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Title: On the local convergence of Kung-Traub's two-point method and its dynamics (English)
Author: Ataei Delshad, Parandoosh
Author: Lotfi, Taher
Language: English
Journal: Applications of Mathematics
ISSN: 0862-7940 (print)
ISSN: 1572-9109 (online)
Volume: 65
Issue: 4
Year: 2020
Pages: 379-406
Summary lang: English
Category: math
Summary: In this paper, the local convergence analysis of the family of Kung-Traub's two-point method and the convergence ball for this family are obtained and the dynamical behavior on quadratic and cubic polynomials of the resulting family is studied. We use complex dynamic tools to analyze their stability and show that the region of stable members of this family is vast. Numerical examples are also presented in this study. This method is compared with several widely used solution methods by solving test problems from different chemical engineering application areas, e.g. Planck's radiation law problem, natch distillation at infinite reflux, van der Waal's equation, air gap between two parallel plates and flow in a smooth pipe, in order to check the applicability and effectiveness of our proposed methods. (English)
Keyword: local convergence
Keyword: Kung-Traub's method
Keyword: complex dynamics
Keyword: parameter space
Keyword: basins of attraction
Keyword: stability
MSC: 37Fxx
MSC: 37P40
MSC: 65F10
MSC: 65H04
idZBL: 07250668
idMR: MR4134140
DOI: 10.21136/AM.2020.0322-18
Date available: 2020-09-07T09:45:14Z
Last updated: 2020-11-18
Stable URL:
Reference: [1] Ahmad, F., Soleymani, F., Haghani, F. Khaksar, Serra-Capizzano, S.: Higher order \hbox{derivative}-free iterative methods with and without memory for systems of nonlinear equations.Appl. Math. Comput. 314 (2017), 199-211. Zbl 1426.65071, MR MR3683867, 10.1016/j.amc.2017.07.012
Reference: [2] Ahmad, F., Tohidi, E., Ullah, M. Z., Carrasco, J. A.: Higher order multi-step Jarratt-like method for solving systems of nonlinear equations: Application to PDEs and ODEs.Comput. Math. Appl. 70 (2015), 624-636. MR 3372047, 10.1016/j.camwa.2015.05.012
Reference: [3] Amat, S., Busquier, S., Plaza, S.: Review of some iterative root-finding methods from a dynamical point of view.Sci., Ser. A, Math. Sci. (N.S.) 10 (2004), 3-35. Zbl 1137.37316, MR 2127479
Reference: [4] Amat, S., Busquier, S., Plaza, S.: A construction of attracting periodic orbits for some classical third-order iterative methods.J. Comput. Appl. Math. 189 (2006), 22-33. Zbl 1113.65047, MR 2202961, 10.1016/
Reference: [5] Argyros, I. K.: A unifying local-semilocal convergence analysis and applications for two-point Newton-like methods in Banach space.J. Math. Anal. Appl. 298 (2004), 374-397. Zbl 1057.65029, MR 2086964, 10.1016/j.jmaa.2004.04.008
Reference: [6] Argyros, I. K.: Computational Theory of Iterative Methods.Studies in Computational Mathematics 15. Elservier, Amsterdam (2007). Zbl 1147.65313, MR 2356038, 10.1016/S1570-579X(13)60006-3
Reference: [7] Argyros, I. K.: Convergence and Applications of Newton-Type Iterations.Springer, New York (2008). Zbl 1153.65057, MR 2428779, 10.1007/978-0-387-72743-1
