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. |
Here is an example of sensitivity to initial conditions for the
tent map. |