On the Numerical Behavior of Matrix Splitting Iteration Methods for Solving Linear Systems

We study the numerical behavior of stationary one-step or two-step matrix splitting iteration methods for solving large sparse systems of linear equations. We show that inexact solutions of inner linear systems associated with the matrix splittings may considerably influence the accuracy of the approximate solutions computed in finite precision arithmetic. For a general stationary matrix splitting iteration method, we analyze two mathematically equivalent implementations and discuss the conditions when they are componentwise or normwise forward or backward stable. We show that a stationary iteration scheme in the residual-updating form is significantly more accurate than in its direct-splitting form when employing inexact inner solves. Theoretical results are illustrated by numerical experiments with the PMHSS method and with the HSS method representing the classes of inexact one-step and two-step splitting iteration methods, respectively.