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The Second Stage: Finding the Weights

In the cases where the calculated nodes can be shown to be spherical designs (N=2,4,6,12), equal weights are optimal. This can be seen with the aid of Theorems 3 and 4. In all other cases appropriate weights have to be calculated.

With the aid of multivariate interpolation theory this is at least possible if the number N of nodes equals the dimension of the linear space of polynomials tex2html_wrap_inline3133. For r=3 this means that tex2html_wrap_inline3377.

Let tex2html_wrap_inline3379 and let tex2html_wrap_inline3381 be the symmetric matrix being composed of the evaluation of the kernel tex2html_wrap_inline3163 at the nodes tex2html_wrap_inline3385, tex2html_wrap_inline3387.

If tex2html_wrap_inline3389 are a fundamental system, i. e. tex2html_wrap_inline3391, then the interpolation property
and the following consequence of the reproducing property of tex2html_wrap_inline3163
are valid. Here, tex2html_wrap_inline3151 denote the Lagrange fundamental polynomials. Equations (6) and (7) lead to
which means that the matrices tex2html_wrap_inline3401 and tex2html_wrap_inline3403 are inverse to each other.

Consider now a cubature formula
The function f can be substituted in the usual way by its interpolation polynomial p with regard to the nodes tex2html_wrap_inline3389. Then
that is
Because tex2html_wrap_inline3419 and tex2html_wrap_inline3421 this yields
which is the row-sum of the matrix L.

Thus, because of (1), the weights of the cubature formula can be computed as the row-sum of the symmetric matrix
The Gegenbauer polynomials of degree m with index r/2 occuring in this composition have the representation
with tex2html_wrap_inline3431.

They can be evaluated with the aid of their recurrence relation (see e. g. Reimer [16])

The inversion of the matrix tex2html_wrap_inline3403 can be avoided. With
we obtain the vector tex2html_wrap_inline3435 of weights as solution of the linear system of equations
In the following, we study the case r=3, that is tex2html_wrap_inline3377, in detail, but the theory can be extended to higher dimensions without any difficulties.

The cases tex2html_wrap_inline3441 cannot be handled yet except for those cases in which the nodes are spherical designs. These are the cases N = 2,4,6,12 . The weights then can be chosen to be tex2html_wrap_inline3445. The cases N=3,5,7 also show a regular behaviour, but up to now it could not be decided if these nodes are spherical designs. As interpolatory formulae our cubatures have a degree of exactness equal to m, whereas the spherical designs yield a degree of exactness equal to 2m or 2m+1, respectively.

next up previous
Next: Implementation Details Up: A Two-Stage Approach Previous: The First Stage: Finding

Joerg Fliege
Thu Dec 23 19:39:35 CET 1999