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Convergence time of a recursive function

  1. Aug 14, 2013 #1
    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:

    pt+1 = f(a0,...,a2r+1;pt)
    f(x) = [itex]\sum[/itex]ai(pt)i(1-pt)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.
    Last edited: Aug 14, 2013
  2. jcsd
  3. Aug 14, 2013 #2


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    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?
  4. Aug 14, 2013 #3
    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.
    Last edited: Aug 14, 2013
  5. Aug 14, 2013 #4


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    If you are summing a finite number of finite terms, then the answer will always be finite no matter what the terms are.
  6. Aug 14, 2013 #5
    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.
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