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Journal of Mathematical Chemistry

, Volume 54, Issue 7, pp 1404–1416 | Cite as

Local convergence and a chemical application of derivative free root finding methods with one parameter based on interpolation

  • Ioannis K. Argyros
  • Á. Alberto MagreñánEmail author
  • Lara Orcos
Original Paper

Abstract

We present a local convergence analysis of a derivative free fourth order method with one parameter based on rational interpolation in order to approximate a locally unique root of a function. The method is optimal in the sense of Traub. In earlier studies such as Steffensen (Scand Actuar J 16(1):64–72, 1933) and Zafer et al. (Sci World J, 2015. doi: 10.1155/2015/934260) the convergence was based on hypotheses on the third derivative or even higher. We extend the applicability of theses methods using only the first derivative. Moreover, we provide computable radii and error bounds based on Lipschitz constants. Furthermore, the dynamics of this method are studied in order to find the best choice of the parameter in terms of convergence. An application is also presented in this study.

Keywords

Fourth order method Rational interpolation Local convergence Divided difference Dynamics 

Mathematics Subject Classification

65D10 65D99 

Notes

Acknowledgments

This work is partially funded by UNIR Research (http://research.unir.net), Universidad Internacional de La Rioja (UNIR, http://www.unir.net), under the Research Support Strategy 3 [2015-2017], Research Group: MOdelación Matemática Aplicada a la INgeniería (MOMAIN), by the Grant SENECA 19374/PI/14 and by the project MTM2014-52016-C2-1-P of the Spanish Ministry of Economy and Competitiveness.

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Ioannis K. Argyros
    • 1
  • Á. Alberto Magreñán
    • 2
    Email author
  • Lara Orcos
    • 2
  1. 1.Department of Mathematics SciencesCameron UniversityLawtonUSA
  2. 2.Universidad Internacional de La RiojaLogroñoSpain

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