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A two-step method adaptive with memory with eighth-order for solving nonlinear equations and its dynamic. (English) Zbl 1524.65207

Summary: In this work, we have constructed the with memory two-step method with four convergence degrees by entering the maximum self-accelerator parameter(three parameters). Then, using Newton’s interpolation, a with-memory method with a convergence order of 7.53 is constructed. Using the information of all the steps, we will improve the convergence order by one hundred percent, and we will introduce our method with convergence order 8. Numerical examples demonstrate the exceptional convergence speed of the proposed method and confirm theoretical results. Finally, we have presented the dynamics of the adaptive method and other without-memory methods for complex polynomials of degrees two, three, and four. The basins of attraction of existing with-memory methods are present and compared to illustrate their performance.

MSC:

65H05 Numerical computation of solutions to single equations
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[1] I. K. Argyros and L. U. Uko, An improved convergence analysis of a one-step intermediate Newton iterative scheme for nonlinear equations, Journal of Applied Mathematics and Computing., 38 (2012), 243-256. · Zbl 1295.65065
[2] I. K. Argyros and S. George, Extended Kung-Traub-Type method for solving equations, TWMS journal of pure and applied mathematics, 12(2) (2021), 193-198. · Zbl 1491.65036
[3] R. Behl, A. Cordero, S. S. Motsa, and J. R. Torregrosa, Multiplicity anomalies of an optimal fourth-order class of iterative methods for solving nonlinear equations, Nonlinear Dynamics., 91(1) (2018), 81-112. · Zbl 1390.37077
[4] B. Campos, A. Cordero, J. R. Torregrosa, and P. Vindel, Stability of King’s family of iterative methods with memory, Journal of Computational and Applied Mathematics., 318 (2017), 504-514. · Zbl 1373.37194
[5] A. Cordero, T. Lotfi, P. Bakhtiari, and J. R. Torregrosa, An efficient two-parametric family with memory for nonlinear equations, Numerical Algorithm., 68(2) (2014), 323-335. · Zbl 1309.65053
[6] A. Cordero, T. Lotfi, A. Khoshandi, and J. R. Torregrosa, An efficient Steffensen-like iterative method with memry, Bull. Math. Soc. Sci. Math. Roum, Tome., 58(1) (2015), 49-58. · Zbl 1340.65091
[7] C. Chun and M. Y. Lee, A new optimal eighth-order family of iterative methods for the solution of nonlinear equations, Applied Mathematics and Computation., 223 (2013), 506-519. · Zbl 1329.65099
[8] C. Chun and B. Neta, An analysis of a new family of eighth-order optimal methods, Applied Mathematics and Computation, 245 (2014), 86-107. · Zbl 1336.65081
[9] C. Chun, M. Y. Lee, B. Neta, and J. Dzunic, On optimal fourth-order iterative methods free from second derivative and their dynamics, Applied Mathematics and Computation., 218(11) (2012), 6427-6438. · Zbl 1277.65031
[10] P. A. Delshad and T. Lotfi, On the local convergence of Kung-Traub’s two-point method and its dynamics, Appli-cations of Mathematics., 65(4) (2020), 379-406. · Zbl 07250668
[11] J. Dzunic, On efficient two-parameter methods for solving nonlinear equations, Numerical Algorithm., 63 (2013), 549-569. · Zbl 1280.65043
[12] M. A. Fariborzi Araghi, T. Lotfi, and V. Torkashvand, A general efficient family of adaptive method with memory for solving nonlinear equations, Bull. Math. Soc. Sci. Math. Roumanie Tome., 62(1) (2019), 37-49. · Zbl 1463.65112
[13] J. Herzberger, Uber Matrixdarstellungen fur Iterationverfahren bei nichtlinearen Gleichungen, Computing., 12 (1974), 215-222. · Zbl 0278.65054
[14] J. P. Jaiswal, Two Bi-Accelerator Improved with Memory Schemes for Solving Nonlinear Equations, Discrete Dynamics in Nature and Society., 2015 (2015), 1-7.
[15] M. Kansal, V. Kanwar, and S. Bhatia, Efficient derivative-free variants of Hansen-Patrick’s family with memory for solving nonlinear equations, Numerical Algorithms., 73(4) (2016), 1017-1036. · Zbl 1357.65058
