Error bounds for Gaussian quadrature formulae with Bernstein-Szego weights that are rational modifications of Chebyshev weight functions of the second kind
Abstract
For analytic functions we study the kernel of the remainder terms of Gaussian quadrature rules with respect to Bernstein-Szego weight functions lt graphic xlink:href="drr044eq1" xmlns:xlink="http://www.w3.org/1999/xlink"/> where 0 lt alpha lt beta, beta not equal 2 alpha, vertical bar delta vertical bar lt beta-alpha, and whose denominator is an arbitrary polynomial of exact degree 2 that remains positive on [-1, 1]. The subcase alpha=1, beta=2/(1+gamma), -1 lt gamma lt 0 and delta=0 has been considered recently by Spalevic, M. M. & Pranic, M. S. ((2010) Error bounds of certain Gaussian quadrature formulae. J. Comput. Appl. Math., 234, 1049-1057). The location on the elliptic contours where the modulus of the kernel attains its maximum value is investigated. This leads to effective error bounds for the corresponding Gaussian quadratures. The approach we use in this paper, which is different from the one adopted in Spalevic, M. M. & Pranic, M. S. ((2010) Error bounds of certain G...aussian quadrature formulae. J. Comput. Appl. Math., 234, 1049-1057), ensures that the actual conditions for determining the locations on the elliptic contours where the modulus of the kernel attains its maximum value are approximated very precisely.
Keywords:
remainder term / kernel / Gaussian quadrature / error bound / elliptic contour / analytic functionSource:
Ima Journal of Numerical Analysis, 2012, 32, 4, 1733-1754Publisher:
- Oxford Univ Press, Oxford
Funding / projects:
- Methods of Numerical and Nonlinear Analysis with Applications (RS-MESTD-Basic Research (BR or ON)-174002)
DOI: 10.1093/imanum/drr044
ISSN: 0272-4979
WoS: 000309923300017
Scopus: 2-s2.0-84867501860
Collections
Institution/Community
Mašinski fakultetTY - JOUR AU - Pejčev, Aleksandar AU - Spalević, Miodrag PY - 2012 UR - https://machinery.mas.bg.ac.rs/handle/123456789/1489 AB - For analytic functions we study the kernel of the remainder terms of Gaussian quadrature rules with respect to Bernstein-Szego weight functions lt graphic xlink:href="drr044eq1" xmlns:xlink="http://www.w3.org/1999/xlink"/> where 0 lt alpha lt beta, beta not equal 2 alpha, vertical bar delta vertical bar lt beta-alpha, and whose denominator is an arbitrary polynomial of exact degree 2 that remains positive on [-1, 1]. The subcase alpha=1, beta=2/(1+gamma), -1 lt gamma lt 0 and delta=0 has been considered recently by Spalevic, M. M. & Pranic, M. S. ((2010) Error bounds of certain Gaussian quadrature formulae. J. Comput. Appl. Math., 234, 1049-1057). The location on the elliptic contours where the modulus of the kernel attains its maximum value is investigated. This leads to effective error bounds for the corresponding Gaussian quadratures. The approach we use in this paper, which is different from the one adopted in Spalevic, M. M. & Pranic, M. S. ((2010) Error bounds of certain Gaussian quadrature formulae. J. Comput. Appl. Math., 234, 1049-1057), ensures that the actual conditions for determining the locations on the elliptic contours where the modulus of the kernel attains its maximum value are approximated very precisely. PB - Oxford Univ Press, Oxford T2 - Ima Journal of Numerical Analysis T1 - Error bounds for Gaussian quadrature formulae with Bernstein-Szego weights that are rational modifications of Chebyshev weight functions of the second kind EP - 1754 IS - 4 SP - 1733 VL - 32 DO - 10.1093/imanum/drr044 ER -
@article{ author = "Pejčev, Aleksandar and Spalević, Miodrag", year = "2012", abstract = "For analytic functions we study the kernel of the remainder terms of Gaussian quadrature rules with respect to Bernstein-Szego weight functions lt graphic xlink:href="drr044eq1" xmlns:xlink="http://www.w3.org/1999/xlink"/> where 0 lt alpha lt beta, beta not equal 2 alpha, vertical bar delta vertical bar lt beta-alpha, and whose denominator is an arbitrary polynomial of exact degree 2 that remains positive on [-1, 1]. The subcase alpha=1, beta=2/(1+gamma), -1 lt gamma lt 0 and delta=0 has been considered recently by Spalevic, M. M. & Pranic, M. S. ((2010) Error bounds of certain Gaussian quadrature formulae. J. Comput. Appl. Math., 234, 1049-1057). The location on the elliptic contours where the modulus of the kernel attains its maximum value is investigated. This leads to effective error bounds for the corresponding Gaussian quadratures. The approach we use in this paper, which is different from the one adopted in Spalevic, M. M. & Pranic, M. S. ((2010) Error bounds of certain Gaussian quadrature formulae. J. Comput. Appl. Math., 234, 1049-1057), ensures that the actual conditions for determining the locations on the elliptic contours where the modulus of the kernel attains its maximum value are approximated very precisely.", publisher = "Oxford Univ Press, Oxford", journal = "Ima Journal of Numerical Analysis", title = "Error bounds for Gaussian quadrature formulae with Bernstein-Szego weights that are rational modifications of Chebyshev weight functions of the second kind", pages = "1754-1733", number = "4", volume = "32", doi = "10.1093/imanum/drr044" }
Pejčev, A.,& Spalević, M.. (2012). Error bounds for Gaussian quadrature formulae with Bernstein-Szego weights that are rational modifications of Chebyshev weight functions of the second kind. in Ima Journal of Numerical Analysis Oxford Univ Press, Oxford., 32(4), 1733-1754. https://doi.org/10.1093/imanum/drr044
Pejčev A, Spalević M. Error bounds for Gaussian quadrature formulae with Bernstein-Szego weights that are rational modifications of Chebyshev weight functions of the second kind. in Ima Journal of Numerical Analysis. 2012;32(4):1733-1754. doi:10.1093/imanum/drr044 .
Pejčev, Aleksandar, Spalević, Miodrag, "Error bounds for Gaussian quadrature formulae with Bernstein-Szego weights that are rational modifications of Chebyshev weight functions of the second kind" in Ima Journal of Numerical Analysis, 32, no. 4 (2012):1733-1754, https://doi.org/10.1093/imanum/drr044 . .