On numerical integration algorithms. (English. Russian original) Zbl 0918.41028

Voronin, S. M.; Skalyga, V. I.

Izv. Math. 60, No. 5, 887-891 (1996); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 60, No. 5, 13-18 (1996).

An effective method for constructing formulas of the form \[ \int^1_0 \cdots\int^1_0 f(\overline x)d_1 \dots dx_{q-1}= {1\over p} \sum_{\nu \bmod p}f \left({\overline b\over p} \cdot\nu \right) \] is given, where \(p\) is a prime number, \(a\) is an integer, \(\overline b=\left(1,a^{p-1}, \dots, a^{{(q-2) (p-1) \over 2}} \right)\), \(a^{(p-1)/2} \not\equiv 1\pmod p\), \(\overline n=(n_1, \dots, n_{q-1})\), \(\overline x=(x_1,\dots,x_{q-1})\in\mathbb{R}^{q-1}\), \[ f (\overline x)=\sum_{\overline n\in\mathbb{Z}^{q-1}}c_n\exp\bigl\{2\pi i(\overline n, \overline x)\bigr\}. \]

Reviewer: Tibor Šalát (Bratislava)

MSC:

41A55	Approximate quadratures
11R04	Algebraic numbers; rings of algebraic integers
11R18	Cyclotomic extensions