Reference: [8] Argyros, I. K., Cordero, A., Magreñán, Á. A., Torregrosa, J. R.: Third-degree anomalies of Traub's method.J. Comput. Appl. Math. 309 (2017), 511-521. Zbl 06626266, MR 3539801, 10.1016/
Reference: [9] Argyros, I. K., Hilout, S.: An improved local convergence analysis for a two-step Steffensen-type method.J. Comput. Appl. Math. 30 (2009), 237-245. Zbl 1180.65067, MR 2496614, 10.1007/s12190-008-0169-6
Reference: [10] Argyros, I. K., Hilout, S.: Computational Methods in Nonlinear Analysis: Efficient Algorithms, Fixed Point Theory and Applications.World Scientific, Hackensack (2013). Zbl 1279.65062, MR 3134688, 10.1142/8475
Reference: [11] Argyros, I. K., Kansal, M., Kanwar, V.: Ball convergence for two optimal eighth-order methods using only the first derivative.Int. J. Appl. Comput. Math. 3 (2017), 2291-2301. Zbl 1397.65071, MR 3680702, 10.1007/s40819-016-0196-1
Reference: [12] Argyros, I. K., Kansal, M., Kanwar, V., Bajaj, S.: Higher-order derivative-free families of Chebyshev-Halley type methods with or without memory for solving nonlinear equations.Appl. Math. Comput. 315 (2017), 224-245. Zbl 1426.65064, MR 3693467, 10.1016/j.amc.2017.07.051
Reference: [13] Argyros, I. K., Magreñán, Á. A., Orcos, L.: Local convergence and a chemical application of derivative free root finding methods with one parameter based on interpolation.J. Math. Chem. 54 (2016), 1404-1416 \99999DOI99999 10.1007/s10910- 016-0605-z. Zbl 1360.65141, MR 3516898, 10.1007/s10910-016-0605-z
Reference: [14] Argyros, I. K., Ren, H.: On an improved local convergence analysis for the Secant method.Numer. Algorithms 52 (2009), 257-271. Zbl 1176.65068, MR 2563704, 10.1007/s11075-009-9271-6
Reference: [15] Beardon, A. F.: Iteration of Rational Functions: Complex Analytic Dynamical Systems.Graduate Texts in Mathematics 132. Springer, New York (1991). Zbl 0742.30002, MR 1128089, 10.1007/978-1-4612-4422-6
Reference: [16] Behl, R., Cordero, A., Motsa, S. S., Torregrosa, J. R.: An eighth-order family of optimal multiple root finders and its dynamics.Numer. Algorithms 77 (2018), 1249-1272. Zbl 1402.65042, MR 3779086, 10.1007/s11075-017-0361-6
Reference: [17] Chicharro, F. I., Cordero, A., Torregrosa, J. R.: Drawing dynamical and parameters planes of iterative families and methods.Sci. World J. 2013 (2013), Article ID 780153, 11 pages. 10.1155/2013/780153
Reference: [18] Chun, C.: Some variants of King's fourth-order family of methods for nonlinear equations.Appl. Math. Comput. 290 (2007), 57-62. Zbl 1122.65328, MR 2335430, 10.1016/j.amc.2007.01.006
Reference: [19] Chun, C., Lee, M. Y., Neta, B., Džunić, J.: On optimal fourth-order iterative methods free from second derivative and their dynamics.Appl. Math. Comput. 218 (2012), 6427-6438. Zbl 1277.65031, MR 2879123, 10.1016/j.amc.2011.12.013
Reference: [20] Cordero, A., Feng, L., Magreñán, Á. A., Torregrosa, J. R.: A new fourth-order family for solving nonlinear problems and its dynamics.J. Math. Chem. 53 (2015), 893-910. Zbl 1318.65028, MR 3311927, 10.1007/s10910-014-0464-4
Reference: [21] Cordero, A., García-Maimó, J., Torregrosa, J. R., Vassileva, M. P., Vindel, P.: Chaos in King's iterative family.Appl. Math. Lett. 26 (2013), 842-848. Zbl 1370.37155, MR 3066701, 10.1016/j.aml.2013.03.012
