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Homework Help: Numerical Analysis, sensitivity root finding.

  1. Oct 10, 2012 #1
    1. The problem statement, all variables and given/known data

    [itex]Let f(x) = x^{n} - ax^{n-1},\: and \: set\: g(x) = x^{n} \\
    (a) \: Use\, the\, Sensitivity\, Formula\, to\, give\, a\, prediction\, for\, the\, nonzero\, \\
    root \, of\,\; f_{\epsilon }(x) = x^{n} -ax^{n-1} + \epsilon x^{n} \, for\,small\,\epsilon.\\
    \\ (b) \: Find \, the \, nonzero \, root \, and \, compare \, with \, the \, prediction. [/itex]

    2. Relevant equations

    Sensitivity Formula for Roots:
    [itex]\Delta r \approx -\frac{\epsilon g(r)}{f'(r)}[/itex]

    [itex]f'(x) = x^{n-2} (a + xn - an)[/itex]

    3. The attempt at a solution

    Some thoughts, and tries...

    One of the roots for f(x) (without the g(x) and Epsilon) is obviously the constant A.
    [itex]a = \frac{x^{n}}{x^{n-1}}[/itex]

    So if I now plugg terms into the sensitivity root

    \Delta r \approx -\frac{\epsilon a^{n}}{x^{n-2} (a + an - an)}
    \Delta r \approx - \epsilon \frac{a^{n}}{a^{n-1}}

    So a prediction of the nonzero root would simply then be

    [itex]r + \Delta r
    nonzero root \approx a - \epsilon \frac{a^{n}}{a^{n-1}}

    Would that be correct?
  2. jcsd
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