Journal of the Chungcheong Mathematical Society (충청수학회지)

The Chungcheong Mathematical Society (충청수학회)
권호 표지
DOI QR코드

DOI QR Code

LOCAL CONVERGENCE RESULTS FOR NEWTON'S METHOD

  • Argyros, Ioannis K. (Cameron University Department of Mathematical Sciences Lawton) ;
  • Hilout, Said (Poitiers University Laboratoire de Mathematiques et Applications Bd.)
  • Published : 2012.05.15
⟨ Previous Next ⟩

Abstract

We present new results for the local convergence of Newton's method to a unique solution of an equation in a Banach space setting. Under a flexible gamma-type condition [12], [13], we extend the applicability of Newton's method by enlarging the radius and decreasing the ratio of convergence. The results can compare favorably to other ones using Newton-Kantorovich and Lipschitz conditions [3]-[7], [9]-[13]. Numerical examples are also provided.

Keywords

References

  1. E. L. Allgower, K. Bohmer, F. A. Potra, W.C. Rheinboldt, A mesh independence principle for operator equations and their discretizations, SIAM J. Numer. Anal. 23 (1986), 160-169. https://doi.org/10.1137/0723011
  2. S. Amat, S. Busquier, Convergence and numerical analysis of a family of two-step Steffensen's method, Comput. Math. Appl. 49 (2005), 13-22. https://doi.org/10.1016/j.camwa.2005.01.002
  3. I. K. Argyros, On the Newton-Kantorovich hypothesis for solving equations, J. Comput. Appl. Math. 169 (2004), 315-332. https://doi.org/10.1016/j.cam.2004.01.029
  4. I. K. Argyros, On the secant method for solving nonsmooth equations, J. Math. Anal. Appl. 322 (2006), 146-157. https://doi.org/10.1016/j.jmaa.2005.09.006
  5. I. K. Argyros, A unifying local-semilocal convergence analysis and applications for two{point Newton-like methods in Banach space, J. Math. Anal. Appl. 298 (2004), 374-397. https://doi.org/10.1016/j.jmaa.2004.04.008
  6. I. K. Argyros, Computational Theory of Iterative Methods, Studies in Computa- tional Mathematics, 15, Elsevier, 2007, New York, U.S.A.
  7. P. N. Brown, A local convergence theory for combined inexact-Newton/finite dif- ference projection methods, SIAM J. Numer. Anal. 24 (1987), 407-434. https://doi.org/10.1137/0724031
  8. S. Chandrasekhar, Radiative Transfer, Dover Publ. New York, 1960.
  9. J. M. Gutierrez, M. A. Hernanadez, M. A. Salanova, Accessibility of solutions by Newton's method, Inter. J. Comput. Math. 57 (1995), 239-241. https://doi.org/10.1080/00207169508804427
  10. L. V. Kantorovich, G. P. Akilov, Functional Analysis in Normed Spaces, Perga- mon Press, Oxford, 1982.
  11. W. C. Rheinboldt, An adaptive continuation process for solving systems of non- linear equations, Banach Center Publ. 3 (1975), 129-142.
  12. X. H. Wang, Convergence on the iteration of Halley family in weak conditions, Chinese Science Bulletin 42 (1997), 552-556. https://doi.org/10.1007/BF03182614
  13. D. Wang, F. Zhao, The theory of Smale's point estimation and its applications, J. Comput. Appl. Math. 60 (1995), 253-269. https://doi.org/10.1016/0377-0427(94)00095-I
  14. T. J. Ypma, Local convergence of inexact Newton methods, SIAM J. Numer. Anal. 21 (1984), 583-590. https://doi.org/10.1137/0721040