Iterating

Iterating refers to the act of repeating a process, applying a function or operation to its own output, in order to produce a sequence of values or to refine a result. Each repetition updates the current state, producing a next state or estimate. Iteration contrasts with a single pass; many practical problems are solved by iterative refinement until a termination condition is met.

In mathematics, iteration usually means composing a function with itself: x_{n+1} = f(x_n). The sequence {x_n} is called the orbit of x_0 under f. Analysis focuses on convergence to a fixed point x*, where f(x*) = x*, or on chaotic or complex behavior. Convergence depends on properties of f near the limit, such as derivative bounds.

In computer science, iterative algorithms repeatedly apply steps until a condition is satisfied. They use loops (for, while) rather than recursion. Iteration can offer predictable memory usage and easier reasoning about termination; it avoids potential stack overflow associated with deep recursion.

Numerical methods commonly use iterative schemes, such as fixed-point iteration, Newton's method, Gauss-Seidel and Jacobi iterations. They start from an initial guess and refine estimates. Convergence criteria depend on the problem, and a stopping tolerance is typically specified (for example, when the change between iterations falls below a threshold).

In design and project management, iterative development emphasizes cycles of planning, implementation, testing, and feedback, allowing gradual improvement and adapting to new information. This approach contrasts with rigid, linear processes and aims to mitigate risk through early validation.