Finiteness of the Area of Basins of Attraction of Relaxed Newton Method for Certain Holomorphic Functions

Abstract

For a nonconstant function F and a real number h is an element of] 0, 1] the relaxed Newton's method N-F,(h) of F is an iterative algorithm for finding the zeroes of F. We show that when relaxed Newton's method is applied to complex function F(z) = P(z)e(Q(z)), where P and Q are polynomials, the basin of attraction of a root of F has finite area if the degree of Q exceeds or equals 3. The key point is that N-F,(h) is a rational map with a parabolic fixed point at infinity.