Alright, so we all know how single variable functions work with iteration:
If f(x) = x^2
Then f^2 (x) = (x^2)^2
This works really well because in with only one independent variable there is one input and one output. So it’s not so hard to understand something like Cos(x) being iterated 24 times.
With all this in mind, how does multivariate function iteration work.
f(x,y) = x+y
f^2 (x,y) = ? There are three combinations for n=2 (number of compositions or iterations) as far as I can tell:
= f( x, f(x,y) ) = x + (x+y) = 2x+y
= f( f(x,y), f(x,y) ) = (x+y) + (x+y)= 2x+2y
= f( f(x,y), y ) = (x+y) + y = x+2y
Unsure if placing y and x in for the first and second argument respectively would still achieve the goal I am looking for or not...
In any case, is it possible to generalize (or even notate at this point) the many combinations that arise as n increases? Has this been investigated before? If not, any ideas?