The error bounds of Gauss–Radau quadrature formulae with Bernstein–Szegő weight functions

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• 作者：Aleksandar V. Pejčev ; Miodrag M. Spalević
• 刊名：Numerische Mathematik
• 出版年：2016
• 出版时间：May 2016
• 年：2016
• 卷：133
• 期：1
• 页码：177-201
• 全文大小：570 KB
• 参考文献：1.Gautschi, W.: On the remainder term for analytic functions of Gauss–Lobatto and Gauss–Radau quadratures. Rocky Mountain J. Math. 21, 209–226 (1991)MathSciNet CrossRef MATH
  2.Gautschi, W., Varga, R.S.: Error bounds for Gaussian quadrature of analytic functions. SIAM J. Numer. Anal. 20, 1170–1186 (1983)MathSciNet CrossRef MATH
  3.Gautschi, W., Tychopoulos, E., Varga, R.S.: A note on the contour integral representation of the remainder term for a Gauss–Chebyshev quadrature rule. SIAM J. Numer. Anal. 27, 219–224 (1990)MathSciNet CrossRef MATH
  4.Gradshteyn, I.S., Ryzhik, I.M.: Tables of integrals, series and products. Academic Press, New York (1965)
  5.Hunter, D.B.: Some error expansions for Gaussian quadrature. BIT 35, 64–82 (1995)MathSciNet CrossRef MATH
  6.Hunter, D.B., Nikolov, G.: On the error term of symmetric Gauss–Lobatto quadrature formulae for analytic functions. Math. Comp. 69, 269–282 (2000)MathSciNet CrossRef MATH
  7.Milovanović, G.V., Spalević, M.M., Pranić, M.S.: On the remainder term of Gauss–Radau quadratures for analytic functions. J. Comput. Appl. Math. 218, 281–289 (2008)MathSciNet CrossRef MATH
  8.Notaris, S.E.: The error norm of Gaussian quadrature formulae for weight functions of Bernstein–Szegő type. Numer. Math. 57, 271–283 (1990)MathSciNet CrossRef MATH
  9.Notaris, S.E.: The error norm of quadrature formulae. Numer. Algorithm 60, 555–578 (2012)MathSciNet CrossRef MATH
  10.Notaris, S.E.: The error norm of Gauss-Radau quadrature formulae with Bernstein–Szegő weight functions. Math. Comp. (2015). doi:10.​1090/​mcom/​2944
  11.Ossicini, A., Rosati, F.: Funzioni caratteristiche nelle formule di quadratura gaussiane con nodi multipli. Boll. Un. Mat. Ital. 11, 224–237 (1975)MathSciNet MATH
  12.Peherstorfer, F.: On the remainder of Gaussian quadrature formulas for Bernstein–Szegő weight functions. Math. Comp. 60, 317–325 (1993)MathSciNet MATH
  13.Pejčev, A.V., Spalević, M.M.: On the remainder term of Gauss–Radau quadrature with Chebyshev weight of the third kind for analytic functions. Appl. Math. Comput. 219, 2760–2765 (2012)MathSciNet MATH
  14.Pejčev, A.V., Spalević, M.M.: Error bounds for Gaussian quadrature formulae with Bernstein–Szegő weights that are rational modifications of Chebyshev weight functions of the second kind. IMA J. Numer. Anal. 32, 1733–1754 (2012)MathSciNet CrossRef MATH
  15.Pejčev, A.V., Spalević, M.M.: Error bounds of Micchelli–Rivlin quadrature formula for analytic functions. J. Approx. Theory 169, 23–34 (2013)MathSciNet CrossRef MATH
  16.Pejčev, A.V., Spalević, M.M.: Error bounds of the Micchelli–Sharma quadrature formula for analytic functions. J. Comput. Appl. Math. 259, 48–56 (2014)MathSciNet CrossRef MATH
  17.Schira, T.: The remainder term for analytic functions of Gauss–Lobatto quadratures. J. Comput. Appl. Math. 76, 171–193 (1996)MathSciNet CrossRef MATH
  18.Schira, T.: The remainder term for analytic functions of symmetric Gaussian quadratures. Math. Comp. 66, 297–310 (1997)MathSciNet CrossRef MATH
  19.Scherer, R., Schira, T.: Estimating quadrature errors for analytic functions using kernel representations and biorthogonal systems. Numer. Math. 84, 497–518 (2000)MathSciNet CrossRef MATH
  20.Spalević, M.M.: Error bounds of Gaussian quadrature formulae for one class of Bernstein–Szegő weights. Math. Comp. 82, 1037–1056 (2013)MathSciNet CrossRef MATH
  21.Spalević, M.M.: Error bounds and estimates for Gauss–Turán quadrature formulae of analytic functions. SIAM J. Numer. Anal. 52, 443–467 (2014)MathSciNet CrossRef MATH
  22.Spalević, M.M., Pranić, M.S.: Error bounds of certain Gaussian quadrature formulae. J. Comput. Appl. Math. 234, 1049–1057 (2010)MathSciNet CrossRef MATH
  23.Spalević, M.M., Pranić, M.S., Pejčev, A.V.: Maximum of the modulus of kernels of Gaussian quadrature formulae for one class of Bernstein–Szegő weight functions. Appl. Math. Comput. 218, 5746–5756 (2012)MathSciNet MATH
  
