A usual fixed point problem has the form $x_{k+1}=f(x_k)$, and you can efficiently solve it by finding the root to $f(x)-x$. What if I now have several dependent inputs $x_{k+1}=f(x_k, y_k)$, and $y_k=g(x_{k-1}, y_{k-1})$. I can certainly run many iterations until $f$ converges (assume a fixed point exists), but is there a more clever algorithm for this?

  • $\begingroup$ If you can find a root of $(x,y)\mapsto (x-f(x,y), y-g(x,y))$, say, by the Newton method, then yes. What's the difference? $\endgroup$
    – fedja
    Aug 21, 2020 at 18:04
  • $\begingroup$ @fedja, thanks for the comment. Actually this is right, but I guess my real question was whether there is a way to decouple this, especially when the input has very large dimensions. That should probably be a new post. $\endgroup$ Aug 28, 2020 at 19:41


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