Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A corrected quadrature formula and applications

Nenad Ujevic, A. J. Roberts

TL;DR

The paper tackles the accuracy limits of Simpson's rule by introducing a corrected, endpoint-derivative quadrature that achieves sixth-order accuracy in the grid spacing. It derives a three-point closed rule via finite-difference ideas, formulates a composite version with a single endpoint-derivative term, and establishes sharp Peano-like error bounds that often outperform standard estimates. The corrected composite rule is exact for polynomials up to degree 5, with a leading error term of order $h^6$, and the authors demonstrate practical performance on problems such as the erf integral, showing rapid convergence. This approach offers higher-accuracy numerical integration with similar computational cost and broader applicability when interior derivatives are limited, by leveraging information at the interval endpoints.

Abstract

A straightforward 3-point quadrature formula of closed type is derived that improves on Simpson's rule. Just using the additional information of the integrand's derivative at the two endpoints we show the error is sixth order in grid spacing. Various error bounds for the quadrature formula are obtained to quantify more precisely the errors. Applications in numerical integration are given. With these error bounds, which are generally better than the usual Peano bounds, the composite formulas can be applied to integrands with lower order derivatives.

A corrected quadrature formula and applications

TL;DR

The paper tackles the accuracy limits of Simpson's rule by introducing a corrected, endpoint-derivative quadrature that achieves sixth-order accuracy in the grid spacing. It derives a three-point closed rule via finite-difference ideas, formulates a composite version with a single endpoint-derivative term, and establishes sharp Peano-like error bounds that often outperform standard estimates. The corrected composite rule is exact for polynomials up to degree 5, with a leading error term of order , and the authors demonstrate practical performance on problems such as the erf integral, showing rapid convergence. This approach offers higher-accuracy numerical integration with similar computational cost and broader applicability when interior derivatives are limited, by leveraging information at the interval endpoints.

Abstract

A straightforward 3-point quadrature formula of closed type is derived that improves on Simpson's rule. Just using the additional information of the integrand's derivative at the two endpoints we show the error is sixth order in grid spacing. Various error bounds for the quadrature formula are obtained to quantify more precisely the errors. Applications in numerical integration are given. With these error bounds, which are generally better than the usual Peano bounds, the composite formulas can be applied to integrands with lower order derivatives.

Paper Structure

This paper contains 7 sections, 46 equations, 1 figure.

Table of Contents

  1. Keywords:
  2. Introduction
  3. Finite differences derive the modified Simpson's rule
  4. Example:
  5. Further error analysis
  6. Applications in numerical integration
  7. Example:

Figures (1)

  • Figure 1: log-log plot of the errors of our integration rule (\ref{['eq:mmsimp']}), $+$'s, as a function showing the ${\cal O}(h^6)$ rate of convergence to the erf integral (\ref{['eq:myerf']}), compared to the ${\cal O}(h^4)$ convergence of the normal Simpson's rule, $\times$'s.