Exhaustive search for higher-order Kronrod-Patterson Extensions

Abstract

Gauss points are not nested and for this reason one searches for quadrature rules with nested points and similar efficiency. 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.