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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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