Chaos and convergence of a family generalizing Homeier’s method with damping parameters
In this paper, a family of parametric iterative methods for solving nonlinear equations, including Homeier’s scheme, is presented. Its local convergence is obtained and the dynamical behavior on quadratic polynomials of the resulting family is studied in order to choose those values of the parameter that ensure stable behavior. To get this aim, the analysis of fixed and critical points and the associated parameter plane show the dynamical richness of the family and allow us to find members of this class with good numerical properties and also other ones with pathological conduct. To check the stable behavior of the good selected ones, the discretized planar 1D-Bratu problem is solved. Some of those chosen members of the family achieve good results when Homeier’s scheme fails.
KeywordsNonlinear equations Iterative methods Dynamical behavior Parameter plane Convergence regions Bratu problem
The authors would like to thank the anonymous reviewers for their valuable suggestions and comments that have substantially improved the final version of this manuscript.
- 1.Amat, S., Busquier, S., Bermúdez, C., Magreñán, Á.A.: On the election of the damped parameter of a two-step relaxed Newton-type method. Nonlinear Dyn. doi: 10.1007/s11071-015-2179-x
- 5.Babajee, D.K.R., Thukral, R.: On a 4-point sixteenth-order king family of iterative methods for solving nonlinear equations. Int. J. Math. Math. Sci. 2012, ID 979245, 13 (2012)Google Scholar
- 10.Chicharro, F., Cordero, A., Torregrosa, J.R.: Drawing dynamical and parameter planes of iterative families and methods. Sci. World J. 2013, Article ID 780153 (2013)Google Scholar
- 15.Fatou, P.: Sur les équations fonctionnelles. Bull. Soc. Math. Fr. 47, 161–271 (1919); 48, 33–94; 208–314 (1920)Google Scholar
- 16.Gelfand, I.M.: Some problems in the theory of quasi-linear equations. Transl. Am. Math. Soc. Ser. 2, 295–381 (1963)Google Scholar
- 18.Homeier, H.H.H.: On Newton-type methods with cubic convergence. J. Comput. Appl. Math. 176, 425–432 (2005)Google Scholar
- 30.Wan, Y.Q., Guo, Q., Pan, N.: Thermo-electro-hydrodynamic model for electrospinning process. Int. J. Nonlinear Sci. Numer. Simul. 5, 5–8 (2004)Google Scholar