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Topic: Help with formula (revised question)
Replies: 7   Last Post: Nov 20, 2013 8:15 PM

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Rock Brentwood

Posts: 121
Registered: 6/18/10
Re: Help with formula (revised question)
Posted: Nov 20, 2013 8:15 PM
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On Wednesday, November 20, 2013 1:50:59 AM UTC-6, Robert Crandal wrote:
> However, I'd now
> like to revise the data and requirements.
> Here is the new data set:

[etc.]

> Can this data be represented with a formula
> that only uses either addition, subtraction,
> multiplication, division, modulus, or the power (^)
> function, or any combination of these?


All finite data sets can be represented by polynomials. So that means: only constants, addition, subtraction and multiplication.

If the points are A, B, C, ..., Z, then the unique polynomial of minimal degree that is equal to 1 at A and 0 everywhere else is
p_A(x) = ((x-B)/(A-B)) ((x-C)/(A-C)) ... ((x-Z)/(A-Z)).
Similar considerations apply to the other points, yielding definitions for p_B(x), p_C(x), ..., p_Z(x).

The unique polynomial of minimal degree that matches the value a at A, b at B, c at C, ..., z at Z is then p(x) = a p_A(x) + b p_B(x) + c p_C(x) + ... + z p_Z(x).

All other polynomials matching these points have p(x) as a factor.

Any other function that matches the points must necessarily involve one of the other operations: division, modulus, exponentiation, or something else besides those.



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