• 作者单位：Aleksandar V. Pejčev (1)
  Miodrag M. Spalević (1)
  
  1. Department of Mathematics, Faculty of Mechanical Engineering, University of Beograd, Kraljice Marije 16, 11120, Belgrade 35, Serbia
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Numerical Analysis
  Mathematics
  Mathematical and Computational Physics
  Mathematical Methods in Physics
  Numerical and Computational Methods
  Applied Mathematics and Computational Methods of Engineering
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：0945-3245

文摘

We consider the Gauss–Radau quadrature formulae $$\begin{aligned} \int _{-1}^1f(t)w(t)\,dt=\sum _{\nu =1}^n\lambda _\nu f(\tau _\nu )+\lambda _{n+1}f(c)+R_n(f), \end{aligned}$$with \(c=-1\) or \(c=1\), for the Bernstein–Szegő weight functions consisting of anyone of the four Chebyshev weights divided by the polynomial \(\rho (t)=1-\frac{4\gamma }{(1+\gamma )^2}\,t^2, t\in (-1,1),\ -1<\gamma \le 0\). For analytic functions the remainder term of this quadrature formula can be represented as a contour integral with a complex kernel. We study the kernel, on elliptic contours with foci at the points \(\mp 1\) and a sum of semi-axes \(\rho >1\), for the given quadrature formula. Starting from the explicit expression of the kernel, we determine the locations on the ellipses where maximum modulus of the kernel is attained. So we derive effective error bounds for this quadrature formula. An alternative approach, which has initiated this research, has been proposed recently by Notaris (Math Comp 10.​1090/​mcom/​2944, 2015). Mathematics Subject Classification 41A55 65D32 65D30 Page %P Close Plain text Look Inside Reference tools Export citation EndNote (.ENW) JabRef (.BIB) Mendeley (.BIB) Papers (.RIS) Zotero (.RIS) BibTeX (.BIB) Add to Papers Other actions Register for Journal Updates About This Journal Reprints and Permissions Share Share this content on Facebook Share this content on Twitter Share this content on LinkedIn Related Content Supplementary Material (0) References (23) References1.Gautschi, W.: On the remainder term for analytic functions of Gauss–Lobatto and Gauss–Radau quadratures. Rocky Mountain J. Math. 21, 209–226 (1991)MathSciNetCrossRefMATH2.Gautschi, W., Varga, R.S.: Error bounds for Gaussian quadrature of analytic functions. SIAM J. Numer. Anal. 20, 1170–1186 (1983)MathSciNetCrossRefMATH3.Gautschi, W., Tychopoulos, E., Varga, R.S.: A note on the contour integral representation of the remainder term for a Gauss–Chebyshev quadrature rule. SIAM J. Numer. Anal. 27, 219–224 (1990)MathSciNetCrossRefMATH4.Gradshteyn, I.S., Ryzhik, I.M.: Tables of integrals, series and products. Academic Press, New York (1965)5.Hunter, D.B.: Some error expansions for Gaussian quadrature. BIT 35, 64–82 (1995)MathSciNetCrossRefMATH6.Hunter, D.B., Nikolov, G.: On the error term of symmetric Gauss–Lobatto quadrature formulae for analytic functions. Math. Comp. 69, 269–282 (2000)MathSciNetCrossRefMATH7.Milovanović, G.V., Spalević, M.M., Pranić, M.S.: On the remainder term of Gauss–Radau quadratures for analytic functions. J. Comput. Appl. Math. 218, 281–289 (2008)MathSciNetCrossRefMATH8.Notaris, S.E.: The error norm of Gaussian quadrature formulae for weight functions of Bernstein–Szegő type. Numer. Math. 57, 271–283 (1990)MathSciNetCrossRefMATH9.Notaris, S.E.: The error norm of quadrature formulae. Numer. Algorithm 60, 555–578 (2012)MathSciNetCrossRefMATH10.Notaris, S.E.: The error norm of Gauss-Radau quadrature formulae with Bernstein–Szegő weight functions. Math. Comp. (2015). doi:10.