Newton's Method and Fractals

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 \[ |N'(a + i b)| = 1, \] 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 \( N(1/2) = 5/3 \).

We can find exactly where the level curve intersects the positive real axis by solving \( N'(x) = -1 \), which gives \( 5^{2/3}/5 \). So the largest circle around the roots such that \( N(x) \) stays inside the circle has radius \[ 1 - \frac{5^{2/3}}{5}. \]

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Last updated: Mon Aug 26
Ralph Freese