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Iteration/Root finding algorithm

  1. Aug 25, 2011 #1
    1. The problem statement, all variables and given/known data

    ea1b2s.jpg

    3. The attempt at a solution

    I've managed to do part a), by factorising the cubic you get when you rearrange the terms. I'm mostly stumped for part b). I know the sequence has to be contractive, otherwise it wouldn't converge. Also, how do I show it's the only solution? Thank you very much!
     
  2. jcsd
  3. Aug 25, 2011 #2

    HallsofIvy

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    I'm not sure what you mean by "show it's the only solution". There are, in fact, three (complex) solutions to the equation. What you are asked to do is show that the limit (which, assuming the sequence converges, is unique) does, in fact, satisfy the given equation, not that it is the "only" solution.

    Your recursion equation is [itex]x_{n+1}= 2x_n/3+ \lambda/(3x_n^2)[/itex]. Take the limit of both sides as n goes to infinity and you have [itex]x= 2x/3+ \lambda/3x^2[/itex], where [itex]x= \lim_{n\to\infty} x_n[/itex]. Multiply both sides of that by [itex]3x^2[/itex].
     
  4. Aug 25, 2011 #3
    Sorry, I meant how to show that the only possible value for limn→∞xn is cbrt(λ)! Also, if you could explain how to solve ii) of b) that would be great!
     
  5. Aug 27, 2011 #4
    Bump: Any ideas about b) are welcome.
     
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