Generating Root-Finder Iterative Methods of Second Order: Convergence and Stability

In this paper, a simple family of one-point iterative schemes for approximating the solutions of nonlinear equations, by using the procedure of weight functions, is derived. The convergence analysis is presented, showing the sufficient conditions for the weight function. Many known schemes are members of this family for particular choices of the weight function. The dynamical behavior of one of these choices is presented, analyzing the stability of the fixed points and the critical points of the rational function obtained when the iterative expression is applied on low degree polynomials. Several numerical tests are given to compare different elements of the proposed family on non-polynomial problems.

[1]  Muhammad Aslam Noor,et al.  On iterative methods for nonlinear equations , 2006, Appl. Math. Comput..

[2]  R. Devaney.  An Introduction to Chaotic Dynamical Systems , 1990 .

[3]  Jeremy E. Kozdon,et al.  Choosing weight functions in iterative methods for simple roots , 2014, Appl. Math. Comput..

[4]  Sergio Amat Plata,et al.  Advances in Iterative Methods for Nonlinear Equations , 2016 .

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

[6]  S. Amat,et al.  Chaotic dynamics of a third-order Newton-type method , 2010 .

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

[8]  Young Ik Kim,et al.  An optimal family of eighth-order simple-root finders with weight functions dependent on function-to-function ratios and their dynamics underlying extraneous fixed points , 2017, J. Comput. Appl. Math..

[9]  Janak Raj Sharma,et al.  On efficient weighted-Newton methods for solving systems of nonlinear equations , 2013, Appl. Math. Comput..

[10]  Yitian Li,et al.  A family of new Newton-like methods , 2007, Appl. Math. Comput..

[11]  J. L. Varona,et al.  Graphic and numerical comparison between iterative methods , 2002 .

[12]  Alicia Cordero,et al.  Choosing the most stable members of Kou's family of iterative methods , 2018, J. Comput. Appl. Math..

[13]  Sergio Plaza Salinas,et al.  Dinámica del método de Newton , 2013 .

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

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

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