# Fixed point iteration algorithm when the inputs have dependencies

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?

• 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? Aug 21, 2020 at 18:04
• @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. Aug 28, 2020 at 19:41