[16] T. Lotfi and P. Assari, Two new three and four parametric with memory methods for solving nonlinear equations, International Journal Industrial Mathematics., 7(3) (2015), 269-276.
[17] T. Lotfi, F. Soleymani, Z. Noori, A. Kilicman, and F. Khaksar Haghani, Efficient iterative methods with and without memory possessing high efficiency indices, Discrete Dynamics in Nature and Society., 2014 (2014), 1-9.
[18] T. Lotfi, S. Sharifi, M. Salimi, and S. Siegmund, A new class of three-point methods with optimal convergence order eight and its dynamics, Numerical Algorithms., 68(2) (2015), 261-288. · Zbl 1309.65054
[19] M. Mohamadi Zadeh, T. Lotfi, and M. Amirfakhrian, Developing two efficient adaptive Newton-type methods with memory, Mathematical Methods in the Applied Sciences., 42(17) (2019), 5687-5695. · Zbl 1433.65095
[20] M. Moccari and T. Lotfi, On a two-step optimal Steffensen-type method: Relaxed local and semi-local convergence analysis and dynamical stability, Journal of Mathematical Analysis and Applications., 468(1) (2018), 240-269. · Zbl 06937851
[21] A. M Maheshwari, A fourth order iterative method for solving nonlinear equations, Applied Mathematics and Computation., 211 (2009), 383-391. · Zbl 1162.65346
[22] B. Neta, A new family of higher order methods for solving equations, International Journal of Computer Mathe-matics., 14 (1983), 191-195. · Zbl 0514.65029
[23] M. S. Petkovic, J. Dzunic, and B. Neta, Interpolatory multipoint methods with memory for solving nonlinear equations, Applied Mathematics and Computation., 218 (2011), 2533-2541. · Zbl 1243.65054
[24] M. S. Petkovic, S. Ilic, and J. Dzunic, Derivative free two-point methods with and without memory for solving nonlinear equations, Applied Mathematics and Computation., 217 (2010), 1887-1895. · Zbl 1200.65034
[25] M. S. Petkovic, B. Neta, L. D. Petkovic, and J. Dzunic, Multipoint methods for solving nonlinear equations, Elsevier, Amsterdam, 2013. · Zbl 1286.65060
[26] J. Raj Sharma and R. Sharma, A new family of modified Ostrowski’s methods with accelerated eighth order convergence, Numerical Algorithms., 54 (2010), 445-458. · Zbl 1195.65067
[27] M. Salimi, T. Lotfi, S. Sharifi, and S. Siegmund, Optimal Newton-Secant like methods without memory for solving nonlinear equations with its dynamics, International Journal of Computer Mathematics., 94(9) (2017), 1759-1777. · Zbl 1391.65135
[28] F. Soleymani, Some optimal iterative methods and their with memory variants, Journal of the Egyptian Mathe-matical Society., 21(2) (2013), 133-141. · Zbl 1315.65047
[29] F. Soleymani, Some High-Order Iterative Methods for Finding All the Real Zeros, Thai Journal of Mathematics., 12(2) (2014), 313-327. · Zbl 1306.41006
[30] F. Soleymani, New class of eighth-order iterative zero-finders and their basins of attraction, Afrika Matematika., 25(1) (2014), 67-79. · Zbl 1301.41013
[31] J. F. Traub, Iterative Methods for the Solution of Equations, Prentice Hall, New York, USA, 1964. · Zbl 0121.11204
[32] V. Torkashvand, Structure of an Adaptive with Memory Method with Efficiency Index 2, International Journal of Mathematical Modelling Computations., 9(4) (2019), 239-252.
[33] V. Torkashvand and M. Kazemi, On an Efficient Family with Memory with High Order of Convergence for Solving Nonlinear Equations, International Journal Industrial Mathematics., 12(2) (2020), 209-224.
[34] H. Veiseh, T. Lotfi, and T. Allahviranloo, A study on the local convergence and dynamics of the two-step and derivative-free Kung-Traub’s method, Computational and Applied Mathematics., 37(3) (2018), 2428-2444. · Zbl 1451.65069
[35] X. Wang, A new accelerating technique applied to a variant of Cordero-Torregrosa method, Journal of Computa-tional and Applied Mathematics., 330 (2018), 695-709. · Zbl 1376.65082
[36] F. Zafar, A. Cordero, J. R. Torregrosa, and A. Rafi, A class of four parametric with-and without-memory root finding methods, Computational and Mathematical Methods., 1(3) (2019), 1-13.
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