Steffensen interpolation formula

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