Abstract
Abstract The use of complex dynamics tools in order to deepen the knowledge of qualitative behaviour of iterative methods for solving non-linear equations is a growing area of research in the last few years with fruitful results. Most of the studies dealt with the analysis of iterative schemes for solving non-linear equations with simple roots; however, the case involving multiple roots remains almost unexplored. The main objective of this paper was to discuss the dynamical analysis of the rational map associated with an existing class of iterative procedures for multiple roots. This study was performed for cases of double and triple multiplicities, giving as a conjecture that the wideness of the convergence regions of the multiple roots increases when the multiplicity is higher and also that this family of parametric methods includes some specially fast and stable elements with global convergence.
Highlights
With the advancement of computer algebra, finding higher-order multi-point methods, not requiring the computation of second-order derivative for multiple roots, becomes very important and is an interesting task from the practical point of view
The basin of attraction of any fixed point belongs to the Fatou set and the boundaries of these basins of attraction belong to the Julia set. By using these tools of complex dynamics, we study the general convergence of family (1) on polynomials with multiple roots of multiplicity 2 and 3
As we will see in the following, the number and the stability of the fixed points depend on the parameter of the family
Summary
With the advancement of computer algebra, finding higher-order multi-point methods, not requiring the computation of second-order derivative for multiple roots, becomes very important and is an interesting task from the practical point of view. In the Fatou set of the rational function R, F (R) is the set of points z ∈ Cwhose orbits tend to an attractor (fixed point, periodic orbit or infinity). The basin of attraction of any fixed point belongs to the Fatou set and the boundaries of these basins of attraction belong to the Julia set By using these tools of complex dynamics, we study the general convergence of family (1) on polynomials with multiple roots of multiplicity 2 and 3.
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