Iteration

Iteration is the process of repeating a set of operations to arrive at a result, producing successive approximations rather than a single computed value.

In mathematics, iteration commonly means applying a function to its own output: x_{n+1} = f(x_n). A point x* with f(x*) = x* is a fixed point, and the sequence {x_n} is studied for convergence toward x*. When f is a contraction on a closed interval, the contraction mapping theorem guarantees convergence to a unique fixed point. Convergence behavior depends on properties of f and the starting value.

Common examples include the Babylonian method for square roots, x_{n+1} = (x_n + a/x_n)/2, and Newton's method for root finding. These iterates aim to reduce error and often achieve rapid convergence under suitable conditions.

In numerical linear algebra, iterative methods solve linear systems without direct factorization. Jacobi and Gauss-Seidel methods update components in place. Their convergence depends on properties of the system matrix, such as spectral radius or diagonal dominance; poor conditioning can hinder convergence and may require preconditioning or alternative formulations.

In programming, iteration refers to looping structures (for, while) that repeat a block of code until a termination condition is met. Iteration contrasts with recursion and can be more space-efficient. Proper termination criteria and bounds on iterations help control performance and ensure termination.

The term derives from Latin iteratio, iterare meaning to repeat. Iteration is used across disciplines to describe progressive refinement, not only in numerical computation but also in design, data analysis, and software development, reflecting a general preference for incremental improvement.