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Difference between revisions of "Stirling interpolation formula"

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The half-sum of the [[Gauss interpolation formula|Gauss interpolation formula]] for forward interpolation with respect to the nodes  $  x _ {0} , x _ {0} + h, x _ {0} - h \dots x _ {0} + nh, x _ {0} - nh $
+
The half-sum of the [[Gauss interpolation formula|Gauss interpolation formula]] for forward interpolation with respect to the nodes  $  x _ {0} , x _ {0} + h, x _ {0} - h ,\dots, x _ {0} + nh, x _ {0} - nh $
 
at the point  $  x = x _ {0} + th $:
 
at the point  $  x = x _ {0} + th $:
  
 
$$  
 
$$  
 
G _ {2n} ( x _ {0} + th)  = \  
 
G _ {2n} ( x _ {0} + th)  = \  
f _ {0} + f _ {1/2} ^ { 1 } t + f _ {0} ^ { 2 } t( t-  
+
f _ {0} + f _ {1/2} ^ { 1 } t + f _ {0} ^ { 2 }\frac{ t( t-  
\frac{1)}{2!}
+
1)}{2!}
 
  +
 
  +
 
$$
 
$$
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+  
 
+  
 
f _ {0} ^ { 2n }  
 
f _ {0} ^ { 2n }  
\frac{t( t  ^ {2} - 1  ^ {2} ) {} \dots [ t  ^ {2} -( n- 1)  ^ {2} ]( t- n) }{(}
+
\frac{t( t  ^ {2} - 1  ^ {2} ) {} \dots [ t  ^ {2} -( n- 1)  ^ {2} ]( t- n) }{(2n)!}
  2n)!
+
   
 
$$
 
$$
  
and Gauss' formula of the same order for backward interpolation with respect to the nodes  $  x _ {0} , x _ {0} - h, x _ {0} + h \dots x _ {0} - nh, x _ {0} + nh $:
+
and Gauss' formula of the same order for backward interpolation with respect to the nodes  $  x _ {0} , x _ {0} - h, x _ {0} + h, \dots ,x _ {0} - nh, x _ {0} + nh $:
  
 
$$  
 
$$  
 
G _ {2n} ( x _ {0} + th)  = \  
 
G _ {2n} ( x _ {0} + th)  = \  
f _ {0} + f _ {-} 1/2 ^ { 1 } t + f _ {0} ^ { 2 } t( t+  
+
f _ {0} + f _ {-1/2 } ^ { 1 } t + f _ {0} ^ { 2 } \frac{t( t+  
\frac{1)}{2!}
+
1)}{2!}
 
  + \dots +
 
  + \dots +
 
$$
 
$$
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+  
 
+  
 
f _ {0} ^ { 2n }  
 
f _ {0} ^ { 2n }  
\frac{t( t  ^ {2} - 1) \dots [ t  ^ {2} -( n- 1)  ^ {2} ]( t+ n) }{(}
+
\frac{t( t  ^ {2} - 1) \dots [ t  ^ {2} -( n- 1)  ^ {2} ]( t+ n) }{( 2n)! }
2n)! .
+
.
 
$$
 
$$
  
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\frac{1}{2}
 
\frac{1}{2}
  [ f _ {1/2} ^ { 2k- 1 } + f _ {-} 1/2 ^ { 2k- 1 } ] ,
+
  [ f _ {1/2} ^ { 2k- 1 } + f _ {- 1/2} ^ { 2k- 1 } ] ,
 
$$
 
$$
  
Line 74: Line 74:
 
+  
 
+  
  
\frac{t( t  ^ {2} - 1) \dots [ t  ^ {2} -( n- 1)  ^ {2} ] }{(}
+
\frac{t( t  ^ {2} - 1) \dots [ t  ^ {2} -( n- 1)  ^ {2} ] }{( 2n- 1)!}
2n- 1)! f _ {0} ^ { 2n- 1 } +  
+
f _ {0} ^ { 2n- 1 } +  
\frac{t( t  ^ {2} - 1) \dots [ t  ^ {2} -( n- 1)  ^ {2} ] }{(}
+
\frac{t( t  ^ {2} - 1) \dots [ t  ^ {2} -( n- 1)  ^ {2} ] }{( 2n)! }
2n)! f _ {0} ^ { 2n } .
+
f _ {0} ^ { 2n } .
 
