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Math Help - Convergent iteration

  1. #1
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    Convergent iteration

    Can someone help me with this problem?
    Consider the fixed point iteration
    x_{k+1}=(m+1)x_k-{x_k}^2, k=0,1,2,...where 1\leq m \leq 2
    a) Show that the iteration converges for any initial guess x_0 satisfying m-1/5 \leq x_0 \leq m+1/5
    b) Assume that the iteration converges, find m such that the method converges quadratically.

    I think I got part a) by letting g(x)=(m+1)x-x^2, then take the derivative g'(x)=2x+m+1. I then show that there exists 0<k<1 such that \mid g'(x) \mid <k, and by a theorem this implies the iteration converges for any initial guess in the specified interval.
    For part b), I need to find m such that lim_{k\to \infty} \frac{|x_{k+1}-x|}{|x_k-x|^2}= lim_{k\to \infty} \frac{|(m+1)x_k-{x_k}^2-x|}{|x_k-x|^2} exists, but I don't get anywhere from here. Hope someone can give a hand.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by namelessguy View Post
    Can someone help me with this problem?
    Consider the fixed point iteration
    x_{k+1}=(m+1)x_k-{x_k}^2, k=0,1,2,...where 1\leq m \leq 2
    a) Show that the iteration converges for any initial guess x_0 satisfying m-1/5 \leq x_0 \leq m+1/5
    b) Assume that the iteration converges, find m such that the method converges quadratically.

    I think I got part a) by letting g(x)=(m+1)x-x^2, then take the derivative g'(x)=2x+m+1. I then show that there exists 0<k<1 such that \mid g'(x) \mid <k, and by a theorem this implies the iteration converges for any initial guess in the specified interval.
    For part b), I need to find m such that lim_{k\to \infty} \frac{|x_{k+1}-x|}{|x_k-x|^2}= lim_{k\to \infty} \frac{|(m+1)x_k-{x_k}^2-x|}{|x_k-x|^2} exists, but I don't get anywhere from here. Hope someone can give a hand.
    Assuming that the itteration converges it is obvious it must converge to either x=0 or x=m, but it is easy to show that itteration will not converge to x=0.

    So suppose x_n=m+\varepsilon

    Then:

    x_{n+1}=(m+1)(m+\varepsilon)-(m_\varepsilon)^2=m+\varepsilon(1-m)-\varepsilon^2

    So if m=1 the itteration converges quadraticaly to x=m=1

    CB
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