Convergence of Halley’s method for operators with the bounded second Fréchet-derivative in Banach spaces
© Argyros et al.; licensee Springer 2013
Received: 29 November 2012
Accepted: 4 May 2013
Published: 23 May 2013
In this paper, we present a semi-local convergence analysis of Halley’s method for approximating a locally unique solution of a nonlinear equation in a Banach space setting, where we assume that the second Fréchet-derivative is bounded. Numerical examples are used to show that the new convergence criteria can provide better information than those provided by the convergence criteria developed earlier.
MSC:65G99, 65J15, 65H10, 47H17, 49M15.
KeywordsHalley’s method Banach space semi-local convergence Fréchet-derivative convergence ball
where F is twice Fréchet-differentiable operator defined on a nonempty open and convex subset of a Banach space X with values in a Banach space Y.
Many problems from computational sciences and other disciplines can be brought in a form similar to equation (1.1) using mathematical modeling [1–3]. The solutions of these equations can rarely be found in a closed form. That is why most solution methods for these equations are iterative. The study about convergence matter of iterative procedures is usually based on two types: semi-local and local convergence analysis. The semi-local convergence matter is, based on the information around an initial point, to give conditions ensuring the convergence of the iterative procedure; while the local one is, based on the information around a solution, to find estimates of the radii of convergence balls.
The convergence of Halley’s method has a long history and has been studied by many authors (cf. [1, 2, 4–7] and the references therein). The most popular conditions for the semi-local convergence of Halley’s method are given as follows.
(C1) There exists such that , the space of bounded linear operator from Y into X;
(C2) for any ;
(C3) for each x in D and ;
(C4) for each and .
Therefore, there is no constant K satisfying (C4). Other examples where (C4) is not satisfied can be found in .
In the present study, we show how to expand even further the applicability of Halley’s method using (C1), (C2), (C3) and the center-Lipschitz condition:
(C5) for each and .
for each is obtained. This modification can lead to a tighter convergence analysis and weaker sufficient convergence conditions or error bounds or the location of the solution for Halley’s method (see numerical examples). The new approach has already led to such advantages in the case of Newton-type methods [1, 2, 4–6].
We use a different approach than recurrent relations in our semi-local convergence analysis. The paper is organized as follows. Section 2 contains the semi-local convergence of Halley’s method, whereas the numerical examples are presented in the concluding section.
2 Semi-local convergence
We present the semi-local convergence analysis of Halley’s method. We shall use an additional condition.
Let us introduce another condition needed in our main theorem.
We refer to (C1), (C2), (C3), (C5), (C6) and (C7) as the (C) conditions. Let , stand, respectively, for the open and closed balls in X with center x and radius . Then we can show the following semi-local convergence result for Halley’s method.
then is the only solution of equation in .
Hence we have .
- (a)exists and
- (d)exists and
- (e)is well defined and
for each . The rest will be shown by induction.
Hence we deduce that , which completes the induction for (a)-(f).
we deduce . This completes the proof. □
3 Numerical examples
In this section, we give some examples to show the application of our theorem.
Then we get . That is, we provide better information on the location of the solution. Moreover, by (2.7) and (2.8), we can set , which extends the uniqueness ball from to .
for any , respectively.
On the other hand, it follows from (3.6) that . Then we get .
i.e., and , where and are the positive roots of the real equation . Consequently, if we look for a solution such that , we can consider , with , as a nonempty open convex subset of X. For example, choose .
So, all the conditions in Theorem 2.1 are satisfied and Theorem 2.1 applies. Consequently, the sequence generated by Halley’s method (1.2) with initial point converges to the unique solution of equation (1.1) on . Moreover, by (2.7) and (2.8), we can set , which extends the uniqueness ball from to .
First, the authors would like to thank the anonymous referee for valuable suggestions that have been implemented in the final version of this paper. The second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant Number: 2012-0008170) and the third author was supported by the National Natural Science Foundation of China (Grant No. 10871178).
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