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A method for the approximate calculation of a definite integral
 
A method for the approximate calculation of a definite integral
  
<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/l/l060/l060820/l0608201.png" /></td> </tr></table>
+
$$
 +
= \int\limits _ { a } ^ { b }  f ( x)  dx ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l0608202.png" /> has exactly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l0608203.png" /> roots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l0608204.png" /> inside the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l0608205.png" />,
+
where $  f $
 +
has exactly $  n $
 +
roots $  x _ {i} $
 +
inside the interval $  [ a, b] $,
  
<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/l/l060/l060820/l0608206.png" /></td> </tr></table>
+
$$
 +
x _ {0}  =  a  < x _ {1}  < \dots < x _ {n}  < = x _ {n+} 1 ,
 +
$$
  
 
and satisfies the conditions stated below. Let
 
and satisfies the conditions stated below. Let
  
<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/l/l060/l060820/l0608207.png" /></td> </tr></table>
+
$$
 +
v _ {i}  = ( - 1 )  ^ {i} \int\limits _ {x _ {i} } ^ { {x _ i+} 1 } f( x)  dx ,\ \
 +
i = 0 \dots n ,
 +
$$
  
then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l0608208.png" />, where
+
then $  I = S $,  
 +
where
  
<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/l/l060/l060820/l0608209.png" /></td> </tr></table>
+
$$
 +
= \sum _ { j= } 0 ^ { n }  ( - 1 )  ^ {j} v _ {j} .
 +
$$
  
It is assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l06082010.png" /> preserves its sign on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l06082011.png" />, has different signs on adjacent intervals, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l06082012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l06082013.png" />. Such a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l06082014.png" /> is said to be oscillatory. The calculation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l06082015.png" /> by means of a quadrature formula for large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l06082016.png" /> is difficult, since a good approximation of an oscillatory function on the whole interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l06082017.png" /> is impossible in practice. The use of the equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l06082018.png" /> leads to the need to calculate all integrals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l06082019.png" />, which is also inadvisable in the case of large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l06082020.png" />.
+
It is assumed that $  f $
 +
preserves its sign on the interval $  [ x _ {i} , x _ {i+} 1 ] $,  
 +
has different signs on adjacent intervals, and $  v _ {i} \neq 0 $,  
 +
$  i = 0 \dots n $.  
 +
Such a function $  f $
 +
is said to be oscillatory. The calculation of $  I $
 +
by means of a quadrature formula for large $  n $
 +
is difficult, since a good approximation of an oscillatory function on the whole interval $  [ a , b ] $
 +
is impossible in practice. The use of the equality $  I = S $
 +
leads to the need to calculate all integrals $  v _ {j} $,  
 +
which is also inadvisable in the case of large $  n $.
  
The approximate calculation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l06082021.png" /> in Longman's method is based on the equality (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l06082022.png" />)
+
The approximate calculation of $  I $
 +
in Longman's method is based on the equality ( $  n \geq  p $)
  
<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/l/l060/l060820/l06082023.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
= \sum _ { k= } 0 ^ { p- }  1 ( - 1 ) ^ {k} 2 ^ {- k - 1 } \Delta  ^ {k} v _ {0} +
 +
$$
  
<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/l/l060/l060820/l06082024.png" /></td> </tr></table>
+
$$
 +
+
 +
( - 1)  ^ {n} \sum _ { k= } 0 ^ { p- }  1 2 ^ {- k - 1 } \Delta  ^ {k} v _ {n-} k + 2  ^ {-} p ( - 1 )  ^ {p} \sum _ { k= } 0 ^ { n- }  p ( - 1 )  ^ {k} \Delta  ^ {p} v _ {k} .
 +
$$
  
In (1) the finite differences of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l06082025.png" /> as functions of the discrete argument <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l06082026.png" /> occur:
+
In (1) the finite differences of $  v _ {j} $
 +
as functions of the discrete argument $  j $
 +
occur:
  
<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/l/l060/l060820/l06082027.png" /></td> </tr></table>
+
$$
 +
\Delta v _ {j}  = v _ {j+} 1 - v _ {j} ,\  j= 0 \dots n- 1,
 +
$$
  
<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/l/l060/l060820/l06082028.png" /></td> </tr></table>
+
$$
 +
\Delta  ^ {r+} 1 v _ {j}  = \Delta  ^ {r} v _ {j+} 1 - \Delta  ^ {r} v _ {j} ,
 +
$$
  
<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/l/l060/l060820/l06082029.png" /></td> </tr></table>
+
$$
 +
= 1 \dots p- 1; \  j  = 0 \dots n- r- 1 .
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l06082030.png" /> is such that on the right-hand side of (1) one can neglect terms containing finite differences of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l06082031.png" />, then the approximate equality
+
If $  v _ {j} $
 +
is such that on the right-hand side of (1) one can neglect terms containing finite differences of order $  p $,  
 +
then the approximate equality
  
<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/l/l060/l060820/l06082032.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
S  \cong  \sum _ { k= } 0 ^ { p- }  1 ( - 1 )  ^ {k} 2 ^ {- k - 1 } \Delta  ^ {k} v _ {0} + ( - 1 )  ^ {n} \sum _ { k= } 0 ^ { p- }  1 2 ^ {- k - 1 } \Delta  ^ {k} v _ {n-} k
 +
$$
  
can be used to calculate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l06082033.png" />. To calculate the right-hand side of (2) it is sufficient to know the first <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l06082034.png" /> values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l06082035.png" />, that is, the values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l06082036.png" />, and the last <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l06082037.png" /> values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l06082038.png" />. Longman's method consists in the use of (2) for an approximate calculation of the sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l06082039.png" />.
+
can be used to calculate $  S $.  
 +
To calculate the right-hand side of (2) it is sufficient to know the first $  p $
 +
values $  v _ {j} $,  
 +
that is, the values $  v _ {0} \dots v _ {p-} 1 $,  
 +
and the last $  p $
 +
values $  v _ {n} \dots v _ {n-} p+ 1 $.  
 +
Longman's method consists in the use of (2) for an approximate calculation of the sum $  S $.
  
