Basin of Attraction Example

To illustrate finding the basins of attraction, we consider the real version of the first problem Cayley solved for the complex Newton's method.

The function f(x) = x2 - 1 has two roots, x = 1 and x = -1. We find the basin of attraction of Newton's method for each root.

graph of the function
Take x0 > 1
Take x0 = 1
Take 1 > x0 > 0
Take x0 = 0
Take -1 < x0 < 0
Take x0 = -1
Take x0 < -1

Consequently,

Return to Complex Newton's Method.