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Central Difference

The central difference for a function tabulated at equal intervals f_n is defined by

delta(f_n)=delta_n=delta_n^1=f_(n+1/2)-f_(n-1/2).
(1)

First and higher order central differences arranged so as to involve integer indices are then given by

delta_(n+1/2)=delta_(n+1/2)^1
(2)
=f_(n+1)-f_n
(3)
delta_n^2=delta_(n+1/2)^1-delta_(n-1/2)^1
(4)
=f_(n+1)-2f_n+f_(n-1)
(5)
delta_(n+1/2)^3=delta_(n+1)^2-delta_n^2
(6)
=f_(n+2)-3f_(n+1)+3f_n-f_(n-1)
(7)

(Abramowitz and Stegun 1972, p. 877).

Higher order differences may be computed for even and odd powers,

delta_n^(2k)=sum_(j=0)^(2k)(-1)^j(2k; j)f_(n+k-j)
(8)
delta_(n+1/2)^(2k+1)=sum_(j=0)^(2k+1)(-1)^j(2k+1; j)f_(n+k+1-j)
(9)

(Abramowitz and Stegun 1972, p. 877).

See also

Backward Difference, Divided Difference, Forward Difference

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References

Abramowitz, M. and Stegun, I. A. (Eds.). "Differences." §25.1 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 877-878, 1972.Jeffreys, H. and Jeffreys, B. S. "Central Differences Formula." §9.084 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 284-286, 1988.Sheppard, W. F. "Central-Difference Formulæ." Proc. London Math. Soc. 31, 449-488, 1899.Whittaker, E. T. and Robinson, G. "Central-Difference Formulae." Ch. 3 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 35-52, 1967.

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Central Difference

Cite this as:

Weisstein, Eric W. "Central Difference." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CentralDifference.html

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