I had my exam about a week ago, and here's one problem from it.
For which values of can we use the fixed-point iteration to solve the following equation,
I know the fixed-point iteration will converge if for all . I get,
, and so . From this I conclude that .
I now define two function, and . When , a solution exists when , that is when .
Something smells here, but I don't know what. Hints are very welcome!
Experiment suggests that for the fixed point iteration that you are using this is indeed the case.
(However this is only for the fixed point iteration that you have proposed, I have not checked what happens with other formulations of fixed point iteration for this problem such as .)
This looks good, except for the last two lines. You define function but it was previously defined differently.
I checked with a graphing calculator.I now define two function, and . When , a solution exists when , that is when .
It does converge, although very slowly, for a = 0.9 .
Similarly, it does diverge (also very slowly) for a = 1.1 .