​1090/​mcom/​2944 11.Ossicini, A., Rosati, F.: Funzioni caratteristiche nelle formule di quadratura gaussiane con nodi multipli. Boll. Un. Mat. Ital. 11, 224–237 (1975)MathSciNetMATH12.Peherstorfer, F.: On the remainder of Gaussian quadrature formulas for Bernstein–Szegő weight functions. Math. Comp. 60, 317–325 (1993)MathSciNetMATH13.Pejčev, A.V., Spalević, M.M.: On the remainder term of Gauss–Radau quadrature with Chebyshev weight of the third kind for analytic functions. Appl. Math. Comput. 219, 2760–2765 (2012)MathSciNetMATH14.Pejčev, A.V., Spalević, M.M.: Error bounds for Gaussian quadrature formulae with Bernstein–Szegő weights that are rational modifications of Chebyshev weight functions of the second kind. IMA J. Numer. Anal. 32, 1733–1754 (2012)MathSciNetCrossRefMATH15.Pejčev, A.V., Spalević, M.M.: Error bounds of Micchelli–Rivlin quadrature formula for analytic functions. J. Approx. Theory 169, 23–34 (2013)MathSciNetCrossRefMATH16.Pejčev, A.V., Spalević, M.M.: Error bounds of the Micchelli–Sharma quadrature formula for analytic functions. J. Comput. Appl. Math. 259, 48–56 (2014)MathSciNetCrossRefMATH17.Schira, T.: The remainder term for analytic functions of Gauss–Lobatto quadratures. J. Comput. Appl. Math. 76, 171–193 (1996)MathSciNetCrossRefMATH18.Schira, T.: The remainder term for analytic functions of symmetric Gaussian quadratures. Math. Comp. 66, 297–310 (1997)MathSciNetCrossRefMATH19.Scherer, R., Schira, T.: Estimating quadrature errors for analytic functions using kernel representations and biorthogonal systems. Numer. Math. 84, 497–518 (2000)MathSciNetCrossRefMATH20.Spalević, M.M.: Error bounds of Gaussian quadrature formulae for one class of Bernstein–Szegő weights. Math. Comp. 82, 1037–1056 (2013)MathSciNetCrossRefMATH21.Spalević, M.M.: Error bounds and estimates for Gauss–Turán quadrature formulae of analytic functions. SIAM J. Numer. Anal. 52, 443–467 (2014)MathSciNetCrossRefMATH22.Spalević, M.M., Pranić, M.S.: Error bounds of certain Gaussian quadrature formulae. J. Comput. Appl. Math. 234, 1049–1057 (2010)MathSciNetCrossRefMATH23.Spalević, M.M., Pranić, M.S., Pejčev, A.V.: Maximum of the modulus of kernels of Gaussian quadrature formulae for one class of Bernstein–Szegő weight functions. Appl. Math. Comput. 218, 5746–5756 (2012)MathSciNetMATH About this Article Title The error bounds of Gauss–Radau quadrature formulae with Bernstein–Szegő weight functions Journal Numerische Mathematik Volume 133, Issue 1 , pp 177-201 Cover Date2016-05 DOI 10.1007/s00211-015-0740-7 Print ISSN 0029-599X Online ISSN 0945-3245 Publisher Springer Berlin Heidelberg Additional Links Register for Journal Updates Editorial Board About This Journal Manuscript Submission Topics Numerical Analysis Mathematics, general Theoretical, Mathematical and Computational Physics Mathematical Methods in Physics Numerical and Computational Physics Appl.Mathematics/Computational Methods of Engineering Keywords 41A55 65D32 65D30 Industry Sectors Aerospace Electronics IT & Software Telecommunications Authors Aleksandar V. Pejčev (1) Miodrag M. Spalević (1) Author Affiliations 1. Department of Mathematics, Faculty of Mechanical Engineering, University of Beograd, Kraljice Marije 16, 11120, Belgrade 35, Serbia Continue reading... To view the rest of this content please follow the download PDF link above.