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 W. Li and 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.
Reviewer: Radek Kucera (Ostrava)
MSC:
| 65F20 | Numerical solutions to overdetermined systems, pseudoinverses |
| 15A09 | Theory of matrix inversion and generalized inverses |
| 65F10 | Iterative numerical methods for linear systems |
Citations:
Zbl 1185.65057References:
| [1] | Ben-Israel, A.; Charnes, A., Contributions to the theory of generalized inverses, SIAM Journal on Applied Mathematics, 11, 3, 667-699 (1963) · Zbl 0116.32202 |
| [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 |
| [3] | Horn, Roger A.; Johnson, Charles R., Matrix Analysis (1991), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0729.15001 |
| [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 |
| [5] | W.G. Li, J. Li, T.T. Qiao, A note on computing the approximate generalized inverses of matrix, 2011, submitted for publishing.; 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, 6, 1604-1613 (2010) · Zbl 1206.65139 |
| [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 |
| [8] | Zhang, X.; Cai, J.; Wei, Y., Interval iterative methods for computing Moore-Penrose inverse, Applied Mathematics and Computation, 183, 1, 522-532 (2006) · Zbl 1115.65039 |
| [9] | Wei, Y., Recurrent neural networks for computing weighted Moore-Penrose inverse, Applied Mathematics and Computation, 116, 279-287 (2000) · Zbl 1023.65030 |
| [10] | Wei, Y.; Wu, H.; Wei, J., Successive matrix squaring algorithm for parallel computing the weighted generalized inverse \(A_{MN}^\dagger \), Applied Mathematics and Computation, 116, 289-296 (2000) · Zbl 1023.65031 |
| [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 |
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.
