Algorithm for Constructing the Best Uniform Approximation Polynomial from Experimental Data
https://doi.org/10.56304/S2304487X1905002X
Abstract
Approximation of a set of experimental points using a set of functions is actual for many engineering studies. To solve such problems, the concept of a linear normed space, whose elements are bounded real functions, is introduced, and the concept of a metric (norm), i.e., a measure of proximity between the elements of the space is used. In many cases, it is required to approximate a complex function by a polynomial of a given order and, at the same time, to ensure the maximum deviation of the polynomial from the function by no more than a certain specified error. In this case, it is reasonable to use the Chebyshev norm and look for a polynomial of the best uniform approximation. However, there are no universal effective algorithms for finding the best-dimensional approximation polynomial. In this work, a simple and efficient algorithm is proposed for constructing the best uniform approximation polynomial for continuously differentiable functions. The algorithm consists of three stages. At the first stage, a polynomial of the degree n is constructed using the least squares method. At the second stage, a special system of nonlinear equations is obtained. At the third stage, the coefficients of the best uniform approximation polynomial and the Chebyshev alternance point are found by solving a system of nonlinear equations by any iterative method. This algorithm is implemented in the SciLab 6.0 system and is experimentally tested.
About the Author
K. Ya. Kudryavtsev
National Research Nuclear University MEPhI (Moscow Engineering Physics Institute)
Russian Federation
Russian Federation
115409
Moscow
References
1. Lebedev V. I. Funktsionalnyy analiz i vychislitelnaya matematika [Functional analysis and computational mathematics]. M.: FIZMATLIT, 2005. 296 p. – ISBN 5-9221-0092-0
2. Babenko V.F. On best uniform approximations by splines in the presence of restrictions on their derivatives. – Mathematical notes of the Academy of Sciences of the USSR December 1991, Volume 50, Issue 6, pp.1227–1232.
3. Dzyadyk V. K. Vvedenie v teoriyu ravnomernogo priblizheniya funktsiy polinomami [Introduction to the theory of uniform approximation of functions by polynomials]. M .: Science, 1977. 512 p.
4. Remez E. Ya. Osnovy chislennykh metodov chebyshevskogo priblizheniya [Fundamentals of numerical methods for the Chebyshev approximation]. – Kiev: Naukova Dumka, 1969.
5. Laurent P.-J. Approksimatsiya i optimizatsiya [Approximation and optimization]. M.: MIR, 1975.
6. Bakhvalov N. S., Zhidkov N. P., Kobelkov G. M. Chislennye metody [Numerical methods]. M.: BINOM. Laboratory of Knowledge, 2008. 636 pp.
7. Scilab – free and open source software. Available at: https://www.scilab.org/ (accessed 18. 01. 2019)
Review
For citations:
Kudryavtsev K.Ya. Algorithm for Constructing the Best Uniform Approximation Polynomial from Experimental Data. Vestnik natsional'nogo issledovatel'skogo yadernogo universiteta "MIFI". 2019;8(5):480-486. (In Russ.) https://doi.org/10.56304/S2304487X1905002X
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