Families of Structured Quadrature Rules and Some Ramifications of Reducibility

  • Christopher T. H. Baker
Part of the ISNM 57: International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série internationale d’Analyse numérique book series (ISNM, volume 57)


We shall study methods for discretizing integrals of the form \(\int\limits_0^x \Phi\left({x,y}\right)dy\), a problem which arises in the numerical treatment of integral equations of Volterra or Abel type. The structure and reducibility properties of these methods have interest in their own right but also provide essential background in the analysis of numerical methods for such equations.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    ALBRECHT, P. Explicit, optimal stability functionals and their application to cyclic discretization methods. Computing 19 (1978) pp. 233–249.CrossRefGoogle Scholar
  2. 2.
    BAKER, C.T.H. The numerical treatment of integral equations. Clarendon Press, Oxford (1977) (reprinted 1978).Google Scholar
  3. 3.
    BAKER, C.T.H. Structure of recurrence relations in the study of stability in the numerical treatment of Volterra equations. J. Integral Equations 2 (1980) pp. 11–29.Google Scholar
  4. 4.
    BAKER, C.T.H. and KEECH, M.S. Stability regions in the numerical treatment of Volterra integral equations. Siam J. Numer. Anal. 15 (1978) pp. 394–417.CrossRefGoogle Scholar
  5. 5.
    BAKER, C.T.H. and WILKINSON, J.C. Stability analysis of Runge-Kutta methods applied to a basic Volterra integral equation. J. Austral. Math. Soc (B) 22 (1981) pp. 515–538.CrossRefGoogle Scholar
  6. 6.
    DIENES, P. The Taylor series — an introduction to the theory of functions of a complex variable. Clarendon Press, Oxford (1931), and Dover, New York (1957).Google Scholar
  7. 7.
    GENIN, Y. A new approach to the synthesis of stiffly-stable linear multistep formulae. IEEE Trans. Circ. Th. Vol. CT20 (1973) pp. 352–360.CrossRefGoogle Scholar
  8. 8.
    HENRICI, P. Applied and computational complex analysis Vol. 1. Wiley, New York (1974).Google Scholar
  9. 9.
    HOLYHEAD, P.A.W. Direct methods for the numerical solution of Volterra integral equations of the first kind. Ph.D. thesis, University of Southampton (1974).Google Scholar
  10. 10.
    KOBAYASI, M. On numerical solution of the Volterra integral equations of the second kind by linear multistep methods. Rep. Stat. Appl. Res., Un. Japan Sci. Engrs. 13. (1966) pp. 1–21.Google Scholar
  11. 11.
    MATTHYS, J. A-stable linear multistep methods for Volterra integro-differential equations. Numer. Math. 27 (1976) pp. 85–94.CrossRefGoogle Scholar
  12. 12.
    NEVANLINNA, O. Positive quadratures for Volterra equations. Computing 16 (1976) pp. 349–357.CrossRefGoogle Scholar
  13. 13.
    STETTER, H. Analysis of discretization methods for ordinary differential equations. Springer-Verlag, Berlin (1973).CrossRefGoogle Scholar
  14. 14.
    WOLKENFELT, P.H.M. Linear multistep methods and the construction of quadrature formulae for Volterra integral and integro-differential equations. Report NW 76/79 Math. Centrum, Amsterdam (1979).Google Scholar
  15. 15.
    WOLKENFELT, P.H.M. The numerical analysis of reducible quadrature methods for Volterra integral and integro-differential equations. (Academisch Proefschrift.) Math. Centrum, Amsterdam (1981).Google Scholar
  16. 16.
    WOLKENFELT, P.H.M. The construction of reducible quadrature rules for Volterra integral and integro-differential equations. Preprint (1981).Google Scholar

Copyright information

© Springer Basel AG 1982

Authors and Affiliations

  • Christopher T. H. Baker

There are no affiliations available

Personalised recommendations