The Pure and Applied Mathematics (한국수학교육학회지시리즈B:순수및응용수학)

Korean Society of Mathematical Education (한국수학교육학회)
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CONVERGENCE THEOREMS FOR NEWTON'S AND MODIFIED NEWTON'S METHODS

  • Published : 2009.11.30
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Abstract

In this study we are concerned with the problem of approximating a locally unique solution of an equation in a Banach space setting using Newton's and modified Newton's methods. We provide weaker convergence conditions for both methods than before [5]-[7]. Then, we combine Newton's with the modified Newton's method to approximate locally unique solutions of operator equations. Finer error estimates, a larger convergence domain, and a more precise information on the location of the solution are obtained under the same or weaker hypotheses than before [5]-[7]. The results obtained here improve our earlier ones reported in [4]. Numerical examples are also provided.

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References

  1. 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
  2. I.K. Argyros: Computational theory of iterative methods. Studies in Computational Mathematics, 15 (Editors: C.K. Chui and L. Wuytack), Elsevier Publ. Co., New York, 2007.
  3. 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), no. 2, 374-397. https://doi.org/10.1016/j.jmaa.2004.04.008
  4. I.K. Argyros: A comparison between convergence theorems for Newton’s and modified Newton’s methods. submitted for publication.
  5. J.M. Gutierrez, M.A. Hernandez & M.A. Salanova: Accessibility of solutions by Newton’s method. Intern. J. Comput. Math. 57 (1995), 239-247. https://doi.org/10.1080/00207169508804427
  6. L.V. Kantorovich & G.P. Akilov: Functional Analysis. Pergamon Press, Oxford, 1982.
  7. M.A. Krasnosel’skii, G.M. Vainikko, P.P. Zabreiko, Ya.B. Rutitskii & V.Ya. Stetsenko: Approximate Solution of Operator Equations. Wolters-Noordhoff Publ., Groningen, 1972.