# Fix point iteration, rate of convergence

Determine whether iterations based on the following formulae converge toward the fixed point $x=\sqrt a$, $a>0$ and for each iteration that converges , find the rate of convergence (i.e. linear, quadratic etc.)

(a) $\phi(x) = {a\over x}$

(b) $\phi(x) = 2x-{a\over x}$

(c) $\phi(x) = {1\over2}\Big( x+{a\over x} \Big)$

When I look at the first one, I can see $x_1=\phi(x_0)={a\over x_0}$, and $x_2=\phi(x_1)= {a\over{a\over x_0}}=x_0$, so it is not convergence, but I am not sure if I am right. and (b), (c), seems i should use the formula $$\lim_{x\to \infty}{|\phi(x_{n+1})-\sqrt a |\over |\phi(x_n)-\sqrt a |^\alpha}=\lambda$$

but i dont know how to do it and what exactly is the rate of convergence.

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