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Steffensen interpolation formula

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A form of notation of the interpolation polynomial obtained from the Stirling interpolation formula by means of the nodes 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
How to Cite This Entry:
Steffensen interpolation formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Steffensen_interpolation_formula&oldid=48830
This article was adapted from an original article by M.K. Samarin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article