Difference between revisions of "Cotes formulas"
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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 | |
+ | \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 | + | 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 [[ | + | 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 & 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> | + | <table> |
+ | <TR><TD valign="top">[a1]</TD> <TD valign="top"> H. Brass, "Quadraturverfahren" , Vandenhoeck & 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> |
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) |
Cotes formulas. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cotes_formulas&oldid=11375