Cotes formulas

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2010 Mathematics Subject Classification: Primary: 65D32 [MSN][ZBL]

Formulas for the approximate computation of definite integrals, given the values of the integrand at finitely many equidistant points, i.e. quadrature formulas with equidistant interpolation points. Cotes' formulas are \begin{equation}\label{eq:1} \int_0^1 f(x) \, dx \approx \sum_{k=0}^n a_k^{(n)} f\left({ \frac{k}{n} }\right)\,,\ \ \ n=1,2,\ldots \end{equation}

The numbers $a_k^{(n)}$ are known as Cotes' coefficients; they are determined from the condition that formula \eqref{eq:1} be exact if $f(x)$ is a polynomial of degree at most $n$.

The formulas were proposed by R. Cotes (1722) and considered in a more general form by I. Newton. See Newton–Cotes quadrature formula.


Cotes' formulas were published in [a2] after Cotes' death. In the Western literature these formulas are known as the Newton–Cotes formulas. A detailed analysis of them can be found in [a1], [a3], [a4].


[a1] H. Brass, "Quadraturverfahren" , Vandenhoeck & Ruprecht (1977)
[a2] R. Cotes, "Harmonia Mensurarum" , 1–2 , London (1722) (Published by R. Smith after Cotes' death)
[a3] P.J. Davis, P. Rabinowitz, "Methods of numerical integration" , Acad. Press (1984)
[a4] H. Engels, "Numerical quadrature and cubature" , Acad. Press (1980)
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This article was adapted from an original article by BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article