In this manuscript, by using the weight-function technique, a new class of iterative methods for solving nonlinear problems is constructed, which includes many known schemes that can be obtained by choosing different weight functions. This weight function, depending on two different evaluations of the derivative, is the unique difference between the two steps of each method, which is unusual. As it is proven that all the members of the class are optimal methods in the sense of Kung-Traub's conjecture, the dynamical analysis is a good tool to determine the best elements of the family in terms of stability. Therefore, the dynamical behavior of this class on quadratic polynomials is studied in this work. We analyze the stability of the presented family from the multipliers of the fixed points and critical points, along with their associated parameter planes. In addition, this study enables us to select the members of the class with good stability properties.
In this manuscript, we analyze the dynamical anomalies of a parametric family of iterative schemes designed by Kou et al. It is known that its order of convergence is three for any arbitrary value of the parameter, but it has order four (and it is optimal in the sense of Kung-Traub's conjecture) when an specific value is selected. Among all the elements of this family, one can choose this fourth-order element or any of the infinite members of third order of convergence, if only the speed of convergence is considered. However, the stability of the methods plays an important role in their reliability when they are applied on different problems. This is the reason why we analyze in this paper the dynamical behavior on quadratic polynomials of the mentioned family. The study of fixed points and their stability, joint with the critical points and their associated parameter planes, show the richness of the class and allows us to find members of it with excellent numerical properties, as well as other ones with very unstable behavior. Some test functions are analyzed for confirming the theoretical results.
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