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Accurate fourteenth-order methods for solving nonlinear equations

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Abstract

We establish new iterative methods of local order fourteen to approximate the simple roots of nonlinear equations. The considered three-step eighth-order construction can be viewed as a variant of Newton’s method in which the concept of Hermite interpolation is used at the third step to reduce the number of evaluations. This scheme includes three evaluations of the function and one evaluation of the first derivative per iteration, hence its efficiency index is 1.6817. Next, the obtained approximation for the derivative of the Newton’s iteration quotient is again taken into consideration to furnish novel fourteenth-order techniques consuming four function and one first derivative evaluations per iteration. In providing such new fourteenth-order methods, we also take a special heed to the computational burden. The contributed four-step methods have 1.6952 as their efficiency index. Finally, various numerical examples are given to illustrate the accuracy of the developed techniques.

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References

  1. Chun, C., Ham, Y.: Some sixth-order variants of Ostrowski root finding methods. Appl. Math. Comput. 193, 389–394 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Cordero, A., Hueso, J.L., Martinez, M., Torregrosa, J.R.: Efficient three-step iterative methods with sixth order convergence for nonlinear equations. Numer. Algorithms 53, 485–495 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Frontini, M., Sormani, E.: Modified Newton’s method with third order convergence and multiple roots. J. Comput. Appl. Math. 156, 345–354 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Galantai, A., Hegedus, C.J.: A study of accelerated Newton methods for multiple polynomial roots. Numer. Algorithms 54, 219–243 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Geum, Y.H., Kim, Y.I.: A multi-parameter family of three-step eighth-order iterative methods locating a simple root. Appl. Math. Comput. 215, 3375–3382 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Grau-Sanchez, M.: Improving order and efficiency: composition with a modified Newton’s method. J. Comput. Appl. Math. 231, 592–597 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Haijun, W.: On new third-order convergent iterative formulas. Numer. Algorithms 48, 317–325 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Kou, J., Li, Y., Wang, X.: A family of fifth-order iterations composed of Newton and third-order methods. Appl. Math. Comput. 186, 1258–1262 (2007)

    Article  MathSciNet  MATH  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)

    MathSciNet  MATH  Google Scholar 

  10. Kyurkchiev, N., Iliev, A.: A note on the constructing of nonstationary methods for solving nonlinear equations with raised speed of convergence. Serdica J. Comput. 3, 47–74 (2009)

    MathSciNet  MATH  Google Scholar 

  11. Lancaster, P.: Error analysis for the Newton–Raphson method. Numer. Math. 9, 55–68 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  12. Petkovic, M.S.: On a general class of multipoint root-finding methods of high computational efficiency. SIAM J. Numer. Anal. 47, 4402–4414 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Sauer, T.: Numerical Analysis. Pearson, Boston (2006)

    Google Scholar 

  14. Sharma, J.R., Sharma, R.: A new family of modified Ostrowski’s methods with accelerated eighth order convergence. Numer. Algorithms 54, 445–458 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  16. Wang, X., Kou, J., Li, Y.: Modified Jarratt method with sixth-order convergence. Appl. Math. Lett. 22, 1798–1802 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Wang, X., Liu, L.: Modified Ostrowski’s method with eighth-order convergence and high efficiency index. Appl. Math. Lett. 23, 549–554 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  19. Yun, B.I., Petkovic, M.S.: Iterative methods based on the Signum function approach for solving nonlinear equations. Numer. Algorithms 52, 649–662 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran

    Parviz Sargolzaei

  2. Young Researchers Club, Zahedan Branch, Islamic Azad University, Zahedan, Iran

    Fazlollah Soleymani

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  1. Parviz Sargolzaei
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  2. Fazlollah Soleymani
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Correspondence to Fazlollah Soleymani.

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Sargolzaei, P., Soleymani, F. Accurate fourteenth-order methods for solving nonlinear equations. Numer Algor 58, 513–527 (2011). https://doi.org/10.1007/s11075-011-9467-4

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  • DOI: https://doi.org/10.1007/s11075-011-9467-4

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