Skip to main content
SpringerLink
Log in

Modified Stieltjes polynomials and Gauss–Kronrod quadrature rules

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

Modified Stieltjes polynomials are defined and used to construct suboptimal extensions of Gaussian rules with one or two degrees less of polynomial exactness than the corresponding Kronrod extension. We prove that, for wide classes of weight functions and a sufficiently large number of nodes, the extended quadratures have positive weights and simple nodes on the interval \([-1,1]\). The classes of weight functions considered complement those for which the Gauss–Kronrod rule is known to exist. Also, strong asymptotic representations on the whole interval \([-1,1]\) are given for the modified Stieltjes polynomials, which prove that they behave asymptotically as orthogonal polynomials. Finally, we provide some numerical examples.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Baratella, P.: Un’estensione ottimale della formula di quadratura di Radau. Rend. Sem. Mat. Univ. Politec. Torino 37, 147–158 (1979)

    MathSciNet  MATH  Google Scholar 

  2. Bary, N.K.: A Treatise on Trigonometric Series. Pergamon Press, New York (1964)

    MATH  Google Scholar 

  3. Begumisa, A., Robinson, I.: Suboptimal Kronrod extension formulae for numerical quadrature. Numer. Math. 58, 807–818 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  4. Berezin, I.S., Zhidkov, N.P.: Computing Methods, vol. I. Pergamon Press, Oxford (1965)

    MATH  Google Scholar 

  5. Bojanov, B., Petrova, G.: Quadrature formulas for Fourier coefficients. J. Comput. Appl. Math. 231, 378–391 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bultheel, A., Daruis, L., González-Vera, P.: A connection between quadrature formulas on the unit circle and the interval \([-1,1]\). J. Comput. Appl. Math. 132, 1–14 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  7. Calvetti, D., Golub, G.H., Gragg, W.B., Reichel, L.: Computation of Gauss–Kronrod quadrature rules. Math. Comput. 69, 1035–1052 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  8. de la Calle Ysern, B.: Optimal extension of the Szegő quadrature. IMA J. Numer. Anal. 35, 722–748 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. de la Calle Ysern, B., Lo\(\acute{{\rm p}}\)ez Lagomasino, G., Reichel, L.: Stieltjes-type polynomials on the unit circle. Math. Comput. 78, 969–997 (2009)

  10. de la Calle Ysern, B., Peherstorfer, F.: Ultraspherical Stieltjes polynomials and Gauss-K-ronrod quadrature behave nicely for \(\lambda <0\). SIAM J. Numer. Anal. 45, 770–786 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Djukić, DLj, Reichel, L., Spalević, M.M.: Truncated generalized averaged Gauss quadrature rules. J. Comput. Appl. Math. 308, 408–418 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Djukić, DLj, Reichel, L., Spalević, M.M., Tomanović, J.D.: Internality of the averaged Gaussian quadratures and their truncated variants with Bernstein–Szegő weight functions. Electron. Trans. Numer. Anal. 45, 405–419 (2016)

    MathSciNet  MATH  Google Scholar 

  13. Duren, P.L.: Theory of \(H^p\) Spaces. Dover, Mineola (2000)

    Google Scholar 

  14. Gautschi, W.: Gauss–Kronrod quadrature—a survey. In: Milovanović, G.V. (ed.) Numerical Methods and Approximation Theory III, pp. 39–66. NIS, Novi Sad (1988)

    Google Scholar 

  15. Gautschi, W.: Orthogonal Polynomials: Computation and Approximation. Oxford University Press, Oxford (2004)

    MATH  Google Scholar 

  16. Gautschi, W.: A historical note on Gauss–Kronrod quadrature. Numer. Math. 100, 483–484 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gautschi, W.: OPQ: A MATLAB suite of programs for generating orthogonal polynomials and related quadrature rules. http://www.cs.purdue.edu/archives/2001/wxg/codes

  18. Grenander, U., Szegő, G.: Toeplitz Forms and Their Applications. Chelsea Publishing Company, New York (1984)

    MATH  Google Scholar 

  19. Hermite, Ch., Stieltjes, T.J.: Correspondance d’Hermite et de Stieltjes. Prentice-Hall, Englewood Cliffs (1966)

    Google Scholar 

  20. Jagels, C., Reichel, L., Tang, T.: Generalized averaged Szegő quadrature rules. J. Comput. Appl. Math. 311, 645–654 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  21. Jones, W.B., Njåstad, O., Thron, W.J.: Moment theory, orthogonal polynomials, quadrature, and continued fractions associated with the unit circle. Bull. Lond. Math. Soc. 21, 113–152 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kahaner, D.K., Waldvogel, J., Fullerton, L.W.: Addition of points to Gauss–Laguerre quadrature formulas. SIAM J. Sci. Stat. Comput. 5, 42–55 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kronrod, A.S.: Integration with control of accuracy. Soviet Phys. Dokl. 9, 17–19 (1964)

