Difference between revisions of "Steffensen interpolation formula"
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| − | + | A form of notation of the interpolation polynomial obtained from the [[Stirling interpolation formula|Stirling interpolation formula]] by means of the nodes $ x _ {0} , x _ {0} + h, x _ {0} - h \dots x _ {0} + nh, x _ {0} - nh $ | |
| + | at a point $ x = x _ {0} + th $: | ||
| − | + | $$ | |
| + | L _ {2n} ( x _ {0} + th) = \ | ||
| + | f _ {0} + tf _ {0} ^ { 1 } + | ||
| + | \frac{t ^ {2} }{2!} | ||
| + | f _ {0} ^ { 2 } + \dots + | ||
| + | $$ | ||
| + | |||
| + | $$ | ||
| + | + | ||
| + | |||
| + | \frac{t( t ^ {2} - 1) \dots [ t ^ {2} - ( n- 1) ^ {2} ] }{(} | ||
| + | 2n- 1)! f _ {0} ^ { 2n- 1 } + | ||
| + | $$ | ||
| + | |||
| + | $$ | ||
| + | + | ||
| + | |||
| + | \frac{t ^ {2} ( t ^ {2} - 1) \dots [ t ^ {2} -( n- 1) ^ {2} ] }{(} | ||
| + | 2n)! f _ {0} ^ { 2n } , | ||
| + | $$ | ||
using the relations | using the relations | ||
| − | + | $$ | |
| + | f _ {0} ^ { 2k- 1 } = | ||
| + | \frac{1}{2} | ||
| + | ( f _ {1/2} ^ { 2k- 1 } + f _ {-} 1/2 ^ { 2k- 1 } ),\ \ | ||
| + | f _ {0} ^ { 2k } = f _ {1/2} ^ { 2k- 1 } - f _ {-} 1/2 ^ { 2k- 1 } . | ||
| + | $$ | ||
After collecting similar terms, the Steffensen interpolation formula can be written in the form | After collecting similar terms, the Steffensen interpolation formula can be written in the form | ||
| − | + | $$ | |
| + | L _ {2n} ( x) = L _ {2n} ( x _ {0} + th) = | ||
| + | $$ | ||
| − | + | $$ | |
| + | = \ | ||
| + | f _ {0} + t( t+ | ||
| + | \frac{1)}{2!} | ||
| + | f _ {1/2} ^ { 1 } | ||
| + | - t( t- | ||
| + | \frac{1)}{2!} | ||
| + | f _ {- 1/2 } ^ { 1 } + \dots + | ||
| + | $$ | ||
| − | + | $$ | |
| + | + | ||
| + | \dots + | ||
| + | \frac{t( t ^ {2} - 1) \dots [ t ^ {2} - | ||
| + | ( n- 1) ^ {2} ]( t+ n) }{(} | ||
| + | 2n)! f _ {1/2} ^ { 2n- 1 } + | ||
| + | $$ | ||
| − | + | $$ | |
| + | - | ||
| + | \frac{t( t ^ {2} - 1) \dots [ t ^ {2} - ( n- 1) ^ {2} ]( t- n) }{(} | ||
| + | 2n)! f _ {-} 1/2 ^ { 2n- 1 } . | ||
| + | $$ | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> G.A. Korn, T.M. Korn, "Mathematical handbook for scientists and engineers" , McGraw-Hill (1968)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> G.A. Korn, T.M. Korn, "Mathematical handbook for scientists and engineers" , McGraw-Hill (1968)</TD></TR></table> | ||
| − | |||
| − | |||
====Comments==== | ====Comments==== | ||
| − | The central differences | + | The central differences $ f _ {i + 1/2 } ^ { 2m+ 1 } $, |
| + | $ f _ {i} ^ { 2m } $( | ||
| + | $ m = 0, ,1 \dots $ | ||
| + | $ i = \dots, - 1, 0, 1,\dots $) | ||
| + | are defined recursively from the (tabulated values) $ f _ {i} ^ { 0 } = f ( x _ {0} + i h ) $ | ||
