Convergence time of a recursive function

[wolfpax50]

I have a recursive function that will eventually converge to either a fixed value or a limit cycle. Depending on the inputs, it will converge to different values (or cycles) at different rates. How could I go about measuring the rate of convergence for different inputs, regardless of what type of limit it ends up at?

To be specific, the relevant equation is:

p^t+1 = f(a₀,...,a_2r+1;p^t)
f(x) = $\sum$a_i(p^t)ⁱ(1-p^t)^2r+1-i

The solution must be numerical as it will be part of a computer program. The program will be used to search for inputs that produce long convergence times.

 

[chiro]

Hey wolfpax50.

Is the summand i over a finite number of integers or an infinite number?

If its infinite you will want to check whether |ai| and the other term is a summable series. If they are both summable in some norm (typically 2-norm or |.|^2) then your series will converge.

The result is due to the relationship between |a||b| and |a|^2 and |b|^2.

Have you got a book or resource on these sorts of things?

 

[wolfpax50]

The summation in the second function is finite, from i=0 to i=2r+1. Sorry if that was unclear.

I don't have a book in front of me but I'm educated through college calculus. If you could point me in the right direction the internet is usually a good resource.

 

[chiro]

If you are summing a finite number of finite terms, then the answer will always be finite no matter what the terms are.

 

[wolfpax50]

No, it's an infinite recursion of a finite summation. There is a function f which is a finite summation, and a function p which is the recursive iteration of function f.