# Numerical Analysis, sensitivity root finding.

1. Oct 10, 2012

### dreamspace

1. The problem statement, all variables and given/known data

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

2. Relevant equations

Sensitivity Formula for Roots:
$\Delta r \approx -\frac{\epsilon g(r)}{f'(r)}$

$f'(x) = x^{n-2} (a + xn - an)$

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.
$a = \frac{x^{n}}{x^{n-1}}$

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

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

Would that be correct?