File:Newton z3-2z+2 iterates R.png

English: Color coded analysis of the dynamics of Newton's method for the polynomial

${\displaystyle z\mapsto z^{3}-2z+2}$

on the real axis in the range from ${\displaystyle -4}$ to 4. Each row shows the color-coded value of z after n-fold applying of the method. The uppermost line (n=0) shows the colors of the numbers ${\displaystyle -4}$ to 4 themselves. This polynomial has one real root at z≈-1.8 whose color code is medium blue. Other colors code for ${\displaystyle -1}$ (cyan), zero (black), one (red), infinity (white) etc.

Examples: starting from any z less than ${\displaystyle -1}$ (left part of the image) converges to the real root (medium blue). Starting at z=0 produces an endless cycle between 0 and 1 (black, red, black, red, ...)