Consider-the-iteration-x-k-1-x-k-f-x-2-f-x-k-f-x-k-f-x-k-k-0-1-2-for-the-solution-of-f-x-0-Explain-the-connection-with-Newton-s-method-and-show-that-x-k-converges-qu

Question Number 18349 by Yozzzzy last updated on 19/Jul/17

C o n s i d e r t h e i t e r a t i o n

x_{k + 1} = x_{k} - \frac{{[f (x)]}^{2}}{f (x_{k} + f (x_{k})) - f (x_{k})}, k = 0, 1, 2, \dots

f o r t h e s o l u t i o n o f f (x) = 0 . E x p l a i n t h e

c o n n e c t i o n w i t h {N e w t o n}^{'} s m e t h o d, a n d s h o w

t h a t (x_{k}) c o n v e r g e s q u a d r a t i c a l l y i f x_{0} i s

s u f f i c i e n t l y c l o s e t o t h e s o l u t i o n .