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Math Help - Newton method help

  1. #1
    MHF Contributor arbolis's Avatar
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    Newton method help

    I have an exam on Tuesday 27th. I need to know well the bisection method, secant method, newton method and a lot more.
    I have a sheet of around 25 exercises. I could do some pretty easy, but others are too hard for me.
    I have a lot of questions in relation with these methods. (That I will post in other threads if I feel the need to).
    Here comes an exercise I couldn't do.
    1)Demonstrate that if f is a function that has a double root in a, then the Newton method applied to f has an order of convergence of 1.
    Demonstrate also that the modified Newton has a convergence order of 2.
    2)Study what happens if p>2.
    As p is not defined, I guess it's the number of multiple roots.
    End of the problem.
    My approach was to write f as f(x)=(x-a)^2. f'(x)=2x-2a<br />
and f^{(2)}(x)=2. Therefore the Newton method will converges regardless of the starting point.
    From it, I supposed x_0=0. I calculated x_1, which is worth  \frac{a}{2}. Also x_2=\frac{3a}{4} and lastly x_3=\frac{7a}{8}.
    So our intuition tells us that the method converges to a. But how can I determine the order? And how can I prove that it effectively converges to a?
    I'd be glad if you could start me up for the common Newton method, this way I hope to finish the rest alone.
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  2. #2
    MHF Contributor arbolis's Avatar
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    I calculated x_4 and guessed well x_5. In fact I noticed the coefficient in front of a can be written as a sequence that tends to 1 when n tends to positive infinite.
    I found that x_{n+1}=2x_n+\frac{1}{2^n}. I set x_0=0. I'm completely despaired. I need help on this problem. How can I prove that the sequence x_n tends to 1? The first terms are 0, \frac{1}{2}, \frac{3}{4}, \frac{7}{8}, etc. I know it's a famous sequence, where each step is half closer to 1 than the precedent.
    But I need to be rigorous, I can't say " I'm seeing that the coefficients of the method are like this sequence". I must prove it and I'm not able.
    Is there any other way I'm not seeing to do the first question? I'm lost.
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