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Numerische Mathematik

, Volume 107, Issue 1, pp 147–174 | Cite as

On sensitivity of Gauss–Christoffel quadrature

  • Dianne P. O’LearyEmail author
  • Zdeněk Strakoš
  • Petr Tichý
Article

Abstract

In numerical computations the question how much does a function change under perturbations of its arguments is of central importance. In this work, we investigate sensitivity of Gauss–Christoffel quadrature with respect to small perturbations of the distribution function. In numerical quadrature, a definite integral is approximated by a finite sum of functional values evaluated at given quadrature nodes and multiplied by given weights. Consider a sufficiently smooth integrated function uncorrelated with the perturbation of the distribution function. Then it seems natural that given the same number of function evaluations, the difference between the quadrature approximations is of the same order as the difference between the (original and perturbed) approximated integrals. That is perhaps one of the reasons why, to our knowledge, the sensitivity question has not been formulated and addressed in the literature, though several other sensitivity problems, motivated, in particular, by computation of the quadrature nodes and weights from moments, have been thoroughly studied by many authors. We survey existing particular results and show that even a small perturbation of a distribution function can cause large differences in Gauss–Christoffel quadrature estimates. We then discuss conditions under which the Gauss–Christoffel quadrature is insensitive under perturbation of the distribution function, present illustrative examples, and relate our observations to known conjectures on some sensitivity problems.

Keywords

Jacobi Matrix Orthogonal Polynomial Conjugate Gradient Method Quadrature Formula Jacobi Matrice 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  • Dianne P. O’Leary
    • 1
    Email author
  • Zdeněk Strakoš
    • 2
  • Petr Tichý
    • 3
  1. 1.Department of Computer Science and Institute for Advanced Computer StudiesUniversity of MarylandCollege ParkUSA
  2. 2.Institute of Computer Science and Faculty of Mathematics and PhysicsAcademy of Sciences of the Czech Republic and Charles UniversityPragueCzech Republic
  3. 3.Institute of Computer ScienceAcademy of Sciences of the Czech RepublicPragueCzech Republic

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