On the election of the damped parameter of a two-step relaxed Newton-type method

In this paper, we are interested to justified two typical hypotheses that appear in the convergence analysis, | λ | ≤ 2 and z 0 sufficient close to z ∗ . In order to proof these ideas, the dynamics of a damped two-step Newton-type method for solving nonlinear equations and systems is presented. We present the parameter space for values of the damping factor in the complex plane, focusing our attention in such values for which the fixed points related to the roots are attracting. Moreover, we study the stability of the strange fixed points, showing that there exists attracting cycles and chaotical behavior for some choices of the damping factor.