Search

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]