Kronrod and other quadratures

  • Giovanni Monegato
Part of the Contemporary Mathematicians book series (CM)


This section is devoted to Gautschi’s work on Kronrod quadratures and other special quadrature rules. For Gauss-type quadrature rules, see Section 15.


  1. 1.
    Pierre Barrucand. Intégration numérique, abscisse de Kronrod–Patterson et polynômes de Szegő. C. R. Acad. Sci. Paris Sér. A–B, 270:A336–A338, 1970.Google Scholar
  2. 2.
    Terje O. Espelid. Algorithm 868: globally doubly adaptive quadrature — reliable Matlab codes. ACM Trans. Math. Software, 33(3)21.1–21.21, 2007.Google Scholar
  3. 3.
    Pedro Gonnet. Increasing the reliability of adaptive quadrature using explicit interpolants. ACM Trans. Math. Software, 37(3):26.1–26.32, 2010.Google Scholar
  4. 4.
    D. K. Kahaner and G. Monegato. Nonexistence of extended Gauss–Laguerre and Gauss–Hermite quadrature rules with positive weights. Z. Angew. Math. Phys., 29(6):983–986, 1978.MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Aleksandr Semenovich Kronrod. Nodes and weights of quadrature formulas. Sixteen-place tables, “Nauka”, Moscow, 1964 (Russian). English translation, Consultants Bureau, New York, 1965.Google Scholar
  6. 6.
    Ravindra Kumar and M. K. Jain. Quadrature formulas for semi-infinite integrals. Math. Comp., 28(126):499–503, 1974. [Table errata: ibid. 56(193):407, 1991].Google Scholar
  7. 7.
    Giovanni Monegato. A note on extended Gaussian quadrature rules. Math. Comp., 30(136):812–817, 1976.Google Scholar
  8. 8.
    Giovanni Monegato. Positivity of the weights of extended Gauss–Legendre quadrature rules. Math. Comp., 32(141):243–245, 1978.Google Scholar
  9. 9.
    Giovanni Monegato. Stieltjes polynomials and related quadrature rules. SIAM Rev., 24(2):137–158, 1982.Google Scholar
  10. 10.
    G. Monegato and L. Scuderi. Quadrature rules for unbounded intervals and their application to integral equations. In Approximation and Computation 185–208, Springer Optim. Appl., 42, Springer, New York, 2011.Google Scholar
  11. 11.
    I. P. Mysovskih. A special case of quadrature formulae containing preassigned nodes (Russian). Vesci Akad. Navuk BSSR Ser. Fiz.-Tehn. Navuk, 4:125–127, 1964.MathSciNetGoogle Scholar
  12. 12.
    Sotirios E. Notaris. Gauss–Kronrod quadrature formulae for weight functions of Berstein–Szegő type. II. J. Comput. Appl. Math., 29(2):161–169, 1990.Google Scholar
  13. 13.
    Franz Peherstorfer. Weight functions admitting repeated positive Kronrod quadrature. BIT 30(1):145–151, 1990.MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Franz Peherstorfer. On Stieltjes polynomials and Gauss–Kronrod quadrature. Math. Comp., 55(192):649–664, 1990.Google Scholar
  15. 15.
    Franz Peherstorfer and Knut Petras. Ultraspherical Gauss–Kronrod quadrature is not possible for λ > 3. SIAM J. Numer. Anal., 37(3):927–948, 2000.Google Scholar
  16. 16.
    B. de la Calle Ysern and F. Peherstorfer. Ultraspherical Stieltjes polynomials and Gauss–Kronrod quadrature behave nicely for λ < 0. SIAM J. Numer. Anal., 45(2):770–786, 2007.MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Ju. S. Ramskiǐ. The improvement of a certain quadrature formula of Gauss type (Russian). Vyčisl. Prikl. Mat. (Kiev), 22:143–146, 1974.Google Scholar
  18. 18.
    L.F. Shampine. Vectorized adaptive quadrature in Matlab. J. Comput. Appl. Math., 211(2):131–140, 2008.Google Scholar
  19. 19.
    T. J. Stieltjes. Correspondance d’Hermite et de Stieltjes. II, 439–441. Gauthier-Villars, Paris, 1905.Google Scholar
  20. 20.
    G. Szegő. Über gewisse orthogonale Polynome, die zu einer oszillierenden Belegungsfunktion gehören. Math. Ann., 110:501–513, 1935.MathSciNetCrossRefGoogle Scholar
  21. 21.
    Walter Van Assche and Ingrid Vanherwegen. Quadrature formulas based on rational interpolation. Math. Comp., 61(204):765-783, 1993.Google Scholar

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© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Giovanni Monegato
    • 1
  1. 1.Dipartimento di MatematicaPolitecnico di TorinoTorinoItaly

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