The Remainder Term of Gauss–Turán Quadratures for Analytic Functions

  • Miodrag M. Spalević
  • Miroslav S. Pranić
Part of the Springer Optimization and Its Applications book series (SOIA, volume 42)


We consider quadrature rules with multiple nodes over a finite interval, taken to be [-1, 1]
$$\int^{1}_{-1} f(t) w(t) dt = \sum \limits ^n_{v=1} \sum \limits ^{2s}_{i=0}\, {\rm A_i},v f^(i) \,(\tau v) \,+\, {\rm R}_n,s(f), $$
involving a positive weight function w, assumed integrable over [-1, 1].


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bernstein, S: Sur les polynomes orthogonaux relatifs à un segment fini. J. Math. Pure Appl. 9, 127–177 (1930)Google Scholar
  2. 2.
    Gautschi, W.: Orthogonal Polynomials: Computation and Approximation. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford (2004)MATHGoogle Scholar
  3. 3.
    Gautschi, W., Varga, R.S.: Error bounds for Gaussian quadrature of analytic functions. SIAM J. Numer. Anal. 20, 1170–1186 (1983)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Gautschi, W., Tychopoulos, E., Varga, R.S.: A note on the contour integral representation of the remainder term for a Gauss–Chebyshev quadrature rule. SIAM J. Numer. Anal. 27, 219–224 (1990)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Gori, L., Micchelli, C.A.: On weight functions which admit explicit Gauss–Turán quadrature formulas. Math. Comp. 65, 1567–1581 (1996)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products. Academic Press, New York (1980)MATHGoogle Scholar
  7. 7.
    Heine, E.: Anwendungen der Kugelfunctionen und der verwandten Functionen. 2nd ed. Reimer, Berlin (1881) [I. Theil: Mechanische Quadratur, 1–31]Google Scholar
  8. 8.
    Hermite, C.: Sur la formule d’interpolation de Lagrange. J. Reine Angew. Math. 84, 70–79 (1878) [Oeuvres III, 432–443]Google Scholar
  9. 9.
    Hunter, D.B.: Some error expansions for Gaussian quadrature. BIT 35, 64–82 (1995)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Kronrod, A.S.: Nodes and Weights for Quadrature Formulae. Sixteen Place Tables. Nauka, Moscow (1964) (Translation by Consultants Bureau, New York, 1965)Google Scholar
  11. 11.
    Kroó, A., Peherstorfer, F.: Asymptotic representation of L p-minimal polynimials, 1 < p < . Constr. Approx. 25, 29–39 (2007)Google Scholar
  12. 12.
    Laurie, D.P.: Anti-Gaussian quadrature formulas. Math. Comp. 65, 739–747 (1996)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Li, S.: Kronrod extension of Turán formula. Studia Sci. Math. Hungar. 29, 71–83 (1994)MATHMathSciNetGoogle Scholar
  14. 14.
    Milovanović, G.V.: Quadratures with multiple nodes, power orthogonality, and moment-preserving spline approximation. J. Comput. Appl. Math. 127, 267–286 (2001)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Milovanović, G.V., Spalević, M.M.: Error bounds for Gauss–Turán quadrature formulae of analytic functions. Math. Comp. 72, 1855–1872 (2003)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Milovanović, G.V., Spalević, M.M: Error analysis in some Gauss–Turán–Radau and Gauss–Turán-Lobatto quadratures for analytic functions. J. Comput. Appl. Math. 164–165, 569–586 (2004)Google Scholar
  17. 17.
    Milovanović, G.V., Spalević, M.M.: An error expansion for Gauss–Turán quadratures and L 1-estimates of the remainder term. BIT 45, 117–136 (2005)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Milovanović, G.V., Spalević, M.M.: Bounds of the error of Gauss–Turán-type quadratures. J. Comput. Appl. Math. 178, 333–346 (2005)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Milovanović, G.V., Spalević, M.M.: Gauss–Turán quadratures of Kronrod type for generalized Chebyshev weight functions. Calcolo 43, 171–195 (2006)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Milovanović, G.V., Spalević, M.M.: Quadrature rules with multiple nodes for evaluating integrals with strong singularities. J. Comput. Appl. Math. 189, 689–702 (2006)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Milovanović, G.V., Spalević, M.M.: On monotony of the error in Gauss–Turán quadratures for analytic functions. ANZIAM J. 48, 567–581 (2007)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Milovanović, G.V., Spalević, M.M., Cvetković, A.S.: Calculation of Gaussian type quadratures with multiple nodes. Math. Comput. Modelling 39, 325–347 (2004)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Milovanović, G.V, Spalević, M.M., Pranić, M.S.: Maximum of the modulus of kernels in Gauss–Turán quadratures. Math. Comp. 77, 985–994 (2008)Google Scholar
  24. 24.
    Milovanović, G.V., Spalević, M.M., Pranić, M.S.: Error estimates for Gauss–Turán quadratures and their Kronrod extensions. IMA J. Numer. Anal. 29, 486–507 (2009)MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Milovanović, G.V., Spalević, M.M., Pranić, M.S.: Bounds of the error of Gauss–Turán-type quadratures II. Appl. Numer. Math. 60, 1–9 (2010)MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Monegato, G.: Stieltjes polynomials and related quadrature rules. SIAM Rev. 24, 137–158 (1982)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Monegato, G.: An overview of the computational aspects of Kronrod quadrature rules. Numer. Algorithms 26, 173–196 (2001)MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Ossicini, A., Rosati, F.: Funzioni caratteristiche nelle formule di quadratura gaussiane con nodi multipli. Bull. Un. Mat. Ital. 11, 224–237 (1975)MathSciNetGoogle Scholar
  29. 29.
    Peherstorfer, F.: Gauss–Turán quadrature formulas: asymptotics of weights. SIAM J. Numer. Anal. 47, 2638–2659 (2009)MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Peherstorfer, F., Petras, K.: Ultraspherical Gauss–Kronrod quadrature is not possible for λ > 3. SIAM J. Numer. Anal 37, 927–948 (2000)MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Shi, Y.G.: Christoffel type functions for m-orthogonal polynomials. J. Approx. Theory 137, 57–88 (2005)MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Shi, Y.G., Xu, G.: Construction of σ-orthogonal polynomials and Gaussian quadrature formulas. Adv. Comput. Math. 27, 79–94 (2007)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Mechanical EngineeringUniversity of BeogradBeogradSerbia

Personalised recommendations