Extending the applicability of Newton’s method by improving a local result due to Dennis and Schnabel

Abstract

We present a tighter local convergence result for Newton’s method under generalized conditions of Kantorovich type than the one given by Dennis and Schnabel (Numerical methods for unconstrained optimization and nonlinear equations. SIAM, Philadelphia, 1996) and by Ezquerro et al. (J Comput Appl Math 236:2246–2258, 2012) by using more precise majorizing functions and sequences. These improvements are obtained under the same computational cost as in the earlier studies. Numerical examples are also provided to show that the new convergence radii are larger and the new error bounds are tighter than the older ones.

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References

  1. 1.

    Argyros, I.K.: A Newton–Kantorovich theorem for equations involving \(m\)-Fréchet differentiable operators and applications in radiative transfer. J. Comput. Appl. Math. 131, 149–159 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  2. 2.

    Argyros, I.K.: On the Newton–Kantorovich hypothesis for solving equations. J. Comput. Appl. Math. 169(2), 315–332 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  3. 3.

    Argyros, I.K., Cho, Y.J., Hilout, S.: Numerical methods for equations and its applications. CRC Press/Taylor and Francis Publications, New York (2012)

    MATH  Google Scholar 

  4. 4.

    Dennis, J.E., Schnabel, R.B.: Numerical methods for unconstrained optimization and nonlinear equations. SIAM, Philadelphia (1996)

    Book  MATH  Google Scholar 

  5. 5.

    Ezquerro, J.A., González, D., Hernández, M.A.: Majorizing sequences for Newton’s method from initial value problems. J. Comput. Appl. Math. 236, 2246–2258 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  6. 6.

    Ezquerro, J.A., González, D., Hernández, M.A.: On the local convergence of Newton’s method under generalized conditions of Kantorovich. Appl. Math. Lett. 26, 566–570 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  7. 7.

    Ezquerro, J.A., Hernández, M.A.: Generalized differentiability conditions for Newton’s method. IMA J. Numer. Anal. 26, 566–570 (2013)

    MATH  Google Scholar 

  8. 8.

    Kantorovich, L.V.: The majorant principle and Newton’s method. Doklady Akademii Nauk SSSR 76, 17–20 (1951) (In Russian)

    Google Scholar 

  9. 9.

    Potra, F.A., Pták, V.: Nondiscrete induction and iterative processes. Pitman Publishing Limited, London (1984)

    MATH  Google Scholar 

Download references

Acknowledgments

This scientific work has been supported by the ‘Proyecto Prometeo’ of the Ministry of Higher Education Science, Technology and Innovation of the Republic of Ecuador.

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Correspondence to D. González.

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Argyros, I.K., González, D. Extending the applicability of Newton’s method by improving a local result due to Dennis and Schnabel. SeMA 63, 53–63 (2014). https://doi.org/10.1007/s40324-014-0011-z

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Keywords

  • Newton’s method
  • Local convergence
  • Order convergence

Mathematics Subject Classification (2000)

  • 47H99
  • 65J15