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On a Steffensen-like method for solving nonlinear equations

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Abstract

We study a generalization of Steffensen’s method in Banach spaces. Our main aim is to obtain similar convergence as Newton’s method, but without evaluating the first derivative of the operator involved. As motivation, we analyse numerical solutions of boundary-value problems approximated by the multiple shooting method that uses the proposed iterative scheme. Sufficient conditions for the semilocal convergence analysis of the method, including error estimates and the \(R\)-order of convergence, are provided. Finally, the theoretical results are applied to a nonlinear system of equations related with the approximation of a Hammerstein-type integral equation.

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Authors and Affiliations

  1. Department of Applied Mathematics and Statistics, Polytechnic University of Cartagena, Paseo Alfonso XIII 52, 30203, Cartagena, Spain

    S. Amat

  2. Department of Mathematics and Computation, University of La Rioja, Calle Luis de Ulloa s/n, 26004, Logroño, Spain

    J. A. Ezquerro & M. A. Hernández-Verón

Authors
  1. S. Amat
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  2. J. A. Ezquerro
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  3. M. A. Hernández-Verón
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Corresponding author

Correspondence to S. Amat.

Additional information

This work was supported in part by the project MTM2011-28636-C02-01 of the Spanish Ministry of Science and Innovation.

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Amat, S., Ezquerro, J.A. & Hernández-Verón, M.A. On a Steffensen-like method for solving nonlinear equations. Calcolo 53, 171–188 (2016). https://doi.org/10.1007/s10092-015-0142-3

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  • DOI: https://doi.org/10.1007/s10092-015-0142-3

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