A family of iterative methods for computing Moore-Penrose inverse of a matrix. (English) Zbl 1258.65035

Summary: This paper improves on generalized properties of a family of iterative methods to compute the approximate inverses of square matrices originally proposed in [the author and Z. Li, Appl. Math. Comput. 215, No. 9, 3433–3442 (2010; Zbl 1185.65057)]. And while the methods of [loc. cit.] can be used to compute the inner inverses of any matrix, it has not been proved that these sequences converge (in norm) to a fixed inner inverse of the matrix. In this paper, it is proved that the sequences indeed are convergent to a fixed inner inverse of the matrix which is the Moore-Penrose inverse of the matrix. The convergence proof of these sequences is given by fundamental matrix calculus, and numerical experiments show that the third-order iterations are as good as the second-order iterations.


65F20 Numerical solutions to overdetermined systems, pseudoinverses


Zbl 1185.65057
Full Text: DOI


[1] Li, W.; Li, Z., A family of iterative methods for computing the approximate inverse of a square matrix and inner inverse of a non-square matrix, Appl. Math. Comput., 215, 3433-3442 (2010) · Zbl 1185.65057
[3] Ben-Israel, A.; Greville, T. N.E., Generalized Inverses, Theory and Applications (2003), Springer: Springer New York, NY · Zbl 1026.15004
[4] Wang, S.; Yang, Z., Generalized Inverse Matrix and Its Applications (1996), Beijing University of Technology Press, (in Chinese)
[5] Gautschi, W., Numerical Analysis: An Introduction (1997), Birkhäuser: Birkhäuser Boston · Zbl 0877.65001
[6] Wei, Y., Successive matrix squaring algorithm for computing Drazin inverse, Appl. Math. Comput., 108, 67-75 (2000) · Zbl 1022.65043
[7] Stanimirovic, P. S.; Cvetkovic-Ilic, D. S., Successive matrix squaring algorithm for computing outer inverses, Appl. Math. Comput., 203, 19-29 (2008) · Zbl 1158.65028
[8] Wei, Y.; Cai, J.; Ng, M. K., Computing Moore-Penrose inverses of Toeplitz matrices by Newton’s iteration, Math. Comput. Modelling, 40, 1-2, 181-191 (2004) · Zbl 1069.65045
[9] Zhang, X.; Cai, J.; Wei, Y., Interval iterative methods for computing Moore-Penrose inverse, Appl. Math. Comput., 183, 1, 522-553 (2006) · Zbl 1115.65039
[10] Cai, J.; Ng, M. K.; Wei, Y., Modified Newton’s algorithm for computing the group inverses of singular Toeplitz matrices, J. Comput. Math., 24, 5, 647-656 (2006) · Zbl 1113.65035
[11] Chen, L.; Krishnamurthy, E. V.; Macleod, I., Generalized matrix inversion and rank computation by successive matrix powering, Parallel Comput., 20, 297-311 (1994) · Zbl 0796.65055
[12] Djordjevic, D. S.; Stanimirovic, P. S.; Wei, Y., The representation and approximation of outer generalized inverses, Acta Math. Hungar., 104, 1-26 (2004) · Zbl 1071.65075
[13] Wei, Y.; Wu, H., The representation and approximation for Drazin inverse, J. Comput. Appl. Math., 126, 417-423 (2000) · Zbl 0979.65030
[14] Wei, Y., A characterization and representation for the generalized inverse \(A_{T \text{;} S}^2\) and its applications, Linear Algebra Appl., 280, 87-96 (1998) · Zbl 0934.15003
[15] 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. Funct. Anal. Optim., 30, 1-2, 168-182 (2009) · Zbl 1165.47004
[16] Najafi, H. S.; Solary, M. S., Computational algorithms for computing the inverse of a square matrix, quasi-inverse of a nonsquare matrix and block matrices, Appl. Math. Comput., 183, 539-550 (2006) · Zbl 1104.65309
[17] Wu, X., A note on computational algorithm for the inverse of a square matrix, Appl. Math. Comput., 187, 962-964 (2007) · Zbl 1121.65027
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