Dynamics of a family of rational operators of arbitrary degree
Abstract
In this paper we analyse the dynamics of a family of rational operators coming from a fourth-order family of root-finding algorithms. We first show that it may be convenient to redefine the parameters to prevent redundancies and unboundedness of problematic parameters. After reparametrization, we observe that these rational maps belong to a more general family Oa,n,k of degree n+k operators, which includes several other families of maps obtained from other numerical methods. We study the dynamics of Oa,n,k and discuss for which parameters n and k these operators would be suitable from the numerical point of view.
Keyword : iterative methods, parameter planes, complex dynamics of rational functions
How to Cite
Campos, B., Canela, J., Garijo, A., & Vindel, P. (2021). Dynamics of a family of rational operators of arbitrary degree. Mathematical Modelling and Analysis, 26(2), 188-208. https://doi.org/10.3846/mma.2021.12642
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
A. Argyros and A. Magreñán. On the convergence of an optimal fourth-order family of methods and its dynamics. Appl. Math. Comp., 252:336–346, 2015. https://doi.org/10.1016/j.amc.2014.11.074
A. Argyros and A. Magreñán. On the local convergence and the dynamics of Chebyshev-Halley methods with six and eight order of convergence. J. Comput. Appl. Math, 298:236–251, 2016. https://doi.org/10.1016/j.cam.2015.11.036
A.F. Beardon. Iteration of rational function. Graduate Texts in Mathematics. Springer-Verlag, New York, 1991. https://doi.org/10.1007/978-1-4612-4422-6
R. Behl. Development and analysis of some new methods for numerical solutions of non linear equations (PhD thesis). Punjab University, 2013.
P. Blanchard. Complex analytic dynamics on the Riemann sphere. Bull. AMS, 11(1):85–141, 1984. https://doi.org/10.1090/S0273-0979-1984-15240-6
P. Blanchard. The dynamics of Newton’s method. Complex dynamical systems (Cincinnati, OH, 1994), Amer. Math. Soc., Proc. Sympos. Appl. Math., 49:139– 154, 1994.
B. Campos, J. Canela and P. Vindel. Convergence regions for the ChebyshevHalley family. Commun. Nonlinear Sci. Numer. Simul., 56:508–525, 2018. https://doi.org/10.1016/j.cnsns.2017.08.024
B. Campos, J. Canela and P. Vindel. Connectivity of the Julia set for the Chebyshev-Halley family on degree n polynomials. Commun. Nonlinear Sci. Numer. Simul., 82:105–126, 2020. https://doi.org/10.1016/j.cnsns.2019.105026
A. Cordero, L. Feng, A. Magreñán and J.R. Torregrosa. A new fourth-order family for solving nonlinear problems and its dynamics. J. Math. Chem., 53:893– 910, 2015. https://doi.org/10.1007/s10910-014-0464-4
A. Cordero, J. García-Maimó, J.R. Torregrosa, M-P. Vassilieva and P. Vindel. Chaos in King’s iterative family. Appl. Math. Lett., 26:842–848, 2013. https://doi.org/10.1016/j.aml.2013.03.012
A. Cordero, J. M. Gutiérrez, A.Á. Magreñán and J.R. Torregrosa. Stability analysis of a parametric family of iterative methods for solving nonlinear models. App. Math. Comput., 285:26–40, 2016. https://doi.org/10.1016/j.amc.2016.03.021
A. Cordero, A. Magreñán, C. Quemada and J.R. Torregrosa. Stability study of eighth-order iterative methods for solving nonlinear equations. J. Comput. Appl. Math., 291:348–357, 2016. https://doi.org/10.1016/j.cam.2015.01.006
A. Cordero, J.R. Torregrosa and P. Vindel. Dynamics of a family of Chebyshev-Halley type methods. App. Math. Comput., 219:8568–8583, 2013. https://doi.org/10.1016/j.amc.2013.02.042
J.M. Gutiérrez, M.A. Hernández and N. Romero. Dynamics of a new family of iterative processes for quadratic polynomials. J. Comput. Appl. Math., 233:2688–2695, 2010. https://doi.org/10.1016/j.cam.2009.11.017
