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Improved semilocal convergence analysis in Banach space with applications to chemistry


We present a new semilocal convergence analysis for Secant methods in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Our analysis includes the computation of the bounds on the limit points of the majorizing sequences involved. Under the same computational cost on the parameters involved our convergence criteria are weaker and the error bounds more precise than in earlier studies. A numerical example is also presented to illustrate the theoretical results obtained in this study.

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This research was supported by Universidad Internacional de La Rioja (UNIR,, under the Plan Propio de Investigación, Desarrollo e Innovación [2015–2017]. Research group: Modelación matemática aplicada a la ingeniería(MOMAIN), by the the grant SENECA 19374/PI/14 and by Ministerio de Ciencia y Tecnología MTM2014-52016-C2-\(\{01\}\)-P.

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Correspondence to Á. A. Magreñán.

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This is one of several papers published together in Journal of Mathematical Chemistry on the “Special Issue: CMMSE 2017”.

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Argyros, I.K., Giménez, E., Magreñán, Á.A. et al. Improved semilocal convergence analysis in Banach space with applications to chemistry. J Math Chem 56, 1958–1975 (2018).

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  • Secant method
  • Banach space
  • Majorizing sequence
  • Divided difference
  • Local convergence
  • Semilocal convergence