If in the integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l06082040.png" /> the upper limit of integration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l06082041.png" /> and
+
If in the integral $  I $
 +
the upper limit of integration $  b = + \infty $
 +
and
  
<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/l/l060/l060820/l06082042.png" /></td> </tr></table>
+
$$
 +
= \sum _ { i= } 0 ^  \infty  ( - 1 )  ^ {i} v _ {i} ,
 +
$$
  
 
then instead of (1) one must use the equality
 
then instead of (1) one must use the equality
  
<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/l/l060/l060820/l06082043.png" /></td> </tr></table>
+
$$
 +
\sum _ { i= } 0 ^  \infty  ( - 1 )  ^ {i} v _ {i}  = \sum _ { i= } 0 ^  \infty  ( - 1
 +
)  ^ {i} 2  ^ {-} i- 1 \Delta  ^ {i} v _ {0}  $$
  
 
(the Euler transform) and replace the series on the right-hand side by a partial sum.
 
(the Euler transform) and replace the series on the right-hand side by a partial sum.

Revision as of 04:11, 6 June 2020


A method for the approximate calculation of a definite integral

$$ I = \int\limits _ { a } ^ { b } f ( x) dx , $$

where $ f $ has exactly $ n $ roots $ x _ {i} $ inside the interval $ [ a, b] $,

$$ x _ {0} = a < x _ {1} < \dots < x _ {n} < b = x _ {n+} 1 , $$

and satisfies the conditions stated below. Let

$$ v _ {i} = ( - 1 ) ^ {i} \int\limits _ {x _ {i} } ^ { {x _ i+} 1 } f( x) dx ,\ \ i = 0 \dots n , $$

then $ I = S $, where

$$ S = \sum _ { j= } 0 ^ { n } ( - 1 ) ^ {j} v _ {j} . $$

It is assumed that $ f $ preserves its sign on the interval $ [ x _ {i} , x _ {i+} 1 ] $, has different signs on adjacent intervals, and $ v _ {i} \neq 0 $, $ i = 0 \dots n $. Such a function $ f $ is said to be oscillatory. The calculation of $ I $ by means of a quadrature formula for large $ n $ is difficult, since a good approximation of an oscillatory function on the whole interval $ [ a , b ] $ is impossible in practice. The use of the equality $ I = S $ leads to the need to calculate all integrals $ v _ {j} $, which is also inadvisable in the case of large $ n $.

The approximate calculation of $ I $ in Longman's method is based on the equality ( $ n \geq p $)

$$ \tag{1 } S = \sum _ { k= } 0 ^ { p- } 1 ( - 1 ) ^ {k} 2 ^ {- k - 1 } \Delta ^ {k} v _ {0} + $$

$$ + ( - 1) ^ {n} \sum _ { k= } 0 ^ { p- } 1 2 ^ {- k - 1 } \Delta ^ {k} v _ {n-} k + 2 ^ {-} p ( - 1 ) ^ {p} \sum _ { k= } 0 ^ { n- } p ( - 1 ) ^ {k} \Delta ^ {p} v _ {k} . $$

In (1) the finite differences of $ v _ {j} $ as functions of the discrete argument $ j $ occur:

$$ \Delta v _ {j} = v _ {j+} 1 - v _ {j} ,\ j= 0 \dots n- 1, $$

$$ \Delta ^ {r+} 1 v _ {j} = \Delta ^ {r} v _ {j+} 1 - \Delta ^ {r} v _ {j} , $$

$$ r = 1 \dots p- 1; \ j = 0 \dots n- r- 1 . $$

If $ v _ {j} $ is such that on the right-hand side of (1) one can neglect terms containing finite differences of order $ p $, then the approximate equality

$$ \tag{2 } S \cong \sum _ { k= } 0 ^ { p- } 1 ( - 1 ) ^ {k} 2 ^ {- k - 1 } \Delta ^ {k} v _ {0} + ( - 1 ) ^ {n} \sum _ { k= } 0 ^ { p- } 1 2 ^ {- k - 1 } \Delta ^ {k} v _ {n-} k $$

can be used to calculate $ S $. To calculate the right-hand side of (2) it is sufficient to know the first $ p $ values $ v _ {j} $, that is, the values $ v _ {0} \dots v _ {p-} 1 $, and the last $ p $ values $ v _ {n} \dots v _ {n-} p+ 1 $. Longman's method consists in the use of (2) for an approximate calculation of the sum $ S $.

If in the integral $ I $ the upper limit of integration $ b = + \infty $ and

$$ I = \sum _ { i= } 0 ^ \infty ( - 1 ) ^ {i} v _ {i} , $$

then instead of (1) one must use the equality

$$ \sum _ { i= } 0 ^ \infty ( - 1 ) ^ {i} v _ {i} = \sum _ { i= } 0 ^ \infty ( - 1 ) ^ {i} 2 ^ {-} i- 1 \Delta ^ {i} v _ {0} $$

(the Euler transform) and replace the series on the right-hand side by a partial sum.

The method was proposed by I.M. Longman [1].

References

[1] I.M. Longman, "A method for the numerical evaluation of finite integrals of oscillatory functions" Math. Comput. , 14 : 69 (1960) pp. 53–59
[2] P.J. Davis, P. Rabinowitz, "Methods of numerical integration" , Acad. Press (1984)
How to Cite This Entry:
Longman method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Longman_method&oldid=47713
This article was adapted from an original article by I.P. Mysovskikh (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article