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Mat. Sb., 2004, Volume 195, Number 10, Pages 21–66 (Mi msb852)  

This article is cited in 6 scientific papers (total in 7 papers)

Extremal polynomials and methods of optimization of numerical algorithms

V. I. Lebedev

Russian Research Centre "Kurchatov Institute"

Abstract: Chebyshëv–Markov–Bernstein–Szegö polynomials $C_n(x)$ extremal on $[-1,1]$ with weight functions $w(x)=(1+x)^\alpha(1- x)^\beta/\sqrt{S_l(x)}$ where $\alpha,\beta=0,\frac12$ and $S_l(x)=\prod_{k=1}^m(1-c_kT_{l_k}(x))>0$ are considered. A universal formula for their representation in trigonometric form is presented. Optimal distributions of the nodes of the weighted interpolation and explicit quadrature formulae of Gauss, Markov, Lobatto, and Rado types are obtained for integrals with weight $p(x)=w^2(x)(1-x^2)^{-1/2}$. The parameters of optimal Chebyshëv iterative methods reducing the error optimally by comparison with the initial error defined in another norm are determined. For each stage of the Fedorenko–Bakhvalov method iteration parameters are determined which take account of the results of the previous calculations. Chebyshëv filters with weight are constructed. Iterative methods of the solution of equations containing compact operators are studied.

DOI: https://doi.org/10.4213/sm852

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English version:
Sbornik: Mathematics, 2004, 195:10, 1413–1459

Bibliographic databases:

UDC: 517.518.8+519.6
MSC: Primary 41A50, 65D32, 65F10; Secondary 26C05
Received: 26.02.2004

Citation: V. I. Lebedev, “Extremal polynomials and methods of optimization of numerical algorithms”, Mat. Sb., 195:10 (2004), 21–66; Sb. Math., 195:10 (2004), 1413–1459

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. M. K. Kerimov, “V. I. Lebedev (on the occasion of his 75th birthday)”, Comput. Math. Math. Phys., 45:11 (2005), 1833–1844  mathnet  mathscinet
    2. A. L. Lukashov, “Ratsionalnye interpolyatsionnye protsessy na neskolkikh otrezkakh”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 5:1-2 (2005), 34–48  mathnet
    3. Yu. N. Karamzin, S. V. Polyakov, I. V. Popov, G. M. Kobelkov, S. G. Kobelkov, Jun Ho Choy, “Modelirovanie protsessov obrazovaniya i migratsii por v mezhsoedineniyakh elektricheskikh skhem”, Matem. modelirovanie, 19:10 (2007), 29–43  mathnet  mathscinet  zmath  elib
    4. G. I. Kurchenkova, V. I. Lebedev, “Solving reactor problems to determine the multiplication: A new method of accelerating outer iterations”, Comput. Math. Math. Phys., 47:6 (2007), 962–969  mathnet  crossref  elib  elib
    5. V. I. Lebedev, “Finding polynomials of best approximation with weight”, Sb. Math., 199:2 (2008), 207–228  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    6. Alexey Lukashov, Dmitri Prokhorov, “Approximation of sgn
      $$(x)$$
      ( x ) on Two Symmetric Intervals by Rational Functions with Fixed Poles”, Comput. Methods Funct. Theory, 2015  crossref  mathscinet
    7. A. G. Babenko, Yu. V. Kryakin, “Modifitsirovannaya funktsiya Bernshteina i ravnomernoe priblizhenie nekotorykh ratsionalnykh drobei polinomami”, Tr. IMM UrO RAN, 23, no. 3, 2017, 43–57  mathnet  crossref  elib
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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