 
Summary: FIXED POINT ITERATION
We begin with a computational example. Consider
solving the two equations
E1: x = 1 + .5 sin x
E2: x = 3 + 2 sin x
Graphs of these two equations are shown on accom
panying graphs, with the solutions being
E1: = 1.49870113351785
E2: = 3.09438341304928
We are going to use a numerical scheme called `fixed
point iteration'. It amounts to making an initial guess
of x0 and substituting this into the right side of the
equation. The resulting value is denoted by x1; and
then the process is repeated, this time substituting x1
into the right side. This is repeated until convergence
occurs or until the iteration is terminated.
In the above cases, we show the results of the first 10
iterations in the accompanying table. Clearly conver
gence is occurring with E1, but not with E2. Why?
x
