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Math Help - Deriving Newton's root-finding method with Taylor polynomials.

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    Deriving Newton's root-finding method with Taylor polynomials.

    Hi,

    when deriving Newton's method using Taylor polynomials my book considers the first Taylor polynomial expanded about \overline{x} where \overline{x} is an approximation to the root p such that f'(\overline{x})\neq 0 and |p-\overline{x}| is small.

    f(x) = f(\overline{x}) + (x-\overline{x})f'(\overline{x}) + \frac{(x-\overline{x})^2}{2}f''(\xi(x)),

    where \xi(x) lies between x and \overline{x}.

    To obtain Newton's method from this, I need to assume |p-\overline{x}| small such that (p-\overline{x})^2 is much smaller so that I can ignore it. Does this mean that my initial guess \overline{x} has to be so close to the root p that 0<|p-\overline{x}|<1 ?

    Thanks!
    Last edited by Mollier; October 4th 2010 at 12:39 AM.
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