Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-28T19:02:42.230Z Has data issue: false hasContentIssue false
  • English
  • Français

A general semilocal convergence result for Newton’s methodunder centered conditions for the second derivative

Published online by Cambridge University Press:  31 July 2012

José Antonio Ezquerro ,
Daniel González  and
Miguel Ángel Hernández
José Antonio Ezquerro
Affiliation:
University of La Rioja, Department of Mathematics and Computation, C/ Luis de Ulloa s/n, 26004 Logroño, Spain. @unirioja.es
Daniel González
Affiliation:
University of La Rioja, Department of Mathematics and Computation, C/ Luis de Ulloa s/n, 26004 Logroño, Spain. @unirioja.es
Miguel Ángel Hernández
Affiliation:
University of La Rioja, Department of Mathematics and Computation, C/ Luis de Ulloa s/n, 26004 Logroño, Spain. @unirioja.es
Get access

Abstract

From Kantorovich’s theory we present a semilocal convergence result for Newton’s methodwhich is based mainly on a modification of the condition required to the second derivativeof the operator involved. In particular, instead of requiring that the second derivativeis bounded, we demand that it is centered. As a consequence, we obtain a modification ofthe starting points for Newton’s method. We illustrate this study with applications tononlinear integral equations of mixed Hammerstein type.

Keywords

Newton’s methodthe Newton–Kantorovich theoremsemilocal convergencemajorizing sequencea priori error estimatesHammerstein’s integral equation
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 1 , January 2013 , pp. 149 - 167
Copyright
© EDP Sciences, SMAI, 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Amat, S. and Busquier, S., Third-order iterative methods under Kantorovich conditions. J. Math. Anal. Appl. 336 (2007) 243–261. Google Scholar
Amat, S., Bermúdez, C., Busquier, S. and Mestiri, D., A family of Halley-Chebyshev iterative schemes for non-Fréechet differentiable operators. J. Comput. Appl. Math. 228 (2009) 486–493. Google Scholar
Argyros, I.K., A Newton–Kantorovich theorem for equations involving m-Fréchet differentiable operators and applications in radiative transfer. J. Comput. Appl. Math. 131 (2001) 149–159. Google Scholar
Argyros, I.K., An improved convergence analysis and applications for Newton-like methods in Banach space, Numer. Funct. Anal. Optim. 24 (2003) 653–572. Google Scholar
Argyros, I.K., On the Newton-Kantorovich hypothesis for solving equations. J. Comput. Appl. Math. 169 (2004) 315–332. Google Scholar
Bruns, D.D. and Bailey, J.E., Nonlinear feedback control for operating a nonisothermal CSTR near an unstable steady state. Chem. Eng. Sci. 32 (1977) 257–264. Google Scholar
K. Deimling, Nonlinear functional analysis. Springer-Verlag, Berlin (1985).
Ezquerro, J.A. and Hernández, M.A., Generalized differentiability conditions for Newton’s method. IMA J. Numer. Anal. 22 (2002) 187–205. Google Scholar
Ezquerro, J.A. and Hernández, M.A., On an application of Newton’s method to nonlinear operators with ω-conditioned second derivative. BIT 42 (2002) 519–530. Google Scholar
Ezquerro, J.A. and Hernández, M.A., Halley’s method for operators with unbounded second derivative. Appl. Numer. Math. 57 (2007) 354–360. Google Scholar
J.A. Ezquerro, D. González and M.A. Hernández, Majorizing sequences for Newton’s method from initial value problems. J. Comput. Appl. Math. (submitted).
Ganesh, M. and Joshi, M.C., Numerical solvability of Hammerstein integral equations of mixed type. IMA J. Numer. Anal. 11 (1991) 21–31. Google Scholar
Gutiérrez, J.M., A new semilocal convergence theorem for Newton’s method. J. Comput. Appl. Math. 79 (1997) 131–145. Google Scholar
Kantorovich, L.V., On Newton’s method for functional equations. Dokl Akad. Nauk SSSR 59 (1948) 1237–1240 (in Russian). Google Scholar
Kantorovich, L.V., The majorant principle and Newton’s method. Dokl. Akad. Nauk SSSR 76 (1951) 17–20 (in Russian). Google Scholar
L.V. Kantorovich and G.P. Akilov, Functional analysis. Pergamon Press, Oxford (1982).
A.M. Ostrowski, Solution of equations in Euclidean and Banach spaces. London, Academic Press (1943).
Potra, F.A. and Pták, V., Sharp error bounds for Newton process. Numer. Math. 34 (1980) 63–72. Google Scholar
Rashidinia, J. and Zarebnia, M., New approach for numerical solution of Hammerstein integral equations. Appl. Math. Comput. 185 (2007) 147–154. Google Scholar
Rheinboldt, W.C., A unified convergence theory for a class of iterative processes. SIAM J. Numer. Anal. 5 (1968) 42–63. Google Scholar
Yamamoto, T., Convergence theorem for Newton-like methods in Banach spaces. Numer. Math. 51 (1987) 545–557. Google Scholar
Zhang, Z., A note on weaker convergence conditions for Newton iteration. J. Zhejiang Univ. Sci. Ed. 30 (2003) 133–135, 144 (in Chinese).Google Scholar