On adaptive extended Filon-Clenshaw-Curtis rules

A class of adaptive extended Filon-Clenshaw-Curtis rules is introduced for computing oscillatory integrals. Also, an efficient algorithm for constructing the rules of this class is developed that is rather simple and fast. An explicit relation of the adaptive extended Filon-Clenshaw-Curtis rules and the corresponding plain Filon-Clenshaw-Curtis rule is derived. This relation enables us to extend a Filon-Clenshaw-Curtis rule to the corresponding adaptive extended Filon-Clenshaw-Curtis rules naturally.