Algorithms for the construction of high-order Kronrod rule extensions with application to sparse-grid integration

Gauss quadrature points are not nested so search for quadrature rules with nested points and similar efficiency are important. A well-studied source of candidates are the Kronrod-Patterson extensions. Under suitable conditions, it is possible to build towers of nested rules. We investigate this topic further and give a detailed description of the algorithms used for constructing such iterative extensions. Our new implementation combines several important ideas spread out in theoretical research papers. We apply the resulting algorithms to the classical orthogonal polynomials and build sparse high-dimensional quadrature rules for each class.