Error bounds for Gauss-type quadratures with Bernstein-Szego weights
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2014
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The paper is concerned with the derivation of error bounds for Gauss-type quadratures with Bernstein-Szego weights, integral(1)(-1) f(t)w(t) dt = G(n)[f] + R-n(f), G(n)[f] = Sigma(n)(nu=1) lambda(nu)f(tau(nu)) (n is an element of N), where f is an analytic function inside an elliptical contour epsilon(rho) with foci at -/+ 1 and sum of semi-axes rho > 1, and w is a nonnegative and integrable weight function of Bernstein-Szego type. The derivation of effective bounds on vertical bar R-n(f)vertical bar is possible if good estimates of max(z is an element of epsilon rho) vertical bar K-n(z)vertical bar are available, especially if one knows the location of the extremal point eta is an element of epsilon(rho) at which vertical bar K-n vertical bar attains its maximum. In such a case, instead of looking for upper bounds on max(z is an element of epsilon rho) vertical bar K-n(z)vertical bar, one can simply try to calculate vertical bar Kn(eta, w)vertical bar. In the case under consideration,... i.e. when w(t) = (1 - t(2))(-1/2)/beta(beta - 2 alpha)t(2) + 2 delta(beta - alpha)t + alpha(2) + delta(2), t is an element of (-1, 1), for some alpha, beta, delta, which satisfy 0 lt alpha lt beta, beta not equal 2 alpha, vertical bar delta vertical bar lt beta - alpha, the location on the elliptical contours where the modulus of the kernel attains its maximum value is investigated. This leads to effective bounds on vertical bar R-n(f)vertical bar. The quality of the derived bounds is analyzed by a comparison with other error bounds proposed in the literature for the same class of integrands.
Ključne reči:
Remainder term / Kernel / Gaussian quadrature / Error bound / Elliptical contour / Analytic functionIzvor:
Numerical Algorithms, 2014, 66, 3, 569-590Izdavač:
- Springer, Dordrecht
Finansiranje / projekti:
- Metode numeričke i nelinearne analize sa primenama (RS-MESTD-Basic Research (BR or ON)-174002)
DOI: 10.1007/s11075-013-9749-0
ISSN: 1017-1398
WoS: 000338336900008
Scopus: 2-s2.0-84903478361
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Institucija/grupa
Mašinski fakultetTY - JOUR AU - Pejčev, Aleksandar PY - 2014 UR - https://machinery.mas.bg.ac.rs/handle/123456789/1884 AB - The paper is concerned with the derivation of error bounds for Gauss-type quadratures with Bernstein-Szego weights, integral(1)(-1) f(t)w(t) dt = G(n)[f] + R-n(f), G(n)[f] = Sigma(n)(nu=1) lambda(nu)f(tau(nu)) (n is an element of N), where f is an analytic function inside an elliptical contour epsilon(rho) with foci at -/+ 1 and sum of semi-axes rho > 1, and w is a nonnegative and integrable weight function of Bernstein-Szego type. The derivation of effective bounds on vertical bar R-n(f)vertical bar is possible if good estimates of max(z is an element of epsilon rho) vertical bar K-n(z)vertical bar are available, especially if one knows the location of the extremal point eta is an element of epsilon(rho) at which vertical bar K-n vertical bar attains its maximum. In such a case, instead of looking for upper bounds on max(z is an element of epsilon rho) vertical bar K-n(z)vertical bar, one can simply try to calculate vertical bar Kn(eta, w)vertical bar. In the case under consideration, i.e. when w(t) = (1 - t(2))(-1/2)/beta(beta - 2 alpha)t(2) + 2 delta(beta - alpha)t + alpha(2) + delta(2), t is an element of (-1, 1), for some alpha, beta, delta, which satisfy 0 lt alpha lt beta, beta not equal 2 alpha, vertical bar delta vertical bar lt beta - alpha, the location on the elliptical contours where the modulus of the kernel attains its maximum value is investigated. This leads to effective bounds on vertical bar R-n(f)vertical bar. The quality of the derived bounds is analyzed by a comparison with other error bounds proposed in the literature for the same class of integrands. PB - Springer, Dordrecht T2 - Numerical Algorithms T1 - Error bounds for Gauss-type quadratures with Bernstein-Szego weights EP - 590 IS - 3 SP - 569 VL - 66 DO - 10.1007/s11075-013-9749-0 ER -
@article{ author = "Pejčev, Aleksandar", year = "2014", abstract = "The paper is concerned with the derivation of error bounds for Gauss-type quadratures with Bernstein-Szego weights, integral(1)(-1) f(t)w(t) dt = G(n)[f] + R-n(f), G(n)[f] = Sigma(n)(nu=1) lambda(nu)f(tau(nu)) (n is an element of N), where f is an analytic function inside an elliptical contour epsilon(rho) with foci at -/+ 1 and sum of semi-axes rho > 1, and w is a nonnegative and integrable weight function of Bernstein-Szego type. The derivation of effective bounds on vertical bar R-n(f)vertical bar is possible if good estimates of max(z is an element of epsilon rho) vertical bar K-n(z)vertical bar are available, especially if one knows the location of the extremal point eta is an element of epsilon(rho) at which vertical bar K-n vertical bar attains its maximum. In such a case, instead of looking for upper bounds on max(z is an element of epsilon rho) vertical bar K-n(z)vertical bar, one can simply try to calculate vertical bar Kn(eta, w)vertical bar. In the case under consideration, i.e. when w(t) = (1 - t(2))(-1/2)/beta(beta - 2 alpha)t(2) + 2 delta(beta - alpha)t + alpha(2) + delta(2), t is an element of (-1, 1), for some alpha, beta, delta, which satisfy 0 lt alpha lt beta, beta not equal 2 alpha, vertical bar delta vertical bar lt beta - alpha, the location on the elliptical contours where the modulus of the kernel attains its maximum value is investigated. This leads to effective bounds on vertical bar R-n(f)vertical bar. The quality of the derived bounds is analyzed by a comparison with other error bounds proposed in the literature for the same class of integrands.", publisher = "Springer, Dordrecht", journal = "Numerical Algorithms", title = "Error bounds for Gauss-type quadratures with Bernstein-Szego weights", pages = "590-569", number = "3", volume = "66", doi = "10.1007/s11075-013-9749-0" }
Pejčev, A.. (2014). Error bounds for Gauss-type quadratures with Bernstein-Szego weights. in Numerical Algorithms Springer, Dordrecht., 66(3), 569-590. https://doi.org/10.1007/s11075-013-9749-0
Pejčev A. Error bounds for Gauss-type quadratures with Bernstein-Szego weights. in Numerical Algorithms. 2014;66(3):569-590. doi:10.1007/s11075-013-9749-0 .
Pejčev, Aleksandar, "Error bounds for Gauss-type quadratures with Bernstein-Szego weights" in Numerical Algorithms, 66, no. 3 (2014):569-590, https://doi.org/10.1007/s11075-013-9749-0 . .