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Consider the quadratic polynomial $f(x) = x^2 − 6x + 2$. The two roots of this function are $R_1 = 3 + \sqrt 7$, $R_2 = 3 − \sqrt 7$. Consider the following 4 different iterative processes.

a)$X_{n+1}=6-\frac{2}{X_n}$;

b)$X_{n+1}=\frac{1}{6}X_n^2+\frac{1}{3}$;

c)$X_{n+1}=\sqrt{6X_n-2}$;

d)$X_{n+1}=\frac{X_n^2-2} {2X_n-6}$;

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    Now the edits make sense... if we set $X_n=X_{n+1}=x$ we get the equation back.2012-05-31

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I will take a leap here and assume that the question is why those processes should find a zero of the given polynomial. Well, if they converge then they converge to a zero of the polynomial since for the limit $x$ (and the case a)) we have

$x=6-\frac 2x\Leftrightarrow x^2-6x+2=0$

and similarly for the other cases. Whether they converge and to which zero they converge depends on your starting point.

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    If you show $|\frac {dX_{n+1}}{dX_n}| \lt 1$ you know it converges if you start close enough and you can estimate the speed of convergence.2012-07-08