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2016 Update

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Tina Müller 2016-12-05 22:15:40 +01:00
parent 948b86eafa
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38
Task/Forward-difference/00DESCRIPTION
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@ -1,6 +1,14 @@
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 tdefmodule Diff do
;Task:
Provide code that produces a list of numbers which is the &nbsp; <big>n<sup>th</sup></big> &nbsp;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 &nbsp; <big>'''A'''</big> &nbsp; is a new list &nbsp; <big>'''B'''</big>, &nbsp; where &nbsp; <big><b>B</b><sub>n</sub> = <b>A</b><sub>n+1</sub> - <b>A</b><sub>n</sub></big>.
List &nbsp; <big>'''B'''</big> &nbsp; should have one fewer element as a result.
The second-order forward difference of &nbsp; <big>'''A'''</big> &nbsp; will be:
<pre>
tdefmodule Diff do
def forward(arr,i\\1) do
forward(arr,[],i)
end
@ -16,14 +24,22 @@ The second-order forward difference of A will be tdefmodule Diff do
def forward([val1|[val2|vals]],diffs,i) do
forward([val2|vals],diffs++[val2-val1],i)
end
endhe same as the first-order forward difference of B.
That new list will have two fewer elements than A and one less than B.
end
</pre>
The same as the first-order forward difference of &nbsp; <big>'''B'''</big>.
That new list will have two fewer elements than &nbsp; <big>'''A'''</big> &nbsp; and one less than &nbsp; <big>'''B'''</big>.
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].
For a more formal description, see the related &nbsp; [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)
;Algorithmic options:
* Iterate through all previous forward differences and re-calculate a new array each time.
* Use this formula (from [[wp:Forward difference|Wikipedia]]):
<big><big>
::: <math>\Delta^n [f](x)= \sum_{k=0}^n {n \choose k} (-1)^{n-k} f(x+k)</math>
</big></big>
::: ([[Pascal's Triangle]] &nbsp; may be useful for this option.)
<br><br>