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Difference between revisions of "Longman method"

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m (tex encoded by computer)
(latex details)
 
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$$  
 
$$  
x _ {0}  =  a  <  x _ {1}  < \dots <  x _ {n}  <  b  =  x _ {n+} 1 ,
+
x _ {0}  =  a  <  x _ {1}  < \dots <  x _ {n}  <  b  =  x _ {n+1} ,
 
$$
 
$$
  
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$$  
 
$$  
v _ {i}  =  ( - 1 )  ^ {i} \int\limits _ {x _ {i} } ^ { {x _ i+} 1 } f( x)  dx ,\ \  
+
v _ {i}  =  ( - 1 )  ^ {i} \int\limits _ {x _ {i} } ^ { {x _ i+1} } f( x)  dx ,\ \  
 
i = 0 \dots n ,
 
i = 0 \dots n ,
 
$$
 
$$
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$$  
 
$$  
S  =  \sum _ { j= } 0 ^ { n }  ( - 1 )  ^ {j} v _ {j} .
+
S  =  \sum_{j=0}^ { n }  ( - 1 )  ^ {j} v _ {j} .
 
$$
 
$$
  
 
It is assumed that  $  f $
 
It is assumed that  $  f $
preserves its sign on the interval  $  [ x _ {i} , x _ {i+} 1 ] $,  
+
preserves its sign on the interval  $  [ x _ {i} , x _ {i+1} ] $,  
 
has different signs on adjacent intervals, and  $  v _ {i} \neq 0 $,  
 
has different signs on adjacent intervals, and  $  v _ {i} \neq 0 $,  
 
$  i = 0 \dots n $.  
 
$  i = 0 \dots n $.  
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$$ \tag{1 }
 
$$ \tag{1 }
S  =  \sum _ { k= } 0 ^ { p- } 1 ( - 1 )  ^ {k} 2 ^ {- k - 1 } \Delta  ^ {k} v _ {0} +
+
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} .
+
( - 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} .
 
$$
 
$$
  
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$$  
 
$$  
\Delta v _ {j}  =  v _ {j+} 1 - v _ {j} ,\  j= 0 \dots n- 1,
+
\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} ,
+
\Delta  ^ {r+1} v _ {j}  =  \Delta  ^ {r} v _ {j+1} - \Delta  ^ {r} v _ {j} ,
 
$$
 
$$
  
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$$ \tag{2 }
 
$$ \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
+
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}
 
$$
 
$$
  
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To calculate the right-hand side of (2) it is sufficient to know the first  $  p $
 
To calculate the right-hand side of (2) it is sufficient to know the first  $  p $
 
values  $  v _ {j} $,  
 
values  $  v _ {j} $,  
that is, the values  $  v _ {0} \dots v _ {p-} 1 $,  
+
that is, the values  $  v _ {0} \dots v _ {p-1} $,  
 
and the last  $  p $
 
and the last  $  p $
values  $  v _ {n} \dots v _ {n-} p+ 1 $.  
+
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 $.
 
Longman's method consists in the use of (2) for an approximate calculation of the sum  $  S $.
  
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$$  
 
$$  
I  =  \sum _ { i= } 0 ^  \infty  ( - 1 )  ^ {i} v _ {i} ,
+
I  =  \sum_{i=0}^  \infty  ( - 1 )  ^ {i} v _ {i} ,
 
$$
 
$$
  
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$$  
 
$$  
\sum _ { i= } 0 ^  \infty  ( - 1 )  ^ {i} v _ {i}  =  \sum _ { i= } 0 ^  \infty   ( - 1
+
\sum_{i=0}^  \infty  ( - 1 )  ^ {i} v _ {i}  =  \sum_{i=0}^  \infty (-1)  ^ {i} 2^{-i-1} \Delta  ^ {i} v _ {0}
)  ^ {i} 2 ^ {-} i- 1 \Delta  ^ {i} v _ {0} $$
+
$$
  
 
(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.
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====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.M. Longman,  "A method for the numerical evaluation of finite integrals of oscillatory functions"  ''Math. Comput.'' , '''14''' :  69  (1960)  pp. 53–59</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.J. Davis,  P. Rabinowitz,  "Methods of numerical integration" , Acad. Press  (1984)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  I.M. Longman,  "A method for the numerical evaluation of finite integrals of oscillatory functions"  ''Math. Comput.'' , '''14''' :  69  (1960)  pp. 53–59</TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top">  P.J. Davis,  P. Rabinowitz,  "Methods of numerical integration" , Acad. Press  (1984)</TD></TR>
 +
</table>

Latest revision as of 19:56, 19 January 2024


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=55220
This article was adapted from an original article by I.P. Mysovskikh (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article