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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 $\bbfR^n$, $e_j$ 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)

65F35Matrix norms, conditioning, scaling (numerical linear algebra)
65F10Iterative methods for linear systems
65F50Sparse matrices (numerical linear algebra)
65Y05Parallel computation (numerical methods)
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