Convergence, efficiency and dynamics of new fourth and sixth order families of iterative methods for nonlinear systems

In this work we present a new family of iterative methods for solving nonlinear systems that are optimal in the sense of Kung and Traub's conjecture for the unidimensional case. We generalize this family by performing a new step in the iterative method, getting a new family with order of convergence six. We study the efficiency of these families for the multidimensional case by introducing a new term in the computational cost defined by Grau-Sanchez et al. A comparison with already known methods is done by studying the dynamics of these methods in an example system.

[1]  Ali Barati,et al.  A fourth-order method from quadrature formulae to solve systems of nonlinear equations , 2007, Appl. Math. Comput..

[2]  Miguel Ángel Hernández,et al.  Dynamics of a new family of iterative processes for quadratic polynomials , 2010, J. Comput. Appl. Math..

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

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

[5]  Dierk Schleicher,et al.  Immediate and Virtual Basins of Newton’s Method for Entire Functions , 2004, math/0403336.

[6]  Alicia Cordero,et al.  A family of modified Ostrowski's methods with optimal eighth order of convergence , 2011, Appl. Math. Lett..

[7]  Alicia Cordero,et al.  Variants of Newton's Method using fifth-order quadrature formulas , 2007, Appl. Math. Comput..

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

[9]  G. Nedzhibov.  A family of multi-point iterative methods for solving systems of nonlinear equations , 2008 .

[10]  Miquel Grau-Sánchez,et al.  On the computational efficiency index and some iterative methods for solving systems of nonlinear equations , 2011, J. Comput. Appl. Math..

[11]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[12]  Alicia Cordero,et al.  Iterative methods of order four and five for systems of nonlinear equations , 2009, J. Comput. Appl. Math..

[13]  Alicia Cordero,et al.  Increasing the convergence order of an iterative method for nonlinear systems , 2012, Appl. Math. Lett..

[14]  Miodrag S. Petkovic,et al.  Construction of optimal order nonlinear solvers using inverse interpolation , 2010, Appl. Math. Comput..

[15]  Yitian Li,et al.  Modified Jarratt method with sixth-order convergence , 2009, Appl. Math. Lett..

[16]  Sergio Amat,et al.  Maximum efficiency for a family of Newton-like methods with frozen derivatives and some applications , 2013, Appl. Math. Comput..

[17]  Alicia Cordero,et al.  Increasing the order of convergence of iterative schemes for solving nonlinear systems , 2013, J. Comput. Appl. Math..

[18]  Miodrag S. Petković,et al.  A family of optimal three-point methods for solving nonlinear equations using two parametric functions , 2011, Appl. Math. Comput..

[19]  Rajni Sharma,et al.  An efficient fourth order weighted-Newton method for systems of nonlinear equations , 2012, Numerical Algorithms.

[20]  Sergio Amat,et al.  A construction of attracting periodic orbits for some classical third-order iterative methods , 2006 .