I have a general question about fixed point iteration.

I have used this method several times in my Numerical Analysis course and sometimes it won't converge to certain root even if the start guess is relatively close to that actual root.

Is the fixed point method only for some roots or is it a general to find ALL roots?

  • 1
    $\begingroup$ Hi! What type of fixed point iterations are you referring to? The convergence depends on which fixed point method you are using. $\endgroup$ – Riccardo Sven Risuleo Feb 13 at 14:44
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    $\begingroup$ The map $x\mapsto 2x$ has $0$ as fixed point, but any point nearby is moved away from it. So the usual condition of $|g'(x^*)|<1$ to have convergence in $x_{n+1}=g(x_n)$ is really necessary (not strictly, more in the sense of "nice to have") to have convergence from close-by points. $\endgroup$ – LutzL Feb 13 at 14:47
  • $\begingroup$ Also have in mind that when you start from an equation $f(x)=0$ and rewrite it in the form $g(x)=x$, function $g$ can be chosen in an infinite number of ways... so, the same root can be obtained with the fixed point method when you use a certain $g$ and not obtained if you choose another. $\endgroup$ – PierreCarre Feb 13 at 15:24
  • $\begingroup$ The fixpoint method does not always find a root. If several roots exist, the root that is found can depend on the starting value. Banach's fix point theorem describes a special case with a unique solution that will be found with the method. $\endgroup$ – Peter Feb 13 at 16:09

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