wirte a matlab code

Fractal Basins of Attraction The applicability of Newton's method for finding complex roots is one of its outstanding strengths. One need only program Newton's method using complex arithmetic The frontiers of numerical analysis and nonlinear dynamics overlap in some intriguing ways. Computer-generated displays with fractal patterns, such as in Figure 3.8, can easily be created with the help of the Newton iteration. The resulting pictures show intricately FIGURE 3.8 Basins of attraction Chapter 3 Locating Roots of Equations interwoven sets in the plane that are quite beautiful if displayed on a color computer monitor. One begins with a polynomial in the complex variable z. For example, p(z) = z4-1 is suitable. This polynomial has four zeros, which are the fourth roots of unity. Each of these zeros has a basin of attraction, that is, the set of all points Zo such that Newton's iteration, started at Zo, will converge to that zero. These four basins of attraction are disjoint from each other, because if the Newton iteration starting at Zo converges to one zero, then it cannot also converge to another zero. One would naturally expect each basin to be a simple set surrounding the zero in the complex plane. But they turn out to be far from simple. To see what they are, we can systematically determine, for a large number of points, which zero of p the Newton iteration converges to if started at zo. Points in each basin can be assigned different colors. The (rare) points for which the Newton iteration does not converge can be left uncolored. Computer Problem 3.2.27 suggests how to do this. Fractal Basins of Attraction The applicability of Newton's method for finding complex roots is one of its outstanding strengths. One need only program Newton's method using complex arithmetic The frontiers of numerical analysis and nonlinear dynamics overlap in some intriguing ways. Computer-generated displays with fractal patterns, such as in Figure 3.8, can easily be created with the help of the Newton iteration. The resulting pictures show intricately FIGURE 3.8 Basins of attraction Chapter 3 Locating Roots of Equations interwoven sets in the plane that are quite beautiful if displayed on a color computer monitor. One begins with a polynomial in the complex variable z. For example, p(z) = z4-1 is suitable. This polynomial has four zeros, which are the fourth roots of unity. Each of these zeros has a basin of attraction, that is, the set of all points Zo such that Newton's iteration, started at Zo, will converge to that zero. These four basins of attraction are disjoint from each other, because if the Newton iteration starting at Zo converges to one zero, then it cannot also converge to another zero. One would naturally expect each basin to be a simple set surrounding the zero in the complex plane. But they turn out to be far from simple. To see what they are, we can systematically determine, for a large number of points, which zero of p the Newton iteration converges to if started at zo. Points in each basin can be assigned different colors. The (rare) points for which the Newton iteration does not converge can be left uncolored. Computer Problem 3.2.27 suggests how to do this