An Optimally Convergent Three-step Class of Derivative-free Methods

In this paper, a new optimally convergent eighth-order class of three-step without memory methods is suggested. We here pursue derivative-free algorithms, i.e., algorithms requiring only the ability to evaluate the (objective) function. Since the types of problems that these algorithms can solve are extremely diverse in nature. The analysis of convergence shows that each derivative-free method of our class requires four pieces of information per full iteration to obtain the optimal efficiency index 1.682. Numerical experiments with comparison to some existing derivative-free methods are furnished to support the underlying theory. AMS classification: 65H05 • 65B99 • 41A25

[1]  A CLASS OF NEWTON-LIKE METHODS WITH CUBIC CONVERGENCE FOR NONLINEAR EQUATIONS , 2010 .

[2]  Young Hee Geum,et al.  A family of optimal sixteenth-order multipoint methods with a linear fraction plus a trivariate polynomial as the fourth-step weighting function , 2011, Comput. Math. Appl..

[3]  M. T. Darvishi.  SOME THREE-STEP ITERATIVE METHODS FREE FROM SECOND ORDER DERIVATIVE FOR FINDING SOLUTIONS OF SYSTEMS OF NONLINEAR EQUATIONS , 2009 .

[4]  F. Soleymani,et al.  A Novel Computational Technique for Finding Simple Roots of Nonlinear Equations , 2011 .

[5]  Changbum Chun,et al.  Some fourth-order iterative methods for solving nonlinear equations , 2008, Appl. Math. Comput..

[6]  Ravi P. Agarwal,et al.  Derivative-Free Optimal Iterative Methods , 2010, Comput. Methods Appl. Math..

[7]  Rajni Sharma,et al.  Some variants of Hansen-Patrick method with third and fourth order convergence , 2009, Appl. Math. Comput..

[8]  M. Hosseini.  A Note on One -step Iteration Methods for Solving Nonlinear Equations , 2009 .

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

[10]  Ioannis K. Argyros,et al.  Sixth order derivative free family of iterative methods , 2011, Appl. Math. Comput..

[11]  F. Soleymani,et al.  New Third- and Sixth-Order Derivative-Free Techniques for Nonlinear Equations , 2011 .

[13]  J. F. Steffensen.  Remarks on iteration. , 1933 .

[14]  Iman Mansouri,et al.  Non-linear analysis of structures using two-point method , 2011 .

[15]  A. Householder.  Solution of Equations and Systems of Equations (A. M. Ostrowski) , 1967 .

[16]  Fazlollah Soleymani,et al.  Regarding the accuracy of optimal eighth-order methods , 2011, Math. Comput. Model..

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

[18]  F. Soleymani.  A Novel and Precise Sixth-Order Method for Solving Nonlinear Equations , 2011 .

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