J. Milnor. Dynamics in one complex variable. Annals of Mathematics Studies, Princeton University Press, 2006.
P. Roesch. On local connectivity for the Julia set of rational maps: Newton’s famous example. Ann. of Math., 168(1):127–174, 2008. https://doi.org/10.4007/annals.2008.168.127
D. Schleicher. On the number of iterations of Newton’s method for complex polynomials. Ergodic Theory Dynam. Systems, 22(3):935–945, 2002. https://doi.org/10.1017/S0143385702000482
M. Shishikura. On the quasiconformal surgery of rational functions. Ann. Sci. Ecolé Norm. Sup., Ser. 4, 20(1):1–29, 1987. https://doi.org/10.24033/asens.1522
M. Shishikura. Branched coverings and cubic Newton maps. Fund. Math., 154(3):207–260, 1997. https://doi.org/10.4064/fm-154-3-207-260
F. Yang. Rational maps without Herman rings. Proc. Amer. Math. Soc., 145(4):1649–1659, 2017. https://doi.org/10.1090/proc/13336
A. Argyros and A. Magreñán. On the local convergence and the dynamics of Chebyshev-Halley methods with six and eight order of convergence. J. Comput. Appl. Math, 298:236–251, 2016. https://doi.org/10.1016/j.cam.2015.11.036
A.F. Beardon. Iteration of rational function. Graduate Texts in Mathematics. Springer-Verlag, New York, 1991. https://doi.org/10.1007/978-1-4612-4422-6
R. Behl. Development and analysis of some new methods for numerical solutions of non linear equations (PhD thesis). Punjab University, 2013.
P. Blanchard. Complex analytic dynamics on the Riemann sphere. Bull. AMS, 11(1):85–141, 1984. https://doi.org/10.1090/S0273-0979-1984-15240-6
P. Blanchard. The dynamics of Newton’s method. Complex dynamical systems (Cincinnati, OH, 1994), Amer. Math. Soc., Proc. Sympos. Appl. Math., 49:139– 154, 1994.
B. Campos, J. Canela and P. Vindel. Convergence regions for the ChebyshevHalley family. Commun. Nonlinear Sci. Numer. Simul., 56:508–525, 2018. https://doi.org/10.1016/j.cnsns.2017.08.024
B. Campos, J. Canela and P. Vindel. Connectivity of the Julia set for the Chebyshev-Halley family on degree n polynomials. Commun. Nonlinear Sci. Numer. Simul., 82:105–126, 2020. https://doi.org/10.1016/j.cnsns.2019.105026
A. Cordero, L. Feng, A. Magreñán and J.R. Torregrosa. A new fourth-order family for solving nonlinear problems and its dynamics. J. Math. Chem., 53:893– 910, 2015. https://doi.org/10.1007/s10910-014-0464-4
A. Cordero, J. García-Maimó, J.R. Torregrosa, M-P. Vassilieva and P. Vindel. Chaos in King’s iterative family. Appl. Math. Lett., 26:842–848, 2013. https://doi.org/10.1016/j.aml.2013.03.012
A. Cordero, J. M. Gutiérrez, A.Á. Magreñán and J.R. Torregrosa. Stability analysis of a parametric family of iterative methods for solving nonlinear models. App. Math. Comput., 285:26–40, 2016. https://doi.org/10.1016/j.amc.2016.03.021
A. Cordero, A. Magreñán, C. Quemada and J.R. Torregrosa. Stability study of eighth-order iterative methods for solving nonlinear equations. J. Comput. Appl. Math., 291:348–357, 2016. https://doi.org/10.1016/j.cam.2015.01.006
A. Cordero, J.R. Torregrosa and P. Vindel. Dynamics of a family of Chebyshev-Halley type methods. App. Math. Comput., 219:8568–8583, 2013. https://doi.org/10.1016/j.amc.2013.02.042
J.M. Gutiérrez, M.A. Hernández and N. Romero. Dynamics of a new family of iterative processes for quadratic polynomials. J. Comput. Appl. Math., 233:2688–2695, 2010. https://doi.org/10.1016/j.cam.2009.11.017
J. Milnor. Dynamics in one complex variable. Annals of Mathematics Studies, Princeton University Press, 2006.
P. Roesch. On local connectivity for the Julia set of rational maps: Newton’s famous example. Ann. of Math., 168(1):127–174, 2008. https://doi.org/10.4007/annals.2008.168.127
D. Schleicher. On the number of iterations of Newton’s method for complex polynomials. Ergodic Theory Dynam. Systems, 22(3):935–945, 2002. https://doi.org/10.1017/S0143385702000482
M. Shishikura. On the quasiconformal surgery of rational functions. Ann. Sci. Ecolé Norm. Sup., Ser. 4, 20(1):1–29, 1987. https://doi.org/10.24033/asens.1522
M. Shishikura. Branched coverings and cubic Newton maps. Fund. Math., 154(3):207–260, 1997. https://doi.org/10.4064/fm-154-3-207-260
F. Yang. Rational maps without Herman rings. Proc. Amer. Math. Soc., 145(4):1649–1659, 2017. https://doi.org/10.1090/proc/13336