Geodesic
Convergence of Traub's iteration under $\omega$ continuity condition in Banach spaces
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 9 (2021), pp. 61-79

Voir la notice de l'article provenant de la source Math-Net.Ru

We provide the local and semi-local convergence analysis of the most celebrated cubically convergent Traub's iterative method (TM) for obtaining solutions of Banach space valued nonlinear operator equations. The significance of our work is that the convergence study only needs the $\omega$ continuity condition on the first-order Fréchet derivative and avoids the use of higher order derivatives, which do not occur in this scheme. Also, the proposed local analysis extends the domain of convergence and applicability of this scheme. Using basins of attraction technique the complex dynamics of the scheme are also explored when it is applied on various complex polynomials. Finally, convergence radii for benchmark numerical problems are computed applying our analytical results. From these numerical tests, it is confirmed that the proposed analysis provides a larger convergence domain in comparison with the earlier work.
Keywords: Traub's iterative method, Semi-local convergence, $\omega$ continuity condition, Basin of attraction.
Mots-clés : Local convergence 
@article{IVM_2021_9_a6,
     author = {D. Sharma and S. K. Sunanda and S. K. Parhi},
     title = {Convergence of {Traub's} iteration under $\omega$ continuity condition in {Banach} spaces},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {61--79},
     publisher = {mathdoc},
     number = {9},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2021_9_a6/}
}
TY  - JOUR
AU  - D. Sharma
AU  - S. K. Sunanda
AU  - S. K. Parhi
TI  - Convergence of Traub's iteration under $\omega$ continuity condition in Banach spaces
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2021
SP  - 61
EP  - 79
IS  - 9
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IVM_2021_9_a6/
LA  - ru
ID  - IVM_2021_9_a6
ER  - 
%0 Journal Article
%A D. Sharma
%A S. K. Sunanda
%A S. K. Parhi
%T Convergence of Traub's iteration under $\omega$ continuity condition in Banach spaces
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2021
%P 61-79
%N 9
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IVM_2021_9_a6/
%G ru
%F IVM_2021_9_a6
D. Sharma; S. K. Sunanda; S. K. Parhi. Convergence of Traub's iteration under $\omega$ continuity condition in Banach spaces. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 9 (2021), pp. 61-79. http://geodesic.mathdoc.fr/item/IVM_2021_9_a6/