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Semilocal convergence and R-order for modified Chebyshev-Halley methods

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Abstract

In this paper, we study the semilocal convergence and R-order for a class of modified Chebyshev-Halley methods for solving non-linear equations in Banach spaces. To solve the problem that the third-order derivative of an operator is neither Lipschitz continuous nor Hölder continuous, the condition of Lipschitz continuity of third-order Fréchet derivative considered in Wang et al. (Numer Algor 56:497–516, 2011) is replaced by its general continuity condition, and the latter is weaker than the former. Furthermore, the R-order of these methods is also improved under the same condition. By using the recurrence relations, a convergence theorem is proved to show the existence-uniqueness of the solution and give a priori error bounds. We also analyze the R-order of these methods with the third-order Fréchet derivative of an operator under different continuity conditions. Especially, when the third-order Fréchet derivative is Lipschitz continuous, the R-order of the methods is at least six, which is higher than the one of the method considered in Wang et al. (Numer Algor 56:497–516, 2011) under the same condition.

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References

  1. Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equation in Several Variables. Academic Press, New York (1970)

    Google Scholar 

  2. Gutiérrez, J.M., Hernández, M.A.: A family of Chebyshev-Halley type methods in Banach spaces. Bull. Aust. Math. Soc. 55, 113–130 (1997)

    Article  MATH  Google Scholar 

  3. Amat, S., Busquier, S., Gutiérrez, J.M.: Geometric constructions of iterative functions to solve nonlinear equations. J. Comput. Appl. Math 157, 197–205 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Hernández, M.A.: Second-derivative-free variant of the Chebyshev method for nonlinear equations. J. Optim. Theory Appl. 104(3), 501–515 (2000)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  7. Hernández, M.A.: Reduced recurrence relations for the Chebyshev method. J. Optim. Theory Appl. 98, 385–397 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hernández, M.A., Salanova, M.A.: Modification of the Kantorovich assumptions for semilocal convergence of the Chebyshev method. J. Comput. Appl. Math. 126, 131–143 (2000)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. Argyros, I.K., Chen, D.: Results on the Chebyshev method in Banach spaces. Proyecciones 12(2), 119–128 (1993)

    MathSciNet  MATH  Google Scholar 

  12. Alefeld, G.: On the convergence of Halley’s method. Amer. Math. Monthly 88, 530–536 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chen, D., Argyros, I.K., Qian, Q.S.: A note on the Halley method in Banach spaces. App. Math. Comput. 58, 215–224 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ezquerro, J.A., Hernández, M.A.: On the R-order of the Halley method. J. Math. Anal. Appl. 303, 591–601 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. Xu, X., Ling, Y.: Semilocal convergence for Halley’s method under weak Lipschitz condition. Appl. Math. Comput. 215, 3057–3067 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gutiérrez, J.M., Hernández, M.A.: Third-order iterative methods for operators with bounded second derivative. J. Comput. Appl. Math. 82, 171–183 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  17. 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 

  18. Wang, X., Gu, C., Kou, J.: Semilocal convergence of a multipoint fourth-order super-Halley method in Banach spaces. Numer. Algor. 56, 497–516 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ganesh, M., Joshi, M.C.: Numerical solvability of Hammerstein integral equations of mixed type. IMA J. Numer. Anal. 11, 21–31 (1991)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  21. Bruns, D.D., Bailey, J.E.: Nonlinear feedback control for operating a nonisothermal CSTR near an unstable steady state. Chem. Eng. Sci. 32, 257–264 (1977)

    Article  Google Scholar 

  22. Ezquerro, J.A., Hernández, M.A.: On the R-order of the Halley method. J. Math. Anal. Appl. 303, 591–601 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kou, J., Li, Y.: Some variants of Chebyshev-Halley methods with fifth-order convergence. Appl. Math. Comput. 189, 49–54 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kou, J.: Some new sixth-order methods for solving non-linear equations. Appl. Math. Comput. 189, 647–651 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Zhengzhou University, Zhengzhou, 450052, China

    Xiuhua Wang

  2. School of Mathematics and Statistics, Hubei Engineering University, Xiaogan, 432100, Hubei, China

    Jisheng Kou

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  1. Xiuhua Wang
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  2. Jisheng Kou
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Correspondence to Jisheng Kou.

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Wang, X., Kou, J. Semilocal convergence and R-order for modified Chebyshev-Halley methods. Numer Algor 64, 105–126 (2013). https://doi.org/10.1007/s11075-012-9657-8

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  • DOI: https://doi.org/10.1007/s11075-012-9657-8

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