Fixed Point Iteration

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.

Ifis near the fixed pointpandthen the iteration will converge top.

Ifis near the fixed pointpandthen the iteration will not converge top.

The distinguishing property for determining convergence is the size of.

Ifis near the fixed pointandin the neighborhood, then the iteration will converge to.

Ifis near the fixed pointandin the neighborhood, then the iteration will not converge to.

Back.