How is 'iteration' typically used in numerical methods?

Iteration in numerical methods refers to the process of refining solutions through successive approximations. This approach is fundamental to many algorithms used for solving mathematical problems, particularly those that involve equations or optimization tasks.

In the context of solving nonlinear equations, for example, methods such as Newton-Raphson or fixed-point iteration make use of initial guesses and then repeatedly improve upon these guesses based on the results from the previous iteration. Each successive approximation ideally brings the solution closer to the actual root or optimal value, achieving better accuracy with each cycle until a specified level of precision is met.

This iterative process is essential because many problems cannot be solved analytically, or the solutions are complex. Therefore, using a sequence of approximations allows for practical solutions that may converge to an answer even when the exact solution is impossible to achieve directly. The convergence of these methods is often studied to ensure that the solutions stabilize as iterations progress, which underscores the importance of iteration in numerical analysis.