Stability analysis of fourth-order iterative method for finding multiple roots of nonlinear equations

[EN] 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.This research was partially supported by Ministerio de Ciencia, Innovación y Universidades PGC2018-095896-B-C22 and Generalitat Valenciana PROMETEO/2016/089Cordero Barbero, A.; Jaiswal, J.; Torregrosa Sánchez, JR. (2019). Stability analysis of fourth-order iterative method for finding multiple roots of nonlinear equations. Applied Mathematics and Nonlinear Sciences. 4(1):43-56. https://doi.org/10.2478/AMNS.2019.1.00005435641Blanchard, P. (1984). Complex analytic dynamics on the Riemann sphere. Bulletin of the American Mathematical Society, 11(1), 85-142. doi:10.1090/s0273-0979-1984-15240-6Kung, H. T., & Traub, J. F. (1974). Optimal Order of One-Point and Multipoint Iteration. Journal of the ACM, 21(4), 643-651. doi:10.1145/321850.32186