A new class of third-order iterative methods for multiple roots and their geometric construction

Abstract

This work aims to provide a class of third-order iterative methods for solving univariate nonlinear equations with multiple roots when the multiplicity is unknown. To obtain the class of methods mentioned above, a univariate nonlinear equation is used that has the same roots as the original equation. However, the roots of this equivalent equation are simple; that is, they have multiplicity one. Therefore, Gander’s theorem can be used to construct a new class of methods. Then the geometric construction of the elements of said class is done, which satisfies the osculance condition. In addition, some families of methods belonging to said class are presented, as well as their geometric construction. Finally, numerical examples are presented where the behavior of some methods belonging to the families within the method class is observed.