TITLE: Finding the all roots of a polynomial by using Newton-Raphson method.
QUESTION [15 upvotes]: Is there a general formulation for finding all roots of a polynomial, especially the complex ones, by using the Newton-Raphson Method?

REPLY [5 votes]: As has been pointed out, the Newton-Raphson, as well as most other such methods, converge to a single isolated root. There are, however, methods to find all roots of a polynomial.
One I know of, is creating the companion matrix, and then finding (numerically) that matrix's eigenvalues, which are precisely the roots of your polynomial. 
If you use Matlab, this is precisely what the 'roots' routine does. From personal experience with that routine, it runs very fast for polynomial of both low and high degrees.