Abstract
Iterative methods, play an important role in computational sciences. In this chapter, we present new semilocal and local convergence results for the Newton-Kantorovich method. These new results extend the applicability of the Newton-Kantorovich method on approximate zeros by improving the convergence domain and ratio given in earlier studies.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Amat, S., Busquier, S., & Negra, M. (2004). Adaptive approximation of nonlinear operators. Numerical Functional Analysis and Optimization, 25, 397–405.
Argyros, I. K., & Szidarovszky, F. (1993). The theory and applications of iteration methods, Systems Engineering Series. Boca Ratón, Florida: CRC Press.
Argyros, I. K. (2004). On the Newton-Kantorovich hypothesis for solving equations. Journal Computational Applied Mathematics, 169, 315–332.
Argyros, I. K. (2007). In Chui, C. K. & Wuytack, L. (Eds.), Computational theory of iterative methods (Vol. 15), Series: Studies in Computational Mathematics. New York: Elsevier Publishing Company.
Argyros, I. K., & Hilout, S. (2010a). A convergence analysis of Newton-like methods for singular equations unsig recurrent functions. Numerical Functional Analysis and Optimization, 31(2), 112–130.
Argyros, I. K., & Hilout, S. (2010b). Extending the Newton-Kantorovich hypothesis for solving equations. Journal of Computational Applied Mathematics, 234, 2993–3006.
Argyros, I. K., & Hilout, S. (2012). Weaker conditions for the convergence of Newton’s method. Journal of Complexity, 28, 364–387.
Argyros, I. K., Cho, Y. J., & Hilout, S. (2012). Numerical method for equations and its applications. New York: CRC Press/Taylor and Francis.
Argyros, I. K., & George, S. (2015). Ball convergence for Steffensen-type fourth-order methods. IJIMAI, 3(4), 27–42.
Argyros, I. K., & Magreñán, Á. A. (2015). Extended convergence of Newton-Kantorovich method to an approximate zero. Journal of Computational and Applied Mathematics, 286, 54–67.
Cianciaruso, F. (2007). Convergence of Newton-Kantorovich approximations to an approximate zero. Numerical Functional Analysis and Optimization, 28(5–6), 631–645.
Ezquerro, J. A., Gutiérrez, J. M., Hernández, M. A., Romero, N., & Rubio, M. J. (2010). The Newton method: from Newton to Kantorovich. (Spanish), Gac. R. Soc. Mat. Esp., 13(1), 53–76.
Kantorovich, L. V., & Akilov, G. P. (1982). Functional analysis. Oxford: Pergamon Press.
Magreñán, Á. A., & Argyros, I. K. (2015). An extension of a theorem by Wang for Smale’s \(\alpha \)-theory and applications. Numerical Algorithms, 68(1), 47–60.
Magreñán, Á. A. (2014a). Different anomalies in a Jarratt family of iterative root-finding methods. Applied Mathematics and Computation, 233, 29–38.
Magreñán, Á. A. (2014b). A new tool to study real dynamics: The convergence plane. Applied Mathematics and Computation, 248, 215–224.
Potra, F. A., & Pták, V. (1984). Nondiscrete induction and iterative processes (Vol. 103). Research Notes in Mathematics. Boston, Massachusetts: Pitman (Advanced Publishing Program).
Proinov, P. D. (2010). New general convergence theory for iterative processes and its applications to Newton-Kantorovich type theorems. Journal of Complexity, 26, 3–42.
Rheinboldt, W. C. (1988). On a theorem of S. Smale about Newton’s method for analytic mappings. Applied Mathematics Letter, 1, 69–72.
Smale, S. (1986). newton’s method estimates from data at one point. In R. Ewing, K. Gross, & C. Martin (Eds.), The merging of disciplines: new directions in pure (pp. 185–196). Applied and Computational Mathematics. New York: Springer.
Wang, X. H. (1999). Convergence of Newton’s method and inverse function theorem in Banach spaces. Mathematics of Compattion, 68, 169–186.
Zabrejko, P. P., & Nguen, D. F. (1987). The majorant method in the theory of Newton-Kantorovich approximations and the Pták error estimates. Numerical Functional Analysis Optimization, 9, 671–684.
Acknowledgements
This research was supported by Universidad Internacional de La Rioja (UNIR, http://www.unir.net), under the Plan Propio de Investigación, Desarrollo e Innovación [2015–2017]. Research group: Modelación matemática aplicada a la ingeniería(MOMAIN), by the grant SENECA 19374/PI/14 and by Ministerio de Ciencia y Tecnología MTM2014-52016-C2-\(\{01\}\)-P.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer Nature Singapore Pte Ltd.
About this chapter
Cite this chapter
Argyros, I.K., Magreñán, Á.A., Sicilia, J.A. (2017). Developments on the Convergence of Some Iterative Methods. In: Matsumoto, A. (eds) Optimization and Dynamics with Their Applications. Springer, Singapore. https://doi.org/10.1007/978-981-10-4214-0_1
Download citation
DOI: https://doi.org/10.1007/978-981-10-4214-0_1
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-10-4213-3
Online ISBN: 978-981-10-4214-0
eBook Packages: Economics and FinanceEconomics and Finance (R0)