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 $ | |
| + | 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 | 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: | 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 | + | For small $ t $, |
| + | Stirling's interpolation formula is more exact than other interpolation formulas. | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> I.S. Berezin, N.P. Zhidkov, "Computing methods" , Pergamon (1973) (Translated from Russian)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> I.S. Berezin, N.P. Zhidkov, "Computing methods" , Pergamon (1973) (Translated from Russian)</TD></TR></table> | ||
| − | |||
| − | |||
====Comments==== | ====Comments==== | ||
| − | The central differences | + | 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==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> F.B. Hildebrand, "Introduction to numerical analysis" , Dover, reprint (1987) pp. 139</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> F.B. Hildebrand, "Introduction to numerical analysis" , Dover, reprint (1987) pp. 139</TD></TR></table> | ||
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 |
Stirling interpolation formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stirling_interpolation_formula&oldid=12181