Numerische Mathematik

, Volume 57, Issue 1, pp 271–283

The error norm of Gaussian quadrature formulae for weight functions of Bernstein-Szegö type

  • Sotirios E. Notaris
Article

Summary

We consider the Gaussian quadrature formulae for the Bernstein-Szegö weight functions consisting of any one of the four Chebyshev weights divided by an arbitrary quadratic polynomial that remains positive on [−1, 1]. Using the method in Akrivis (1985), we compute the norm of the error functional of these quadrature formulae. The quality of the bounds for the error functional, that can be obtained in this way, is demonstrated by two numerical examples.

Subject Classifications

AMS (MOS): Primary 65D32 Secondary 33A65 CR: G1.4 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Akrivis, G.: Fehlerabschätzungen für Gauss-Quadraturformeln. Numer. Math.44, 261–278 (1984)Google Scholar
  2. 2.
    Akrivis, G., Burgstaller, A.: Fehlerabschätzungen für nichtsymmetrische Gauss-Quadraturformeln. Numer. Math.47, 535–543 (1985)Google Scholar
  3. 3.
    Akrivis, G.: The error norm of certain Gaussian quadrature formulae. Math. Comp.45, 513–519 (1985)Google Scholar
  4. 4.
    Gautschi, W., Notaris, S.E.: Gauss-Kronrod quadrature formulae for weight functions of Bernstein-Szegö type. J. Comput. Appl. Math.25(2), 199–224 (1989); erratum in: J. Comput. Appl. Math.27 (3), 429 (1989)Google Scholar
  5. 5.
    Gautschi, W., Varga, R.S.: Error bounds for Gaussian quadrature of analytic functions. SIAM J. Numer. Anal.20, 1170–1186 (1983)Google Scholar
  6. 6.
    Gröbner, W., Hofreiter, N. (eds.) Integraltafel, II Teil. Wien: Springer 1961Google Scholar
  7. 7.
    Hämmerlin, G.: Fehlerabschätzungen bei numerischer Integration nach Gauss. In: Brosowski, B., Martensen, E. (eds.) Methoden und Verfahren der mathematischen Physik vol. 6, pp. 153–163. Mannheim, Wien, Zürich: Bibliographisches Institut 1972Google Scholar
  8. 8.
    Rivlin, T.J.: The Chebyshev Polynormials. New York: Wiley, 1974Google Scholar
  9. 9.
    Szegö, G.: Orthogonal Polynomials. Colloquium Publications, vol. 23, 4th ed., American Mathematical Society. Providence, RI, 1975Google Scholar

Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • Sotirios E. Notaris
    • 1
  1. 1.Department of Mathematical SciencesIndiana University-Purdue University at IndianapolisIndianapolisUSA

Personalised recommendations