An analysis of a Khattri's 4th order family of methods

In this paper we analyze an optimal fourth-order family of methods suggested by Khattri and Babajee, (2013). We analyze the family using the information on the extraneous fixed points. Two measures of closeness to the imaginary axis of the set of extraneous points are considered and applied to the members of the family to find its best performer. The results are compared to three best members of King's family of methods.

[1]  S. Amat,et al.  Review of some iterative root-finding methods from a dynamical point of view , 2004 .

[2]  Changbum Chun,et al.  On optimal fourth-order iterative methods free from second derivative and their dynamics , 2012, Appl. Math. Comput..

[3]  Jeremy E. Kozdon,et al.  Choosing weight functions in iterative methods for simple roots , 2014, Appl. Math. Comput..

[4]  Changbum Chun,et al.  Basins of attraction for Zhou-Chen-Song fourth order family of methods for multiple roots , 2015, Math. Comput. Simul..

[5]  Alicia Cordero,et al.  Chaos in King's iterative family , 2013, Appl. Math. Lett..

[6]  Changbum Chun,et al.  Corrigendum to "Basins of attraction for optimal eighth-order methods to find simple roots of nonlinear equations" , 2014, Appl. Math. Comput..

[7]  H Wozniakowski,et al.  MAXIMAL ORDER OF MULTIPOINT ITERATIONS USING n EVALUATIONS , 1976 .

[8]  Gerald S. Shedler,et al.  Parallel numerical methods for the solution of equations , 1967, CACM.

[9]  Beny Neta,et al.  On a family of multipoint methods for non-linear equations , 1981 .

[10]  B. D. Stewart.  Attractor basins of various root-finding methods , 2001 .

[11]  Changbum Chun,et al.  Basin attractors for various methods for multiple roots , 2012, Appl. Math. Comput..

[12]  Ángel Alberto Magreñán,et al.  Different anomalies in a Jarratt family of iterative root-finding methods , 2014, Appl. Math. Comput..

[13]  Beny Neta,et al.  A sixth-order family of methods for nonlinear equations , 1979 .

[14]  E. Vrscay,et al.  Extraneous fixed points, basin boundaries and chaotic dynamics for Schröder and König rational iteration functions , 1987 .

[15]  S. Amat,et al.  Dynamics of the King and Jarratt iterations , 2005 .

[16]  Bahman Kalantari,et al.  On Extraneous Fixed-Points of the Basic Family of Iteration Functions , 2003 .

[17]  Changbum Chun,et al.  Basins of attraction for several optimal fourth order methods for multiple roots , 2014, Math. Comput. Simul..

[18]  Miodrag S. Petkovic,et al.  A family of three-point methods of optimal order for solving nonlinear equations , 2010, J. Comput. Appl. Math..

[19]  Alicia Cordero,et al.  Complex dynamics of derivative-free methods for nonlinear equations , 2013, Appl. Math. Comput..

[20]  Alicia Cordero,et al.  Real qualitative behavior of a fourth-order family of iterative methods by using the convergence plane , 2014, Math. Comput. Simul..

[21]  Changbum Chun,et al.  ON JARRATT'S FAMILY OF OPTIMAL FOURTH-ORDER ITERATIVE METHODS AND THEIR DYNAMICS , 2014 .

[22]  J. Traub.  Iterative Methods for the Solution of Equations , 1982 .

[23]  Changbum Chun,et al.  Basins of attraction for several methods to find simple roots of nonlinear equations , 2012, Appl. Math. Comput..

[24]  Beny Neta,et al.  Multipoint Methods for Solving Nonlinear Equations , 2012 .

[25]  Changbum Chun,et al.  Basin attractors for various methods , 2011, Appl. Math. Comput..

[26]  Sergio Amat,et al.  Dynamics of a family of third-order iterative methods that do not require using second derivatives , 2004, Appl. Math. Comput..

[27]  H. T. Kung,et al.  Optimal Order of One-Point and Multipoint Iteration , 1974, JACM.

[28]  Xia Wang,et al.  Modified Ostrowski's method with eighth-order convergence and high efficiency index , 2010, Appl. Math. Lett..

[29]  R. F. King.  A Family of Fourth Order Methods for Nonlinear Equations , 1973 .

[30]  Changbum Chun,et al.  On a family of Laguerre methods to find multiple roots of nonlinear equations , 2013, Appl. Math. Comput..