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The basin of attraction is a method to visually comprehend how an algorithm behaves as a function of the various starting points. In this section, we consider the stopping criterion for convergence to be less than 10 4 wherein the maximum number of full cycles for each method is considered to be In this way, we distinguish the attraction basins by their colors for different methods.

Based on Figs.

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Note that 18 contains lower number of diverging points in contrast to the other compared methods. The roots are the five roots of unity. The results are presented in Figs. The following methods performed acceptably: 18 and The methods 3 and 5 did not perform well. Note that the method 5 shows chaotic behavior. The roots are 0. The following methods performed good: 21 and We find out in this section that some algorithms have a smooth convergence pattern and others have a rather chaotic pattern, which leads the algorithm to converge to an unwanted root or diverge the black areas.

Overall, the method 18 does not contain very chaotic behavior for the test problems. Furthermore, considering Table 1, the efficiency of the methods , one can use 18 as an improvement over 1 - 5. We test the without memory method 18 numerically in this section. Apart from this method, some other iterative methods 3 - 5 for multiple roots, which also require three functional evaluations, have been chosen.

The list of test nonlinear functions including multiple roots with their multiplicities are presented in Table 2. The results are summarized in Tables after two and three full iterations, respectively, for two different initial guesses. As they show, the novel scheme is comparable with all of the methods. All numerical instances were performed using digits floating point arithmetic. As can be seen, the obtained results in Tables are in harmony with the analytical procedure given in Section 3. Although the two-point method 18 mostly produces the best approximations in the case of considered functions after three full cycles, we cannot claim that, in general, it is better than other methods without memory for multiple roots of the same order.

The tests show that the considered methods generate results of approximately the same accuracy. Note that for calculating the multiplicity m of a root, it is necessary to use one of the following ways:. An important concern regarding the implementation of multi-point multiple zero-finders is in the choice of the starting value. Fortunately, this need can easily be lifted using the ways recently discussed by Soleymani and colleagues in [26,27], which provide very robust initial points for starting the iteration processes.

This paper has contributed an optimal method for solving nonlinear equations in the form of a two-point root solver, when the multiplicity of the zero is known or given. The derived method 18 comprises three functional evaluations, which shows its optimality in the sense of Kung-Traub.

Mandelbrot set

Further investigation has been made on the complex plane for such methods to reveal their basins of attraction for solving nonlinear equations by presenting their corresponding fractals. The dynamic study of the method also supports the theoretical aspects. The proposed method along with the existing multiple zero-finders were tested through numerical test problems. From Tables and the test problems, we can conclude that the new multiple root solver work acceptably. The authors are grateful to the two referees for their valuable comments and suggestions that improved the quality and presentation of this paper.

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Abstract of research paper on Earth and related environmental sciences, author of scientific article — F. Babajee, Taher Lotfi Abstract An optimal method is developed for approximating the multiple zeros of a nonlinear function, when the multiplicity is known. Similar topics of scientific paper in Earth and related environmental sciences , author of scholarly article — F. Babajee, Taher Lotfi Some optimal iterative methods and their with memory variants. Introduction There is a vast literature on the solution of nonlinear equations or systems, see e.

E-mail address: fazlollah. Peer review under responsibility of Egyptian Mathematical Society. Brief review It is known that when the multiplicity m of the root is available then the modified Newton's method, a. Later, this technology has been used by many researchers for constructing optimal in the sense of Kung-Traub iterative algorithms of high order.

Even when a polynomiograph is a fractal image it does not It is known that the basin of attraction of an iterative diminish its uniqueness. We use CAS to help with he remaining sections of this paper are organized as calculations too time consuming. Mathematica 8 is one of follows. Section 2 provides some basic concepts in order the most popular CASs available today [23]. Due to its wide to deal with dynamics associated with iterative functions. Section 4 gives the list of methods that we are going to compare and their dynamics on quadratic 3.

Basins of Attractions for the Uniparametric polynomials is analyzed. Next in Section 5, we present the Steffensen Scheme basins of attraction for various high-order Stefensen-type methods. In this paper, it was showed that no Scaling heorem is possible for this derivative-free scheme and, therefore, the 2. Basic Concepts analysis was made in particular polynomials.

In general, it We need some basic deinitions and notions before going to can be concluded that the stability of derivative-free schemes Section 3. If this happens, the ininity as any other derivative is zero then the point is called super-attracting [20]. In this way, we distinguish Lemma 2 see [20].

Every attracting periodic orbit is con- the attraction basins by their colors for diferent methods. In what follows, we deine the test problems in this paper. Method 3 has diferent radii of 1. Behavior of Several Steffensen-Like Methods method for solving systems of nonlinear equations, while on Quadratic Polynomials there is no need to compute the Jacobian matrix which is costly.

Another interesting point which should be in hard problems, the calculations of the derivatives take included is the computational time required for obtaining the some deal of time. In fact, in our numerical methods are better root solvers and are in focus recently. In fact, we want the methods to have very few black points on all examples not just one.

A classical result establishes 11 that there is at least one critical point associated with each invariant Fatou component. To be more components, but there exist in the family, some free critical precise see [31, 32] , a given method is generally convergent points, that is, critical points diferent from the roots, some of if the scheme converges to a root for almost every starting them depending on the value of the parameter. If any of these point and for almost every polynomial of a given degree.

So, only the behavior on speciic polynomials can be analyzed.

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In this paper, the Lemma 6. It is denoted by L4. Lemma 8. Lemma 7. Journal of Applied Mathematics 9 1. Cordero et al. We will denote it by C7. In Figure 15, the stability function of these specific fixed point is shown. As we know, if both elements coexist, basins of attraction of ixed points diferent from the roots appear. Lemma Moreover, Lemma Let us remark that tive and we expect to have small basins of attraction for them. Furthermore, it should be 3. To avoid this chaotic behavior, one for scheme with chaotic behavior alongside the convergence may follow Remark 4.

Conclusions for the worst. Notice again that the igures show how fast the method In this paper, we have analyzed the dynamics of diferent converges to a root based on shading to indicate speed of Stefensen-type methods, irstly on quadratic polynomials convergence. Amat, S. Busquier, and S. Chicharro, A. Cordero, J. Torregrosa, and for drawing the basins of attraction. Choosing 12, pp.

Torregrosa, M. Vassileva, us the ability to avoid computation of the Jacobian matrix and P. Although we have [14] C. Chun, M. Lee, B. Neta, and J. Torregrosa, and P. Soleimani, F.