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On the number of nodes in self-contained integration formulas of degree two for compact planar regions

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Abstract

Approximate integration formulas of precision two for compact planar regions are studied. We restrict our attention to formulas for which the nodes lie in the region and consider the problem of determining the minimum number of nodes attainable. While Stroud has shown that this minimum number is always at least 3, we find a two-parameter family of symmetric star-shaped regions for which it is equal to 4. That is, the conditions of symmetry and star-shapedness are not sufficient for the existence of such formulas with the minimal number of nodes.

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References

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Authors and Affiliations

  1. Computation Department, Lawrence Radiation Laboratory, P. O. Box 808, 94550, Livermore, Cal., USA

    F. N. Fritsch

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  1. F. N. Fritsch
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Work performed under the auspices of the U.S. Atomic Energy Commission.

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Fritsch, F.N. On the number of nodes in self-contained integration formulas of degree two for compact planar regions. Numer. Math. 16, 224–230 (1970). https://doi.org/10.1007/BF02219774

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