Efficient two-step with memory methods and their dynamics.

Document Type : Original Article

Author

Department of Mathematics, Farhangian University Tehran, Iran

Abstract
In this work,a fourth-order without-memory method is proposed that has a self-
accelerator parameter.This method doesn’t need to compute a derivative function for
solving nonlinear equations.We have approximated the self-accelerator parameter and
have increased the convergence order to %50 without increase function evaluation.The
efficiency index of the with-memory method sixth-order is equal to 1.81712. Which is
higher than one-, two-, three-, and four-step optimal methods.The attraction basin of
the proposed methods is compared by the famous Newton’s method and Kung-Traub’s
method.

Keywords

Subjects


[1] Ahmad, F., Soleymani, F., Haghani, F. K., Serra-Capizzano, S. (2017). Higher order derivative-free iterative methods with and without memory for systems of nonlinear equations, Applied Mathematics and Computation, 314, 199-211.
[2] Argyros, I.K., George, S. (2015). Ball convergence theorems for eighth-order variants of Newton’s method under weak conditions, Arabian Journal of Mathematics, 4(2), 81-90.
[3] Babajee, D.K.R., Madhu, K. (2019). Comparing two techniques for developing higher order two-point iterative methods for solving quadratic equations, SeMA Journal, 76(2), 227-248.
[4] Bisheh-Niasar, M. (2023). The Effect of the Caputo Fractional Derivative on Polynomiography, Mathematics Interdisciplinary Research, 8(4), 347-358.
[5] Choubey, N., Panday, B., Jaiswal, J.P. (2018). Several two-point with memory iterative methods for solving nonlinear equations, Afrika matematika, 29(3), 435-449.
[6] Chun, C., Lee, M. Y. (2013). A new optimal eighth-order family of iterative methods for the solution of nonlinear equations, Applied Mathematics and Computation, 223, 506-519.
[7] Chicharro, F. I., Cordero, A., Garrido, N., Torregrosa, J. R. (2020). On the choice of the best members of the Kim family and the improvement of its convergence, Mathematical Methods in the Applied Sciences, 43(14), 8051-8066,
[8] Cordero, A., Hueso, J. L., Martinez, E., Torregrosa, J.R. (2010). New modifications of Potra-Ptak’s method with optimal fourth and eighth orders of convergence, Journal of computational and applied mathematics, 234(10), 2969-2976.
[9] Cordero, A., Rojas-Hiciano, R. V., Torregrosa, J. R., Vassileva, M. P., (2024). A highly efficient class of optimal fourth-order methods for solving nonlinear systems, Numerical Algorithms, 95(4), 1879-1904.
[10] Darvishi, H., Darvishi, M. T. (2023). An analytical study on two high-order hybrid methods to solve systems of nonlinear equations, Journal of Mathematics, 2023, 1-18.
[11] Geum, Y. H., Kim, Y. I. (2011). A biparametric family optimally convergent sixteenth-order multipoint methods with their fourth-step weighting function as a sum of a rational and a generic twovariable function, Journal of Computational and Applied Mathematics. 235, 3178-3188.
[12] Jarratt, P. (1966). Some fourth order multipoint methods for solving equations, Mathematics of computation, 20, 434-437.
[13] Liu, C. S., Lee, T. L. (2021). A New Family of Fourth-Order Optimal Iterative Schemes and Remark on Kung and Traub’s Conjecture, Journal of Mathematics, 2021, 1-9.
[14] Kung, H. T., Traub, J. F. (1974). Optimal order of one-point and multipoint iteration, J. Assoc. Comput. Mach. 21 (4), 643-651.
[15] Maheshwari, A. K. (2009). A fourth order iterative method for solving nonlinear equations, Applied Mathematics and Computation, 211, 383-391.
[16] Moccari, M., Lotfi, T., Torkashvand, V. (2023). On Stability of a Two-Step Method for a Fourth-degree Family by Computer Designs along with Applications, International Journal of Nonlinear Analysis and Applications (IJNAA), 14 (4) , 261-282.
[17] McDougall, T. J., Wotherspoon, S. J., Barker, P. M. (2019). An accelerated version of Newton’s method with convergence order √3+1, Results in Applied Mathematics, 4(100078), 1-10.
[18] Ostrowski, A. M. Solution of equations and systems of equations, Academic press, New York, 1960.
[19] Petkovic, M. S., Neta, B., Petkovic, L. D., Dzunic, J. Multipoint methods for solving nonlinear equations ,Elsevier, Amsterdam, 2013.
[20] Petkovic, M.S., Ilic, S., Dzunic, J. (2010). Derivative free two-point methods with and without memory for solving nonlinear equations, Applied Mathematics and Computation, 217, 1887-1895.
[21] Ren, H., Wu, Q., Bi, W. (2009). A class of two-step Steffensen type methods with fourth-order convergence, Applied Mathematics and Computation, 209, 206-210.
