Dynamics of a new family of iterative processes for quadratic polynomials

AbstractIn this work we show the presence of the well-known Catalan numbers in the study of the convergence and the dynamical behavior of a family of iterative methods for solving nonlinear equations. In fact, we introduce a family of methods, depending on a parameter m∈N∪{0}. These methods reach the order of convergence m+2 when they are applied to quadratic polynomials with different roots. Newton’s and Chebyshev’s methods appear as particular choices of the family appear for m=0 and m=1, respectively. We make both analytical and graphical studies of these methods, which give rise to rational functions defined in the extended complex plane. Firstly, we prove that the coefficients of the aforementioned family of iterative processes can be written in terms of the Catalan numbers. Secondly, we make an incursion into its dynamical behavior. In fact, we show that the rational maps related to these methods can be written in terms of the entries of the Catalan triangle. Next we analyze its general convergence, by including some computer plots showing the intricate structure of the Universal Julia sets associated with the methods

Orbits of period two in the family of a multipoint variant of Chebyshev-Halley family

[EN] The study of the dynamical behaviour of the operators defined by iterative methods help us to know more deeply the regions where these methods have a good performance. In this paper, we follow the dynamical study of a multipoint variant of the known Chebyshev-Halley's family, showing the existence of attractive periodic orbits of period 2 for some values of the parameter.This research was partially supported by Ministerio de Econom´ı a y Competitividad MTM2014-52016-C02-2-PCampos, B.; Cordero Barbero, A.; Torregrosa Sánchez, JR.; Vindel Cañas, P. (2016). Orbits of period two in the family of a multipoint variant of Chebyshev-Halley family. Numerical Algorithms. 73(1):141-156. https://doi.org/10.1007/s11075-015-0089-0141156731Blanchard, P.: Complex analytic dynamics on the Riemann sphere. Bull. AMS 11(1), 85–141 (1984)Beardon, A.F.: Iteration of Rational Functions, Graduate Texts in Mathematics. Springer-Verlag, New York (1991)Behl, R., Kanwar, V.: Variants of Chebyshev’s method with optimal order of convergence. Tamsui Oxf. J. Inf. Math. Sci. 29(1), 39–53 (2013)Campos, B., Cordero, A., Magreñan, A., Torregrosa, J.R., Vindel, P.: Study of a bi-parametric family of iterative methods. Abstr. Appl. Anal. 2014. Art. ID 141643, 12 ppCampos, B., Cordero, A., Torregrosa, J.R., Vindel, P.: Bifurcations in the dynamics of a variant of Chebyshev method. In: Proceedings of the 15th International Conference on Computational and Mathematical Methods in Science and Engineering CMMSE 2015, ISBN 978-84-617-2230-3, pp. 291–299 (2015)Chicharro, F., Cordero, A., Torregrosa, J.R.: Drawing dynamical and parameter planes of iterative families and methods. The Scientific World Journal Volume 2013 Article ID 780153Varona, J.L.: Graphic and numerical comparison between iterative methods. Math. Intelligencer 24, 37–46 (2002)Amat, S., Busquier, S., Plaza, S.: Review of some iterative root-finding methods from a dynamical point of view. Sci. Ser. A: Math. Sci. 10, 3–35 (2004)Cordero, A., García-Maimó, J., Torregrosa, J.R., Vassileva, M.P., Vindel, P.: Chaos in King’s iterative family. Appl. Math. Lett. 26, 842–848 (2013)Cordero, A., Torregrosa, J.P., Vindel, P.: Dynamics of a family of Chebyshev-Halley type method. Appl. Math. Comput. 219, 8568–8583 (2013)Gutiérrez, J.M., Hernández, M.A., Romero, N.: Dynamics of a new family of iterative processes for quadratic polynomials. J. Comput. Appl. Math. 233, 2688–2695 (2010)Neta, B., Chun, C., Scott, M.: Basins of attraction for optimal eighth order methods to find simple roots of nonlinear equation. App. Math. Comput. 227, 567–592 (2014)Scott, M., Neta, B., Chun, C.: Basin attractors for various methods. Appl. Math. Comput. 218, 2584–2599 (2011)Blanchard, P.: The dynamics of Newton’s method. Proc. Symp. Appl. Math. 49, 139–154 (1994)Cordero, A., Torregrosa, J.R.: Variants of Newton’s method using fifth-order quadrature formulas. Appl. Math. Comput. 190, 686–698 (2007

Stability analysis of a parametric family of seventh-order iterative methods for solving nonlinear systems

