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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Error bounds of Gaussian quadrature formulae for one class of Bernstein-Szegő weights


Author: Miodrag M. Spalević
Journal: Math. Comp. 82 (2013), 1037-1056
MSC (2010): Primary 41A55; Secondary 65D30, 65D32
Published electronically: December 14, 2012
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Abstract: The kernels $ K_n(z)$ in the remainder terms $ R_n(f)$ of the Gaussian quadrature formulae for analytic functions $ f$ inside elliptical contours with foci at $ \mp 1$ and a sum of semi-axes $ \rho >1$, when the weight function $ w$ is of Bernstein-Szegő type

$\displaystyle w(t)\equiv w_\gamma ^{(-1/2,1/2)}(t)=\displaystyle \sqrt {\frac {... ...ac {4\gamma }{(1+\gamma )^2}\,t^2}, \quad t\in (-1,1),\quad \gamma \in (-1,0), $

are studied. Sufficient conditions are found ensuring that the kernel attains its maximal absolute value at the intersection point of the contour with the positive real semi-axis. This leads to effective error bounds of the corresponding Gauss quadratures. The quality of the derived bounds is analyzed by a comparison with other error bounds intended for the same class of integrands. In part our analysis is based on the well-known Cardano formulas, which are not very popular among mathematicians.

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Additional Information

Miodrag M. Spalević
Affiliation: Department of Mathematics, University of Beograd, Faculty of Mechanical Engineering, Kraljice Marije 16, 11120 Belgrade 35, Serbia
Email: mspalevic@mas.bg.ac.rs

DOI: http://dx.doi.org/10.1090/S0025-5718-2012-02667-6
PII: S 0025-5718(2012)02667-6
Keywords: Maximum, modulus, kernel, Gaussian quadrature formula, Bernstein-Szegő weight function
Received by editor(s): September 22, 2010
Received by editor(s) in revised form: March 6, 2011
Published electronically: December 14, 2012
Additional Notes: The author was supported in part by the Serbian Ministry of Education and Science (Research Project: “Methods of numerical and nonlinear analysis with applications”, No. #174002)
Article copyright: © Copyright 2012 American Mathematical Society