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Let $f(x)=\frac{1}{5}x^5+\frac{1}{3}x^3+x-1$

  • Show that $f$ has only one zero $r$ in interval $(0,1)$
  • To find approximation of $r$ we apply Newton's method $x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$. Show that for every $x_0\in(0,1)$ this method converges.
  • Show that order of convergence is exactly quadratic

First one is trivial, but I don't know how to solve second and third. For second I was trying to show that error $|x_{n+1}-r|=e_{n+1}=e_n-\frac{f(x_n)}{f'(x_n)}$ is approaching $0$ when $n$ approaches $+\infty$ but it didn't lead me to anywhere.

Also showing the third seems very complicated. Can anybody help?

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  • $\begingroup$ Try computing a few iterates, or better still, draw a picture and see if you can conclude something about the iterates. $\endgroup$ – copper.hat Nov 6 '13 at 17:23
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For the second part, note that $f'(x) > 0 \; \forall x$. From part one, we know that $f$ has only one root. If you start with $x_0$ to the left of $r$, then the quantity $-f(x)/f'(x)$ is positive and approaches $0$ as $n->\infty$. If you start with $x_0$ to the right of $r$, then $-f(x)/f'(x)$ is negative and approaches $0$ as $n->\infty$. Does this help?

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