General algorithm for the numerical integration of functions of several variables

Abstract

An algorithm is proposed for the numerical integration of an arbitrary function represent-able 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 NlnlnN. Solutions of related problems are also given.

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Correspondence to E. A. Bailov.

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Original Russian Text © E.A. Bailov, M.B. Sikhov, N. Temirgaliev, 2014, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2014, Vol. 54, No. 7, pp. 1059–1077.

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Bailov, E.A., Sikhov, M.B. & Temirgaliev, N. General algorithm for the numerical integration of functions of several variables. Comput. Math. and Math. Phys. 54, 1061–1078 (2014). https://doi.org/10.1134/S0965542514070045

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Keywords

  • discrepancy
  • uniformly distributed grids
  • Korobov grids
  • optimal coefficients
  • quadrature formulas
  • divisor theory
  • lattice
  • ideal