A geometrical interpretation of the addition of nodes to an interpolatory quadrature rule while preserving positive weights

A novel mathematical framework is derived for the addition of nodes to univariate and 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 are explicit. Besides addition of nodes these inequalities also yield an algorithmic description of the replacement and removal of nodes. It is shown that it is not always possible to add a single node while preserving positive weights. 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. In case of addition of multiple nodes the inequalities describing the regions where nodes can be added become implicit. It is shown that the well-known Patterson extension of quadrature rules is a special case that forms the boundary of these regions and various examples of the applicability of the framework are discussed. By exploiting the framework, two new sets of quadrature rules are proposed. Their performance is compared with the well-known Gaussian and Clenshaw–Curtis quadrature rules, demonstrating the advantages of our proposed nested quadrature rules with positive weights and fine granularity.