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 $x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$
Best Answer
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.