Summary
In order to compute an integralI[f], one needs at least two cubature formulaeQ j ,j∈{1, 2}. |Q 1[f]−Q 2[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 forQ 1, inQ 2. The easiest way to construct embedded cubature formulae is: start with a high degree formulaQ 1, drop (at least) one knot and calculate the weights such that the new formulaQ 2 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⊂ℝn,n≧2. We will also show the influence of structure on the cubature formulae.
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Cools, R., Haegemans, A. On the construction of multi-dimensional embedded cubature formulae. Numer. Math. 55, 735–745 (1989). https://doi.org/10.1007/BF01389339
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DOI: https://doi.org/10.1007/BF01389339