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It is known that the immediate basins of attraction of zeros of a complex polynomial f(z) with simple zeros under the iteration of family of iterations, i.e., Schröder's formula of the second kind (König's formula or the Basic Family) with the convergence degree m ≥ 2, converge to the Voronoi diagram of the zeros as m → ∞ . In a previous work, we consider a region, i.e., the intersection of the regions surrounded by the circles of Apollonius for zeros and show that for each zero, this region included in the immediate basin of attraction of the zero converges to the Voronoi cell of the zero as m → ∞ . Here, for a polynomial f(z) with multiple zeros, by extending our previous work, we show how the immediate basins of attraction of the family to the rational function f (z) / f ′ (z) , whose zeros are simple, converge to the Voronoi diagram. Honorato has shown a similar result, but our approach is more direct and quantitative. Numerical examples illustrate the relationship between the regions above, the immediate basins of attraction and the Voronoi cells, and show that the region above monotonically enlarges to the Voronoi cell. [ABSTRACT FROM AUTHOR]

Schröder's iterative formula of the second kind (S2 formula) for finding zeros of a function f(z) is a generalization of Newton's formula to an arbitrary order m of convergence. For iterative formulae, convergence regions of initial values to zeros in the complex plane z are essential. From numerical experiments, it is suggested that as order m of the S2 formula grows, the complicated fractal structure of the boundary of convergence regions gradually diminishes. We propose a method of estimating the convergence regions with the circles of Apollonius to verify this result for polynomials f(z) with simple zeros. We indeed show that as m grows, each region surrounded by the circles of Apollonius monotonically enlarges to the Voronoi cell of a zero of f(z). Numerical examples illustrate convergence regions for several values of m and some polynomials. [ABSTRACT FROM AUTHOR]