International Journal of Nonlinear Analysis and Applications
Dynamics of the modified Halley's method
Document Type : Research Paper
Author
Departamento of Matematica, Facultad de Ciencias y Tecnolog'ia, Universidad de Carabobo, Naguanagua 2005, Venezuela
Abstract
In this work, the dynamics of the Modified Halley's method to multiple roots are established. We find the fixed and critical points. The stable and unstable behaviors are studied. The parameter space associated with the method is studied and finally, some dynamical planes that show different aspects of the dynamics of this method are presented.
Keywords
[1] S. Amat, S. Busquier and S. Plaza , Review of some iterative root-finding methods from a dynamical point of view, Sci. A: Math. Sci. 10 (2004), 3–35.
[2] S. Amat, S. Busquier and S. Plaza, A construction of attracting periodic orbits for some classical third-order iterative methods, J. Comput. Appl. Math. 189 (2006), 22–33.
[3] S. Amat, S. Busquier and S. Plaza, On the dynamics of a family of third-order iterative functions, ANZIAM J. 48 (2007), no. 3, 343–359.
[4] S. Amat, C. Berm´udez, S. Busquier, J. Carrasco and S. Plaza, Super-attracting periodic orbits for a classic third order method, J. Comput. Appl. Math. 206 (2007), 599–602.
[5] S. Amat, C. Berm´udez, S. Busquier and S. Plaza, On the dynamics of the Euler iterative function, Appl. Math. Comput. 197 (2008), 725–732.
[6] S. Amat, S. Busquier and S. Plaza, Chaotic dynamics of a third-order Newton-type method, J. Math. Anal. Appl. 366 (2010), 24–32.
[7] S. Amat, S. Busquier, E. Navarro and S. Plaza, Superattracting cycles for some Newton type iterative methods, Sci. China Math. 54 (2011), no. 3, 539–544.
[8] I.K. Argyros and A.A. Magre˜nan, ´ On the convergence of an optimal fourth-order family of methods and its dynamics, Appl. Math. Comput. 252 (2015), 336–346.
[9] A.F. Beardon, Iteration of Rational Functions, vol. 132 of Graduate Texts in Mathematics, Springer, New York, NY. USA, 1991.
[10] P. Blanchard, Complex analytic dynamics on the Riemann sphere, Bull. Amer. Math. Soc. 11 (1984), no. 1, 85–141.
[11] P. Blanchard, The dynamics of Newton’s method, Proc. Symp. Appl. Math. 49 (1994), 139–154.
[12] C.E. Cadenas, On Several Gander’s theorem based third-order iterative methods for solving nonlinear equations and their geometric constructions, J. Numer. Math. Stoch. 9 (2017), no. 1, 1–19.
[13] C.E. Cadenas, A family of Newton-Halley type methods to find simple roots of nonlinear equations and their dynamics, Commun. Numer. Anal. 2017 (2017), no. 2, 157–171.
[14] C.E. Cadenas, A family of Newton-Chebyshev type methods to find simple roots of nonlinear equations and their dynamics, Commun. Numer. Anal. 2017 (2017), no. 2, 172–185.
[15] C.E. Cadenas, Cuencas de atracci´on usando MatLab, Rev. MATUA, 4 (2017), no. 2, 1–15.
[16] C.E. Cadenas, On Geometric Constructions of third order methods for multiple roots of nonlinear equations, Commun. Numer. Anal. 2018 (2018), 42–55.
[17] C.E. Cadenas, A new approach to study the dynamics of the modified Newton’s method to multiple roots, Bull. Math. Soc. Sci. Math. Roumanie.Tome 62 110 (2019), no. 1, 67—75.
[18] B. Campos, A. Cordero, A. Magre, J.R. Torregrosa and P. Vindel, Bifurcations of the roots of a 6-degree symmetric polynomial coming from the fixed point operator of a class of iterative methods, Proc. CMMSE, 2014, pp. 253–264.
[19] B. Campos, A. Cordero, A.A. Magre˜n´an, J.R. Torregrosa and P. Vindel, ´ Study of a biparametric family of iterative methods, Abstr. Appl. Anal. 2014 (2014), ID 141643, 12 pages.
[20] B. Campos, A. Cordero, J.R. Torregrosa and P. Vindel, Dynamics of the family of c-iterative methods, Int. J. Comp. Math. 92 (2015), no. 9, 1815–1825.
