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Thread: function that corresponds to Newton's method

  1. #1
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    Question function that corresponds to Newton's method

    Question is if $\displaystyle f(x) = x^4 + x - 19$ then which of the following functions correspond to Newton's method $\displaystyle x_{n + 1} = g(x_n)$

    Answer is given as $\displaystyle
    g(x) = \frac{{3x^4 + 19}}{{4x^3 + 1}}
    $

    But I would have thought it would be
    $\displaystyle
    g(x) = \frac{{ - x^4 + 4x^3 - x + 20}}{{4x^3 + 1}}
    $

    considering

    $\displaystyle
    \begin{array}{l}
    x_{n + 1} = x_n - \frac{{f(x)}}{{f'(x)}} \\
    x_{n + 1} = x_n - \frac{{x^4 + x - 19}}{{4x^3 + 1}} \\
    x_{n + 1} = \frac{{4x^3 + 1}}{{4x^3 + 1}} - \frac{{x^4 + x - 19}}{{4x^3 + 1}} \\
    x_{n + 1} = \frac{{ - x^4 + 4x^3 - x + 20}}{{4x^3 + 1}} \\
    \end{array}
    $

    Could someone clarify what I am doing wrong or not interpreting correctly. Thanks
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  2. #2
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    earboth's Avatar
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    Quote Originally Posted by Craka View Post
    Question is if $\displaystyle f(x) = x^4 + x - 19$ then which of the following functions correspond to Newton's method $\displaystyle x_{n + 1} = g(x_n)$

    Answer is given as $\displaystyle
    g(x) = \frac{{3x^4 + 19}}{{4x^3 + 1}}
    $

    But I would have thought it would be
    $\displaystyle
    g(x) = \frac{{ - x^4 + 4x^3 - x + 20}}{{4x^3 + 1}}
    $

    considering

    $\displaystyle
    \begin{array}{l}
    x_{n + 1} = x_n - \frac{{f(x)}}{{f'(x)}} \\
    x_{n + 1} = x_n - \frac{{x^4 + x - 19}}{{4x^3 + 1}} \\
    x_{n + 1} = \frac{{4x^3 + 1}}{{4x^3 + 1}} - \frac{{x^4 + x - 19}}{{4x^3 + 1}} \\
    x_{n + 1} = \frac{{ - x^4 + 4x^3 - x + 20}}{{4x^3 + 1}} \\
    \end{array}
    $

    Could someone clarify what I am doing wrong or not interpreting correctly. Thanks
    Your mistake is hidden in the third row of your transformations:

    According to Newtons formula you are looking for:

    $\displaystyle g(x)=x-\dfrac{x^4+x-19}{4x^3+1}=\dfrac{x(4x^3+1)}{4x^3+1}-\dfrac{x^4+x-19}{4x^3+1}=\dfrac{4x^4+x-(x^4+x-19)}{4x^3+1} =$$\displaystyle \dfrac{4x^4+x-x^4-x+19}{4x^3+1}$

    Collect like terms and you'll get the given answer.
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  3. #3
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    Thanks, just a little silly algebraic mistake on my behalf
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