Perturbation analysis of singular linear systems with arbitrary index. (Q1412447)

The authors consider linear systems of equations \(Ax=b\), where \(A \in\mathbb{C}^{n \times n}\) is a singular matrix with arbitrary index. The following convergence result for semi-iterative methods [see \textit{J. J. Climent}, \textit{M. Neumann} and \textit{A. Sidi}, J. Comput. Appl. Math. 87, 21--38 (1997; Zbl 0899.65020)] is presented. If \(A\) is singular and \(\text{Ind}(A) = \alpha\), then the semi-iterative sequence \(\{x_m\}\) converges to \(A^Db + (I -AA^D)x_0\) for an arbitrary initial guess \(x_0\). \(A^D\) denotes the Drazin inverse of the matrix \(A\). Furthermore, a perturbation analysis for the system of equations \(A^\alpha A x = A^\alpha b\) is given. Hereby, consistent as well as inconsistent perturbed systems are considered.