Newton's method

Concept

x_{k + 1} = x_{k} - \frac{f ( x _{k} )}{f ^{'} ( x _{k} )}

sometime we use $g (x)$

g (x) = x - \frac{f ( x )}{f ^{'} ( x )}

and

g^{'} (x) = \frac{f ( x ) f ^{''} ( x )}{f ^{'} ( x ) ^{2}}

f (x) = A - \frac{1}{x}

then $f (\frac{1}{A}) = 0$ .

g (x) = x - \frac{f}{f} = x - (A x^{2} - x) = 2 x - A x^{2}

f (x) = x^{2} - A

and

g (x) = x - \frac{x ^{2} - A}{2 x} = \frac{1}{2} (x + \frac{A}{x})

convergence rate for simple root of Newton’s method is quadratic (r = 2).
for multiple root(multiplicity $m > 1$ ), it has linear rate.