[21] B. Campos, A. Cordero, J.R. Torregrosa and P. Vindel, Behaviour of fixed and critical points of the (α, c)-family of iterative methods, J. Math. Chem. 53 (2015), 807-–827.
[22] C. Chun, M.Y. Lee, B. Neta and J. Dˇzuni´c, On optimal fourth-order iterative methods free from second derivative and their dynamics, Appl. Math. Comput., 218 (2012), no. 11, 6427–6438.
[23] A. Cordero, J.R. Torregrosa and P. Vindel, On complex dynamics of some third-order iterative methods, Proc.Int. Conf. Comput. Math. Meth. Sci. Engin. I, 2011, pp. 374–383.
[24] A. Cordero and J.R. Torregrosa, Study of the dynamics of third-order iterative methods on quadratic polynomials, Int. J. Comp. Math. 89 (2012), 1826–1836.
[25] A. Cordero, J.R. Torregrosa and P. Vindel, Dynamics of a family of Chebyshev-Halley type methods. Appl. Math. Comput. 219 (2013), no. 16, 8568–8583.
[26] A. Cordero, J.R. Torregrosa and P. Vindel, Period-doubling bifurcation in the family Chebyshev-Halley type methods, Int. J. of Comp. Math. 90 (2013), no. 10, 2061–2071.
[27] A. Cordero, J.R. Torregrosa and P. Vindel, Bulbs of period two in the family Chebyshev-Halley type methods on quadratic polynomials, Abstr. Appl. Anal. 2013 (2013), ID 536910, 10 pages.
[28] A. Cordero, J. Garc´ıa-Maim´o, J.R. Torregrosa, M. Vassileva and P. Vindel, Chaos in King’s iterative family, Appl. Math. Lett. 26 (2013), no. 8, 842–848.
[29] A. Cordero, M. Fardi, M. Ghasemi, J.R. Torregrosa, Accelerated iterative methods for finding solutions of nonlinear equations and their dynamical behavior, Calcolo 51 (2014), 17–30.
[30] R.L. Devaney, The Mandelbrot set, the Farey tree and the Fibonacci sequence, Amer. Math. Mon., 106 (1999), no. 4, 289–302.
[31] B.I. Epureanu, H. S. Greenside, Fractal basins of attraction associated with a damped Newton’s method, SIAM REV 40 (1998), no. 1, 102–109.
[32] J.M. Guti´errez, M.A. Hern´andez and N. Romero, Dynamics of a new family of iterative processes for quadratic polynomials, J. Comput. Appl. Math. 233 (2010), 2688–2695.
[33] J.M. Guti´errez, A.A. Magre˜nan and J.L. Varona, ´ Fractal dimension of the universal Julia sets for the ChebyshevHalley family of methods, ICNAAM 2011. AIP Conf. Proc., 1389 (2011), no. 1, 1061–1064.
[34] G. Honorato, S. Plaza and N. Romero, Dynamics of a high-order family of iterative methods, J. Complexity 27 (2011), 221–229.
[35] K. Kneisl, Julia sets for the super-Newton method, Cauchy’s method and Halley’s method, Chaos 11 (2001), no. 2, 359–370.
[36] M.Y. Lee and C. Chun, Attracting periodic cycles for an optimal fourth-order nonlinear solver, Abstr. Appl. Anal. 2012 (2012), ID 263893, 8 pages.
[37] T. Lotfi et al, A variant of Steffensen–King’s type family with accelerated sixth-order convergence and high efficiency index: Dynamic study and approach, Appl. Math. Comput. 252 (2015), 347–353.
[38] A.A. Magre˜nan, ´ Estudio de la din´amica del m´etodo de Newton amortiguado (Ph.D. thesis), Servicio de Publicaciones, Universidad de La Rioja, 2013.
[39] B. Neta, M. Scott and C. Chun, Basins of attraction for several methods to find simple roots of nonlinear equations, Appl. Math. Comput. 218 (2012), 10548–10556.
[40] B. Neta, M. Scott and C. Chun, Basin attractors for various methods for multiple roots, Appl. Math. Comput. 218 (2012), no. 9, 5043–5066.
[41] C.A. Pickover, A note on chaos and Halley’s method, Commun. ACM 31 (1988), no. 11, 1326–1329.
[42] S. Plaza and N. Romero, Attracting cycles for the relaxed Newton’s method, J. Comput. Appl. Math. 235 (2011), 3238–3244.
