Dynamics of a fifth-order iterative method

References

• Amat , S. , Busquier , S. and Gutiérrez , J. M. 2003 . Geometric constructions of iterative functions to solve nonlinear equations . J. Comput. Appl. Math , 157 ( 1 ) : 197 – 205 .
   Web of Science ®Google Scholar
• Amat , S. , Busquier , S. and Plaza , S. 2004 . Dynamics of a family of third-order iterative methods that do not require using second derivatives . Appl. Math. Comput , 154 ( 3 ) : 735 – 746 .
   Web of Science ®Google Scholar
• Amat , S. , Busquier , S. and Plaza , S. 2005 . Dynamics of the King and Jarratt iterations . Aequationes Math , 69 ( 3 ) : 212 – 223 .
   Google Scholar
• Amat , S. , Busquier , S. and Plaza , S. 2007 . On the dynamics of a family of third-order iterative functions . ANZIAM J , 48 ( 3 ) : 343 – 359 .
   Web of Science ®Google Scholar
• Amat , S. , Bermúdez , C. , Busquier , S. and Plaza , S. 2008 . On the dynamics of the Euler iterative function . Appl. Math. Comput , 197 ( 2 ) : 725 – 732 .
   Web of Science ®Google Scholar
• Amat , S. , Busquier , S. and Plaza , S. 2010 . Chaotic dynamics of a third-order Newton-type method . J. Math. Anal. Appl , 366 : 24 – 32 .
   Web of Science ®Google Scholar
• Beardon , A. F. 1991 . Iteration of Rational Functions: Complex Analytic Dynamical Systems , New York : Springer-Verlag .
   Google Scholar
• Buff , X. and Henriksen , C. 2003 . On König's root-finding algorithms . Nonlinearity , 16 : 989 – 1015 .
   Web of Science ®Google Scholar
• Carleson , L. and Gamelin , T. 1993 . Complex Dynamics , New York : Springer-Verlag .
   Google Scholar
• Curry , J. H. , Garnett , L. and Sullivan , D. 1983 . On the iteration of a rational function: Computer experiment with Newton's method . Commun. Math. Phys , 91 : 267 – 277 .
   Web of Science ®Google Scholar
• Grau-Sánchez , M. and Gutiérrez , J. M. 2010 . Some variants of Chebychev-Halley family of methods with fifth order of convergence . Int. J. Comput. Math , 87 ( 4 ) : 818 – 833 .
   Web of Science ®Google Scholar
• Kneisl , K. 2001 . Julia sets for the super-Newton method, Cauchy's method and Halley's method . Chaos , 11 ( 2 ) : 359 – 370 .
   PubMed Web of Science ®Google Scholar
• McMullen , C. 1987 . Families of rational maps and iterative root-finding algorithms . Ann. Math , 125 : 467 – 493 .
   Web of Science ®Google Scholar
• McMullen , C. 1994 . “ Complex dynamics and renormalization ” . In Annals of Mathematics Studies , Vol. 135 , Princeton , NJ : Princeton University Press .
   Google Scholar
• Milnor , J. 2006 . Dynamics in One Complex Variable , 3 , Princeton , NJ : Princeton University Press .
   Google Scholar
• Morosawa , S. , Nishimura , Y. , Taniguchi , M. and Ueda , T. 2000 . Holomorphic Dynamics , Cambridge , , UK : Cambridge University Press .
   Google Scholar
• Plaza , S. 2001 . Conjugacy classes of some numerical methods . Proyecciones , 20 ( 1 ) : 1 – 17 .
   Google Scholar
• Smale , S. 1985 . On the efficiency of algorithms of analysis . Bull. AMS , 13 ( 2 ) : 87 – 121 .
   Web of Science ®Google Scholar
• Steinmetz , N. 1993 . Rational Iteration: Complex Analytic Dynamical Systems , Berlin : Walter de Gruyter .
   Google Scholar
• Traub , J. F. 1982 . Iterative Methods for the Solution of Equations , 2 , New York : Chelsea Publishing Company .
   Google Scholar
• Varona , J. L. 2002 . Graphic and numerical comparison between iterative methods . Math. Intelligencer , 24 ( 1 ) : 37 – 46 .
   Web of Science ®Google Scholar
• Vrscay , E. 1986 . Julia sets and Mandelbrot-like sets associated with higher order Schröder rational iteration functions: A computer assisted study . Math. Comput , 46 ( 173 ) : 151 – 169 .
   Web of Science ®Google Scholar
• Vrscay , E. and Gilbert , W. 1988 . Extraneous fixed points, basin boundary and chaotic dynamics for Schröder and König rational iteration functions . Numer. Math , 52 : 1 – 16 .
   Web of Science ®Google Scholar