Efficient construction of FCC+ rules

A comparative study of Filon-type rules for oscillatory integrals

“…, N , is negligible. Finally, having the Lagrange interpolant at the Clenshaw-Curtis, one can efficiently construct the corresponding Hermite-Birkhoff interpolant p that satisfies (7), too (see [14]).…”

“…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 contrast to the common ideas in the literature, we see that the FCC+ rules A c c e p t e d M a n u s c r i p t have no advantages over the plain FCC rules in practice. At first, we give a brief review of the FCC+ rules (see [7,14]).…”

“…), where p N +2s is the (N + 2s) th degree Hermite interpolation polynomial satisfying the following interpolation conditions (see [14]):…”