A novel bi-parametric sixth order iterative scheme for solving nonlinear systems and its dynamics

Abstract In this paper, we propose a general bi-parametric family of sixth order iterative methods to solve systems of nonlinear equations. The presented scheme contains some well known existing methods as special cases. The stability of the proposed class, presented as an appendix, is used for selecting the most stable members of the family with optimum numerical performance. From the comparison with some existing methods of similar nature, it is observed that the presented methods show robust and efficient character.

[1]  Herbert H. H. Homeier.  A modified Newton method with cubic convergence: the multivariate case , 2004 .

[2]  M. Frontini,et al.  Third-order methods from quadrature formulae for solving systems of nonlinear equations , 2004, Appl. Math. Comput..

[3]  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..

[4]  Alicia Cordero,et al.  A modified Newton-Jarratt’s composition , 2010, Numerical Algorithms.

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

[6]  Muhammad Aslam Noor,et al.  Some iterative methods for solving a system of nonlinear equations , 2009, Comput. Math. Appl..

[7]  Alicia Cordero,et al.  Some new efficient multipoint iterative methods for solving nonlinear systems of equations , 2015, Int. J. Comput. Math..

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

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

[10]  Vinay Kanwar,et al.  New two-parameter Chebyshev-Halley-like family of fourth and sixth-order methods for systems of nonlinear equations , 2016, Appl. Math. Comput..

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

[12]  J. Sharma,et al.  Efficient Jarratt-like methods for solving systems of nonlinear equations , 2014 .

[13]  Rajni Sharma,et al.  An improved Newton-Traub composition for solving systems of nonlinear equations , 2016, Appl. Math. Comput..

[14]  Miodrag S. Petkovic,et al.  Remarks on "On a General Class of Multipoint Root-Finding Methods of High Computational Efficiency" , 2011, SIAM J. Numer. Anal..

[15]  Miodrag S. Petkovic,et al.  On a General Class of Multipoint Root-Finding Methods of High Computational Efficiency , 2010, SIAM J. Numer. Anal..

[16]  Alicia Cordero,et al.  Variants of Newton's method for functions of several variables , 2006, Appl. Math. Comput..

[17]  Puneet Gupta,et al.  An efficient fifth order method for solving systems of nonlinear equations , 2014, Comput. Math. Appl..

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

[19]  Á. Magreñán,et al.  A new fourth-order family for solving nonlinear problems and its dynamics , 2015, Journal of Mathematical Chemistry.

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

[21]  Ali Barati,et al.  A third-order Newton-type method to solve systems of nonlinear equations , 2007, Appl. Math. Comput..

[22]  L. Jay,et al.  A Note on Q-order of Convergence , 2001 .

[23]  D. K. R. Babajee.  On a two-parameter Chebyshev-Halley-like family of optimal two-point fourth order methods free from second derivatives , 2015 .

[24]  Rajni Sharma,et al.  New third and fourth order nonlinear solvers for computing multiple roots , 2011, Appl. Math. Comput..

[25]  Rajni Sharma,et al.  A novel family of composite Newton-Traub methods for solving systems of nonlinear equations , 2015, Appl. Math. Comput..

[26]  Mehdi Ahmadi,et al.  An efficient three-step method to solve system of nonlinear equations , 2015, Appl. Math. Comput..

[27]  Xiaoyong Xiao,et al.  A new class of methods with higher order of convergence for solving systems of nonlinear equations , 2015, Appl. Math. Comput..

[28]  Alicia Cordero,et al.  Drawing Dynamical and Parameters Planes of Iterative Families and Methods , 2013, TheScientificWorldJournal.