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3 results

  • 14 - Numerical Quadrature

  • from Part II - Constructive Approximation Theory
  • Abner J. Salgado, University of Tennessee, Knoxville, Steven M. Wise, University of Tennessee, Knoxville
  • Book:
    Classical Numerical Analysis
    Published online:
    29 September 2022
    Print publication:
    20 October 2022, pp 372-416

  • Summary

    We study the problem of approximating the value of a (weighted) integral of a function. We introduce the concepts of a quadrature rule and its consistency. Interpolatory quadrature rules and their construction are then discussed. Their error analysis, based on notions of previous chapters, is then presented. Then an error analysis based on the Peano Kernel Theorem and scaling arguments is developed. Newton-Cotes formulas then are developed and analyzed. Then, composite quadrature rules are presented and their use is illustrated. Their analysis is presented then, on the basis of Euler-Maclaurin formulas. The chapter concludes with a discussion of Gaussian quadrature formulas, their properties and optimality.

  • Generalized Hadamard's inequalities based on general Euler 4-point formulae

  • M. Klaričić Bakula, J. Pečarić
  • Journal:
    The ANZIAM Journal / Volume 48 / Issue 3 / January 2007
    Published online by Cambridge University Press:
    17 February 2009, pp. 387-404
    • Article
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  • We present a general closed 4-point quadrature rule based on Euler-type identities. We use this rule to prove a generalization of Hadamard's inequalities for (2r)-convex functions (r ≥ 1).