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A family of higher-order convergent iterative methods for computing the Moore-Penrose inverse. (English) Zbl 06045802
The paper describes an iterative method for computing the Moore-Penrose inverse that is an extension of the Li and Li method. 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 30x30 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 Li and Li paper. It seems also that the method is not suitable for large-scale problems, since each iteration requires multiplications of matrices - the time consuming operation.
##### MSC:
 65F20 Overdetermined systems, pseudoinverses (numerical linear algebra) 15A09 Matrix inversion, generalized inverses 65F10 Iterative methods for linear systems
##### Keywords:
Moore-Penrose inverse; iterative method; convergence rate
##### References:
 [1] Ben-Israel, A.; Charnes, A.: Contributions to the theory of generalized inverses, SIAM journal on applied mathematics 11, No. 3, 667-699 (1963) · Zbl 0116.32202 · doi:10.1137/0111051 [2] Ben-Israel, A.; Cohen, D.: On iterative computation of generalized inverses and associated projections, SIAM journal on numerical analysis 3, 410-419 (1966) · Zbl 0143.37402 · doi:10.1137/0703035 [3] Horn, Roger A.; Johnson, Charles R.: Matrix analysis, (1991) [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. [6] Petković, Marko D.; Stanimirović, Predrag S.: Iterative method for computing the Moore-Penrose inverse based on Penrose equations, Journal of computational and applied mathematics 235, No. 6, 1604-1613 (2010) · Zbl 1206.65139 · doi:10.1016/j.cam.2010.08.042 [7] Stanimirović, P. S.; Cvetković-Llić, D. S.: Successive matrix squaring algorithm for computing outer inverses, Applied mathematics and computation 203, 19-29 (2008) · Zbl 1158.65028 · doi:10.1016/j.amc.2008.04.037 [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$†$, 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