• Title/Summary/Keyword: Newton-Kantorovich method

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CHEYSHEFF-HALLEY-LIKE METHODS IN BANACH SPACES

  • Argyros, Ioannis-K.
    • Journal of applied mathematics & informatics
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    • v.4 no.1
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    • pp.83-108
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    • 1997
  • Chebysheff-Halley methods are probably the best known cubically convergent iterative procedures for solving nonlinear equa-tions. These methods however require an evaluation of the second Frechet-derivative at each step which means a number of function eval-uations proportional to the cube of the dimension of the space. To re-duce the computational cost we replace the second Frechet derivative with a fixed bounded bilinear operator. Using the majorant method and Newton-Kantorovich type hypotheses we provide sufficient condi-tions for the convergence of our method to a locally unique solution of a nonlinear equation in Banach space. Our method is shown to be faster than Newton's method under the same computational cost. Finally we apply our results to solve nonlinear integral equations appearing in radiative transfer in connection with the problem of determination of the angular distribution of the radiant-flux emerging from a plane radiation field.

APPROXIMATING SOLUTIONS OF EQUATIONS BY COMBINING NEWTON-LIKE METHODS

  • Argyros, Ioannis K.
    • The Pure and Applied Mathematics
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    • v.15 no.1
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    • pp.35-45
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    • 2008
  • In cases sufficient conditions for the semilocal convergence of Newtonlike methods are violated, we start with a modified Newton-like method (whose weaker convergence conditions hold) until we stop at a certain finite step. Then using as a starting guess the point found above we show convergence of the Newtonlike method to a locally unique solution of a nonlinear operator equation in a Banach space setting. A numerical example is also provided.

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AN IMPROVED UNIFYING CONVERGENCE ANALYSIS OF NEWTON'S METHOD IN RIEMANNIAN MANIFOLDS

  • Argyros, Ioannis K.
    • Journal of applied mathematics & informatics
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    • v.25 no.1_2
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    • pp.345-351
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    • 2007
  • Using more precise majorizing sequences we provide a finer convergence analysis than before [1], [7] of Newton's method in Riemannian manifolds with the following advantages: weaker hypotheses, finer error bounds on the distances involved and a more precise information on the location of the singularity of the vector field.

SEMILOCAL CONVERGENCE THEOREMS FOR A CERTAIN CLASS OF ITERATIVE PROCEDURES

  • Ioannis K. Argyros
    • Journal of applied mathematics & informatics
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    • v.7 no.1
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    • pp.29-40
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    • 2000
  • We provide semilocal convergence theorems for Newton-like methods in Banach space using outer and generalized inverses. In contrast to earlier results we use hypotheses on the second instead of the first Frechet-derivative. This way our Newton-Kantorovich hypotheses differ from earlier ones. Our results can be used to solve undetermined systems, nonlinear least square problems and ill-posed nonlinear operator equations.

CONVERGENCE THEOREMS FOR NEWTON'S AND MODIFIED NEWTON'S METHODS

  • Argyros, Ioannis K.
    • The Pure and Applied Mathematics
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    • v.16 no.4
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    • pp.405-416
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    • 2009
  • 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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ON THE "TERRA INCOGNITA" FOR THE NEWTON-KANTROVICH METHOD WITH APPLICATIONS

  • Argyros, Ioannis Konstantinos;Cho, Yeol Je;George, Santhosh
    • Journal of the Korean Mathematical Society
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    • v.51 no.2
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    • pp.251-266
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    • 2014
  • In this paper, we use Newton's method to approximate a locally unique solution of an equation in Banach spaces and introduce recurrent functions to provide a weaker semilocal convergence analysis for Newton's method than before [1]-[13], in some interesting cases, provided that the Fr$\acute{e}$chet-derivative of the operator involved is p-H$\ddot{o}$lder continuous (p${\in}$(0, 1]). Numerical examples involving two boundary value problems are also provided.

LOCAL CONVERGENCE RESULTS FOR NEWTON'S METHOD

  • Argyros, Ioannis K.;Hilout, Said
    • Journal of the Chungcheong Mathematical Society
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    • v.25 no.2
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    • pp.267-275
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    • 2012
  • 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.