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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
DGMRES
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