Local Convergence for an Improved Jarratt-type Method in Banach Space

20 DOI: 10.9781/ijimai.2015.344 Abstract — We present a local convergence analysis for an improved Jarratt-type methods of order at least five to approximate a solution of a nonlinear equation in a Banach space setting. The convergence ball and error estimates are given using hypotheses up to the first Fréchet derivative in contrast to earlier studies using hypotheses up to the third Fréchet derivative. Numerical examples are also provided in this study, where the older hypotheses are not satisfied to solve equations but the new hypotheses are satisfied.


I. Introduction
I n this study we are concerned with the problem of approximating a solution * x of the equation where F is a Fréchet-differentiable operator defined on a convex subset D of a Banach space X with values in a Banach space Y . Many problems in computational sciences and other disciplines can be brought in a form like (1) using mathematical modelling [11,12,28,30]. Moreover, artificial intelligence and e-learning are topics of increasing interest in recent years. Other authors and people from various other areas of expertise can follow these techniques to serve a community of learners. The solutions of these equations can rarely be found in 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. In particular, the practice of Numerical Functional Analysis for finding solution * x of equation (1) is essentially connected to variants of Newton's method. This method converges quadratically to * x if the initial guess is close enough to the solution. Iterative methods of convergence order higher than two such as Chebyshev-Halley-type methods [5, 6, 11, 12, 19-27, 29, 30, 32] require the evaluation of the second Fréchet-derivative, which is very expensive in general. However, there are integral equations, where the second Fréchet-derivative is diagonal by blocks and inexpensive or for quadratic quations the second Fréchet-derivative is constant. Moreover, in some applications involving stiff systems, high order methods are usefull. That is why in a unified way we study the local convergenve of the improved Jarratt-type method (IJTM) defined for each  0,1,2, = n by where 0 x is an initial point and I is the identity operator. If we This method has been shown to be of convergence order between 5 and 6 [28,32]. The usual conditions for the semilocal convergence of these methods are (  ): • There exists is a non-decreasing function.
The local convergence conditions are similar but 0 x is * x in ( 1  ) and ( 2  ). There is a plethora of local and semilocal convergence results under the (  ) conditions . These conditions restrict the applicability of these methods. That is why, in our study we assume the conditions (  ): Local convergence for an improved Jarratt-type method in Banach space

Ioannis K. Argyros and Daniel González
Cameron University and Escuela Politécnica Nacional Notice that the (  ) conditions are weaker than the (  ) conditions. Hence, the applicability of (IJTM) is expanded under the (  ) conditions.
As a motivational example, let us define function f on ) 2 are satisfied for The paper is organized as follows: In Section 2 we present the local convergence of these methods. The numerical examples are given in the concluding Section 3.
In the rest of this study, ) , ( q w U and ) , ( q w U stand, respectively, for the open and closed ball in X with center X w ∈ and of radius 0 > q .

II. Local convergence
In this section we present the local convergence of IJTM under the ) ( conditions. It is convenient for the local convergence of IJTM to introduce some funcitons and parameters.
Notice that Then, we have that Moreover, define functions 4 f and 5 f on the interval Furthermore, define functions 4 f and 5 f on the interval ) [0, 0 r by  (17) Then, we have by the choice of r that Next, we present the main local convergence for IJTM under the ) ( conditions. where function 5 f is defined by (14).
Proof. We shall use induction to show that estimates (22) hold for each  0,1,2, = n Using ) ( 2  and the hypothesis ) , by the choice of . It follows from (24) and the Banach lemma on invertible operators that [11,12,27] ) Using the first substep of IJTM for and the choice of r we get that . Using the second substep of IJTM, we get by (27) and (19) Next, we shall find upper bounds on Then, from the fourth substep of IJTM for 0 = n , (27), (28), (29), Notice that we used , respectively to arrive at (23), which complete the induction. can be arbitrarily large [2][3][4][5][6].
2. In view of condition ( 2  ) and the estimate can be dropped and K can be replaced by 3. It is worth noticing that r is such that The convergence ball of radius A r was given by us in [2,3,5] for Newton's method under conditions ( 1  )-( 3  ). Estimate (24) shows that the convergence ball of higher than two IJTM methods is smaller than the convergence ball of the quadratically convergent Newton's method. The convergence ball given by Rheinboldt [30] for Newton's method is 4. The local results can be used for projection methods such as Arnoldi's method, the generalized minimum residual method (GMREM), the generalized conjugate method (GCM) for combined Newton/finite projection methods and in connection to the mesh independence principle in order to develop the cheapest and most efficient mesh refinement strategy [11,12,30]. 5. The results can also be used to solve equations where the operator F ′ satisfies the autonomous differential equation [11,12,28,30]: where T is a known continuous operator. Since

III. Numerical examples
We present numerical examples where we compute the radii of the convergence balls.

IV. Acknowledgements
This scientific work has been supported by the 'Proyecto Prometeo' of the Ministry of Higher Education, Science, Technology and Innovation of the Republic of Ecuador.