iterative

Iterative is an adjective describing a process that is repeated in sequence to approach a desired result. In mathematics, computer science, and related fields, an iterative method generates a sequence of approximations that converge toward a solution. Each iteration applies a procedure to the current estimate to produce a new estimate, and the process continues until a stopping criterion is met, such as a small change between successive estimates or reaching a maximum number of iterations.

Common iterative techniques include fixed-point iteration, where a problem is rewritten as x = g(x) and repeated as x_{n+1} = g(x_n); iterative methods for solving linear systems, such as Jacobi and Gauss-Seidel, which refine the solution step by step; and Newton-Raphson, which generates successive approximations to a root via x_{n+1} = x_n - f(x_n)/f'(x_n). Convergence behavior depends on the method and problem properties; some methods require a good initial guess and may converge rapidly, while others may fail to converge or converge slowly.

In computer science, iterative algorithms use loops to process data or refine results, in contrast to recursive algorithms that rely on repeated function calls with a stack. Iteration is pervasive in numerical analysis, graphics, optimization, data processing, and simulations.

Beyond technical computation, iterative processes are foundational in design and development, where solutions are incrementally refined through prototypes, testing, and feedback, as seen in iterative design and agile methodologies. The core idea across contexts is repeated refinement aimed at increasingly accurate or effective outcomes.