[EN] In this paper, a parametric family of seventh-order of iterative method to solve systems of nonlinear equations is presented. Its local convergence is studied and quadratic polynomials are used to investigate its dynamical behavior. The study of the fixed and critical points of the rational function associated to this class allows us to obtain regions of the complex plane where the method is stable. By depicting parameter planes and dynamical planes we obtain complementary information of the analytical results. These results are used to solve some nonlinear problems. (C) 2017 Elsevier Inc. All rights reserved.This research was partially supported by Ministerio de Economia y Competitividad MTM2014-52016-C2-2-P and by Generalitat Valenciana PROMETEO/2016/089.Amiri, A.; Cordero Barbero, A.; Darvishi, M.; Torregrosa Sánchez, JR. (2018). Stability analysis of a parametric family of seventh-order iterative methods for solving nonlinear systems. Applied Mathematics and Computation. 323:43-57. https://doi.org/10.1016/j.amc.2017.11.040S435732

Study of the dynamics of third-order iterative methods on quadratic polynomials

In this paper, we analyse the dynamical behaviour of the operators associated with multi-point interpolation iterative methods and frozen derivative methods, for solving nonlinear equations, applied on second-degree complex polynomials. We obtain that, in both cases, the Julia set is connected and separates the basins of attraction of the roots of the polynomial. Moreover, the Julia set of the operator associated with multi-point interpolation methods is the same as the Newton operator, although it is more complicated for the frozen derivative operator. We explain these differences by obtaining the conjugacy function of each method and by showing that the operators associated with Newton's method and multi-point interpolation methods are both conjugate to powers of z.The authors thank Professors X. Jarque and A. Garijo for their help. The authors also thank the referees for their valuable comments and suggestions that have improved the content of this paper. This research was supported by Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-02 and by Vicerrectorado de Invetigacion, Universitat Politecnica de Valencia, PAID-06-2010-2285Cordero Barbero, A.; Torregrosa Sánchez, JR.; Vindel Cañas, P. (2012). Study of the dynamics of third-order iterative methods on quadratic polynomials. International Journal of Computer Mathematics. 89(13):1826-1836. https://doi.org/10.1080/00207160.2012.687446S182618368913Amat, S., Busquier, S., & Plaza, S. (2006). A construction of attracting periodic orbits for some classical third-order iterative methods. Journal of Computational and Applied Mathematics, 189(1-2), 22-33. doi:10.1016/j.cam.2005.03.049Amat, S., Bermúdez, C., Busquier, S., & Plaza, S. (2008). On the dynamics of the Euler iterative function. Applied Mathematics and Computation, 197(2), 725-732. doi:10.1016/j.amc.2007.08.086Amat, S., Busquier, S., & Plaza, S. (2010). Chaotic dynamics of a third-order Newton-type method. Journal of Mathematical Analysis and Applications, 366(1), 24-32. doi:10.1016/j.jmaa.2010.01.047Blanchard, P. (1995). The dynamics of Newton’s method. Proceedings of Symposia in Applied Mathematics, 139-154. doi:10.1090/psapm/049/1315536Cordero, A., & Torregrosa, J. R. (2010). On interpolation variants of Newton’s method for functions of several variables. Journal of Computational and Applied Mathematics, 234(1), 34-43. doi:10.1016/j.cam.2009.12.002Cordero, A., Hueso, J. L., Martínez, E., & Torregrosa, J. R. (2009). Multi-Point Iterative Methods for Systems of Nonlinear Equations. Lecture Notes in Control and Information Sciences, 259-267. doi:10.1007/978-3-642-02894-6_25Cordero, A., Hueso, J. L., Martínez, E., & Torregrosa, J. R. (2010). Iterative methods for use with nonlinear discrete algebraic models. Mathematical and Computer Modelling, 52(7-8), 1251-1257. doi:10.1016/j.mcm.2010.02.028Curry, J. H., Garnett, L., & Sullivan, D. (1983). On the iteration of a rational function: Computer experiments with Newton’s method. Communications in Mathematical Physics, 91(2), 267-277. doi:10.1007/bf01211162Douady, A., & Hubbard, J. H. (1985). On the dynamics of polynomial-like mappings. Annales scientifiques de l’École normale supérieure, 18(2), 287-343. doi:10.24033/asens.1491Frontini, M., & Sormani, E. (2003). Some variant of Newton’s method with third-order convergence. Applied Mathematics and Computation, 140(2-3), 419-426. doi:10.1016/s0096-3003(02)00238-2Gutiérrez, J. M., Hernández, M. A., & Romero, N. (2010). Dynamics of a new family of iterative processes for quadratic polynomials. Journal of Computational and Applied Mathematics, 233(10), 2688-2695. doi:10.1016/j.cam.2009.11.017Özban, A. . (2004). Some new variants of Newton’s method. Applied Mathematics Letters, 17(6), 677-682. doi:10.1016/s0893-9659(04)90104-8PLAZA, S. (2001). CONJUGACIES CLASSES OF SOME NUMERICAL METHODS. Proyecciones (Antofagasta), 20(1). doi:10.4067/s0716-09172001000100001Plaza, S., & Romero, N. (2011). Attracting cycles for the relaxed Newton’s method. Journal of Computational and Applied Mathematics, 235(10), 3238-3244. doi:10.1016/j.cam.2011.01.010F.A. Potra and V. Pták,Nondiscrete Introduction and Iterative Processes, Research Notes in Mathematics Vol. 103, Pitman, Boston, MA, 1984

