# How do you call the following iterative solving method

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.

• 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) $f$. – hardmath Mar 15 '14 at 17:47
• 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. – Giuseppe Negro Mar 15 '14 at 17:56
• @GiuseppeNegro: Newton's method for root-finding $f(x)=0$ is a fixed point iteration for $x = x - f(x)/f'(x)$. – hardmath Mar 15 '14 at 18:05