An efficient approach for calculating a definite integral with about a dozen of sampling points

Abstract

Introduction/purpose: An approximate approach to definite integral calculation has been an attractive problem continuously since the creation of integration due to practical needs in scientific and engineering areas. In most practical cases, the integrand is complex, which leads to a difficulty of obtaining an exact value of integration, so an approximate value of the definite integral with certain accuracy is satisfactory for practical applications. In this paper, an efficient approach for calculating a definite integral with a small number of sampling points is proposed based on the uniform design method from the viewpoint of practical application. Methods: The distribution of sampling points in its single peak domain is deterministic and uniform, which follows the rule of the uniform design method and good lattice points. Results: The efficient evaluation of a definite integral for a periodical function in its single peak domain can be obtained by using 11 sampling points in one dimension, 17 sampling points in two dimensions, and 19 sampling points in three dimensions. Conclusion: The efficient approach for a definite integral developed here on the basis of the uniform test design method is promised from the viewpoint of practical application; the sampling points are deterministically and uniformly distributed according to the rule of the uniform design method and “good lattice points”. The efficient approach developed in this article will be beneficial to relevant research and application.

Keywords

• uniform design method
• good lattice point
• definite integral
• single peak domain
• finite sampling points