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On the Error of Extended Gaussian Quadrature Formulae for Functions of Bounded Variation

  • Sven Ehrich
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 119)

Abstract

We investigate the remainder R 2n+1 of (minimum node) extended Gaussian quadrature formulae Q 2n+1 by means of the constants ϱv (R 2n+1), which are best possible in the error bound
$$\left| {{R_{2n + 1}}\left[ f \right]} \right| \leqslant \varrho v({R_{2n + 1}})Var(f)$$
for all functions f of bounded variation Var(f). For the most often used Gauss-Kronrod extensions Q 2n+1 GK we prove that there holds
$$\mathop {\lim }\limits_{n \to \infty } (2n + 1)\varrho v(R_{2n + 1}^{GK}) = \frac{\pi }{2}$$
.

As a consequence, we obtain that, among all extended Gaussian formulae whose additional nodes interlace with the Gaussian ones, the Gauss-Kronrod formula Q 2n+1 GK is asymptotically optimal with respect to ϱv.

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Copyright information

© Birkhäuser 1994

Authors and Affiliations

  • Sven Ehrich
    • 1
  1. 1.Institut für MathematikUniversität HildesheimHildesheimGermany

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