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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 (cf. [[Quadrature formula|Quadrature formula]]). Cotes' formulas are
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{{TEX|done}}{{MSC|65D32}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026700/c0267001.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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Formulas for the approximate computation of definite integrals, given the values of the integrand at finitely many equidistant points, i.e. [[quadrature formula]]s with equidistant interpolation points. Cotes' formulas are
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\begin{equation}\label{eq:1}
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\int_0^1 f(x) \, dx \approx \sum_{k=0}^n a_k^{(n)} f\left({ \frac{k}{n} }\right)\,,\ \ \ n=1,2,\ldots
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\end{equation}
  
The numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026700/c0267002.png" /> are known as Cotes' coefficients; they are determined from the condition that formula (*) be exact if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026700/c0267003.png" /> is a polynomial of degree at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026700/c0267004.png" />.
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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|Newton–Cotes quadrature formula]].
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The formulas were proposed by R. Cotes (1722) and considered in a more general form by I. Newton. See [[Newton–Cotes quadrature formula]].
  
  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Brass,  "Quadraturverfahren" , Vandenhoeck &amp; Ruprecht  (1977)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Cotes,  "Harmonia Mensurarum" , '''1–2''' , London  (1722)  (Published by R. Smith after Cotes' death)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  P.J. Davis,  P. Rabinowitz,  "Methods of numerical integration" , Acad. Press  (1984)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H. Engels,  "Numerical quadrature and cubature" , Acad. Press  (1980)</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Brass,  "Quadraturverfahren" , Vandenhoeck &amp; Ruprecht  (1977)</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Cotes,  "Harmonia Mensurarum" , '''1–2''' , London  (1722)  (Published by R. Smith after Cotes' death)</TD></TR>
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<TR><TD valign="top">[a3]</TD> <TD valign="top">  P.J. Davis,  P. Rabinowitz,  "Methods of numerical integration" , Acad. Press  (1984)</TD></TR>
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<TR><TD valign="top">[a4]</TD> <TD valign="top">  H. Engels,  "Numerical quadrature and cubature" , Acad. Press  (1980)</TD></TR>
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</table>

Latest revision as of 21:28, 18 January 2018

2020 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.


Comments

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].

References

[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)
How to Cite This Entry:
Cotes formulas. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cotes_formulas&oldid=11375
This article was adapted from an original article by BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article