## 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
Full Text:

### 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 University Press Cambridge, New York, New Rochelle, Melbourne, Sydney [15] Ortega, James M., Numerical Analysis: A Second Course (1973), Academic Press: Academic Press New York · Zbl 0701.65002 [16] Wang, G.; Wei, Y.; Qiao, S., Generalized Inverses: Theory and Computations (2004), Science Press
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.