Can someone help me with this problem?
Consider the fixed point iteration
, k=0,1,2,...where
a) Show that the iteration converges for any initial guess
satisfying
b) Assume that the iteration converges, find m such that the method converges quadratically.
I think I got part a) by letting
, then take the derivative
. I then show that there exists
such that
, and by a theorem this implies the iteration converges for any initial guess in the specified interval.
For part b), I need to find m such that
exists, but I don't get anywhere from here. Hope someone can give a hand.