Polynomial Root-Finding Methods Whose Basins of Attraction Approximate Voronoi Diagram

Given a complex polynomial p(z) with at least three distinct roots, we first prove that no rational iteration function exists where the basin of attraction of a root coincides with its Voronoi cell. In spite of this negative result, we prove that the Voronoi diagram of the roots can be well approximated through a high order sequence of iteration functions, the Basic Family, B (m) (z), ma parts per thousand yen2. Let theta be a simple root of p(z), V(theta) its Voronoi cell, and A (m) (theta) its basin of attraction with respect to B (m) (z). We prove that given any closed subset C of V(theta), including any homothetic copy of V(theta), there exists m (0) such that for all ma parts per thousand yenm (0), C is also a subset of A (m) (theta). This implies that when all roots of p(z) are simple, the basins of attraction of B (m) (z) uniformly approximate the Voronoi diagram of the roots to within any prescribed tolerance. Equivalently, the Julia set of B (m) (z), and hence the chaotic behavior of its iterations, will uniformly lie to within prescribed strip neighborhood of the boundary of the Voronoi diagram. In a sense, this is the strongest property a rational iteration function can exhibit for polynomials. Next, we use the results to define and prove an infinite layering within each Voronoi cell of a given set of points, whether known implicitly as roots of a polynomial equation, or explicitly via their coordinates. We discuss potential application of our layering in computational geometry.