BIT Numerical Mathematics

, Volume 22, Issue 3, pp 339–352 | Cite as

The number of positive weights of a quadrature formula

  • G. Sottas
  • G. Wanner
Part II Numerical Mathematics

Abstract

In this paper we show that the number of positive weights of a quadrature formula is related to the number of rotations of a certain path in the plane. Necessary and sufficient conditions for all weights to be positive can then be obtained. Also, much of classical theory appears in a new light.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    R. Askey,Positivity of the Cotes numbers for some Jacobi abscissas, Numer. Math.19 (1972), 46–48.Google Scholar
  2. 2.
    R. Askey,Positivity of the Cotes numbers for some Jacobi abscissas (II), J. Inst. Math. Applics.24 (1979), 95–98.Google Scholar
  3. 3.
    R. Askey, J. Fitch,Positivity of the Cotes numbers for some ultraspherical abscissas, SIAM J. Numer. Anal.5 (1968), 199–201.Google Scholar
  4. 4.
    K. Burrage,Stability and efficiency properties of implicit Runge-Kutta methods, PhD Thesis, University of Auckland, Auckland, New Zealand, 1978.Google Scholar
  5. 5.
    K. Burrage, J. C. Butcher,Stability criteria for implicit Runge-Kutta methods, SIAM J. Numer. Anal.16 (1979), 46–57.Google Scholar
  6. 6.
    G. Dahlquist, R. Jeltsch,Generalized disks of contractivity for explicit and implicit Runge-Kutta methods, TRITA-NA Report 7906, Dept of Numer. Anal., Institute of Technology, Stockholm 1979.Google Scholar
  7. 7.
    L. Fejer, Mechanische Quadraturen mit positiven Coteschen Zahlen, Math. Zeitschr.37 (1933), 287–309.Google Scholar
  8. 8.
    F. R. Gantmacher,Theoria Matriz, Moskva 1954. (English, German, French translations).Google Scholar
  9. 9.
    S. P. Nørsett, G. Wanner,Perturbed collocation and Runge-Kutta methods, Numer. Math.38 (1981), 193–208.Google Scholar
  10. 10.
    F. Peherstorfer,Characterization of positive quadrature formulas, SIAM J. Math. Anal.12 (1981), 935–942.Google Scholar
  11. 11.
    J. Shohat,On mechanical quadratures, in particular, with positive coefficients, Trans. AMS42 (1937), 461–496.Google Scholar
  12. 12.
    G. Sottas,Quadrature formulas with positive weights, BIT21 (1981), 491–504.Google Scholar
  13. 13.
    V. Steklov, Remarques sur les quadratures, Bull. de l'Acad. des Sciences de Russie (6), 12 (1918), 99–118.Google Scholar
  14. 14.
    G. Szegö,Orthogonal Polynomials, AMS, Colloquium Publications, Volume XXIII, New York, 1939.Google Scholar

Copyright information

© BIT Foundations 1982

Authors and Affiliations

  • G. Sottas
    • 1
  • G. Wanner
    • 1
  1. 1.Section de mathématiquesUniversité de GenèveGenève 24Switzerland

Personalised recommendations