Numerische Mathematik

, Volume 55, Issue 6, pp 735–745 | Cite as

On the construction of multi-dimensional embedded cubature formulae

  • Ronald Cools
  • Ann Haegemans


In order to compute an integralI[f], one needs at least two cubature formulaeQ j ,j∈{1, 2}. |Q1[f]−Q2[f]| can be used as an error estimate for the less precise cubature formula. In order to reduce the amount of work, one can try to reuse some of the function evaluations needed forQ1, inQ2. The easiest way to construct embedded cubature formulae is: start with a high degree formulaQ1, drop (at least) one knot and calculate the weights such that the new formulaQ2 is exact for as much monomials as possible. We describe how such embedded formulae with positive weights can be found. The disadvantage of such embedded cubature formulae is that there is in general a large difference in the degree of exactness of the two formulae. In this paper we will explain how the high degree formula can be chosen to obtain an embedded pair of cubature formulae of degrees 2m+1/2m−1. The method works for all regionsΩ⊂ℝ n ,n≧2. We will also show the influence of structure on the cubature formulae.

Subject Classifications

AMS(MOS): 65D32 41A55 41A63 CR: G.1.4 


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Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • Ronald Cools
    • 1
  • Ann Haegemans
    • 1
  1. 1.Department of Computer ScienceKatholieke Universiteit LeuvenHeverleeBelgium

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