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A special case of the [[Newton–Cotes quadrature formula|Newton–Cotes quadrature formula]], in which three nodes are specified:
 
A special case of the [[Newton–Cotes quadrature formula|Newton–Cotes quadrature formula]], in which three nodes are specified:
  
<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/s/s085/s085450/s0854501.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
\int\limits _ { a } ^ { b }  f ( x)  dx  \cong \
 +
{
 +
\frac{b - a }{6}
 +
}
 +
\left [ f ( a) + 4f \left ( {
 +
\frac{a + b }{2}
 +
} \right ) + f ( b) \right ] .
 +
$$
  
Let the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085450/s0854502.png" /> be broken up into an even number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085450/s0854503.png" /> of subintervals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085450/s0854504.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085450/s0854505.png" />, of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085450/s0854506.png" />, and calculate the integral over the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085450/s0854507.png" /> by the quadrature formula (1):
+
Let the interval $  [ a, b] $
 +
be broken up into an even number $  n $
 +
of subintervals $  [ x _ {i} , x _ {i + 1 }  ] $,  
 +
$  i = 0 \dots n - 1 $,  
 +
of length $  h = ( b - a)/n $,  
 +
and calculate the integral over the interval $  [ x _ {2k} , x _ {2k + 2 }  ] $
 +
by the quadrature formula (1):
  
<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/s/s085/s085450/s0854508.png" /></td> </tr></table>
+
$$
 +
\int\limits _ {x _ {2k} } ^ { {x _ {2k}  + 2 } }
 +
f ( x)  dx  \cong  {
 +
\frac{h}{3}
 +
}
 +
[ f ( x _ {2k} ) + 4f ( x _ {2k + 1 }  ) + f ( x _ {2k + 2 }  )].
 +
$$
  
Summation over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085450/s0854509.png" /> from 0 to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085450/s08545010.png" /> on both sides leads to the composite Simpson formula
+
Summation over $  k $
 +
from 0 to $  ( n/2) - 1 $
 +
on both sides leads to the composite Simpson formula
  
<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/s/s085/s085450/s08545011.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
\int\limits _ { a } ^ { b }  f ( x) dx \cong
 +
$$
  
<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/s/s085/s085450/s08545012.png" /></td> </tr></table>
+
$$
 +
\cong \
 +
{
 +
\frac{h}{3}
 +
} \{ f ( a) + f ( b) + 2 [ f ( x _ {2} ) + f ( x _ {4} ) +
 +
\dots
 +
+ f ( x _ {n - 2 }  )] +
 +
$$
  
<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/s/s085/s085450/s08545013.png" /></td> </tr></table>
+
$$
 +
+
 +
{} 4 [ f ( x _ {1} ) +
 +
f ( x _ {3} ) + \dots + f ( x _ {n - 1 }  )] \} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085450/s08545014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085450/s08545015.png" />. The quadrature formula (2) is also called Simpson's formula (that is, the word composite is dropped). The algebraic degree of accuracy of (2), and of (1), is equal to 3.
+
where $  x _ {j} = a + jh $,  
 +
$  j = 0 \dots n $.  
 +
The quadrature formula (2) is also called Simpson's formula (that is, the word composite is dropped). The algebraic degree of accuracy of (2), and of (1), is equal to 3.
  
If the integrand <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085450/s08545016.png" /> has a continuous derivative of the fourth order on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085450/s08545017.png" />, then the error <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085450/s08545018.png" /> of the quadrature formula (2) — the difference between the left-hand and right-hand members of the approximate equation (2) — can be written as
+
If the integrand $  f $
 +
has a continuous derivative of the fourth order on $  [ a, b] $,  
 +
then the error $  R ( f  ) $
 +
of the quadrature formula (2) — the difference between the left-hand and right-hand members of the approximate equation (2) — can be written as
  
<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/s/s085/s085450/s08545019.png" /></td> </tr></table>
+
$$
 +
R ( f  )  = - {
 +
\frac{b - a }{180}
 +
} h  ^ {4} f ^ { ( 4) } ( \xi ),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085450/s08545020.png" /> is some point in the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085450/s08545021.png" />.
+
where $  \xi $
 +
is some point in the interval $  [ a, b] $.
  
 
Simpson's formula was named after Th. Simpson, who obtained it in 1743, although the formula was already known, for example to J. Gregory, in 1668.
 
Simpson's formula was named after Th. Simpson, who obtained it in 1743, although the formula was already known, for example to J. Gregory, in 1668.
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 08:14, 6 June 2020


A special case of the Newton–Cotes quadrature formula, in which three nodes are specified:

$$ \tag{1 } \int\limits _ { a } ^ { b } f ( x) dx \cong \ { \frac{b - a }{6} } \left [ f ( a) + 4f \left ( { \frac{a + b }{2} } \right ) + f ( b) \right ] . $$

Let the interval $ [ a, b] $ be broken up into an even number $ n $ of subintervals $ [ x _ {i} , x _ {i + 1 } ] $, $ i = 0 \dots n - 1 $, of length $ h = ( b - a)/n $, and calculate the integral over the interval $ [ x _ {2k} , x _ {2k + 2 } ] $ by the quadrature formula (1):

$$ \int\limits _ {x _ {2k} } ^ { {x _ {2k} + 2 } } f ( x) dx \cong { \frac{h}{3} } [ f ( x _ {2k} ) + 4f ( x _ {2k + 1 } ) + f ( x _ {2k + 2 } )]. $$

Summation over $ k $ from 0 to $ ( n/2) - 1 $ on both sides leads to the composite Simpson formula

$$ \tag{2 } \int\limits _ { a } ^ { b } f ( x) dx \cong $$

$$ \cong \ { \frac{h}{3} } \{ f ( a) + f ( b) + 2 [ f ( x _ {2} ) + f ( x _ {4} ) + \dots + f ( x _ {n - 2 } )] + $$

$$ + {} 4 [ f ( x _ {1} ) + f ( x _ {3} ) + \dots + f ( x _ {n - 1 } )] \} , $$

where $ x _ {j} = a + jh $, $ j = 0 \dots n $. The quadrature formula (2) is also called Simpson's formula (that is, the word composite is dropped). The algebraic degree of accuracy of (2), and of (1), is equal to 3.

If the integrand $ f $ has a continuous derivative of the fourth order on $ [ a, b] $, then the error $ R ( f ) $ of the quadrature formula (2) — the difference between the left-hand and right-hand members of the approximate equation (2) — can be written as

$$ R ( f ) = - { \frac{b - a }{180} } h ^ {4} f ^ { ( 4) } ( \xi ), $$

where $ \xi $ is some point in the interval $ [ a, b] $.

Simpson's formula was named after Th. Simpson, who obtained it in 1743, although the formula was already known, for example to J. Gregory, in 1668.

Comments

Simpson's formula is also called Simpson's rule.

References

[a1] R. Courant, "Vorlesungen über Differential- und Integralrechnung" , 1 , Springer (1971)
[a2] D.M. Young, R.T. Gregory, "A survey of numerical mathematics" , Dover, reprint (1988) pp. §7.4
How to Cite This Entry:
Simpson formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Simpson_formula&oldid=17966
This article was adapted from an original article by I.P. Mysovskikh (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article