Adaptive FCC+ rules for oscillatory integrals

A comparative study of Filon-type rules for oscillatory integrals

“…Therefore, without doubt, Filon-Clenshaw-Curtis (FCC) rules and their extensions are the most efficient Filon-type rules for computing the oscillatory integral (1). Extended FCC (FCC+) rules and their adaptive versions can be constructed efficiently by the algorithms of [14] and [13], respectively. To the best of our knowledge, they are the most efficient algorithms for constructing the FCC+ and adaptive FCC+ rules so far.…”

“…In comparison with the algorithm of [5] for constructing the FCC rules, the algorithms of [14] and [13] are more complex. This difference in the computational costs may be negligible when the maximum order of derivatives s in the FCC+ rules is rather small.…”

“…In Section 3, the same is done for the FCC and the adaptive FCC+ rules. We also modify Theorem 2.2 of [13] and give a more practical error bound for the adaptive FCC+ rules that is explicit in k and the rule parameters. In Section 4, we account several other advantages of the FCC rules over the (adaptive) FCC+ ones.…”

“…In this section, we compare the efficiency in practice of the FCC and the adaptive FCC+ rules of [13]. An adaptive FCC+ rule and its corresponding FCC+ rule always have the same asymptotic order while the former is derivative-free and less complex.…”

“…We will see, however, that in the race to achieve a given accuracy, the adaptive FCC+ rules are still overtaken by the corresponding FCC rules. At first, we give a brief review of [13], where the concept of adaptive FCC+ rules has been introduced for the first time.…”