Iteration

From CasGroup

Iteration is related to feedback, it is the mathematical counterpart of feedback:

"Iteration is the mathematical counterpart of feedback. Feedback in general is any response to something sent out. In mathematics, that translates as "what goes out comes back in again." It is output that returns to serve as input. In temporal processes, feedback is that part of the past that influences the present, or that part of the present that influences the future. Positive feedback amplifies or accelerates the output. It causes an event to become magnified over time. Negative feedback dampens or inhibits output, or causes an event to die away over time. Feedback shows up in climate, biology, electrical engineering, and probably in most other fields in which processes continue over time." (from Chaos Theory Tamed by G. P. Williams)

Recursion vs Iteration

We know that iteration of functions or L-systems can lead to self-similar fractals. Both iteration and recursion run a part of an algorithm repeatedly, a recursive function calls itself with different parameters, while an iterative function includes a loop, which iterates a pre-determined number of times. Therefore iteration is the analog of recursion.

The network analog of a recursive function

What happens if we add interaction to iteration? What is the network analog of a recursive function? (This was one of the ambitious questions in the Wolfram Science Conference 2006) Is iteration in a network of connected nodes the counterpart of a certain recursive function? Iteration in a network of connected nodes leads to nested propagation loops. The patterns in CA and RBN are an example of such nested propagation loops.

Links

• Wikipedia entry for Fractal
• Wikipedia entry for L-system
• Wikipedia entry for Iterated function system