Numerische Mathematik

, Volume 41, Issue 2, pp 147–163

Calculation of Gauss quadratures with multiple free and fixed knots

  • G. H. Golub
  • J. Kautsky


Algorithms are derived for the evaluation of Gauss knots in the presence of fixed knots by modification of the Jacobi matrix for the weight function of the integral. Simple Gauss knots are obtained as eigenvalues of symmetric tridiagonal matrices and a rapidly converging simple iterative process, based on the merging of free and fixed knots, of quadratic convergence is presented for multiple Gauss knots. The procedures also allow for the evaluation of the weights of the quadrature corresponding to the simple Gauss knots. A new characterization of simple Gauss knots as a solution of a partial inverse eigenvalue problem is derived.

Subject Classifications

AMS (MOS): 65F15 65D30 CR: 5.14 5.16 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Galant, D.: An implementation of Christoffel's theorem in the theory of orthogonal polynomials. Math. Comput.25, 111–113 (1971)Google Scholar
  2. 2.
    Gautschi, W.: On generating Gaussian quadrature rules. In: Numerische Integration, ISNM 45, Hämmerlin, G. (ed.), Birkhäuser, Basel, pp. 147–154, 1979Google Scholar
  3. 3.
    Gautschi, W.: A survey of Gauss-Christoffel quadrature formulae. In: Christoffel, E.B.: The Influence of his work on Mathematics and the Physical Sciences, Butzer, P.L., Fehér, F. (eds.), Birkhäuser, Basel, pp. 72–147, 1981Google Scholar
  4. 4.
    Golub, G.H., Welsch, J.H.: Calculation of Gauss Quadrature Rules. Math. Comput.23, 221–230 (1969)Google Scholar
  5. 5.
    Golub, G.H.: Some modified matrix eigenvalue problems. SIAM Rev.15, 318–334 (1973)Google Scholar
  6. 6.
    Kautsky, J.: Matrices related to interpolatory quadratures. Numer. Math.36, 309–318 (1981)Google Scholar
  7. 7.
    Kautsky, J., Elhay, S.: Calculation of the weights of interpolatory quadratures. Numer. Math.40, 407–422 (1982)Google Scholar
  8. 8.
    Kautsky, J., Golub, G.H.: Evaluation of Jacobi matrices. Report, School of Mathematical Sciences, Flinders University of South Australia, 1982Google Scholar
  9. 9.
    Martin, R.S., Wilkinson, J.H.: The ImplicitQL Algorithm. Numer. Math.12, 277–383 (1968)Google Scholar
  10. 10.
    Stancu, D.D.: Sur quelques formules generales de quadrature du type Gauss-Christoffel. Mathematica (Cluj)1 (24), 167–182 (1959)Google Scholar
  11. 11.
    Stancu, D.D., Stroud, A.H.: Quadrature formulas with simple Gaussian nodes and multiple fixed nodes. Math. Comput.17, 384–394 (1963)Google Scholar
  12. 12.
    Stroud, A.H., Stancu, D.D.: Quadrature formulas with multiple Gaussian nodes. SIAM J. Numer. Anal. Series B2, 129–143 (1965)Google Scholar
  13. 13.
    Turán, P.: On the theory of the mechanical quadrature. Acta Sci. Math. (Szeged)12, 30–37 (1950)Google Scholar
  14. 14.
    Wilf, H.S.: Mathematics for the Physical Sciences. (Chapter 2). New York: Wiley, 1962Google Scholar
  15. 15.
    Wilkinson, J.H.: The Algebraic Eigenvalue Problem. Oxford: Clarendon Press, 1965Google Scholar

Copyright information

© Springer-Verlag 1983

Authors and Affiliations

  • G. H. Golub
    • 1
  • J. Kautsky
    • 2
  1. 1.Department of Computer ScienceStanford UniversityStanfordUSA
  2. 2.School of Mathematical SciencesFlinders UniversityBedford ParkAustralia

Personalised recommendations