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.
J. P. Bertrandias,Calcul d’une intégrale au moyen de la suite X n =An. Évaluation de l’erreur. Publ. Inst. Statist. Univ. Paris,9 (1960), pp. 335–357.
C. B. Haselgrove,A method for numerical integration, Mathematics of computation,15 (1961), pp. 323–337.
E. Hlawka,Zur angenäherten Berechnung mehrfacher Integrale, Monatsh. Math.,66 (1962), pp. 140–151.
—— ——,Funktionen von beschränkter Variation in der Theorie der Gleichverteilung, Ann. Mat. Pura Appl., (IV)54 (1961), pp. 325–334.
A. Ya. Khinchin,Continued Fractions, Phoenix Books, University of Chicago Press, 1964.
J. F. Koksma,Some theorems on Diophantine inequalities, Math. Centrum Amsterdam, Scriptum5 (1960).
-- --,A general theorem from the theory of uniform distribution modulo 1, Mathematica, Zutphen B, pp.11, 7–11 (1942) (Dutch; quoted after Mathematical Reviews).
A. Ostrowski,Bemerkungen zur Theorie der Diophantischen Approximationen, Abh. Math. Sem. Math. Univ. Hamburg,1 (1922), pp. 77–98.
O. Perron,Die Lehre von den Kettenbrüchen, 2. Aufl., B. G. Teubner, Leipzig-Berlin, 1929.
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
- Stated Condition
- High Power
- Lattice Point
- Practical Method
- Finite Sequence