Abstract
This paper focuses on the importance of center conditions on the first derivative of the operator involved in the solution of nonlinear equations by Newton’s method when the semilocal convergence of the method is established from the technique of recurrence relations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Andronow, A.A., Chaikin, C.E.: Theory of oscillations. Princenton University Press, New Jersey (1949)
Cianciaruso, F., De Pascale, E.: Newton-Kantorovich Approximations when the Derivative is Hölderian: Old and New Results. Numer. Funct. Anal. Optim. 24 (7-8), 713–723 (2003)
Cianciaruso, F., De Pascale, E.: Estimates of Majorizing Sequences in The Newton-Kantorovich Method. Numer. Funct. Anal. Optim. 27(5-6), 529–538 (2006)
Cianciaruso, F., De Pascale, E.: Estimates of Majorizing Sequences in the Newton-Kantorovich Method: a Further Improvement. J. Math. Anal. Appl. 322, 329–335 (2006)
Ezquerro, J.A., Hernández, M. A.: Generalized differentiability conditions for Newton’s method, IMA. J. Numer. Anal. 22(2), 187–205 (2002)
Ezquerro, J.A., Gutiérrez, J. M., Hernández, M. A., Romero, N., Rubio, M.J.: The Newton method: from Newton to Kantorovich, (Spanish). Gac. R. Soc. Mat. Esp. 13(1), 53–76 (2010)
Ezquerro, J.A., Hernández-Verón, M. A.: On the Accessibility of Newton’s Method under a Hölder Condition on the First Derivative. Algorithms 8(3), 514–528 (2015)
Hernández, M. A.: The Newton method for operators with Hölder continuous first derivative. J. Optim. Theory Appl. 109(3), 631–648 (2001)
Kantorovich, L.V., Akilov, G.P.: Functional analysis. Pergamon Press, Oxford (1982)
Keller, H.: Numerical methods for two-point boundary value problems. Dover Publications, New York (1992)
Ortega, J.M.: The Newton-Kantorovich theorem. Amer. Math. Monthly 75, 658–660 (1968)
Rokne, J.: Newton’s method under mild differentiability conditions with error analysis. Numer. Math. 18, 401–412 (1972)
Stoker, J.J.: Nonlinear vibrations. Interscience-Wiley, New York (1950)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by: Raymond H. Chan
Rights and permissions
About this article
Cite this article
Ezquerro, J.A., Hernández-Verón, M.A. A study of the influence of center conditions on the domain of parameters of Newton’s method by using recurrence relations. Adv Comput Math 43, 1103–1129 (2017). https://doi.org/10.1007/s10444-017-9518-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10444-017-9518-z