Skip to main content
SpringerLink
Log in

A note on generalized averaged Gaussian formulas for a class of weight functions

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

In the recent paper Notaris (Numer. Math., 142:129–147, 2019) it has been introduced a new and useful class of nonnegative measures for which the well-known Gauss–Kronrod quadrature formulae coincide with the generalized averaged Gaussian quadrature formulas. In such a case, the given generalized averaged Gaussian quadrature formulas are of the higher degree of precision, and can be numerically constructed by an effective and simple method; see Spalević (Math. Comp., 76:1483–1492, 2007). Moreover, as almost immediate consequence of our results from Spalević (Math. Comp.,76:1483–1492, 2007) and that theory, we prove the main statements in Notaris (Numer. Math.,142:129–147, 2019) in a different manner, by means of the Jacobi tridiagonal matrix approach.

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  4. Djukić, D.Lj., Reichel, L., Spalević, M.M., Tomanović, J.D.: Internality of generalized averaged Gaussian quadrature rules and their truncated variants for modified Chebyshev measures of the second kind. J. Comput. Appl. Math. 345, 70–85 (2019)

    Article  MathSciNet  Google Scholar 

  5. Djukić, D.Lj., Reichel, L., Spalević, M.M.: Internality of generalized averaged Gaussian quadratures and their truncated variants for measures induced by Chebyshev polynomials. Appl. Numer. Math. 142, 190–205 (2019)

    Article  MathSciNet  Google Scholar 

  6. Gauss, C.F.: Methodus nova integralium valores per approximationem inveniendi. Commentationes Societatis Regiae Scientiarum Göttingensis Recentiores, 3. Also in Werke III, 163–196 (1814)

  7. Gautschi, W.: On generating orthogonal polynomials. SIAM J. Sci. Stat. Comput. 3, 289–317 (1982)

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  9. 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

  10. Gautschi, W., Notaris, S.E.: Stieltjes polynomials and related quadrature formulae for a class of weight functions. Math. Comp. 65, 1257–1268 (1996)

    Article  MathSciNet  Google Scholar 

  11. Golub, G.H., Welsch, J.H.: Calculation of Gauss quadrature rules. Math. Comp. 23, 221–230 (1969)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  13. Kahaner, D.K., Monegato, G.: Nonexistence of extended Gauss-Laguerre and Gauss-Hermite quadrature rules with positive weights. Z. Angew. Math. Phys. 29, 983–986 (1978)

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  17. Máté, A., Nevai, P., Van Assche, W.: The supports of measures associated with orthogonal polynomials and the spectra of the related self-adjoint operators. Rocky Mountain J. Math. 21, 501–527 (1991)

    Article  MathSciNet  Google Scholar 

  18. Notaris, S.E.: Anti-Gaussian quadrature dormulae based on the zeros of Stieltjes polynomials. BIT 58, 179–198 (2018)

    Article  MathSciNet  Google Scholar 

  19. Notaris, S.E.: Stieltjes polynomials and related quadrature formulae for a class of weight functions, II. Numer. Math. 142, 129–147 (2019)

    Article  MathSciNet  Google Scholar 

  20. Peherstorfer, F.: On positive quadrature formulas. In: Brass, H., Hämmerlin, G. (eds.) Numerical Integration IV, Intern. Ser. Numer. Math. # 112, pp 297–313. Basel, Birkhäuser (1993)

  21. Peherstorfer, F.: Positive quadrature formulas III: Asymptotics of weights. Math. Comp. 77, 2241–2259 (2008)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  24. 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  Google Scholar 

  25. 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  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  29. Wilf, H.S.: Mathematics for the Physical Sciences. Wiley, New York (1962)

    MATH  Google Scholar 

Download references

Funding

This work was supported in part by the Serbian Ministry of Education, Science and Technological Development (Research Project: “Methods of numerical and nonlinear analysis with applications” (#174002)).

Author information

Authors and Affiliations

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

    Miodrag M. Spalević

Authors
  1. Miodrag M. Spalević
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Miodrag M. Spalević.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Spalević, M.M. A note on generalized averaged Gaussian formulas for a class of weight functions. Numer Algor 85, 977–993 (2020). https://doi.org/10.1007/s11075-019-00848-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-019-00848-x

Keywords

Mathematics Subject Classification (2010)

Search

Navigation