Local convergence of a family of iterative methods for Hammerstein equations

[EN] In this paper we give a local convergence result for a uniparametric family of iterative methods for nonlinear equations in Banach spaces. We assume boundedness conditions involving only the first Fr,chet derivative, instead of using boundedness conditions for high order derivatives as it is usual in studies of semilocal convergence, which is a drawback for solving some practical problems. The existence and uniqueness theorem that establishes the convergence balls of these methods is obtained. We apply this theory to different examples, including a nonlinear Hammerstein equation that have many applications in chemistry and appears in problems of electro-magnetic fluid dynamics or in the kinetic theory of gases. With these examples we illustrate the advantages of these results. The global convergence of the method is addressed by analysing the behaviour of the methods on complex polynomials of second degree.This research was supported by Ministerio de Ciencia y Tecnologia MTM2014-52016-C2-02.This research was supported by Ministerio de Ciencia y Tecnología MTM2014-52016-C2-02.Martínez Molada, E.; Singh, S.; Hueso Pagoaga, JL.; Gupta, D. (2016). Local convergence of a family of iterative methods for Hammerstein equations. Journal of Mathematical Chemistry. 54(7):1370-1386. https://doi.org/10.1007/s10910-016-0602-2S13701386547I.K. Argyros, S. Hilout, M.A. Tabatabai, Mathematical Modelling with Applications in Biosciences and Engineering (Nova Publishers, New York, 2011)J.F. Traub, Iterative Methods for the Solution of Equations (Prentice-Hall, Englewood Cliffs, New Jersey, 1964)A.M. Ostrowski, Solutions of Equations in Euclidean and Banach Spaces (Academic Press, New York, 1973)I.K. Argyros, J.A. Ezquerro, J.M. Gutiárrez, M.A. Hernández, S. Hilout, On the semilocal convergence of efficient ChebyshevSecant-type methods. J. Comput. Appl. Math. 235, 3195–3206 (2011)José L. Hueso, E. Martínez, Semilocal convergence of a family of iterative methods in Banach spaces. Numer. Algorithms 67, 365–384 (2014)X. Wang, C. Gu, J. Kou, Semilocal convergence of a multipoint fourth-order super-Halley method in Banach spaces. Numer. Algorithms 54, 497–516 (2011)J. Kou, Y. Li, X. Wang, A variant of super Halley method with accelerated fourth-order convergence. Appl. Math. Comput. 186, 535–539 (2007)L. Zheng, C. Gu, Recurrence relations for semilocal convergence of a fifth-order method in Banach spaces. Numer. Algorithms 59, 623–638 (2012)S. Amat, M.A. Hernández, N. Romero, A modified Chebyshevs iterative method with at least sixth order of convergence. Appl. Math. Comput. 206, 164–174 (2008)X. Wang, J. Kou, C. Gu, Semilocal convergence of a sixth-order Jarratt method in Banach spaces. Numer. Algorithms 57, 441–456 (2011)A. Cordero, J.A. Ezquerro, M.A. Hernández-Verón, J.R. Torregrosa, On the local convergence of a fifth-order iterative method in Banach spaces. Appl. Math. Comput. 251, 396–403 (2015)I.K. Argyros, S. Hilout, On the local convergence of fast two-step Newton-like methods for solving nonlinear equations. J. Comput. Appl. Math. 245, 1–9 (2013)S. Weerakoon, T.G.I. Fernando, A variant of Newton’s method with accelerated third-order convergence. Appl. Math. Lett. 13(8), 87–93 (2000)X. Feng, Y. He, High order oterative methods without derivatives for solving nonlinear equations. Appl. Math. Comput. 186, 1617–1623 (2007)X. Wang, J. Kou, Y. Li, Modified Jarratt method with sixth-order convergence. Appl. Math. Lett. 22, 1798–1802 (2009)A.D. Polyanin, A.V. Manzhirov, Handbook of Integral Equations (CRC Press, Boca Raton, 1998)S. Plaza, N. Romero, Attracting cycles for the relaxed Newton’s method. J. Comput. Appl. Math. 235(10), 3238–3244 (2011)A. Cordero, J.R. Torregrosa, P. Vindel, Study of the dynamics of third-order iterative methods on quadratic polynomials. Int. J. Comput. Math. 89(13–14), 1826–1836 (2012)Gerardo Honorato, Sergio Plaza, Natalia Romero, Dynamics of a higher-order family of iterative methods. J. Complex. 27(2), 221–229 (2011)J.M. Gutirrez, M.A. Hernández, N. Romero, Dynamics of a new family of iterative processes for quadratic polynomials. J. Comput. Appl. Math. 233(10), 2688–2695 (2010)I.K. Argyros, A.A. Magreñan, A study on the local convergence and dynamics of Chebyshev-Halley-type methods free from second derivative. Numer. Algorithms. doi: 10.1007/s11075-015-9981-xI.K. Argyros, S. George, Local convergence of modified Halley-like methods with less computation of inversion (Novi Sad J. Math, Draft version, 2015

