I have just started numerical analysis so this question probably seems trivial.
Say I have a function $f(x) = x^2 - x - 3$
I let $g(x) = x^2 - 3$
Then I want to find the roots of $f(x)$ so I have $f(x) = g(x) - x = 0$
So I can find the roots by finding the fixed points of $x = g(x)$
So far so good yes?
Now I use the fixed point iteration formula - $x_{n+1} = g(x_{n}) = x_{n}^2 - 3$
Say I pick 3 for $x_{1}$, then I get $x_{2} = 6$ and $x_{3} = 33$...so it is diverging
I then tried picking 1 for $x_{1}$, then I get $x_{2} = -2$ and $x_{3} = 1$...so it is going to infinitely alternate...
So the fixed point iteration method isn't working. Why is it not working in this situation and what are the conditions it needs to work?