Numerical Elimination, Newton Method and Multiple Roots

Newton’s iteration has quadratic convergence for simple roots. We present a Newton-based iteration scheme with quadratic convergence for multiple roots of systems of analytic functions. This is a report on work in progress. 1. Newton Iteration, Approximate Roots and γ-Theorems 1.1. Newton Iteration. Let f : Cn → Cn be an analytic function. Newton’s method for solving f = 0 consists in approximating f by its linearization at a given point z, whence the equation (1) f(z) + f ′(z)(y − z) = 0 from where solving for y yields the following iteration (2) zk+1 = Nf (zk) := zk − f (zk)f(zk). For this method to converge to a root ζ, it is necessary that f ′(ζ) be invertible. The exact domain from where the iteration converges to a solution can have a very complicated fractal structure. However, convergence is usually very fast provided the initial point z0 be chosen sufficiently close to ζ. This is made more precise in the following [1, Ch. 8]. Theorem 1 (Smale). Let f be analytic and γ(f, z) := sup k>1 (∥∥f ′(z)−1f (k)(z)∥∥ k! ) 1 k−1