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I have the following implicit equation

$$ x= f(x) $$

which I solve by starting with some value for $x$, then setting $x$ to the new value $f(x)$ and so forth until convergence.

How is that method called? What are its convergence properties, what are caveats one should be aware of, especially wrt the convexity of $f$? In my case $f$ is infinitely derivable.

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  • $\begingroup$ This is called fixed point iteration. You'll find there have been a number of Questions at Math.SE about its convergence properties, esp. for smooth (differentiable) $f$. $\endgroup$ – hardmath Mar 15 '14 at 17:47
  • $\begingroup$ AFAIK, this method is not really used in numerical analysis, because its convergence is rather slow. One usually goes for the Newton method, which requires a lot more assumptions on $f$ (namely, differentiability and some control over derivatives), but converges much faster. $\endgroup$ – Giuseppe Negro Mar 15 '14 at 17:56
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    $\begingroup$ @GiuseppeNegro: Newton's method for root-finding $f(x)=0$ is a fixed point iteration for $x = x - f(x)/f'(x)$. $\endgroup$ – hardmath Mar 15 '14 at 18:05
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It is called the fixed point method. There is a whole theory, called Fixed Point Theory, part of an nonlinear analysis, but it's results are applied in many other fields such as numerical analysis (your question is about that I suppose).

The most known result about this theory is called Banach's contraction principle, and states: In a complete space, for a contractive mapping, there exist a unique fixed point.

So your method is convergent if the space you're working in is complete and if the function is contractive.

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