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Graphic and numerical comparison between iterative methods. (English) Zbl 1003.65046

From the introduction: There are two motives for studying convergence of iterative methods: (a) to find roots of nonlinear equations, and to know the accuracy and stability of the numerical algorithms; (b) to show the beauty of the graphics that can be generated with the aid of computers. Generally, there are three strategies to obtain graphics from Newton’s method:
(i) We take a rectangle \(D\subset\mathbb{C}\) and we assign a color (or a gray level) to each point \(z_0\in D\) according to the root at which Newton’s method starting from \(z_0\) converges; and we mark the point as black (for instance) if the method does not converge. In this way, we distinguish the attraction basins by their colors.
(ii) Instead of assigning the color according to the root reached by the method, we assign the color according to the number of iterations required to reach some root with a fixed precision. Again, black is used if the method does not converge. This does not single out the Julia sets, but it does generate nice pictures.
(iii) This is a combination of the two previous strategies. Here, we assign a color to each attraction basin of a root. But we make the color lighter or darker according to the number of iterations needed to reach the root with the fixed precision required. As before, we use black if the method does not converge. In my opinion, this generates the most beautiful pictures.
All these strategies have been extensively used for polynomials, mainly for polynomials of the form \(z^n-1\) whose roots are well known. Of course, many other families of functions have been studied.
Although Newton’s method is the best known, in the literature there are many other iterative methods devoted to finding roots of nonlinear equations. Thus, my aim in this article is to study some of these iterative methods for solving \(f(z)= 0\), where \(f:\mathbb{C}\to \mathbb{C}\), and to show the fractal pictures that they generate (mainly, in the sense described in (iii)). Not to neglect numerical analysis, I will compare the regions of convergence of the methods and their speeds.

MSC:

65H05 Numerical computation of solutions to single equations
65E05 General theory of numerical methods in complex analysis (potential theory, etc.)
65D18 Numerical aspects of computer graphics, image analysis, and computational geometry
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
68U05 Computer graphics; computational geometry (digital and algorithmic aspects)

Software:

Mathematica
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References:

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