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Best Chebyshev approximation of functions of one and many variables

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Cybernetics and Systems Analysis Aims and scope

The problem of the best Chebyshev approximation is discussed. The advantages of approximation algorithms associated with tradeoff between speed and accuracy are substantiated.

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References

  1. P. L. Chebyshev, Complete Collection of Works [in Russian], Vols. 2, 3, Izd. AN SSSR, Moscow (1947).

    Google Scholar 

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

    Google Scholar 

  3. I. P. Natanson, Constructive Theory of Functions [in Russian], Gostekhizdat, Moscow–Leningrad (1949).

    Google Scholar 

  4. N. I. Akhiezer, Lectures on the Theory of Approximation [in Russian], Nauka, Moscow (1965).

    Google Scholar 

  5. S. B. Stechkin and Yu. N. Subbotin, Splines in Computational Mathematics [in Russian], Nauka, Moscow (1976).

    Google Scholar 

  6. V. F. Dem’yanov and V. N. Malozemov, Introduction to Minimax [in Russian], Nauka, Moscow (1972).

    Google Scholar 

  7. V. I. Berdyshev and L. V. Petrak, Approximation of Functions, Compression of Numerical Information, Applications [in Russian], UrO RAN, Ekaterinburg (1999).

    Google Scholar 

  8. A. A. Kalenchuk-Porkhanova, “Best Chebyshev approximation: Algorithms and their Application,” in: Proc. Intern. Symp. “Computation Optimization Problems (POO–XXXV)” [in Ukrainian], V. M. Glushkov Inst. of Cybernetics (2009), pp. 279–284.

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

    Google Scholar 

  10. A. Ralston, “Rational Chebyshev approximation by Remes’ algorithm,” Numerische Mathematik, 7, No. 4, 322–330 (1965).

    Article  MATH  MathSciNet  Google Scholar 

  11. H. Werner, J. Stoer, and W. Bommas, “Rational Chebyshev approximation,” Numerische Mathematik, 10, N 4, 342–352 (1967).

    Article  MathSciNet  Google Scholar 

  12. A. M. Ostrowsky, Solution of Equations and Systems of Equations, Acad. Press, New York (1960).

    Google Scholar 

  13. E. Stiefel, “Note on Jordan elimination, linear programming and Chebysheff approximation,” Numerische Mathematik, 2, 1–17 (1960).

    Article  MATH  MathSciNet  Google Scholar 

  14. V. L. Aleksandrenko, “An algorithm to construct an approximate uniformly best solution of a system of inconsistent linear equations,” in: Algorithms and Algorithmic Languages [in Russian], Issue 3, VTs AN SSSR, Moscow (1968), pp. 57–74.

    Google Scholar 

  15. V. V. Voevodin, Roundoff Errors and Stability in Direct Methods of Linear Algebra [in Russian], Izd. MGU i VTs, Moscow (1969).

    Google Scholar 

  16. V. L. Aleksandrenko and A. A. Porkhanova, Algorithm for approximating functions of one variable with polynomials by the Remez second method,” in: Vych. Prikl. Mat., Issue 3, Izd. KGU, Kyiv (1967).

    Google Scholar 

  17. A. A. Kalenchuk-Porkhanova, “An algorithm of polynomial Chebyshev approximation,” in: Optimization of Computational Methods [in Russian], Inst. Kibern. AN USSR, Kyiv (1974), pp. 45–51.

    Google Scholar 

  18. A. A. Porkhanova, “Chebyshev approximation by rational-fractional expressions,” in: Computational Mathematics in Modern Scientific and Technical Progress [in Russian], Znanie, Kyiv (1974), pp. 300–308.

    Google Scholar 

  19. A. A. Kalenchuk-Porkhanova, “Algorithms and analysis of the error of the best one-variable Chebyshev approximation,” in: Proc. Intern. Conf. on the Theory of Function Approximation, Kaluga, 1975 [in Russian], Nauka, Moscow (1977), pp. 213–218.

    Google Scholar 

  20. V. V. Ivanov and A. A. Kalenchuk, “Efficiency of algorithms of polynomial and rational-fractional Chebyshev approximations,” in: Proc. Intern. Conf. on the Constructive Theory of Functions, Izd. Bolg. AN, Sofia (1983), 72–77.

    Google Scholar 

  21. A. A. Kalenchuk-Porkhanova, “Approximation of functions of one and many variables,” in: Numerical Methods for the Multiprocessor Computer Complex ES [in Russian], Izd. VVIA im. N. E. Zhukovskogo (1987), pp. 366–395.

  22. A. A. Kalenchuk-Porkhanova, “Approximation apparatus for the analysis and synthesis of complex systems,” in: Proc. Intern. Conf. “50th anniversary of the V. M. Glushkov Institute of Cybernetics NASU” [in Ukrainian], Kyiv (2008), 354–361.

  23. A. A. Kalenchuk-Porkhanova, “Algorithms for implementing the best Chebyshev approximations — increase of their efficiency,” in: Proc. Intern. Symp. “Computation Optimization Problems (POO–XXXVI)” [in Ukrainian], V. M. Glushkov Inst. of Cybernetics (2009), pp. 285–290.

  24. A. A. Kalenchuk-Porkhanova and L. P. Vakal, “Best Chebyshev approximation for compressing numerical information,” Komp. Matematika, No. 1, 3–9 (2009).

  25. A. A. Kalenchuk-Porkhanova and L. P. Vakal, “Approximation apparatus within the software for a supercomputer with cluster architecture,” Iskusstv. Intellekt, No. 1, 52–60 (2009).

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Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 155–164, November–December 2009.

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Kalenchuk-Porkhanova, A.A. Best Chebyshev approximation of functions of one and many variables. Cybern Syst Anal 45, 988–996 (2009). https://doi.org/10.1007/s10559-009-9163-8

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