A classical way to construct a quadrature formula is to approximate the integral \(\int\limits_a^b {f\left( t \right)} \)dt by \(\int\limits_a^b {H\left( {,f;t} \right)dt} \) where H(x, f; t) is the Hermite interpolation polynomial for the function f based on a given system of nodes x = {(x1, v1),..., (xn, vn)} (i.e. x1,...,xn of multiplicities v1,...,vn, respectively). The error E(x; f) of this quadrature rule is expressed by the divided difference f[x,x] of f at the points x and x. We have
$$E\left( {;f} \right) = \int\limits_a^b {f\left[ {,x} \right]{{\left( {x - {x_1}} \right)}^{{v_1}}}...{{\left( {x - {x_n}} \right)}^{{v_n}}}dx} $$


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Barrow, D., On multiple node Gaussian quadrature formulae. Math. Comp. 32 (1978), 431–439.CrossRefGoogle Scholar
  2. 2.
    Bojanov, B.D., Extremal problems in the set of polynomials with fixed multiplicities of zeros. Comp. Rend. Acad. Bulgare Sci. 31 (1978), 377–400.Google Scholar
  3. 3.
    Bojanov, B.D., A generalization of Chebyshev polynomials. J. Approximation Theory 26 (1979), 293–300.CrossRefGoogle Scholar
  4. 4.
    Schwartz, J.T., Nonlinear Functional Analysis. Gordon and Breach, New York, 1969.Google Scholar
  5. 5.
    Tschakaloff, L., General quadrature formulae of Gaussian type, Bulgar. Akad. Nauk Izv. Mat. Inst. 1,2 (1954), 67–84.Google Scholar

Copyright information

© Springer Basel AG 1982

Authors and Affiliations

  • Borislav Bojanov

There are no affiliations available

Personalised recommendations