The primary focus of this study is to introduce a new three-step iterative method without memory for root finding by merging two different existing techniques. Based on the computational cost, the proposed method acquires optimal eight-order convergence with four functional evaluations (three evaluations for the function and one computation of the first derivative). Furthermore, the suggested scheme supports Kung-Traub’s Conjecture with an efficiency index of $8^\frac{1}{4}=1.682$. We also established the convergence criteria developed for the root-finding technique and demonstrated the fact that the suggested approach is eighth-order convergent. In order to demonstrate the efficacy as well as application of the constructed root-finding technique, we addressed a few practical engineering models and some non-linear functions. In contrast to several existing approaches, this particular method converges more quickly. Finally, several forms of complex functions are taken into consideration under basins of attraction in order to observe the overall fractal behavior of the proposed technique.
[1] S. Abbasbandy, Improving newton–raphson method for nonlinear equations by modified adomian decomposition method, Applied mathematics and computation, 145(2-3) (2003), 887–893.
[2] S. Abdullah, N. Choubey, and S. Dara, Two novel with and without memory multipoint iterative methods for solving non-linear equations, Communications in Mathematics and Applications, 3 (2024), 9-31.
[3] S. Abdullah, N. Choubey, and S. Dara, An efficient two-point iterative method with memory for solving non-linear equations and its dynamics, Journal of Applied Mathematics and Computing, 70(1) (2024), 285–315.
[4] S. Abdullah, N. Choubey, and S. Dara, Optimal fourth-and eighth-order iterative methods for solving nonlinear equations with basins of attraction, Journal of Applied Mathematics and Computing, 70 (2024), 3477–3507.
[5] S. Abdullah, N. Choubey, and S. Dara, Dynamical analysis of optimal iterative methods for solving nonlinear equations with applications, Journal of Applied Analysis and Computation, 14(6) (2024), 3349–3376.
[6] H. A. Abro and M. M. Shaikh, A new time-efficient and convergent nonlinear solver Applied Mathematics and Computation, 355 (2019), 516–536.
[7] N. Choubey and J. P. Jaiswal, An improved optimal eighth-order iterative scheme with its dynamical behaviour, International Journal of Computing Science and Mathematics, 7(4) (2016), 361–370.
[8] A. Cordero, J. L. Hueso, E. Mart´ınez, and J. R. Torregrosa, A modified Newton-Jarratt’s composition, Numerical Algorithms, 55 (2010) , 87–99.
[9] A. Cordero, T. Lotfi, K. Mahdiani, and J. R. Torregrosa, Two optimal general classes of iterative methods with eighth-order, Acta applicandae mathematicae, 134(1) (2014), 61–74.
[10] J. Dˇzuni´c and M. S. Petkovi´c, A family of three-point methods of ostrowski’s type for solving nonlinear equations, Journal of Applied Mathematics, 2012(2) (2012).
[11] J. H. He and X. H. Wu, Variational iteration method: new development and applications, Computers and Mathematics with Applications, 54(7-8) (2007), 881–894.
[12] S. K. Khattri, Quadrature based optimal iterative methods with applications in high-precision computin, Numerical Mathematics: Theory, Methods and Applications, 5(4) (2012), 592–601.
[13] B. Kong-ied, Two new eighth and twelfth order iterative methods for solving nonlinear equations, International Journal of Mathematics and Computer Science, 16 (2021), 333–344.
[14] J. Kou, Y. Li, and X. Wang, A composite fourth-order iterative method for solving non-linear equations, Applied Mathematics and Computation, 184(2) (2007), 471–475.
[15] H. Kung and J. F. Traub, Optimal order of one-point and multipoint iteration, Journal of the ACM (JACM), 21(4) (1974), 643–651.
[16] L. Liu and X. Wang, Eighth-order methods with high efficiency index for solving nonlinear equations, Applied Mathematics and Computation, 215(9) (2010), 3449–3454.
[17] N. A. Mir and T. Zaman, Some quadrature based three-step iterative methods for non-linear equations, Applied Mathematics and Computation 193(2) (2007), 366–373.
