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On numerical integration algorithms. (English. Russian original) Zbl 0918.41028
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, $$\overline b=\left(1,a^{p-1}, \dots, a^{{(q-2) (p-1) \over 2}} \right)$$, $$a^{(p-1)/2} \not\equiv 1\pmod p$$, $$\overline n=(n_1, \dots, n_{q-1})$$, $$\overline x=(x_1,\dots,x_{q-1})\in\mathbb{R}^{q-1}$$, $f (\overline x)=\sum_{\overline n\in\mathbb{Z}^{q-1}}c_n\exp\bigl\{2\pi i(\overline n, \overline x)\bigr\}.$

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