Convergence Analysis and Complex Geometry of an Efﬁcient Derivative-Free Iterative Method

: To locate a locally-unique solution of a nonlinear equation, the local convergence analysis of a derivative-free ﬁfth order method is studied in Banach space. This approach provides radius of convergence and error bounds under the hypotheses based on the ﬁrst Fréchet-derivative only. Such estimates are not introduced in the earlier procedures employing Taylor’s expansion of higher derivatives that may not exist or may be expensive to compute. The convergence domain of the method is also shown by a visual approach, namely basins of attraction. Theoretical results are endorsed via numerical experiments that show the cases where earlier results cannot be applicable.


Introduction
Banach [1] or complete normed vector spaces constantly bring new solving strategies for real problems in domains dealing with numerical methods (see for example [2][3][4][5]). In this context, development of new methods [6] and their convergence analysis [7] are of growing interest.
Let B 1 , B 2 be Banach spaces and Ω ⊆ B 1 be closed and convex. In this study, we locate a solution x * of the nonlinear equation where F : Ω ⊆ B 1 → B 2 is a Fréchet-differentiable operator. In computational sciences, many problems can be transformed into form (1). For example, see the References [8][9][10][11]. The solution of such nonlinear equations is hardly attainable in closed form. Therefore, most of the methods for solving such equations are usually iterative. The important issue addressed to an iterative method is its domain of convergence since it gives us the degree of difficulty for obtaining initial points. This domain is generally small. Thus, it is necessary to enlarge the domain of convergence but without any additional hypotheses. Another important problem related to convergence analysis of an iterative method is to find precise error estimates on x n+1 − x n or x n − x * . A good reference for the general principles of functional analysis is [12]. Recurrence relations for rational cubic methods are revised in [13] (for Halley method) and in [14] (for Chebyshev method). A new iterative modification of Newton's method for solving nonlinear scalar equations was proposed in [15], while a modification of a variant of it with accelerated third order convergence was where u n = x n + βF(x n ), β ∈ R − {0} has a quadratic order of convergence. Based on (2), Sharma et al. [32] have recently proposed a derivative-free method with fifth order convergence for approximating a solution of F(x) = 0 using the weight-function scheme defined for each n = 0, 1 . . . by wherein H(x n ) = 2I − [u n , x n ; F] −1 [z n , y n ; F]. The computational efficiency of this method was discussed in detail and performance was favorably compared with existing methods in [32]. To prove the local convergence order, the authors used Taylor expansions with hypotheses based on a Fréchet-derivative up to the fifth order. It is quite clear that these hypotheses restrict the applicability of methods to the problems involving functions that are at least five times Fréchet-differentiable. For example, let us define a function g on Ω = [− 1 2 , 5 2 ] by We have that g (t) = 3t 2 ln t 2 + 5t 4 − 4t 3 + 2t 2 , and Then, g is unbounded on Ω. Notice also that the proofs of convergence use Taylor expansions.
In this work, we study the local convergence of the methods (3) using the hypotheses on the first Fréchet-derivative only taking advantage of the Lipschitz continuity of the first Fréchet-derivative. Moreover, our results are presented in the more general setting of a Banach space. We summarize the contents of the paper. In Section 2, the local convergence analysis of method (3) is presented. In Section 3, numerical examples are performed to verify the theoretical results. Basins of attraction showing convergence domain are drawn in Section 4. Concluding remarks are reported in Section 5.
We will study the local convergence of method (3) in a Banach space setting under the following hypotheses (collectively called by the name 'A'): There exists a continuous and nondecreasing function w 0 : where r has been defined before. There exists continuous and Theorem 1. Suppose that the hypotheses (A) hold. Then, the sequence {x n } generated by method (3) for converges to x * . Moreover, the following conditions hold: and where the functions g i , i = 1, 2, 3 are defined as above. Furthermore, the vector x * is the only solution of F(x) = 0 in Ω 1 .
Proof. We shall show estimates (9)-(11) using mathematical induction. By hypothesis (a3) and for x ∈ U(x * , r 3 ), we have that By (12) and the Banach Lemma [9], we have that [u n , x n ; F] −1 ∈ L(B 2 , B 1 ) and We show that y n is well defined by the method (3) for n = 0. We have Then, using (8) (for i = 1), the conditions (a4) and (13), we have in turn that which implies (9) for n = 0 and y 0 ∈ U(x * , r 3 ).

Numerical Examples
We illustrate the theoretical results shown in Theorem 1. For the computation of divided difference, let us choose [x, y; F] = 1 0 F (y + θ(x − y))dθ . Consider the following three numerical examples: Example 1. Assume that the motion of a particle in three dimensions is governed by a system of differential equations: A solution of the system is given for u = (x, y, z) T by function F : Its Fréchet-derivative F (u) is given by Then, for x * = (0, 0, 0) T , we deduce that w 0 (s, t) = w 1 (s, t) = L 0 2 (s + t) and v 0 (t) = 1 2 (1 + e 1 L 0 ), Then, using a definition of parameters, the calculated values are displayed as r * = min{r 1 , r 2 , r 3 } = min{0.313084, 0.165881, 0.0715631} = 0.0715631.

Basins of Attraction
The basin of attraction is a useful geometrical tool for assessing convergence regions of the iterative methods. These basins show us all the starting points that converge to any root when we apply an iterative method, so we can see in a visual way which points are good choices as starting points and which are not. We take the initial point as z 0 ∈ R, where R is a rectangular region in C containing all the roots of a poynomial p(z) = 0. The iterative methods starting at a point z 0 in a rectangle can converge to the zero of the function p(z) or eventually diverge. In order to analyze the basins, we consider the stopping criterion for convergence as 10 −3 up to a maximum of 25 iterations. If the mentioned tolerance is not attained in 25 iterations, the process is stopped with the conclusion that the iterative method starting at z 0 does not converge to any root. The following strategy is taken into account: A color is assigned to each starting point z 0 in the basin of attraction of a zero. If the iteration starting from the initial point z 0 converges, then it represents the basins of attraction with that particular color assigned to it and, if it fails to converge in 25 iterations, then it shows the black color.
Observing the behavior of the method, we say that the divergent zones (black zones) are becoming smaller with the decreasing value of β.

Conclusions
In this paper, the local convergence analysis of a derivative-free fifth order method is studied in Banach space. Unlike other techniques that rely on higher derivatives and Taylor series, we have used only derivative of order one in our approach. In this way, we have extended the usage of the considered method since the method can be applied to a wider class of functions. Another advantage of analyzing the local convergence is the computation of a convergence ball, uniqueness of the ball where the iterates lie and estimation of errors. Theoretical results of convergence thus achieved are confirmed through testing on some practical problems.
The basins of attraction have been analyzed by applying the method on some polynomials. From these graphics, one can easily visualize the behavior and suitability of any method. If we choose an initial guess x 0 in a domain where different basins of the roots meet each other, it is uncertain to predict which root is going to be reached by the iterative method that begins from x 0 . Thus, the choice of initial guess lying in such a domain is not suitable. In addition, black zones and the zones with different colors are not suitable to take the initial guess x 0 when we want to achieve a particular root. The most attractive pictures appear when we have very intricate boundaries of the basins. Such pictures belong to the cases where the method is more demanding with respect to the initial point.