Abstract
We present a general class of derivative free iterative methods with optimal order of convergence for solving nonlinear equations. The methodology is based on quadratically convergent Traub–Steffensen scheme and inverse Padé approximation. Unlike that of existing higher order techniques the proposed technique is attractive since it leads to a simple implementation. Numerical examples are provided to confirm the theoretical results and to show the feasibility and efficacy of the new methods. The performance is compared with well established methods in literature. Computational results, including the elapsed CPU-time, confirm the accurate and efficient character of proposed techniques. Moreover, the stability of the methods is checked through complex geometry shown by drawing basins of attraction.
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References
Burden, R.L., Faires, J.D.: Numerical Analysis. Brooks/Cole, Boston (2005)
Chapra, S.C., Canale, R.P.: Numerical Methods for Engineers. McGraw-Hill Book Company, New York (1988)
Cordero, A., Torregrosa, J.R.: A class of Steffensen type methods with optimal order of convergence. Appl. Math. Comput. 217, 7653–7659 (2011)
Cordero, A., Hueso, J.L., Martínez, E., Torregrosa, J.R.: Generating optimal derivative free iterative methods for nonlinear equations by using polynomial interpolation. Math. Comput. Model. 57, 1950–1956 (2013)
Cordero, A., Hueso, J.L., Martínez, E., Torregrosa, J.R.: A new technique to obtain derivative-free optimal iterative methods for solving nonlinear equations. J. Comput. Appl. Math. 252, 95–102 (2013)
Danby, J.M.A., Burkardt, T.M.: The solution of Kepler’s equation, I. Celest. Mech. 40, 95–107 (1983)
Džunić, J., Petković, M.S., Petković, L.D.: Three-point methods with and without memory for solving nonlinear equations. Appl. Math. Comput. 218, 4917–4927 (2012)
Hoffman, J.D.: Numerical Methods for Engineers and Scientists. McGraw-Hill Book Company, New York (1992)
Jay, L.O.: A note on Q-order of convergence. BIT 41, 422–429 (2001)
Kung, H.T., Traub, J.F.: Optimal order of one-point and multipoint iteration. J. ACM 21, 643–651 (1974)
Liu, Z., Zheng, Q., Zhao, P.: A variant of Steffensen’s method of fourth-order convergence and its applications. Appl. Math. Comput. 216, 1978–1983 (2010)
Lotfi, T., Sharifi, S., Salimi, M., Siegmund, S.: A new class of three-point methods with optimal convergence order eight and its dynamics. Numer. Algorithms 68, 261–288 (2015)
Ostrowski, A.M.: Solutions of Equations and Systems of Equations. Academic Press, New York-London (1966)
Ren, H., Wu, Q., Bi, W.: A class of two-step Steffensen type methods with fourth-order convergence. Appl. Math. Comput. 209, 206–210 (2009)
Scott, M., Neta, B., Chun, C.: Basin attractors for various methods. Appl. Math. Comput. 218, 2584–2599 (2011)
Sharma, J.R., Arora, H.: Some novel optimal eighth order derivative-free root solvers and their basins of attraction. Appl. Math. Comput. 284, 149–161 (2016)
Sharma, J.R., Gupta, P.: On some highly efficient derivative free methods with and without memory for solving nonlinear equations. Int. J. Comput. Methods 12, 1350093 (2015). https://doi.org/10.1142/S021987621350093X
Soleymani, F.: On a bi-parametric class of optimal eighth-order derivative-free methods. Int. J. Pure Appl. Math. 72, 27–37 (2011)
Soleymani, F., Vanani, S.K.: Optimal Steffensen-type method with eighth order of convergence. Comput. Math. Appl. 62, 4619–4626 (2011)
Steffensen, J.F.: Remarks on iteration. Skand. Aktuarietidskr 16, 64–72 (1933)
Traub, J.F.: Iterative Methods for the Solution of Equations. Prentice-Hall, Englewood Cliffs (1964)
Varona, J.L.: Graphic and numerical comparison between iterative methods. Math. Intell. 24, 37–46 (2002)
Vrscay, E.R., Gilbert, W.J.: Extraneous fixed points, basin boundaries and chaotic dynamics for Schröder and König rational iteration functions. Numer. Math. 52, 1–16 (1988)
Wolfram, S.: The Mathematica Book, 5th edn. Wolfram Media, Champaign (2003)
Yun, B.I.: A non-iterative method for solving non-linear equations. Appl. Math. Comput. 198, 691–699 (2008)
Zheng, Q., Li, J., Huang, F.: An optimal Steffensen-type family for solving nonlinear equations. Appl. Math. Comput. 217, 9592–9597 (2011)
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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 80, 333–352 (2023). https://doi.org/10.1007/s40324-022-00288-z
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DOI: https://doi.org/10.1007/s40324-022-00288-z