I'm trying to find the condition number on the function $A$ for the iterative method below however I'm struggling to begin.
$$p_{n+1}=p_n-A(p_n)\frac{f(p_n)}{f'(p_n)}=g(p_n)$$
In particular, the problem is: Given $f(x)$ has a simple root at $x=p$ and $f$ is sufficiently differentiable, derive a condition on function $A$ which will ensure convergence from a sufficiently accurate $p_0$.
I know that the relative condition number is defined as
$$K(d) = \frac {\|\delta x\| /\|x\|}{\|\delta d\|/\|d\|}=\|G'(d)\| \frac{\|d\|}{\|G(d)\|}$$
or the ratio of the perturbations in $x$ relative to the perturbations in data $d$. Also, $x:=G(d)$. But how do I begin to apply this definition the the iterative method above. Is it simply $p_{n+1}=G(A)$?
Some indication on how to begin would be appreciated!
That question is not about the condition number, but about a condition on the function $A(p)$. You need to apply the fixed point theorem to the mapping $p \mapsto g(p) = p - A(p) \frac{f(p)}{f'(p)}$. For sufficiently accurate initial guess a sufficient condition would be $|g'(p)| \leq q < 1$.