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Articles

An optimal eighth-order class of three-step weighted Newton's methods and their dynamics behind the purely imaginary extraneous fixed points

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Pages 2174-2211 | Received 01 Feb 2017, Accepted 17 Jul 2017, Published online: 29 Aug 2017

References

  • L.V. Ahlfors, Complex Analysis, McGraw-Hill, New York, 1979.
  • S. Amat, S. Busquier, and S. Plaza, Review of some iterative root-finding methods from a dynamical point of view, Scientia 10 (2004), pp. 3–35.
  • S. Amat, S. Busquier, and S. Plaza, Dynamics of the King and Jarratt iterations, Aeq. Math. 69 (2005), pp. 212–223. doi: 10.1007/s00010-004-2733-y
  • C. Andreu, N. Cambil, A. Cordero, and J.R. Torregrosa, A class of optimal eighth-order derivative-free methods for solving the Danchick-Gauss problem, Appl. Math. Comput. 232 (2014), pp. 237–246.
  • I.K. Argyros and Á.A. Magreñán, On the convergence of an optimal fourth-order family of methods and its dynamics, Appl. Math. Comput. 252 (2015), pp. 336–346.
  • A.F. Beardon, Iteration of Rational Functions, Springer, New York, 1991.
  • W. Bi, Q. Wu, and H. Ren, A new family of eighth-order iterative methods for solving nonlinear equations, Appl. Math. Comput. 214(4) (2009), pp. 236–245.
  • F. Chicharro, A. Cordero, J.M. Gutiérrez, and J.R. Torregrosa, Complex dynamics of derivative-free methods for nonlinear equations, Appl. Math. Comput. 219 (2013), pp. 7023–7035.
  • C. Chun and B. Neta, Basins of attraction for Zhou-Chen-Song fourth order family of methods for multiple roots, Math. Comput. Simul. 109 (2015), pp. 74–91. doi: 10.1016/j.matcom.2014.08.005
  • C. Chun and B. Neta, Comparison of several families of optimal eighth order methods, Appl. Math. Comput. 274 (2016), pp. 762–773.
  • C. Chun and B. Neta, Comparative study of eighth order methods for finding simple roots of nonlinear equations, Numer. Algor. 74(4) (2017), pp. 1169–1201. doi: 10.1007/s11075-016-0191-y
  • C. Chun, M.Y. Lee, B. Neta, and J. Dunić, On optimal fourth-order iterative methods free from second derivative and their dynamics, Appl. Math. Comput. 218 (2012), pp. 6427–6438.
  • C. Chun, B. Neta, J. Kozdon, and M. Scott, Choosing weight functions in iterative methods for simple roots, Appl. Math. Comput. 227 (2014), pp. 788–800.
  • A. Cordero, J.R. Torregrosa, and M.P. Vassileva, Three-step iterative methods with optimal eighth-order convergence, J. Comput. Appl. Math. 235 (2011), pp. 3189–3194. doi: 10.1016/j.cam.2011.01.004
  • A. Cordero, J. García-Maimó, J.R. Torregrosa, M.P. Vassileva, and P. Vindel, Chaos in King's iterative family, Appl. Math. Lett. 26 (2013), pp. 842–848. doi: 10.1016/j.aml.2013.03.012
  • R.L. Devaney, An Introduction to Chaotic Dynamical Systems, Addison-Wesley, Redwood City, CA, 1987.
  • J. Džunić, M.S. Petković, and L.D. Petković, A family of optimal three-point methods for solving nonlinear equations using two parametric functions, Appl. Math. Comput. 217 (2011), pp. 7612–7619.
  • Y.H. Geum and Y.I. Kim, A uniparametric family of three-step eighth-order multipoint iterative methods for simple roots, Appl. Math. Lett. 24 (2011), pp. 929–935. doi: 10.1016/j.aml.2011.01.002
  • Y.H. Geum, Y.I. Kim, and B. Neta, A class of two-point sixth-order multiple-zero finders of modified double-Newton type and their dynamics, Appl. Math. Comput. 270 (2015), pp. 387–400.
  • Y.H. Geum, Y.I. Kim, and Á.A. Magreñán, A biparametric extension of King's fourth-order methods and their dynamics, Appl. Math. Comput. 282 (2016), pp. 254–275.
