Chinese Journal of Computers   Full Text
Title	Solving Numerical Integration Based on Evolution Strategy Method
Authors	ZHOU Yong-Quan1) ZHANG Ming2) ZHAO Bin3)
Address	1)(College of Mathematics and Computer Science, Guangxi University for Nationalities, Nanning 530006) 2)(School of Science, Dalian Fisheries University, Dalian, Liaoning 116023) 3)(School of Science, Central University for Nationalities, Beijing 100081)
Year	2008
Issue	No.2(196—206)
Abstract &   Background	Abstract In this paper, two new kinds of calculating numerical integration methods based on evolution strategy algorithm are proposed. One of which is based on mixing basis function evolution strategies for solving numerical integration, and the other is based on inequality point’s segmentation for solving numerical integration. Both of these two algorithms adopt single-gene mutation evolution strategies that suit for high dimensional optimization, which can not only compute usual definite integral for any functions, but also compute singular integral and oscillatory integral. Finally, several experimental results show that the two proposed numerical integration methods are more efficient and feasible in computing the arbitrary functions numerical integration compared with traditional numerical integration methods. keywords hybrid basis functions; inequality point’s segmentation; fitness; evolution strategies; single-gene mutation; numerical integration background This work is supported by the National Natural Science Foundation of China under grant No.60461001 and the Natural Science Foundation of Guangxi under grant No.0542048. In engineering technology, the numerical integration is a very basic computation problem. So far, the method which people usually adopt is to solve the primitive function of the integrated function at first, then compute the definite integral by the Newton-Leibniz formula. In fact, it is not easy or direct to compute the solutions of many definite integral by using the above method, as some integrated functions’ explicit primitive functions can not be shown or the primitive functions which are so complex are not suitable for computation. Therefore, it is necessary to propose a novel computation method to solve numerical integration, in order to solve the defect of the approximate definite integral computing method. According to these problems, an approximate numerical integral evolution strategy computing method was proposed based on hybrid basis functions, whose basic idea is to use hybrid basis functions evolution strategies algorithm to approximate integrated function in order to accomplish the definite integral’s approximate computing. On the other hand, nowadays people usually use the numerical integral formula based on equally spaced nodes Summed-division or the regulated equally spaced nodes which are invariable in the whole computing process. All the integral formulas having equally spaced nodes can not be divided according to the form of the integrated functions, we need more nodes to obtain higher precision, the best division is not the equally spaced previously chosen, but to choose division points randomly in terms of the form of the integral function which means the concave and protruding form of the integral function curve. Thus, divide first, sum next, the precision of the function’s integral value is higher than the traditional method. In the light of the idea, another Evolution Strategies numerical integral method based on unequally spaced nodes division is proposed, not only this Algorithm can calculate the usual significant functions’ definite integral but also calculate the singular integral and oscillating integral. The two algorithms in this paper proposed can not only calculate the usual significant functions’ definite integral but also calculate the singular integral and oscillating integral can be regarded as the improvement of the traditional numerical integral method, they have widespread application in the engineering technology and science computation.