[43] G.E. Roberts and J. Horgan-Kobelski, Newton’s versus Halley’s method: a dynamical system approach, Int. J. Bifurcat. Chaos 14 (2004), no. 10, 3459–3475.
[44] J.F. Traub, Iterative methods for resolution of equations, Prectice Hall, NJ 1964.
[2] S. Amat, S. Busquier and S. Plaza, A construction of attracting periodic orbits for some classical third-order iterative methods, J. Comput. Appl. Math. 189 (2006), 22–33.
[3] S. Amat, S. Busquier and S. Plaza, On the dynamics of a family of third-order iterative functions, ANZIAM J. 48 (2007), no. 3, 343–359.
[4] S. Amat, C. Berm´udez, S. Busquier, J. Carrasco and S. Plaza, Super-attracting periodic orbits for a classic third order method, J. Comput. Appl. Math. 206 (2007), 599–602.
[5] S. Amat, C. Berm´udez, S. Busquier and S. Plaza, On the dynamics of the Euler iterative function, Appl. Math. Comput. 197 (2008), 725–732.
[6] S. Amat, S. Busquier and S. Plaza, Chaotic dynamics of a third-order Newton-type method, J. Math. Anal. Appl. 366 (2010), 24–32.
[7] S. Amat, S. Busquier, E. Navarro and S. Plaza, Superattracting cycles for some Newton type iterative methods, Sci. China Math. 54 (2011), no. 3, 539–544.
[8] I.K. Argyros and A.A. Magre˜nan, ´ On the convergence of an optimal fourth-order family of methods and its dynamics, Appl. Math. Comput. 252 (2015), 336–346.
[9] A.F. Beardon, Iteration of Rational Functions, vol. 132 of Graduate Texts in Mathematics, Springer, New York, NY. USA, 1991.
[10] P. Blanchard, Complex analytic dynamics on the Riemann sphere, Bull. Amer. Math. Soc. 11 (1984), no. 1, 85–141.
[11] P. Blanchard, The dynamics of Newton’s method, Proc. Symp. Appl. Math. 49 (1994), 139–154.
[12] C.E. Cadenas, On Several Gander’s theorem based third-order iterative methods for solving nonlinear equations and their geometric constructions, J. Numer. Math. Stoch. 9 (2017), no. 1, 1–19.
[13] C.E. Cadenas, A family of Newton-Halley type methods to find simple roots of nonlinear equations and their dynamics, Commun. Numer. Anal. 2017 (2017), no. 2, 157–171.
[14] C.E. Cadenas, A family of Newton-Chebyshev type methods to find simple roots of nonlinear equations and their dynamics, Commun. Numer. Anal. 2017 (2017), no. 2, 172–185.
[15] C.E. Cadenas, Cuencas de atracci´on usando MatLab, Rev. MATUA, 4 (2017), no. 2, 1–15.
[16] C.E. Cadenas, On Geometric Constructions of third order methods for multiple roots of nonlinear equations, Commun. Numer. Anal. 2018 (2018), 42–55.
[17] C.E. Cadenas, A new approach to study the dynamics of the modified Newton’s method to multiple roots, Bull. Math. Soc. Sci. Math. Roumanie.Tome 62 110 (2019), no. 1, 67—75.
[18] B. Campos, A. Cordero, A. Magre, J.R. Torregrosa and P. Vindel, Bifurcations of the roots of a 6-degree symmetric polynomial coming from the fixed point operator of a class of iterative methods, Proc. CMMSE, 2014, pp. 253–264.
[19] B. Campos, A. Cordero, A.A. Magre˜n´an, J.R. Torregrosa and P. Vindel, ´ Study of a biparametric family of iterative methods, Abstr. Appl. Anal. 2014 (2014), ID 141643, 12 pages.
[20] B. Campos, A. Cordero, J.R. Torregrosa and P. Vindel, Dynamics of the family of c-iterative methods, Int. J. Comp. Math. 92 (2015), no. 9, 1815–1825.
[21] B. Campos, A. Cordero, J.R. Torregrosa and P. Vindel, Behaviour of fixed and critical points of the (α, c)-family of iterative methods, J. Math. Chem. 53 (2015), 807-–827.
[22] C. Chun, M.Y. Lee, B. Neta and J. Dˇzuni´c, On optimal fourth-order iterative methods free from second derivative and their dynamics, Appl. Math. Comput., 218 (2012), no. 11, 6427–6438.