    MathSciNet  MATH  Google Scholar 

  24. Kronrod, A.S.: Nodes and Weights for Quadrature Formulae. Sixteen-Place Tables. Consultants Bureau, New York (1965)

    MATH  Google Scholar 

  25. Laurie, D.P.: Anti-Gaussian quadrature formulas. Math. Comput. 65, 739–747 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  26. Laurie, D.P.: Calculation of Gauss–Kronrod quadrature rules. Math. Comput. 66, 1133–1145 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  27. Laurie, D.P.: Calculation of Radau–Kronrod and Lobatto–Kronrod quadrature formulas. Numer. Algorithms 45, 139–152 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  28. Monegato, G.: Stieltjes polynomials and related quadrature rules. SIAM Rev. 24, 137–158 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  29. Monegato, G.: An overview of the computational aspects of Kronrod quadrature rules. Numer. Algorithms 26, 173–196 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  30. Notaris, S.: Gauss–Kronrod quadrature formulae—a survey of fifty years of research. Electron. Trans. Numer. Anal. 45, 371–404 (2016)

    MathSciNet  MATH  Google Scholar 

  31. Patterson, T.N.L.: The optimum addition of points to quadrature formulae. Math. Comput. 22, 847–856 (1968) Errata. ibid. 23, 892 (1969)

  32. Patterson, T.N.L.: An algorithm for generating interpolatory quadrature rules of the highest degree of precision with preassigned nodes for general weight functions. ACM Trans. Math. Softw. 15, 123–136 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  33. Patterson, T.N.L.: Algorithm 672: generation of interpolatory quadrature rules of the highest degree of precision with preassigned nodes for general weight functions. ACM Trans. Math. Softw. 15, 137–143 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  34. Patterson, T.N.L.: Modified optimal quadrature extensions. Numer. Math. 64, 511–520 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  35. Peherstorfer, F.: On the asymptotic behaviour of functions of second kind and Stieltjes polynomials, and on Gauss–Kronrod quadrature formulas. J. Approx. Theory 70, 156–190 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  36. Peherstorfer, F.: Stieltjes polynomials and functions of second kind. J. Comput. Appl. Math. 65, 319–338 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  37. Peherstorfer, F., Petras, K.: Ultraspherical Gauss–Kronrod quadrature is not possible for \(\lambda > 3\). SIAM J. Numer. Anal. 37, 927–948 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  38. Peherstorfer, F., Petras, K.: Stieltjes polynomials and Gauss–Kronrod quadrature for Jacobi weight functions. Numer. Math. 95, 689–706 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  39. Piessens, R., de Doncker-Kapenga, E., Überhuber, C.W., Kahaner, D.K.: QUADPACK: a subroutine package for automatic integration. Springer Series in Computational Mathematics, vol. 1. Springer, Berlin (1983)

    MATH  Google Scholar 

  40. Reichel, L., Rodriguez, G., Tang, T.: New block quadrature rules for the approximation of matrix functions. Linear Algebra Appl. 502, 299–326 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  41. Reichel, L., Spalević, M.M., Tang, T.: Generalized averaged Gauss quadrature rules for the approximation of matrix functionals. BIT 56, 1045–1067 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  42. Spalević, M.M.: On generalized averaged Gaussian formulas. Math. Comput. 76, 1483–1492 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  43. Spalević, M.M.: A note on generalized averaged Gaussian formulas. Numer. Algorithms 46, 253–264 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  44. Spalević, M.M.: On generalized averaged Gaussian formulas. II. Math. Comput. 86, 1877–1885 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  45. Szegő, G.: Über gewisse orthogonale Polynome, die zu einer oszillierenden Belegungsfunktion gehören. Math. Ann. 110, 501–513 (1935)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors wish to thank the anonymous referees and the associate editor for their valuable suggestions that helped to improve a previous version of this paper.

Author information

Authors and Affiliations

  1. Departamento de Matemáticas del Área Industrial, E.T.S. de Ingenieros Industriales, Universidad Politécnica de Madrid, José G. Abascal 2, 28006, Madrid, Spain

    B. de la Calle Ysern

  2. Department of Mathematics, Faculty of Mechanical Engineering, University of Beograd, Kraljice Marije 16, 11120, Belgrade 35, Serbia

    M. M. Spalević

Authors
  1. B. de la Calle Ysern
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. M. Spalević
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to B. de la Calle Ysern.

Additional information

The work of the authors was partially supported by the Serbian Ministry of Education, Science and Technological Development through Research Project #174002: “Methods of Numerical and Nonlinear Analysis with Applications”. The first author received support from Dirección General de Investigación under grant MTM2014-54053-P and from Universidad Politécnica de Madrid under Grant GI-1505440113.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

de la Calle Ysern, B., Spalević, M.M. Modified Stieltjes polynomials and Gauss–Kronrod quadrature rules. Numer. Math. 138, 1–35 (2018). https://doi.org/10.1007/s00211-017-0901-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-017-0901-y

Mathematics Subject Classification

Search

Navigation