Оптимальне чисельне інтегрування

Abstract

Побудовано оптимальні за точністю та близькі до них квадратурні формули наближеного обчислення перетворення Фур`є, вейвлет-перетворень, перетворення Бесселя для деяких класів підінтегральних функцій. Застосовано теорію тестування якості прикладного програмного забезпечення до ви-значення якості запропонованих квадратур-них формул та оцінок їх характеристик.

The purpose of the article is to use the example of constructing optimal in accuracy (and close to them) quadrature formulas for calculating integrals for integrands of various degrees of smoothness and for oscillat-ing factors of different types and constructing a priori estimates of their total error, as well as applying to them of the theory of testing the quality of algorithms-programs to create a theory of optimal numerical integration. Results. The optimal in accuracy (and close to them) quadrature formulas for calculating the Fourier transform, wavelet transforms, and Bessel transform were constructed both in the classical formulation of the problem and for interpolation classes of functions corresponding to the case when the information operator about the integrand is given by a fixed table of its values. The paper considers a passive pure minimax strategy for solving the problem. Within the framework of this strategy, we used the method of “caps” by N.S. Bakhvalov and the method of boundary functions developed at the V.M. Glushkov Institute of Cybernet-ics of the NAS of Ukraine. Great attention is paid to the quality of the error estimates and the methods to obtain them. The article describes some aspects of the theory of algorithms-programs testing and presents the results of testing the constructed quadrature formulas for calculating integrals of rapidly oscillating functions and esti-mates of their characteristics. The problem of determining the ranges of admissible values of control parame-ters of programs for calculating integrals with the required accuracy, as well as their best values for integration with the minimum possible error, is considered for programs calculating a priori estimates of characteristics.