STABILITY, CONVERGENCE, AND CONDITIONING OF STATIONARY ITERATIVE METHODS OF THE FORM X(I+1) = PX(I)+Q FOR THE SOLUTION OF LINEAR-SYSTEMS

A statistical approach to the study of the stability of a stationary iterative method for solving a linear system x = Px + q is studied. An asymptotic stability factor is introduced. The relations between this stability measure, the spectral radius of the iteration matrix, and the condition number of the system are studied. The special case when the iteration matrix is normal is treated separately from the general one. For iteration matrices that are normal, the following logical implications are found: large condition number double-line arrow pointing right large asymptotic stability factor double-line arrow pointing left and right poor convergence. In the general case, a large asymptotic stability factor does not imply poor convergence, i.e.: large condition number double-line arrow pointing right large asymptotic stability factor double line arrow pointing left poor convergence.