
A Density Theorem for Purely Iterative Zero Finding Methods
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Abstract:
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.
