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Ingy döt Net 2013-04-11 12:07:39 -07:00
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Provide code that produces a list of numbers which is the n-th order forward difference, given a non-negative integer (specifying the order) and a list of numbers. The first-order forward difference of a list of numbers (A) is a new list (B) where B<sub>n</sub> = A<sub>n+1</sub> - A<sub>n</sub>. List B should have one fewer element as a result. The second-order forward difference of A will be the same as the first-order forward difference of B. That new list will have two fewer elements than A and one less than B. The goal of this task is to repeat this process up to the desired order.
For a more formal description, see the related [http://mathworld.wolfram.com/ForwardDifference.html Mathworld article].
Algorithmic options:
*Iterate through all previous forward differences and re-calculate a new array each time.
*Use this formula (from [[wp:Forward difference|Wikipedia]]):
:<math>\Delta^n [f](x)= \sum_{k=0}^n {n \choose k} (-1)^{n-k} f(x+k)</math>
:([[Pascal's Triangle]] may be useful for this option)