Numerical Algorithms

, Volume 68, Issue 1, pp 47–60 | Cite as

An extension of a theorem by Wang for Smale’s α-theory and applications

  • Ángel Alberto MagreñánEmail author
  • Ioannis K. Argyros
Original Paper


We present an extension of a Theorem by Wang (see Math. Comp. 68, 169–186 1999) for Smale’s α-theory [16, 17] in order to approximate a locally unique solution of a nonlinear equation using Newton’s method. The advantages of our approach over the studies such as Argyros (Atti. Semin. Mat. Fis. Univ. Modena Reggio Emilia 56, 31–40 2008/09), Cianciaruso (Numer. Funct. Anal. Optim. 28, 631–645 2007), Rheinboldt (Appl. Math. Lett. 1, 3–42 1988), Shen and Li (J. Math. Anal. Appl. 369, 29–42 2010), [16, 17], Wang and Zhao (J. Comput. Appl. Math. 60, 253–269 1995), Wang (Math. Comp. 68, 169–186 1999) under the same computational cost are: weaker sufficient convergence condition; more precise error estimates on the distances involved and an at least as precise information on the location of the solution. These advantages are obtained by introducing the notion of the center γ 0-condition and using this condition in combination with the γ-condition in the convergence analysis of Newton’s method. Numerical examples and applications are also provided to show that the older convergence criteria are not satisfied but the new convergence criteria are satisfied.


Newton’s method Banach space Semi-local convergence Smale’s α-theory Fréchet-derivative 

AMS Subject Classification Codes

65J15 65H05 90C30 47H17 49M15 


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  1. 1.
    Amat, S., Busquier, S., Negra, M.: Adaptive approximation of nonlinear operators. Numer. Funct. Anal. Optim. 25, 397–405 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Argyros, I.K.: Computational theory of iterative methods. In: Chui, C.K., Wuytack, L. (eds.) Series: Studies in Computational Mathematics, 15. Elsevier, New York (2007)Google Scholar
  3. 3.
    Argyros, I.K.: A new semilocal convergence theorem for Newton’s method under a gamma-type condition. Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia 56, 31–40 (2008/09)Google Scholar
  4. 4.
    Argyros, I.K.: A semilocal convergence analysis for directional Newton methods. Math. Comput. 80, 327–343 (2011)CrossRefzbMATHGoogle Scholar
  5. 5.
    Argyros, I.K., Hilout, S.: Weaker conditions for the convergence of Newton’s method. J. Complex. 28, 364–387 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Argyros, I.K., Hilout, S.: Convergence of Newton’s method under weak majorant condition. J. Comput. Appl. Math. 236, 1892–1902 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Argyros, I.K., Cho, Y.J., Hilout, S.: Numerical methods for equations and its applications. CRC Press/Taylor and Francis Publ., New York (2012)zbMATHGoogle Scholar
  8. 8.
    Cianciaruso, F.: Convergence of Newton-Kantorovich approximations to an approximate zero. Numer. Funct. Anal. Optim. 28, 631–645 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Dedieu, J.P.: Points fixes, zéros et la méthode de Newton, 54. Springer, Berlin (2006)Google Scholar
  10. 10.
    Ezquerro, J.A., Gutiérrez, J.M., Hernández, M.A., Romero, N., Rubio, M.J.: The Newton method: from Newton to Kantorovich. (Spanish) Gac. R. Soc. Math. Esp. 13, 53–76 (2010)zbMATHGoogle Scholar
  11. 11.
    Kantorovich, L.V., Akilov, G.P.: Functional Analysis. Pergamon Press, Oxford (1982)zbMATHGoogle Scholar
  12. 12.
    Ortega, L.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic press, New York (1970)zbMATHGoogle Scholar
  13. 13.
    Proinov, P.D.: New general convergence theory for iterative processes and its applications to Newton-Kantorovich type theorems. J. Complex. 26, 3–42 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Rheinboldt, W.C.: On a theorem of S. Smale about Newton’s method for analytic mappings. Appl. Math. Lett. 1, 3–42 (1988)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Shen, W., Li, C.: Smale’s α-theory for inexact Newton methods under the γ-condition. J. Math. Anal. Appl. 369, 29–42 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Smale, S.: Newton’s Method Estimates from Data at One Point. The Merging of Disciplines: New Directions in Pure, Applied, and Computational Mathematics (Laramie, Wyo., 1985), pp. 185–196. Springer, New York (1986)Google Scholar
  17. 17.
    Smale, S.: Algorithms for solving equations. Proceedings of the International Congress of Mathematicians, (Berkeley, Calif., 1986), vol. 1–2, pp. 172195. Am. Math. Soc., Providence, RI (1987)Google Scholar
  18. 18.
    Wang, D.R., Zhao, F.G.: The theory of Smale’s point estimation and its applications, Linear/nonlinear iterative methods and verification of solution (Matsuyama, 1993). J. Comput. Appl. Math. 60, 253–269 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Wang, X.H.: Convergence of Newton’s method and inverse function theorem in Banach space. Math. Comput. 68, 169–186 (1999 )CrossRefzbMATHGoogle Scholar
  20. 20.
    Yakoubsohn, J.C.: Finding zeros of analytic functions: α–theory for Secant type method. J. Complex. 15, 239–281 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Zabrejko, P.P., Nguen, D.F.: The majorant method in the theory of Newton-Kantorovich approximations and the Pták error estimates. Numer. Funct. Anal. Optim. 9, 671–684 (1987)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Ángel Alberto Magreñán
    • 1
    Email author
  • Ioannis K. Argyros
    • 2
  1. 1.Departamento de Matemáticas y ComputaciónUniversidad de La RiojaLogroñoSpain
  2. 2.Department of Mathematics SciencesCameron UniversityLawtonUSA

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