Reference: [22] Cordero, A., Guasp, L., Torregrosa, J. R.: CMMSE 2017: On two classes of fourth- and seventh-order vectorial methods with stable behavior.J. Math. Chem. 56 (2018), 1902-1923. Zbl 1407.65052, MR 3825965, 10.1007/s10910-017-0814-0
Reference: [23] Cordero, A., Lotfi, T., Mahdiani, K., Torregrosa, J. R.: Two optimal general classes of iterative methods with eighth-order.Acta Appl. Math. 134 (2014), 61-74. Zbl 1305.65142, MR 3273685, 10.1007/s10440-014-9869-0
Reference: [24] Cordero, A., Lotfi, T., Mahdiani, K., Torregrosa, J. R.: A stable family with high order of convergence for solving nonlinear equations.Appl. Math. Comput. 254 (2015), 240-251. Zbl 1410.65154, MR 3314451, 10.1016/j.amc.2014.12.141
Reference: [25] Cordero, A., Lotfi, T., Torregrosa, J. R., Assari, P., Mahdiani, K.: Some new bi-accelerator two-point methods for solving nonlinear equations.Comput. Appl. Math. 35 (2016), 251-267. Zbl 1342.65126, MR 3489900, 10.1007/s40314-014-0192-1
Reference: [26] Cordero, A., Soleymani, F., Torregrosa, J. R., Haghani, F. Khaksar: A family of Kurchatov-type methods and its stability.Appl. Math. Comput. 294 (2017), 264-279. Zbl 1411.65071, MR 3558276, 10.1016/j.amc.2016.09.021
Reference: [27] Fatou, P.: Sur les équations fonctionelles.Bull. Soc. Math. Fr. 47 (1919), 161-271 French \99999JFM99999 47.0921.02. MR 1504787, 10.24033/bsmf.998
Reference: [28] Fatou, P.: Sur les équations fonctionelles.Bull. Soc. Math. Fr. 48 (1920), 208-314 French \99999JFM99999 47.0921.02. MR 1504797, 10.24033/bsmf.1008
Reference: [29] Gutiérrez, J. M., Hernández, M. A., Romero, N.: Dynamics of a new family of iterative processes for quadratic polynomials.J. Comput. Appl. Math. 233 (2010), 2688-2695. Zbl 1201.65071, MR 2577854, 10.1016/
Reference: [30] Jarratt, P.: Some fourth order multipoint iterative methods for solving equations.Math. Comput. 20 (1966), 434-437. Zbl 0229.65049, MR 0191085, 10.1090/S0025-5718-66-99924-8
Reference: [31] Jay, L. O.: A note on $Q$-order of convergence.BIT 41 (2001), 422-429. Zbl 0973.40001, MR 1837404, 10.1023/A:1021902825707
Reference: [32] Julia, G.: Mémoire sur l'itération des fonctions rationnelles.Journ. de Math. 8 (1918), 47-245 French \99999JFM99999 46.0520.06.
Reference: [33] King, R. F.: A family of fourth order methods for nonlinear equations.SIAM J. Numer. Anal. 10 (1973), 876-879. Zbl 0266.65040, MR 0343585, 10.1137/0710072
Reference: [34] Kung, H. T., Traub, J. F.: Optimal order of one-point and multipoint iteration.J. Assoc. Comput. Mach. 21 (1974), 643-651. Zbl 0289.65023, MR 0353657, 10.1145/321850.321860
Reference: [35] Li, D., Liu, P., Kou, J.: An improvement of Chebyshev-Halley methods free from second derivative.Appl. Math. Comput. 235 (2014), 221-225. Zbl 1334.65086, MR 3194598, 10.1016/j.amc.2014.02.083
Reference: [36] Lotfi, T., Magreñán, Á. A., Mahdiani, K., Rainer, J. Javier: A variant of Steffensen-King's type family with accelerated sixth-order convergence and high efficiency index: Dynamic study and approach.Appl. Math. Comput. 252 (2015), 347-353. Zbl 1338.65130, MR 3305113, 10.1016/j.amc.2014.12.033
