogle Scholar
  3. 3.
    Battles, Z., Trefethen, L.N.: An extension of Matlab to continuous functions and operators. SIAM J. Sci. Comput. 25, 1743–1770 (2004) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Berrut, J.P.: Rational functions for guaranteed and experimentally well-conditioned global interpolation. Comput. Math. Appl. 15, 1–16 (1988) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Berrut, J.P., Baltensperger, R.: The linear rational pseudospectral method for boundary value problems. BIT Numer. Math. 41, 868–879 (2001) MathSciNetCrossRefGoogle Scholar
  6. 6.
    Berrut, J.P., Baltensperger, R., Mittelmann, H.D.: Recent developments in barycentric rational interpolation. In: de Bruin, M.G., Mache, D.H., Szabados, J. (eds.) Trends and Applications in Constructive Approximation. International Series of Numerical Mathematics, vol. 151, pp. 27–51. Birkhäuser, Basel (2005) CrossRefGoogle Scholar
  7. 7.
    Berrut, J.P., Floater, M.S., Klein, G.: Convergence rates of derivatives of a family of barycentric rational interpolants. Appl. Numer. Math. 61, 989–1000 (2011) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Berrut, J.P., Mittelmann, H.D.: Lebesgue constant minimizing linear rational interpolation of continuous functions over the interval. Comput. Math. Appl. 33, 77–86 (1997) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Berrut, J.P., Mittelmann, H.D.: Matrices for the direct determination of the barycentric weights of rational interpolation. J. Comput. Appl. Math. 78, 355–370 (1997) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Berrut, J.P., Trefethen, L.N.: Barycentric Lagrange interpolation. SIAM Rev. 46, 501–517 (2004) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Bos, L., De Marchi, S., Hormann, K., Klein, G.: On the Lebesgue constant of barycentric rational interpolation at equidistant nodes. Technical Report 2011-2, Department of Mathematics, University of Fribourg, May (2011) Google Scholar
  12. 12.
    Brass, H.: Quadraturverfahren. Studia Mathematica, vol. 3. Vandenhoeck & Ruprecht, Göttingen (1977) MATHGoogle Scholar
  13. 13.
    Davis, P.J., Rabinowitz, P.: Methods of Numerical Integration, 2nd edn. Computer Science and Applied Mathematics. Academic Press, Orlando (1984) MATHGoogle Scholar
  14. 14.
    Floater, M.S., Hormann, K.: Barycentric rational interpolation with no poles and high rates of approximation. Numer. Math. 107, 315–331 (2007) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Gautschi, W.: Orthogonal Polynomials. Computation and Approximation. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford (2004) MATHGoogle Scholar
  16. 16.
    Glaser, A., Liu, X., Rokhlin, V.: A fast algorithm for the calculation of the roots of special functions. SIAM J. Sci. Comput. 29, 1420–1438 (2007) MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Henrici, P.: Essentials of Numerical Analysis with Pocket Calculator Demonstrations. Wiley, New York (1982) MATHGoogle Scholar
  18. 18.
    Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985) MATHGoogle Scholar
  19. 19.
    Huybrechs, D.: Stable high-order quadrature rules with equidistant points. J. Comput. Appl. Math. 231, 933–947 (2009) MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Isaacson, E., Keller, H.B.: Analysis of Numerical Methods. Wiley, New York (1966) MATHGoogle Scholar
  21. 21.
    Krommer, A.R., Ueberhuber, C.W.: Computational Integration. SIAM, Philadelphia (1998) MATHCrossRefGoogle Scholar
  22. 22.
    Platte, R.B., Trefethen, L.N., Kuijlaars, A.B.J.: Impossibility of fast stable approximation of analytic functions from equispaced samples. SIAM Rev 53, 308–318 (2011) MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Pólya, G.: Über die Konvergenz von Quadraturverfahren. Math. Z. 37, 264–286 (1933) MathSciNetCrossRefGoogle Scholar
  24. 24.
    Rivlin, T.: An Introduction to the Approximation of Functions. Blaisdell Publishing Company, Waltham (1969) MATHGoogle Scholar
  25. 25.
    Schneider, C., Werner, W.: Some new aspects of rational interpolation. Math. Comput. 47, 285–299 (1986) MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Schwarz, H.R., Köckler, N.: Numerische Mathematik, 5th ed. Teubner, Stuttgart (2004) MATHGoogle Scholar
  27. 27.
    Trefethen, L.N.: Spectral Methods in Matlab. SIAM, Philadelphia (2000) MATHCrossRefGoogle Scholar
  28. 28.
    Trefethen, L.N.: Computing numerically with functions instead of numbers. Math. Comput. Sci. 1, 9–19 (2007) MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Trefethen, L.N.: Is Gauss quadrature better than Clenshaw–Curtis? SIAM Rev. 50, 67–87 (2008) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2011

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of FribourgPérollesSwitzerland

Personalised recommendations