Let r be the zero of multiplicity 2 of the polynomial p(x), how do I prove that $x_{n}$ converges quadratically to r? I only know the basic Newton's Method which is $x_{n+1} = x_{n} - \frac{f(x_{n})}{f'(x_{n})}$, I can't think other way to prove that from this. I tried to googled, but couldn't find any reference for this. Thanks in advance.
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The basic Newton's method does not converge quadratically for multiple roots, as you can see on the example $f(x)=x^2$ for which $x_{n+1}=x_n/2$. This is why the modified Newton's method is used. – Oct 26 '14 at 20:39