Tuesday, September 23, 2014

somewhat generalized recurrence

I'm not sure what class of curves this works on, I've tried it on a few exponential looking curves.
Suppose you have 2*n numbers indexed to the counting numbers:
You'd like to know a logical way it could continue...
A graph of the series with where you might guess the next point would be if it follows a smooth curve is:
I found that you can project to a solution solving a set of n-1 equations of degree n-1 like so:

The points are in order 1,17,54,166,332,680. The first equation is set up so the last point is a^2 times the  point before it plus b times the point before that and 1 times the point before that. And this same type of recurrence is applied for the second equation to 332. 
So you solve for a and b and plug the values of those coefficients in for a next term 
x= a^2*680+b*332+166
So now the plot with that point added looks like:
It seems to fit well. 
In the above I was using exponents on the coefficients, but since each variable only occurred to the same power I think it's sufficient to just eliminate the powers. And I was always just plainly adding the last term but it works out nicely that every point in the whole list counts in the calculation if that last term has a coefficient as well and it should be more flexible as far as which curves can be evaluated. 
For an even number n points in a list of L values at the counting numbers solve n/2 equations in n/2 unknowns with x being the n/2th letter of the alphabet like:
L[n] = a*L[n-1]+b*L[n-2] + ... x*L[n/2]
L[n-1] = a*L[n-2]+b*L[n-3]+...x*L[n/2 -1]
L[n/2+1] = a*L[n/2] + b*L[n/2-1] +...x*L[1]

Then use those coefficients to find the next point... 

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