Iteracions

Iteracions are the repeated application of a process or operation to generate a sequence of results. In mathematics, computing, and related fields, iteration is a core technique for approximating solutions, simulating dynamics, and implementing algorithms. An iteration typically starts from an initial value x0 and produces successive values x1, x2, … according to a rule such as x_{n+1} = F(x_n).

Convergence is a central concern: the sequence is said to converge when it approaches a limit x*, often a fixed point where x* = F(x*). Stopping rules are used in practice, including a maximum number of steps and tolerance-based criteria like |x_{n+1}-x_n| < ε or |F(x_n)-x_n| < ε.

Common types include fixed-point iteration, where the goal is to solve x = g(x) by iterating x_{n+1} = g(x_n); and iterative root finding, such as Newton's method, which updates x_{n+1} = x_n - f(x_n)/f'(x_n). Iterations also form the basis of many algorithms (for example, gradient descent) and of dynamical systems, where discrete time steps model evolution. The Mandelbrot and Julia sets use complex iterations to classify points.

Applications span numerical analysis, engineering, computer science, and economics. The term is used across languages; in Catalan, iteracions means iterations.