Stability, convergence and conditioning estimates of iterative methods for the solution of linear systems

The asymptotic stability of an iterative method for the solution of linear systems is defined as the limit of the mean roundoff error when the number of iterations tends to infinity. The relations among this stability measure, the spectral radius of the iteration matrix and the condition number of the system are studied. The special case of normal iteration matrtx is treated separately from the general one. For normal iteration matrices poor convergence and numerical instability are equivalent properties and both of them are implied by ill-conditioning. Weaker results hold in the generel case.