Tutorial 6, q6

[walke362]

I’m a bit confused as to where the g(x)=x-f(x)/f’(x) came from? I assumed the following lines were setting up a contraction for using theorem 4b, but I’m not sure since g resembles the formula for x_{k+1}. Do we have to show g(x)=x iff f(x)=0?

This also relates to another question, which in the slides, it says Newton’s method always converges if f is twice differentiable and x_0 is close to the root. What might be considered close…?

[ccc]

The g given is Newton’s iteration function; see lecture notes on Newton’s method.
To get guaranteed convergence, we use one of the theorems, in this case Thm 4b,
and we need to do the usual study (contraction, self-mapping) for Newton’s g.
The fact that Newton’s always converges if f twice diff and x0 close to the root, while a fact, it does not give you a specific interval for the convergence. It just tells you there is an interval.
You still have to use Thm 4b in some interval where conditions are satisfied, so you still need to do the usual study (contraction, self-mapping).