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Math Help - Newton-Raphson Approximation

  1. #1
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    Newton-Raphson Approximation

    A student, asked to find a root of the equation
    f(x)\equiv x^3-14x^2+49x-8=0
    did not notice the solution x=8 but chose, instead, to use Newton's Method taking x=7.2 as the first approximation. He then calculated correctly, f(7.2)=-7.712, f'(7.2)=2.02, and deduced (again correctly) the second approximation 9.84. By means of a graph, or otherwise, explain why Newton's Method failed to give a better approximation in this case. Prove that using Newton's Method, a first approximation \alpha, for a value of \alpha in the interval 7.2<\alpha<8, would give a second approximation which is closer to the root x=8 provided that 2(8-\alpha)f'(\alpha)+f(\alpha)>0 and deduce that any value of \alpha in the above range exceeding 5+\sqrt{5.6} would in fact give improvement.
    [You may, if you wish, assume without proof that f'(x) is positive and increasing forx>7.2; also that 2(8-\alpha)f'(\alpha)+f(\alpha)=(8-\alpha)(5\alpha^2-50\alpha+97).]

    For the first part, I found
    f'(x)=3x^2-28x+49
    f''(x)=6x-28
    There are three reasons given in the examples of this book for why the method doesn't work.
    1) h is too large
    2) f'(a) is too small
    3) f''(a) is too large
    I thought it would be that f''(a) is too large, f''(7.2)=15.2, but in the next portion, f''(7.3), say, is 15.8, so that can't be the case.
    f'(a)= 2.92, which isn't too small, the other explanation then is that h=0.8 is too large, the problem is I'm given in the book that 0<h<1.
    This is where I'm stuck
    Thanks!
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  2. #2
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    Look at the graph of y=f(x) and you should be able to see immediately what the problem is.
    Notice that y=f(x) has a minimum at x=7 and that 7.2 is just to the right of this.
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  3. #3
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    so what exactly is the reason, the one given sounds a bit vague to me. is it because its only a little to the right, hence its gradient is too small?
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  4. #4
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    Yes, that's pretty much it. The graph provides you with an explanation of why you might be kicked further away from the root than you started. In this case it isn't a problem, the iteration will converge quite nicely from the RHS, (starting from 9.84).
    If you are looking for something analytical, check out the value of the expression
    2(8-\alpha)f^{\prime}(\alpha)+f(\alpha)
    that you see later in the question.
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