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)

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