View source for Newton-Cotes quadrature formula

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The interpolation [[Quadrature formula|quadrature formula]]

$$ 
\int\limits _ { a } ^ { b }  f ( x)  dx  \cong \ 
( b - a)
\sum _ {k = 0 } ^ { n } 
B _ {k}  ^ {(} n)
f ( x _ {k}  ^ {(} n) )
$$

for the computation of an integral over a finite interval  $  [ a, b] $, 
with nodes  $  x _ {k}  ^ {(} n) = a + kh $, 
$  k = 0 \dots n $, 
where  $  n $
is a natural number,  $  h = ( b - a)/n $, 
and the number of nodes is  $  N = n + 1 $. 
The coefficients are determined by the fact that the quadrature formula is interpolational, that is,

$$ 
B _ {k}  ^ {(} n)  = \ 

\frac{(- 1) ^ {n - k } }{k! ( n - k)! n }

\int\limits _ { 0 } ^ { n } 

\frac{t ( t - 1) \dots ( t - n) }{t - k }
  dt.
$$

For  $  n = 1 \dots 7 , 9 $
all coefficients are positive, for  $  n = 8 $
and  $  n \geq  10 $
there are both positive and negative ones among them. The algebraic degree of accuracy (the number  $  d $
such that the formula is exact for all polynomials of degree at most  $  d $
and not exact for  $  x ^ {d + 1 } $) 
is  $  n $
for odd  $  n $
and  $  n + 1 $
for even  $  n $. 
The simplest special cases of the Newton–Cotes quadrature formula are:  $  n = 1 $, 
$  h = b - a $, 
$  N = 2 $,

$$ 
\int\limits _ { a } ^ { b }  f ( x)  dx  \cong \ 
{
\frac{b - a }{2}
 } [ f ( a) + f ( b)],
$$

the [[Trapezium formula|trapezium formula]];  $  n = 2 $, 
$  h = ( b - a)/2 $, 
$  N = 3 $,

$$ 
\int\limits _ { a } ^ { b } 
f ( x)  dx  \cong \ 
{
\frac{b - a }{6}
 }
\left [ f ( a) + 4f \left ( {
\frac{a + b }{2}
 } \right ) + f ( b)
\right ] ,
$$

the [[Simpson formula|Simpson formula]];  $  n = 3 $, 
$  h = ( b - a)/3 $, 
$  N = 4 $,

$$ 
\int\limits _ { a } ^ { b } 
f ( x)  dx  \cong \ 
{
\frac{b - a }{8}
 }
\left [ f ( a) + 3f ( a + h) + 3f ( a + 2h) + f ( b) \right ] ,
$$

the  "three-eighths"  quadrature formula. For large  $  n $
the Newton–Cotes formula is seldom used (because of the property of the coefficients for  $  n \geq  10 $
mentioned above). One prefers to use for small  $  n $
the compound Newton–Cotes quadrature formulas, namely, the trapezium formula and Simpson's formula.

The coefficients of the Newton–Cotes quadrature formula for  $  n $
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  $  n $
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>