Breakdowns and stagnation in iterative methods

One of the disadvantages of Krylov subspace iterative methods is the possibility of breakdown. This occurs when it is impossible to get the next approximation of the solution to the linear system of equationsAu=f. There are two different situations: lucky breakdown, when we have found the solution and hard breakdown, when the next Krylov subspace cannot be generated and/or the next approximate solution (iterate) cannot be computed. We show that some breakdowns depend on the chosen method of generating the basis vectors. Another undesirable feature of the iterative methods is stagnation. This occurs when the error does not change for several iterative steps. We investigate when iterative methods can stagnate and describe conditions which characterize stagnation. We show that in some cases stagnation can imply breakdown.