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Optimality parameter of Korobov parallelepipedal grids for cubature formulas

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In Blessed Memory of Academician N.S. Bakhvalov and Professor N.M. Korobov

Abstract

When multiple integrals are approximately evaluated using Korobov cubature formulas, it is necessary to introduce a parameter characterizing the uniform distribution of the grid nodes. A new parameter for Korobov parallelepipedal grids is proposed, and an algorithm for its computation is described.

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References

  1. N. M. Korobov, “Approximate Evaluation of Multiple Integrals,” Dokl. Akad. Nauk SSSR 124, 1207–1210 (1959).

    MathSciNet  MATH  Google Scholar 

  2. N. M. Korobov, Number-Theoretic Methods in Approximate Analysis (MTsNMO, Moscow, 2004) [in Russian].

    MATH  Google Scholar 

  3. S. Haber, Experiments on Optimal Coefficients Reprinted from Applications of Number Theory to Numerical Analysis (Academic, New York, 1972).

    Google Scholar 

  4. Hua Loo Keng and Wang Yuan, Applications of Number Theory to Numerical Analysis (Springer-Verlag, Berlin, 1981).

    MATH  Google Scholar 

  5. N. S. Bakhvalov, “Approximate Evaluation of Multiple Integrals,” Vestn. Mosk. Univ., No. 4, 3–18 (1959).

  6. V. A. Bykovskii, “An Algorithm for Computing Local Minima of Lattices,” Dokl. Akad. Nauk 399, 585–589 (2004) [Dokl. Math. 70, 928–930 (2004)].

    MathSciNet  Google Scholar 

  7. G. F. Voronoi, Selected Works (Kiev, 1952), Vol. 1 [in Russian].

  8. H. Minkowski, “Generalization de la theorie des fraction continues,” Ann. Sci. Ecole Norm. Super. 13(2), 41–60 (1896).

    MathSciNet  Google Scholar 

  9. J. W. S. Cassels, An Introduction to the Geometry of Numbers (Springer-Verlag, Berlin, 1959; Mir, Moscow, 1965).

    MATH  Google Scholar 

  10. V. A. Bykovskii, “On the Error of Number-Theoretic Quadrature Formulas,” Dokl. Akad. Nauk 389, 154–155 (2003) [Dokl. Math. 67, 175–176 (2003)].

    MathSciNet  Google Scholar 

  11. Ch. Hermite, “Sur l’introduction des variables continues dans la theorie des nombres,” J. Math. 41, 191–216 (1851); (Qeuvres, Paris, 1905), Vol. I, pp. 164–192.

    MATH  Google Scholar 

  12. H. A. Cohen, Course in Computational Algebraic Number Theory (Springer-Verlag, Berlin, 1993). 0965–5425, Computational Mathematics and Mathematical Physics, 2011, Vol. 51, No. 8, pp. 1280–1285.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Institute of Applied Mathematics (Khabarovsk Branch), Far East Branch, Russian Academy of Sciences, ul. Dzerzhinskogo 54, Khabarovsk, 680000, Russia

    V. A. Bykovskii

  2. Institute of Applied Mathematics, Far East Branch, Russian Academy of Sciences, ul. Radio 7, Vladivostok, 690041, Russia

    S. V. Gassan

Authors
  1. V. A. Bykovskii
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  2. S. V. Gassan
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Correspondence to V. A. Bykovskii.

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Original Russian Text © V.A. Bykovskii, S.V. Gassan, 2011, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2011, Vol. 51, No. 8, pp. 1363–1369.

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Bykovskii, V.A., Gassan, S.V. Optimality parameter of Korobov parallelepipedal grids for cubature formulas. Comput. Math. and Math. Phys. 51, 1273–1279 (2011). https://doi.org/10.1134/S0965542511080057

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  • DOI: https://doi.org/10.1134/S0965542511080057

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