Reference: [37] Lotfi, T., Soleymani, F., Ghorbanzadeh, M., Assari, P.: On the construction of some tri-parametric iterative methods with memory.Numer. Algorithms 70 (2015), 835-845. Zbl 1337.65040, MR 3428683, 10.1007/s11075-015-9976-7
Reference: [38] Magreñán, Á. A.: Different anomalies in a Jarratt family of iterative root-finding methods.Appl. Math. Comput. 233 (2014), 29-38. Zbl 1334.65083, MR 3214960, 10.1016/j.amc.2014.01.037
Reference: [39] Maheshwari, A. K.: A fourth order iterative method for solving nonlinear equations.Appl. Math. Comput. 211 (2009), 383-391. Zbl 1162.65346, MR 2524167, 10.1016/j.amc.2009.01.047
Reference: [40] Neta, B., Chun, C., Scott, M.: Basins of attraction for optimal eighth order methods to find simple roots of nonlinear equations.Appl. Math. Comput. 227 (2014), 567-592. Zbl 1364.65110, MR 3146342, 10.1016/j.amc.2013.11.017
Reference: [41] Ostrowski, A. M.: Solutions of Equations and System of Equations.Pure and Applied Mathematics 9. Academic Press, New York (1966). Zbl 0222.65070, MR 0216746, 10.1016/s0079-8169(08)x6158-6
Reference: [42] Petković, M. S., Neta, B., Petković, L. D., Džunić, J.: Multipoint Methods for Solving Nonlinear Equations.Elsevier, Amsterdam (2013). Zbl 1286.65060, MR 3293985, 10.1016/B978-0-12-397013-8.00001-7
Reference: [43] Qasim, S., Ali, Z., Ahmad, F., Serra-Capizzano, S., Ullah, M. Z., Mahmood, A.: Solving systems of nonlinear equations when the nonlinearity is expensive.Comput. Math. Appl. 71 (2016), 1464-1478. MR 3477716, 10.1016/j.camwa.2016.02.018
Reference: [44] Rheinboldt, W. C.: An adaptive continuation process for solving systems of nonlinear equations.Mathematical Models and Numerical Methods Banach Center Publications 3. Banach Center, Warsaw (1978), 129-142. Zbl 0378.65029, MR 0514377, 10.4064/-3-1-129-142
Reference: [45] Roberts, G. E., Horgan-Kobelski, J.: Newton's versus Halley's method: A dynamical systems approach.Int. J. Bifurcation Chaos Appl. Sci. Eng. 14 (2004), 3459-3475. Zbl 1129.37332, MR MR2107558, 10.1142/S0218127404011399
Reference: [46] Scott, M., Neta, B., Chun, C.: Basin attractors for various methods.Appl. Math. Comput. 218 (2011), 2584-2599. Zbl 06043881, MR 2838167, 10.1016/j.amc.2011.07.076
Reference: [47] Shacham, M.: An improved memory method for the solution of a nonlinear equation.Chem. Eng. Sci. 44 (1989), 1495-1501. 10.1016/0009-2509(89)80026-0
Reference: [48] Soleymani, F., Vanani, S. Karimi: Optimal Steffensen-type methods with eighth order of convergence.Comput. Math. Appl. 62 (2011), 4619-4626. Zbl 1236.65056, MR 2855607, 10.1016/j.camwa.2011.10.047
Reference: [49] Soleymani, F., Lotfi, T., Tavakoli, E., Haghani, F. Khaksar: Several iterative methods with memory using self-accelerators.Appl. Math. Comput. 254 (2015), 452-458. Zbl 1410.65177, MR 3314466, 10.1016/j.amc.2015.01.045
Reference: [50] Traub, J. F.: Iterative Methods for the Solution of Equations.Series in Automatic Computation. Prentice-Hall, Englewood Cliffs (1964). Zbl 0121.11204, MR 0169356
Reference: [51] Veiseh, H., Lotfi, T., Allahviranloo, T.: A study on the local convergence and dynamics of the two-step and derivative-free Kung-Traub's method.Comput. Appl. Math. 37 (2018), 2428-2444. Zbl 06973223, MR 3825991, 10.1007/s40314-017-0458-5

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