# Section1.3Exercises

This set of exercises will be mostly experimental. So, fire up your favorite computational environment. This text will include examples using both Python and Mathematica.

##### 1

Continuing with the example of $f(x)=x^2-2$ explored in Example 1.1.3, compute ten Newton iterations for several values of $x_0\text{.}$ Be sure to choose both positive and negative values and values that are both large and small in magnitude.

##### 2

In the previous exercise, what happens when $x_0=0\text{?}$ Draw a graph to illustrate the situation.

##### 3

Let $f$ be a quadractic function that has two, distinct, real roots but that is otherwise arbitrary. Using a geometrical understanding of the real Newton's method, show why an initial seed $x_0$ always leads to a sequence that converges to the closer of the two roots of $f\text{.}$

##### 4

Let's modify Newton's original example just a little bit to consider

\begin{equation*} f(x) = x^3-2x-2. \end{equation*}
1. Compute the corresponding Newton's method iteration function, $N\text{.}$
2. Iterate $N$ from the initial point $x_0 = 0\text{.}$ What behavior do you see?
3. Iterate $N$ from several initial points $x_0$ close to zero. Now, what behavior do you see?
##### 5

Figure 8 shows the graph of the function

\begin{equation*} f(x) = \frac{1}{3} x (x + 1) (x - 3) (x^2 - 2). \end{equation*}

The green dots represent points on the graph with $x$-coordinates that we might consider as initial seeds for Newton's method.

1. Suppose we start at the green dot whose $x$ coordinate is just slightly larger than 1. To which root do you think the process will converge?
2. Suppose we start at the green dot whose $x$ coordinate is between 2 and 3. To which root do you think the process will converge?
3. Find a specific value of the initial seed $x_0$ between 2 and 3 with the property that the process converges to the smallest root of the function.
4. Find a specific value of the initial seed $x_0$ between 2 and 3 with the property that the process converges to the value 1.
##### 6

Launch the interactive tool for generating the basins of attraction of Newton's method for polynomials here: https://marksmath.org/visualization/complex_newton/.

Now, use the tool to generate images for the following polynomials and answer any additional questions that are asked.

1. $f(z) = z^4 - 1$
1. What are the four roots of the function? Where do they fit into the picture?
2. Click on the picture. How do you interpret the line that is drawn?
2. $f(z) = (z^2-1)(z-10)$
1. What initial step should you take to enter your input?
2. What are the roots of the function? How could you account for this when generating the picture?
3. $f(z) = z^3 - 2z - 2$
1. You should see some black regions. What's up with that?