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well, could any one tell me how to prove this in the relation with newton raphson method? if $y$ is a root of $f(x)=0$ with multiplicity $p$,then iterative formula becomes $x_{n+1}=x_n-p[f(x_n)/f'(x_n)]$

we know in general for simple root the iteration formula is $x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$

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    Yes, I understand that. I was merely pointing out that you haven't told us what you want to see "proved", so all you can expect as an answer is an informal explanation why this formula is useful for finding multiple roots; if you want a proof, you have to first provide a formal statement to be proved.2012-08-05

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Your formula is discussed in William J Gilbert, Newton's method for multiple roots, Comput. & Graphics 18 (1994) 227-229, available here. Gilbert cites equations 8.6-13 in Ralston and Rabinowitz, A First Course in Numerical Analysis, McGraw-Hill 1978.

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    See that book [here](http://books.google.com/books?hl=en&id=czHV-1bEFl0C&pg=PA354).2012-08-06