I want to numerically find all the roots of a system of polynomials (n equations in n variables). Since I can compute the Jacobian for the system (analytically or otherwise), I can use the Newton Raphson method to find a single root (e.g. as described in the Numerical Recipes book).
How do I find the other roots (if they exist), either using a different algorithm or extending the Newton Raphson method? If this is not always possible, what about finding all the roots in a bounded interval [a,b]? Other roots may exist, but I only need to find the ones that lie in the interval - however, it is possible for multiple roots to lie in the interval?