Newton's method and graphical iteration

On the left is the graph of the logistic map f(x) = 4x(1-x).

On the right is graph of the Newton function Nf(x) = x - (4x(1-x))/(4 - 8x) = x2/(2x - 1).

Newton's method is iterated in (a), graphical iteration is applied in (b).

To emphasize that both generate the same sequence of points, below are the two graphs superimposed, Newton's method in red, graphical iteration in blue.

Return to real Newton's method.