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 Zh. Vychisl. Mat. Mat. Fiz., 2014, Volume 54, Number 7, Pages 1059–1077 (Mi zvmmf10059)

General algorithm for the numerical integration of functions of several variables

E. A. Bailova, M. B. Sikhovb, N. Temirgalieva

a Institute of Theoretical Mathematics and Scientific Computations, Eurasian National University, ul. Mirzoyana 2, Astana, 010008, Kazakhstan
b Kazakh National University, pr. Al-Farabi 71, Almaty, Kazakhstan

Abstract: An algorithm is proposed for the numerical integration of an arbitrary function representable as a sum of an absolutely converging multiple trigonometric Fourier series. The resulting quadrature formulas have identical weights, and the nodes form a Korobov grid that is completely defined by two positive integers, of which one is the number of nodes. In the case of classes of functions with dominant mixed smoothness, it is shown that the algorithm is almost optimal in the sense that the construction of a grid of $N$ nodes requires far fewer elementary arithmetic operations than $N\ln\ln N$. Solutions of related problems are also given.

Key words: discrepancy, uniformly distributed grids, Korobov grids, optimal coefficients, quadrature formulas, divisor theory, lattice, ideal.

DOI: https://doi.org/10.7868/S0044466914070047

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English version:
Computational Mathematics and Mathematical Physics, 2014, 54:7, 1061–1078

Bibliographic databases:

UDC: 519.644.7
MSC: 65D15
Revised: 21.01.2014

Citation: E. A. Bailov, M. B. Sikhov, N. Temirgaliev, “General algorithm for the numerical integration of functions of several variables”, Zh. Vychisl. Mat. Mat. Fiz., 54:7 (2014), 1059–1077; Comput. Math. Math. Phys., 54:7 (2014), 1061–1078

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Zh. N. Temirgaliyeva, N. Temirgaliyev, “Rapid “algebraic” Fourier transforms on uniformly distributed meshes”, Russian Math. (Iz. VUZ), 60:5 (2016), 81–85
2. Zh. N. Temirgaliyeva, N. Temirgaliyev, ““Geometry of numbers” in a context of algebraic theory of numbers”, Russian Math. (Iz. VUZ), 60:10 (2016), 77–81
3. N. Zh. Nauryzbaev, A. A. Shomanova, N. Temirgaliyev, “On some special effects in theory on numerical integration and functions recovery”, Russian Math. (Iz. VUZ), 62:3 (2018), 84–88
4. N. Temirgaliyev, Sh. K. Abikenova, Sh. U. Azhgaliev, G. E. Taugynbaeyva, “The Radon transform in the scheme C(N)D-inverstigations and the quasi-Monte Carlo theory”, Russian Math. (Iz. VUZ), 64:3 (2020), 87–92
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