Study of the dynamics of third-order iterative methods on quadratic polynomials

In this paper we analyze the dynamical behaviour of the operators associated to multi-point iterative methods and frozen derivative methods, for solving nonlinear equations, applied on second degree complex polynomials. We obtain that, in both cases, the Julia set is a connected set that separates the basins of attraction of the roots of the polynomial. Moreover, the Julia set of the operator associated to multi-point iteration methods is the same as the Newton operator, although it is more complicated for the frozen derivative operator. We explain these differences by obtaining the conjugacy function of each method and by showing that the operators associated to Newton’s method and multi-point iteration methods are both conjugate to powers of z.