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

  • V. K. Zadiraka
  • S. S. Melnikova
  • L. V. Luts


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.


integrals of rapidly oscillating functions quadrature formulas optimal algorithms quality characteristic testing ε-solution of a problem 


  1. 1.
    V. K. Zadiraka and S. S. Melnikova, Digital Signal Processing [in Russian], Naukova Dumka, Kyiv (1993).Google Scholar
  2. 2.
    J. F. Traub and H. Wozniakowski, A General Theory of Optimal Algorithms, Academic Press (1980).Google Scholar
  3. 3.
    V. V. Ivanov, Metods for Computer Calculations: A Handbook [in Russian], Naukova Dumka, Kyiv (1986).Google Scholar
  4. 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).Google Scholar
  5. 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).Google Scholar
  6. 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).Google Scholar
  7. 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.Google Scholar
  8. 8.
    J. H. Wilkinson, The Algebraic Eigenvalue Problem, Oxford University Press, London (1965).MATHGoogle Scholar
  9. 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).Google Scholar
  10. 10.
    N. S. Bakhvalov, Numerical Methods [in Russian], Vol. 1, Nauka, Moscow (1973).Google Scholar
  11. 11.
    Ya. M. Zhileikin and A. B. Kukarkin, Approximate Integration of Rapidly Oscillating Functions: A Handbook [in Russian], MGU, Moscow (1987).Google Scholar
  12. 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).MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    L. V. Luts, “Estimating the quality of some quadrature formulas to integrate rapidly oscillating functions,” Shtuch. Intelekt, No. 4, 671–682 (2008).Google Scholar
  14. 14.
    A. I. Berezovskii and O. S. Kondratenko, “Revealing and revising a priori information,” USiM, No. 6, 17–22 (1997).Google Scholar
  15. 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).Google Scholar
  16. 16.
    D. Evans (ed.), Parallel Processing Systems, Cambridge Univ. Press (1982).Google Scholar
  17. 17.
    G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw Hill, New York (1968).Google Scholar
  18. 18.
    M. D. Babich, “An approximation-iteration method for solving nonlinear operator equations,” Cybern. Syst. Analysis, 43, No. 1, 26–38 (1991).MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2011

Authors and Affiliations

  • V. K. Zadiraka
    • 1
  • S. S. Melnikova
    • 1
  • L. V. Luts
    • 1
  1. 1.V. M. Glushkov Institute of Cybernetics, National Academy of Sciences of UkraineKyivUkraine

Personalised recommendations