Graphical Iteration

Given an initial value x₀ and a function f, we generate the sequence

x₀, x₁ = f(x₀), x₂ = f(x₁), x₃ = f(x₂), ...

The algebraic mechanism is clear, though perhaps unenlightening.

Graphical iteration is a geometrical method for visualizing this process.

Briefly, we use an alternating collection of vertical and horizontal segments to connect these points.

(x0,0) to (x0,f(x0)) = (x0,x1)

(x0,x1) to (x1,x1)

(x1,x1) to (x1,f(x1)) = (x1,x2)

(x1,x2) to (x2,x2)

and so on.

Here is an illustration of the first few steps of graphical iteration.

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