Finite Differencing

Finite differencing[Paige, R. and Koenig, S., ``Finite Differencing of Computable Expressions'', ACM Transactions on Programming Languages and Systems, vol. 4, no. 3 (July 1982), pp. 402-454] is a general technique for optimizing expensive computations f(i) that occur in a loop:

• Maintain local variables that hold previous values of the expensive computation f(i) and perhaps some auxiliary values.

• Incrementally compute a new value f(i + &delta) using:
  • the previous value f(i)
  • a difference from f(i).

Example: f(i) = i²

i	0		1		2		3		4		5
i2	0		1		4		9		16		25
first difference:		1		3		5		7		9
second difference:			2		2		2		2

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