On a numerical technique for finding multiple zeros and its dynamic

Abstract An optimal method is developed for approximating the multiple zeros of a nonlinear function, when the multiplicity is known. Analysis of convergence for the proposed technique is studied to reveal the fourth-order convergence. We further investigate the dynamics of such multiple zero finders by using basins of attraction and their corresponding fractals in the complex plane. A fourth-order method will also be presented, when the multiplicity m is not known. Numerical comparisons will be made to support the underlying theory of this paper.

[1]  Beny Neta,et al.  A higher order method for multiple zeros of nonlinear functions , 1983 .

[2]  M. A. Hafiz,et al.  Solving nonsmooth equations using family of derivative-free optimal methods , 2013 .

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

[4]  Changbum Chun,et al.  Basin attractors for various methods for multiple roots , 2012, Appl. Math. Comput..

[5]  M. Frontini,et al.  Hermite interpolation and a new iterative method¶for the computation of the roots¶of non-linear equations , 2003 .

[6]  Jafar Biazar,et al.  A new third-order family of nonlinear solvers for multiple roots , 2010, Comput. Math. Appl..

[7]  E. Hansen,et al.  A family of root finding methods , 1976 .

[8]  J. Traub.  Iterative Methods for the Solution of Equations , 1982 .

[9]  Dhiman Basu,et al.  From third to fourth order variant of Newton's method for simple roots , 2008, Appl. Math. Comput..

[10]  Aurél Galántai,et al.  A study of accelerated Newton methods for multiple polynomial roots , 2010, Numerical Algorithms.

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

[12]  Fazlollah Soleymani,et al.  Construction of Optimal Derivative-Free Techniques without Memory , 2012, J. Appl. Math..

[13]  Stephen Wolfram,et al.  The Mathematica Book , 1996 .

[14]  Fazlollah Soleymani,et al.  Computing Simple Roots by an Optimal Sixteenth-Order Class , 2012, J. Appl. Math..

[15]  F. Soleymani,et al.  Some optimal iterative methods and their with memory variants , 2013 .

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

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

[18]  Fazlollah Soleymani,et al.  Finding the solution of nonlinear equations by a class of optimal methods , 2012, Comput. Math. Appl..

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

[20]  A. Galántai.  The theory of Newton's method , 2000 .

[21]  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 .

[22]  Janet Helmstedt,et al.  Graphics with Mathematica: Fractals, Julia Sets, Patterns and Natural Forms , 2004 .