Symbolicnumeric Gaussian cubature rules
Faculty of Sciences. Mathematics and Computer Science
Applied numerical mathematics. - Amsterdam
, p. 929-945
University of Antwerp
It is well known that Gaussian cubature rules are related to multivariate orthogonal polynomials. The cubature rules found in the literature use common zeroes of some linearly independent set of products of basically univariate polynomials. We show how a new family of multivariate orthogonal polynomials, so-called spherical orthogonal polynomials, leads to symbolicnumeric Gaussian cubature rules in a very natural way. They can be used for the integration of multivariate functions that in addition may depend on a vector of parameters and they are exact for multivariate parameterized polynomials. Purely numeric Gaussian cubature rules for the exact integration of multivariate polynomials can also be obtained. We illustrate their use for the symbolicnumeric solution of the partial differential equations satisfied by the Appell function F2, which arises frequently in various physical and chemical applications. The advantage of a symbolicnumeric formula over a purely numeric one is that one obtains a continuous extension, in terms of the parameters, of the numeric solution. The number of symbolicnumeric nodes in our Gaussian cubature rules is minimal, namely m for the exact integration of a polynomial of homogeneous degree 2m−1. In Section 1 we describe how the symbolicnumeric rules are constructed, in any dimension and for any order. In Sections 2, 3 and 4 we explicit them on different domains and for different weight functions. An illustration of the new formulas is given in Section 5 and we show in Section 6 how numeric cubature rules can be derived for the exact integration of multivariate polynomials. From Section 7 it is clear that there is a connection between our symbolicnumeric cubature rules and numeric cubature formulae with a minimal (or small) number of nodes.