On the stability of a two-step method for a fourth-degree family by computer designs along with applications

Document Type : Research Paper


1 Department of Mathematics, Hamedan Branch, Islamic Azad University, Hamedan, Iran

2 Young Researchers and Elite Club, Shahr-e-Qods Branch, Islamic Azad University, Tehran, Iran

3 Farhangian University, Tehran, Iran


In this paper, some important features of Traub's method are studied: Analysis of the stability behavior, obtaining the 4th root of a matrix, semi-local convergence, and local convergence. The stability of Traub's method is studied by using the dynamic behavior of a family of 4th-degree polynomials. The obtained equations are very complex and do not solve with the software. Therefore, we find the results by plotting diagrams and pictures, and then we show the very stable behavior of Traub's method. Then Traub's method is extended to a matrix iterative method for calculating the 4th root of a square matrix. We also present the local and semi-local convergence of the method based on the divided differences, and therefore, the benefits of our approach are more precise error estimation in semi-local convergence and a large ball of convergence in local convergence. We confirm our theoretical results by some numerical examples such as the nonlinear integral equation of mixed Hammerstein type.


[1] S. Amat, C. Berm´udez, M.A. Hern´andez-Ver´on, and E. Mart´─▒nez, On an efficient k-step iterative method for nonlinear equations, J. Comput. Appl. Math. 302 (2016), 258–271.
[2] S. Amat, S. Busquier, and S. Plaza, Review of some iterative root-finding methods from a dynamical point of view, Scientia 10 (2004), no. 3, 35.
[3] , Chaotic dynamics of a third-order Newton-type method, J. Math. Anal. Appl. 366 (2010), no. 1, 24–32.
[4] S. Amat, J.A. Ezquerro, and M.A. Hern´andez-Ver´on, On a new family of high-order iterative methods for the matrix pth root, Numer. Linear Alg. Appl. 22 (2015), no. 4, 585–595.
[5] S. Amat, M.A. Hern´andez, and N. Romero, Semilocal convergence of a sixth order iterative method for quadratic equations, Appl. Numer. Math. 62 (2012), no. 7, 833–841.
[6] S. Amat, A.A. Magre˜n´an, and N. Romero, ´ On a two-step relaxed Newton-type method, Appl. Math. Comput. 219 (2013), no. 24, 11341–11347.
[7] I.K. Argyros, R. Behl, and S.S. Motsa, Unifying semilocal and local convergence of Newton’s method on Banach space with a convergence structure, Appl. Numer. Math. 115 (2017), 225–234.
[8] I.K. Argyros, Y.J. Cho, and S. Hilout, On the midpoint method for solving equations, Appl. Math. Comput. 216 (2010), no. 8, 2321–2332.
[9] I.K. Argyros, A. Cordero, A.A. Magre˜n´an, and J.R. Torregrosa, ´ Third-degree anomalies of Traub’s method, J. Comput. Appl. Math. 309 (2017), 511–521.
[10] I.K. Argyros and S. Hilout, Extending the Newton–Kantorovich hypothesis for solving equations, J. Comput. Appl. Math. 234 (2010), no. 10, 2993–3006.
[11] , Weaker convergence conditions for the Secant method, Appl. Math. 59 (2014), no. 3, 265–284.
[12] P. Bakhtiari, A. Cordero, T. Lotfi, K. Mahdiani, and J.R. Torregrosa, Widening basins of attraction of optimal iterative methods, Nonlinear Dyn. 87 (2017), no. 2, 913–938.
[13] R. Behl, S. Amat, A.A. Magre˜n´an, and S.S. Motsa, ´ An efficient optimal family of sixteenth order methods for nonlinear models, J. Comput. Appl. Math. 354 (2019), 271–285.
