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Calcolo
Published in: Calcolo 2/2016

01-06-2016

On a Steffensen-like method for solving nonlinear equations

Authors: S. Amat, J. A. Ezquerro, M. A. Hernández-Verón

Published in: Calcolo | Issue 2/2016

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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.
previous article A self-adaptive three-term conjugate gradient method for monotone nonlinear equations with convex constraints
next article Solving nonlinear equations by a derivative-free form of the King’s family with memory
Literature
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Alarcón, V., Amat, S., Busquier, S., López, D.J.: A Steffensen’s type method in Banach spaces with applications on boundary-value problems. J. Comput. Appl. Math. 216(1), 243–250 (2008)MathSciNetCrossRefMATH
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Amat, S., Busquier, S.: A two-step Steffensen’s method under modified convergence conditions. J. Math. Anal. Appl. 324(2), 1084–1092 (2006)MathSciNetCrossRefMATH
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Ezquerro, J.A., Hernández, M.A., Romero, N., Velasco, A.I.: On Steffensen’s method on Banach spaces. J. Comput. Appl. Math. 249, 9–23 (2013)MathSciNetCrossRefMATH
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Metadata
Title
On a Steffensen-like method for solving nonlinear equations
Authors
S. Amat
J. A. Ezquerro
M. A. Hernández-Verón
Publication date
01-06-2016
Publisher
Springer Milan
Published in
Calcolo / Issue 2/2016
Print ISSN: 0008-0624
Electronic ISSN: 1126-5434
DOI
https://doi.org/10.1007/s10092-015-0142-3

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