Fixed point iteration: finding sufficient conditions

Asked Feb 15 '16 at 23:59

Active Feb 16 '16 at 10:37

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1) Suppose it is known that the fpi $x_n :=g(x_{n-1})$ satisfies the estimate $|x_n-\alpha| \leq c|x_{n-1}-\alpha|^2$

a) Show by induction that $c|x_n-\alpha| \leq (c|x_0-\alpha|)^{2^n} $ and give some condition that is sufficient for the convergence of $x_n$ to $\alpha$

edited Feb 16 '16 at 10:37

the_candyman

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asked Feb 15 '16 at 23:59

user314580

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So, where is the inductive argument? They give you the base case. – Moo Feb 16 '16 at 01:58
This is all, I'm not missing anything – user314580 Feb 16 '16 at 04:38
You just need to prove $$c|x_n-\alpha| <= (c|x_0-\alpha|)^{2^n}$$ for n > 1 using induction. The convergence condition comes from this inequality. – runaround Feb 16 '16 at 05:10

Fixed point iteration: finding sufficient conditions

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