0

I've just started calculus, so the phi function and such are perhaps a year away or such. I realize there are similar questions and perhaps duplicates. I'm don't know enough to do that. But here is my question.

If only allowed pre- or very early calculus like limits how can one show which of the following two functions converges faster

$f\left(x\right)=x^{3}-x-1$

$g\left(x\right)=x^{2}$

The newton method, which I understand:

$x_{1}= x_{0}-\frac{f\left(x_{0}\right)}{f'\left(x_{0}\right)}$

$x_0$ is not provided, if I'm to do the iteration, I'll guess a point close to the root of the function in question.

This is a old exam question, it does not have and solution, but is in the suggested questions to be able to solve. There will probably be a variation of this question on the my exam.

My take as of yet is that $g$ has a two solutions $x_0=x_1=0$ and converges to this one and only point where $g$ intersects the x-axis. I don't know what to make of this information, but $g'$ will plain out as $x\rightarrow0$, since the metod will yield this result.

The function $f$ intersects somewhere between $(1,2)$ and it's $f'$ not leveling out.

So I wonder, can I from this somehow conclude that $f$ conversion rate is faster?

Simon
  • 175
  • I think it's the newton method. I use the formula supplied to iteratively find the roots. – Simon Oct 08 '16 at 21:54
  • No such point is provided. I'll have to "guess" a point. Updated my question. Thanks. – Simon Oct 08 '16 at 21:59
  • See page 5: http://homepage.divms.uiowa.edu/~atkinson/ftp/ENA_Materials/Overheads/sec_3-5.pdf. When we have a multiple root, Newton's Method converges much more slowly (we have a double zero root for $g(x)$). If I use $x_0 = 3$ on both cases, the first converges in seven steps and the second in twenty-nine steps. The first only has one real root. So, you can show the error from steps or show the number of iterations or use the theory. For multiple roots, we lose the awesome quadratic convergence of Newton's: http://math.stackexchange.com/questions/389368/convergence-rate-of-newtons-method – Moo Oct 08 '16 at 22:05
  • Well thank you! This was exactly the kind of think I was looking for. How did you find this? – Simon Oct 08 '16 at 22:08
  • I see. Practice makes perfect! – Simon Oct 08 '16 at 22:12

0 Answers0