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Approximate inverse preconditioners via sparse-sparse iterations. (English) Zbl 0922.65034
The authors consider iteration methods for finding approximate inverse preconditioners $$M$$ to the inverse $$A^{-1}$$ of a given $$(n\times n)$$-matrix $$A$$. The iterative methods aim at the minimization of the functional $F(M):= \| I- AM\|^2_F= \sum^n_{j=1}\| e_j- Am_j\|^2_2$ on the space of all $$(n\times n)$$-matrices, where $$\|\cdot\|_F$$ denotes the Frobenius norm, $$\|\cdot\|_2$$ is the Euclidean norm in $$\mathbb{R}^n$$, $$e_j$$ and $$m_j$$ are the $$j$$th columns of the identity matrix $$I$$ and of the matrix $$M$$, respectively. The authors propose and analyze several iterative methods (Newton, MR, GMRES) with several modifications (numerical dropping in the iterates or in the search directions, self-preconditioning etc.). The different techniques are compared numerically on several examples taken from the well-known Harwell-Boeing collection and from matrices generated by the fluid dynamics analysis package FIDAP.
Reviewer: U.Langer (Linz)

##### MSC:
 65F35 Numerical computation of matrix norms, conditioning, scaling 65F10 Iterative numerical methods for linear systems 65F50 Computational methods for sparse matrices 65Y05 Parallel numerical computation
FIDAP; ILUS
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