A general semilocal convergence result for Newton's method under centered conditions for the second derivative
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 1, pp. 149-167.

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

DOI : 10.1051/m2an/2012026
Classification : 45G10, 47H99, 65J15
Mots-clés : Newton's method, the Newton-Kantorovich theorem, semilocal convergence, majorizing sequence, a priori error estimates, Hammerstein's integral equation 
@article{M2AN_2013__47_1_149_0,
     author = {Ezquerro, Jos\'e Antonio and Gonz\'alez, Daniel and Hern\'andez, Miguel \'Angel},
     title = {A general semilocal convergence result for {Newton's} method under centered conditions for the second derivative},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {149--167},
     publisher = {EDP-Sciences},
     volume = {47},
     number = {1},
     year = {2013},
     doi = {10.1051/m2an/2012026},
     mrnumber = {2968699},
     zbl = {1271.65092},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2012026/}
}
TY  - JOUR
AU  - Ezquerro, José Antonio
AU  - González, Daniel
AU  - Hernández, Miguel Ángel
TI  - A general semilocal convergence result for Newton's method under centered conditions for the second derivative
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2013
SP  - 149
EP  - 167
VL  - 47
IS  - 1
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2012026/
DO  - 10.1051/m2an/2012026
LA  - en
ID  - M2AN_2013__47_1_149_0
ER  - 
%0 Journal Article
%A Ezquerro, José Antonio
%A González, Daniel
%A Hernández, Miguel Ángel
%T A general semilocal convergence result for Newton's method under centered conditions for the second derivative
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2013
%P 149-167
%V 47
%N 1
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2012026/
%R 10.1051/m2an/2012026
%G en
%F M2AN_2013__47_1_149_0
Ezquerro, José Antonio; González, Daniel; Hernández, Miguel Ángel. A general semilocal convergence result for Newton's method under centered conditions for the second derivative. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 1, pp. 149-167. doi : 10.1051/m2an/2012026. http://www.numdam.org/articles/10.1051/m2an/2012026/

[1] S. Amat and S. Busquier, Third-order iterative methods under Kantorovich conditions. J. Math. Anal. Appl. 336 (2007) 243-261. | MR | Zbl

[2] S. Amat, C. Bermúdez, S. Busquier and D. Mestiri, A family of Halley-Chebyshev iterative schemes for non-Fréechet differentiable operators. J. Comput. Appl. Math. 228 (2009) 486-493. | MR | Zbl

[3] I.K. Argyros, A Newton-Kantorovich theorem for equations involving m-Fréchet differentiable operators and applications in radiative transfer. J. Comput. Appl. Math. 131 (2001) 149-159. | MR | Zbl

[4] I.K. Argyros, An improved convergence analysis and applications for Newton-like methods in Banach space, Numer. Funct. Anal. Optim. 24 (2003) 653-572. | MR | Zbl

[5] I.K. Argyros, On the Newton-Kantorovich hypothesis for solving equations. J. Comput. Appl. Math. 169 (2004) 315-332. | MR | Zbl

[6] D.D. Bruns and J.E. Bailey, Nonlinear feedback control for operating a nonisothermal CSTR near an unstable steady state. Chem. Eng. Sci. 32 (1977) 257-264.

[7] K. Deimling, Nonlinear functional analysis. Springer-Verlag, Berlin (1985). | MR | Zbl

[8] J.A. Ezquerro and M.A. Hernández, Generalized differentiability conditions for Newton's method. IMA J. Numer. Anal. 22 (2002) 187-205. | MR | Zbl

[9] J.A. Ezquerro and M.A. Hernández, On an application of Newton's method to nonlinear operators with ω-conditioned second derivative. BIT 42 (2002) 519-530. | MR | Zbl

[10] J.A. Ezquerro and M.A. Hernández, Halley's method for operators with unbounded second derivative. Appl. Numer. Math. 57 (2007) 354-360. | MR | Zbl

[11] J.A. Ezquerro, D. González and M.A. Hernández, Majorizing sequences for Newton's method from initial value problems. J. Comput. Appl. Math. (submitted). | Zbl

[12] M. Ganesh and M.C. Joshi, Numerical solvability of Hammerstein integral equations of mixed type. IMA J. Numer. Anal. 11 (1991) 21-31. | MR | Zbl

[13] J.M. Gutiérrez, A new semilocal convergence theorem for Newton's method. J. Comput. Appl. Math. 79 (1997) 131-145. | MR | Zbl

[14] L.V. Kantorovich, On Newton's method for functional equations. Dokl Akad. Nauk SSSR 59 (1948) 1237-1240 (in Russian).

[15] L.V. Kantorovich, The majorant principle and Newton's method. Dokl. Akad. Nauk SSSR 76 (1951) 17-20 (in Russian).

[16] L.V. Kantorovich and G.P. Akilov, Functional analysis. Pergamon Press, Oxford (1982). | MR | Zbl

[17] A.M. Ostrowski, Solution of equations in Euclidean and Banach spaces. London, Academic Press (1943). | MR | Zbl

[18] F.A. Potra and V. Pták, Sharp error bounds for Newton process. Numer. Math. 34 (1980) 63-72. | MR | Zbl

[19] J. Rashidinia and M. Zarebnia, New approach for numerical solution of Hammerstein integral equations. Appl. Math. Comput. 185 (2007) 147-154. | MR | Zbl

[20] W.C. Rheinboldt, A unified convergence theory for a class of iterative processes. SIAM J. Numer. Anal. 5 (1968) 42-63. | MR | Zbl

[21] T. Yamamoto, Convergence theorem for Newton-like methods in Banach spaces. Numer. Math. 51 (1987) 545-557. | MR | Zbl

[22] Z. Zhang, A note on weaker convergence conditions for Newton iteration. J. Zhejiang Univ. Sci. Ed. 30 (2003) 133-135, 144 (in Chinese). | MR | Zbl

Cité par Sources :