Accurate computation of component centers in the degree- n bifurcation set

Abstract

A highly accurate computation of component centers in the degree- n bifurcation set is presented via Newton's method applied to the transformed polynomial equation that governs component centers. The transformed polynomial as well as some properties of the degree- n bifurcation set are investigated and efficiently used in Newton's method. Since the initial values are taken from the approximated roots found by fsolve routine of Maple V, the Newton's method shows a great accuracy with a confirmed convergence of order two and with at least 48 significant digits after the decimal point in both real and imaginary parts of the computed component centers. Although many cases are studied for 2 ⪯ n ⪯ 25 and 1 ⪯ k ⪯ 10, the limited space allows us to list only typical cases for n = 3, 4,12 and 25 with 2 ⪯ k ⪯ 5 reflecting highly numerical accuracy. Our study extends the results given by Peitgen and Richter [1].