I am using Newton's Method to solve approximate complex zeros of polynomials.
$$x_{n+1}=x_{n}-\frac{f(x_{0})}{f'(x_{0})}$$
Take the case of a cubic, with three solutions.
My issue is, that three different, random initial points, $(x_{0},y_{0})$ , $(x_{1},y_{1})$, and $(x_{2},y_{2})$, may all converge to one single point, or two different points, but not find all three complex zeros.
Are there ways to optimize the initial points chosen such that it is guaranteed they will converge to the three different zeros of the function.
Failing that, are there other methods that can be used to approximate zeros of polynomials.