Weighted averaged Gaussian quadrature rules for modified Chebyshev measures
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2023
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This paper is concerned with the approximation of integrals of a real-valued integrand over
the interval [−1, 1] by Gauss quadrature. The averaged and optimal averaged quadrature
rules ([13,21]) provide a convenient method for approximating the error in the Gauss
quadrature. However, they are applicable to all integrands that are continuous on the
interval [−1, 1] only if their nodes are internal, i.e. if they belong to this interval.
We discuss two approaches to determine averaged quadrature rules with nodes in
[−1, 1]: (i) truncating the Jacobi matrix associated with the optimal averaged rule, and
(ii) weighting the optimal averaged quadrature rule. We consider Chebyshev measures of
the first, second, and third kinds that are modified by a linear over linear rational factor,
and discuss the internality of averaged, optimal averaged, and truncated optimal averaged
quadrature rules. Moreover, we show that the weighting yields internal averaged rules
if a weighting parameter ...is properly chosen, and we provide bounds for this parameter
that guarantee internality. Finally, we illustrate that the weighted averaged rules give more
accurate estimates of the quadrature error than the truncated optimal averaged rules.
Izvor:
Applied Numerical Mathematics, 2023Izdavač:
- Elsevier
Finansiranje / projekti:
- Ministarstvo nauke, tehnološkog razvoja i inovacija Republike Srbije, institucionalno finansiranje - 200105 (Univerzitet u Beogradu, Mašinski fakultet) (RS-MESTD-inst-2020-200105)
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Institucija/grupa
Mašinski fakultetTY - JOUR AU - Đukić, Dušan AU - Mutavdžić Đukić, Rada AU - Reichel, Lothar AU - Spalević, Miodrag PY - 2023 UR - https://machinery.mas.bg.ac.rs/handle/123456789/7066 AB - This paper is concerned with the approximation of integrals of a real-valued integrand over the interval [−1, 1] by Gauss quadrature. The averaged and optimal averaged quadrature rules ([13,21]) provide a convenient method for approximating the error in the Gauss quadrature. However, they are applicable to all integrands that are continuous on the interval [−1, 1] only if their nodes are internal, i.e. if they belong to this interval. We discuss two approaches to determine averaged quadrature rules with nodes in [−1, 1]: (i) truncating the Jacobi matrix associated with the optimal averaged rule, and (ii) weighting the optimal averaged quadrature rule. We consider Chebyshev measures of the first, second, and third kinds that are modified by a linear over linear rational factor, and discuss the internality of averaged, optimal averaged, and truncated optimal averaged quadrature rules. Moreover, we show that the weighting yields internal averaged rules if a weighting parameter is properly chosen, and we provide bounds for this parameter that guarantee internality. Finally, we illustrate that the weighted averaged rules give more accurate estimates of the quadrature error than the truncated optimal averaged rules. PB - Elsevier T2 - Applied Numerical Mathematics T1 - Weighted averaged Gaussian quadrature rules for modified Chebyshev measures DO - 10.1016/j.apnum.2023.05.014 ER -
@article{ author = "Đukić, Dušan and Mutavdžić Đukić, Rada and Reichel, Lothar and Spalević, Miodrag", year = "2023", abstract = "This paper is concerned with the approximation of integrals of a real-valued integrand over the interval [−1, 1] by Gauss quadrature. The averaged and optimal averaged quadrature rules ([13,21]) provide a convenient method for approximating the error in the Gauss quadrature. However, they are applicable to all integrands that are continuous on the interval [−1, 1] only if their nodes are internal, i.e. if they belong to this interval. We discuss two approaches to determine averaged quadrature rules with nodes in [−1, 1]: (i) truncating the Jacobi matrix associated with the optimal averaged rule, and (ii) weighting the optimal averaged quadrature rule. We consider Chebyshev measures of the first, second, and third kinds that are modified by a linear over linear rational factor, and discuss the internality of averaged, optimal averaged, and truncated optimal averaged quadrature rules. Moreover, we show that the weighting yields internal averaged rules if a weighting parameter is properly chosen, and we provide bounds for this parameter that guarantee internality. Finally, we illustrate that the weighted averaged rules give more accurate estimates of the quadrature error than the truncated optimal averaged rules.", publisher = "Elsevier", journal = "Applied Numerical Mathematics", title = "Weighted averaged Gaussian quadrature rules for modified Chebyshev measures", doi = "10.1016/j.apnum.2023.05.014" }
Đukić, D., Mutavdžić Đukić, R., Reichel, L.,& Spalević, M.. (2023). Weighted averaged Gaussian quadrature rules for modified Chebyshev measures. in Applied Numerical Mathematics Elsevier.. https://doi.org/10.1016/j.apnum.2023.05.014
Đukić D, Mutavdžić Đukić R, Reichel L, Spalević M. Weighted averaged Gaussian quadrature rules for modified Chebyshev measures. in Applied Numerical Mathematics. 2023;. doi:10.1016/j.apnum.2023.05.014 .
Đukić, Dušan, Mutavdžić Đukić, Rada, Reichel, Lothar, Spalević, Miodrag, "Weighted averaged Gaussian quadrature rules for modified Chebyshev measures" in Applied Numerical Mathematics (2023), https://doi.org/10.1016/j.apnum.2023.05.014 . .