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 quadratically if x


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

$C o n s i d e r t h e i t e r a t i o n$ $x_{k + 1} = x_{k} - \frac{{[f (x)]}^{2}}{f (x_{k} + f (x_{k})) - f (x_{k})}, k = 0, 1, 2, . . .$ $f o r t h e s o l u t i o n o f f (x) = 0 . E x p l a i n t h e$ $c o n n e c t i o n w i t h {N e w t o n}^{'} s m e t h o d, a n d s h o w$ $t h a t (x_{k}) c o n v e r g e s q u a d r a t i c a l l y i f x_{0} i s$ $s u f f i c i e n t l y c l o s e t o t h e s o l u t i o n .$
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