A geometric approach for the addition of nodes to an interpolatory quadrature rule with positive weights

A novel mathematical framework is derived for the addition of nodes to interpolatory quadrature rules. The framework is based on the geometrical interpretation of the Vandermonde-matrix describing the relation between the nodes and the weights and can be used to determine all nodes that can be added to an interpolatory quadrature rule with positive weights such that the positive weights are preserved. In the case of addition of a single node, the derived inequalities that describe the regions where nodes can be added or replaced are explicit. It is shown that, depending on the location of existing nodes and moments of the distribution, addition of a single node and preservation of positive weights is not always possible. On the other hand, addition of multiple nodes and preservation of positive weights is always possible, although the minimum number of nodes that need to be added can be as large as the number of nodes of the quadrature rule. Moreover, in this case the inequalities describing the regions where nodes can be added become implicit. An algorithm is presented to explore these regions and it is shown that the well-known Patterson extension of quadrature rules forms the boundary of these regions. Two new quadrature rules, based on the framework, are proposed and their efficiency is numerically demonstrated by using Genz test functions.

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