Graphical Iteration
If
is a real function then an iteration sequence of the form
can be represented graphically using the graphs
and
plotted on the same axes. Any intersection of the curveand the linerepresents a fixed point of
The diagram illustrates the iteration process.
Given the point
draw a vertical line to meetat
Draw a horizontal line to meetat
Given any initial point
we can apply the above steps repeatedly to construct the sequence
The diagram above suggest the sequence converges to the value of
given by the intersection ofand
We can illustrate this for the function
There is a fixed point at
(there is also a fixed point at
) .Start with
Plot
and
Plot
and
Plot
and
and so on.