Bulbs of period two in the family of Chebyshev-Halley iterative methods on quadratic polynomials

The parameter space associated to the parametric family of Chebyshev-Halley on quadratic polynomials shows a dynamical richness worthy of study. This analysis has been initiated by the authors in previous works. Every value of the parameter belonging to the same connected component of the parameter space gives rise to similar dynamical behavior. In this paper, we focus on the search of regions in the parameter space that gives rise to the appearance of attractive orbits of period two.This research was supported by Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-02, by Vicerrectorado de Investigacion, Universitat Politecnica de Valencia PAID SP20120498 and by Vicerrectorado de Investigacion, Universitat Jaume I P11B2011-30. The authors would like to thank Mr. Francisco Chicharro for his valuable help with the numerical and graphic tools for drawing the dynamical planes.Cordero Barbero, A.; Torregrosa Sánchez, JR.; Vindel Cañas, P. (2013). Bulbs of period two in the family of Chebyshev-Halley iterative methods on quadratic polynomials. Abstract and Applied Analysis. 2013. https://doi.org/10.1155/2013/536910S2013Amat, S., Bermúdez, C., Busquier, S., & Plaza, S. (2008). On the dynamics of the Euler iterative function. Applied Mathematics and Computation, 197(2), 725-732. doi:10.1016/j.amc.2007.08.086Amat, S., Busquier, S., & Plaza, S. (2006). A construction of attracting periodic orbits for some classical third-order iterative methods. Journal of Computational and Applied Mathematics, 189(1-2), 22-33. doi:10.1016/j.cam.2005.03.049Gutiérrez, J. M., Hernández, M. A., & Romero, N. (2010). Dynamics of a new family of iterative processes for quadratic polynomials. Journal of Computational and Applied Mathematics, 233(10), 2688-2695. doi:10.1016/j.cam.2009.11.017Plaza, S., & Romero, N. (2011). Attracting cycles for the relaxed Newton’s method. Journal of Computational and Applied Mathematics, 235(10), 3238-3244. doi:10.1016/j.cam.2011.01.010Chicharro, F., Cordero, A., Gutiérrez, J. M., & Torregrosa, J. R. (2013). Complex dynamics of derivative-free methods for nonlinear equations. Applied Mathematics and Computation, 219(12), 7023-7035. doi:10.1016/j.amc.2012.12.075Chun, C., Lee, M. Y., Neta, B., & Džunić, J. (2012). On optimal fourth-order iterative methods free from second derivative and their dynamics. Applied Mathematics and Computation, 218(11), 6427-6438. doi:10.1016/j.amc.2011.12.013Neta, B., Scott, M., & Chun, C. (2012). Basins of attraction for several methods to find simple roots of nonlinear equations. Applied Mathematics and Computation, 218(21), 10548-10556. doi:10.1016/j.amc.2012.04.017Cordero, A., Torregrosa, J. R., & Vindel, P. (2013). Dynamics of a family of Chebyshev–Halley type methods. Applied Mathematics and Computation, 219(16), 8568-8583. doi:10.1016/j.amc.2013.02.042Blanchard, P. (1984). Complex analytic dynamics on the Riemann sphere. Bulletin of the American Mathematical Society, 11(1), 85-142. doi:10.1090/s0273-0979-1984-15240-6Kneisl, K. (2001). Julia sets for the super-Newton method, Cauchy’s method, and Halley’s method. Chaos: An Interdisciplinary Journal of Nonlinear Science, 11(2), 359-370. doi:10.1063/1.1368137Cordero, A., Torregrosa, J. R., & Vindel, P. (2013). Period-doubling bifurcations in the family of Chebyshev–Halley-type methods. International Journal of Computer Mathematics, 90(10), 2061-2071. doi:10.1080/00207160.2012.745518Devaney, R. L. (1999). The Mandelbrot Set, the Farey Tree, and the Fibonacci Sequence. The American Mathematical Monthly, 106(4), 289. doi:10.2307/258955

