An efficient twelfth-order iterative method for finding all the solutions of nonlinear equations

In this study, the sixteenth-order iterative scheme of Li et al. [X. Li, C. Mu, J. Ma, C. Wang, Sixteenth-order method for nonlinear equations, Appl. Math. Com. 215 2010, 3754--3758] is considered. We increase its efficiency index from 1.587 to 1.644, by reducing the number of evaluations from six to five per iteration. This goal is achieved by providing an approximation for the first-order derivative of the function in the fourth step. Error analysis will also be studied. In the sequel, some numerical instances are given to show the accuracy of the new obtained twelfth-order technique. Therein, another objective of this paper is achieved by proposing a hybrid method for finding all the real solutions of nonlinear equations.

[1]  Miodrag S. Petkovic,et al.  A class of three-point root-solvers of optimal order of convergence , 2010, Appl. Math. Comput..

[2]  Yitian Li,et al.  A variant of Jarratt method with sixth-order convergence , 2008, Appl. Math. Comput..

[3]  Fazlollah Soleymani,et al.  Two new classes of optimal Jarratt-type fourth-order methods , 2012, Appl. Math. Lett..

[4]  R. Baker Kearfott,et al.  Introduction to Interval Analysis , 2009 .

[5]  Jinwen Ma,et al.  Sixteenth-order method for nonlinear equations , 2010, Appl. Math. Comput..

[6]  J. Miller.  Numerical Analysis , 1966, Nature.

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

[9]  Fazlollah Soleymani,et al.  Optimal Steffensen-type methods with eighth order of convergence , 2011, Comput. Math. Appl..

[10]  Stan Wagon.  Mathematica in action , 1991 .

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

[12]  F Soleymani.  ON A NOVEL OPTIMAL QUARTICALLY CLASS OF METHODS , 2011 .

[13]  Fazlollah Soleymani,et al.  Accurate fourteenth-order methods for solving nonlinear equations , 2011, Numerical Algorithms.

[14]  Fazlollah Soleymani,et al.  An optimized derivative-free form of the Potra-Pták method , 2012, Math. Comput. Model..

[15]  Roozbeh Hazrat.  Mathematica: A Problem-Centered Approach , 2010 .

[16]  Qingbiao Wu,et al.  New variants of Jarratt’s method with sixth-order convergence , 2009, Numerical Algorithms.

[17]  Beny Neta,et al.  High order nonlinear solver , 2008, J. Comput. Methods Sci. Eng..

[18]  J. Keiper.  Interval Arithmetic in Mathematica , 1993 .

[19]  Fazlollah Soleymani.  Revisit of Jarratt method for solving nonlinear equations , 2010, Numerical Algorithms.

[20]  F. Soleymani.  ON A BI-PARAMETRIC CLASS OF OPTIMAL EIGHTH-ORDER DERIVATIVE-FREE METHODS , 2011 .

[21]  F. Soleymani.  Optimal fourth-order iterative method free from derivative , 2011 .

[22]  Fazlollah Soleymani,et al.  Some modifications of King's family with optimal eighth order of convergence , 2012, Math. Comput. Model..