# The answer to Fixed-point iteration

If the fixed-point iteration is applied to the function F(x) = 2 + (x-2)^4, starting at x = 2.5, what order of convergence results? Find the range of starting values for which this fixed-point iteration converges. Note that 2 is a fixed point.

Please provide as much detail as possible so that I can understand how the solution is derived. I would like to understand this problem so that I can apply the techniques used to similar problems. I really appreciate your help!

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#### Solution Preview

Assume that is the root of the fixed point equation

(1.1)

In addition is continuously differentiable for all x near and for some .

Furthermore, assume that

(1.2)

Then if the initial guess is sufficiently close to , the iteration

(1.3)

Will ...

#### Solution Summary

Fixed-point iterations are analyzed.