Drawing dynamical and parameters planes of iterative families and methods

The complex dynamical analysis of the parametric fourth-order Kim s iterative family is made on quadratic polynomials, showing the MATLAB codes generated to draw the fractal images necessary to complete the study. The parameter spaces associated with the free critical points have been analyzed, showing the stable (and unstable) regions where the selection of the parameter will provide us the excellent schemes (or dreadful ones).The authors thank the anonymous referees for their valuable comments and for the suggestions to improve the readability of the paper. This research was supported by Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-02.Chicharro López, FI.; Cordero Barbero, A.; Torregrosa Sánchez, JR. (2013). Drawing dynamical and parameters planes of iterative families and methods. The Scientific World Journal. 2013. https://doi.org/10.1155/2013/780153S2013Douady, A., & Hubbard, J. H. (1985). On the dynamics of polynomial-like mappings. Annales scientifiques de l’École normale supérieure, 18(2), 287-343. doi:10.24033/asens.1491Curry, J. H., Garnett, L., & Sullivan, D. (1983). On the iteration of a rational function: Computer experiments with Newton’s method. Communications in Mathematical Physics, 91(2), 267-277. doi:10.1007/bf01211162Varona, J. L. (2002). Graphic and numerical comparison between iterative methods. The Mathematical Intelligencer, 24(1), 37-46. doi:10.1007/bf03025310Gutiérrez, J. M., Hernández, M. A., & Romero, N. (2010). Dynamics of a new family of iterative processes for quadratic polynomials. Journal of Computational and Applied Mathematics, 233(10), 2688-2695. doi:10.1016/j.cam.2009.11.017Honorato, G., Plaza, S., & Romero, N. (2011). Dynamics of a higher-order family of iterative methods. Journal of Complexity, 27(2), 221-229. doi:10.1016/j.jco.2010.10.005Chicharro, F., Cordero, A., Gutiérrez, J. M., & Torregrosa, J. R. (2013). Complex dynamics of derivative-free methods for nonlinear equations. Applied Mathematics and Computation, 219(12), 7023-7035. doi:10.1016/j.amc.2012.12.075Artidiello, S., Chicharro, F., Cordero, A., & Torregrosa, J. R. (2013). Local convergence and dynamical analysis of a new family of optimal fourth-order iterative methods. International Journal of Computer Mathematics, 90(10), 2049-2060. doi:10.1080/00207160.2012.748900Scott, M., Neta, B., & Chun, C. (2011). Basin attractors for various methods. Applied Mathematics and Computation, 218(6), 2584-2599. doi:10.1016/j.amc.2011.07.076Chun, C., Lee, M. Y., Neta, B., & Džunić, J. (2012). On optimal fourth-order iterative methods free from second derivative and their dynamics. Applied Mathematics and Computation, 218(11), 6427-6438. doi:10.1016/j.amc.2011.12.013Neta, B., Scott, M., & Chun, C. (2012). Basin attractors for various methods for multiple roots. Applied Mathematics and Computation, 218(9), 5043-5066. doi:10.1016/j.amc.2011.10.071Cordero, A., García-Maimó, J., Torregrosa, J. R., Vassileva, M. P., & Vindel, P. (2013). Chaos in King’s iterative family. Applied Mathematics Letters, 26(8), 842-848. doi:10.1016/j.aml.2013.03.012Devaney, R. L. (1999). The Mandelbrot Set, the Farey Tree, and the Fibonacci Sequence. The American Mathematical Monthly, 106(4), 289. doi:10.2307/2589552Ik Kim, Y. (2012). A triparametric family of three-step optimal eighth-order methods for solving nonlinear equations. International Journal of Computer Mathematics, 89(8), 1051-1059. doi:10.1080/00207160.2012.673597Cordero, A., Torregrosa, J. R., & Vindel, P. (2013). Dynamics of a family of Chebyshev–Halley type methods. Applied Mathematics and Computation, 219(16), 8568-8583. doi:10.1016/j.amc.2013.02.04