[22] Sharifi, S., Salimi, M., Siegmund, S., Lotfi, T. (2016). A new class of optimal four-point methods with convergence order 16 for solving nonlinear equations, Math. Comput. Simul. 119(c), 69-90.
[23] Shams, M., Rafiq, N., Kausar, N., Agarwal, P., Mir, N.A,. Li, Y. M. (2022). On highly efficient simultaneous schemes for finding all polynomial roots, Fractals, 30(10), 2240198.
[24] Shams, M., Rafiq, N., Kausar, N., Agarwal, P., Mir, N. A., El-Kanj, N. (2022). On Inverse Iteration process for finding all roots of nonlinear equations with applications, Fractals, 30(10), 2240265.
[25] Shams, M., Rafiq, N., Kausar, N., Ahmed, S. F., Mir, N. A., Chandra Saha, S. (2021). Inverse family of numerical methods for approximating all simple and roots with multiplicity of nonlinear polynomial equations with engineering applications, Mathematical Problems in Engineering, 2021(1), 3124615.
[26] Shams, M., Kausar, N., Araci, S., Kong, L., Carpentieri, B. (2024). Highly efficient family of two-step simultaneous method for all polynomial roots, AIMS Mathematics, 9(1), 1755-1771.
[27] Shams, M., Kausar, N., Araci, S., Kong, L. (2024). On the stability analysis of numerical schemes for solving non-linear polynomials arises in engineering problems, American Institute of Mathematical Sciences.
[28] Sivakumar, P., Madhu, K., Jayaraman, J. (2021). Optimal Eighth And Sixteenth Order Iterative Methods For Solving Nonlinear Equation With Basins Of Attraction, Applied Mathematics E-Notes, 21, 320-343.
[29] Steffensen, J. F. (1933). Remarks on iteration, Scandinavian Aktuarietidskr, 16, 64-72.
[30] Torkashvand, V. (2022). A two-step method adaptive with memory with eighth-order for solving nonlinear equations and its dynamic, Computational Methods for Differential Equations, 50(1), 1007-1026.
[31] Torkashvand, V., Kazemi, M. (2020). On an Efficient Family with Memory with High Order of Convergence for Solving Nonlinear Equations, International Journal Industrial Mathematics, 12(2), 209-224.
[32] Torkashvand, V., Lotfi, T., Fariborzi Araghi, M. A. (2019). A new family of adaptive methods with memory for solving nonlinear equations, Mathematical Sciences, 13 (1), 1-20.
[33] Torkashvand, V. (2023). Transforming Ostrowski’s method into a derivative-free method and its dynamics, Computational Mathematics and Computer Modeling with Applications, 2(1), 1-10.
[34] Torkashvand, V., Kazemi, M. (2023). Structure A Family Of Three-Step With-Memory Methods For Solving Nonlinear Equations And Their Dynamics, The 1st National Conference on Smart systems, Soft Computing, and Applied Mathematics.
[35] Sivakumar, P., Madhu, K., Jayakumar. J. (2021). Optimal eighth and sixteenth order iterative methods for solving nonlinear equation with basins of attraction, Journal Appl. Math. E-Notes, 21, 320-343.
Thukral, R. (2012). New sixteenth-order derivative-free methods for solving nonlinear equations, American. J. Comput. Appl. Math. 2(3), 112-118.
[36] Traub, J.F. Iterative methods for the molution of equations, Prentice Hall, New York, New Jersey, USA, 1964.
[37] Wang, X., Fan, Q. (2020). A modified Ren’s method with memory using a simple self-accelerating parameter, Mathematics, 8(5040), 1-12.
[38] Wang, X. (2018). A family of Newton-type iterative methods using some special self-accelerating parameters, International Journal of Computer Mathematics, 95(10), 2112-2127.
[39] Weerakoon, S., Fernando, T.G.I. (2000). A variant of Newton’s method with accelerated third-order convergence, Applied mathematics letters, 13, 87-93.
[40] Zafar, F., Cordero, A., Torregrosa, J. R. (2019). Stability analysis of a family of optimal fourth-order methods for multiple roots, Numerical Algorithms, 81, 947-981.
[41] Zhanlav, T., Otgondorj, K., Chuluunbaatar, O. (2019). Families of optimal derivative-free two-and three-point iterative methods for solving nonlinear equations, Computational Mathematics and Mathematical Physics, 59(6), 864-880.
[42] Zheng, Q., Li, J., Huang, F. (2011). An optimal Steffensen-type family for solving nonlinear equations, Applied Mathematics and Computation, 217, 9592-9597.
[43] Zheng, Q., Zhao, X., Liu, Y. (2015). An optimal biparametric multipoint family and its selfacceleration with memory for solving nonlinear equations, Algorithms, 8, 1111-1120.
[44] Zheng, Q., Zhao, P., Zhang, L., Ma, W. (2010). Variants of Steffensen-secant method and applications, Applied Mathematics and Computation, 216, 3486-3496.
Volume 5, Issue 3
Summer 2024
Pages 80-92

  • Receive Date 15 June 2024
  • Revise Date 07 October 2024
  • Accept Date 08 October 2024