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Peano-like bounds for some Newton-Cotes formulae

Nenad Ujevic

Abstract

An error analysis for some Newton-Cotes quadrature formulae is presented. Peano-like error bounds are obtained. They are generally, but not always, better than the usual Peano bounds.

Peano-like bounds for some Newton-Cotes formulae

Abstract

An error analysis for some Newton-Cotes quadrature formulae is presented. Peano-like error bounds are obtained. They are generally, but not always, better than the usual Peano bounds.

Paper Structure

This paper contains 5 sections, 6 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Simpson's rule
  4. 3/8 Simpson rule
  5. Boole's rule

Key Result

Lemma 1

Let $I\subset R$ be an open interval and $a,b\in I,$$a<b.$ Let $f:I\rightarrow R$ be a twice differentiable function and $\ $let $x\in \left[ a,b\right]$ be a fixed element. Then we have where

Theorems & Definitions (9)

  • Lemma 1
  • Lemma 2
  • Corollary 3
  • Theorem 4
  • Remark 5
  • Theorem 6
  • Remark 7
  • Theorem 8
  • Remark 9