[18] A. Naseem, M. Rehman, S. Qureshi, and N. A. D. Ide, Graphical and numerical study of a newly developed root-finding algorithm and its engineering applications, IEEE Access, 11 (2023), 2375–2383.
[19] M. A. Noor and M. Waseem, Some iterative methods for solving a system of nonlinear equations. Computers and Mathematics with Applications, 57(1) (2009), 101–106.
[20] J. M. Ortega, Numerical Analysis, A Second Course. SIAM, (1990).
[21] A. Y. Ozban, Some new variants of newton’s method, Applied Mathematics Letters, 17(6) (2004), 677–682.
[22] S. Parhi and D. K. Gupta, A sixth order method for nonlinear equations, Applied Mathematics and Computation, 203(1) (2008), 50–55.
[23] S. Parimala, K. Madhu, and J. Jayaraman, A new class of optimal eighth order method with two weight functions for solving nonlinear equation, J. Nonlinear Anal. Appl, (2018), 83–94.
[24] S. Qureshi, H. Ramos, and A. K. Soomro, A new nonlinear ninth-order root-finding method with error analysis and basins of attraction, Mathematics, 9(16) (2021).
[25] S. Qureshi, A. Soomro, A. A. Shaikh, E. Hinca,l and N. Gokbulut, A novel multistep iterative technique for models in medical sciences with complex dynamics, Computational and Mathematical Methods in Medicine, (2022).
[26] M. Rafiullah, A fifth-order iterative method for solving nonlinear equations, Numerical Analysis and Applications, 4(3) (2011), 239.
[27] J. R. Sharma and H. Arora, An efficient family of weighted-newton methods with optimal eighth order convergence, Applied Mathematics Letters, 29 (2014), 1–6.
[28] A. Singh and J. P. Jaiswal, An efficient family of optimal eighth-order iterative methods for solving nonlinear equations and its dynamics, Journal of Mathematics, (2014).
[29] O. Solaiman and A. IshakHashim, The attraction basins of several root finding methods, with a note about optimal methods, Proceedings the 6th International Arab Conference on Mathematics and Computations, (2019), 68.
[30] H. Tari, D. Ganji, and H. Babazadeh, The application of he’s variational iteration method to nonlinear equations arising in heat transfer, Physics Letters A, 363(3) (2007), 213–217.
[31] A. Tassaddiq, S. Qureshi, A. Soomro, E. Hincal, D. Baleanu, and A. A Shaikh, A new three-step root-finding numerical method and its fractal global behavior, Fractal and Fractional, 5(4) (2021), 204.
[32] F. Zafar and N. A. Mir, A generalized family of quadrature based iterative methods, General Math, 18(4) (2010), 43–51.
Adullah, S. , Choubey, N. and Dara, S. (2025). A new three-step optimal without memory iterative scheme for solving non-linear equations with basins of attraction. Computational Methods for Differential Equations, 13(4), 1289-1305. doi: 10.22034/cmde.2024.59161.2514
MLA
Adullah, S. , , Choubey, N. , and Dara, S. . "A new three-step optimal without memory iterative scheme for solving non-linear equations with basins of attraction", Computational Methods for Differential Equations, 13, 4, 2025, 1289-1305. doi: 10.22034/cmde.2024.59161.2514
HARVARD
Adullah, S., Choubey, N., Dara, S. (2025). 'A new three-step optimal without memory iterative scheme for solving non-linear equations with basins of attraction', Computational Methods for Differential Equations, 13(4), pp. 1289-1305. doi: 10.22034/cmde.2024.59161.2514
CHICAGO
S. Adullah , N. Choubey and S. Dara, "A new three-step optimal without memory iterative scheme for solving non-linear equations with basins of attraction," Computational Methods for Differential Equations, 13 4 (2025): 1289-1305, doi: 10.22034/cmde.2024.59161.2514
VANCOUVER
Adullah, S., Choubey, N., Dara, S. A new three-step optimal without memory iterative scheme for solving non-linear equations with basins of attraction. Computational Methods for Differential Equations, 2025; 13(4): 1289-1305. doi: 10.22034/cmde.2024.59161.2514
)