Originally Posted by

**namelessguy** Can someone help me with this problem?

Consider the fixed point iteration

$\displaystyle x_{k+1}=(m+1)x_k-{x_k}^2$, k=0,1,2,...where $\displaystyle 1\leq m \leq 2$

a) Show that the iteration converges for any initial guess $\displaystyle x_0$ satisfying $\displaystyle m-1/5 \leq x_0 \leq m+1/5$

b) Assume that the iteration converges, find m such that the method converges quadratically.

I think I got part a) by letting $\displaystyle g(x)=(m+1)x-x^2$, then take the derivative $\displaystyle g'(x)=2x+m+1$. I then show that there exists $\displaystyle 0<k<1$ such that $\displaystyle \mid g'(x) \mid <k$, 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 $\displaystyle lim_{k\to \infty} \frac{|x_{k+1}-x|}{|x_k-x|^2}= lim_{k\to \infty} \frac{|(m+1)x_k-{x_k}^2-x|}{|x_k-x|^2}$ exists, but I don't get anywhere from here. Hope someone can give a hand.