The problem of the best Chebyshev approximation is discussed. The advantages of approximation algorithms associated with tradeoff between speed and accuracy are substantiated.
Similar content being viewed by others
References
P. L. Chebyshev, Complete Collection of Works [in Russian], Vols. 2, 3, Izd. AN SSSR, Moscow (1947).
E. Ya. Remez, Fundamentals of Numerical Methods of Chebyshev Approximations [in Russian], Naukova Dumka, Kyiv (1969).
I. P. Natanson, Constructive Theory of Functions [in Russian], Gostekhizdat, Moscow–Leningrad (1949).
N. I. Akhiezer, Lectures on the Theory of Approximation [in Russian], Nauka, Moscow (1965).
S. B. Stechkin and Yu. N. Subbotin, Splines in Computational Mathematics [in Russian], Nauka, Moscow (1976).
V. F. Dem’yanov and V. N. Malozemov, Introduction to Minimax [in Russian], Nauka, Moscow (1972).
V. I. Berdyshev and L. V. Petrak, Approximation of Functions, Compression of Numerical Information, Applications [in Russian], UrO RAN, Ekaterinburg (1999).
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.
N. S. Bakhvalov, Numerical Methods [in Russian], Nauka, Moscow (1973).
A. Ralston, “Rational Chebyshev approximation by Remes’ algorithm,” Numerische Mathematik, 7, No. 4, 322–330 (1965).
H. Werner, J. Stoer, and W. Bommas, “Rational Chebyshev approximation,” Numerische Mathematik, 10, N 4, 342–352 (1967).
A. M. Ostrowsky, Solution of Equations and Systems of Equations, Acad. Press, New York (1960).
E. Stiefel, “Note on Jordan elimination, linear programming and Chebysheff approximation,” Numerische Mathematik, 2, 1–17 (1960).
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.
V. V. Voevodin, Roundoff Errors and Stability in Direct Methods of Linear Algebra [in Russian], Izd. MGU i VTs, Moscow (1969).
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).
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.
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.
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.
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.
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.
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.
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.
A. A. Kalenchuk-Porkhanova and L. P. Vakal, “Best Chebyshev approximation for compressing numerical information,” Komp. Matematika, No. 1, 3–9 (2009).
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).
Author information
Authors and Affiliations
Additional information
Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 155–164, November–December 2009.
Rights and permissions
About this article
Cite this article
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
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10559-009-9163-8