Computers in mathematical research: the study of three-point root-finding methods

This paper is motivated by results of extensive comparative study of three-point methods for solving nonlinear equations presented in the paper (C. Chun, B. Neta, Comparative study of eighth-order methods for finding simple roots of nonlinear equations, Numer. Algor. 74 , 1169–1201, 2017 ). In the first part, we show that the best ranking method, constructed by Sharma and Arora in 2016, is actually a special case of the family of three-point methods presented in the paper (J. Džunić, M. S. Petković, L. D. Petković, A family of optimal three-point methods for solving nonlinear equations using two parametric functions, Appl. Math. Comput. 217 , 7612–7619, 2011 ). Since the mentioned Chun-Neta’s comparative study was carried out by testing algebraic polynomials of low degree (not higher than 6), in the second part we have extended comparative study to higher-order polynomials, polynomials with random coefficients, and polynomials with clusters of zeros. The performed experiments confirmed results of Chun and Neta. As in the cited paper of Chun and Neta, we have also applied the quality study based on computer visualization plotting the basins of attraction.

[1]  Miquel Grau-Sánchez,et al.  Ostrowski type methods for solving systems of nonlinear equations , 2011, Appl. Math. Comput..

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

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

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

[5]  Xia Wang,et al.  Modified Ostrowski's method with eighth-order convergence and high efficiency index , 2010, Appl. Math. Lett..

[6]  Wentao Huang,et al.  Linearizability and local bifurcation of critical periods in a cubic Kolmogorov system , 2013, J. Comput. Appl. Math..

[7]  Changbum Chun,et al.  An Analysis of a King-based Family of Optimal Eighth-order Methods , 2015 .

[8]  Young Hee Geum,et al.  A uniparametric family of three-step eighth-order multipoint iterative methods for simple roots , 2011, Appl. Math. Lett..

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

[10]  Xiuhua Wang,et al.  Some eighth-order root-finding three-step methods , 2010 .

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

[12]  Puneet Gupta,et al.  Improved King's methods with optimal order of convergence based on rational approximations , 2013, Appl. Math. Lett..

[13]  B. Kalantari.  Polynomial Root-finding and Polynomiography , 2008 .

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

[15]  Janak Raj Sharma,et al.  A new family of optimal eighth order methods with dynamics for nonlinear equations , 2016, Appl. Math. Comput..

[16]  M. Petkovic,et al.  Two-point methods , 2013 .

[17]  A. Ostrowski.  Solution of equations in Euclidean and Banach spaces , 1973 .

[18]  Changbum Chun,et al.  On the new family of optimal eighth order methods developed by Lotfi et al. , 2015, Numerical Algorithms.

[19]  Janak Raj Sharma,et al.  An efficient family of weighted-Newton methods with optimal eighth order convergence , 2014, Appl. Math. Lett..

[20]  Changbum Chun,et al.  Corrigendum to "Basins of attraction for optimal eighth-order methods to find simple roots of nonlinear equations" , 2014, Appl. Math. Comput..

[21]  Changbum Chun,et al.  Comparative study of eighth-order methods for finding simple roots of nonlinear equations , 2017, Numerical Algorithms.

[22]  Miodrag S. Petkovic,et al.  Multipoint methods for solving nonlinear equations: A survey , 2014, Appl. Math. Comput..

[23]  Changbum Chun,et al.  Some fourth-order iterative methods for solving nonlinear equations , 2008, Appl. Math. Comput..

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

[25]  Changbum Chun,et al.  An analysis of a new family of eighth-order optimal methods , 2014, Appl. Math. Comput..

[26]  Changbum Chun,et al.  Comparison of several families of optimal eighth order methods , 2016, Appl. Math. Comput..

[27]  Beny Neta,et al.  On a family of multipoint methods for non-linear equations , 1981 .

[28]  P. Jarratt.  Some fourth order multipoint iterative methods for solving equations , 1966 .

[29]  Miodrag S. Petković,et al.  Families of optimal multipoint methods for solving nonlinear equations: A survey , 2010 .