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Zbl 1175.65063
Liu, Xiaoji; Yu, Yaoming; Hu, Chunmei
The iterative methods for computing the generalized inverse $A^{(2)}_{T,S}$ of the bounded linear operator between Banach spaces.
(English)
[J] Appl. Math. Comput. 214, No. 2, 391-410 (2009). ISSN 0096-3003

The authors start by noting that many well known generalized inverses such as the Bott-Duffin inverse $A_{(l)}^{-1},$ or the Moore-Penrose inverse $A_{(MN)}^{+}$ and many others are the generalized inverse $A_{(TS)}^{2}$, the (2) inverse with prescribed range T and null-space S. Let $\cal{X}$ and $\cal{Y}$ denote arbitrary Banach spaces and $\cal{B}(\cal{X},\cal{Y})$ the set of all bounded operators. In one of their main results the authors define for $A\in \cal{B}(\cal{X},\cal{Y})$ an approximating sequence $(X_{k})_{k}$ in $\cal{B}(\cal{Y},\cal{X})$ and prove that $$\| A_{(TS)}^{(2)} - X_{k}\| < (q^{p^{k}})\|(1-q)^{-1}\| X_{0}\|$$ for a certain integer $p\geq 2$ and $q<1.$ Analogous results are obtained for the generalized Drazin inverse in Banach algebras. Examples are given for the matrix $$A=\pmatrix 2 & 1 & 1\\ 0 & 2 & 0\\ 0 & 0 & 2\\ 0 & 0 & 0 \endpmatrix \in \cal{C}^{4\times 3}$$ and a matrix from $\cal{C}^{58\times 57}.$
[Erwin Schechter (Moers)]
MSC 2000:
*65J10 Equations with linear operators (numerical methods)
65F20 Overdetermined systems (numerical linear algebra)
47A05 General theory of linear operators

Keywords: iterative method; generalized inverses; error bounds; linear operators; Bott-Duffin inverse; Moore-Penrose inverse; Banach spaces; Drazin inverse

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