What is the difference? Are these the same? If not, can someone please give me an example?
MW:
Iteration – 1 : the action or a process of iterating or repeating: as a : a procedure in which repetition of a sequence of operations yields results successively closer to a desired result b : the repetition of a sequence of computer instructions a specified number of times or until a condition is met
Recursion – 3 : a computer programming technique involving the use of a procedure, subroutine, function, or algorithm that calls itself one or more times until a specified condition is met at which time the rest of each repetition is processed from the last one called to the first
We can distinguish (as is done in SICP) recursive and iterative procedures from recursive and iterative processes. The former are as your definition describes, where recursion is basically the same as mathematical recursion: a recursive procedure is defined in terms of itself. An iterative procedure repeats a block of code with a loop statement. A recursive process, however, is one that takes non-constant (e.g. O(n) or O(lg(n)) space) to execute, while an iterative process takes O(1) (constant) space.
For mathematical examples, the Fibonacci numbers are defined recursively:
Sigma notation is analogous to iteration:
as is Pi notation. Similar to how some (mathematical) recursive formulae can be rewritten as iterative ones, some (but not all) recursive processes have iterative equivalents. All recursive procedures can be transformed into iterative ones by keeping track of partial results in your own data structure, rather than using the function call stack.