On the efficient of adaptive methods to solve nonlinear equations

Torkashvand, Vali; Ezzati, Reza

doi:10.22075/ijnaa.2021.4799

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	On the efficient of adaptive methods to solve nonlinear equations
International Journal of Nonlinear Analysis and Applications
مقاله 24، دوره 12، شماره 1، مرداد 2021، صفحه 301-316 اصل مقاله (591.33 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22075/ijnaa.2021.4799
نویسندگان
Vali Torkashvand^* ¹^{، 2}؛ Reza Ezzati³
¹‎‎‎Young Researchers and Elite Club, Shahr-e-Qods Branch, Islamic Azad University Tehran Iran
²Farhangian University, Tehran, Iran
³Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran
تاریخ دریافت: 27 اسفند 1397، تاریخ بازنگری: 24 دی 1398، تاریخ پذیرش: 03 بهمن 1399
چکیده
The main goal of this work, obtaining a family of Steffensen-type iterative methods adaptive with memory for solving nonlinear equations, which uses three self-accelerating parameters. For this aim, we present a new scheme to construct the self-accelerating parameters and obtain a family of Steffensen-type iterative methods with memory. The self-accelerating parameters have the properties of simple structure and easy calculation, which do not increase the computational cost of the iterative methods. The convergence order of the new iterative methods has increased from 4 to 8. Also, these methods possess very high computational efficiency. Another advantage of the new method is that they remove the severe condition $f'(x)$ in a neighborhood of the required root imposed on Newton's method. Numerical comparisons have made to show the performance of the proposed methods, as shown in the illustrative examples.‎
کلیدواژه‌ها
Nonlinear equations؛ Newton's interpolatory polynomial؛ Adaptive method with memory؛ The order of convergence؛ Self accelerating parameter‎‎

