## 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:

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