Example 1.  Use fixed point iteration to find the fixed point(s) for the function   .

Solution 1.

Plot the function and determine graphically that there are two solutions to the equation  .

Will iteration prove successful as a computational tool to numerically find these two fixed points.

Use fixed point iteration to find a numerical approximation.
First, do the iteration one step at a time.  Type each of the following commands in a separate cell and execute them one at a time.

Now use the FixedPointIteration subroutine to perform the computations.

Are 7 iterations enough to locate the fixed point?

How many decimal places agreement do you see to the equation   ?

If it is not sufficient for your purposes you must request more iterations!

Comparison 1. Compare with Mathematica's built in "FindRoot" subroutine for numerically finding solutions.

Did Mathematica get all of the digits?  Why?
There are options to every built in Mathematica subroutine. For FindRoot it is:
Options[FindRoot]
{AccuracyGoal->Automatic,Compiled->True,DampingFactor->1, Jacobian->Automatic,MaxIterations->15,WorkingPrecision->16}
It is the purpose of numerical analysis to study the last two items "MaxIterations,"  "WorkingPrecision."

Comparison 2. Compare with Mathematica's built in "Solve" subroutine for symbolically finding solutions.

Mathematica's answers are "mathematically perfect."

Now investigate fixed point iteration near the other fixed point,  .

Now use the FixedPointIteration subroutine to perform the computations.

Remark. You need to look carefully at the above output and determine what it means !  Did it converge ?

The distinguishing property for determining convergence is the size of  .

If    is near the fixed point  p  and      then the iteration will converge to  p.

If    is near the fixed point  p  and      then the iteration will not converge to  p.

The distinguishing property for determining convergence is the size of  .

If    is near the fixed point    and    in the neighborhood, then the iteration will converge to  .

If    is near the fixed point    and    in the neighborhood, then the iteration will not converge to  .

(c) John H. Mathews 2004