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Efficient optimal eighth-order derivative-free methods for nonlinear equations

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Abstract

This paper opens the issue of solving a nonlinear equation by iterative methods in which no derivative is required per iteration. Hence, an optimal class of three-step methods without memory is demonstrated, where each of its elements (methods) requires only four function evaluations per iteration to achieve the eighth order of convergence. In order to verify the effectiveness of the proposed methods, we employ numerical examples. The attained results are consistent with the developed theory. We finally present an algorithm to capture all the real simple solutions of nonlinear functions in an interval.

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References

  1. Alefeld G.: Verified numerical computation for nonlinear equations. Jpn. J. Ind. Appl. Math. 26, 297–315 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Dayton B.H., Li T.-Y., Zeng Z.: Multiple zeros of nonlinear systems. Math. Comput. 80, 2143–2168 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Iliev A., Kyurkchiev N., Fang Q.: On a generalization of the Euler–Chebyshev method for simultaneous extraction of only a part of all roots of polynomials. Jpn. J. Ind. Appl. Math. 23, 63–73 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Johnson T., Tucker W.: Enclosing all zeros of an analytic function—a rigorous approach. J. Comput. Appl. Math. 228, 418–423 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Kalantari B.: Polynomial root-finding and polynomiograpghy. World Scientific, Singapore (2009)

    Google Scholar 

  6. Kanno S., Kjurkchiev N.V., Yamamoto T.: On some methods for the simultaneous determination of polynomial zeros. Jpn. J. Ind. Appl. Math. 13, 267–288 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  7. Khattri S.K., Argyros I.K.: Sixth order derivative free family of iterative methods. Appl. Math. Comput. 217, 5500–5507 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Keiper J.B.: Interval arithmetic in Mathematica. Math. J. 5, 66–71 (1995)

    Google Scholar 

  9. Kung H.T., Traub J.F.: Optimal order of one-point and multipoint iteration. J. Assoc. Comput. Math. 21, 643–651 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  10. McClure, M.: Newton’s method for complex polynomials. Math. Edu. Res. 11 (2006)

  11. Petkovic M.S., Ilic S., Dzunic J.: Derivative free two-point methods with and without memory for solving nonlinear equations. Appl. Math. Comput. 217, 1887–1895 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Polyak B.T.: Newton’s method and its use in optimization. Eur. J. Oper. Res. 181, 1086–1096 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Rahimian S.K., Jalali F., Seader J.D., White R.E.: A new homotopy for seeking all real roots of a nonlinear equation. Comput. Chem. Eng. 35, 403–411 (2011)

    Article  Google Scholar 

  14. 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)

    Article  MathSciNet  MATH  Google Scholar 

  15. Schroder E.: Uber unendlich viele algorithmen zur Auflösung der Gleichungen. Math. Ann. 2, 317–365 (1870)

    Article  MathSciNet  Google Scholar 

  16. Soleymani F., Hosseinabadi V.: New third- and sixth-order derivative-free techniques for nonlinear equations. J. Math. Res. 3, 107–112 (2011)

    MATH  Google Scholar 

  17. Soleymani, F.: Optimized Steffensen-type methods with eighth-order convergence and high efficiency index. Int. J. Math. Math. Sci. 2012, Article ID 932420 (2012)

  18. Soleymani F.: Optimal fourth-order iterative methods free from derivative. Miskolc Math. Notes 12, 255–264 (2011)

    MathSciNet  MATH  Google Scholar 

  19. Soleymani F., Mousavi B.S.: On novel classes of iterative methods for solving nonlinear equations. Comput. Math. Math. Phys. 52, 203–210 (2012)

    Article  MathSciNet  Google Scholar 

  20. Soleymani, F., Karimi Vanani, S., Jamali Paghaleh M.: A class of three-step derivative-free root-solvers with optimal convergence order, J. Appl. Math. 2012, Article ID 568740 (2012)

  21. Soleymani F.: Optimal eighth-order simple root-finders free from derivative. WSEAS Tran. Inf. Sci. Appl. 8, 293–299 (2011)

    Google Scholar 

  22. Soleymani F., Soleimani F.: Novel computational derivative-free methods for simple roots. Fixed Point Theory 13, 247–258 (2012)

    MathSciNet  Google Scholar 

  23. Steffensen J.F.: Remarks on iteration. Skand. Aktuarietidskr 16, 64–72 (1933)

    Google Scholar 

  24. Traub, J.F.: Iterative Methods for the Solution of Equations. Prentice Hall, New York (1964)

  25. Wolfram, S.: The Mathematica Book, 5th edition. Wolfram Media, Illinois (2003)

  26. Zheng Q., Zhao P., Huang F.: A family of fourth-order Steffensen-type methods with the applications on solving nonlinear ODEs. Appl. Math. Comput. 217, 8196–8203 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  27. http://mathematica.stackexchange.com/questions/5663/aboutmultirootsearchinmathematicafortranscendentalequations?lq=1

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Authors and Affiliations

  1. Department of Mathematics, Islamic Azad University, Zahedan Branch, Zahedan, Iran

    Fazlollah Soleymani

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  1. Fazlollah Soleymani
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Correspondence to Fazlollah Soleymani.

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Soleymani, F. Efficient optimal eighth-order derivative-free methods for nonlinear equations. Japan J. Indust. Appl. Math. 30, 287–306 (2013). https://doi.org/10.1007/s13160-013-0103-7

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  • DOI: https://doi.org/10.1007/s13160-013-0103-7

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