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Stopping when progress is no longer being made

In addition to limiting the total amount of work by limiting the maximum number of iterations one is willing to do, it is also natural to consider stopping when no apparent progress is being made. Some methods, such as Jacobi and SOR, often exhibit nearly monotone linear convergence, at least after some initial transients, so it is easy to recognize when convergence degrades. Other methods, like the conjugate gradient method, exhibit ``plateaus'' in their convergence, with the residual norm stagnating at a constant value for many iterations before decreasing again; in principle there can be many such plateaus (see Greenbaum and Strakos [109]) depending on the problem. Still other methods, such as CGS, can appear wildly nonconvergent for a large number of steps before the residual begins to decrease; convergence may continue to be erratic from step to step.

In other words, while it is a good idea to have a criterion that stops when progress towards a solution is no longer being made, the form of such a criterion is both method and problem dependent.