Finding the solution of nonlinear equations by a class of optimal methods

This paper is devoted to the study of an iterative class for numerically approximating the solution of nonlinear equations. In fact, a general class of iterations using two evaluations of the first order derivative and one evaluation of the function per computing step is presented. It is also proven that the class reaches the fourth-order convergence. Therefore, the novel methods from the class are Jarratt-type iterations, which agree with the optimality hypothesis of Kung-Traub. The derived class is further extended for multiple roots. That is to say, a general optimal quartic class of iterations for multiple roots is contributed, when the multiplicity of the roots is available. Numerical experiments are employed to support the theory developed in this work.

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

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

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

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

[5]  Yongzhong Song,et al.  Constructing higher-order methods for obtaining the multiple roots of nonlinear equations , 2011, J. Comput. Appl. Math..

[6]  P. Jarratt.  Some efficient fourth order multipoint methods for solving equations , 1969 .

[7]  Rajni Sharma,et al.  Modified Jarratt method for computing multiple roots , 2010, Appl. Math. Comput..

[8]  Fazlollah Soleymani.  NEW OPTIMAL ITERATIVE METHODS IN SOLVING NONLINEAR EQUATIONS , 2011 .

[9]  José Luis Díaz-Barrero,et al.  Zero-finder methods derived using Runge-Kutta techniques , 2011, Appl. Math. Comput..

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

[11]  Miodrag S. Petkovic,et al.  A note on some recent methods for solving nonlinear equations , 2007, Appl. Math. Comput..

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

[13]  Vejdi I. Hasanov,et al.  On some families of multi-point iterative methods for solving nonlinear equations , 2006, Numerical Algorithms.

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

[15]  Fazlollah Soleymani,et al.  An Improvement of Ostrowski’s and King’s Techniques with Optimal Convergence Order Eight , 2012, J. Optim. Theory Appl..