Julia sets for the standard Newton’s method, Halley’s method, and Schröder’s method

摘要

We analyze the Julia sets theory for the standard Newton’s method, Halley’s method, and Schröder’s method, study structural characteristics of Julia sets, show the properties and conditions of fixed points for the standard Newton’s method, Halley’s method, and Schröder’s method. We observe the relation between roots of polynomial and Julia sets structure, namely if keeping the relative position of the roots of a polynomial invariable, then the topological structure is also invariable. If there is an extraneous fixed point, then the extraneous fixed point is also invariable. Otherwise, the structures of Julia sets and fixed points would be changed.