Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

The error norm of certain Gaussian quadrature formulae


Author: G. Akrivis
Journal: Math. Comp. 45 (1985), 513-519
MSC: Primary 65D32
DOI: https://doi.org/10.1090/S0025-5718-1985-0804940-0
MathSciNet review: 804940
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider Gauss quadrature formulae $ {Q_n}$, $ n \in {\mathbf{N}}$, approximating the integral $ I(f): = \smallint _{ - 1}^1w(x)f(x)\;dx$, $ w = W/{p_i}$, $ i = 1,2$, with $ W(x) = {(1 - x)^\alpha }{(1 + x)^\beta }$, $ \alpha ,\beta = \pm 1/2$ and $ {p_1}(x) = 1 + {a^2} + 2ax$, $ {p_2}(x) = (2b + 1){x^2} + {b^2}$, $ b > 0$. In certain spaces of analytic functions the error functional $ {R_n}: = I - {Q_n}$ is continuous. In [1] and [2] estimates for $ \left\Vert {{R_n}} \right\Vert$ are given for a wide class of weight functions. Here, for a restricted class of weight functions, we calculate the norm of $ {R_n}$ explicitly.


References [Enhancements On Off] (What's this?)

  • [1] G. Akrivis, "Fehlerabschätzungen für Gauss-Quadraturformeln," Numer. Math., v. 44, 1984, pp. 261-278. MR 753958 (85m:65023)
  • [2] G. Akrivis & A. Burgstaller, "Fehlerabschätzungen für nichtsymmetrische Gauss-Quadraturformeln," Numer. Math. (To appear.) MR 812618 (87e:65017)
  • [3] H. Brass, Quadraturverfahren, Vandenhoeck and Ruprecht, Göttingen, Zürich, 1977. MR 0443305 (56:1675)
  • [4] P. J. Davis & P. Rabinowitz, Methods of Numerical Integration, Academic Press, New York, 1975. MR 0448814 (56:7119)
  • [5] W. Gautschi, "On Padé approximants associated with Hamburger series," Calcolo, v. 20, 1983, pp. 111-127. MR 746349 (85f:41012)
  • [6] W. Gautschi & R. S. Varga, "Error bounds for Gaussian quadrature of analytic functions," SIAM J. Numer. Anal., v. 20, 1983, pp. 1170-1186. MR 723834 (85j:65010)
  • [7] W. Gröbner & N. Hofreiter (editors), Integraltafel, II Teil, Springer-Verlag, Wien, 1961.
  • [8] G. Hämmerlin, "Fehlerabschätzungen bei numerischer Integration nach Gauss," Methoden und Verfahren der mathematischen Physik, Vol. 6 (B. Brosowski and E. Martensen, eds.), Bibliographisches Institut, Mannheim, Wien, Zürich, 1972, pp. 153-163. MR 0359277 (50:11732)
  • [9] R. Kumar, "A class of quadrature formulas," Math. Comp., v. 28, 1974, pp. 769-778. MR 0373240 (51:9441)
  • [10] R. Kumar, "Certain Gaussian quadratures," J. Inst. Math. Appl., v. 14, 1974, pp. 175-182. MR 0356452 (50:8922)
  • [11] T. J. Rivlin, The Chebyshev Polynomials, Wiley, New York, 1974. MR 0450850 (56:9142)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65D32

Retrieve articles in all journals with MSC: 65D32


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1985-0804940-0
Article copyright: © Copyright 1985 American Mathematical Society

American Mathematical Society