Newton-Cotes Quadrature

Newton-Cotes rules approximate an integral by integrating the polynomial that interpolates the integrand at equally spaced points, giving familiar formulas such as the trapezoidal and Simpson's rules.

Definition

A Newton-Cotes quadrature rule is an interpolatory quadrature rule whose nodes are equally spaced across the integration interval, with weights obtained by integrating the corresponding interpolating polynomial.

Scope

This topic covers closed and open Newton-Cotes formulas, their degrees of exactness and error terms, the composite trapezoidal and Simpson rules obtained by subdividing the interval, Romberg integration via Richardson extrapolation, and the instability of high-order Newton-Cotes rules that limits their practical degree.

Core questions

• How are the trapezoidal and Simpson rules derived as integrated interpolants?
• What are the error terms of these rules, and why does Simpson's rule gain an extra order from symmetry?
• How do composite rules and Romberg extrapolation improve accuracy systematically?
• Why do high-order Newton-Cotes rules become unstable, and what limits their use?

Key theories

Degree of exactness and error terms

The trapezoidal rule is exact for linear integrands with error proportional to the second derivative, while Simpson's rule, by symmetry, is exact for cubics with error proportional to the fourth derivative, gaining an order beyond its interpolation degree.

Composite rules and Romberg integration

Applying a basic rule on many subintervals yields a composite rule whose error decreases polynomially in the step size; Richardson extrapolation of the composite trapezoidal rule produces the rapidly convergent Romberg scheme.

Mechanisms

Each basic rule integrates the equispaced interpolant exactly: the trapezoidal rule integrates a straight-line fit, Simpson's rule a parabola. Composite rules partition the interval and sum basic rules on each piece, so halving the step size predictably reduces error. Romberg integration tabulates composite trapezoidal estimates at successively halved step sizes and applies repeated Richardson extrapolation, cancelling leading error terms to achieve high-order accuracy for smooth integrands. High-order single-interval Newton-Cotes rules acquire large oscillatory weights of mixed sign, mirroring the Runge phenomenon, which causes cancellation and instability.

Clinical relevance

Newton-Cotes rules, especially the composite trapezoidal and Simpson forms, are the default low-cost quadrature tools when integrand samples are naturally equally spaced — for example tabulated experimental data, time-series integration, and simple simulation post-processing — and Romberg integration provides accurate results for smooth functions with minimal coding.

History

These rules originate with Newton and Cotes in the early eighteenth century and with Thomas Simpson, whose rule bears his name; Werner Romberg's 1955 extrapolation scheme turned the elementary trapezoidal rule into a high-accuracy method and remains a standard teaching and computing tool.

Key figures

• Isaac Newton
• Roger Cotes
• Thomas Simpson
• Werner Romberg

Seminal works

• davis1984
• quarteroni2007

Frequently asked questions

Why is Simpson's rule more accurate than the trapezoidal rule?

Simpson's rule fits a parabola through three points rather than a line through two, and because of symmetry it integrates cubic polynomials exactly, so its error depends on the fourth derivative and shrinks much faster as the step size decreases.

Why not just use a very high-order Newton-Cotes rule?

High-order Newton-Cotes rules on equally spaced nodes develop large weights with alternating signs, causing numerical cancellation and instability. In practice one uses composite low-order rules, Romberg extrapolation, or Gaussian quadrature instead.