## The iterative methods for computing the generalized inverse $$A^{(2)}_{T,S}$$ of the bounded linear operator between Banach spaces.(English)Zbl 1175.65063

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 $$\mathcal{X}$$ and $$\mathcal{Y}$$ denote arbitrary Banach spaces and $$\mathcal{B}(\mathcal{X},\mathcal{Y})$$ the set of all bounded operators. In one of their main results the authors define for $$A\in \mathcal{B}(\mathcal{X},\mathcal{Y})$$ an approximating sequence $$(X_{k})_{k}$$ in $$\mathcal{B}(\mathcal{Y},\mathcal{X})$$ 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 $A=\begin{pmatrix} 2 & 1 & 1\\ 0 & 2 & 0\\ 0 & 0 & 2\\ 0 & 0 & 0 \end{pmatrix} \in \mathcal{C}^{4\times 3}$ and a matrix from $$\mathcal{C}^{58\times 57}.$$

### MSC:

 65J10 Numerical solutions to equations with linear operators 65F20 Numerical solutions to overdetermined systems, pseudoinverses 47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
Full Text:

### References:

 [1] Ben-Israel, A.; Greville, T. N.E., Generalized Inverses: Theory and Applications (1974), John Wiley: John Wiley New York · Zbl 0305.15001 [2] Chen, X.; Hartwig, R. E., The hyperpower iteration revisited, Linear Algebra Appl., 233, 207-229 (1996) · Zbl 0848.65021 [3] Chen, Y., Iterative methods for computing the generalized inverses $$A_{T, S}^{(2)}$$ of a matrix A, Appl. Math. Comput., 75, 2-3, 207-222 (1996) · Zbl 0853.65044 [4] Chen, Y.; Chen, X., Representation and approximation of the outer inverse $$A_{T, S}^{(2)}$$ of a matrix A, Linear Algebra Appl., 308, 85-107 (2000) · Zbl 0957.15002 [5] Djordjević, D. S., Iterative methods for computing generalized inverses, Appl. Math. Comput., 189, 1, 101-104 (2007) · Zbl 1125.65046 [6] Djordjević, D. S.; Stanimirović, P. S., On the generalized Drazin inverse and generalized resolvent, Czechoslovak. Math. J., 126, 617-634 (2001) · Zbl 1079.47501 [7] Djordjević, D. S.; Stanimirović, P. S., Splittings of operators and generalized inverses, Publ. Math. Debrecen, 59, 147-159 (2001) · Zbl 0981.47001 [8] Djordjević, D. S.; Stanimirović, P. S.; Wei, Y., The representation and approximations of outer generalized inverses, Acta Math. Hungar., 104, 1-2, 1-26 (2004) · Zbl 1071.65075 [9] Djordjević, D. S.; Stanimirović, P. S., Iterative methods for computing generalized inverses related with optimization methods, J. Aust. Math. Soc., 78, 2, 257-272 (2005) · Zbl 1102.46035 [10] Djordjević, D. S.; Wei, Y., Outer generalized inverses in rings, Comm. Algebra, 33, 3051-3060 (2005) · Zbl 1111.15007 [11] Li, X.; Wei, Y., Iterative methods for the Drazin inverse of a matrix with a complex spectrum, Appl. Math. Comput., 147, 855-862 (2004) · Zbl 1038.65037 [12] Nacevska, B., Iterative methods for computing generalized inverses and splittings of operators, Appl. Math. Comput., 228, 1, 186-188 (2009) · Zbl 1160.65312 [13] Sheng, X.; Chen, G., Several representations of generalized inverse $$A^{(2)}_{T,S}$$ and their application, Int. J. Comput. Math., 85, 9, 1441-1453 (2008) · Zbl 1151.15010 [14] Sheng, X.; Chen, G.; Gong, Y., The representation and computation of generalized inverse $$A_{T, S}^{(2)}$$, J. Comput. Appl. Math., 213, 1, 248-257 (2008) · Zbl 1135.65021 [15] Wang, G.; Wei, Y.; Qiao, S., Generalized Inverses: Theory and Computations (2004), Science Press: Science Press Beijing [16] Wei, Y., A characterization and representation of the generalized inverse $$A_{T, S}^{(2)}$$ and its applications, Linear Algebra Appl., 280, 87-96 (1998) · Zbl 0934.15003 [17] Wei, Y., Integral representation of the generalized inverse $$A_{T, S}^{(2)}$$ and its applications, (Recent Research on Pure and Applied Algebra (2003), Nova Science Publisher: Nova Science Publisher Hauppage, NY), 59-62 · Zbl 1062.15003 [18] Wei, Y., Recent results on the generalized inverse $$A_{T, S}^{(2)}$$, (Ling, G. D., Linear Algebra Research Advances (2007), Nova Science Publisher), 231-250 [19] Wei, Y.; Wu, H., The representation and approximation for the generalized inverse $$A_{T, S}^{(2)}$$, Appl. Math. Comput., 135, 263-276 (2003) · Zbl 1027.65048 [20] Wei, Y.; Wu, H., $$\{T, S \}$$ splitting methods for computing the generalized inverse $$A_{T, S}^{(2)}$$ and regular systems, Int. J. Comput. Math., 77, 3, 401-424 (2001) · Zbl 0986.65038 [21] Wei, Y.; Wu, H., On the perturbation and subproper splitting for the generalized inverse $$A_{T, S}^{(2)}$$ of regular matrix A, J. Comput. Appl. Math., 137, 2, 317-329 (2001) · Zbl 0993.15003 [22] Wei, Y.; Zhang, N., A note on the representation and approximation of the outer inverse $$A_{T, S}^{(2)}$$ of a matrix A, Appl. Math. Comput., 147, 837-841 (2004) · Zbl 1040.15007 [23] Yu, Y., PCR algorithm for parallel computing the solution of the general restricted linear equations, J. Appl. Math. Comput., 27, 125-136 (2008) · Zbl 1155.65031 [24] Yu, Y.; Wei, Y., The representation and computational procedures for the generalized inverse $$A_{T, S}^{(2)}$$ of an operator A in Hilbert spaces, Numer. Func. Anal. Optimiz., 30, 1-2, 168-182 (2009) · Zbl 1165.47004 [25] Y. Yu, Y. Wei, Determinantal representation of the generalized inverse $$A_{T , S}^{( 2 )}$$ doi:10.1080/03081080701871665; Y. Yu, Y. Wei, Determinantal representation of the generalized inverse $$A_{T , S}^{( 2 )}$$ doi:10.1080/03081080701871665 · Zbl 1182.15007 [26] Zheng, B.; Wang, G., Representation and approximation for generalized inverse $$A_{T, S}^{(2)}$$, J. Appl. Math. Comput., 22, 3, 225-240 (2006) · Zbl 1112.15013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.