Best Chebyshev approximation of functions of one and many variables
- PDF / 125,391 Bytes
- 9 Pages / 595.276 x 793.701 pts Page_size
- 102 Downloads / 213 Views
BEST CHEBYSHEV APPROXIMATION OF FUNCTIONS OF ONE AND MANY VARIABLES UDC 519.651.2
A. A. Kalenchuk-Porkhanova
The problem of the best Ñhebyshev approximation is discussed. The advantages of approximation algorithms associated with tradeoff between speed and accuracy are substantiated. Keywords: approximation, algorithms, Chebyshev alternance, error estimates, optimization, compression of data files. INTRODUCTION The difficulties arising in processing data files when solving applied problems such as mathematical modeling and forecasting, storage of great amount of data and their high-speed transmission through communication channels, and obtaining the necessary additional data on functional dependences at “cites lacking measurements” necessitates the analytical processing of large information flows and make this problem urgent. The problem can be solved by using different methods of analytical processing of files in order to represent them approximately (to approximate) as analytical expressions (approximants) with a small number of parameters. As compared with interpolation and root-mean-square approximations, the best Chebyshev approximation is most efficient, universal, and has an especial property of not only obtaining a high approximation accuracy at points of discrete representation of functional dependences but also providing such an accuracy at all the points of the continuous interval of their definition. This allows solving with high accuracy both the direct problem of data file compression with large compaction ratio and the inverse problem of obtaining new values of discrete representation of initial functional dependences. The fundamentals of the theory of best uniform approximation of functions were laid down by P. L. Chebyshev (1821–1894) [1] and developed in the beginning of the last century by S. N. Bernstein, P. Kirchberg, and C. J. de la Vallee Pousin. A systematic development of general numerical approaches to solve the Chebyshev approximation problem began only in 1933–1934 with fundamental studies by E. Ya. Remez [2] who proposed two theoretically justified methods (the first and the second) based on sequential Chebyshev approximations. However, the computational complexity of the powerful apparatus of the Chebyshev approximation (which covered such divisions of the numerical analysis as the solution of systems of linear, nonlinear, and differential equations, matrix algebra, operations with continued fractions and polynomials, and Boolean algebra) on the one hand and limited capacities of computational means existing for that period on the other hand did not allow obtaining the best approximants in practice. The numerical implementation of the apparatus as a whole became possible only with the creation of computers. In 1951, Small electronic accounting machine MESM, first in the Europe, was created in Kyiv, and since the late 1950s, the park of domestic computers proliferated: “Kiev,” “Strela,” “Ural,” M-20, M-220, a series of BESM computers, etc. By this time, the programming department, fir
Data Loading...
