I am implementing a function that approximates a root of a complex function, say $f(z)$. As we know, at iteration $i$ we ave
$$z_i = z_{i-1} - \frac{f(z_{i-1})}{f'(z_{i-1})}$$
The division of complex implies that we should have a division by the norm (modulus) of $f'(z_{i-1})$ but sometimes this norm gets little enough so it is so close to zero that it passes the double precision and it is zero for the computer. I won't like to deal with numbers with more precision than that. I was wondering if there is any motive to a point that gives a derivtive with norm 0 to not convert to a root. I mean, is there any geometric motive or something like that? What is the interpretation of the modulus of the derivative of a complex function be 0?
I am sorry for the question formulated like that, but I am not used to work with complex number, further derivatives of complex.
Thanks for your help!