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Semilocal convergence of a sixth-order Jarratt method in Banach spaces

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Abstract

In this paper, we study the semilocal convergence for a sixth-order variant of the Jarratt method for solving nonlinear equations in Banach spaces. The semilocal convergence of this method is established by using recurrence relations. We derive the recurrence relations for the method, and then prove an existence-uniqueness theorem, along with a priori error bounds which demonstrates the R-order of the method. Finally, we give some numerical applications to demonstrate our approach.

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References

  1. Ahmad, F., Hussain, S., Mir, N.A., Rafiq, A.: New sixth order jarratt method for solving nonlinear equations. Int. J. Appl. Math. Mech 5(5), 27–35 (2009)

    Google Scholar 

  2. Argyros, I.K., Hilout, S.: A convergence analysis for directional two-step Newton methods. Numer. Algor. 55(4), 503–528 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Babajee, D.K.R., Dauhoo, M.Z.: Spectral analysis of the errors of some families of multi-step Newton-like methods. Numer. Algor. 52(1), 25–46 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Candela, V., Marquina, A.: Recurrence relations for rational cubic methods I: the Halley method. Computing 44(2), 169–184 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  5. Candela, V., Marquina, A.: Recurrence relations for rational cubic methods II: the Chebyshev method. Computing 45(4), 355–367 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chun, C.: Some improvements of Jarratt’s method with sixth-order convergence. Appl. Math. Comput. 190(2), 1432–1437 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ezquerro, J.A., Hernández, M.A.: Recurrence relations for Chebyshev-type methods. Appl. Math. Optim. 41(2), 227–236 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ezquerro, J.A., Hernández, M.A.: New iterations of R-order four with reduced computational cost. BIT Numer. Math. 49(2), 325–342 (2009)

    Article  MATH  Google Scholar 

  9. Gutiérrez, J.M., Hernández, M.A.: Recurrence relations for the super-Halley method. Comput. Math. Appl. 36(7), 1–8 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hernández, M.A.: Chebyshev’s approximation algorithms and applications. Comput. Math. Appl. 41(3–4), 433–445 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hernandez, M.A., Salanova, M.A.: Sufficient conditions for semilocal convergence of a fourth order multipoint iterative method for solving equations in Banach spaces. Southwest J. Pure Appl. Math. 1, 29–40 (1999)

    MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  13. Kou, J., Li, Y.: An improvement of the Jarratt method. Appl. Math. Comput. 189(2), 1816–1821 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Parhi, S.K., Gupta, D.K.: Semilocal convergence of a stirling-like method in banach spaces. Int. J. Comput. Methods 7(02), 215–228 (2010)

    Article  MathSciNet  Google Scholar 

  15. Parida, P.K., Gupta, D.K.: Recurrence relations for a Newton-like method in Banach spaces. J. Comput. Appl. Math. 206(2), 873–887 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Rall, L.B.: Computational Solution of Nonlinear Operator Equations. Robert E. Krieger, New York (1979)

    MATH  Google Scholar 

  17. Ren, H., Wu, Q., Bi, W.: New variants of Jarratt’s method with sixth-order convergence. Numer. Algor. 52(4), 585–603 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Wang, X., Gu, C., Kou, J.: Semilocal convergence of a multipoint fourth-order super-Halley method in Banach spaces. Numer. Algor. doi:10.1007/s11075-010-9401-1 (2010)

    Google Scholar 

  19. Wang, X., Kou, J., Li, Y.: A variant of Jarratt method with sixth-order convergence. Appl. Math. Comput. 204(1), 14–19 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Wang, X., Kou, J., Li, Y.: Modified Jarratt method with sixth-order convergence. Appl. Math. Lett. 22(12), 1798–1802 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ye, X., Li, C.: Convergence of the family of the deformed Euler-Halley iterations under the Hölder condition of the second derivative. J. Comput. Appl. Math. 194(2), 294–308 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ye, X., Li, C., Shen, W.: Convergence of the variants of the Chebyshev-Halley iteration family under the Hölder condition of the first derivative. J. Comput. Appl. Math. 203(1), 279–288 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  23. Zhao, Y., Wu, Q.: Newton-Kantorovich theorem for a family of modified Halley’s method under Hölder continuity conditions in Banach space. Appl. Math. Comput. 202(1), 243–251 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Shanghai University, Shanghai, 200444, People’s Republic of China

    Xiuhua Wang, Jisheng Kou & Chuanqing Gu

  2. Department of Mathematics, Xiaogan University, Xiaogan, 432100, Hubei, People’s Republic of China

    Xiuhua Wang

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  1. Xiuhua Wang
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  2. Jisheng Kou
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  3. Chuanqing Gu
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Correspondence to Jisheng Kou.

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Wang, X., Kou, J. & Gu, C. Semilocal convergence of a sixth-order Jarratt method in Banach spaces. Numer Algor 57, 441–456 (2011). https://doi.org/10.1007/s11075-010-9438-1

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  • DOI: https://doi.org/10.1007/s11075-010-9438-1

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