Adaptive FCC+ rules for oscillatory integrals

Adaptive extended Filon–Clenshaw–Curtis rules are introduced for computing oscillatory integrals, and efficient algorithms are developed for their construction. Also, an error estimate has been obtained that is explicit in the wave number and the rule parameters. In addition, we rigorously prove that an adaptive extended Filon–Clenshaw–Curtis rule preserves the asymptotic order of the corresponding extended Filon–Clenshaw–Curtis rule. Meanwhile, the connection between these rules and the Filon–Clenshaw–Curtis rules is declared. The connection enables one to construct an adaptive extended Filon–Clenshaw–Curtis rule from the corresponding Filon–Clenshaw–Curtis rule naturally. Also, we estimate complexity of the proposed construction algorithms asymptotically. In some cases of the proposed construction algorithms, one encounters bad-conditioned dense linear systems. We carry out some numerical experiments, which suggest efficient tools for achieving reliable solutions of the systems. In our numerical experiments, the adaptive extended Filon–Clenshaw–Curtis rules reveal themselves more efficient than the corresponding extended Filon–Clenshaw–Curtis rules.