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Chebyshev Approximation of Multivariable Functions by a Logarithmic Expression

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Abstract

The method for constructing the Chebyshev approximation of multivariable functions by a logarithmic expression with an absolute error is proposed. It implies constructing an intermediate Chebyshev approximation of the values of the exponent of an approximated function by a polynomial with the relative error. The construction of the Chebyshev approximation by a polynomial is based on calculating the boundary mean-power approximation by an iterative scheme based on the least squares method with properly formed values of variable weight function. The presented results of test examples’ solving confirm the fast convergence of the method in calculating the parameters of the Chebyshev approximation of the functions of one, two, and three variables by the logarithmic expression.

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References

  1. R. Gutierrez and J. Valls, “Low cost hardware implementation of logarithm approximation,” IEEE Trans. Very Large Scale Integr. (VLSI) Syst., Vol. 19, No. 12, 2326–2330 (2011). https://doi.org/10.1109/TVLSI.2010.2081387.

  2. A. F. Hajjar and M. H. Awedh, “Efficient logarithmic function approximation,” Int. J. Sci. Eng. Technol., Vol. 4, Iss.7, 387–391 (2015).

  3. Y. L. Luke, Mathematical Functions and their Approximations, 1st ed., Academic Press (1975).

  4. B. A. Popov and G. S. Tesler, Computer Calculation of Functions: A Reference Book [in Russian], Naukova Dumka, Kyiv (1984).

    Google Scholar 

  5. B. A. Popov and G. S. Tesler, Approximation of Functions for Engineering Applications [in Russian], Naukova Dumka, Kyiv (1980).

    MATH  Google Scholar 

  6. P. S. Malachivskyy, Ya. V. Pizyur, N. V. Danchak, and E. B. Orazov, “Chebyshev approximation by exponential-power expression,” Cybern. Syst. Analysis, Vol. 49, No. 6, 877–881 (2013). https://doi.org/10.1007/s10559-013-9577-1.

    Article  MathSciNet  MATH  Google Scholar 

  7. P. S. Malachivskyy, Ya. V. Pizyur, N. V. Danchak, and E. B. Orazov, “Chebyshev approximation by exponential expression with relative error,” Cybern. Syst. Analysis, Vol. 51, No. 2, 286–290 (2015). https://doi.org/10.1007/s10559-015-9720-2.

    Article  MathSciNet  MATH  Google Scholar 

  8. A. O. Kalenchuk-Porkhanova and L. P. Vakal, “Recovering functional dependences based on nonlinear approximations of some types,” Abstr. of Intern. Conf. “Problems of Decision Making under Uncertainties” (Chernivtsi, Ukraine, May 21–25, 2007), Chernivtsi (2007), pp. 135–137.

  9. P. S. Malachivskyy, Ya. V. Pizyur, R. P. Malachivsky, “Chebyshev approximation by a rational expression for functions of many variables,” Cybern. Syst. Analysis, Vol. 56, No. 5, 811–819 (2020). https://doi.org/10.1007/s10559-020-00302-0.

    Article  MATH  Google Scholar 

  10. O. A. Gerashchenko, A. I. Gordov, A. K. Eremina, et al., Temperature Measurements: A Reference Book [in Russian], Naukova Dumka, Kyiv (1989).

    Google Scholar 

  11. S. Rudtsch and C. von Rohden, “Calibration and self-validation of thermistors for high-precision temperature measurements,” Measurement, Vol. 76, 1–6 (2015). https://doi.org/10.1016/j.measurement.2015.07.028.

    Article  Google Scholar 

  12. Y. Huang and M. B. Shahabadi, “Why logarithmic? A note on the dependence of radiative forcing on gas concentration,” J. Geophys. Res. Atmos., Vol. 119, Iss. 24, 13683–13689 (2014). https://doi.org/10.1002/2014JD022466.

  13. M. B. Shahabadi and Y. Huang, “Logarithmic radiative effect of water vapor and spectral kernels,” J. Geophys. Res. Atmos., Vol. 119, Iss. 10, 6000–6008 (2014). https://doi.org/10.1002/2014JD021623.

  14. J. Majewski, “Low humidity characteristics of polymer-based capacitive humidity sensors,” Metrol. Meas. Syst., Vol. 24, No. 4, 607–616 (2017). DOI: https://doi.org/10.1515/MMS-2017-0048

    Article  MathSciNet  Google Scholar 

  15. A. Bomba, S. Baranovsky, O. Blavatska, and L. Bachyshyna, “Infectious disease model generalization based on diffuse perturbations under conditions of body’s temperature reaction,” Comput. Biol. Med., Vol. 146, 105561 (2022). https://doi.org/10.1016/j.compbiomed.2022.105561.

  16. A. Ya. Bomba, S. V. Baranovsky, M. S. Pasichnyk, and O. V. Pryshchepa, “Modeling small-scale spatial distributed influences on the development of infectious disease process,” Math. Model. Comput., Vol. 7, No. 2, 310–321 (2020). https://doi.org/10.23939/mmc2020.02.310.

  17. P. S. Malachivskyy, Ya. N. Matviychuk, Ya. V. Pizyur, and R. P. Malachivskyi, “Uniform approximation of functions of two variables,” Cybern. Syst. Analysis, Vol. 53, No. 3, 426–431. (2017). https://doi.org/10.1007/s10559-017-9943-5.

    Article  MathSciNet  MATH  Google Scholar 

  18. P. S. Malachivskyy, Ya. V. Pizyur, R. P. Malachivskyi, and O. M. Ukhanska, “Chebyshev approximation of functions of several variables,” Cybern. Syst. Analysis, Vol. 56, No. 1, 76–86 (2020). https://doi.org/10.1007/s10559-020-00227-8.

    Article  MATH  Google Scholar 

  19. P. S. Malachivskyy, L. S. Melnychok, and Ya. V. Pizyur, “Chebyshev approximation of multivariable functions by the exponential expression,” Cybern. Syst. Analysis, Vol. 57, No. 3, 429–435 (2021). https://doi.org/10.1007/s10559-021-00367-5.

    Article  MathSciNet  MATH  Google Scholar 

  20. 20 L. Collatz and W. Krabs, Approximationstheorie: Tschebyscheffsche Approximation mit Anwendungen (Teubner Studienbucher Mathematik), Vieweg+Teubner Verlag,Wiesbaden (1973). https://doi.org/10.1007/978-3-322-94885-4.

    Article  MATH  Google Scholar 

  21. E. Ya. Remez, Fundamentals of the Numerical Methods of Chebyshev Approximation [in Russian], Naukova Dumka, Kyiv (1969).

    MATH  Google Scholar 

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Authors and Affiliations

  1. Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, National Academy of Sciences of Ukraine, Lviv, Ukraine

    P. S. Malachivskyy & L. S. Melnychok

  2. Lviv Polytechnic National University, Lviv, Ukraine

    Ya. V. Pizyur

Authors
  1. P. S. Malachivskyy
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  2. L. S. Melnychok
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  3. Ya. V. Pizyur
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Correspondence to P. S. Malachivskyy.

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Translated from Kibernetyka ta Systemnyi Analiz, No. 2, March–April, 2023, pp. 158–165.

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Malachivskyy, P.S., Melnychok, L.S. & Pizyur, Y.V. Chebyshev Approximation of Multivariable Functions by a Logarithmic Expression. Cybern Syst Anal 59, 317–324 (2023). https://doi.org/10.1007/s10559-023-00565-3

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  • DOI: https://doi.org/10.1007/s10559-023-00565-3

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