Homework #5
Instructions. This is due on November 8
at the beginning of class. It is worth 20 points. Late papers lose 4 points if
turned in Monday plus 3 points per
additional day or partial day late. You must work by yourself, but feel free to
consult with me. Explain what you did and why you did it. Your paper should
start with an introduction and finish with a conclusion. Start early, ask
questions and do well.
Introduction. This assignment explores an unusual aspect
of Newton’s method: art! As you will see in class, Newton’s method can be used
to generate fractals, complex geometric shapes that contain miniature
copies of themselves. This assignment provides many of the details behind the
fractal that you will be shown in class. Throughout the assignment, we will
work with
f(x) = x3 - 3x2 + 2x = x(x-1)(x-2)
Note that the three solutions of the equation f(x) = 0 are x = 0, x = 1 and x = 2. Since there are three solutions, one issue is which solution will be produced by Newton’s method. That is the focus of this assignment.
Problems. Determine which of the three solutions is produced using (a) x0
= 0.1, (b) x0 = 1.1, (c) x0 = 2.1. Based on
this, describe the logic that would tell you in advance which solution you will
get.
Now, determine which solution is produced using (d) x0
= 0.54, (e) x0 = 0.55, (f) x0 = 0.56. Would
this have been predicted by your logic from steps (a)-(c)?
It turns out that this problem is extremely
difficult to solve. You have seen that starting values from 0.54 to 0.56
produce all three solutions. Now, find starting values between 0.552 and 0.553
that produce all three solutions. In fact, you could keep going, finding
numbers closer and closer together containing all three solutions.
To take this one step further, we can use complex
numbers in Newton’s method (same formula, no changes). Determine which solution is produced using
(g) x0 = 1 + i, (h) x0 = 0.48 + 0.02i, (i) x0 = 0.48 + 0.03i.
When problems get this complicated, often a picture
helps us visualize any patterns that might exist. We could create a picture as
follows: pick a starting number x0. We can use complex
numbers, so x0 could have the form a + bi.
Determine which solution Newton’s method converges to. If it converges to 0,
color the point (a,b) purple; if it converges to 1, color the
point (a,b) green; if it converges to 2, color the point (a,b)
red. You will be shown some pictures in class. Briefly describe what these
pictures look like. Do you think they qualify as art?