Hi, I am trying to program the Newton iteration for polynomials. Everything works fine but now I have no idea how to choose the starting values properly to find all zeros of a polynomial in an efficient way.
There is no way to do it. You have to pick it arbitrarily.
Moreover, Newton method is chaotic and that means that the method can "jump" over some zeros. Don't bother if you're technic isn't perfect, I know a McGill professor working on this problem and it isn't solved yet.