## A family of iterative methods for computing the approximate inverse of a square matrix and inner inverse of a non-square matrix.(English)Zbl 1185.65057

If $$V_0$$ is an approximate inverse of a nonsingular square matrix $$A$$ such that $$\|I-AV_0\|<1$$, the authors show that the iterative formula
$V_{q+1}=V_q\left[kI-(k(k-1)/2)AV_q+\dots+(-1)^{k-1}(AV_q)^{k-1}\right],\quad k=2,3,\dots,$
converges to $$A^{-1}$$ with order of convergence equal to $$k$$. An easy method is presented to find an initial approximate inverse $$V_0$$. A matrix $$V$$ is called a generalized inner inverse of a rectangular matrix $$A$$ if $$AVA=A$$. The above iterative formula is shown to converge weakly to an inner inverse of $$A$$ under appropriate conditions since the inner inverse is not unique. Numerical examples illustrate the efficiency of the new iterative method with the proposed new initial approximation.

### MSC:

 65F10 Iterative numerical methods for linear systems 15A09 Theory of matrix inversion and generalized inverses
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### References:

 [1] Saberi Najafi, H.; Shams Solary, M., Computational algorithms for computing the inverse of a square matrix, quasi-inverse of a nonsquare matrix and block matrices, Applied mathematics and computation, 183, 539-550, (2006) · Zbl 1104.65309 [2] Phillips, G.M.; Taylor, P.J., Theory and applications of numerical analysis, (1980), Academic Press · Zbl 0312.65002 [3] Wu, Xinyuan, A note on computational algorithm for the inverse of a square matrix, Applied mathematics and computation, 187, 962-964, (2007) · Zbl 1121.65027 [4] Wei, Y., Successive matrix squaring algorithm for computing Drazin inverse, Applied mathematics and computation, 108, 67-75, (2000) · Zbl 1022.65043 [5] Stanimirovic, Predrag S.; Cvetkovic-Ilic, Dragana S., Successive matrix squaring algorithm for computing outer inverses, Applied mathematics and computation, 203, 19-29, (2008) · Zbl 1158.65028 [6] Wei, Yimin; Cai, Jianfeng; Ng, Michael K., Computing Moore-Penrose inverses of Toeplitz matrices by newton’s iteration, Mathematical and computer modelling, 40, 1-2, 181-191, (2004) · Zbl 1069.65045 [7] Zhang, Xian; Cai, Jianfeng; Wei, Yimin, Interval iterative methods for computing Moore-Penrose inverse, Applied mathematics and computation, 183, 1, 522-532, (2006) · Zbl 1115.65039 [8] Cai, Jian-feng; Ng, Michael K.; Wei, Yi-min, Modified newton’s algorithm for computing the group inverses of singular Toeplitz matrices, Journal of computational mathematics, 24, 5, 647-656, (2006) · Zbl 1113.65035 [9] Chen, L.; Krishnamurthy, E.V.; Macleod, I., Generalized matrix inversion and rank computation by successive matrix powering, Parallel computing, 20, 297-311, (1994) · Zbl 0796.65055 [10] Djordjevic, D.S.; Stanimirovic, P.S.; Wei, Y., The representation and approximation of outer generalized inverses, Acta Mathematica hungar, 104, 1-26, (2004) · Zbl 1071.65075 [11] Wei, Y.; Wu, H., The representation and approximation for Drazin inverse, Journal of computational and applied mathematics, 126, 417-423, (2000) · Zbl 0979.65030 [12] Wei, Y., A characterization and representation for the generalized inverse $$A_{T \text{;} S}^2$$ and its applications, Linear algebra and its applications, 280, 87-96, (1998) · Zbl 0934.15003 [13] Yu, Yaoming; Wei, Yimin, The representation and computational procedures for the generalized inverse $$A(2) T, S$$ of an operator A in Hilbert spaces, Numerical functional analysis and optimization, 30, 1-2, 168-182, (2009) · Zbl 1165.47004 [14] Horn, R.A.; Johnson, C.R., Matrix analysis, (1986), Cambridge University Press Cambridge, New York, New Rochelle, Melbourne, Sydney [15] Ortega, James M., Numerical analysis: A second course, (1973), Academic Press New York · Zbl 0701.65002 [16] Wang, G.; Wei, Y.; Qiao, S., Generalized inverses: theory and computations, (2004), Science Press
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