Brief Explanation
What you see here is the complex plane. The center is the origin and the upper right
corner is the complex point 2 + 2i. The lower left corner is the complex point -2 -
2i. The three black points are the complex roots of x3 - 1. The
colors indicate to which root Newton's method will gravitate. You can click anywhere
and see 5 iterations of Newton's method. (Some might be off the screen.) More tools
will be added later.
f(x) = x3 - 1
(The equation)
N(x) = (2x3+1)/(3x2)
(Newton's iterate function)
N'(x) = 2(x3-1)/(3x3)
(The derivative of N)
A DERIVE Plot of the Level Curves: |N'(a + ib)| = c
I wanted to answer the following question: if a point x is in a circle of radius 1/2 of
one of the three roots, is N(x) also in that circle? Looking at the level curve |N'(a
+ ib)| = 1, we see that it gets closer than 1/2 from the roots, suggesting that the
answer to the problem is no. Using the applet above we can verify this by clicking at the
point 1/2 or just calculating N(1/2) = 5/3.
