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Good lattice points, discrepancy, and numerical integration

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Summary

Methods based on Diophantine approximations lead to a simple and easy construction of rational vectors the multiple of which, reduced modulo 1, form finite sequences of points with certain properties of equipartition over the unit square. Given a function of bounded variation over this square, it is suggested that computing the average of its values at the points of such a sequence can be a practical method of numerical integration. Precise bounds for the error ore obtained. In the general case, these are of the order of the product of the reciprocal and of the logarithm of the number of points; in the case of a function satisfying stated conditions of regularity and periodicity, they are of the order of the logarithm of the number of points divided by an appropriately high power of this number. A slight sharpening of some well-known results on equipartiton is obtained incidentally.

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A part of this paper was read at a meeting of the Accademia Nazionale di Scienze Lettere e Arti in Modena on April 7th. 1965.

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Zaremba, S.K. Good lattice points, discrepancy, and numerical integration. Annali di Matematica 73, 293–317 (1966) doi:10.1007/BF02415091

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Keywords

  • Stated Condition
  • High Power
  • Lattice Point
  • Practical Method
  • Finite Sequence