Newton's Method in the Complex Plane

by Jesse Smith, Leif Johnson, and Alex Rosefielde

December 1995

for Calculus with Topics

professor Helen Compton

NC School of Science and Mathematics

Newton's Method is a technique used to find the roots of any continuous, differentiable function, called f(x). To start, an initial x value is chosen, called x_0, and the tangent line to the function at that x value is plotted. From our earlier investigations in calculus and local linearity, we concluded that this tangent line is locally linear to the function at that point, so its equation is

A new x value, x_1, is generated as shown in the illustration when the tangent line intersects the x axis, and therefore where y value of the tangent line equals zero. Since y = 0, the equation of the line can now be rearranged algebraically to form the iterative equation for Newton's Method. This equation takes the general form of

To find any x_n, all that is necessary is a knowledge of x_{n-1} and the function f(x).

The complex plane is a way to represent complex numbers graphically using two dimensions. The horizontal, or real, axis represents the first part (a) of an imaginary number in the form a + bi, and the vertical, or imaginary, axis represents the second part (b) of the imaginary number. The point 1 + 2i, for example, is located at the point (1, 2) in the complex plane. When an imaginary number is chosen as x_0, then it is iterated through the equation given above until the iterations approach a root of f(x). If a different color is assigned to each root of an equation, then each point in the complex plane that is iterated using Newton's Method is colored according to the root that the point approaches. Using Newton's Method allows us to find roots of functions that are both real and imaginary.

Suppose f(z) = z^2 - 1. We know that the two roots of this function are real and located at the points (1,0) and (-1,0). These two roots are expressed on the complex plane at the points 1 + 0i and -1 + 0i. If all the points in the complex plane are iterated using Newton's Method and colored according to the root that they approach, then we get a colored picture of the complex plane that seems to be evenly divided along the imaginary axis into two halves. Any imaginary number that has a positive real part goes to the root at 1 + 0i, and any number that has a negative real part will go to the root at -1 + 0i. These two areas of the complex plane surrounding the two roots are called basins of attraction because initial values used in the iterations that are close together tend to approach the same root. Numbers that do not have real parts do not go to either root; instead, they move randomly along the imaginary axis and never approach a root. That is, numbers of the form 0 + bi are not in a basin of attraction, and so they never approach a root.

If the function takes the form f(z) = z^3 - 1, whose roots are located at the points 1 + 0i, -1/2 + (sqrt(3)/2)i, and -1/2 - (sqrt(3)/2)i, then the plane becomes more complicated, and it has more chaos and instability where the distance between two roots is equal. Since the function has three roots, the graph of the complex plane is divided into three parts, each of which is a basin of attraction for a root. However, areas where basins meet become turbulent but predictable. The turbulence generally takes the form of a teardrop shape, and the narrow part of the teardrop points toward the root that the turbulent area approaches. The large teardrops, as shown in the illustration, are surrounded by smaller teardrops of turbulence, and the narrow end of each of these smaller teardrops points toward a larger teardrop, which points toward the root that that teardrop goes to. In turn, the smaller teardrops are surrounded by even tinier teardrops, and the pattern of similarity continues.

When f(z) = z^4 - 1, more patterns and similarities emerge that link general trends together. The roots of this equation are located at the points 1 + 0i, 1 - 0i, 0 + i, and 0 - i, and areas of turbulence again occur where two basins of attraction meet. This time, however, the turbulence is much more complicated. Although the teardrop shape stays the same, the drops are more narrow and numerous. The narrow end of the teardrop still points toward the root that the drop approaches, however, and that pattern seems to stay constant throughout our investigation.

For f(z) = z^5 - 1, the general trends become more apparent and easy to identify. The turbulent areas between basins of attraction get more complicated as the degree of the equation increases, and the turbulent area surrounding the center of the imaginary plane becomes bigger.

Through our investigations of the functions presented, we made a general conclusion about the roots and basins of attraction for any function of the form f(z) = z^n - 1. Each of the n roots of functions of this form has a basin of attraction which occupies (at maximum) an area that is 1/n of the size of the plane. The greater n gets, however, the smaller each basin becomes because the turbulent area between basins becomes larger, and more basins split the plane into smaller pieces. In addition, we found that the roots of functions of this form are located at points evenly spaced about the unit circle, and the first root is located at the point 0 on the unit circle, which is 1 + 0i in the complex plane. The other roots are located at points on the unit circle represented by integer multiples of the angle 2pi/n, and in the complex plane the roots are located at the points cos(2pi/n) + i sin(2pi/n).

We discovered several interesting anomalies during our investigation of these functions. The largest and most puzzling one of these is the conclusion that the root located at the point 1 + 0i is "stronger" than the other roots, meaning the color that represents this root occurs more often than the other colors in the complex plane. This is an interesting discovery mathematically because it shows that if a random number were chosen from anywhere in the complex plane, there is a higher probability of that number iterating to the root at 1 + 0i than to any other root. To find some conclusive evidence, we zoomed in on the graph of the basins of attraction for f(z) = z^2 - 1 near the point 0 + 0i. The boundary for this graph appears to be a straight line, but when we zoomed in, we discovered that a very small portion of the basin for the root at - 1 + 0i actually iterated to the root at 1 + 0i, as shown in the picture. We decided that this could not be a technology error because the small area appeared on two separate computers running the same program, and the area was too regularly shaped to be an error. Further support of this theory came when we observed general trends in all of the functions that we tested. We found that none of the basins of attraction contacted each other except for the basin of the root at 1 + 0i. Graphically speaking, none of the colors corresponding to other roots touch each other, but the color that corresponds to the root at 1 + 0i separates all of the other colors, as shown in the zoomed image of f(z) = z^4 - 1. This means that the color corresponding to the root at 1 + 0i touches all of the other colors, but none of the other colors touch each other.

One of the other interesting trends that we discovered in this investigation dealt with the small, relatively calm circle that appeared in the center of the graph (surrounding the origin of the complex plane). We discovered that this circle existed for all values of n greater than 2 in functions of the form f(z) = z^n - 1, and we also found that odd values of n produced a single color circle. While exploring the even values of n, we found that only half of these values had circles that were a single color; the other half of the even values produced a circle that was two colors, divided vertically by the imaginary axis. Our general conclusion regarding the circles in the center of the graph is that for all odd values of n, the circle is a single color that corresponds to the root at 1 + 0i. For all even values of n that are not evenly divisible by four, the same result occurs. However, for even values of n that are evenly divisible by four, the circle is two colors divided along the imaginary axis. The color on the left side of the circle corresponds to the root at 1 + 0i, and the root on the right side of the circle corresponds to the root at -1 + 0i. We also observed that the circle of relative calmness in the center of the graph gets larger as the values of n increase.

This calculus investigation led us through a lot of algebra and personal exploration into an absolutely fascinating realm of fractals, apparent randomness, and chaotic graphs and trends. The calculus involved only worked in the background, but without it, the entire investigation would have been impossible. The entire concept of derivatives and local linearity that Newton's Method is based on is necessary and the core of the investigation, but it is far less interesting to investigate by itself than the fractals and graphs. As of this writing, our investigations into these fractals only dealt with zooming further into the graphs and pictures, but recently we began exploring the concept of zooming out. Despite the incredibly interesting trends that appeared when we zoomed out instead of in, we could not include them as well in this paper due to hardware limitations and time restrictions ! At any rate, the investigation was very illuminating and led to many new understandings of trends involved with Newton's Method and the complex plane.