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An efficient two-point iterative method with memory for solving non-linear equations and its dynamics

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Abstract

In this paper, we present a novel class of two-step iterative methods with memory for solving non-linear equations. By transforming an existing sixth-order scheme without memory into with memory method, we elevate both the order of convergence and computational efficiency. To attain an accelerated order of convergence, we explore several distinct approximations of self-accelerated parameters, calculated based on the current and previous iterations using Hermite interpolation polynomials. Additionally, we eliminate the need for the second order derivative in the existing without memory method by employing a third-degree Hermite interpolating polynomial. Specifically, the proposed two-step method with memory enhances the R-order of convergence from 6 to 6.7015, 7, and 7.2749 without the need for additional function evaluations. The efficiency index of our method increases from 1.37 to 1.64. Notably, our proposed approach remains effective even when the derivative approaches extremely small values near the desired root or when \(f'(u)\) equals 0. We validate and demonstrate the effectiveness of our proposed approach by conducting numerical comparisons with several existing methods across a range of application-based problems. Finally, we employ basin of attraction plots to visualize the fractal behavior and dynamic characteristics of our proposed method in comparison to some existing methods.

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References

  1. Traub, J.F.: Iterative Methods for the Solution of Equations, vol. 312. American Mathematical Society (1982)

    Google Scholar 

  2. Ostrowki, S.H.A.: AM-Solution of Equations and Systems of Equations. SIAM Publications, Philadelphia (1967)

    Google Scholar 

  3. Choubey, N., Jaiswal, J., Choubey, A.: Family of multipoint with memory iterative schemes for solving nonlinear equations. Int. J. Appl. Comput. Math. 8(2), 1–16 (2022)

    Article  MathSciNet  Google Scholar 

  4. Ortega, J., Rheinboldt, W.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York (1970)

    Google Scholar 

  5. Kumar, S., Kanwar, V., Tomar, S.K., Singh, S.: Geometrically constructed families of newton’s method for unconstrained optimization and non-linear equations. Int. J. Math. Math. Sci. 2011 (2011)

  6. Petković, M.S., Neta, B., Petković, L.D., Džunić, J.: Multipoint methods for solving non-linear equations: a survey. Appl. Math. Comput. 226, 635–660 (2014)

    MathSciNet  Google Scholar 

  7. Chen, J.: Some new iterative methods with three-order convergence. Appl. Math. Comput. 181(2), 1519–1522 (2006)

    MathSciNet  Google Scholar 

  8. Choubey, N., Jaiswal, J.P.: An improved optimal eighth-order iterative scheme with its dynamical behaviour. Int. J. Comput. Sci. Math. 7(4), 361–370 (2016)

    Article  MathSciNet  Google Scholar 

  9. Choubey, N., Jaiswal, J.P.: Improving the order of convergence and efficiency index of an iterative method for nonlinear systems. Proc. Natl. Acad. Sci. India Sect. A 86, 221–227 (2016)

    Article  MathSciNet  Google Scholar 

  10. Jarratt, P.: Some fourth order multipoint iterative methods for solving equations. Math. Comput. 20(95), 434–437 (1966)

    Article  Google Scholar 

  11. Neta, B., et al.: A sixth-order family of methods for nonlinear equations. Int. J. Comput. Math. 7(1997), 157–161 (1979)

    Article  MathSciNet  Google Scholar 

  12. Džunić, J.: On efficient two-parameter methods for solving nonlinear equations. Numer. algorithms 63(3), 549–569 (2013)

    Article  MathSciNet  Google Scholar 

  13. Džunić, J., Petković, M., Petković, L.: Three-point methods with and without memory for solving nonlinear equations. Appl. Math. Comput. 218(9), 4917–4927 (2012)

    MathSciNet  Google Scholar 

  14. Cordero, A., Lotfi, T., Bakhtiari, P., Torregrosa, J.R.: An efficient two-parametric family with memory for nonlinear equations. Numer. Algorithms 68(2), 323–335 (2015)

    Article  MathSciNet  Google Scholar 

  15. Cordero, A., Lotfi, T., Mahdiani, K., Torregrosa, J.R.: Two optimal general classes of iterative methods with eighth-order. Acta Appl. Math. 134(1), 61–74 (2014)

    Article  MathSciNet  Google Scholar 

  16. Petković, M.S., Džunić, J., Neta, B.: Interpolatory multipoint methods with memory for solving nonlinear equations. Appl. Math. Comput. 218(6), 2533–2541 (2011)

    MathSciNet  Google Scholar 

  17. Petković, M.S., Sharma, J.R.: On some efficient derivative-free iterative methods with memory for solving systems of nonlinear equations. Numer. Algorithms 71(2), 457–474 (2016)

