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On cubature formulae with a minimal number of knots

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Summary

In this paper an approach is outlined to the two-dimensional analogon of the Gaussian quadrature problem. The main results are necessary and sufficient conditions for the existence of cubature formulae which are exact for all polynomials of degree ≦m and which have a minimal number of 1/2k(k+1) knots,k=[m/2]+1. Ifm is odd, similar results are due to I.P. Mysovskikh ([5, 6]) which will be derived in a new way as a special case of the general characterization given here. Furthermore, it will be shown how this characterization can be used to construct minimal formulae of even degree.

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References

  1. Fritsch, F.N.: On the existence of regions with minimal third degree integration formulas. Math. Comput.24, 855–861 (1970)

    Google Scholar 

  2. Möller, H.M.: Kubaturformeln mit minimaler Knotenzahl. Numer. Math.25, 185–200 (1976)

    Google Scholar 

  3. Mysovskikh, I.P.: On the construction of cubature formulas for the simplest regions [in Russ.]. Ž. Vyčisl. Mat. i Mat. Fiz.4, 3–14 (1964)

    Google Scholar 

  4. Mysovskikh, I.P.: On the construction of cubature formulas with fewest nodes. Soviet Math. Dokl.9, 277–280 (1968)

    Google Scholar 

  5. Mysovskikh, I.P.: Numerical characteristics of orthogonal polynomials in two variables [in Russ.]. Vestnik Leningrad Univ. Math.19, 46–53 (1970)

    Google Scholar 

  6. Mysovskikh, I.P.: Orthogonal polynomials in several variables [in Russ.]. Metody Vyčisl.10, zdvo Len. Univ. 26–35 (1976)

    Google Scholar 

  7. Radon, J.: Zur mechanischen Kubatur. Monatsh. Math.52, 286–300 (1948)

    Google Scholar 

  8. Risler, J.J.: Unc charactérisation des idéaux des variétés algébriques réelles. Note aux CRAS. Paris271, 1171–1173 (1970)

    Google Scholar 

  9. Stroud, A.H.: Approximate calculation of multiple integrals. Englewood Cliffs, New Jersey: Prentice Hall 1971

    Google Scholar 

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Authors and Affiliations

  1. Mathematisches Institut der Universität Erlangen-Nürnberg, Bismarckstr. 1 1/2, D-8520, Erlangen, Germany (Fed. Rep.)

    Hans Joachim Schmid

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  1. Hans Joachim Schmid
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Schmid, H.J. On cubature formulae with a minimal number of knots. Numer. Math. 31, 281–297 (1978). https://doi.org/10.1007/BF01397880

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  • DOI: https://doi.org/10.1007/BF01397880

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