##
**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.

(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) |

### Keywords:

iterative methods; roots of nonlinear equations; Newton’s method; Julia sets; fractal pictures; regions of convergence### Software:

Mathematica
Full Text:
DOI

### References:

[1] | Alexander, C.; Giblin, I.; Newton, D., Symmetry groups on fractals, The Mathematical Intelligencer, 14, 2, 32-38 (1992) · Zbl 0752.58020 |

[2] | Argyros, I. K.; Chen, D.; Qian, Q., The Jarratt method in Banach space setting, J. Comput. Appl. Math., 51, 103-106 (1994) · Zbl 0809.65054 |

[3] | Argyros, I. K.; Szidarovszky, F., The Theory and Applications of Iteration Methods (1993), Boca Raton, FL: CRC Press, Boca Raton, FL · Zbl 0844.65052 |

[4] | Bergweiler, W., Iteration of meromorphic functions, Bull. Am. Math. Soc. (N. S.), 29, 151-188 (1993) · Zbl 0791.30018 |

[5] | Blanchard, P., Complex analytic dynamics on the Riemann sphere, Bull. Am. Math. Soc. (N. S.), 11, 85-141 (1984) · Zbl 0558.58017 |

[6] | Ezquerro, J. A.; Gutierrez, J. M.; Hernandez, M. A.; Salanova, M. A., The application of an inverse-free Jarratt-type approximation to nonlinear integral equations of Hammerstein-type, Comput. Math. Appl., 36, 9-20 (1998) · Zbl 0932.65060 |

[7] | Ezquerro, J. A.; Hernández, M. A., On a convex acceleration of Newton’s method, J. Optim. Theory Appl., 100, 311-326 (1999) · Zbl 0915.90235 |

[8] | Gander, W., On Halley’s iteration method, Am. Math. Monthly, 92, 131-134 (1985) · Zbl 0574.65041 |

[9] | Gautschi, W., Numerical Analysis: An Introduction (1997), Boston: Birkhäuser, Boston · Zbl 0877.65001 |

[10] | Gutiérrez, J. M.; Hernández, M. A., A family of Chebyshev- Halley type methods in Banach spaces, Bull. Austral. Math. Soc., 55, 113-130 (1997) · Zbl 0893.47043 |

[11] | Hernández, M. A., An acceleration procedure of the Whittaker method by means of convexity, Zb. Rad. Phrod.-Mat. Fak. Ser. Mat., 20, 27-38 (1990) · Zbl 0748.65044 |

[12] | Jarratt, P., Some fourth order multipoint iterative methods for solving equations, Math. Comp., 20, 434-437 (1966) · Zbl 0229.65049 |

[13] | Kincaid, D.; Cheney, W., Numerical Analysis: Mathematics of Scientific Computing (1996), Pacific Grove, CA: Brooks/Cole, Pacific Grove, CA · Zbl 0877.65002 |

[14] | King, R. F., A family of fourth order methods for nonlinear equations, SIAM J. Numer. Anal., 10, 876-879 (1973) · Zbl 0266.65040 |

[15] | Ortega, J. M.; Rheinboldt, W. C., Iterative Solution of Nonlinear Equations in Several Variables, Monographs Textbooks Comput. Sci. Appl. Math. (1970), New York: Academic Press, New York · Zbl 0241.65046 |

[16] | Peitgen, H. O.; Richter, P. H., The Beauty of Fractals (1986), New York: Springer-Verlag, New York · Zbl 0601.58003 |

[17] | Shub, M., Mysteries of mathematics and computation, The Mathematical Intelligencer, 16, 2, 10-15 (1994) · Zbl 0806.68037 |

[18] | Traub, J. F., Iterative Methods for the Solution of Equations (1964), Englewood Cliffs, NJ: Prentice-Hall, Englewood Cliffs, NJ · Zbl 0121.11204 |

[19] | W. Werner, Some improvements of classical iterative methods for the solution of nonlinear equations, inNumerical Solution of Nonlinear Equations (Proa, Bremen, 1980), E. L. Allgower, K. Glashoff and H. O. Peitgen, eds., Lecture Notes in Math. 878 (1981), 427-440. |

[20] | S. Wolfram,The Mathematica Book, 3rd ed., Wolfram Media/Cambridge University Press, 1996. |

[21] | Neuberger, J. W., The Mathematical Intelligencer, 21, 3, 18-23 (1999) · Zbl 1052.30502 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.