Backward Difference

The backward difference is a Finite Difference defined by

$$\nabla_p \equiv \nabla f_p\equiv f_p-f_{p-1}.$$ (1)

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

$$\nabla^2_p = \nabla(\nabla p)=\nabla(f_p-f_{p-1})=\nabla f_p-\nabla f_{p-1}$$ (2)

$$= (f_p-f_{p-1})-(f_{p-1}-f_{p-2})$$

$$= f_p-2f_{p-1}+f_{p-2}$$ (3)

In general,

$$\nabla^k_p \equiv \nabla^k f_p \equiv \sum_{m=0}^k (-1)^m {k\choose m} f_{p-k+m},$$ (4)

where ${k\choose m}$ is a Binomial Coefficient.

Newton's Backward Difference Formula expresses $f_p$ as the sum of the $n$th backward differences

$$f_p=f_0+p\nabla_0+{\textstyle{1\over 2!}}p(p+1)\nabla_0^2+{\textstyle{1\over 3!}}p(p+1)(p+2)\nabla_0^3+\ldots,$$ (5)

where $\nabla_0^n$ is the first $n$th difference computed from the difference table.

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

References

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