Modular Technology of Definite Multiple Integrals Calculation: Analytical Analysis and Experimental Verification of Efficiency

Analyzed is a new approach to the use of enumeration procedures for calculation of definite multiple integrals. It is shown that the effectiveness of this approach depends on the following factors: on requirements for the accuracy of calculations, on complexity of the integrand and on the number of variables of the problem to be solved. Presented are the conditions that allow us to adapt the modular enumeration algorithms in relation to the numerical parameters, characterizing these factors. The aim of adaptation is to minimize the search time. Analytical analysis of the comparative efficiency of brute force and modular enumeration algorithms for calculating multiple definite integrals leads to the following results: 1. The optimal number of modules is really equal to two. This result coincides with the optimal search mode by modular enumeration of globally optimal solutions to discrete programming problems. 2. The gain in time η in the time of calculating a definite integral by modular technology compared with the time of calculating the same definite integral by brute force exceeds value of the number of variables n. 3. The greater is the number of variables n and the greater is the complexity of the integrand, the greater is the gain in computation time. Experimental verification of the analytical results obtained confirms their validity.