A new family of methods for single and multiple roots

Abstract We present a new family of iterative methods for multiple and single roots of nonlinear equations. This family contains as a special case the authors’ family for finding simple roots from Herceg and Herceg (2015). Some well-known classical methods for simple roots, for example Newton, Potra–Ptak, Newton–Steffensen, King and Ostrowski’s methods, belong to this family, which implies that our new family contains modifications of these methods suitable for finding multiple roots. Convergence analysis shows that our family contains methods of convergence order from 2 to 4. The new methods require two function evaluations and one evaluation of the first derivative per iteration, so all our fourth order methods are optimal in terms of the Kung and Traub conjecture. Several examples are presented and compared. Through various test equations, relevant numerical experiments strongly support the claimed theory in this paper. Extraneous fixed points of the iterative maps associated with the proposed methods are also investigated. Their dynamics is explored along with illustrated basins of attraction for various polynomials.

[1]  R. F. King.  A Family of Fourth Order Methods for Nonlinear Equations , 1973 .

[2]  Baoqing Liu,et al.  A new family of fourth-order methods for multiple roots of nonlinear equations , 2013 .

[3]  Jisheng Kou.  Some variants of Cauchy's method with accelerated fourth-order convergence , 2008 .

[4]  Yitian Li,et al.  Fourth-order iterative methods free from second derivative , 2007, Appl. Math. Comput..

[5]  J. R. Sharma,et al.  A composite third order Newton-Steffensen method for solving nonlinear equations , 2005, Appl. Math. Comput..

[6]  Xin Chen,et al.  Families of third and fourth order methods for multiple roots of nonlinear equations , 2013, Appl. Math. Comput..

[7]  Changbum Chun,et al.  Basins of attraction for several optimal fourth order methods for multiple roots , 2014, Math. Comput. Simul..

[8]  Changbum Chun,et al.  New families of nonlinear third-order solvers for finding multiple roots , 2009, Comput. Math. Appl..

[9]  Jisheng Kou.  Second-derivative-free variants of Cauchy's method , 2007, Appl. Math. Comput..

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

[11]  E. Vrscay,et al.  Extraneous fixed points, basin boundaries and chaotic dynamics for Schröder and König rational iteration functions , 1987 .

[12]  Beny Neta,et al.  Author's Personal Copy Computers and Mathematics with Applications Some Fourth-order Nonlinear Solvers with Closed Formulae for Multiple Roots , 2022 .

[13]  R. Thukral.  A new family of multi-point iterative methods for finding multiple roots of nonlinear equations , 2014 .

[14]  A. Serghini,et al.  A general family of third order method for finding multiple roots , 2014, Appl. Math. Comput..

[15]  Djordje Herceg,et al.  A family of methods for solving nonlinear equations , 2015, Appl. Math. Comput..

[16]  Ernst Schröder,et al.  Ueber unendlich viele Algorithmen zur Auflösung der Gleichungen , 1870 .

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

[18]  Chen Dong,et al.  A family of multiopoint iterative functions for finding multiple roots of equations , 1987 .

[19]  Jisheng Kou.  Fourth-order variants of Cauchy's method for solving non-linear equations , 2007, Appl. Math. Comput..

[20]  A. Ostrowski.  Solution of equations and systems of equations , 1967 .

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

[22]  Changbum Chun,et al.  On a family of Laguerre methods to find multiple roots of nonlinear equations , 2013, Appl. Math. Comput..

[23]  Beny Neta,et al.  High-order nonlinear solver for multiple roots , 2008, Comput. Math. Appl..

[24]  Changbum Chun,et al.  Basins of attraction for Zhou-Chen-Song fourth order family of methods for multiple roots , 2015, Math. Comput. Simul..

[25]  Changbum Chun,et al.  Basins of attraction for several third order methods to find multiple roots of nonlinear equations , 2015, Appl. Math. Comput..