Iteration

TL;DR

Repeating a set of instructions or calculations until a condition is satisfied.
Common in programming (e.g., for loops) and mathematics (e.g., recursive definitions).
Used to break complex tasks into repeated steps or smaller subproblems.

Definition

Iteration is the repetition of a certain process or algorithm in computer programming and mathematics, performed by repeating a set of instructions or calculations until a specific condition is met.

Explanation

Iteration repeats instructions to achieve the efficient completion of complex tasks. In programming, iteration typically uses constructs like for loops with a counter variable to traverse sequences or arrays; the loop continues while a condition holds and stops when the condition is no longer true. Recursive algorithms implement iteration by calling the algorithm itself with reduced inputs until a base case is reached, at which point the recursion stops.

Examples

For loop example

A for loop iterating through numbers 1 to 10:

for (int i = 1; i <= 10; i++) {

System.out.println(i);

}

In this example, the counter variable i is incremented by 1 with each iteration. The loop continues until the condition i <= 10 is no longer true.

Recursive factorial example

Mathematical definition: the factorial of 5 is 5!, which is equal to 5 * 4 * 3 * 2 * 1 = 120.

A recursive algorithm to compute factorial n:

int factorial(int n) {

if (n == 1) {

return 1;

}

else {

return n * factorial(n - 1);

}

}

In this example, the function calls itself with a reduced value of n on each iteration until the base case n == 1 is reached.

Use cases

Iterating through a sequence of numbers or elements in an array (for loops).
Implementing recursive algorithms for problems such as search algorithms and complex data structures.

Notes or pitfalls

A loop continues until its condition becomes false; ensure the loop condition will eventually be false to avoid infinite loops.
Recursive algorithms require a base case (for example, n == 1 in the factorial example) to terminate the recursion.

For loop
Recursive algorithm / Recursion
Counter variable
Sequence
Array
Base case
Factorial
Search algorithms
Data structures