Newton’s Method in the Complex Plane
In this investigation we extend Newton’s method to the problem of finding that complex zeros of polynomials. In
particular, we are interested in finding the basins of attraction of these zeros, which will be regions in the complex
plane rather than intervals on the real number line. This investigation requires the use of technology that can
operate on complex numbers and generate images in the complex plane.
We begin by using Newton’s method to investigate the basins of attraction of the zeros of
f ( z )  z 2  1 . The zeros of f ( z ) are complex numbers z for which z 2  1 , that is 1and -1. These numbers
called the square roots of unity, are written as 1  0i and 1  0i in complex number notation. Newton’s method
generates a sequence of estimates of the zeros of f  z  using the iterative equation
zn  zn 1 
Since
f  zn 1 
.
f   zn 1 
f  z   z 2  1 and f   z   2 z , Newton’s method generates a sequence using the recursive equation
zn  zn 1 
zn21  1
2 zn 1
Which can be written as
zn 
zn21  1
.
2 zn 1
()
Because the numbers z  1  0i and z  1  0i both have an imaginary part of 0i , the zeros are both located on
the real axis. This can be demonstrated by graphing the numbers in the complex plane. We can investigate the
basins of attraction for these zeros by iterating Newton’s method for various starting values z0 . For complex
values of z0 with positive real parts (such as 1  i, 0.5  0.75i , and 0.1  2i ), Newton’s method converges to 1.
For complex values of z 0 with negative real parts, Newton’s method converges to -1. Every starting value z 0 that
converges to 1 is in the basin of attraction of 1. Likewise, every starting z 0 that converges to -1 is in the basin of
attraction of -1. The imaginary axis in the complex plane appears to form a boundary between the two basins of
attraction for the zeros of f  z  .
To see the basins more easily, we can assign the color teal to each point in the basin of attraction of 1 and assign the
color orange to each point in the basin of attraction of -1. (Technology that allows you to color points on a graph
would be helpful.)
Then a picture of the basins of attraction will be teal to the right of the imaginary axis and orange to the left of the
imaginary axis. What happens to points located exactly on the imaginary axis?
The values generated by using equation () and initial values z 0 on the imaginary axis (that is, z0 of the form
0  ki ) do not belong to either basin of attraction. Some of these values remain on the imaginary axis and never
converge to any value. Others lead to failure of Newton’s method. To understand why this occurs, consider what
happens if we choose 0 as the starting value. Newton’s method fails due to division by 0 in equation (). In
addition, if we start with  i , one iteration will take us to 0, and then Newton’s method fails on the second iteration.
By extension, a starting value that yields  i in one iteration will go to 0 in two iterations, and again the method
fails. The starting values that go to 0 in two iterations are the solutions to
i
z2 1
z2 1
.
and i 
2z
2z
Each equation is quadratic and has two solutions of the form 0  ki , so a total of four starting values on the
imaginary axis go to 0 in two iterations. Each of these four values can be reached in one iteration by two other
values, so eight starting values go to 0 in three iterations. Putting all this together, observe that 2 starting values go
to 0 in one iteration, 4 starting values go to 0 in 2 iterations, and 8 starting values go to 0 in 3 iterations. In general
2 n starting values go to 0 in n iterations. In fact, an infinite number of points on the imaginary axis will eventually
iterate to 0, thus failing, under Newton’s method. Other than 0, i , and  i , all of these numbers have irrational
imaginary parts. As we noted in the situation with basins of attraction on the real number line, we cannot represent
irrational numbers exactly with technology; we are able to work only with finite decimal approximations of these
numbers. In practice, then, these numbers will not appear in the sequence of iterates.
Exercises for Investigations
These questions assume the use of technology to iterate Newton’s method in the complex plane and to generate
images of the basins of attraction.
1. Verify with technology that the basins of attraction for the zeros of f  z   z 2  1 are the left and right halfplanes divided by the imaginary axis, as described in the text.
2. For initial values on the imaginary axis, how do the iterates of Newton’s method for f  z   z 2  1 move around
on the imaginary axis?
3. Let
f  z   z 3  1.
a. Find the zeros of f  z  , These numbers are also the roots of
z 3  1 , and they are called the cube
roots of unity. Graph these numbers in the complex plane.
b. Write the iterative equation to use with Newton’s method to find the zeros of f  z  .
c. If the technology you are using allows it, generate a three-color image of the basins of attraction
in the complex plane. What do you notice about the three colors near the boundaries of the basin?
The image generated in problem 3c is an example of a fractal. Because of the infinitely complicated boundaries of
the basins of attraction, the dimension of the boundaries is not q, as it would be if the boundary were a smooth
curve. On the other hand, the dimension of each boundary is a fraction between 1 and 2, hence the name fractal.
Fractals often possess the characteristic of self-similarity, meaning the image looks the same at varying levels of
magnification. The Newton’s method fractal in problem 3 is self-similar, as zooming in on the boundaries of the
basins of attraction reveals.
4. Let
f  z   z 4 1
a. Find the zeros of f  z  , which are the fourth roots of unity. Graph these numbers in the complex plane.
b. Write the iterative equation to use with Newton’s method to find the zeros of f  z  .
c. If the technology you are using allows it, generate a four-color image of the basins of attraction in the
complex plane. What do you notice about the four colors near the boundaries of the basins?
5
5. Repeat problem 4 for f  z   z  1 (the fifth roots of unity). Use five colors in part c.
6. If n is an integer greater than 2, where do you think the nth roots of unity are located in the complex plane?
n
The nth roots of unity are zeros of the function f  z   z  1 . Describe the basins of attraction of these
numbers.
7. Suppose a fifth-degree polynomial has complex zeros given by 2, 2i , and 1  i .
Find a formula for the polynomial and generate an image of the basins of attraction of the zeros.