View source for Newton-Cotes quadrature formula

The interpolation [[Quadrature formula|quadrature formula]]

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for the computation of an integral over a finite interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066510/n0665102.png" />, with nodes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066510/n0665103.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066510/n0665104.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066510/n0665105.png" /> is a natural number, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066510/n0665106.png" />, and the number of nodes is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066510/n0665107.png" />. The coefficients are determined by the fact that the quadrature formula is interpolational, that is,

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For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066510/n0665109.png" /> all coefficients are positive, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066510/n06651010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066510/n06651011.png" /> there are both positive and negative ones among them. The algebraic degree of accuracy (the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066510/n06651012.png" /> such that the formula is exact for all polynomials of degree at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066510/n06651013.png" /> and not exact for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066510/n06651014.png" />) is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066510/n06651015.png" /> for odd <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066510/n06651016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066510/n06651017.png" /> for even <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066510/n06651018.png" />. The simplest special cases of the Newton–Cotes quadrature formula are: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066510/n06651019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066510/n06651020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066510/n06651021.png" />,

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the [[Trapezium formula|trapezium formula]]; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066510/n06651023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066510/n06651024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066510/n06651025.png" />,

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the [[Simpson formula|Simpson formula]]; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066510/n06651027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066510/n06651028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066510/n06651029.png" />,

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the  "three-eighths"  quadrature formula. For large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066510/n06651031.png" /> the Newton–Cotes formula is seldom used (because of the property of the coefficients for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066510/n06651032.png" /> mentioned above). One prefers to use for small <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066510/n06651033.png" /> the compound Newton–Cotes quadrature formulas, namely, the trapezium formula and Simpson's formula.

The coefficients of the Newton–Cotes quadrature formula for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066510/n06651034.png" /> from 1 to 20 are listed in [[#References|[3]]].

The formula first appeared in a letter from I. Newton to G. Leibniz in 1676 (see [[#References|[1]]]) and later in the book [[#References|[2]]] by R. Cotes, where the coefficients of the formula are given for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066510/n06651035.png" /> from 1 to 10.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I. Newton,   "Mathematical principles of natural philosophy"  A.N. Krylov (ed.) , ''Collected works'' , '''7''' , Moscow-Leningrad  (1936)  (In Russian; translated from Latin)</TD></TR><TR><TD valign="top">[2]</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">[3]</TD> <TD valign="top">  V.I. Krylov,   L.T. Shul'gina,   "Handbook on numerical integration" , Moscow  (1966)  (In Russian)</TD></TR></table>



====Comments====
The formulas above are often referred to as closed Newton–Cotes formulas, in contrast to open Newton–Cotes formulas, which do not include the end points as nodes.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Engels,   "Numerical quadrature and cubature" , Acad. Press  (1980)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Brass,   "Quadraturverfahren" , Vandenhoeck &amp; Ruprecht  (1977)</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">  A.H. Stroud,   "Numerical quadrature and solution of ordinary differential equations" , Springer  (1974)</TD></TR></table>