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Introduction. In this paper we shall give an account of some methods developed for the numerical evaluation of multidimensional integrals. These methods are based on the theory of Diophantine approximation. They are suitable for some problems for which the Monte Carlo method is commonly used and, like the Monte Carlo method, are well fitted for use with an electronic digital computer. We shall show, however, that they are superior to the Monte Carlo method provided that the integrand satisfies certain conditions. We shall also show that they are superior, for integrals in space of several dimensions, to formulas typified by those of Gauss and Simpson; they may be superior even to certain new integration formulas specially constructed for the evaluation of multiple integrals (see for example Hammer 2, who gives a bibliography, and Miller 5, 6, 7). The method of antithetic varites which is described by Hammersley and others 3, 4 may be used to obtain better estimates than the Monte Carlo method but the author thinks that the method described in the present paper is simpler to apply and gives better results. Various authors have suggested methods which are particular cases of those described in this paper but without the underlying theory. See for example Davis and Rabinowitz 1. In this section we shall give a short account of the behavior of the error in the Monte Carlo method and the direct-product Gauss-type methods so that we can compare these with the errors of the new methods. We shall not give an account of the method of antithetic varites. Suppose that we wish to estimate the integral 1 = / / " ' / f (Xl >Xl> " ' ' x*) dxi d%2 dxk. Jo Jo JoWe shall denote the vector (xi, x2, , xk) by x. Numerical methods for the evaluation of I involve the calculation of / (x) at a number N of points x. The most desirable of such methods for use on an electronic computer are those which require the evaluation of/ (x) at the smallest number of points x, in order to obtain an estimate with a sufficiently small error. The Monte Carlo method gives as an estimate for I the sum where the points x are chosen at random in the range of integration. The error of such an estimate has standard deviation 0{N~m) provided that the function / (x) satisfies certain conditions. It is sufficient that the function be bounded. A Gauss-type formula for a one-dimensional integral takes the form r>a+h v \ g (y) dy = hCi g (a +) + R.
C. B. Haselgrove (Sun,) studied this question.