iteration

Repetition of a sequence of instructions. A fundamental part of many algorithms. Iteration is characterised by a set of initial conditions, an iterative step and a termination condition.

A well known example of iteration in mathematics is Newton-Raphson iteration. Iteration in programs is expressed using a loop, e.g. in C:

	new_x = n/2;
	do
	{
	  x = new_x;
	  new_x = 0.5 * (x + n/x);
	} while (abs(new_x-x) > epsilon);

Iteration can be expressed in functional languages using recursion:

	solve x n = if abs(new_x-x) > epsilon
		    then solve new_x n
		    else new_x
		    where new_x = 0.5 * (x + n/x)

        solve n/2 n