What is the purpose of the convergence criterion in iterative methods?

The purpose of the convergence criterion in iterative methods is to define when an iterative algorithm should stop. As iterative methods are designed to find an approximate solution to equations, they involve repeated calculations to improve that approximation. The convergence criterion provides a specific condition based on the behavior of the solutions generated through each iteration, which signals when the algorithm has sufficiently approached the desired accuracy or when further iterations would yield negligible improvements.

This is crucial because it helps in balancing the need for accuracy with computational efficiency. If the algorithm were to continue indefinitely without a stopping rule, it would waste computational resources, especially if the solution is already close enough to the true value. Thus, by establishing a clear convergence criterion, one can ensure that the iterations cease when it is reasonable to consider the solution adequate for practical purposes, leading to effective use of time and resources during computation.