| + | by the formulas | ||
| − | + | $$ | |
| + | f _ {i+ 1/2 } ^ { 2m+ 1 } = \ | ||
| + | f _ {i+} 1 ^ { 2m } - f _ {i} ^ { 2m } ; \ \ | ||
| + | f _ {i} ^ { 2m } = f _ {i+ 1/2 } ^ { 2m- 1 } - | ||
| + | f _ {i - 1/2 } ^ { 2m- 1 } . | ||
| + | $$ | ||
The Steffensen interpolation formula is also known as Everett's second formula. | The Steffensen interpolation formula is also known as Everett's second formula. | ||
Latest revision as of 08:23, 6 June 2020
A form of notation of the interpolation polynomial obtained from the Stirling interpolation formula by means of the nodes $ x _ {0} , x _ {0} + h, x _ {0} - h \dots x _ {0} + nh, x _ {0} - nh $
at a point $ x = x _ {0} + th $:
$$ L _ {2n} ( x _ {0} + th) = \ f _ {0} + tf _ {0} ^ { 1 } + \frac{t ^ {2} }{2!} f _ {0} ^ { 2 } + \dots + $$
$$ + \frac{t( t ^ {2} - 1) \dots [ t ^ {2} - ( n- 1) ^ {2} ] }{(} 2n- 1)! f _ {0} ^ { 2n- 1 } + $$
$$ + \frac{t ^ {2} ( t ^ {2} - 1) \dots [ t ^ {2} -( n- 1) ^ {2} ] }{(} 2n)! f _ {0} ^ { 2n } , $$
using the relations
$$ f _ {0} ^ { 2k- 1 } = \frac{1}{2} ( f _ {1/2} ^ { 2k- 1 } + f _ {-} 1/2 ^ { 2k- 1 } ),\ \ f _ {0} ^ { 2k } = f _ {1/2} ^ { 2k- 1 } - f _ {-} 1/2 ^ { 2k- 1 } . $$
After collecting similar terms, the Steffensen interpolation formula can be written in the form
$$ L _ {2n} ( x) = L _ {2n} ( x _ {0} + th) = $$
$$ = \ f _ {0} + t( t+ \frac{1)}{2!} f _ {1/2} ^ { 1 } - t( t- \frac{1)}{2!} f _ {- 1/2 } ^ { 1 } + \dots + $$
$$ + \dots + \frac{t( t ^ {2} - 1) \dots [ t ^ {2} - ( n- 1) ^ {2} ]( t+ n) }{(} 2n)! f _ {1/2} ^ { 2n- 1 } + $$
$$ - \frac{t( t ^ {2} - 1) \dots [ t ^ {2} - ( n- 1) ^ {2} ]( t- n) }{(} 2n)! f _ {-} 1/2 ^ { 2n- 1 } . $$
References
| [1] | G.A. Korn, T.M. Korn, "Mathematical handbook for scientists and engineers" , McGraw-Hill (1968) |
Comments
The central differences $ f _ {i + 1/2 } ^ { 2m+ 1 } $, $ f _ {i} ^ { 2m } $( $ m = 0, ,1 \dots $ $ i = \dots, - 1, 0, 1,\dots $) are defined recursively from the (tabulated values) $ f _ {i} ^ { 0 } = f ( x _ {0} + i h ) $ by the formulas
$$ f _ {i+ 1/2 } ^ { 2m+ 1 } = \ f _ {i+} 1 ^ { 2m } - f _ {i} ^ { 2m } ; \ \ f _ {i} ^ { 2m } = f _ {i+ 1/2 } ^ { 2m- 1 } - f _ {i - 1/2 } ^ { 2m- 1 } . $$
The Steffensen interpolation formula is also known as Everett's second formula.
References
| [a1] | F.B. Hildebrand, "Introduction to numerical analysis" , McGraw-Hill (1956) pp. 103–105 |
| [a2] | J.F. Steffensen, "Interpolation" , Chelsea, reprint (1950) |
| [a3] | C.-E. Froberg, "Introduction to numerical analysis" , Addison-Wesley (1965) pp. 157 |
Steffensen interpolation formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Steffensen_interpolation_formula&oldid=13445