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Local convergence theorems for Newton methods

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Affine invariant sufficient conditions are given for two local convergence theorems involving inexact Newton-like methods. The first uses conditions on the first Fréchet-derivative whereas the second theorem employs hypotheses on themth (m ≥ 2 an integer). Radius of convergence as well as rate of convergence results are derived. Results involving superlinear convergence and known to be true for inexact Newton methods are extended here. Moreover, we show that under hypotheses on the mth Fréchet-derivative our radius of convergence can sometimes be larger than the corresponding one in [10]. This allows a wider choice for the initial guess. A numerical example is also provided to show that our radius of convergence is larger than the one in [10].

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Correspondence to Ioannis K. Argyros.

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Argyros, I.K. Local convergence theorems for Newton methods. Korean J. Comput. & Appl. Math. 8, 253–268 (2001). https://doi.org/10.1007/BF02941964

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AMS Mathematics Subject Classification

  • 65B05
  • 47H17
  • 49D15

Key words and phrases

  • Inexact Newton-like method
  • Banach space
  • radius of convergence
  • rate of convergence
  • Fréchet-derivative
  • superlinear
  • strong
  • weak convergence