Will the Newton's method be convergent to the root of the following function: $f(x)=\frac{-x}{x^2-1}$?

Question

Will the Newton's method be convergent to the root of the following function, if the starting point $x_0>1$ will be chosen?

$$ f(x)=\frac{-x}{x^2-1} $$

Answer 1

The Newton's method iteration function is

$$N(x)=x-f(x)/f'(x) = \frac{2 x^3}{x^2+1}.$$

Why don't you plug in $x=2$ and see what comes out? You could then plug that result back in and try a few more iterates to see what's going on. If you understand the follow the slope description of Newton's method, then the results make perfect sense.

Answer 2

I would say he shape of the graph of function $\frac{-x}{x^2-1}$ the Newton's method converges only for |x|<1, see http://www.wolframalpha.com/input/?i=%5Cfrac%7B-x%7D%7Bx%5E2-1%7D