Theory of integrals computing from fast oscillating functions

Authors:

Valeriy K. Zadiraka, Head of the Department of Numerical Methods for Optimization at the V.M. Glushkov Institute of Cybernetics of the NAS of Ukraine, Academician of NASU, Doctor of Physical and Mathematical Sciences, Professor; V. M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine, Kyiv, Ukraine.

https://orcid.org/0000-0001-9628-0454

Scopus Author ID: 14062655100

http://iscopt.com.ua/index.php/uk/zadiraka-valerij-kostyantinovich8

 

Liliya V. Luts, Senior Researcher of the Department of Numerical Methods for Optimization at the V.M. Glushkov Institute of Cybernetics of the NAS of Ukraine, Candidate of Physical and Mathematical Sciences (Ph. D.); V. M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine, Kyiv, Ukraine.

https://orcid.org/0000-0003-0746-9701

Scopus Author ID: 23025089400

http://www.webofscience.com/wos/author/record/AAF-8037-2021

http://iscopt.com.ua/index.php/uk/luts-liliya-volodimirivna

 

Inna V. Shvidchenko, Leading Researcher of the Department of Numerical Methods for Optimization at the V.M. Glushkov Institute of Cybernetics of the NAS of Ukraine, Candidate of Physical and Mathematical Sciences (Ph. D.), Senior Research Officer; V. M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine, Kyiv, Ukraine.

https://orcid.org/0000-0002-5434-2845

Scopus Author ID: 23394005400

http://www.webofscience.com/wos/author/record/M-4280-2018

http://iscopt.com.ua/index.php/uk/shvidchenko-inna-vitalijivna

 

Reviewers:

Oleksandr M. Khimich is Deputy Director for Research, Head of the Department of Department of Numerical Methods and Computer Modeling at the V.M. Glushkov Institute of Cybernetics of the NAS of Ukraine, Academician of the NASU, Doctor of Physical and Mathematical Sciences, Professor; V. M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine, Kyiv, Ukraine.

https://orcid.org/0000-0002-8103-4223

Scopus Author ID: 6602740665

https://scholar.google.com.ua/citations?hl=ru&user=b8JVCQ8AAAAJ&view_op=list_works&sortby=pubdate

http://incyb.kiev.ua/employee/khimich-oleksandr-mykolayovych

 

Sergey I. Lyashko is Head of the Department of  Computational Mathematics of Faculty of Computer Sciences and Cybernetics of Taras Shevchenko National University of Kyiv, Correspondent Member of the NASU, Doctor of Physical and Mathematical Sciences, Professor; Taras Shevchenko National University of Kyiv, Kyiv, Ukraine.

https://orcid.org/0000-0003-1016-5231

http://www.webofscience.com/wos/author/record/F-4429-2017

https://science.knu.ua/en/researchgroups/research.php?ELEMENT_ID=2515

 

Bohdan M. Shevchuk is Leading Researcher of the Department of Numerical Methods for Optimization at the V.M. Glushkov Institute of Cybernetics of the NAS of Ukraine, Doctor of Technical Sciences, Senior Research Officer; V. M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine, Kyiv, Ukraine.

Scopus Author ID: 56431179800

https://scholar.google.com.ua/citations?hl=ru&user=1IMCDysAAAAJ

 

Affiliation:

Project: Scientific book

Year: 2023

Publisher: PH "Naukova Dumka"

Pages: 472

DOI:

https://doi.org/10.15407/978-966-00-1843-3

ISBN: 978-966-00-1843-3

Language: Ukrainian

How to Cite:

Zadiraka, V.K., Luts, L.V., Shvidchenko, I.V. (2023) Theory of integrals computing from fast oscillating functions. Kyiv, Naukova Dumka. 472p. [in Ukrainian].

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.

Keywords:

integrals of highly oscillatory functions, quadrature formula, cubature formula, optimal algorithm, quality characteristics, algorithms-programs testing.

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