A Density Theorem for Purely Iterative Zero Finding Methods

In this paper a wide class of purely iterative root finding methods is proved to work for all complex valued polynomials with a positive probability depending only, on the method and the degree of the polynomial. More precisely, if the set of polynomials with roots in the knit ball is considered, then for fixed degree the area of convergent points in the ball of radius 2 is bounded below by some constant for any purely iterative method $z_{i + 1} \leftarrow T_f (z_i )$ where $T_f (z)$ is a rational function of z and $f_{i}$ and its derivatives, for which (1) $\infty $ is repelling fixed point for all f of degree greater than 1 and (2) $T_f (z)$ depends only on z and f’s roots and commutes with linear maps on the complex plane.