Skip to main content
SpringerLink
Log in

Using reserves for computation optimization to improve the integration of rapidly oscillating functions

  • Published:
Cybernetics and Systems Analysis Aims and scope

Abstract

We present a theory of evaluating integrals of rapidly oscillating functions in various classes of subintegral functions with the use of a mesh information operator on subintegral functions. The theory allows us to derive and prove optimal (with respect to accuracy and (or) performance) and nearly optimal quadrature formulas and to test their quality against well-known and proposed numerical integration algorithms and to determine their efficiency domains. A technique is proposed to determine the optimal parameters of computational algorithms that obtain the ε-solution of the problem.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. V. K. Zadiraka and S. S. Melnikova, Digital Signal Processing [in Russian], Naukova Dumka, Kyiv (1993).

    Google Scholar 

  2. J. F. Traub and H. Wozniakowski, A General Theory of Optimal Algorithms, Academic Press (1980).

  3. V. V. Ivanov, Metods for Computer Calculations: A Handbook [in Russian], Naukova Dumka, Kyiv (1986).

    Google Scholar 

  4. V. V. Ivanov, M. D Babich, A. I. Berezovskii, et al., Characteristics of Problems, Algorithms, and Computers in Program Systems of Computational Mathematics [in Russian], Prepr. V. M. Glushkov Institute of Cybernetics AS USSR, 84–36, Kyiv (1984).

  5. V. S. Mikhalevich, I. V. Sergienko, V. K. Zadiraka, et al., To Develop a Quality Testing System for Application Software [in Russian], Dep. in VNTNTs, No. 0290.037707, Kyiv (1989).

  6. V. K. Zadiraka, M. D Babich, A. I. Berezovskii, et al., T-Eficient Algorithms for Approximate Solution of Problems in Computational and Applied Mathematics [in Ukrainian], Zbruch, Ternopil (2003).

  7. M. D. Babich, A. I. Berezovskii, P. M. Besarab, et al., “A technology to solve problems in applied and computational mathematics with prescribed quality characteristics,” in: Theory of Computations [in Ukrainian], V. M. Glushkov Institute of Cybernetics AS USSR, Kyiv (1999), pp. 16–20.

  8. J. H. Wilkinson, The Algebraic Eigenvalue Problem, Oxford University Press, London (1965).

    MATH  Google Scholar 

  9. M. D. Babich, A. I. Berezovskii, P. N. Besarab, et al., “T-efficient calculation of the ε-solutions to problems of calculus and applied mathematics. I; II,” Cybern. Syst. Analysis, 37, No. 2, 187–202 (2001); No. 3, 381–397 (2001).

  10. N. S. Bakhvalov, Numerical Methods [in Russian], Vol. 1, Nauka, Moscow (1973).

    Google Scholar 

  11. Ya. M. Zhileikin and A. B. Kukarkin, Approximate Integration of Rapidly Oscillating Functions: A Handbook [in Russian], MGU, Moscow (1987).

  12. V. K. Zadiraka, S. S. Mel’nikova, and L. V. Luts, “Optimal quadrature and cubature formulas for computing Fourier transform of finite functions of one class. Case of strong oscillation,” Cybern. Syst. Analysis, 43, No. 5, 731–748 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  13. L. V. Luts, “Estimating the quality of some quadrature formulas to integrate rapidly oscillating functions,” Shtuch. Intelekt, No. 4, 671–682 (2008).

  14. A. I. Berezovskii and O. S. Kondratenko, “Revealing and revising a priori information,” USiM, No. 6, 17–22 (1997).

  15. L. V. Luts, “Testing the quality of quadrature formulas for integrating rapidly oscillating functions of class W 2, L, N ,” Komp. Matematika, No. 2, 107–116 (2007).

  16. D. Evans (ed.), Parallel Processing Systems, Cambridge Univ. Press (1982).

  17. G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw Hill, New York (1968).

    Google Scholar 

  18. M. D. Babich, “An approximation-iteration method for solving nonlinear operator equations,” Cybern. Syst. Analysis, 43, No. 1, 26–38 (1991).

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. V. M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine, Kyiv, Ukraine

    V. K. Zadiraka, S. S. Melnikova & L. V. Luts

Authors
  1. V. K. Zadiraka
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. S. Melnikova
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. L. V. Luts
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. K. Zadiraka.

Additional information

Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 125–145, July–August 2011.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zadiraka, V.K., Melnikova, S.S. & Luts, L.V. Using reserves for computation optimization to improve the integration of rapidly oscillating functions. Cybern Syst Anal 47, 613–630 (2011). https://doi.org/10.1007/s10559-011-9342-2

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10559-011-9342-2

Keywords

Search

Navigation