مراجع
[1] S. Abbasbandy, Modified homotopy perturbation method for nonlinear equations and comparison with Adomian decomposition method, Appl. Math. Comput. 172 (2006) 431–438. [2] S. Artidiello, A. Cordero, J.R. Torregrosa and M.P. Vassileva, Two weighted eight-order classes of iterative rootfinding methods, Int. J. Comput. Math. 92(9) (2015) 1790–1805. [3] D.K.R. Babajee, A. Cordero, F. Soleymani and J.R. Torregrosa, On improved three-step schemes with high-efficiency index and their dynamics, Numer. Algor. 65 (2014) 153–169. [4] R. Behl and S.S. Motsa, Geometric construction of eighth-order optimal families of Ostrowski’s method, Sci. World J. 2015 (2015) 1–11. [5] C. Chun, Cunstruction of Newton-like iteration methods for solving nonlinear equations, Numer. Math. 104 (2006) 297–315. [6] C. Chun and B. Neta, An analysis of a new family of eighth-order optimal methods, Appl. Math. Comput. 245 (2014) 86–107. [7] A. Cordero, T. Lotfi, A. Khoshandi and J.R. Torregrosa, An efficient Steffensen-like iterative method with memory, Bull. Math. Soc. Sci. Math. Roum, Tome 58(106)(1) (2015) 49–58. [8] A. Cordero, T. Lotfi, J.R. Torregrosa, P. Assari and K. Mahdiani, Some new bi-accelarator two-point methods for solving nonlinear equations, Comput. Appl. Math. 35 (2016) 251–267. [9] A. Cordero, M. Fardi, M. Ghasemi and J.R. Torregrosa, Accelerated iterative methods for finding solutions of nonlinear equations and their dynamical behavior, Calcolo 51(1) (2014) 17–30. [10] N. Choubey and J.P. Jaiswal, An improved optimal eighth-order iterative scheme with its dynamical behaviour, Int. J. Comput. Science. Math. 7(4) (2016) 361–370. [11] J. Dzunic, On efficient two-parameter methods for solving nonlinear equations, Numer. Algor. 63 (2013) 549–569. [12] J. Dzunic and M.S. Petkovic, A cubicaly convergent Steffensen-like method for solving nonlinear equations, Appl. Math. Lett. 25 (2012) 1881–1886. [13] M. Fardi, M. Ghasemi and A. Davari, New iterative methods with seventh-order convergence for solving nonlinear equations, Int. J. Nonlinear Anal. Appl. 3(2) (2012) 31–37. [14] Y. H. Geum and Y.I. Kim, A biparametric family of four-step sixteenth-order root-finding methods with the optimal efficiency index, Appl. Math. Lett. 24 (2011) 1336–1342. [15] L.O. Jay, A note on Q-order of convergence, BIT. 41(2) (2001) 422–429. [16] P. Jarratt, Some efficient fourth-order multipoint methods for solving equations, BIT. 9(2) (1969) 119–124. [17] R.F. King, A family of fourth order methods for nonlinear equations, Siam. J. Numer. Anal. 10(5) (1973) 876–879. [18] J. Kou, Y. Li and X. Wang, A family of fourth-order methods for solving non-linear equations, Appl. Math. Comput. 118(1) (2007) 1031–1036. [19] H.T. Kung and J.F. Traub, Optimal order of one-point and multipoint iteration, J. Assoc. Comput. Mach. 21(4) (1974) 643–651. [20] M.Y. Lee and Y.I. Kim, A family of fast derivative-free fourth-order multipoint optimal methods for nonlinear equations, Int. J. Comput. Math. 89(15) (2012) 2081–2093. [21] T. Lotfi and P. Assari, New three-and four-parametric iterative with memory methods with efficiency index near 2, Appl. Math. Comput. 270 (2015) 1004–1010. [22] T. Lotfi, F. Soleymani, M. Ghorbanzadeh and P. Assari, On the construction of some tri-parametric iterative methods with memory, Numer. Algor. 70(4) (2015) 835–845. [23] T. Lotfi, F. Soleymani, Z. Noori, A. Kilicman and F. Khaksar Haghani, Efficient iterative methods with and without memory possessing high-efficiency indices, Dis. Dyn. Nat. Soc. 2014 (2014) 1–9. [24] A.K. Maheshwari, A fourth-order iterative method for solving nonlinear equations, Appl. Math. Comput. 211 (2009) 383–391. [25] B. Neta and M. Scott, On a family of Halley-like methods to find simple roots of nonlinear equations, Appl. Math. Comput. 219 (2013) 7940–7944. [26] J.M. Ortega and W.G. Rheinboldt, Iterative Solutions of Nonlinear Equations in Several Variables, Academic Press, New York, 1970. [27] A.M. Ostrowski, Solution of Equations and Systems of Equations, Academic Press, New York, 1960. [28] M.S. Petkovic, J. Dzunic and L.D. Petkovic, A family of two-point with memory for solving nonlinear equations, Appl. Anal. Disc. Math. 5 (2011) 298–317. [29] M.S. Petkovic, B. Neta, L.D. Petkovic and J. Dzunic, Multipoint Methods for Solving Nonlinear Equations, Elsevier, Amsterdam, 2013. [30] J.Raj. Sharma and H. Arora, An efficient family of weighted-Newton methods with optimal eighth order convergence, Appl. Math. Lett. 29 (2014) 1–6. [31] H. Ren, Q. Wu and W. Bi, A class of two-step Steffensen type methods with fourth-order convergence, Appl. Math. Comput. 209 (2009) 206–210. [32] M. Salimi, T. Lotfi, S. Sharifi and S. Siegmund, Optimal Newton–Secant like methods without memory for solving nonlinear equations with its dynamics, Int. J. Comput. Math. 94(9) (2017) 1759–1777. [33] S. Sharifi, M. Salimi, S. Siegmund and T. Lotfi, A new class of optimal four-point methods with convergence order 16 for solving nonlinear equations, Math. Comput. Simul. 119 (c) (2016) 69-90. [34] S. Sharifi, M. Ferrara, M. Salimi and S. Siegmund, New modification of Maheshwari’s method with optimal eighth order of convergence for solving nonlinear equations, Open Math. 14 (2016) 443–451. [35] S. Sharifi, S. Siegmund and M. Salimi, Solving nonlinear equations by a derivative-free form of the King’s family with memory, Calcolo 53 (2016) 201–215. [36] F. Soleymani, On a bi-parametric class of optimal eighth-order derivative-free methods, Int. J. Pure Appl. Math. 72 (1) (2011) 27–37. [37] F. Soleymani, Some optimal iterative methods and their with memory variants, J. Egyp. Math. Soc. (2013) 1–9. [38] F. Soleymani, T. Lotfi, E. Tavakoli and F. Khaksar Haghani, Several iterative methods with memory using self accelerators, Appl. Math. Comput. 254 (2015) 452–458. [39] F. Soleymani, M. Sharifi and B. S. Mousavi, An improvement of Ostrowski’s and King’s techniques with optimal convergence order eight, J. Optim. Theory. Appl. 153 (2012) 225–236. [40] F. Soleymani, S. K. Khattri and S. Karimi Vanani, Two new classes of optimal Jarratt-type fourth-order methods, Appl. Math. Lett. 25 (2012) 847–853. [41] J.F. Steffensen, Remarks on iteration, Scand. Aktuar. 16 (1933) 64–72. [42] R. Thukral and M.S. Petkovic, A family of three-point methods of optimal order for solving nonlinear equations, J. Comput. Appl. Math. 233 (2010) 2278–2284. [43] V. Torkashvand, T. Lotfi and M.A. Fariborzi Araghi, A new family of adaptive methods with memory for solving nonlinear equations, Math. Sci. 13 (2019) 1–20. [44] V. Torkashvand and M. Kazemi, On an Efficient Family with Memory with High Order of Convergence for Solving Nonlinear Equations, Int. J. Industrial Mathematics, 12(2) (2020) 209–224. [45] J.F. Traub, Iterative Methods for the Solution of Equations, Prentice Hall, New York, USA, 1964. [46] X. Wang, An Ostrowski-type method with memory using a novel self-accelerating parameter, J. Comput. Appl. Math. 330 (2017) 1–18. [47] X. Wang and T. Zhang, A new family of Newton-type iterative methods with and without memory for solving nonlinear equations, Calcolo 51 (2014) 1–15. [48] X. Wang, T. Zhang and Y. Qin, Efficient two-step derivative-free iterative methods with memory and their dynamics, Int. J. Comput. Math. 93(8) (2015) 1–27. [49] X. Wang, J. Dzunic and T. Zhang, On an efficient family of derivative-free three-point methods for solving nonlinear equations, Appl. Math. Comput. 219 (2012) 1749–1760. [50] S. Weerakoon and T.G.I. Fernando, A variant of Newton's method with accelerated third-order convergence, J. Appl. Math. Comput. 13(8) (2000) 87–93. [51] Q. Zheng, J. Li and F. Huang, An optimal Steffensen-type family for solving nonlinear equations, Appl. Math. Comput. 217 (2011) 9592–9597. [52] Q. Zheng, X. Zhao and Y. Liu, An optimal biparametric multipoint family and its self-acceleration with memory for solving nonlinear equations, Algorithms 8 (4) (2015) 1111–1120.
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On the efficient of adaptive methods to solve nonlinear equations