Backward Difference

The backward difference is a finite difference defined by

 del _p=del f_p=f_p-f_(p-1).

Higher order differences are obtained by repeated operations of the backward difference operator, so

del _p^2=del (del p)=del (f_p-f_(p-1))=del f_p-del f_(p-1)

In general,

 del _p^k=del ^kf_p=sum_(m=0)^k(-1)^m(k; m)f_(p-m),

where (k; m) is a binomial coefficient.

The backward finite difference are implemented in the Wolfram Language as DifferenceDelta[f, i].

Newton's backward difference formula expresses f_p as the sum of the nth backward differences

 f_p=f_0+pdel _0+1/(2!)p(p+1)del _0^2+1/(3!)p(p+1)(p+2)del _0^3+...,

where del _0^n is the first nth difference computed from the difference table.

See also

Adams' Method, Divided Difference, Finite Difference, Forward Difference, Newton's Backward Difference Formula, Reciprocal Difference

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Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 429 and 433, 1987.

Referenced on Wolfram|Alpha

Backward Difference

Cite this as:

Weisstein, Eric W. "Backward Difference." From MathWorld--A Wolfram Web Resource.

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