Majorizing sequences for Newton's method under centred conditions for the derivative

Abstract

We present semi-local and local convergence results for Newton's method in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Our technique is more flexible than in earlier studies such that [J.A. Ezquerro, D. González, and M.A. Hernández, Majorizing sequences for Newton's method from initial value problems, J. Comput. Appl. Math. 236 (2012), pp. 2246–2258; J.A. Ezquerro, D. González, and M.A. Hernández, A general semi-local convergence result for Newton's method under centred conditions for the second derivative, ESAIM: Math. Model. Numer. Anal. 47 (2013), pp. 149–167]. The operator involved is twice Fréchet-differentiable. We also assume certain centred Lipschitz-type conditions for the derivative which are more precise than the Lipschitz conditions used in earlier works. Numerical examples are used to show that our results apply to solve equations but earlier ones do not in the semi-local case. In the local case we obtain a larger convergence ball. These advantages are obtained under the same computational cost as before [17,18].

Keywords:

• Newton's method
• Banach space
• semi-local convergence
• Fréchet derivative
• majorizing sequence
• generalized Lipschitz condition

2000 AMS Subject Classifications:

• 65G99
• 65J15
• 47H17
• 49M15

Acknowledgements

The research of the third author is supported by the Grant MTM 2011-28636-C02-01.