[14] R. Behl, A. Cordero, S.S. Motsa, and J.R. Torregrosa, Stable high-order iterative methods for solving nonlinear models, Appl. Math. Comput. 303 (2017), 70–88.
[15] N.J. Bini, D.A.and Higham and B. Meini, Algorithms for the matrix pth root, Numer. Algorithms 39 (2005), no. 4, 349–378.
[16] P. Blanchard, Complex analytic dynamics on the Riemann sphere, Bull. Amer. Math. Soc. 11 (1984), no. 1, 85–141.
[17] J. Chen, I.K. Argyros, and R.P. Agarwal, Majorizing functions and two-point Newton-type methods, J. Comput. Appl. Math. 234 (2010), no. 5, 1473–1484.
[18] A. Cordero, F. Soleymani, J.R. Torregrosa, and F.K. Haghani, A family of Kurchatov-type methods and its stability, Appl. Math. Comput. 294 (2017), 264–279.
[19] A. Cordero, F. Soleymani, J.R. Torregrosa, and M.Z. Ullah, Numerically stable improved Chebyshev–Halley type schemes for matrix sign function, J. Comput. Appl. Math. 318 (2017), 189–198.
[20] A. Cordero and J.R. Torregrosa, A sixth-order iterative method for approximating the polar decomposition of an arbitrary matrix, J. Comput. Appl. Math. 318 (2017), 591–598.
[21] J.A. Ezquerro, M.A. Hern´andez, and N. Romero, Newton-type methods of high order and domains of semilocal and global convergence, Appl. Math. Comput. 214 (2009), no. 1, 142–154.
[22] J.A. Ezquerro, M.A. Hern´andez, and M.A. Salanova, Recurrence relations for the midpoint method, Tamkang J. Math. 31 (2000), no. 1, 33–42.
[23] P. Fatou, Sur les ´equations fonctionnelles, Bull. Soc. Math. France 47 (1919), no. 48, 1920.
[24] C.-H. Guo, On Newton’s method and Halley’s method for the principal pth root of a matrix, Linear Alg. Appl. 432 (2010), no. 8, 1905–1922.
[25] N.J. Higham, Functions of matrices: theory and computation, vol. 104, Siam, 2008.
[26] B. Iannazzo, On the Newton method for the matrix pth root, SIAM J. Matrix Anal. Appl. 28 (2006), no. 2, 503–523.
[27] G. Julia, Memoire sur l’iteration des fonctions rationnelles, J. Math. Pures Appl. 8 (1918), 47–245.
[28] L.V. Kantorovich, Functional analysis and applied mathematics, Uspekhi Mate. Nauk 3 (1948), no. 6, 89–185.
[29] S.K. Khattri, How to increase convergence order of the Newton method to 2×m?, Appl. Math. 59 (2014), no. 1,15–24.
[30] T. Liu, X. Qin, and P. Wang, Local convergence of a family of iterative methods with sixth and seventh order convergence under weak conditions, Int. J. Comput. Meth. 16 (2019), no. 08, 1850120.
[31] A.A. Magrenan Ruiz and I.K. Argyros, Two-step Newton methods, J. Complexity 30 (2014), no. 4, 533–553.
[32] M. Moccari and T. Lotfi, On a two-step optimal Steffensen-type method: Relaxed local and semi-local convergence analysis and dynamical stability, J. Math. Anal. Appl. 468 (2018), no. 1, 240–269.
[33] , Using majorizing sequences for the semi-local convergence of a high-order and multipoint iterative method along with stability analysis, J. Math. Exten. 15 (2020).
[34] P.J. Psarrakos, On the mth roots of a complex matrix, Electron. J. Linear Alg. 9 (2002), 32–41.
[35] J.R. Sharma, D. Kumar, I.K. Argyros, and A.A. Magre˜n´an, ´ On a bi-parametric family of fourth order composite newton–jarratt methods for nonlinear systems, Mathematics 7 (2019), no. 6, 492.
[36] W.T. Shaw, Complex analysis with Mathematica, Cambridge University Press, 2006.
[37] J.F. Traub, Iterative methods for the solution of equations, vol. 312, American Mathematical Soc., 1982.
Volume 14, Issue 4
April 2023
Pages 261-282
  • Receive Date: 09 January 2022
  • Accept Date: 26 April 2022