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Number-theoretic Methods in Statistics. Silver Spring pop: 71, Advertisement System V2.
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Centreville pop: 71, Kolambija pop: 99, Howard , Мэриленд. Ellicott City pop: 65, The above discussion shows that if one attempts to provide an appropriate approximation for a periodical function within one single peak domain, 11 sampling points in one dimension , 17 sampling points in two dimensions , or 19 sampling points in three dimensions are needed for the calculation of a definite integral, respectively, while the sampling points are deterministically distributed according to the rule of the uniform design method and GLP.
In the following sections, we will check the applicability of the above descriptions. Efficient approach for numerical integration on the basis of the uniform test design method and GLD for a single peak function. Similarly, the consequences in the last sections present that 17 and 19 uniformly distributed sampling points in one periodical range could provide an appropriate approximation with a relative error of around 0.
In addition, other functions can be expanded as sine or cosine functions generally.
Hence, here in this section, let us conduct some typical definite integrals to show the rationality of the approach. Our first example is the probability integral Navidi, ,. Table 1 The positions of the distribution of the sampling points in the integral domain [0, 4] Point No. The summation of the right-hand side of Eq. A2 Approximation of the elliptic integral calculus for the magnetic induction intensity of an elliptical current-carrying ring.
Take an elliptical current-carrying ring as an example, which is with the major axis a, the minor axis b, the distance between the focal point F and the center O is c; the distance from a point M on the ellipse to the center O is r, see Fig. The problem is to find the magnetic induction intensity at the center of the ellipse. Thus, the expression of the magnetic induction at the center of the current-carrying ellipse can be written as Ju et al, ,. Let us mark the integration part in Eq.
The approximate result of the right-hand side of Eq. Here, let us take the integration of an example. The integration of with the precise value of 0.
The distribution of the sampling points in the integral domain [1. Table 3 Distribution of the sampling points in the integral domain [1. The summation result of the right-hand side of Eq. Chen et al took the integration as an example to study the validity of the integration of multivariate functions by orthogonal arrays Chen et al, Obviously, their amount of simulation calculation is really huge.
The above studies including the analysis and example calculations indicate that the efficient result for a definite integral of a function with an accuracy of around 0. This result is much better than those of classic methods on the one hand; besides, the approach is even better than the MC simulation in the sense of workload of calculation. The novelty and contribution of this study is to use a small number of sampling points to obtain an efficient result for a definite integral with a certain accuracy.
As to this target, the aim is fulfilled. Of course, more sampling points could further improve the accuracy provided the distribution of sampling points follows the rules of uniform design and good lattice points at this stage. Exploration of much better distributions of sampling points might be one of future directions for a more efficient assessment of a definite integral. Applications of the present approach might be another orientation for future studies.
The efficient approach to a definite integral developed here on the basis of the uniform test design method is promising from the viewpoint of practical application.
An efficient result for a definite integral of a function could be obtained by using this approach with 11 sampling points for one dimension, 17 sampling points for two dimensions, and 19 sampling points for three dimensions within its single peak domain.
Byrd, P. Berlin, Heidelberg: Springer-Verlag. Caflisch, R. Valuation of mortgage-backed securities using Brownian bridges to reduce effective dimension.
Journal of Computational Finance , 1 1 , pp. Chen, Z. Integration of Multivariate Functions by Orthogonal Arrays. In Chinese. Fang, K-T. Theory and Application of Uniform Experimental Designs. Number-theoretic Methods in Statistics.
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