$$
 
$$
  

Latest revision as of 00:52, 31 December 2021


The half-sum of the Gauss interpolation formula for forward interpolation with respect to the nodes $ x _ {0} , x _ {0} + h, x _ {0} - h ,\dots, x _ {0} + nh, x _ {0} - nh $ at the point $ x = x _ {0} + th $:

$$ G _ {2n} ( x _ {0} + th) = \ f _ {0} + f _ {1/2} ^ { 1 } t + f _ {0} ^ { 2 }\frac{ t( t- 1)}{2!} + $$

$$ + f _ {1/2} ^ { 3 } \frac{t( t ^ {2} - 1 ^ {2} ) }{3!} + f _ {0} ^ { 4 } \frac{t( t ^ {2} - 1 ^ {2} )( t - 2) }{4!} + \dots + $$

$$ + f _ {0} ^ { 2n } \frac{t( t ^ {2} - 1 ^ {2} ) {} \dots [ t ^ {2} -( n- 1) ^ {2} ]( t- n) }{(2n)!} $$

and Gauss' formula of the same order for backward interpolation with respect to the nodes $ x _ {0} , x _ {0} - h, x _ {0} + h, \dots ,x _ {0} - nh, x _ {0} + nh $:

$$ G _ {2n} ( x _ {0} + th) = \ f _ {0} + f _ {-1/2 } ^ { 1 } t + f _ {0} ^ { 2 } \frac{t( t+ 1)}{2!} + \dots + $$

$$ + f _ {0} ^ { 2n } \frac{t( t ^ {2} - 1) \dots [ t ^ {2} -( n- 1) ^ {2} ]( t+ n) }{( 2n)! } . $$

Using the notation

$$ f _ {0} ^ { 2k- 1 } = \ \frac{1}{2} [ f _ {1/2} ^ { 2k- 1 } + f _ {- 1/2} ^ { 2k- 1 } ] , $$

Stirling's interpolation formula takes the form:

$$ L _ {2n} ( x) = L _ {2n} ( x _ {0} + th) = \ f _ {0} + tf _ {0} ^ { 1 } + \frac{t ^ {2} }{2!} f _ {0} ^ { 2 } + \dots + $$

$$ + \frac{t( t ^ {2} - 1) \dots [ t ^ {2} -( n- 1) ^ {2} ] }{( 2n- 1)!} f _ {0} ^ { 2n- 1 } + \frac{t( t ^ {2} - 1) \dots [ t ^ {2} -( n- 1) ^ {2} ] }{( 2n)! } f _ {0} ^ { 2n } . $$

For small $ t $, Stirling's interpolation formula is more exact than other interpolation formulas.

References

[1] I.S. Berezin, N.P. Zhidkov, "Computing methods" , Pergamon (1973) (Translated from Russian)

Comments

The central differences $ f _ {i+ 1/2 } ^ { 2m+ 1 } $ and $ f _ {i} ^ { 2m } $( $ m = 0, 1 \dots $ $ i = \dots, - 1, 0, 1, . . . $) are defined recursively from the (tabulated values) $ f _ {i} ^ { 0 } = f ( x _ {0} + ih) $ by the formulas

$$ f _ {i + 1/2 } ^ { 2m+ 1 } = \ f _ {i+} 1 ^ { 2m } - f _ {i} ^ { 2m } ; \ \ f _ {i} ^ { 2m } = \ f _ {i + 1/2 } ^ { 2m- 1 } - f _ {i - 1/2 } ^ { 2m - 1 } . $$

References

[a1] F.B. Hildebrand, "Introduction to numerical analysis" , Dover, reprint (1987) pp. 139
How to Cite This Entry:
Stirling interpolation formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stirling_interpolation_formula&oldid=51820
This article was adapted from an original article by M.K. Samarin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article