Newton-type methods of high order and domains of semilocal and global convergence

  1. Ezquerro, J.A. 1
  2. Hernández, M.A. 1
  3. Romero, N. 1
  1. 1 Universidad de La Rioja
    info

    Universidad de La Rioja

    Logroño, España

    ROR https://ror.org/0553yr311

Journal:
Applied Mathematics and Computation

ISSN: 0096-3003

Year of publication: 2009

Volume: 214

Issue: 1

Pages: 142-154

Type: Article

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DOI: 10.1016/J.AMC.2009.03.072 SCOPUS: 2-s2.0-67349140698 WoS: WOS:000267585200018 GOOGLE SCHOLAR

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Abstract

We present the geometric construction of some classical iterative methods that have global convergence and "infinite" speed of convergence when they are applied to solve certain nonlinear equations f (t) = 0. In particular, for nonlinear equations with the degree of logarithmic convexity of f′, Lf′ (t) = f′ (t) f‴ (t) / f″ (t)2, is constant, a family of Newton-type iterative methods of high orders of convergence is constructed. We see that this family of iterations includes the classical iterative methods. The convergence of the family is studied in the real line and the complex plane, and domains of semilocal and global convergence are located. © 2009 Elsevier Inc. All rights reserved.