NEW STOPPING CRITERIA FOR SOME ITERATIVE METHODS FOR A CLASS OF UNSYMMETRIC LINEAR-SYSTEMS

When the iterative procedure x(k+1) = Gx(k) + g for the linear system Ax = b is considered, one of the important items for the method is the stopping criterion. Usually one kind of norm is used as a measure, and if the norm of the pseudoresidual vector delta-k = Gx(k) + g - x(k) is small, then the iterative procedure is terminated. However, this does not guarantee that the norm of the error vector epsilon-k = x(k) - x* is small. In this short note it is shown that if there exists a nonsingular matrix Z such that ZGZ-1 is skew-symmetric, then parallel-to epsilon-k parallel-to Z less-than-or-equal-to parallel-to delta-k parallel-to z where parallel-to y parallel-to z = parallel-to Zy parallel-to 2. The relative error bound is also given.