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

José Antonio Ezquerro; Daniel González; Miguel Ángel Hernández

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2013)

  • Volume: 47, Issue: 1, page 149-167
  • ISSN: 0764-583X

Abstract

top
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.

How to cite

top

Ezquerro, José Antonio, González, Daniel, and 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 - Modélisation Mathématique et Analyse Numérique 47.1 (2013): 149-167. <http://eudml.org/doc/273116>.

@article{Ezquerro2013,
abstract = {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.},
author = {Ezquerro, José Antonio, González, Daniel, Hernández, Miguel Ángel},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {Newton’s method; the Newton–Kantorovich theorem; semilocal convergence; majorizing sequence; a priori error estimates; Hammerstein’s integral equation; Newton's method; Newton-Kantorovich theorem; Hammerstein's integral equation},
language = {eng},
number = {1},
pages = {149-167},
publisher = {EDP-Sciences},
title = {A general semilocal convergence result for Newton’s method under centered conditions for the second derivative},
url = {http://eudml.org/doc/273116},
volume = {47},
year = {2013},
}

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 - Modélisation Mathématique et Analyse Numérique
PY - 2013
PB - EDP-Sciences
VL - 47
IS - 1
SP - 149
EP - 167
AB - 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.
LA - eng
KW - Newton’s method; the Newton–Kantorovich theorem; semilocal convergence; majorizing sequence; a priori error estimates; Hammerstein’s integral equation; Newton's method; Newton-Kantorovich theorem; Hammerstein's integral equation
UR - http://eudml.org/doc/273116
ER -

References

top
  1. [1] S. Amat and S. Busquier, Third-order iterative methods under Kantorovich conditions. J. Math. Anal. Appl.336 (2007) 243–261. Zbl1128.65036MR2348504
  2. [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. Zbl1173.65036MR2514306
  3. [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. Zbl0983.65069MR1835709
  4. [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. Zbl1040.47045MR2011587
  5. [5] I.K. Argyros, On the Newton-Kantorovich hypothesis for solving equations. J. Comput. Appl. Math.169 (2004) 315–332. Zbl1055.65066MR2072881
  6. [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. [7] K. Deimling, Nonlinear functional analysis. Springer-Verlag, Berlin (1985). Zbl0559.47040MR787404
  8. [8] J.A. Ezquerro and M.A. Hernández, Generalized differentiability conditions for Newton’s method. IMA J. Numer. Anal.22 (2002) 187–205. Zbl1006.65051MR1897406
  9. [9] J.A. Ezquerro and M.A. Hernández, On an application of Newton’s method to nonlinear operators with ω-conditioned second derivative. BIT42 (2002) 519–530. Zbl1028.65061MR1931884
  10. [10] J.A. Ezquerro and M.A. Hernández, Halley’s method for operators with unbounded second derivative. Appl. Numer. Math.57 (2007) 354–360. Zbl1252.65098MR2292441
  11. [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). Zbl1241.65051
  12. [12] M. Ganesh and M.C. Joshi, Numerical solvability of Hammerstein integral equations of mixed type. IMA J. Numer. Anal.11 (1991) 21–31. Zbl0719.65093MR1089546
  13. [13] J.M. Gutiérrez, A new semilocal convergence theorem for Newton’s method. J. Comput. Appl. Math.79 (1997) 131–145. Zbl0872.65045MR1437974
  14. [14] L.V. Kantorovich, On Newton’s method for functional equations. Dokl Akad. Nauk SSSR 59 (1948) 1237–1240 (in Russian). 
  15. [15] L.V. Kantorovich, The majorant principle and Newton’s method. Dokl. Akad. Nauk SSSR 76 (1951) 17–20 (in Russian). 
  16. [16] L.V. Kantorovich and G.P. Akilov, Functional analysis. Pergamon Press, Oxford (1982). Zbl0484.46003MR664597
  17. [17] A.M. Ostrowski, Solution of equations in Euclidean and Banach spaces. London, Academic Press (1943). Zbl0304.65002MR359306
  18. [18] F.A. Potra and V. Pták, Sharp error bounds for Newton process. Numer. Math.34 (1980) 63–72. Zbl0434.65034MR560794
  19. [19] J. Rashidinia and M. Zarebnia, New approach for numerical solution of Hammerstein integral equations. Appl. Math. Comput.185 (2007) 147–154. Zbl1110.65126MR2298437
  20. [20] W.C. Rheinboldt, A unified convergence theory for a class of iterative processes. SIAM J. Numer. Anal.5 (1968) 42–63. Zbl0155.46701MR225468
  21. [21] T. Yamamoto, Convergence theorem for Newton-like methods in Banach spaces. Numer. Math.51 (1987) 545–557. Zbl0633.65049MR910864
  22. [22] Z. Zhang, A note on weaker convergence conditions for Newton iteration. J. Zhejiang Univ. Sci. Ed. 30 (2003) 133–135, 144 (in Chinese). Zbl1043.65075MR1973328

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.