The error bounds of Gauss-Kronrod quadrature formulae for weight functions of Bernstein-SzegoIi type
Abstract
We consider the Gauss-Kronrod quadrature formulae for the Bernstein-SzegoIi weight functions consisting of any one of the four Chebyshev weights divided by the polynomial . 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 a" 1 and 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 by S. Notaris (Numer. Math. 103, 99-127, 2006).
Keywords:
Remainder term for analytic functions / Gauss-Kronrod quadrature formulae / Error bound / Contour integral representation / Bernstein-Szego weight functionsSource:
Numerical Algorithms, 2018, 77, 4, 1003-1028Publisher:
- Springer, Dordrecht
Funding / projects:
- Methods of Numerical and Nonlinear Analysis with Applications (RS-MESTD-Basic Research (BR or ON)-174002)
DOI: 10.1007/s11075-017-0351-8
ISSN: 1017-1398
WoS: 000428381200003
Scopus: 2-s2.0-85020543732
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Institution/Community
Mašinski fakultetTY - JOUR AU - Đukić, Dušan AU - Pejčev, Aleksandar AU - Spalević, Miodrag PY - 2018 UR - https://machinery.mas.bg.ac.rs/handle/123456789/2933 AB - We consider the Gauss-Kronrod quadrature formulae for the Bernstein-SzegoIi weight functions consisting of any one of the four Chebyshev weights divided by the polynomial . 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 a" 1 and 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 by S. Notaris (Numer. Math. 103, 99-127, 2006). PB - Springer, Dordrecht T2 - Numerical Algorithms T1 - The error bounds of Gauss-Kronrod quadrature formulae for weight functions of Bernstein-SzegoIi type EP - 1028 IS - 4 SP - 1003 VL - 77 DO - 10.1007/s11075-017-0351-8 ER -
@article{ author = "Đukić, Dušan and Pejčev, Aleksandar and Spalević, Miodrag", year = "2018", abstract = "We consider the Gauss-Kronrod quadrature formulae for the Bernstein-SzegoIi weight functions consisting of any one of the four Chebyshev weights divided by the polynomial . 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 a" 1 and 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 by S. Notaris (Numer. Math. 103, 99-127, 2006).", publisher = "Springer, Dordrecht", journal = "Numerical Algorithms", title = "The error bounds of Gauss-Kronrod quadrature formulae for weight functions of Bernstein-SzegoIi type", pages = "1028-1003", number = "4", volume = "77", doi = "10.1007/s11075-017-0351-8" }
Đukić, D., Pejčev, A.,& Spalević, M.. (2018). The error bounds of Gauss-Kronrod quadrature formulae for weight functions of Bernstein-SzegoIi type. in Numerical Algorithms Springer, Dordrecht., 77(4), 1003-1028. https://doi.org/10.1007/s11075-017-0351-8
Đukić D, Pejčev A, Spalević M. The error bounds of Gauss-Kronrod quadrature formulae for weight functions of Bernstein-SzegoIi type. in Numerical Algorithms. 2018;77(4):1003-1028. doi:10.1007/s11075-017-0351-8 .
Đukić, Dušan, Pejčev, Aleksandar, Spalević, Miodrag, "The error bounds of Gauss-Kronrod quadrature formulae for weight functions of Bernstein-SzegoIi type" in Numerical Algorithms, 77, no. 4 (2018):1003-1028, https://doi.org/10.1007/s11075-017-0351-8 . .