Solving for a factor in a large sum

[IsomaDan]

Dear everyone.

First of all Merry Xmas, when everybody gets to that.

I have a problem solving for a factor within a sum.

My formula looks as follows:

T = Æ© I_t * A₀^t

The sum runs from t=1 to N, and the aim is to solve for A₀, but all my calculations end up extremely messy.

All the best,

Dan

 

[Kosomoko]

What is I_t?

 

[IsomaDan]

Thanks for the reponse.

That is the t'th observation of I. They have no well-defined relation to t. In other words; just a bunch of numbers.

 

[Kosomoko]

In that case, your equation seems to be a general polynomial of order N. I guess N is probably fairly large (as in... not 2 or 3). You will need to use some numerical method to find roots of the function f(A₀) = Æ© I_t * A₀^t - T.

I suggest Newton's method, it should be convenient to implement, because you can easily compute the derivative of the function analytically.

 

[IsomaDan]

Actually it is not that large (N, that is), it is just that it varies a lot from case to case and hence I have written it as a sum.

Thanks so much for the response. I will try to see if it gets me any further!

 

[IsomaDan]

And just to be clear. The A_0 has t as their exponent. It is not a subscript!

 

[scurty]

If N is equal to 3 or 4, the polynomial is still solvable by analytic methods, but in general it is easier to use a numerical technique like Newton's Method (which is very easy to program if you choose to do that).

The N = 3 case isn't this bad, but consider the solutions for N = 4:

http://upload.wikimedia.org/wikipedia/commons/9/99/Quartic_Formula.svg

Of course, if you know $a$ is a solution to the polynomial, you can long divide by $(x-a)$ to reduce the degree by 1.