New Stopping Criteria for Some Iterative Methods for a Class of Unsymmetric Linear Systems
David John Evans, C. Li · SIAM Journal on Matrix Analysis and Applications · 1991
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 $\varepsilon _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 $\| \varepsilon_k \|_Z \leqq \| \delta _k \|_Z $ where $\| y \|_Z = \| Zy \|_2 $. The relative error bound is also given.