# If fixed-point iteration has linear convergence, how can Newton's Method have quadratic convergence?

Newton's Method for finding the roots of a function can be considered a type of fixed point iteration of $g(x) = x - \frac{f(x)}{f'(x)}$, since $f(k) = 0 \rightarrow g(k) = k$. But it is well-known that fixed point iteration has linear convergence while Newton's Method has quadratic convergence, given some assumptions on $f$ and $g$ such that $f'(k) \neq 0$. How is this possible? Shouldn't Newton's Method have the same rate of convergence in this case as fixed point iteration?

• "it is well-known that fixed point iteration has linear convergence" [citation-needed] – Kenny Lau May 23 '18 at 6:44
• Under assumptions of Banach fixed point theorem, the method is known to have at least linear convergence, but it does not contradict that more detailed analysis reveals, in fact, at least quadratic convergence for Newtons's method. Convergence analysis is always the worst case study. – A.Γ. May 23 '18 at 6:55