Stopping criterion

The rule that decides when an iteration has reached sufficient accuracy.

Definition

The usual criteria combine the increment xk+1xk|x_{k+1}-x_k| (absolute or relative), the residual f(xk)|f(x_k)| and a maximum number of iterations as a safety net. Each can mislead on its own: the increment in slow iterations, the residual in ill-conditioned problems.

How it is used

In practice one requires both a small increment and a small residual, with tolerances consistent with machine epsilon and with the data.