[23] A. Cordero, J.R. Torregrosa and P. Vindel, On complex dynamics of some third-order iterative methods, Proc.Int. Conf. Comput. Math. Meth. Sci. Engin. I, 2011, pp. 374–383.
[24] A. Cordero and J.R. Torregrosa, Study of the dynamics of third-order iterative methods on quadratic polynomials, Int. J. Comp. Math. 89 (2012), 1826–1836.
[25] A. Cordero, J.R. Torregrosa and P. Vindel, Dynamics of a family of Chebyshev-Halley type methods. Appl. Math. Comput. 219 (2013), no. 16, 8568–8583.
[26] A. Cordero, J.R. Torregrosa and P. Vindel, Period-doubling bifurcation in the family Chebyshev-Halley type methods, Int. J. of Comp. Math. 90 (2013), no. 10, 2061–2071.
[27] A. Cordero, J.R. Torregrosa and P. Vindel, Bulbs of period two in the family Chebyshev-Halley type methods on quadratic polynomials, Abstr. Appl. Anal. 2013 (2013), ID 536910, 10 pages.
[28] A. Cordero, J. Garc´ıa-Maim´o, J.R. Torregrosa, M. Vassileva and P. Vindel, Chaos in King’s iterative family, Appl. Math. Lett. 26 (2013), no. 8, 842–848.
[29] A. Cordero, M. Fardi, M. Ghasemi, J.R. Torregrosa, Accelerated iterative methods for finding solutions of nonlinear equations and their dynamical behavior, Calcolo 51 (2014), 17–30.
[30] R.L. Devaney, The Mandelbrot set, the Farey tree and the Fibonacci sequence, Amer. Math. Mon., 106 (1999), no. 4, 289–302.
[31] B.I. Epureanu, H. S. Greenside, Fractal basins of attraction associated with a damped Newton’s method, SIAM REV 40 (1998), no. 1, 102–109.
[32] J.M. Guti´errez, M.A. Hern´andez and N. Romero, Dynamics of a new family of iterative processes for quadratic polynomials, J. Comput. Appl. Math. 233 (2010), 2688–2695.
[33] J.M. Guti´errez, A.A. Magre˜nan and J.L. Varona, ´ Fractal dimension of the universal Julia sets for the ChebyshevHalley family of methods, ICNAAM 2011. AIP Conf. Proc., 1389 (2011), no. 1, 1061–1064.
[34] G. Honorato, S. Plaza and N. Romero, Dynamics of a high-order family of iterative methods, J. Complexity 27 (2011), 221–229.
[35] K. Kneisl, Julia sets for the super-Newton method, Cauchy’s method and Halley’s method, Chaos 11 (2001), no. 2, 359–370.
[36] M.Y. Lee and C. Chun, Attracting periodic cycles for an optimal fourth-order nonlinear solver, Abstr. Appl. Anal. 2012 (2012), ID 263893, 8 pages.
[37] T. Lotfi et al, A variant of Steffensen–King’s type family with accelerated sixth-order convergence and high efficiency index: Dynamic study and approach, Appl. Math. Comput. 252 (2015), 347–353.
[38] A.A. Magre˜nan, ´ Estudio de la din´amica del m´etodo de Newton amortiguado (Ph.D. thesis), Servicio de Publicaciones, Universidad de La Rioja, 2013.
[39] B. Neta, M. Scott and C. Chun, Basins of attraction for several methods to find simple roots of nonlinear equations, Appl. Math. Comput. 218 (2012), 10548–10556.
[40] B. Neta, M. Scott and C. Chun, Basin attractors for various methods for multiple roots, Appl. Math. Comput. 218 (2012), no. 9, 5043–5066.
[41] C.A. Pickover, A note on chaos and Halley’s method, Commun. ACM 31 (1988), no. 11, 1326–1329.
[42] S. Plaza and N. Romero, Attracting cycles for the relaxed Newton’s method, J. Comput. Appl. Math. 235 (2011), 3238–3244.
[43] G.E. Roberts and J. Horgan-Kobelski, Newton’s versus Halley’s method: a dynamical system approach, Int. J. Bifurcat. Chaos 14 (2004), no. 10, 3459–3475.
[44] J.F. Traub, Iterative methods for resolution of equations, Prectice Hall, NJ 1964.
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Volume 14, Issue 5
May 2023Pages 17-26
- Receive Date: 04 December 2022
- Revise Date: 17 February 2023
- Accept Date: 19 February 2023