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A family of higher-order convergent iterative methods for computing the Moore-Penrose inverse. (English) Zbl 1298.65068
The paper describes an iterative method for computing the Moore-Penrose inverse that is an extension of the method by {\it W. Li} and {\it Z. Li} [ibid. 215, No. 9, 3433--3442 (2010; Zbl 1185.65057)]. The paper is short, well-written, and relatively clear. It contains basic concept of the method, three auxiliary lemmas, and the main theorem proving the convergence. The promising numerical experiments are performed for at most $30\times 30$ matrices. On the other hand, not all is written in the paper. To understand some parameters of the basic iterative scheme, it is necessary to see the original paper by Li and Li. It seems also that the method is not suitable for large-scale problems, since each iteration requires multiplications of matrices -- the time consuming operation.

65F20Overdetermined systems, pseudoinverses (numerical linear algebra)
15A09Matrix inversion, generalized inverses
65F10Iterative methods for linear systems
Full Text: DOI
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[4] Li, W. G.; Li, Z.: A family of iterative methods for computing the approximate inverse of a square matrix and inner inverse of a non-square matrix, Applied mathematics and computation 215, 3433-3442 (2010) · Zbl 1185.65057 · doi:10.1016/j.amc.2009.10.038
[5] W.G. Li, J. Li, T.T. Qiao, A note on computing the approximate generalized inverses of matrix, 2011, submitted for publishing.
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[8] Zhang, X.; Cai, J.; Wei, Y.: Interval iterative methods for computing Moore -- Penrose inverse, Applied mathematics and computation 183, No. 1, 522-532 (2006) · Zbl 1115.65039 · doi:10.1016/j.amc.2006.05.098
[9] Wei, Y.: Recurrent neural networks for computing weighted Moore -- Penrose inverse, Applied mathematics and computation 116, 279-287 (2000) · Zbl 1023.65030 · doi:10.1016/S0096-3003(99)00147-2
[10] Wei, Y.; Wu, H.; Wei, J.: Successive matrix squaring algorithm for parallel computing the weighted generalized inverse AMN$\dagger $, Applied mathematics and computation 116, 289-296 (2000) · Zbl 1023.65031 · doi:10.1016/S0096-3003(99)00151-4
[11] Wei, Y.; Wu, H.: The representation and approximation for the weighted Moore -- Penrose inverse, Applied mathematics and computation 121, 17-28 (2001) · Zbl 1024.15003 · doi:10.1016/S0096-3003(99)00275-1