A Density Theorem for Purely Iterative Zero Finding Methods
• Postscript version.
• Dvi version.
In this paper we prove that a wide class of purely iterative root finding methods work for all complex valued polynomials with a positive probability depending only on the method and the degree of the polynomial. More precisely, if we consider the set of polynomials with roots in the unit ball, 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 $f$ and its derivatives for which (1) $\infty$ is repelling fixed point for all $f$ of degree $>1$ and (2) $T_f(z)$ depends only on $z$ and $f$'s roots and commutes with linear maps on the complex plane.