Measures of the Basins of Attracting n-Cycles for the Relaxed Newton's Method

Abstract

The relaxed Newton's method modifies the classical Newton's method with a parameter h in such a way that when it is applied to a polynomial with multiple roots and we take as parameter one of these multiplicities, it is increased the order of convergence to the related multiple root.

For polynomials of degree three or higher, the relaxed Newton's method may possess extraneous attracting (even super-attracting) cycles. The existence of such cycles is an obstacle for using the relaxed Newton's method to find the roots of the polynomial. Actually, the basins of these attracting cycles are open subsets of C.

The authors have developed some algorithms and implementations that allow to compute the measure (area or probability) of the basin of a p-cycle when it is taken in the Riemann sphere. In this work, given a non negative integer n, we use our implementations to study the basins of non-repelling p-cycles, for 1 <= p <= n, when we perturb the relaxing parameter h. As a consequence, we quantify the efficiency of the relaxed Newton's method by computing, up to a given precision, the measure of the different attracting basins of non-repelling cycles. In this way, we can compare the measure of the basins of the ordinary fixed points (corresponding to the polynomial roots) with the measure of the basins of the point at infinity and the basins of other non-repelling p-cyclic points for p > 1.