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Perturbation analysis of singular linear systems with arbitrary index. (English) Zbl 1033.65026
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 J. J. Climent, M. Neumann and 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.

MSC:
65F20 Numerical solutions to overdetermined systems, pseudoinverses
65F10 Iterative numerical methods for linear systems
Software:
DGMRES
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References:
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