Theory of integrals computing from fast oscillating functions

Abstract

We present a general theory of computation integrals of highly oscillatory functions (IHOF) in various classes of subintegral functions with the use of a net information operator on subintegral functions. The monograph considers the calculation of integrals involving the following functions as a kernel: exponential (Fourier transform and others), trigonometric, wavelets and Bessel functions. The proposed theory is based on the theory of calculations, theory of computational errors, general theory of optimal accuracy algorithms, algorithms for detecting and refining a priori information about the subintegral function and the theory of testing algorithms—programs. The theory allows us to derive and prove optimal (with respect to accuracy and (or) performance) and nearly optimal quadrature and cubature formulas of calculation of IHOF 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. Great attention is paid to the quality of the error estimates and the methods to obtain them. The monograph describes some aspects of the theory of algorithms-programs testing and presents the results of their quality testing against well-known and proposed numerical integration algorithms and estimations of their characteristics. The problem of determining the ranges of admissible values of control parameters of programs for calculating integrals with the required accuracy, as well as their best values for integration with the minimal possible error, is considered for programs calculating a priori estimates of characteristics. In the last part the developed computer technology of calculation of IHOF with the set values of quality characteristics on accuracy and speed is presented. For researchers, graduate students, senior students and specialists involved in the development of algorithmic and software solutions to problems related to the use of IHOF.We present a general theory of computation integrals of highly oscillatory functions (IHOF) in various classes of subintegral functions with the use of a net information operator on subintegral functions. The monograph considers the calculation of integrals involving the following functions as a kernel: exponential (Fourier transform and others), trigonometric, wavelets and Bessel functions. The proposed theory is based on the theory of calculations, theory of computational errors, general theory of optimal accuracy algorithms, algorithms for detecting and refining a priori information about the subintegral function and the theory of testing algorithms—programs. The theory allows us to derive and prove optimal (with respect to accuracy and (or) performance) and nearly optimal quadrature and cubature formulas of calculation of IHOF 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. Great attention is paid to the quality of the error estimates and the methods to obtain them. The monograph describes some aspects of the theory of algorithms-programs testing and presents the results of their quality testing against well-known and proposed numerical integration algorithms and estimations of their characteristics. The problem of determining the ranges of admissible values of control parameters of programs for calculating integrals with the required accuracy, as well as their best values for integration with the minimal possible error, is considered for programs calculating a priori estimates of characteristics. In the last part the developed computer technology of calculation of IHOF with the set values of quality characteristics on accuracy and speed is presented. For researchers, graduate students, senior students and specialists involved in the development of algorithmic and software solutions to problems related to the use of IHOF.