Behaviour of fixed and critical points of the $$\left( \alpha ,c\right) $$ α , c -family of iterative methods

Abstract

In this paper we study the dynamical behavior of the \((\alpha ,c)\)-family of iterative methods for solving nonlinear equations, when we apply the fixed point operator associated to this family on quadratic polynomials. This is a family of third-order iterative root-finding methods depending on two parameters; so, as we show throughout this paper, its dynamics is really interesting, but complicated. In fact, we have found in the real \((\alpha ,c)\)-plane a line in which the corresponding elements of the family have a lower number of free critical points. As this number is directly related with the quantity of basins of attraction, it is probable to find more stable behavior between the elements of the family in this region.