Iteraci

Iteraci is the process of repeating a set of operations to obtain a result or approximate a solution. It is a general concept used in mathematics, computer science, and engineering. The basic idea is to apply a rule or function repeatedly, using the result of the current step as input for the next.

In mathematics and numerical analysis, iteration is often used to approximate fixed points or solve equations. Fixed-point iteration uses x_{k+1} = g(x_k); convergence depends on properties of g, such as being a contraction. The rate of convergence describes how quickly the sequence approaches the limit.

In computer science, iteration is implemented via loops such as for and while. Iteration contrasts with recursion, though both can compute repeated work. Practical concerns include time and space complexity, and the choice of stopping criteria, e.g., a maximum number of iterations or a tolerance for changes between steps.

Common iterative methods include Jacobi and Gauss-Seidel for linear systems, Newton's method for nonlinear equations, and gradient-based optimization methods that use iterative updates.

Originating from Latin iterare meaning “to repeat,” the concept of iteration is expressed in many languages with forms related to iteraci. In practice, iteration underpins a wide range of algorithms and simulations, enabling progressive refinement and approximation where exact solutions are impractical.