Estimating convergence regions of Schröder’s iteration formula: how the Julia set shrinks to the Voronoi boundary

Abstract

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.

Appendix: Proof of Lemma 4.2

From (4.4) we have

$$ t=\tau(x)=\log(x/B)\big/\log(1-x). $$

(A.1)

In view of x ∈ (0,1) and t > 0, from (A.1) we have log(x/B) < 0. It follows that

$$0<x<B_{1}=\min\{1,B\},\quad\lim_{x\to0}\tau(x)=\infty,\quad \lim_{x\to B_{1}}\tau(x)= 0. $$

Since from (A.1) we have

$$\tau^{\prime}(x)=\{x^{-1}\log(1-x)+(1-x)^{-1}\log(x/B)\}\big/ \{\log(1-x)\}^{2}<0\quad(0<x<B_{1}), $$

τ(x) monotonically decreases on (0,B₁). Therefore, t = τ(x) has the inverse function x = ψ(t) (0 < t < ∞) that monotonically decreases and $\lim _{t\to \infty }\psi (t)= 0$.

It remains to verify (4.5). Expanding log(1 − x) in (A.1) gives

$$ \tau(x)=\frac{-\log(x/B)}{{\sum}_{k = 1}^{\infty}x^{k}/k}<\frac{-\log(x/B)}{x} \quad (0<x< B_{1}). $$

(A.2)

Substituting x = (log t)/t in (A.2), we have

$$\tau\left( \frac{\log t}{t}\right)< \frac{-t\log\{(\log t)/(tB)\}}{\log t} =t-t\frac{\log(\log t)-\log B}{\log t}. $$

So, for log t ≥ B, or t ≥ e^B, we have τ((log t)/t) < t. Recalling that x = ψ(t) is a monotonically decreasing function, we obtain

$$(\log t)/t=\psi\circ\tau((\log t)/t)>\psi(t)\quad(t\ge e^{B}). $$

This establishes the lemma.