    Article  MathSciNet  Google Scholar 

  18. Petković, M.S., Džunić, J., Petković, L.D.: A family of two-point methods with memory for solving nonlinear equations. Appl. Anal. Discrete Math. 8, 298–317 (2011)

    Article  MathSciNet  Google Scholar 

  19. Wang, X.: An Ostrowski-type method with memory using a novel self-accelerating parameter. J. Comput. Appl. Math. 330, 710–720 (2018)

    Article  MathSciNet  Google Scholar 

  20. Wang, X., Zhang, T., Qin, Y.: Efficient two-step derivative-free iterative methods with memory and their dynamics. Int. J. Comput. Math. 93(8), 1423–1446 (2016)

    Article  MathSciNet  Google Scholar 

  21. Choubey, N., Panday, B., Jaiswal, J.: Several two-point with memory iterative methods for solving non-linear equations. Afr. Mat. 29(3), 435–449 (2018)

    Article  MathSciNet  Google Scholar 

  22. Noor, K.I., Noor, M.A.: Predictor-corrector Halley method for nonlinear equations. Appl. Math. Comput. 188(2), 1587–1591 (2007)

    MathSciNet  Google Scholar 

  23. Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. SIAM (2000)

    Book  Google Scholar 

  24. Alefeld, G., Herzberger, J.: Introduction to Interval Computation. Academic Press (2012)

    Google Scholar 

  25. Wang, X., Zhang, T.: A new family of newton-type iterative methods with and without memory for solving nonlinear equations. Calcolo 51(1), 1–15 (2014)

    Article  MathSciNet  CAS  Google Scholar 

  26. Weerakoon, S., Fernando, T.: A variant of Newton’s method with accelerated third-order convergence. Appl. Math. Lett. 13(8), 87–93 (2000)

    Article  MathSciNet  Google Scholar 

  27. Shams, M., Rafiq, N., Kausar, N., Mir, N.A., Alalyani, A.: Computer oriented numerical scheme for solving engineering problems. Comput. Syst. Sci. Eng. 42(2), 689–701 (2022)

    Article  Google Scholar 

  28. Naseem, A., Rehman, M., Qureshi, S., Ide, N.A.D.: Graphical and numerical study of a newly developed root-finding algorithm and its engineering applications. IEEE Access 11, 2375–2383 (2023)

    Article  Google Scholar 

  29. Chapra, S.: EBOOK: Applied Numerical Methods with MATLAB for Engineers and Scientists. McGraw Hill (2011)

    Google Scholar 

  30. Tassaddiq, A., Qureshi, S., Soomro, A., Hincal, E., Baleanu, D., Shaikh, A.A.: A new three-step root-finding numerical method and its fractal global behavior. Fractal Fractional 5(4), 204 (2021)

    Article  Google Scholar 

  31. Sivakumar, P., Jayaraman, J.: Some new higher order weighted newton methods for solving nonlinear equation with applications. Math. Comput. Appl. 24(2), 59 (2019)

    MathSciNet  Google Scholar 

  32. Qureshi, S., Soomro, A., Shaikh, A.A., Hincal, E., Gokbulut, N.: A novel multistep iterative technique for models in medical sciences with complex dynamics. Comput. Math. Methods Med. 2022 (2022)

  33. Sharma, J.R., Kumar, S., Singh, H.: A new class of derivative-free root solvers with increasing optimal convergence order and their complex dynamics. SEMA J. 7, 1–20 (2022)

    CAS  Google Scholar 

  34. Stewart, B.D.: Attractor basins of various root-finding methods. Technical report. Naval Postgraduate School Monterey CA (2001)

  35. Vrscay, E., Gilbert, W.: Extraneous fixed points, basin boundaries and chaotic dynamics for schröder and könig rational iteration functions. Numericshe Mathematik 52, 1–16 (1988)

    Google Scholar 

  36. Zachary, J.: Introduction to Scientific Programming: Computational Problem Solving Using Maple and C. Springer, Berlin (2012)

    Google Scholar 

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  1. Shahid Abdullah, Neha Choubey and Suresh Dara contributed equally to this work.

Authors and Affiliations

  1. School of Advanced Sciences and Languages, VIT Bhopal University, Kothri-Kalan, Sehore, MP, 466114, India

    Shahid Abdullah, Neha Choubey & Suresh Dara

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  1. Shahid Abdullah
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  2. Neha Choubey
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  3. Suresh Dara
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Correspondence to Neha Choubey.

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Abdullah, S., Choubey, N. & Dara, S. An efficient two-point iterative method with memory for solving non-linear equations and its dynamics. J. Appl. Math. Comput. 70, 285–315 (2024). https://doi.org/10.1007/s12190-023-01953-w

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