  • Y.H. Geum, Y.I. Kim, and B. Neta, A sixth-order family of three-point modified Newton-like multiple-root finders and the dynamics behind their extraneous fixed points, Appl. Math. Comput. 283 (2016), pp. 120–140.
  • L. Hörmander, An Introduction to Complex Analysis in Several Variables, North-Holland, 1973.
  • P. Jarratt, Multipoint iterative methods for solving certain equations, Comput. J. 8(4) (1966), pp. 398–400. doi: 10.1093/comjnl/8.4.398
  • H.T. Kung and J.F. Traub, Optimal order of one-point and multipoint iteration, J. Assoc. Comput. Mach. 21 (1974), pp. 643–651. doi: 10.1145/321850.321860
  • S.D. Lee, Y.I. Kim, and B. Neta, An optimal family of eighth-order simple-root finders with weight functions dependent on function-to-function ratios and their dynamics underlying extraneous fixed points, J. Comput. Appl. Math. 317 (2017), pp. 31–54. doi: 10.1016/j.cam.2016.11.036
  • L. Liu and X. Wang, Eighth-order methods with high efficiency index for solving nonlinear equations, Appl. Math. Comput. 215 (2010), pp. 3449–3454.
  • Á.A. Magreñan, Different anomalies in a Jarratt family of iterative root-finding methods, Appl. Math. Comput. 233 (2014), pp. 29–38.
  • Á.A. Magreñan, A new tool to study real dynamics: The convergence plane, Appl. Math. Comput. 248 (2014), pp. 215–224.
  • B. Neta and C. Chun, Basins of attraction for several optimal fourth order methods for multiple roots, Math. Comput. Simul. 103 (2014), pp. 39–59. doi: 10.1016/j.matcom.2014.03.007
  • 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), pp. 10548–10556.
  • B. Neta, M. Scott, and C. Chun, Basin attractors for various methods for multiple roots, Appl. Math. Comput. 218 (2012), pp. 5043–5066.
  • B. Neta, C. Chun, and M. Scott, Basins of attraction for optimal eighth order methods to find simple roots of nonlinear equations, Appl. Math. Comput. 227 (2014), pp. 567–592.
  • M.S. Petković, B. Neta, L.D. Petković, and J. Džunić, Multipoint Methods for Solving Nonlinear Equations, Elsevier, New York, 2012.
  • M.S. Petković, B. Neta, L.D. Petković, and J. Džunić, Multipoint methods for solving nonlinear equations: A survey, Appl. Math. Comput. 226 (2014), pp. 635–660.
  • M. Scott, B. Neta, and C. Chun, Basin attractors for various methods, Appl. Math. Comput. 218 (2011), pp. 2584–2599.
  • B.V. Shabat, Introduction to Complex Analysis PART II, Functions of Several Variables, American Mathematical Society, Providence, 1992.
  • J.R. Sharma and H. Arora, A new family of optimal eighth order methods with dynamics for nonlinear equations, Appl. Math. Comput. 273 (2016), pp. 924–933.
  • M.R. Spiegel, Theory and Problems of Mathematical Handbook of Formulas and Tables, Schaum's Outline Series in Mathematics, McGraw-Hill, New York, 1968.
  • B.D. Stewart, Attractor basins of various root-finding methods, M.S. thesis, Naval Postgraduate School, Department of Applied Mathematics, Monterey, CA, June 2001.
  • J.F. Traub, Iterative Methods for the Solution of Equations, Chelsea Publishing, New York, 1982.
  • J.V. Uspensky, Theory of Equations, McGraw-Hill, New York, 1948.
  • E.R. Vrscay and W.J. Gilbert, Extraneous fixed points, basin boundaries and chaotic dynamics for Shröder and König rational iteration functions, Numer. Math. 52 (1988), pp. 1–16. doi: 10.1007/BF01401018
  • S. Wolfram, The Mathematica Book, 5th ed., Wolfram Media, Champaign, IL, 2003.

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