Chaos and convergence of a family generalizing Homeier's method with damping parameters

[EN] 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.This research was supported by Ministerio de Economia y Competitividad MTM2014-52016-C02-2-P.Cordero Barbero, A.; Franques, A.; Torregrosa Sánchez, JR. (2016). Chaos and convergence of a family generalizing Homeier's method with damping parameters. Nonlinear Dynamics. 85(3):1939-1954. https://doi.org/10.1007/s11071-016-2807-0S19391954853Amat, 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-xAmat, S., Busquier, S., Bermúdez, C., Plaza, S.: On two families of high order Newton type methods. Appl. Math. Lett. 25, 2209–2217 (2012)Amat, S., Busquier, S., Plaza, S.: Review of some iterative root-finding methods from a dynamical point of view. Sci. Ser. A Math. Sci. 10, 3–35 (2004)Babajee, D.K.R., Cordero, A., Torregrosa, J.R.: Study of iterative methods through the Cayley Quadratic Test. J. Comput. Appl. Math. 291, 358–369 (2016)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)Blanchard, P.: Complex analytic dynamics on the Riemann sphere. Bull. AMS 11(1), 85–141 (1984)Blanchard, P.: The dynamics of Newton’s method. Proc. Symp. Appl. Math. 49, 139–154 (1994)Boyd, J.P.: One-point pseudospectral collocation for the one-dimensional Bratu equation. Appl. Math. Comput. 217, 5553–5565 (2011)Bratu, G.: Sur les equation integrals non-lineaires. Bull. Math. Soc. Fr. 42, 113–142 (1914)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)Chun, C., Lee, M.Y.: A new optimal eighth-order family of iterative methods for the solution of nonlinear equations. Appl. Math. Comput. 223, 506–519 (2013)Cordero, A., García-Maimó, J., Torregrosa, J.R., Vassileva, M.P., Vindel, P.: Chaos in King’s iterative family. Appl. Math. Lett. 26, 842–848 (2013)Cordero, A., Torregrosa, J.R.: Variants of Newton’s method using fifth-order quadrature formulas. Appl. Math. Comput. 190, 686–698 (2007)Cordero, A., Torregrosa, J.R., Vindel, P.: Dynamics of a family of Chebyshev–Halley type method. Appl. Math. Comput. 219, 8568–8583 (2013)Fatou, P.: Sur les équations fonctionnelles. Bull. Soc. Math. Fr. 47, 161–271 (1919); 48, 33–94; 208–314 (1920)Gelfand, I.M.: Some problems in the theory of quasi-linear equations. Transl. Am. Math. Soc. Ser. 2, 295–381 (1963)Gutiérrez, J.M., Hernández, M.A., Romero, N.: Dynamics of a new family of iterative processes for quadratic polynomials. J. Comput. Appl. Math. 233, 2688–2695 (2010)Homeier, H.H.H.: On Newton-type methods with cubic convergence. J. Comput. Appl. Math. 176, 425–432 (2005)Jacobsen, J., Schmitt, K.: The Liouville–Bratu–Gelfand problem for radial operators. J. Differ. Equ. 184, 283–298 (2002)Jalilian, R.: Non-polynomial spline method for solving Bratu’s problem. Comput. 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Comput. 256, 119–124 (2015)Varona, J.L.: Graphic and numerical comparison between iterative methods. Math. Intell. 24, 37–46 (2002)Wan, Y.Q., Guo, Q., Pan, N.: Thermo-electro-hydrodynamic model for electrospinning process. Int. J. Nonlinear Sci. Numer. Simul. 5, 5–8 (2004

Convergence regions for the Chebyshev--Halley family

In this paper, we study the dynamical behaviour of the Chebyshev--Halley family applied on a family of degree n polynomials. For n=2 we bound the set of parameters for which the iterative methods have convergence regions which do not correspond to the basins of attraction of the roots. We also study the dynamics of indifferent fixed points on the boundary of the regions of parameters with bad behaviour. Finally, we provide a numerical study on the boundedness of the regions of parameters with bad behaviour for the family of degree n polynomials